Uploaded by weranom948

SOLUTIONS Chemical Biochemical and Engi

advertisement
D 7KHUPRVWDWLFEDWKLPSRVHVLWVWHPSHUDWXUHTRQWKHV\VWHP E &RQWDLQHULPSRVHVFRQVWUDLQWRIFRQVWDQWYROXPH7KHUPDOLVRODWLRQLPSOLHV
WKDWKHDWIORZPXVWEH]HURZKLOHPHFKDQLFDOLVRODWLRQ DQGFRQVWDQWYROXPH LPSOLHV WKHUH LV QR ZRUN IORZ &RQVHTXHQWO\ WKHUH LV QR PHFKDQLVP IRU
DGGLQJ RU UHPRYLQJ HQHUJ\ IURP WKH V\VWHP 7KXV V\VWHP YROXPH DQG
HQHUJ\DUHFRQVWDQW
F 7KHUPDOO\LVRODWHGŸDGLDEDWLF
)ULFWLRQOHVVSLVWRQŸSUHVVXUHRIV\VWHPHTXDOVDPELHQWSUHVVXUH RUDPELHQW
SUHVVXUH wg/A LI SLVWRQF\OLQGHU LQ YHUWLFDO SRVLWLRQ +HUH
w Z HLJKWRISLVWRQ A LWVDUHDDQGgLVWKHIRUFHRIJUDYLW\ G 7KHUPRVWDWLFEDWKŸFRQVWDQWWHPSHUDWXUHT
)ULFWLRQOHVVSLVWRQŸFRQVWDQWSUHVVXUH VHHSDUWFDERYH H 6LQFH SUHVVXUH GLIIHUHQFH LQGXFHVDPDVVIORZSUHVVXUHHTXLOLEUDWHVUDSLGO\
7HPSHUDWXUHHTXLOLEUDWLRQZKLFKLVDUHVXOWRIKHDWFRQGXFWLRQRFFXUVPXFK
PRUH VORZO\ 7KHUHIRUH LI YDOYH EHWZHHQ WDQNV LV RSHQHG IRU RQO\ D VKRUW
WLPHDQGWKHQVKXWWKHSUHVVXUHLQWKHWZRWDQNVZLOOEHWKHVDPHEXWnotWKH
WHPSHUDWXUHV
D :DWHULVLQDSSURSULDWHDVDWKHUPRPHWULFIOXLGEHWZHHQq&DQGq&VLQFH
WKH YROXPH LV QRW D XQLTXH IXQFWLRQ RI WHPSHUDWXUH LQ WKLV UDQJH LH WZR
WHPSHUDWXUHVZLOOFRUUHVSRQGWRWKHVDPHVSHFLILFYROXPH
V T q & a V T q & V T q & a V T q & HWF 9+2
7
TLQ R &DQGVLQFF J &RQVHTXHQWO\ ZKLOH T XQLTXHO\ GHWHUPLQHV V V GRHV QRW XQLTXHO\
GHWHUPLQHT
E $VVXPLQJ WKDW D PHUFXU\ WKHUPRPHWHU LV FDOLEUDWHG DW q& DQG q& DQG
WKDW WKH VSHFLILF YROXPH RI PHUFXU\ YDULHV OLQHDUO\ EHWZHHQ WKHVH WZR
WHPSHUDWXUHV\LHOGV
Chapter 1
V T
&h V cT &h
c h V cT cT &h & &
V R & R
R
R
V
R
R
TV
ZKHUH T LV WKH DFWXDO WHPSHUDWXUH DQG TV LV WKH WHPSHUDWXUH UHDG RQ WKH
WKHUPRPHWHU VFDOH $W q& V T q & FF J +RZHYHU
H[S
WKHVFDOHWHPSHUDWXUHIRUWKLVVSHFLILFYROXPHLVIURPHTQ
TV
VH[S T u u '7 L
F
7
L
7KH WHPSHUDWXUH HUURU SORWWHGKHUH UHVXOWV IURP WKH QRQOLQHDU GHSHQGHQFH RI
WKHYROXPHRIPHUFXU\RQWHPSHUDWXUH,QDUHDOWKHUPRPHWHUWKHUHZLOODOVR
EHDQHUURUDVVRFLDWHGZLWKWKHLPSHUIHFWERUHRIWKHFDSLOODU\WXEH
'L
'V
A
c
h
M wV wT 'T
A
$ VPDOO DUHD A DQG D ODUJH PDVV RI IOXLG M PDJQLILHV 'L REWDLQHG IRU D
JLYHQ 'T 7KXVZHXVHDFDSLOODU\WXEH VPDOOA DQGEXOE ODUJHM WRJHW
DQDFFXUDWHWKHUPRPHWHUVLQFH wV wT LVVRVPDOO
c
:KHQZHXVHDIOXLGILOOHGWKHUPRPHWHUWRPHDVXUH 'T ZHUHDOO\PHDVXUH 'L ZKHUH
q & q & 5HSHDWLQJ FDOFXODWLRQ DW RWKHU
7KXV T TV DW q&
WHPSHUDWXUHV\LHOGVILJXUHEHORZ
DERYH
h
1
2
2.1
d C ˜V
dM =
=M IN -M OUT
dt
dt
a) at steady state
g
M IN =1
M OUT C ˜ volumetric flow rate out
min
g
10m3
1 g
g
1
0.1 3
Cu
C
min
min
10 m3
m
dC
dC
m3
m3
V
M C u 10
V
M C 10
b)
dt
min
min
dt
3
V 6 u 6 u 3 108 m
V 6 u 6 u 3 108 m3
3
dC
g
10m
Cu
108m3
1
min
min
dt
dC
g
1
10 1
C
m
3
dt 108 m min
108
or for C in g 3
m
dC
dC
108
10C 1
10.8
dt
dt
C 0.1
10.8 ln
C-0.1
-0.1
t
§ -0.005 ·
10.8 ln ¨
¸
© -0.1 ¹
t = 32.35 minutes
For C = 0.95 * 0.1 = 0.095
108 ln 20 = t
2.2
dC
dt
kC
If Cf = 0.5 Ci
ln
Cf
Ci
kt
ln 0.5 = -kt or k
ln 2
t
for a half life of seven years
k = 0.099 years-1
so that
C
ln f k 25 0.099 u 25 2.475
22
Cf 22 exp (-2.475) 1.85 ppm
2.3
Basis: 2-liter vessel = 2000 cm3 = 2×103 ×10-6
Initial moles of phosphine = 5
m3
= 2 × 10-3 m3
cm3
kmol
mol
×103
×2×10-5 m3 = 10 mol
kmol
m3
t
2
a)
Mass balance from stoichiometry
Species
Initial
Final
PH3
10
10 – 4X = 7
P4
0
X
H2
0
6X
Since 10 - 4X = 7
3 = 4X
X=¾
Ÿ P4 = 0.75 moles
H2 = 6 moles
10-4X moles
moles
b)
CPH3 at any time =
= 5-2X ×10-3
2×10-3 m3
m3
Therefore
dCPH3
=3.715×10-6 CPH3
dt
dX
-2×10-3
=-3.715×10-6 5-2X ×10-3
dt
dX
=3.715×10-6 2.5-X
dt
dX
d(2.5-X )
=
=-3.715×10-6 dt
2.5 - X
2.5-X
ln 2.5-X t -ln 2.5-X 0 =-3.715×10-6 t
2.5-X -2.715u10 t
=e
2.5
-6
2.5-X =2.5 e 2.715u10 t
-6
-6
X =2.5 1-e-2.715×10 t
Therefore
Moles PH 3 =10-4X =10-10 1-e-2.715u10 t = 10 e-2.715u10 t
-6
-6
Moles PH 4 =2.5 1-e-2.715u10 t
-6
Moles H 2 = 15 1- e-2.715u10 t
-6
Time required for 3 moles to react = time for X = 0.75
-6
X =2.5 1-e-2.715×10 t
= 0.75
T = 1.314 × 105 sec = 36.49 hours
2.4
2 NO + O2 ĺ122
a)
0.5 moles NO produces 0.5 moles NO2
b)
Initial
Species
Conc
NO
O2
NO2
1
3
0
dCNO
2
CO2
=-1.4×10-4 CNO
dt
d 1- 2 X
dX
2
=-1.4×10-4 1-2X 3-X =-2
dt
dt
Conc. at
X
1–2X
3–X
6X
4–X
3
Ÿ
dX
2
=0.7×10-4 1-2X 3-X
dt
2X-1
1
1
0.7 × 10-4 t =
+
ln
5 1-2X 25
X-3
Now for 0.5 mole of NO to react Ÿ 1 – 2X = 0.5 X = 0.25
1
1
0.7×10-4 t =
+
ln 0.5/2.75 =0.332
5×0.5 25
0.332
t=
×104 =0.474×104 sec = 1.317 hrs
0.7
3
3.1
(a) By an energy balance, the bicycle stops when final potential energy equals
initial kinetic energy. Therefore 1 2
mvi
2
vi2
mgh f or h f
F 20 km u 1000 m u 1 hr I
H hr
km 3600 sec K
2g
2 u 9.807
2
m
sec2
or h=1.57 m.
(b) The energy balance now is
1 2 1 2
mv f
mvi mghi or v 2f vi2 2 ghi
2
2
3600 sec 2
km 2
m
km
v 2f
20
2 u 9.807
u
u
u
70
m
1000 m
hr
hr
sec2
v f = 134.88 km/hr. Anyone who has bicycled realizes that this number is much
F
H
I
K
F
H
I
K
too high, which demonstrates the importance of air and wind resistance.
3.2
The velocity change due to the 55 m fall is
3600 sec I
km
u
c'v h 2 u 9.807 secm u 55 m u FH 1000
m
hr K
2
2
2
v f = 118.24 km/hr. Now this velocity component is in the vertical direction. The
initial velocity of 8 km/hr was obviously in the horizontal direction. So the final
velocity is
km
v
vx2 v 2y 11851
.
hr
3.3
(a) System: contents of the piston and cylinder
(closed isobaric = constant pressure)
M.B.: M 2 M1 'M 0 Ÿ M 2 M1 M
0
0
E.B.: M 2U 2 M1U1 'M H
Q Ws PdV
z
d i
c
h Q z PdV Q Pz dV Q PaV V f
M cU U h Q PM cV V h
Q M cU U h M c PV PV h M cU PV h cU PV h
M c H H h
M U 2 U1
2
2
1
2
2
1
2
1
2
1
1
1
2
2
1
1
Solutions to Chemical and Engineering Thermodynamics, 5h ed
P
1013
.
bar | 01
. MPa
V
1.6958
1.9364
T 100
T 150
Linear interpolation
1.8161
T 125q C
Final state
P 01
. MPa , V2
3.565
T 500q C
4.028
T 600q C
Linear interpolation
W
U
2506.7
2582.8
Chapter 3
H
2676.2
2776.4
2544.8
2726.3
3
m /kg
.
36322
3488.1
3704.7
Initial state
T2 500
3565
36322
.
.
T2 514.5q C
4.028 3565
.
600 500
514.5 500
.
H 2 34881
H 2 3519.5
600 500
3704.7 34881
.
Q 1 kg 3519.5 2726.3 kJ kg 7932
. kJ
z
PdV
a
1 bar u V2 V1
1 bar u 100,000
18161
m kg
.
.
f 1 bar u 36322
3
Pa
1 kg
1J
u
u
u 18161
m3 kg
.
bar m ˜ s2 ˜ Pa m2 ˜ s2 ˜ kg
1816
. kJ kg
(b) System is closed and constant volume
M.B.: M 2 M1 M
E.B.: M 2U 2 M1U1
Q
c
M U 2 U1
d i 0 Q W 0 – z PdV 0
'M H
h
s
Here final state is P 2 u 1013
.
bar ~ 0.2 MPa ; V2
(since piston-cylinder volume is fixed)
.
P 0.2 MPa ; V2 18161
T(q C)
V
U
500
1.7814
3130.8
600
2.013
3301.4
17814
.
.
18161
.
2.013 17814
T 500
0.0347
600 500 0.2316
T 515q C
V1
18161
m3 kg
.
.
~ 015
.
01498
U 2 31308
.
.
U 2 3156.4 kJ kg
01498
. 31308
.
33014
Q 1 kg u 3156.4 2544.8 kJ kg 6116
. kJ
(c) Steam as an ideal gas—constant pressure
N
PV
PV
Ÿ 1 1
RT
RT1
P2V 2
but V 2
RT2
2V 1 ; P1
P2
Solutions to Chemical and Engineering Thermodynamics, 5h ed
P12V 1
Ÿ T2
T2
PV
1 1
T1
T1
T2
27315
. 125 39815
. K
2 u T1 796.3 K 52315
. qC
760.9 kJ
z
PdV
W
2 u T1
1000 g kg
1 kJ
u 34.4 J mol K u 796.3 39815
. Ku
18 g mol
1000 J
N' H
Q
Chapter 3
P'V
P
FG NRT NRT IJ NRaT T f
H P P K
2
1
2
1
1
1000
.
u 8.314 u 39815
18
. kJ
1839
(d) Ideal gas - constant volume
PV
1 1
RT1
So again
2 P1 ˜V1
; T2
T2
PV
1 1
T1
Q
P2V 2
here V 1 V 2 ; P2
RT2
N'U
2T1
2 P1
796.3 K .
1
1000 g kg
. u
u 34.4 8.314 u 796.3 39815
1000
18 g mol
CP R ; Q
CV
577.0 kJ
3.4
M wU w, f M wU w, i
Mw
M weight
a
1 kg u CP Tf Ti
3.5
M weight u g u 1 m
1 kg
f 1 kg u 9.807 m s u 1 m u m 1kgJ s
1 kg u 4184
.
J g Ku
'T
Ws
2
2
1000 g
u 'T
kg
9.807
K
4184
.
u 1000
2
9.807 J
9.807
2.344 u 103 K
a
f a
f
From Illustration 3.2-3 we have that H T1, P1 H T2 , P2 for a Joule-Thomson
expansion. On the Mollier diagram for steam, Fig. 3.3-1a, the upstream and
downstream conditions are connected by a horizontal line. Thus, graphically,
we find that T ~ 383 K . (Alternatively, one could also use the Steam Tables
of Appendix III.)
Solutions to Chemical and Engineering Thermodynamics, 5h ed
Chapter 3
For the ideal gas, enthalpy is a function of temperature only.
Thus,
becomes
which
implies
that
H T1, P1 H T2 , P2
H T1 H T2 ,
T1 T2 600q C .
a
3.6
f a
af a f
f
System: Contents of Drum (open system)
mass balance: M t2 M t1 'M
energy balance:
MU MU
t2
z
'MH in Q Ws PdV
t1
steam
but Q
0 by problem statement, Ws
and
PdV
P'V
80q C
.
u 103 m3 kg ). Also from the Steam Tables
1029
V T
z
H in
0
(Note V T
is negligible.
H T
300q C, P
. bar
30
.
u 103 m3 kg ,
1003
25q C
300 kPa
3069.3 kJ kg
and recognizing that the internal energy of a liquid does not depend on pressure
gives
U
t1
U T
25q C, 1.013 bar
U sat., T
25q C
104.88 kJ kg
U T
80q C, 1.013 bar
U sat., T
80q C
334.86 kJ kg
and
U
t2
Now using mass balance and energy balances with M t1
100 kg yields
Solutions to Chemical and Engineering Thermodynamics, 5h ed
M t2 u 334.86 kJ 100 u 104.88 kJ
Chapter 3
M t2 100 u 3069.3 kJ
Thus
M t2 3069.3 334.86
M t2
3.7
108.41 kg , and 'M
100 u 3069.3 104.88
M t2 M t1
8.41 kg of steam added.
(a) Consider a change from a given state 1 to a given state 2 in a closed system.
Since initial and final states are fixed, U1 , U 2 , V1 , V2 , P1 , P2 , etc. are all
fixed. The energy balance for the closed system is
U 2 U1
where W
z
Ws PdV
adiabatic. Thus, U 2 U1
z
Q Ws PdV
total work. Also, Q
Q W
0 since the change of state is
W.
Since U1 and U 2 are fixed (that is, the end states are fixed regardless of the
path), it follows that W is the same for all adiabatic paths. This is not in
contradiction with Illustration 3.5-6, which established that the sum Q W is
the same for all paths. If we consider only the subset of paths for which
Q 0 , it follows, from that illustration that W must be path independent.
(b) Consider two different adiabatic paths between the given initial and final
states, and let W * and W ** be the work obtained along each of these paths,
i.e.,
Path 1: U 2 U1 W * ; Path 2: U 2 U1 W **
Now suppose a cycle is constructed in which path 1 is followed from the
initial to the final state, and path 2, in reverse, from the final state (state 2)
back to state 1. The energy balance for this cycle is
U 2 U1 W *
U 2 U1 W **
0 W * W **
a
f
Thus if the work along the two paths is different, i.e., W * z W ** , we have
created energy!
3.8
System contents of tank at any time
mass balance: M 2 M1 'M
energy balance: MU MU
'MH
c h c h
2
1
(a) Tank is initially evacuated Ÿ M1
in
0
Solutions to Chemical and Engineering Thermodynamics, 5h ed
Chapter 3
H in H 5 bar, 370q C 3209.6 kJ kg
interpolation).
Then
U P 5 bar, T ? 3209.6 kJ kg .
2
interpolation, using the Steam Tables (Appendix A.III) T 548q C
Thus
U 2
U
'M , and
M2
V P
V V
Therefore M
5 bar, T
548q C # 0.756 m3 kg
c
.
kg .
h 13228
1 m3 0.756 m3 kg
(by
By
(b) Tank is initially filled with steam at 1 bar and 150qC
and
Ÿ V1 V P 1 bar, T 150q C 194
. m3 kg
U1 2583 kJ kg ,
M V V 1 V 05155
.
kg . Thus, M
05155
.
'M kg . Energy balance
1
2
is M 2U 2 05155
.
u 2583
T2 , using T2 and P2
M 05155
.
u 3209.6 . Solve by guessing value of
5 bar to find V and U in the Steam Tables
2
2
1 m3 V2 are satisfied. By
trial and error: T2 ~ 425q C and M 2 # 1563
.
kg of which 1.323 kg was
present in tank intially. Thus, 'M M 2 M1 0.24 kg .
(Appendix A.III). See if energy balance and M 2
3.9
a) Use kinetic energy = mv2/2 to find velocity.
1 kg u
v 2 m2
2 sec2
1000 J = 1000
kg
so v= 44.72 m/sec
m2 sec2
b) Heat supplied = specific heat capacity u temperature change, so
1 mol
J
u 2510
.
u 'T 1000 J so 'T=2.225 K.
1000g u
55.85g
mol ˜ K
3.10 System resistor
Energy balance: dU dt Ws Q
where W E ˜ I , and since we are interested only in steady state dU dt
s
Thus
Q
Ws
1 amp u 10 volts
and 1 watt 1 volt u 1 amp 1 J s .
10 watt u 1 J s ˜ watt
25q C
ŸT
0.2 J s ˜ K
3.11 System
0.2 u T 25q C J s
. qC
750
gas contained in piston and cylinder (closed)
0
Energy balance: U t2 U t1 Q Ws PdV
(a) V
constant,
z
z
PdV
0.
0 , Q U t2 U t1
d
N U t2 U t1
i NC aT T f
V
2
1
From ideal gas law
N
Thus
PV
RT
114,367 Pa u 0120
m3
.
3
8.314 Pa ˜ m mol ˜ K u 298 K
5539
mol (see note following)
.
Solutions to Chemical and Engineering Thermodynamics, 5h ed
Q
NCV
298 K 298 630
.
3610
. K
T1 T2
Chapter 3
10,500 J
5.539 mol u 30.1 J mol ˜ K
Since N and V are fixed, we have, from the ideal gas law, that
P2
P1
3610
.
u 114.367 kPa
298.0
T2
P1
T1
T2
or P2
T1
1385
u 105 Pa
.
114367
.
u 105 Pa ;
301
. 8.314 38.414 J mol ˜ K
(b) P constant
CP CV R
Energy balance U t2 U t1
a
Q P'V , since P
constant
f Q PaV V f Q N aRT RT f
Ÿ Q NC aT T f
Ÿ NCV T2 T1
P
2
T2
2
1
2
1
1
T1 Q
NCP
298 10,500
5539
u 38.414
.
347.35 K
and
NR'T
P
'V
5539
.
mol u 8.314 Pa ˜ m3 mol ˜ K u 49.35 K
114,367 Pa
012
. 0.0199
V
Note: The initial pressure P
Patm
1013
.
bar
0.01987 m3
01399
.
m3
Patm Pwt of piston
1013
.
u 105 kPa
200 kg 1 Ns2
u
u 9.8 m s2
2
kg
m
˜
0.15 m
kPa
13067
.
Thus, intial pressure 114.367 kPa .
Pwt piston
13,067 N m2
3.12 System contents of storage tank (open system)
Mass balance: M 2 M1 'M
Energy balance:
'M H
MU MU
c h c h
2
1
in
since Q
13,067 Pa
W
0 and steam
entering is of constant properties.
Initially system contains 0.02 m3 of liquid water and 40 0.02 39.98 m3 of
steam.
Since vapor and liquid are in equilibrium at 50qC, from Steam Tables,
Also from the Steam Tables V L 0.001012 m3 kg ,
P 12.349 Pa .
V V 12.03 m3 kg ,
H V 25921
. kJ kg ,
H L 209.33 kJ kg ,
U L
209.32 kJ kg , and U V
24435
. kJ kg .
Solutions to Chemical and Engineering Thermodynamics, 5h ed
0.02 m3
0.001012 m3 kg
M1L
Chapter 3
U|
|V M
||
W
19.76 kg;
M1L M1V
. kg .
2308
39.98 m
. kg;
332
12.03 m3 kg
. u 24435
. 12,248.6 kJ
U1 19.76 u 209.32 332
1
3
M1V
Also
H in
. 010
. u 419.04
0.90 u 26761
2450.4 kJ kg
Possibilities for final state: 1) vapor-liquid mixture, 2) all vapor, and 3) all liquid.
First possibility is most likely, so we will assume V-L mixture. Since
P 1013
.
bar , T must be 100qC. Thus we can find properties of saturated vapor
and saturated liquid in the Steam Tables:
V L 0.001044 m3 kg ,
and
U L 418.94 kJ / kg ,
V V 16729
.
m3 kg ,
H V 26761
. kJ kg ,
U V 25065
. kJ / kg.
1 x 0.001044
V x 16729
.
2
0.001044 16719
.
x m3 kg , where
x quality
U 2 x 25065
. 1 x 418.94 418.94 2087.56 x kJ kg
Substituting into energy balance
M 2 418.94 2087.56 x 12,248.6
. f ˜ 2450.4
a M 2308
2
where
M2
V
V2
40 m3
0.001044 16719
.
x
(quality), M 2 46.36 kg , and
Solving by trial and error yields x 05154
.
'M 2328
. kg . Also the final state is a vapor-liquid mixture, as assumed.
3.13 System = tank and its contents (open system)
(a) Steady state mass balance
dM
M
M
0 M
1
2
3
dt
M
ŸM
M
10 kg min
3
1
2
b
g
.
.
M1
T1
M2
T2
.
M3 , T3
Steady state energy balance
dU
H M
H M
H
0 M
1 1
2 2
3 3
dt
H 3 H exit stream H at temperature of tank contents
Solutions to Chemical and Engineering Thermodynamics, 5h ed
Also T3
T
Chapter 3
temperature of tank contents
H C T T , assuming C is not a function of temperature
a
f
0 m
a fr m
a
5T 5T
1
ŸT
aT T f 65q C
10
2
Now H
0
P
0
P
a
fr m
5 H 0 CP T1 T0 5 H 0 CP T2 T0 10 H 0 CP T T0
1
2
1
fr
2
dM
M
M
(no useful information here)
0 M
1
2
3
dt
dU
H M
H M
H
energy balance:
M
1 1
2 2
3 3
dt
dU
d
dU
dT
dT
since CP | CV
but
MU
M
MCV
~ MCP
dt
dt
dt
dt
dt
dT
liquids. Thus MCP 3 5CPT1 5CPT2 10CPT3 and M 50 kg .
dt
(b) mass balance:
c h
10
dT3
2T3
dt
At t o f , T3
80 50
130 Ÿ T3
Ae t 5 C t
0 , T3
C 65q C
A C 25q C Ÿ A
40q C
So finally T3
65q C 40q Ce t 5 t
minutes
At t
E.B:
M FU F M iU i 'MH in
M FU F M FU F M i U i M i U i
c
L
L
V
V
Also known is that V
h c
60 m3
L
L
V
V
h
minutes
. H L, in 0.9 H V, in
M LF M VF 01
M LFVLF M VFVVF . Ÿ2 equations and 2 unknowns
V M VFVVF
VLF
M LF
FG V M V U M U IJ c M U M U h
K
H V
LMV M V M OP 01. H 0.9 H
Q
N V
F F
F
V V
L
F
L
F F
V V
F
L
F
V
F
V
F
V
i
L
L, in
i
L
i
V
V, in
3.14 Thermodynamic properties of steam from the Steam Tables
Initial conditions:
Specific volume of liquid and of vapor:
m3
m3
; VVi 08857
.
kg
kg
Specific internal energy of liquid and of vapor
kJ
kJ
U Li 3139
.
; U Vi 24759
.
kg
kg
VLi
.
1061
u 103
M.B: M f M i
'M i
for
i
V
Solutions to Chemical and Engineering Thermodynamics, 5h ed
200 liters
= 194.932 kg;
VLi
Mi
M Li M Vi ; M Li
M Vi
60 m3 200 liters
=14.476 kg and so Mi=209.408 kg
VVi
E.B.
c
M f U f M iU i 'MH in
M f U f M f U f M i U i M i U i
L
V
L
V
Chapter 3
h c
L
L
V
V
h
. H L, in 0.9 H V, in
M Lf M Vf 01
Total internal energy of steam + water in the tank
194.932u313.0 + 14.476u2475.9 = 9.686u104 kJ
Properties of steam entering, 90% quality
Specific volume = Vin = 0.1u1.061u10-3+ 0.9u0.8857 = 0.797 m3/kg
Specific enthalpy = H =0.1u504.70 + 0.9 u 2706.7 = 2.486 u103 kJ/kg
in
60 m3
Also have that V
M LfVLf M VfVVf .
This gives two equations, and two unknowns, M Lf and M Vf .
The solution (using MATHCAD) is M Lf = 215.306 kg and M Vf = 67.485 kg.
Therefore, the fraction of the tank contents that is liquid by weight is 0.761.
3.15 System contents of both chambers (closed, adiabatic system of constant volume.
Also Ws 0 ).
Energy balance: U t2 U t1 0 or U t2 U t1
(a) For the ideal gas u is a function of temperature only.
Thus,
U t2 U t1 Ÿ T t2 T t1 500 K . From ideal gas law
af af
af af
af af af af
PV
1 1
PV
2 2
N1RT1
N 2 RT2
but
N1 N 2 since system is closed
T1 T2 see above
and V2 2V1 see problem statement.
1
. MPa Ÿ T2 500 K, P2 05
. MPa
P1 5 bar 05
2
(b) For steam the analysis above leads to U t2 U t1 or, since the system is
closed U t
U t , V t
2V t . From the Steam Tables, Appendix III,
Ÿ P2
af af
af af af af
U at f U T 500 K, P 1 MPa U T 22685
. q C, P 1 MPa
2
1
2
1
1
# 2669.4 kJ kg
V t1 V T 22685
. q C, P
af
af af
af
1 MPa # 0.2204 m3 kg
Therefore U t2 U t1 2669.4 kg kg and
V t
2V t
0.4408 m3 kg . By, essentially, trial and error, find that
af
2
1
T ~ 216.3q C , P ~ 05
. MPa .
Solutions to Chemical and Engineering Thermodynamics, 5h ed
af af
af
af
Chapter 3
af
af
af
(c) Here U t2 U t1 , as before, except that U t1 U I t1 U II t1 , where
superscript denotes chamber.
Also, M t
M I t1 M II t1 {mass balance} and
2V M t
2V M I t M II t
V t
af
2
1
af
2
af
1
1
af
1
For the ideal gas, using mass balance, we have
a f P V P V Ÿ 2P
I
1 1
T1I
P1I P1II
T1I T1II
(1)
U 0 NCV T T0 , and cancel terms, use N
PV RT and get
P2 2V1
T2
Energy balance: N 2U 2
Substitute U
a
II
1 1
T1II
Eqns.
(1)
T2
N1IU 1I N1IIU 1II
f
2 P2
Using
2
and
(2)
P1I P1II
get
P2
(2)
7.5 u 10 Pa
5
0.75 MPa
and
529.4 K 256.25q C .
T2
(d) For steam, solution is similar to (b). Use Steam Table to get M1I and M1II in
terms of V.
Chamber 1: U1I 2669.4 kJ kg ; V1I 0.2204 m3 kg ;
M I V V I 4.537V
1
1
1
a
Chamber
2:
U1II U T 600 K, P
V II 05483
.
m3 kg ; M II 1824
. V V V II
1
1
2V1
Thus, V2
I
M M II
U
M IU I M IIU II
2
c
1
1
1
2V1
4.537V1 1824
. V1
. MPa
05
f 28459. kJ kg ;
1
0.3144 m3 kg ;
h b M M g 2720.0 kJ kg
I
1
II
1
By trial and error: T2 ~ 302q C 575 K and P ~ 0.76 MPa .
3.16 System: contents of the turbine (open, steady state)
(a) adiabatic
dM
M
ŸM
0 M
M
mass balance:
1
2
2
1
dt
0
dU
H M
H Q 0 W P dV
0 M
energy balance:
1 1
2 2
s
dt
dt
Ÿ Ws M1 H1 H2 M1 3450.9 28656
. kJ kg
c
c
h
5853
M
u 10
1 .
But Ws
5
h J kg
7.5 u 105 watt
7.5 u 105 J s
7.50 u 105 J s
.
kg s 4.613 u 103 kg h
1281
.
5853
u 105 J kg
(b) Energy balance is
0
dU
H M
H Q W P dV
0 M
1 1
2 2
s
dt
dt
M
1
Solutions to Chemical and Engineering Thermodynamics, 5h ed
a
Chapter 3
f
60 kJ kg
H 2 H 150q C, 0.3 MPa 27610
. kJ kg
where Q M
1
Thus
Ws 1281
.
kg s 3450.9 27610
. 60 kJ kg 807 kJ s
8.07 u 105 watt
807 kW
3.17 System: 1 kg of water (closed system).
Work of vaporization
PdV P dV
z
z
P'V since P is constant at 1.013 bar.
Also, from Steam Tables
m3 kg ; 'V 16719
.
m3 kg
V L 0.001044 m3 kg ; V V 16729
.
Energy balance for closed system (1 kg):
U U Q PdV Q 1013
.
.
u 105 Pa u 16719
m3 kg
2
z
1
Q 16945
.
u 105 J kg
U 2
U
1
. kJ kg
25065
2.5065 u 106 J kg
418.94 kJ kg
.
u 105 J kg
41894
Thus
Q U 2 U1 W
u 105 16945
u 105
.
.
2.5065 u 106 41894
2.2570 u 106 J kg
z
PdV
W
16945
u 105 J kg .
.
So heat needed to vaporize liquid
2.2570 u 106 J kg of which 016945
.
u 106 is
recovered as work against the atmosphere. The remainder, 2.088 u 106 kJ kg ,
goes to increase internal energy.
3.18 System = Contents of desuperheater (open, steady state)
Superheated steam
T=500ºC
P=3 MPa
Desuperheater
Water
25ºC
M
1
M
2
500 kg hr ; H 1 34565
. kJ kg
H sat’d liq., T 25q C
?; H
2
Mass B: 0
Saturated steam
2.25 MPa
104.89 kJ kg
M
M
M
1
2
3
H M
H M
H Q 0 W
0 P dV
M
1 1
2 2
3 3
s
dt
500 M 2 kg hr ; H3 H sat’d steam, P 2.25 MPa
0
Energy B: 0
M
3
Thus,
b
g
. kJ kg
28017
Solutions to Chemical and Engineering Thermodynamics, 5h ed
b
0 500 u 34565
. M
2
ŸM
2
Chapter 3
g
u 28017
104.89 500 M
.
2
1214
. kg hr
3.19 The process here is identical to that of Illustration 3.5-3, so that we can use the
equation
P2
P1 T1 CV CP P2 P1 Tin
a
T2
2.0 MPa , Tin
20.99 J mol K .
developed in the illustration.
CP
29.3 J mol K , CV
Here, P2
CP R
CP
Tin 5488
. K
CV
Cylinder 1: P1
0 , T2
Cylinder 2: P1
01
. MPa , T1
f
20q C
Cylinder 3: P1
1 MPa , T1
20q C
39315
. K,
27565
. qC
29315
. K
2.0
. 29315
. 20.99 29.3 2.0 01
. 39315
.
01
T2
120q C
29315
. K ; Ÿ T2
252.7q C
. K
52587
38216
. K 109.01q C
3.20 System: Gas contained in the cylinder (closed system)
M piston g
4000 kg
9.8 m s2
(a) P 01013
10133
u 105 u
.
.
MPa A
2.5 m2 1 kgm Ns2
11701
.
u 105 Pa
moles of
N
gas initially
in system
0117
.
MPa
PV
RT
11701
.
u 105 Pa u 25 m3
8.314 Pa ˜ m3 mol K u 29315
. K
1200
kmol
.
.
u 103 mol 1200
(b) Energy balance: U 2 U1 Q PdV
z
'V 3 m u 2.5 m
Final temperature:
2
T2
PV2
NR
U 2 U1
7.5 m ; P'V
Q P'V since P is constant.
11701
.
u 105 Pa u 7.5 m3
3
11701
u 105 Pa u 25 7.5 m3
.
3
. u 10 mol u 8.314 Pam3 mol K
12
a
N U 2 U1
8.7758 u 105 J
3812
. K 108.05q C
f NC aT T f
V
2
1
12
. u 10 mol u 30 8.314 J mol K u 3812
. 29315
. K
3
2.291 u 106 J
(c)
Q
'U P'V
'T of
gas
work
2.291 u 106 8.7758 u 105
work 27.7% of energy absorbed
'T 72.3%
.
3169
u 106 J
.
3169
MJ
(d) System: Gas contained within Piston + Cylinder (open system).
Solutions to Chemical and Engineering Thermodynamics, 5h ed
Chapter 3
[Note: Students tend to assume dT dt 0 . This is true, but not obvious!]
dN
mass balance:
N
dt
dV
d
0
NU
N H out Q
P
energy balance:
dt
dt
Here (1) P is constant, (2) Ideal Gas Law V NRT P , (3) T and P of Gas
Leaving Cylinder T and P of gas in the system. Thus,
d NRT
dN
dN
dU
N
U
P
H
dt
P
dt
dt
dt
dT
dN
d
NCV
NT
Ÿ H U
R
dt
dt
dt
dN
dT
dT
dN
dT
RT
0
NCV
NR
RT
Ÿ N CV R
dt
dt
dt
dt
dt
dT
0 Q. E.D.
Ÿ
dt
Thus T3 T2 3812
. K
Now going back to
F
H
I
K
a
N
dN
dU
U
dt
dt
H
dN
dt
Ÿ H U
f
dT
dU
dV
dN
and using
P
0
dt
dt
dt
dt
dN
P dV
dV
dN
P
RT
or
dt
dt
dt
RT dt
(**)
Since P and T are constants
N3
N2
V3
V2
25 m3
25 7.5 m3
0.7692
Thus N 3 0.7692 u 1200 mol 923 mol ;
'N 277 mol 0.277 kmol
3.21 (a) System: Gas contained within piston-cylinder (closed system) [neglecting the
potential energy change of gas]
energy balance:
d NU
dt
But T
Thus
PV
dT
Ÿ
NR
dt
N
dU
dt
dV
dT
Q P
; NCV
dt
dt
F I PA dh .
H K NR dt
P dV
NR dt
dh
Q PA
dt
Solutions to Chemical and Engineering Thermodynamics, 5h ed
F
H
Chapter 3
I
K
ACV P dh
dh
C
dh PACP dh
PA V 1
AP
R dt
dt
R
dt
R dt
30 J mol K
.
u 11701
u 105 Pa u 2.5 m2 u 0.2 m s
8.314 J mol K
Q
2111
. u 105 J s
(b) System: Gas contained within piston and cylinder (open system). Start from
result of Part (d), Problem 3.20 (see eqn. (**) in that illustration)
dN
dt
P dV
RT dt
PA dh
with P and T constant
RT dt
(See solution to Problem 3.20)
u 105 Pa u 2.5 m2
11701
.
u 0.2 m s
8.314 J mol K u 3812
. K
0.01846 kmol s
a
dN
dt
[check:
18.46 mol sec u 15 sec
Problem 3.20]
f 18.46 mol s
276.9 mol compare with part d of
3.22 System: gas contained in the cylinder (open system)
Important observation . . . gas leaving the system (That is, entering the exit valve
of the cylinder) has same properties as gas in the cylinder.
dN ½ Note that these are
mass balance
N
°°
dt
¾ Eqns. (d) and (e) of
d NU
° Illustration 3.5-5
energy balance
NH
°¿
dt
dN
Note that these are
N
mass balance
dt
Eqns. (d) and (e) of
d NU
energy balance
N H Illustration 2.5- 5
dt
Proceeding as in that illustration we get Eqn. (f)
|VU
|W
FG T t IJ
HT 0 K
CP R
FG P t IJ or T t
HP0K Pt
R CP
320
10a8.314 30f
169.05 K
(1)
where we have used a slightly different notation. Now using the mass balance we
get
dN
dt
F I V d a P T f N
H K R dt
d PV
dt RT
or
a4.5 28f mol s u 8.314 Pa ˜ m mol K
a f NR
8.908 Pa K ˜ s
d PT
dt
3
V
3
. m
015
Solutions to Chemical and Engineering Thermodynamics, 5h ed
Chapter 3
and
P
Tt
P
8.908 u 105 t
Tt 0
Using t 5 minutes
and (2) yields
bar K for P in bar and t in secs.
(2)
300 secs in Eqn. (2) and simultaneously solving Eqns. (1)
T 5 min
152.57 K , P 5 min
0.6907 bar
Computation of rates of change from mass balance
d ln P d ln T
F I 1 F dP Pd ln T I NR
H K T H dt dt K V or dt dt
d P
dt T
NRT
PV
(3)
From energy balance (using 2 eqns. above and eqn. (f) in Illustration (3.5-5))
a f
CV d ln T
R dt
d ln P T
C d ln T
or P
R dt
dt
d ln P
dt
(4)
Now using Eqn. (4) in Eqn. (3). Thus,
CV d ln T CV dT NRT
or
R dt
RT dt
PV
dT
N RT 2
.
K sec
1151
dt t 5 min
PVCV t 5 min
and
CP P dT
RT dt 5 min
dP
dt 5 min
0.0188 bar s
3.23 Consider a fixed mass of gas as the (closed) system for this problem. The energy
balance is:
d NU
dt
N
From the ideal gas law we have P
CV N
dT
dt
dU
dt
NCV
dT
dt
P
dV
dt
NRT V . Thus
NRT dV
C d ln T
Ÿ V
R dt
V
dt
d ln V
dt
or
CV T2
ln
R
T1
FG IJ
H K
V
T
ln 2 Ÿ 2
V1
T1
CV R
FG V IJ
HV K
1
2
(*)
Solutions to Chemical and Engineering Thermodynamics, 5h ed
Chapter 3
or
V2T2 CV R
V1T1CV R
VT CV R
constant
Substituting the ideal gas law gives PV CP CV PV J constant. Note that the
heat capacity must be independent of temperature to do the integration in Eqn. (*)
as indicated.
3.24 System: Contents of the tank (at any time)
(a) Final temperature T 330 K and pressure P
c
h
.
1013
u 105 Pa are known.
Thus, there is no need to use balance equations.
1013
u 105 Pa u 0.3 m3
.
8.314 J mol K ˜ 330 K
PV
RT
Nf
1108
. mol
0.01108 kmol
(b) Assume, as usual, that enthalpy of gas leaving the cylinder is the same as gas
in the cylinder . . . See Illustration 3.5-5. From Eqn. (f) of that illustration we
have
Pf
Pi
Thus
Nf
2136
. mol
FG T IJ
HTK
CP R
f
i
Tf
or
Tf
Ti
FG P IJ
H PK
R CP
f
FG 10133
.
u 10 I
H 10. u 10 JK
5 8.314 29
.
05187
6
i
05187
.
u 330 K 17119
. K,
0.02136 kmol .
Pf
1013
.
bar ,
and
3.25 Except for the fact that the two cylinders have different volumes, this problem is
just like Illustration 3.5-5. Following that illustration we obtain
2 P1i
T1i
2 P1i
2 P1f P2f
f
T1f
T2
2 P1f P2f or P f
for Eqn. (a')
2 i
P1
3
for Eqn. (c')
and again get Eqn. (f)
FG T IJ
HT K
f
1
i
1
Then we obtain P f
1333
. bar , T1f
CP R
FG P IJ
HP K
f
1
i
1
2234
. K , and T2f
328.01 K .
3.26 From problem statement P1f P2f P f and T1f T2f T f .
Mass balance on the composite system of two cylinders
N1f N 2f
N1i or
2 P1f P2f
f
T1f
T2
3P f
Tf
2 Pi
Ti
Solutions to Chemical and Engineering Thermodynamics, 5h ed
Chapter 3
Energy balance on composite system
N1iU 1i
and T f
FI
HK
3P f i
T1
2 P1i
3 2 i
T1
2 3
T1i
2 u 200 bar
3
2 Pi
3
N1f U 1f N 2f U 2f Ÿ P f
1333
. bar (as before)
250 K .
3.27 Even though the second cylinder is not initially evacuated, this problem still bears
many similarities to Illustration 3.5-5). Proceeding as in that illustration, we
obtain
2 P1i P2i
i
T1i
T2
2 P1i P2i
2 P1f P2f
f instead of Eqn (a')
T1f
T2
2 P1f P2f
3 P f instead of Eqn. (c)
2 u 200 1 u 20 / 3 140 bar ] and again recover Eqn. (f) for
[Thus, P f
Cylinder 1
FG T IJ
HT K
f
1
i
1
Solution is P1f
P2f
140 bar , T1f
FG P IJ
HP K
CP R
f
1
i
1
226.47 K , T2f
Eqn. (f)
28651
. K.
3.28 (a) System = Gas contained in room (open system)
dN
N
mass balance:
dt
d NU
dN N H Q H
Q
energy balance:
dt
dt
Thus,
Q
dN
d NU
H
dt
dt
UH
dU
dN
N
dt
dt
F I V d F PI
H K R dt H T K
V d P
NT d F P I
dT
dT
Q RT F I F I NC
RT ˜
NC
H R K dt H T K
P dt H T K
dt
dt
For the ideal gas, H U
PV
RT ;
dN
dt
d PV
dt RT
V
Q
Since P
NRT dP
dT
dT
NR
NCV
dt
dt
P dt
constant,
dP
dt
0 , Q
NCPdT
or
dt
V
Solutions to Chemical and Engineering Thermodynamics, 5h ed
dT
dt
Q RT
CP PV
Chapter 3
15
. u 103 W ˜ 8.314 J mol K ˜ 28315
. K
5
29 J mol K ˜ 10133
u 10 Pa ˜ 35
.
. u 5 u 3 m3
0.0229 K s 137
. K min
(b) System = Gas contained in sealed room (closed system) N
d NU
dU
dT
N
NCv
Q
Energy balance:
dt
dt
dt
Q
NCV
dT
dt sealed
room
CP dT
CV dt unsealed
room
0
29
. K min
u 137
29 8.314
1925
.
K min
3.29 In each case we must do work to get the weights on the piston, either by pushing
the piston down to where it can accept the weights, or by lifting the weights to the
location of the piston. We will consider both alternatives here. First, note that
choosing the gas contained within piston and cylinder as the system, 'U Q W .
Also
But 'U 0 , since the gas is ideal and T constant.
W PdV NRT ln Vf Vi , for the same reasons. Thus, in each case, we have
z
a
f
that the net heat and work flows to the gas are
W work done on gas
u 10
.
FG V IJ 2479 ln 1213
HV K
2.334 u 10
NRT ln
2
f
i
and Q
W
2
1622.5 J
1622.5 J (removed from gas)
If more work is delivered to the piston, the piston will oscillate eventually
dissipating the addition work as heat. Thus, more heat will be removed from the
gas + piston and cylinder than if only the minimum work necessary had been used.
Note that in each case the atmosphere will provide
Watm
P'V
1013
.
u 105 kPa u 2.334 1213
.
u 102 m3
11356
. J
and the change in potential energy of piston
mg'h
5 kg u 9.8 m s2 u
.
2.334 1213
u 102 m3
2
1 u 10 m
54.9 J
The remainder 1622.5 11356
. 54.9 432.0 J must be supplied from other
sources, as a minimum.
(a) One 100 kg weight.
An efficient way of returning the system to its original state is to slowly (i.e.,
at zero velocity) force the piston down by supplying 432.0 J of energy. When
the piston is down to its original location, the 100 kg is slid sideways, onto the
piston, with no energy expenditure.
An inefficient process would be to lift the 100 kg weight up to the present
location of the piston and then put the weight on the piston. In this case we
would supply
Solutions to Chemical and Engineering Thermodynamics, 5h ed
Mg'h
Chapter 3
b
g
2
u 102 m3
.
'V
m 2.334 u 10 1213
100 kg u 9.8 2 u
2
2
A
s
1 u 10 m
2
2
1098.6 kg m s 1098.6 J
Mg
This energy would be transmitted to the gas as the piston moved down. Thus
11356
. J 54.9 J 1098.6 J
W on gas
atmosphere
PE of piston
W J Q J
Efficient
1622.5
Inefficient 2289.1
22891
.
PE of weight
Wcycle Qcycle
1622.5 11905
. 432.0
22891
. 11905
. 1098.6
(b) Two 50 kg weights
In this case we also recover the potential energy of the topmost weight.
mg'h
50 kg u 9.8
1213
u 102 m3
m 1597
.
.
u
2
0.01 m2
s
188.2 J
Thus in an efficient process we need supply only
1622.5 11356
. 54.9 188.2
2438
. J
An efficient process would be to move the lowest weight up to the position of
the piston, by supplying
50 kg u 9.8
u 102 m3
.
m 2.334 1213
u
2
2
1 u 10 m2
s
549.3 J
Slide this weight onto the piston and let go. The total work done in this case
is
11356
. atmosphere
54.9
'PE of piston
2438
.
'PE of weight
549.3
supplied by us
19836
. J
Therefore
W J Q Wcycle Qcycle
1622.5 1378.7 2438
. J
Efficient
1622.5
19836
. 1378.7 604.9 J
Inefficient 1983.6
(c) Four 25 kg weights.
In this case the recovered potential energy of weights is
25 kg u 9.8 m s2 u
FG 1897
1213
1597
1213
1379
1213
.
.
.
.
.
.
u 10 I
JK m
H
1 u 10
302.3 J
Thus in an efficient process we need supply only
2
2
Solutions to Chemical and Engineering Thermodynamics, 5h ed
Chapter 3
1622.5 11356
. 54.9 302.3 129.7 J
An inefficient process would be to raise the lowest weight up to the piston,
expending
25 kg u 9.8 m s2 u
.
u 102 m3
2.334 1213
1 u 102 m
274.6 J
Thus the total work done is
11356
. 54.9 302.3 274.6 1767.4 J
and
W Q Wcycle Qcycle
. 129.5
Efficient
1622.5 1622.5 14930
. 274.4
Inefficient 1767.4 1767.4 14930
(d) Grains of sand
Same analysis as above, except that since one grain of sand has essentially
zero weight W 1622.5 J , Q 1622.5 J , Wcycle Qcycle 0 .
3.30 System = Gas contained in the cylinder (closed system)
d NU
dU
dT
dV
P
N
NCV
energy balance:
dt
dt
dt
dt
ideal gas equation of state}
Since CV and CP are constant
CV 1 dT
R T dt
Ÿ T2
FG IJ FG V IJ
H K HV K
1 dV
T
or 2
V dt
T1
. u
25 27315
P2
NRT dV
V
dt
R CV
1
FG L IJ
HL K
{Using the
R CV
1
2
FG 0.03 m IJ
H 0.03 0.6 u 0.05 m K
3
2
8.314 30 8.314
3
. K 44.58q C and
22857
.
1 22857
V T2
20 u u
7.666 bar
P1 1
.
2 29815
V2 T1
FG IJ FG IJ
H KH K
From the difference (change of state) form of energy balance
'U
Q
0
W
a
NCV T2 T1
f z PdV
PV
20 bar u 0.03 m3 kmol ˜ K
0.0242 kmol
RT 298.15 K u 8.314 u 102 bar ˜ m3
. 22857
. K
Ÿ W 'U 0.0242 kmol u 30 8.314 kJ kmol ˜ K ˜ 29815
3652
. kJ
Where has this work gone?
and N
Solutions to Chemical and Engineering Thermodynamics, 5h ed
Chapter 3
(a) To increase potential energy of piston
(b) To increase kinetic energy of piston
(c) To push back atmosphere so system can expand
(d) Work done against friction (and converted to heat).
To see this, write Newton’s 2nd Law of Motion for the piston
Patm u A
f
Fr
Frictional Force
mg
f
Fr
Pressure of gas (P) u A
a
MA Ÿ PA Patm A mg f fr
f
Thus, P
m dv
mg f fr
Patm A dt
A
A
'U 36,520 J PdV
z
z
PatmdV Now
f m dvdt ; v velocity of piston
1 dV
A dt
dh
dt
v (h
z
z
piston height) and v
36,520 J
z
1
mg
dv
m dv dV
dt dV f fr
dt
A
A
dt
A dt dt
Patm 'V 3000 J
Work against
atmosphere
mv 2
2
since
vinitial 0
dv
dt
(1)
c h
1 d 2
v
2 dt
mg'h
z
f fr vdt
1760 J
Work used to
increase potential
energy of piston
z
mv 2
1760 f fr vdt .
2
(a) If there is no friction f fr 0 then
Thus 36,520 J
3000 36520 3000 1760 J u 2
300 kg
v2
(b) If we assume only sliding friction, f fr
z
f fr vdv
2117
. m2 s2 Ÿ v
14.55 m s
kv
z
k v 2dt Ÿ 36520 3000 1760
z
m 2
v k v 2dt
2
In order to determine the velocity now we need to know the coefficient of
sliding friction k, and then would have to solve the integral equation above (or
integrate successively over small time steps). It is clear, however, that
v with friction v without friction
14.55 m s
System for
part a
25q C,
3.0 u 106 Pa = 3Mpa
125 kg/s
System for
part b
Solutions to Chemical and Engineering Thermodynamics, 5h ed
Chapter 3
3.31 25qC, 30
. u 106 Pa 3 MPa
125 kg s
(a) mass balance (steady-state)
M
0 M
1
2
ŸM
M
125 kg s
1
2
Energy balance (neglecting PE terms)
0
IJ
K
FG
H
FG
H
2
2
H v1 M
H v2
M
1
1
2
2
2
2
IJ
K
UvA mnvA ; U mass density, n molar density,
M
v velocity, A pipe area, m molecular weight.
M
P
vA
m RT
125 kg s
30
. u 106 Pa
u v m s u S u 0.09 m2
Ÿ
16 kg kmol 298.15 K u 8.314 u 103 Pa ˜ m3
Ÿ v 22.83 m s
a f
a
f
mv 2 16 kg kmol u 22.83 m s 2
4170
u 103 J kmol
.
2
2 u 1 kg ˜ m Ns2
Back to energy balance, now on a molar basis
a
mv22 mv12
C p T1 T2
2
2
As a first guess, neglect kinetic energy terms . . .
Cp T1 T2 0 Ÿ T1 T2 25q C
H1 H 2
a
417
. kJ kmol
f
f
Now check this assumption
. u 106 v1
n1v1 Pv
30
1 1
34.24 m s
n2
P2
2.0 u 106
Recalculate including the kinetic energy terms
m 2 2 16
Cp T1 T2
v1 v2
34.242 22.832 5209 J kmol
2
2
5209 J kmol
T2 T1 T1 014
. qC
. J mol u 1000 mol kmol
368
Thus the kinetic energy term makes such a small contribution, we can safely
ignore it.
v2
a
(b)
f
c
h
c
Mass balance on compressor (steady-state) 0
2.0 u 106 Pa
T1=25q C
compressor
h
N 1 N 2
3.0 u 106 Pa
T2= ?
Energy balance on compressor, which is in steady-state operation
Solutions to Chemical and Engineering Thermodynamics, 5h ed
0
Ws Ÿ Ws
N 1 H 1 N 2 H 2 Q
0
Chapter 3
N 1Cp (T2 T1)
25qC
adiabatic compressor
Can compute Ws if T2 is known or vice versa. However, can not compute both
without further information.
2.0 u 106 Pa
T2= ?
Gas cooler
3.0 u 106 Pa
T3=25q C
Analysis as above except that Q z 0 but W
0 N 2 N 3
Here we get Q N 1Cp ( T3 T2 )
R|
S|
T
0.
25q C
Can not compute Q until T2 is known.
See solution to Problem 3.10.
3.32 a) Define the system to be the nitrogen gas. Since a Joule-Thomson expansion is
isenthalpic, H T1, P1 H T2 , P2 . Using the pressure enthalpy diagram for
nitrogen, Figure 3.3-3, we have
H 135 K,20 MPa 153 kJ / kg and then T2 T P2 0.4 MPa, H 153 kJ / kg
a f a
f
c
h
From which we find that T = 90 K, with approximately 55% of the nitrogen as
vapor, and 45% as liquid.
b) Assuming nitrogen to be an ideal gas (poor assumption), then the enthalpy
depends only on temperature. Since a Joule-Thomson expansion is isenthalpic, this
implies that the temperature is unchanged, so that the final state will be all vapor.
3.33 Plant produces 136
. u 109 kwh of energy per year
Ÿ Plant uses 136
. u 109 u 4 544
. u 109 kwh of heat
6
1 kwh 36
. u 10 J
J year
. u 106
u 544
. u 109 kwh 19.584 u 1015 J year
Ÿ Plant uses 36
kwh
'H of rock (total) M ˜ Cp Tf Ti
a
12
10
f
kg u 1 J g K u 1000 g kg u 110 600 K
= 490 u 1015 J
Ÿ 19.58 u 1015 J year u x years 490 u 1015 J
x 2502
. years
3.34
a)
Body temperature is 37oC. From Appendix A.III, ice at 0oC has a volume of
0.0010908 m3/kg and an enthalpy (and internal energy) of -333.43 kJ/kg. For
water at 35oC Û = 146.67 kJ/kg and at 40oC Û = 167.56 kJ/kg, so that at 37oC
Solutions to Chemical and Engineering Thermodynamics, 5h ed
Chapter 3
Û = 155.03 kJ/kg. Also, 1 L = 10-3 m3, so that 1 L of water 10-3 m3/0.0010908
m3/kg = 0.9168 kg. So that the amount of energy needed to melt 1 L of ice is
0.9168 kg ¯(155.03-(-333.43))kJ/kg = 447.8 kJ.
b) 447.8 kJ/(42 kJ/kg fat) = 10.66 g of fat (= 0.023 lbs of fat)
c) For water at 0oC, Û = 0 kJ/kg, and V̂ = 0.001 m3/kg. Therefore, 1 L of
water = 1 kg, and the energy required to warm up the water is
1 kg¯155.03 kJ/kg = 155.03 kJ, and only 155.03/42 = 3.69 g of fat would be
consumed.
3.35
a)
M
2
M
3
M
M
1
2
Mass balance
M
M
10 kg/s M
3
1
2
Energy balance
Stream 120°C (1 atm)
M
3
M
10 kg/s M
1
2
ˆ
H
ˆ
M
1 1 M2H2
10
M
3
10 kq/s
kg ˆ ˆ
ˆ
H
H 3 M1H1 M
2 2
s
10
kq ˆ
H3
s
Ĥ1 = 83.96 J/g
Stream 2450°C, 2.5 MPa Ĥ 2 = 3344.0 J/g
Ĥ 3 = 376.92 J/g
Stream 390°C (1 atm)
M
10-M
u 83.96 10-M
M
1
MB
EB
M
1
M
2
b)
3344 10 x 376.92
9.101 kg/s M
1
9.101 kg/s M
1
9.101 kq/s
0.899 kq/s
M
10 kg/s, T=90°C
10 kg/s, T=20°C
O
N
P
10 kg/s, T=20°C
100 kg/s, quality = 50%
Ĥ
Steady balance:
1
1
2676.1 419.04
2
2
12547.6 J/g
Solutions to Chemical and Engineering Thermodynamics, 5h ed
Chapter 3
H
ˆ
H
ˆ
ˆ
ˆ
M
M
1 1 M2H2
3 3 M4H4
M
M
10 kg/s
But
1
3
M
2
M
4
? kg/s
x3344 = 10 × 376.92 + M
× 1547.6
10 × 83.96 + M
2
2
M ¯ (3344 – 1547.6) = 10 × (376.92 – 83.96)
2
=1.63 kg/s
M
2
3.36 ǻ vap H at 37°C ~ 2412 J/g
Amount evaporated = 4184 u 103
J
hr
1
2412 J
1.735 u 102 g
g
hr
1.735 kg/hr
If only 75% evaporates Ÿ
M
3.37
1.735
0.75
N
1.631 kg/hr of sweat produced
25°C, 0.8 bar
–50°C
0.1 bar
Mass balance – steady state
N
N
O N
N
1
2
2
1
Energy balance – steady state
W
W
H Q
N
H H Q
O N
1
1
2
1
2
W
Q
N
1
H 2 H1
H 2 H1
³ C dT
Cp T2 T1
p
T1
30
W
Q
N1
T2
J
u 75k
mol K
T2
³ C dT
p
2250
J
mol
Cp T2 T1
T1
30
J
u 75k
mol K
PV NRT
100 m3 Ÿ
N
Ÿ
N
1
2250
J
Mol
0.8bar u 100 m3
u 298.15 M
bar m3
8.314 u 105
mol K
3228 mols/min
PV
RT
0.03228 u 105 mol
3228 mols
Solutions to Chemical and Engineering Thermodynamics, 5h ed
W
Q
W
Q
Chapter 3
7.262 u 106 J/min 1.210 u 105 J/s
1.210 u 105 Watts 121.0 kw
$
Cost 0.2
u 121.0 kw 24.2 $/hr
kw ˜ hr
7.262 u 106 J/min 1.210 u 105 J/s
1.210 u 105 Watts 121.0 kw
$
Cost 0.2
u 121.0 kw 24.2 $/hr
kw ˜ hr
3.38 In the folder Aspen for Textbook>Chapter 3>Problem 3.38 (Prob 3.5)
3.39 In the folder Aspen for Textbook>Chapter 3>Problem 3.39 (Prob 3.16a)
3.40 In the folder Aspen for Textbook>Chapter 3>Problem 3.40 (Prob 3.22)
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Chapter 4
4
4.1
Ball (1) + Water (2) Energy balance: M U f M U f M U i M U i
(a)
System
1
c
h
1
2
c
Ÿ M1CV,1 T1 f T1i M 2CV,2 T2f T2i
Tf
M1CV,1T1i M 2CV,2T2i
M1CV,1 M 2CV,2
2
1
1
2
0
2
h 0 ; also T T . Thus
f
f
2
1
. u 75 12 u 103 u 4.2 u 5
5 u 103 u 05
. 12 u 103 u 4.2
5 u 103 u 05
8.31q C
[Note: Since only 'T s are involved, q C were used instead of K)].
dT
(b) For solids and liquids we have (eqn. 4.4-6). That 'S M CP
T
z
which CP is a constant. Thus
Ball: 'S
.
5 u 103 g u 05
RS
T
.
8.31 27315
J
u ln
.
75 27315
g˜K
. s
53161
12 u 103 g u 4.2
Water: 'S
RS
T
T
MCP ln 2 for the case in
T1
J
UV 53161
W . K
.
8.31 27315
J
u ln
.
5 27315
g˜K
UV 596.22 J
K
W
and
J
J
64.61
K
K
Note that the system Ball + Water is isolated. Therefore
J
'S Sgen 64.61
K
Energy balance on the combined system of casting and the oil bath
'S Ball Water
4.2
c
596.22 53161
.
h
c
h 0 since there is a common final temperature.
kJ
kJ
T 450iK 150 kg u 2.6
20 kg u 0.5
d
dT 450iK 0
kg ˜ K
kg ˜ K
M cCV,c T f Tci M oCV,o T f Toi
f
f
This has the solution Tf = 60oC = 313.15 K
Since the final temperature is known, the change in entropy of this system can be calculated
kJ
. 60
. 60
27315
27315
.
. u ln
4135
150 u 2.6 u ln
from 'S 20 u 05
K
. 50
. 450
27315
27315
F
H
4.3
I
K
Closed system energy and entropy balances
dU
dV dS Q ;
Sgen ;
Q Ws P
dt
dt dt T
dS
Thus, in general Q T
TSgen and
dt
F
H
I
K
Solutions to Chemical and Engineering Thermodynamics, 5th ed
dU
dS
dV
T
TSgen P
dt
dt
dt
dU dV
Q P
dt
dt
Ws
c
Reversible work: WsRev
h dUdt T dSdt P dVdt
WsRev Sgen
c
S
W
s gen
(b) System at constant S & P Ÿ
0
dU
dt
(a) System at constant U & V Ÿ
dS
dt
Chapter 4
0 and
dV
dt
h W
0
dS
dt
0
Rev
S
T
0 and
dP
dt
0Ÿ P
dV
dt
d
PV
dt
d
U PV
dt
dH
dt
so that
c
h W
Ws Sgen
rev
S
0
dU d
PV
dt dt
4.4
700 bar, 600oC
10 bar, T = ?
Steady-state balance equations
dM
M
0 M
1
2
dt
0
dU
H M
H
H M
H Q 0 W 0 P dV
M
0 M
1 1
2 2
1 1
2 2
s
dt
dt
or H 1 H 2
Drawing a line of constant enthalpy on Mollier Diagram we find, at P
At 700 bar and 600q C
V 0.003973 m3 kg
H 3063 kJ kg
S 5522
.
kJ kg K
10 bar, T # 308q C
At 10 bar and 308q C
V | 0.2618 m3 kg
H | 3063 kJ kg
S 7145
.
kJ kg K
Also
dS
dt
c
0
S S
M
1 2
1
Ÿ Sgen
h
0
S M
S Q
Sgen 0
M
1 1
2 2
T
Sgen kJ
or
S2 S1 7145
1623
. 5522
.
.
M1
kg ˜ K
4.5
1
System
2
Ws
Energy balance
'U
cU U h cU U h Q
f
2
i
2
f
1
i
1
adiabatic
z
constant
WS PdV volume
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Ws
c
h
c
T2f
Tf Ÿ
MCp T2f T2i MCp T1 f T1i
but T1 f
Ws
MCP
Chapter 4
h MC cT T h cT T h
p
f
2
i
2
f
1
i
1
2T f T1i T2i
Entropy balance
adiabatic
'S
f
2
i
2
f
1
i
1
cS S h cS S h 0
R T T UV 0 ; T T
or lnS
TTT W
Ÿ cT h cT T h or T
f
2
z
cS S h cS S h
i
2
f
1
i
1
f
1
i i
2 1
0
Q
dt Sgen
T
0 for maximum work
T
Tf
MCP ln 2i MCP ln 1 i
T2
T1
f
f
f
2
f 2
1
i i
1 2
f
2
T1i T2i ; but T1 f
f
T1i T2i and
Ws
MCP
4.6
2T f T1i T2i
1 bar
10 bar
290 K
575 K
T2f
Tf
2 T1i T2i T1i T2i
(a) Entropy change per mole of gas
T
P
'S CP ln 2 R ln 2 eqn. (4.4-3)
T1
P1
J
J
J
10
575
Thus 'S 29.3
ln
ln
0.9118
8.314
mol K 290
mol K 1
mol K
(b) System contents of turbine (steady-state system)
dN
Mass balance
0 N 1 N 2 Ÿ N 2 N 1 N
dt
dV 0
dU
0
Energy balance
Ws P
0 N 1 H 1 N 2 H 2 Q
dt
dt
T T
Ws N H 2 H 1 NC
1
P 2
J
Ws
W
CP T2 T1 29.3
u 575 290 K
mol K
N
a
a
f
a
f
f
J
mol
(c) In Illustration 4.5-1, W 7834.8 J mol because of irreversiblitities 'S z 0 , more work is
done on the gas here. What happens to this additional energy input? It appears as an increase of
the internal energy (temperature) of the gas.
83505
.
4.7
Heat loss from metal block
dU
dT
CP
Q
dt
dt
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Chapter 4
RS
T
Q heat out of metal
T T2 Q 1
T
Q heat into heat engine
W
a
T
f W Ÿ z Wdt
C z F1 I dT
H TK
T
T O
L
C MaT T f T ln P
W C aT T f C ˜ T ln
T
TQ
N
LF T I T O
W C T MG1 J ln P
NH T K T Q
F TI
Q z C dT C aT T f C T G1 J
H TK
CP
dT T T2
dt
T
T2
t
2
P
T1
0
2
P
2
1
2
2
P
2
P
1
2
1
1
2
2
1
1
P 2
T2
1
P
P
2
1
P 2
2
T1
Alternate way to solve the problem
T2
System is the metal block + heat engine (closed)
dU
dT
E.B.:
CP
Q W
dt
dt
dS
Q
Sgen
S.B.:
dt T 2
0 for maximum work
Q
W
W
W
4.8
dS dU
;
T2
dt
dt
dU
dS
T2
dt
dt
z
Wdt
z FH
T2
CP
dS dT
W ; dU CPdT ; dS
T
dt
C
T
CPdT T2 P dT CP 1 2 dT
T
T
T2
F
H
CP 1 T1
a
I
K
T2
dT
T
f
T
CP T2 T1 T2CP ln 2
T1
z FH
I
K
I
K
LF T I T O
C T MG1 J ln P
NH T K T Q
T2
CP
T1
1
T2
dT
T
1
2
2
1
P 2
This problem is not well posed since we do not know exactly what is happening. There are several
possibilities:
(1) Water contact is very short so neither stream changes T very much. In this case we have the
Carnot efficiency
K
W
Q
Thigh Tlow
Thigh
22
27 273
22
300
0.0733
7.33%
(2) Both warm surface water (27qC) and cold deep water (5qC) enter work producing device, and
they leave at a common temperature.
TH
TL
TO
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Chapter 4
b
dM
M
M
ŸM
M
0 M
M
H
L
0
0
H
L
dt
dU
H M
H M
H W 0
E.B.:
0 M
H H
L L
0 0
dt
H M
M
H
H M
W M
M.B.:
H
c
H
L
h
L
b
c
H
L
H H M
H H
M
0
0
H
H
L
L
a
f
a
h
g
C T T M
C T T
M
H P 0
H
L P 0
L
S.B.:
g
0
f
0
0
dS
S M
S M S Q
Sgen
0 M
H H
L L
0 0
T
dt
M H SH M L SL M H M L S0 0
b
c h
FG T IJ FG T IJ
HT K HT K
g
c
S S M
S S
M
H H
0
L L
0
M
H
H
H
L
H P
L P
0
M
L
T0 M H M L
f
a
1 or TH M H TL M L
L
0
h 0 Ÿ M C ln TT M C ln TT
0
0
0
M
TH H
b M M gT M b M M g
H
L
L
H
T0
L
From this can calculate T0 . Then
L
a
W
C T T M
C T T
M
H P 0
H
L P 0
L
f
This can be used for any flow rate ratio.
(3) Suppose very large amount of surface water is contacted with a small amount of deep water, i.e.,
!! M
. Then T ~ T
M
H
L
0
H
W
a
a
f
a
f
C T T M
C T T ~ M
C T T
M
H P H
H
L P H
L
L P H
L
f
(4) Suppose very large amount of deep water is contacted with a small amount of surface water, i.e.,
M
, T ~T .
M
H
L
0
L
C T T M
C T T ~ M
C T T
W M
H P
4.9
a
L
H
f
L P
a
L
L
f
H P
a
L
H
f
System
contents of the turbine. This is a steady-state, adiabatic, constant volume system.
dM
M
or M
M
(a) Mass balance
0 M
2
1
1
2
dt
Energy balance
constant
dU
H M
H Q adiabatic W P dV volume
0 M
s
1 1
2 2
dt
dt
Entropy balance
dS
S Q Sgen
S M
0 M
1 1
2 2
T
dt
0, by problem statement
Thus
M
M
4500 kg h
2
1
W M H H
M.B.
S2
S.B.
S
1
S1
c
1
2
h
E.B.
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Chapter 4
State T1 500q C
1
P1 60 bar
Steam
 o
Tables
H 1 3422.2 kJ kg
S1 68803
.
kJ kg
State
2
Steam
 o
Tables
T2 # 240.4q C
P2
10 bar
S2
S1
kJ
kgK
.
68803
H 2 | 29205
. kJ kg
kg
kJ
kJ
2257650
u 29205
. 3422.2
h
kg
h
(b) Same exit pressure P2 10 bar , and still adiabatic
H H .
Ÿ W M
Ws
4500
a
c
s
1
1
2
6271
. kW
f
h
Here, however,
Ws
c
08
. Ws Part a
Ÿ H 2
P
Thus
08
. 2.258 u 106
2
T2 # 286.7 K
S | 7.0677 kJ kg K
Steam
 o
Tables
. kJ kg
30208
10 bar
Sgen
h kJh 4500c H 3422.2h kJh
c
2
kJ
kJ
.
7.0677
8433
.
h 4500 kgh u 68803
kg K
K˜h
S S
M
1 1
2
(c) Flow across valve is a Joule-Thompson (isenthalpic expansion) ... See Illustration 3.4-1.
Thus, H into valve H out of valve , and the inlet conditions to the turbine are
H 1
P1
H out of valve
H into valve
3422.2 kJ kg
30 bar
T1 | 484.8q C
S1 | 71874
.
kJ kg K
Steam
 o
Tables
Flow across turbine is isentropic, as in part (a)
S2
P2
S1 71874
.
kJ kg K
10 bar
Steam
 o
Tables
T2 # 3181
. qC
H | 3090.4 kJ kg
2
kg
kJ
kJ
u 3090.4 3422.2
1493
.
u 106
414.8 kW
h
kg
h
4.10 Since compression is isentropic, and gas is ideal with constant heat capacity, we have
Ws
4500
FG T IJ FG P IJ
HT K H PK
2
2
1
1
FG 3 u 10 IJ
H 2 u 10 K
aT T f
Problem 3.31, that W NC
So that T2
T1
FG P IJ
HPK
6 8.314 36.8
R CP
2
29815
.
6
1
s
R CP
P
2
1
326.75 K . Now using, from solution to
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Ws
125
Chapter 4
kg 1 mol
J
1000 g
u
u 368
u 326.75 29815
.
. Ku
mol K
kg
s
16 g
8.23 u 106 J s
The load on the gas cooler is, from Problem 3.31,
Q
a
T T
NC
p 3
2
f
125 kg s u 1000 g kg
J
.
. 326.75 K
u 368
u 29815
16 g mol
mol K
8.23 u 106 J s
4.11 (a) This is a Joule-Thomson expansion Ÿ H 70 bar, T ? H 10133
.
bar, T 400q C | H 1 bar, T 400q C
3278.2 kJ kg
and T 447q C , S 6.619 kJ kg K
(b) If turbine is adiabatic and reversible S
S
6.619 kJ kg K and P
0 , then S
c
h
gen
out
in
1013
.
bar. This suggests that a two-phase mixture is leaving the turbine
Let x
fraction vapor
S V 7.3594 kJ kg K
.
S L 13026
kJ kg K
Then x 7.3594 1 x 13026
.
fluid leaving turbine is
H
.
u
08788
.
26755
H V sat’d, 1 bar
6.619 kJ kg K or x
.
1 08778
u
.
08778
.
417.46
H L sat’d, 1 bar
2399.6
Energy balance
0
H M
H Q 0 W P dV
M
s
in in
out out
dt
0
but M
M
in
out
Ws
kJ
3278.2 2399.6 878.6
Ÿ
M in
kg
(c) Saturated vapor at 1 bar
S
Ws
M
in Actual
Efficiency %
Sgen
M
in
7.3594 kJ kg K ; H
.
3278.2 26755
602.7 u 100
878.6
7.3594 6.619
. kJ kg
26755
602.7 kJ kg
68.6%
0.740 kJ Kh
Therefore the enthalpy of
kJ
kg
Solutions to Chemical and Engineering Thermodynamics, 5th ed
(d)
M
ŸM
M
1
2
2
0
W
Steam
70 bar
447q C
c
c
M
1
h
H H W Q P dV
M
1
1
2
s
dt
Q
S S S
M
1 1
2
gen
T
0
Water
1 bar
25q C
Chapter 4
0
Q
h
Simplifications to balance equations
dV
Sgen 0 (for maximum work); P
0 (constant volume)
dt
Q Q
where T0 25q C (all heat transfer at ambient temperature)
T T0
kJ kJ
; S sat' d liq, T 25q C 0.3674
H sat' d liq, 25q C 104.89
kg
kg K
Q
Ws
T0 S2 S1 ;
H 1 H 2 T0 S2 S1
H 1 T0 S1 H 2 T0 S2
M
M
c
h
c
h c
h c
h
max
Ws
M
. u 6.619 104.89 29815
. u 0.3674
3278.2 29815
max
1304.75 4.65 1309.4 kJ kg
4.12
Take that portion of the methane initially in the tank that is also in the tank finally to be in the
system. This system is isentropic S f Si .
(a) The ideal gas solution
S i Ÿ Tf
Sf
FP I
TG J
HPK
f
R Cp
i
N=
PV
Ÿ Ni
RT
'N
N f Ni
PV
i
RTi
0.0195
35.90 kg u 1000
28
g
mol
150.2 K
1964.6 mol; N f
Si
m3
, so that mi
kg
Ni
8.314 36
Pf V
RTf
1768.4 mol
(b) Using Figure 2.4-2.
70 bar | 7 MPa, T = 300 K
Vi
F 35. I
H 70 K
300
i
g
kg
505
. kJ kg K
0.7 m3
m3
0.0195
kg
1282 mol
S f
. kg.
3590
196.2 mol
Solutions to Chemical and Engineering Thermodynamics, 5th ed
At 3.5 bar = 0.35 MPa and S f
3
Vf
.
0192
.
3646
kg u 1000
Nf
28
'N
4.13
g
kg
.
3646
kg.
130.2 mol
g
mol
N f Ni
505
. kJ kg K Ÿ T | 138 K. Also,
0.7 m3
m3
.
0192
kg
m
, so that m f
kg
Chapter 4
130.2 1282
11518
. mol
dS
C
dV
dT
eqn. (4.4-1)
R
V
T
'S
z LMN
a R bT cT 2 dT 3 e dT
dV
R
2
T
V
T
f a
2
OP
Q
z
so that
a
S T2 , V 2 S T1, V 1
f a R ln TT baT T f 2c cT T h
2
2
2
1
2
1
1
c
h c
h
V
d 3
e
T2 T13 T22 T12 R ln 2
V1
3
2
Now using
PV
a
f a
S T2 , P2 S T1, P1
RT Ÿ
V2
V1
T2 P1
˜ Ÿ
T1 P2
f a ln TT baT T f 2c cT T h
d
e
P
cT T h cT T h R ln
P
3
2
2
2
2
2
1
2
1
1
3
2
2
2
3
1
2
1
2
1
Finally, eliminating T2 using T2
a
f a
S P2 ,V 2 S P1,V 1
T1 P2 V 2 PV
1 1 yields
f a lnFGH PPVV IJK Rb a P V PV f
c
a P V f a PV f
2R
d
a P V f a PV f
3R
2
2
2
1 1
2
1 1
2
2
2
2
2
3
2
2
1 1
3
3
1 1
eR 2
P
2
P2V 2 2 PV
R ln 2
1 1
P1
2
d
i d
i
Solutions to Chemical and Engineering Thermodynamics, 5th ed
4.14
Chapter 4
System: contents of valve (steady-state, adiabatic, constant volume system)
Mass balance
0
Energy balance
0
N 1 N 2
0
dV
N 1 H 1 N 2 H 2 Q 0 Ws P
dt
Ÿ H1 H 2
0
Q 0
N 1 S 1 N 2 S 2 Sgen T
Sgen
Ÿ 'S S 2 S 1
N
(a) Using the Mollier Diagram for steam (Fig. 3.3-1a) or the Steam Tables
Entropy balance 0
T1
P1
H 1
'S
7 bar
T2 | 293q C
Ÿ . J g
30453
S2 7.277 J g K
H 2 30453
. J g . Thus S1
S S 0.717 J g K
2
H 2 Ÿ T1
a
293q C
p
T2
600 K
f a f C ln TT R ln PP
S T2 , P2 S T1, P1
P
R ln 2
P1
'S
.
65598
J g K ; Texit
1
(b) For the ideal gas, H 1
'S
4.15
600 K P2
35 bar H 2
2
2
1
1
. J mol K Ÿ
1338
0.743 J mol K
From the Steam Tables
P 15538
.
MPa
L
V
0.001157 m3 / kg VV 012736
.
m3 / kg
U L 850.65 kJ / kg
U V 25953
. kJ / kg
At 200oC,
L
L
H
H
852.45 kJ / kg
27932
. kJ / kg
L
V
S
2.3309 kJ / kg ˜ K S
6.4323 kJ / kg ˜ K
.
kJ / kg ˜ K
'H vap 1940.7 kJ / kg 'S vap 41014
(a) Now assuming that there will be a vapor-liquid mixture in the tank at the end, the properties
of the steam and water will be
P 0.4578 MPa
V L 0.001091 m3 / kg VV 0.3928 m3 / kg
o
At 150 C,
U L
H L
63168
. kJ / kg
632.20 kJ / kg
L
S
18418
.
kJ / kg ˜ K
vap
'H
2114.3 kJ / kg
U V
H V
2559.5 kJ / kg
27465
. kJ / kg
V
S
68379
.
kJ / kg ˜ K
vap
'S
4.9960 kJ / kg ˜ K
(b) For simplicity of calculations, assume 1 m3 volume of tank.
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Chapter 4
Then
Mass steam initially =
0.8 m3
0.12736 m3 / kg
6.2814 kg
0.2 m3
172.86 kg
0.001157 m3 / kg
6.2814
Weight fraction of steam initially =
0.03506
179.14
6.2814
Weight fraction of water initially =
0.96494
179.14
The mass, energy and entropy balances on the liquid in the tank (open system) at any time
yields
L L
L L
dM L
L ; dM U
L H V ; and dM S
L SV
M
M
M
dt
dt
dt
L
dU L L dM L
L H V H V dM
ML
M
U
or
dt
dt
dt
L
L
dU
dM
ML
H V U L
dt
dt
Also, in a similar fashion, from the entropy balance be obtain
dS L dM L V L
dM L vap
ML
'S
S S
dt
dt
dt
There are now several ways to proceed. The most correct is to use the steam tables, and to use
either the energy balance or the entropy balance and do the integrals numerically (since the
internal energy, enthalpy, entropy, and the changes on vaporization depend on temperature.
This is the method we will use first. Then a simpler method will be considered.
Using the energy balance, we have
dM L
dU L
, or replacing the derivatives by finite differences
L
V
M
H U L
MiL1 MiL U iL1 U iL
U L U iL
or finally MiL1 MiL 1 i V1
L
V
L
Mi
H i U i
H i U iL
So we can start with the known initial mass of water, then using the Steam Tables and the data
at every 5oC do a finite difference calculation to obtain the results below.
Mass water initially =
c
c
h
h
FG
H
i
1
2
3
4
5
6
7
8
9
10
11
T (oC)
200
195
190
185
180
175
170
165
160
155
150
U iL (kJ/kg K)
850.65
828.37
806.19
784.10
762.09
740.17
718.33
696.56
674.87
653.24
631.68
IJ
K
H iV (kJ/kg K)
2793.2
2790.0
2786.4
2782.4
2778.2
2773.6
2768.7
2763.5
2758.1
2752.4
2746.5
MiL (kg)
172.86
170.88
168.95
167.06
165.22
163.42
161.67
159.95
158.27
156.63
155.02
So the final total mass of water is 155.02 kg; using the specific volume of liquid water at
150oC listed at the beginning of the problem, we have that the water occupies 0.1691 m3
leaving 0.8309 m3 for the steam. Using its specific volume, the final mass of steam is found to
be 2.12 kg. Using these results, we find that the final volume fraction of steam is 83.09%, the
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Chapter 4
final volume fraction of water is 16.91%, and the fraction of the initial steam + water that has
been withdrawn is
(172.86+6.28-155.02-2.12)/(172.86+6.28) = 0.1228 or 12.28%. A total of 22.00 kg of steam
has withdrawn, and 87.7% of the original mass of steam and water remain in the tank.
For comparison, using the entropy balance, we have
dM L
dS L
, or replacing the derivatives by finite differences
L
V
M
S S L
SiL1 SiL
MiL1 MiL SiL1 SiL
L
L
of
finally
1
M
M
1
i
i
L
M
'S vap
'S vap
i
FG
H
i
i
IJ
K
So again we can start with the known initial mass of water, then using the Steam Tables and
the data at every 5oC do a finite difference calculation to obtain the results below.
i
T (oC)
1
2
3
4
5
6
7
8
9
10
11
200
195
190
185
180
175
170
165
160
155
150
SiL (kJ/kg K)
2.3309
2.2835
2.2359
2.1879
2.1396
2.0909
2.0419
1.9925
1.9427
1.8925
1.8418
SiL (kJ/kg K)
6.4323
6.4698
6.5079
6.5465
6.5857
6.6256
6.6663
6.7078
6.7502
6.7935
6.8379
MiL (kg)
172.86
170.86
168.92
167.02
165.17
163.36
161.60
159.87
158.18
156.53
154.91
So the final total mass of water is 154.91 kg; using the specific volume of liquid water at
150oC listed at the beginning of the problem, we have that the water occupies 0.1690 m3
leaving 0.8310 m3 for the steam. Using its specific volume, the final mass of steam is found to
be 2.12 kg. Using these results, we find that the final volume fraction of steam is 83.10%, the
final volume fraction of water is 16.90%, and the fraction of the initial steam + water that has
been withdrawn is
(172.86+6.28-154.91-2.12)/(172.86+6.28) = 0.1234 or 12.34%. A total of 22.11 kg of steam
has withdrawn, and 87.7% of the original mass of steam and water remain in the tank.
These results are similar to that from the energy balance. The differences are the result of
round off errors in the simple finite difference calculation scheme used here (i.e., more
complicated predictor-corrector methods would yield more accurate results.).
A simpler method of doing the calculation, avoiding numerical integration, is to assume that
the heat capacity and change on vaporization of liquid water are independent of temperature.
Since liquid water is a condensed phase and the pressure change is small, we can make the
following assumptions
U L | H L and H V H L 'H vap
dT L
dS L CPL dT L
dU L dH L
|
| CPL
|
; and
dt
dt
T dt
dt
dt
With these substitutions and approximations, we obtain from the energy balance
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Chapter 4
dH L dM L vap
dU L dM L V L
'H
o ML
H U
dt
dt
dt
dt
dT dM L vap
'H
M LCPL
dt
dt
Now using an average value of CPL and 'H vap over the temperature range we obtain
c
h
1 dM L
M L dt
or
ML
CPL dT
'H vap dt
CPL
150 200
'H vap
FG M IJ
HM K
L
f
L
i
ln
and from the entropy balance
dS L dM L vap
C L dT dM L vap
ML
'S
'S
o ML P
dt
dt
T dt
dt
Now using an average value of CPL and 'S vap over the temperature range we obtain
CPL dT
T'S vap dt
F
H
1 dM L
M L dt
150 27315
CPL
.
ln
200 27315
.
'S vap
or
I lnFG M IJ
K HM K
L
f
L
i
From the Steam Table data listed above, we obtain the following estimates:
U (T 200o C) U (T 150o C) 852.45 632.20
kJ
CPL
4.405
kg ˜ K
200o C - 150o C
50
or using the ln mean value (more appropriate for the entropy calculation) based on
FG T IJ SaT f SaT f
HTK
CPL ln
2
2
1
1
S(T
kJ
.
200o C) S(T 150o C) 2.3309 18418
4.3793
.
.
200 27315
47315
kg
˜K
ln
ln
.
.
150 27315
42315
Also, obtaining average values of the property changes on vaporization, yields
1
kJ
1
u 'H vap T 150o C 'H vap T 200o C
u 2114.3 1940.7 2027.5
'H vap
2
2
kg
1
1
kJ
4.5487
u 'S vap T 150o C 'S vap T 200o C
u 4.9960 41014
'S vap
.
2
2
kg ˜ K
With this information, we can now use either the energy of the entropy balance to solve the
problem. To compare the results, we will use both (with the linear average Cp in the energy
balance and the log mean in the entropy balance. First using the energy balance
M fL
CPL
4.405 u 50
ln
150
200
.
010863
L
vap
M
2027.5
'H
CPL
F
H
F
H
I
K
b
g
b
g
I
K
b
b
g
g
FG IJ
H K
i
M
L
f
L
i
exp 010863
.
.
089706
M
Now using the entropy balance
M fL
.
CPL
150 27315
ln
ln
L
vap
.
M
200 27315
'S
FG IJ
. I
. I
42315
F
I 4.3793 lnF 42315
0.9628 lnF
H
K
H
K
H
.
. K
4.5487
47315
47315
H K
M
. I
F 42315
.
089805
H 47315
. K
M
i
L
f
L
i
0.9628
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Chapter 4
Given the approximations, the two results are in quite good agreement. For what follows, the
energy balance result will be used. Therefore, the mass of water finally present (per m3) is
.
. kg
u M L initial 15506
M L final 0897
L
L
o
. u 0.001091 01692
.
occupying V M final u V 150 C 15506
m3
b
g
Therefore, the steam occupies 0.8308 m3 , corresponding to
0.8308 m3
2115
.
kg
m3
0.3928
kg
So the fraction of liquid in the tank by mass at the end is 155.06/(155.06+2.12) = 0.9865,
though the fraction by volume is 0.1692. Similarly the fraction of the tank volume that is
steam is 0.8308, though steam is only 2.12/(155.06+2.12) = 0.0135 of the mass in the tank.
0.8308 m3
V V 150o C
M V final
b
g
(c) Initially there was 6.28 + 172.86 = 179.14 kg of combined steam and water, and finally from
the simpler calculation above there is 155.06 + 2.12 = 157.18 kg. Therefore, 87.7% of the
total amount of steam + water initially in the tank are there finally, or 12.3% has been
withdrawn. This corresponds to 21.96 kg being withdrawn. This is in excellent agreement
with the more rigorous finite difference calculations done above.
4.16 (a)
dN
dt
0
N 1 N 2 ;
dU
dt
0
dV
N 1 H 1 N 2 H 2 WS Q P
dt
dS
dt
0
or N 2
N 1
Q
N 1 S 1 N 1 S 2 Sgen
T0
z
Tf
WS
= H 2 - H1 =
CPdT
N 1
298.15K
Sgen
N
W
WS N 1 H 1 N 1 H 2 or S = H 2 - H 1
N
1
S 2 S1
CP
37151
.
1
c
h
CP ˜ Tf 29815
. K
J
mol ˜ K
if the heat capacity is independent of
temperature. First consider the reversible case,
z
Tf
S 2 S1
0 gives
CP
dT
T
T
i
WSrev
N 1
Wact
CP ˜ 49914
. 29815
. K
125
. WSrev
(b)
dP
P
1
7467
c
The
J
.
mol
The
solution
is
499.14K.
actual
work
is
25%
Then
greater
h
J
CP ˜ Tf 29815
. K
mol
549.39 K
9334
The solultion is Tf
z
10
R
Repeat the calculation with a temperature-dependent heat capacity
CP T 22.243 5977
.
˜ 102 T 3499
.
˜ 105 T 2 7.464 ˜ 109 T 3
Assuming reversibility Tf = 479.44K. Repeating the calculations above with the temperaturedependent heat capacity we find Wact = 9191 J, and Tf =520.92K.
So there is a significant difference between the results for the constant heat capacity and variable
heat capacity cases.
Solutions to Chemical and Engineering Thermodynamics, 5th ed
4.17
Chapter 4
Ti = 300 K, Tf = 800 K, and Pi = 1.0 bar
CP (T ) = 29.088 - 0.192 u 10-2 T + 0.4 u 10-5 T 2 - 0.870 u 10-9 T 3
z
T f 800K
J
mol ˜ K
z
Pf
CP (T )
dT
T
T 300K
P
i
dP
P
P 1
i
Calculated final pressure Pf = 3.092 u 106 Pa.
z
T f 800K
Wrev
1458
u 104
.
CP (T )dT
Ti 300K
4.18
J
mol
Stage 1 is as in the previous problem.
Stage 2
Following the same calculation as above.
Stage 2 allowed pressure Pf ,2 = 9.563 u 107 Pa
Wrev = 1.458 u 10-4
J
= Stage 2 work
mol
Stage 3
Following the same calculation method
Pf ,3 = 2.957 u 10-9 Pa = Stage 3 allowed pressure.
J
= Stage 3 work
mol
Question for the student: Why is the calculated work the same for each stage?
Wrev = 1.458 u 104
4.19
The mass, energy and entropy balances are
dM
M
0, M
M
M
1
2
2
1
dt
dU
H M
H Q W ; M
H H W
0 M
s
s
1 1
2 2
1
1
2
dt
H H
Ws M
1
2
1
c
c
dS
dt
0
300q C, 5 bar
05
. MPa
100q C, 1 bar
01
. MPa
c
gen
M
1
2676.2 3064.2
S2 S1
h
0
S S S
M
gen
1 1
2
h
H 1 3064.2 kJ kg
S1 7.4599 kJ kg K
H
2676.2 kJ kg
2
S2
Ws
M
1
S
0;
h
S M
S Q S
M
gen
1 1
2 2
T
S S
Sgen M
1 2
1
c
h
7.3614 kJ kg K
388 kJ kg satisfied the energy balance.
7.3614 7.4599
0.0985 kJ kg K can not be. Therefore the process is impossible.
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Chapter 4
4.19
N
M
0.5 kg/s 100°C, steam
1 kg/s
e
300°C
1 MPa
Streams
1
2
3
0.5 kg/s 100°C, sat’d liq
T
300°C
100°C
100°C
P
1 MPa
Condition
sup heated
Ĥ kJ/kg
3051.2
Ŝ kJ/kg
7.1229
0.10135 MPa
sat’d liq
419.04
1.3069
Energy balance
kg
ˆ 0.5 u 1344.0 W
0
1 u 3051.2 0.5 u H
2
s
1.15 MW 1500 kJ
but W
s
ˆ 0.5 u 419.04 1500
3051.2 0.5 u H
2
Ĥ 2
2683.4
kJ
kg
ˆV
Now at at 0.1 MPa and T 100o C H
2676.1
kJ
kJ
and Sˆ 7.3549
kg
kg ˜ K
So steam is slightly superheated and Quality is 1
Entropy balance (using Ŝ at saturation)
1 u 7.1229 – 0.5 u 1.3069 – 0.5 u 7.3549 + S gen = 0
S
0.5 1.3069 7.3549 7.1229
gen
2.792 kJ / K S
So process is not possible
(using better estimate, by interpolation, for exit entropy of steam would have no effect on
conclusion).
4.20 Steam 20 bar
2 MPa and 300q C
H 30235
. kJ kg
S 6.7664 kJ kg
U 2772.6 kJ kg
Final pressure = 1 bar. For reference saturation conditions are
P 01
. MPa, T 99.63
L
H L 417.46
U
417.36
S L 13026
.
V
V
25061
26755
S V 7.3594
U
.
H
.
(a) Adiabatic expansion valve W 0 and Q 0
(from Steam Tables)
Solutions to Chemical and Engineering Thermodynamics, 5th ed
dM
;
M
M
0; M
M
2
1
1
2
dt
dU
H M
H
E.B.:
0 ; H 2 H 1
M
1 1
2 2
dt
From Steam Tables
T 250q C
.
Ÿ H2 30235
H 2974.3 kJ kg
Chapter 4
M.B.:
P
By interpolation T
kJ kg K
S 81245
.
dS
dt
S
gen
M
gen
300q C
H 3074.3 kJ/kg
275q C gives H = 3023.5 kJ / kg Ÿ all vapor
T
01
. MPa
S 8.0333 kJ kg K
S 8.2158 kJ/kg K
M
S S
MS
gen
1
2 2
0
S2 S1 = 8.1254 6.7664
kJ kg K
1359
.
(b) Well designed, adiabatic turbine
H M
H W
E.B.: M
1 1
2 2
cH H h
0 ; W
2
1
S M
S 0 ; S S ; S 6.7664 kJ kg K
S.B.: M
1 1
2 2
2
2
1
Ÿ Two-phase mixture. Solve for fraction of liquid using entropy balance.
.
6.7664
x ˜ 7.3594 1 x ˜ 13026
x 0.902 not good for turbine!
. 0.098 u 417.46
0.902 u 26755
H
2
W
M
.
2454.2 30235
569.3 kJ kg
W
M
2454.2 kJ kg
569.3 kJ kg
(c) Isorthermal turbine Ÿ superheated vapor
T 300q C
H 3074.3 kJ kg
final state
P 01
. MPa
S 8.2158 kJ kg K
H M
H Q W 0
E.B.: M
OP
Q
1 1
2
2
s
0
S M
S Q Sgen
S.B.: M
0
1 1
2 2
T
Q
S M
S
S S
M
M
1 1
2 2
1 2
1
T
Q
. 8.2158 6.7664 kJ kg K
T S2 S1
300 27315
M
c
c
W
s
M
h
h
830.7 kJ kg
Q
H 1 H 2
M
c
h 830.7 30235. 3074.3 779.9 kJ kg
Ÿ get more work out than in adiabatic case, but have to put in heat.
Solutions to Chemical and Engineering Thermodynamics, 5th ed
4.21
Chapter 4
contents of the compressor (steady-state, constant volume). Also, gas is ideal. (a) Mass balance
0 N N o N
N
System
1
Energy balance
2
2
1
N 1 H 1 N 2 H 2 0
dV
0
Ws P
Q
dt
adiabatic
0
Entropy balance
reversible
0
Q 0 compressor
Sgen Ÿ S 1
N 1 S 1 N 2 S 2 T
0
From the energy balance Ws
T T or Ws
NC
P 2
1
N
a
f
From the entropy balance S 1
S 2 Ÿ T2
T1
FG P IJ
HPK
S2
a
CP T2 T1
f
R CP
2
1
Thus
Ws
N
CPT1
LMF P I
MNGH P JK
R CP
2
OP
PQ
1
1
(b) Two stage compression, with intercooling, so that gas is returned to initial temperature, before
entering 2nd compressor
LF P I
work in stage 1
NC T MG J
MNH P K
LP
NC T MF I
work in stage 2 W
NH P K
L
T MFG P IJ
Total work W W
NC
MNH P K
R CP
WsI
1
II
s
R CP
2
P 1
*
I
s
R CP
II
1
To find P* for minimum work, set d Ws dP
R R F P I aR C f1 1
T |S G J
b g 0 NC
|T C H P K
Ÿ cP h
aP P f
d Ws
dP
P
P 1
P
P1
1
2 R CP
1 2
R CP
2
P 1
s
OP
PQ
O
1P
Q
P
F I
HP K
1 ; where P
P 1
2
0.
F Ia
H K
R P2
CP P
pressure after 1st compressor.
OP W
PQ
s
f F P I U|
R CP 1
H P K V|W
2
2
R CP
or
P
P1 P2
Students should check that this results in minimum, and not maximum work.
4.22
System: nitrogen contained in both tanks (closed, adiabatic, constant volume)
(1)
Mass balance: M1i M1f M 2f
f f
f f
(2)
Energy balance: M U
M U M U
1 1
1
P1 f
1
P2f
2
2
(3)
Final pressure condition:
For the entropy balance, the nitrogen in the first tank that remains in the tank will be taken as
the system. Then
(4)
S1i S1f
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Chapter 4
Equation (1) –(4), together with eqn. of state information of the form S S T , P , U U T , P
and V V T , P which we can get from Fig. 3.3-3 provides 4 eqns. for the 4 unknowns
T1 f , P1 f , T2f and P2f . Procedure to be followed in solution
(iv)
Guess a final pressure P f
Use eqn. (4) + Fig. 3.3-3 to compute T1 f , caluculate U1f
Use Fig. 2.4-3 to get V1 f , compute M1f V f
V1
f
f
f
f
V M
M
M M , and V
(v)
Use P and V2f to get T2f and U 2f
(i)
(ii)
(iii)
2
2
1
1
2
f
(vi) See if energy balance, Eqn. (2), is satisfied. If it is, guessed P f is correct. If it is not,
guess new p f , go back to (ii), and repeat calculation.
Some preliminaries
Figure
H 1i 368 kJ kg
T1i 250 K
 
o
2.4 - 3
V1i | 0.0037 m3 kg
P1i 200 bar
Thus M1i
V / V1i
0.01 m3 / 0.0037 m3 kg
2.703 kg
As a first guess, use ideal gas solution for pressure. (Also try some neighboring pressures.) My
solution is
P1 f
P2f
. bar same as ideal gas solution
1333
c
285 K cideal gas solution: T
T1
f
226 K ideal gas solution: T1 f
T2
f
f
2
h
330.4 Kh
222.8 K
4.23 (a) Set up just as in Problem 3.22 above. Solution after a number of iterations is P1 f
T1 f
2756
. q C and T2f
P2f | 5 bar .
497.7q C .
(b) Since now there is heat exchange between the two chambers we have T1 f T2f . This equation
is used instead of entropy balance. Solution procedure is to guess a final pressure, and then
compute final temperature using first the mass balance
M1f M 2f Ÿ
M1i
V1
V1i
V1 V2
Ÿ Vf
V f
V1 V2
V1 f V2 f
V1 V2 i
V1
V1
(1)
That is, choose T f until eqn (1) is satisfied. Then compute T f from energy balance i.e.
M1iU1i
M1f U1f M 2f U 2f
f
c M M hU Ÿ U
f
1
f
2
f
2
i
1
U f
f
(2)
When guessed P is correct, T computed from eqns. (1) and (2) using the Steam Tables will
be identical. My solution is P f 5 bar and T f # 366q C .
4.24
System
contents of turbine (open, constrant volume, steady-state)
dN
Mass balance:
0 N 1 N 2 Ÿ N 2 N 1
dt
constant
volume
Energy balance:
dN
dt
0
N 1 H 1 N 2 H 2 dV 0
0
Ws P
Q
dt
adiabatic
Solutions to Chemical and Engineering Thermodynamics, 5th ed
a
Ÿ Ws
N 1 H 1 H 2
Chapter 4
f N C aT T f for the ideal gas
1 P
1
2
Q 0 dN
Entropy balance:
Sgen
0 N 1 S 1 N 2 S 2 T
dt
P
T
N 1 CP ln 2 R ln 2
Ÿ Sgen N 1 S 1 S 2
P1
T1
or
R CP
Sgen
P
exp
T2 T1 2
P
N C
a
LM
N
FG IJ
H K
1
RS FG IJ
T H K
f
1 P
FG IJ UV
H KW
OP
Q
(a) For T2 to be a minimum, since Sgen t 0 and N 1 ! 0 , Sgen must be zero. Thus the minimum
outlet temperature occurs in reversible operation.
(b) Ws N 1CP T2 T1 . Since T1 ! T2 , the maximum work occurs when T2 is a minimum. Thus,
W is a maximum (in magnitude) for a reversible process.
a
f
s
4.25 (a) For any system:
LM M S Q S OP
T
Q
N
dS
dt
i i
gen
S t 0 or ¦ M
S d 0 . Also, S
depending 0n the process Q t 0 or Q d 0 and ¦ M
gen
i i
i i
0 or
Sgen ! 0 , depending on whether or not the process is reversible. Thus, dS dt for a system can
be greater than, less than, or equal to zero.
Since, by definition, the universe contains everything, it must be a closed system and
adiabatic, since there is nothing for the universe to exchange mass or heat with. Therefore
dS
dt
dS
0 0 Sgen Ÿ
dt
Sgen t 0
Thus the entropy of the universe can not decrease, and the statement is true.
(b) Consider the change from any state 1 to any state 2 in a closed system. The energy and entropy
balances for this transformation are:
1
2
U2 U2
S2 S1
Q W
Sgen
RS
T
W Since the process is adiabatic
If the transformation is possible, then Sgen t 0 now consider the transformation from state 2 to
state 1. Here
3
4
U1 U1
S1 S2
W
Sgen
Comparing eqns. (1) and (3) we have W W (This is ok).
Comparing eqns. (2) and (4) we have Sgen Sgen
(5)
Separately we have, if the processes are possible, that Sgen t 0 and Sgen t 0 . The only way that
all these three equations for Sgen and Sgen can be satisfied is if Sgen
Sgen
0 , that is, both
processes are reversible. Generally, processes are not reversible. However, eqn. (5) requires
that only one of Sgen and Sgen can be greater than zero. Thus,
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Chapter 4
If Sgen ! 0 1 o 2 is possible, but 2 o 1 is not possible.
If Sgen ! 0 2 o 1 is possible, but 1 o 2 is not possible.
4.26
W
1 § T Tamb · ¨ c
¸Q
2 © TC
¹
Q
q k Tc Tamb
Heat flow into
collector
Convective heat loss from collector
1 § Tamb ·
¨1 ¸ q k Tc Tamb
2 ©
Tc ¹
W
1
1
1 Tamb
1
1 T2
q k Tc Tamb q kTamb k amb
2
2
2 Tc
2
2 Tc
dW
dTc
1
k
2
kTc2
1 Tamb
1 T2
q k amb
2
2 Tc
2 Tc2
2
qTamb kTamb
Tc
4.27
0 (for a maximum)
amb
qT
2
Tamb
k
q
1
kTamb
Tamb
System: contents of the tank at any time (open, adiabatic, constant volume system).
dN
(a) Mass balance:
N
dt
PV
PV
ideal gas law N
; N
where V volumetric flow rate.
RT
RT
d P
P
d PV
PV
Thus,
ŸV
V since V and V are both constant.
dt T
T
dt RT
RT
d
P
V
P
P
Vt
bar
or
or
exp 1082
.
u 103
ln
dt
T
V
T 5 min T 0
V
K
F I
H K
F I
H K
F I
H K
P 5 min
Energy balance:
or N
dU
dt
FG IJ
H K
1082
.
u 103 T 5 min
d
NU
dt
dN
dU
N
dt
dt
dT
NCV
dt
N H Ÿ U
dN
dN
H U Ÿ RT
dt
dt
^P
K`
bar, T
H
dN
dt
FG IJ
H K
RC
P
P
dP P
dT
or T2 T1 2
CP
P1
dt
T
dt
[Note: could have gotten this result from the entropy balance also!]
using N
PV RT yields R
(1)
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Ÿ T 5 min
Chapter 4
F P 5 min I
H 1 bar K
8.314 39
(2)
340
simultaneously solving equations (1) and (2) yields
0.281 bar and T 5 min
P 5 min
259.3 K
S out or T3
(b) Since pump is adiabatic and reversible, S in
T2
FG P IJ
HPK
R CP
3
since P3
P1 . This
2
equation implies that T3
4.28
T1
340 K .
Number of moles of gas in tank initially
N 0
N0
N 0
15 bar u 0.2 m3 u 105 J m3 bar
8.314 J mol K u 22 27315
. K
P 0V
RT 0
122.26 mol
dN
N 4.5 mol min Ÿ N t 122.26 4.5t mol t min
dt
(a) Entropy balance on an element of gas that remains in the tank (see Illustration 3.5-2) yields
S t S 0
0Ÿ
Tt
T0
FG P t IJ
HP0K
R CP
Ÿ T (t )
FPt I
H 15 K
0.3779
29515
.
(1)
From the ideal gas equation of state
V
R
Tt N t
Pt
PtV
N t RT t Ÿ
Tt
Pt
29515
. K
122.26
u
15 bar
122.26 4.5t
T0N 0
P0
Thus
19.68
1 0.03681t
(2)
Now using eqn. (1) in eqn. (2) to solve for P t and T t yields
Pt
15 u 1 0.03681t 1.6075
Tt
. u 1 0.03681t 0.6075
29515
But T t is temperature in the tank. What about temperature of gas leaving the throttling valve?
Gas going thru valve undergoes a Joule-Thomson expansion Ÿ H in H out . Since gas is ideal,
this implies Tin
Tout . Thus, T t out of valve
29515
. u 1 0.03681t 0.6075
(b) If tank is isothermal, then, instead of eqn. (2), we have
Pt
N t
RT
V
P0
ŸPt
N 0
15 1 0.03681t bar
Solutions to Chemical and Engineering Thermodynamics, 5th ed
and T t
constant
Chapter 4
29515
. K
Summary
T K
P bar
0.6075
. 1 0.03681t
15 1 0.03681t 1.6075
Adiabatic 29515
.
15 1 0.03681t
Isothermal 29515
4.29
This is a tough problem!
Subscript 1 denotes properties in initally filled tank
Subscript 2 denotes properties in initially "evacuated" tank
We will use i and f (superscripts) to denote initial and final properties, and we will assume
negligible mass hold-up in engine.
1) Mass balance on closed system consisting of both tanks
N1i N 2i
N1f N 2f Ÿ
P1iV1 P2iV2
i
T1i
T2
P1 f V1 P2f V2
T1 f T2f
but
P1 f
Pf Ÿ Pf
P2 f
FG 0.3 0.75IJ 14.0 u 0.3 0.35 u 0.75
H T T K 97315
29815
.
.
f
f
1
2
bar m3
u 10
5196
.
K
(1)
3
2) Entropy balance on gas contained in tank 1 initially and finally. This is a closed, adiabatic,
reversible system
dS
dt
0 Ÿ S1i
FG T IJ R lnFG P IJ
HT K HP K
(2)
F P IJ
97315
. G
H 14.0K
(3)
S1f Ÿ CP ln
f
f
1
1
i
1
i
1
Thus
FG P IJ
HPK
f
T1
f
T1i
R CP
f
1
27
1
i
1
Equation (2) implies that T1 and P1 are related as follows
d ln T1
dt1
R d ln P1
CP dt
(4)
3) Mass and energy balances on tank 1
or N1CV
dN1
dt
d
N 1 { N ;
dt
dT1
dt
dN1
H1 U 1
dt
a
a N U f N H
1
1
f RT dNdt
1
1
1
RT1 N
U1
dU 1
dN1
N1
dt
dt
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Chapter 4
using eqn. (4) gives
N1
CV d ln P1
CP dt
dN1
dt
N
(5)
4) balances on the engine: adiabatic, reversible (for maximum work) and since no hold-up of
mass, dN1 dt 0
Subscript eng refers to gas leaving engine and going into tank 2.
0 H 1 H eng N Ws
d
i
0 dS S i N
1
eng
Ÿ S1
S eng ;
Also P2
Peng
c
Ÿ Teng
T1 Peng P1
h
a
R CP
f
T1 P2 P1 R CP
(6)
(Note that Teng z T2 )
and
c
Ws
T T
NC
P 1
eng
h
(7)
5) balances on tank 2 [Note, irreversible mixing occurs unless, fortuitously, Teng
T2 at all times
(can this occur?). Thus, Sgen ! 0 , and entropy balance gives no useful information]
N
Mass balance: dN 2 dt
a
f U dN N dU N H dN H But,
dt
dt
dt
H C aT T f
UV where T reference temperature
U C aT T f RT C T C T W
Energy balance:
P
0
V
0
d N 2U 2
dt
2
2
2
2
2
eng
eng
0
0
Ÿ
dN 2
H eng U 2
dt
or
d
V
P 0
i dNdt mC T C T C T C T r N C dTdt
2
2
P eng
P 0
m
N CPTeng CVT2
V 2
P 0
2 V
r N C dTdt
2
(8)
2 V
and
N
IJ PV dT V dP
a f dtd FGH PV
RT dt
RT K
RT dt
d
N2
dt
2 2
2
2
2 2
2
Ÿ
dT2
dt
2
2
2
2
RT22 N T2 dP2
PV
P2 dt2
2 2
(9)
using eqns. (6) and (9) in eqn. (8)
R| F I
S| GH JK
T
P
N CPT1 2
P1
R CP
CVT2
U| N C ˜ RT N N C T dP
V|
PV
P dt
W
2
2
2 V
2
2
2
2
2 V
2 2
PV
T dP
CVT2 N 2 2 ˜ CV 2 2
RT2
P2 dt2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
FG IJ
H K
T P2
NC
P 1
P1
R CP
Chapter 4
CVV2 dP2
R dt
and using eqn. (5)
N
FG IJ
H K
CV
P
CPT1 2
CP
P1
R CP
d ln P1
dt
CV
V2 dP2
R dt
or, finally
FG P IJ
HPK
R CP
dP1
dt
1
dP
V2 P2 R CP 2
dt
V1
2
V2
dP2
dt
V1 P1 R CP
dP1
dt
or
z
dP2
V2
R CP
P
Pi 2
2
z
P1 f
P2f
V1
dP1
R CP
P
Pi 1
1
Ÿ
{c h
2.5 P f
Ÿ Pf
CV CP
0.35 CV CP
} {c P h
3053
.
bar: using Eqn. (3), T1 f
f CV CP
14.0 CV CP
}
629.8 K .
f
Now using eqn. (1), T2
6119
. K
Finally, to get the total work, we do an overall energy balance (system
closed, constant volume).
N f U 1f N 2f U 2f N1i U 1i N 2i U i2 Ws
Ws
4.30
m a
f
two tanks; adiabatic,
r
CV
P f V1 V2 P1iV1 P2iV2
R
5
^3053
.
0.3 0.75 14 u 0.3 0.35 u 0.75`
2
3142
u 105 J 314.2 kJ
bar m3 3142
.
.
Note: be careful about coordinate system. A mass flow in the negative u direct is negative!
L
Heat exchanger is
in steady-state operation
+x
'L
Case I: Concurrent flow
Mass balance on shaded volume
Solutions to Chemical and Engineering Thermodynamics, 5th ed
mass in
element at
time t 't
mass in
element
at time t
Chapter 4
mass in
at face
at L in 't
b
mass out
at face at
L 'L in 't
g
M
M
0
L
L 'L 't
steady - state
M
M
M
L 'L
L
Energy balance on the shaded volume
energy in
energy in
element element
at t 't
at t
energy flow in
by heat flow
in time 't
energy flow in
energy flow out
by mass flow by mass flow at
at L in 't
L 'L in 't
(steady-state)
H 't M
0 M
L L
L 'L H L 'L 't Q'L't
H
H
T
M
MC
T
Q 'L
a
L 'L
L
f
P
a
L 'L
L
f
dividing by 'L , taking limit as 'L o 0 , and using subscript 1 to denote fluid 1
a
C dT1
M
1 P ,1
dC
Q N T2 T1
f
Q heat flow rate per unit length of exchanger.
Similarly, for fluid 2 (other part of exchanger)
C dT2
M
2 P ,2
dL
a
Q
N T2 T1
f
and M
are both + for concurrent flow)
(M
1
2
Adding the 2 equations
C dT1 M
C dT2 0
M
1 P ,1
2 P ,2
dL
dL
From problem statement, M
1
a
and C
M
2
P ,1
f
dT1 dT2
d
T1 T2 0 or T1 T2
dL dL dL
and T1 C T2 ; T2 C T1
now going back to
Ÿ
dT1
dL
and integrating
N
MC
P
CP,2
constant
C
2T f
aT T f N aCMC
1
2
1
P
Solutions to Chemical and Engineering Thermodynamics, 5th ed
FG C 2T IJ 2NL
H C 2T K MC
f
T1 f
15q C , T1i
1
ln
i
1
35q C , T2f
P
5q C , T2i
Chapter 4
MC
P
2N
L
; L0
L0
15q C .
Also, C T1i T2i T1 f T2f 20q C
[ i initial conditions, conditions at L
L* length of exchanger]
Using this in equation above gives
0; f
final conditions, conditions at L * where
F 20 70I ln 5 1609
.
H 20 30K
L*
L0
ln
And, more generally, at any point in the exchanger
FG L IJ Ÿ T L C FG1 expFG L IJ IJ T expFG L IJ
H L KK
H LK
H LK
2H
F LI
T L 10 25 expG J q C
H LK
F LI
T L C T L 10 25 expG J q C
H LK
C 2T1 L
C 2T1i
exp i
1
1
0
0
0
1
0
2
1
0
Now writing an entropy balance
S 't M
M
L L
L 'L S L 'L 't
0
ŸM
dS
dS
dL
Q
T1
a
N T2 T1
f
Q
'L't
T
N 50 exp L L0
.
10 25 exp L L0 27315
a
T1
a
a
f
f
need absolute
T here
FG IJ
a fH K
25 expa L L f
F LI
dG J
C
28315
. 25 expa L L f H L K
28315
. 25 expa L L f U
Ÿ S L S L 0 C lnRS
VW
30815
.
T
f
L
50NL0
exp L L0
d
M 28315
L0
. 25 exp L L0
0
p
0
0
0
p
Case II Countercurrent flow
M
M
2
1
dT
1
C
(1)
M
N T2 T1
1 P
dL
35
15q C
dT2
(2)
M 2CP
N T2 T1
-15
5q C
dL
C dT2 N T T
(3)
M
1 P
2
1
dL
. Subtracting eqn. (1) from eqn. (3) gives
Eqn. (3) comes from eqn. (2) using M
M
2
1
a
f
a
a
f
f
Solutions to Chemical and Engineering Thermodynamics, 5th ed
a
f
d
T1 T2 0 Ÿ T1 T2
dL
T2 T1 30q C
Chapter 4
constant
30q C
C
Thus
a
f
C dT1 N T T
30N
M
1 P
2
1
dL
30NL
T1i
T1 MC
P
30NL
L
L
35 35 15
T2 5 15
MC
L
L
P
0
0
MC
P
and the entropy balance
2N
where L0
dS1
M
dL
dS
Q
T1
N 30
T1
N 30
27315
M (35 30NL MC
. )
P
1
dL
absolute
temperature
needed here
15C
F LI
FG IJ
dG J
HL K
K
H
L
L
.
30815
15
g
L
30NL0
d
30815
M
L
. 30NL MC
0
P
dS1
b
CP
dx
. x
2054
FG
H
L
S 0 CP ln 1 0.048
L0
Ÿ S L
P
0
IJ
K
Summary
Concurrent flow
FG L IJ
H LK
F LI
T 10 25 expG J
H LK
T1
10 25 exp 0
2
0
S L
S L
0 CP ln
. 25 expa L L f U
RS 28315
VW
.
30815
T
0
Countercurrent flow
T1
35 15
T2
5 15
S L
4.31 (a) L
L0
L
L0
FG
H
L
S 0 CP ln 1 0.048
L0
dU
dt
dV
WS Q P
dt
JIK
a
f
dV
dV
WS Q P0
P P0
dt
dt
0
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Chapter 4
and
dS
dt
Q Sgen
T0
Now let
a
f
dV
dV
P P0
WS P0
dt
dt
W
and
Wu
a
dV
W P0
dt
f
dV
dU
Ÿ
WS P P0
dt
dt
dV
Wu Q P0
dt
or
a
f
U 2 U1
Wu Q P0 V2 V1
S2 S1
Q
Sgen Ÿ Q
T0
U 2 U1
Wu T0 S2 T0 S1 T0 Sgen P0 V2 V1
T0 S2 T0S1 T0Sgen
and
Wu
a
f
0 1
0 gen
aU PV T S f aU PV T S f T S
2
0 2
0 2
1
0 1
since T0 Sgen t 0
A 2 A1, where A
Wumax
(b)
0
0
Here WS
U PV
0 T0 S
H M
H Q W
M
S
1
2
Q
S M
S S
M
gen
1
2
T0
Wu
Ÿ Wu
H M
H Q
M
2
1
Ÿ Wu
H T S M
H T S T S
M
0 2
0 1
0 gen
2
1
a
Since T0 Sgen t 0
Wumax
b
f a
g
H M
H MT
S MT
S T S
M
0 1
0 2
0 gen
2
1
B B where B
M
2
1
f
H T0 S
(c) Using the Steam Tables we find
i)
at 30 bar 3 MPa and 600qC
U 32850
. kJ kg , S 7.5085 kJ kg K , V
013243
.
m3 kg
Solutions to Chemical and Engineering Thermodynamics, 5th ed
1
A
Chapter 4
U PV
0 T0 S
32850
bar 013243
m3 kg u 102 kJ bar ˜ m3 29815
. 1013
.
.
. u 7.5085
1059.76 kJ kg
ii)
. MPa and 300qC
at 5 bar 05
U 2802.9 kJ kg , S 7.4599 kJ kg K , V
2
A
05226
.
m3 kg
2802.9 1013
.
u 05226
.
u 102 29815
. u 7.4599
Wu
2A
1
A
(63167
. 1059.76) kJ kg
63167
. kJ kg
428.09 kJ kg
This is the maximum useful work that can be obtained in the transformation with the
environment at 25qC and 1.013 bar. It is now a problem of clever engineering design to
develop a device which will extract this work from the steam in a nonflow process.
bar, any component which
(d) Since the inlet and exit streams are at 25qC and P 1013
.
passes through the power plant unchanged (i.e., the organic matter, nitrogen and excess
oxygen in the air, etc.) does not contribute to the change in availability, or produce any
useful work. Therefore, for each kilogram of coal the net change is:
0.7 kg of carbon
58.33 mol of C
58.33 mol of O2
to produce 58.33 mol CO2
also
015
. kg of water
8.33 mol of H 2O undergoes a phase change
from liquid to vapor
Therefore
B
in
M
¦ b N i Bi gin
i
B
out
M
58.33 u 0 58.33 u 0 8.33 u 68.317 29815
. u 0.039
carbon
oxygen
liquid water
1976 kJ kg coal
58.33 u 94.052 8.33 u 57.8 29815
. u 0.0106
¦ N B
b
i
i
g
i out
carbon dioxide
water vapor
24858 kJ kg coal
Wumax
Wuactual
24858 1976 kJ kg coal
22882 kJ kg coal
2.2 kW - hr kg coal 7920 kJ kg coal
7920 u 100
Efficiency in %
34.6%
22882
Thus a coal-fired electrical power generation plants converts slightly more than 1/3 of the
useful work obtainable from the coal it consumes. This suggests that it would be useful
to look for another method of generating electrical power from coal . . . for example,
using an electro-chemical fuel cell. Considering the amount of coal consumed each year
in power generation, and the consequences (strip mining, acid rain, greenhouse effect,
etc.) the potential economic savings and environmental impact of using only 1/3 as much
coal is enormous.
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Chapter 4
4.32 Three subsystems: unknowns T1 f , P1 f , T2f , P2f , T3f , P3f (6 unknowns)
After process P1 f P2f P3f (2 equations)
Subsystem 1 has undergone a reversible adiabatic expansion
Ÿ S 1f
FG P IJ
HPK
f
S 1i , or T1 f
T1i
R CP
1
(1 equation)
i
1
(#1)
Subsystem 3 has undergone a reversible adiabatic compression
Ÿ S 3f
S i3 , or T3f
FG P IJ
HPK
f
3
i
3
T3i
R CP
FG P IJ
HPK
f
T3i
R CP
(1 equation)
i
3
(#2)
Mass balance subsystems 1 + 2
N1f N 2f
or
Pf
N1i N 2i Ÿ
P1 f V1 P2f V2
T1 f
T2f
P1iV1 P2iV2i
i
T1i
T2
FG 05. V IJ 10 u 05. 1 u 0.25 0.017909 (1 equation)
29315
.
.
H T T K 29315
f
1
f
2
f
2
(#3)
Energy balance on subsystems 1 + 2 + 3
N1f U 1f N 2f U 2f N 3f U 3f
N1i U 1i N 2i U i2 N 3i U i3
P fV
P fV
P1 f V1
CVT1 f 2 f2 CVT2f 3 f3 CVT3f
f
RT3
RT2
RT1
c
P f V1 V2f V3f
but V1 V2f V3f
V1 V2i V3i
h PV PV PV
i
1 1
05
. 0.25 0.25
i
2 2
in eqn. (1) o T1 f
252.45 K
in eqn. (2) o T3
448.93 K
f
P3i T3f
˜
P3f T3i
0.25 u
i
3 3
1 m3
10 u 05
. 1 u 0.25 1 u 0.25
1
Pf
using this result
PiV
PiV
P1iV1
C T i 2 2i CVT2i 3 i3 CVT3i
i V 1
RT3
RT2
RT1
1 448.93
u
. 29315
.
55
V3f
V3i
V2f
0.25 u 2 0.06961 0.4304 m3
55
. bar
0.06961 m3
Now using Eqn. (#3)
Pf
FG 05. V IJ 55. FG 05. 0.4304 IJ 0.017909 Ÿ T
H T T K H 252.45 T K
f
1
f
2
f
2
f
f
2
2
Thus the state of the system is as follows
T1
P1
Initial
293.15 K
10 bar
Final
252.45 K
5.5 bar
337.41 K
Solutions to Chemical and Engineering Thermodynamics, 5th ed
T2
P2
V2
T3
P3
V3
Chapter 4
293.15 K
1 bar
337.41 K
5.5 bar
0.25 m3
293.15 K
0.4304 m3
448.93 K
1
5.5 bar
3
0.25 m
0.0696 m3
Work done on subsystem 3
Energy balance
N 3f U 3f N 3i U i3
W
z
PdV
P3f V3f
P3iV3i
f
CVT3i
C
T
V 3
RT3i
RT3f
W
P3f V3f
CV
C
P3iV3i V
R
R
W
CP R f f
. u 0.0696 1 u 0.25
P3 V3 P3iV3i 3 55
R
0.3984 bar ˜ m3 39.84 kJ
c
h
From simple statics the change in atmospheric pressure dP accompanying a change in height dh is
dP Ugdh
where U is the local mass density and g is the gravitational constant. Assuming a packet of air undergoes an
altitude change relatively rapidly (compared to heat transfer), the entropy change for this process is
CP
R
dS
dT dP 0 since both Q and Sgen equal zero.
T
P
Combining the two equations above we have
CP
R
R
R N
M
dT
dP Ugdh Mgdh gdh
T
P
P
PV
T
Mg
dT
or
dh
CP
K
dT
dT
For dry air
# 9.7
is referred to as the adiabatic lapse rate.
. Note that
km
dh
dh
Also, its value will be less than that above as the humidity increases.
In fact, if the humidity is 100%, so water will condense as the pressure decreases, the adiabatic lapse rate
will be almost zero.
4.33
4.34
W
-10 kW
M
i
300°C
1 MPa
1 kg/s
Mass balance
100°C
Steady State
N
300°C
Sat’d L
Solutions to Chemical and Engineering Thermodynamics, 5th ed
0
Chapter 4
0.5
1 M
1
M
M
M
i
1
2
0 Ÿ M
1
0.5 kg/s
Energy balance
0
assume no heat loss Q
1344.0 kJ/kg
Ĥ 2
10 kw
10
kJ
s
O
3.2534 kJ/kg ˜ K
Sˆ i
1344.0 kJ/kg Sˆ 2
Ĥ i
0 1
W
H
ˆ
ˆ
ˆ
M
i i 0.5 H1 0.5 H 2 W
3.2534 kJ/kg ˜ K
kg
kJ
ˆ 0.5 u 1344.0 10 kJ
u 1344.0
0.5 u H
1
s
kg
s
ˆ
0.5 H
1
1344.0 0.5 u 1344.0 10
Ĥ1 =1324.0 kJ/kg
662
at 100°C
HV = 2676.1
At 100°C HL = 419.04
1324 = x (2676.1) + (1-x) 419.04
1324 – 419.04 = x (2676.1 – 419.04)
x = 0.401
quality
where x = quality
Ŝ1 = 0.401 × 7.3549 + 0.599 × 1.3069 = 3.7322 kJ/kg.K
Entropy balance
0
Sˆ M
Sˆ M
Sˆ S
M
i i
1 1
2 2
gen
where S gen t 0
1u 3.2534 0.5 u 3.7322 0.5 u 3.2534 S gen
0.2394 S gen
O
Ÿ S gen
kJ
kg ˜ K ˜ s
0.2394
0
so device is possible
4.35 a) Basis: 1 mole air
Energy balance
Uf Ui
WQ
0
C v Tf Ti
Entropy balance
Sf Si 0 Ÿ Sf Si
Ÿ S Tf , Vf S Ti , Vf
ln
C*V ln
Tf
V
R ln f
Tf
Vi
0
8.314 § 1 ·
ln ¨ ¸ 1.2743
21
© 25 ¹
Tf
Ti
Tf
Ti e1.2743
W
8.314
303.15e1.2743
1084.1 K
J
u 1084.1 303.15 K
mol K
6493
J
mol
6.493
kJ
mol
6.493 kJ / mol
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Chapter 4
b)
ln
Tf
Ti
8.314 § 1 ·
ln ¨ ¸
21
© 10 ¹
0.9116
Tf
303.15 u e0.9116
754.33
W
8.314 u 754.33 303.15
3751.1
J
mol
3.751
kJ
mol
4.36
N
M
2 MPa, 200°C
200 kPa, 100°C
Find initial number of moles in each compartment
PV=NRT
N
PV
RT
200 kPa u 100 u 103 cm3 u
N1
kPa m3
8.314 u 10
u 373.15 K
mol K
a)
8.314 u 103
6.447 moles
3
2 u 103 kPa u 2 u 105 cm3 u
N2
1 m3
106 cm3
1 m3
106 cm3
kPa m3
u 473.15 K
mol K
Mass balance: Nf – N1 – N2 = 0
101.68 moles
Nf = N1 + N2 = 6.45 + 101.68
Nf = 108.13
Energy balance: Nf Uf – N1 U1 – N2 U2 = 0
6.448 × CV (Tf – 373.15) + 101.68 × CV (Tf – 473.15) = 0
(6.447 + 101.68) Tf = 6.447 × 373.15 + 101.68 × 473.15
6.447 u 373.15 101.68 u 473.15
= 467.19 K
Tf
6.447 101.68
Final Pressure
kPa m3
u 467.19 K
NRT
mol K
P
V
m3
3 u 105 cm3 u 106
cm3
3
2
2
1.0813 u 10 u 8.314 u 10 u 4.6719 u 10
14.0 u 102 kPa 1.4 MPa
3 u 101
108.16 mol u 8.314 u 103
b)
Entropy Change
N f Sf N1S1 N 2 S2
Sgen t 0
­
­
T
P ½
T
P ½
N1 ®CP ln f R ln f ¾ N 2 ®CP ln f R ln f ¾ Sgen
T1
P1 ¿
T2
P2 ¿
¯
¯
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Chapter 4
467.9
1.4 ½ J
467.9
1.4 ½
­
­
8.314 ln
101.68 ®30 ln
8.314 ln
6.45 mol u ®30 ln
¾
¾ Sgen
373.15
0.2 ¿ mol K
473.15
2 ¿
¯
¯
0
6.45 u ^30 u 0.2263 8.314 u 1.9459` 101.68 ^30 u 0.01116 8.314 0.35667 ` =
60.56
J
J
267.47
K
K
Sgen
Q
4.37
206.9
J
t 0! (as it should be)
K
0
Assume steady state operation
N
M
2 bar = 0.2 MPa
20 bar
400°C
W
M
M
0
1
2
2 bar = 0.2 MPa
M
2
M
1
Entropy balance
0 0 for maximum work
Q
ˆ
ˆ
S M
S S
0 M
1 1
2 2
gen
T
Ÿ Sˆ 2 Sˆ 1
Sˆ 1 Sˆ 2 MPa, 400qC 7.1271
Ĥ1
Ŝ2 0.2MPa, T
at T
?
kJ
kg ˜ K
3247.6 kJ / kg
7.1271 kJ / kg ˜ K
200qC Sˆ 7.5066
150qC Sˆ 7.1272
kJ / kg K
ˆ
H
2870.5 kJ / kg
kJ / kg
Ĥ
2768.8 kJ / kg
Ÿ
T2 = 150°C
Energy Balance
0
ˆ
H
ˆ
0 M
1 1 M1 H 2 W Q
W
5000
kg
kJ
9247.6 2768.8
hr
kg
2.394 u 106
kJ
hr
665
kJ
S
665 kW
4.38
20°C
T=?
1 bar
5 bar
Heater
a) Assume adiabatic and reversible operation of compressor
Ÿ Isentropic
S (20°C, 1 bar) – S (T = ?, 5 bar) = 0
C*p ln
T
5
R ln
293.15
1
0
400°C
5 bar
Solutions to Chemical and Engineering Thermodynamics, 5th ed
R
Chapter 4
§ 5 · Cp
T 293.15 ¨ ¸
462.7 K since C*p C*v R
©1¹
E.B. H (20°C, 1 bar) – H (462.7 K, 5 bar) + W = 0
W = H (462.7 K, 5 bar) – H (293.15 K, 1 bar)
Cpx 462.7 293.15 4971.2 J / mol
*
21 8.314
29.314
J
mol ˜ K
b) Heater
H (462.7 K, 5 bar) – H (673.15 K, 5 bar) + Q = 0
Cpx 673.15 462.7 6169.1 J / mol
4.39 From Fig. 3.3-2
ˆ
At 160 K
H
260 kJ / kg
L
ŜL 2.05 kJ / kg K
ˆ # 0.00295 m3 / kg
V
L
If adiabatic and reversible, Sgen
0
Sˆ initial
Sˆ initial
?, P
? 0.25 Sˆ V T
2.05 kJ / kg
0.75 Sˆ L T
Sˆ final
?, P
?
but since it is an equilibrium V-L mixture T & P are connected by the saturation line
ˆ
Sˆ V 5.03 Ÿ Sˆ
Guess T = 120 K SL 1.03
Ŝ 0.75 u 1.03 0.25 5.03 2.03
ˆ
H
L
ˆ
V
L
ˆ
H
V
3
ˆ
0.0025 m / kg VV
105 kJ / kg
ˆ #H
ˆ
U
L
L
ˆ
U
V
ˆ
U
f
0.3 m3 / kg
P
0.2 MPa
105 kJ / kg
ˆ PV
ˆ
H
L
L
595
kJ
Pa
1J
kJ
0.2 MPa u 106
u 3 u 103
kg
MPa m Pa
J
595
kJ
kJ
60
kg
kg
ˆ 0.25 U
ˆ
0.75 U
L
V
ˆ H
ˆ
U
i
i
595 kJ / kg
Close Enough
535
kJ
kg
212.5 kJ / kg
260 kJ / kg
Energy balance on piston-cylinder (closed system)
o
ˆ U
ˆ
U
Q
W 212.5 260 47.5 kJ / kg
f
i
(Negative sign Ÿ system does work on surroundings)
W
4.40
a)
20 bar
550°C
20 bar = 2 MPa = 2,000 kPa
From the superheated steam tables
1 bar
0.1 MPa
Adiabatic and Isentropic
Ÿ Sˆ IN Sˆ OUT
Ÿ Vertical line of Fig. 3.3-1b
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Ĥ (20 bar, 600°C) = 3690.1
Chapter 4
kJ/kg
kJ/kg K
Ŝ (20 bar, 600°C) = 8.6451
Estimate From Fig. 3.3-1b (vertical line down to 1 bar)
ˆ 2820 kJ / kg Also all vapor
T # 172qC
H
Steady state energy balance
W
H
ˆ
ˆ
ˆ
ˆ
0 M
H
IN IN M OUT H OUT W Ÿ OUT H IN
M IN
W
kJ
870.1 kJ u 5000 kg 1327.2 kJ
3690.16 2820 870.1
W
kg
kg
hr
s
M
1327.2 kW
IN
If only 90% of the work generated, W 0.9 u 870.1 783.1 kW
kJ
All vapor
so Ĥ out 3690.1 783.1 2907.0
at 1 bar (0.1 MPa)
kg
By interpolation using the Superheated Steam Tables
kJ
T 216o C and S=7.8983
kg ˜ K
b)
0
Entropy balance
dS
Sˆ M
Sˆ Q S
0 M
in in
out out
gen
dt
T
S gen
kJ
7.8983 7.7024 0.1959
or S gen
kg ˜ K
M in
979.5
kJ
hr ˜ K
4.41 The gas in the tank undergoes a uniform adiabatic expansion, which is isentropic, so that its
temperature and pressure are related by
R / C*p
§P ·
TT,f TT,i ¨ T,f ¸
¨P ¸
© T,i ¹
The gas in the cylinder to the right of the piston undergoes a uniform adiabatic compression, which is also
isentropic, so its temperature and pressure are related by
TCR ,f
§P
·
TCR ,i ¨ CR ,f ¸
¨P ¸
© CD,i ¹
R / C*p
Also PT,f = Pc,f.
An energy balance on the whole system gives
(UT,f + UCL,f + UCR,f) – (UT,I + UCR,i) = 0
NT,f UT,f + NCL,f UCL,f + NCR,f UCR,f = NT,i UT,i + NCR,i UCR,i
Also, the mass balance
NT,f + NCL,f + NCR,f = NT,i + NCR,i
Now using the ideal gas law
V
V
V
M.B.
Pf TT Pf CL,f Pf CR ,f
RTCL,f
RTCR ,f
R Tf
E.B.
PT,i
V
VT
PCR ,i CR ,i
RTT,i
RTCR ,i
PV
PV
Pf VT
CV TT,f t CL C v TCL,f f CR ,f C V TCR ,f
RTT,f
kTCL,f
RTCR ,i
These equations reduce to
PT,i VT
RTT,i
C V TT,i PCR ,i VCR ,i
RTCR ,i
CV TCR ,i
Solutions to Chemical and Engineering Thermodynamics, 5th ed
ªV
V
V º
Pf « T CL,f CR ,f »
¬« TTf TCL,f TCR ,f ¼»
and
Pf ª¬ VT VCL,f VCR ,f º¼
but
VCL,f VCR ,f
Pf
TT,i
PCR ,i VCR ,i
TCR ,i
PT,i VT PCR ,i VCR ,i
0.5 m3
VCR ,i
Pf ª¬ VT 0.5 m3 º¼
PT,i VT
Chapter 4
0.4 VT 0.1u 0.5 m3
0.4VT 0.1u 0.5
VT 0.5
0.4 u 0.25 0.1u 0.5
0.25 0.5
0.2 MPa
8.314
TT,f
§ 0.2 · 29.3
473.15 ¨
¸
© 0.4 ¹
TCR ,f
§ 0.2 ·
288.15 ¨
¸
© 0.1 ¹
8.314
388.7 K
29.3
350.8 K
Mass balance on region to right of piston
P V
PCR ,f VCR ,f
N CR ,i N CR ,f Ÿ CR ,i CR ,i
RTCR ,i
RTCR ,f
Ÿ
0.1u 0.5
288.15
0.2 u VCR ,f
350.8
VCR ,f
0.1u 0.5 350.8
u
288.15
0.2
0.3043 m3
Ÿ VCL,f = 0.5 – 0.3043 = 0.1557 m3
Moles in tank initially
PT,i VT,i
0.4 u 0.25
N T,i
25.42 moles
RTi
8.314 u 106 u 473.15
Moles in tank finally
0.2 u 0.25
N T,f
15.47 moles
8.314 u 106 u 388.7
Ÿ N CL,f 25.42 15.47 9.95 moles
TCL,f
PCL,f VCL,f
R ˜ N CL,f
0.2 u 0.1557
8.314 u 106 u 9.95
376.43
0.1u 0.5
20.87 moles
8.314 u 106 u 288.15
(QWURS\FKDQJH ǻ6of N2 finally in tank ǻ6 of N2 finally left of piston ǻ6of N2 finally right of piston
N CR
388.7
0.2 ·
376.43
0.2 ·
350.8
0.2 ·
§
§
§
15.47 ¨ C*P ln
R ln
9.95 ¨ C*P ln
R ln
N CR ¨ C*P ln
R ln
¸
¸
473.15
0.4 ¹
473.15
0.4 ¹
288.15
0.1 ¸¹
©
©
©
9.26
[Note that the first and third terms should be zero, since the gas in the tank under went a uniform, adiabatic
expansion, and the gas to the right of the piston under went a uniform, adiabatic compression. So only the
middle term is important.]
4.42 Initial
To the left of the piston
TCL,f = 376.43 K VCL,f = 0.1557
P = 0.2 MPa
J
K
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Chapter 4
NCL,f = 9.95 Mols
TCR,f = 350.8 K VCR,f = 0.3043 m3
NCR,f = 20.87 K
Mass balance after conduction of heat
Pf VCL
Pf VCR
N CL,f 9.95
N CR 20.87
RTf
RTf
but Pf is the same on both sides of piston, and Tf is the same on both sides of piston
Ÿ
9.95
20.87
VCL
VCR
VCL
0.5 VCL
0.5 – VCL = 2.097 VCL
0.5 VCL
20.87
VCL
9.95
0.5
3.097
0.1614 m3
VCL
VCR = 0.5 –0.1614 = 0.3386 m3
Energy balance
9.95 UL,2 + 20.87 UR,2 = 9.95 UL,1 + 20.87 UR,1
9.95 (T–376.43) + 20.87 (T–350.8) = 0
30.82 T = 9.95 u376.43 + 20.87 + 350.8
T = 359.1 K
Pf
9.95 moles u 8.314 u 10 5
9.95 u R u T
VCL
bar m3
u 359.1 K
mol K
0.1614 m3
1.840 bar 184.0 kPa
4.43
110 bar
80oC
V=0.1 m3, initial conditions 25oC and 1 bar
a) Mass balance: N f
N i 'N with N i
PVi
RT
1 bar ˜ 0.1m3
bar ˜ m3
8.314 ˜10-5
˜ 298.15 K
mol ˜ K
4.034 mol
Energy balance:
NfU f
Ni U i 'N H in
For simplicity in the calculation of thermal properties, will use 25oC as Tref so that Uref = 0 and
Href=-298.15R, so that
100 bar ˜ 0.1 m3
1 bar ˜ 0.1 m3
˜ CV ˜ T Tref
˜ CV ˜ Tin Tref
R ˜Tf
R ˜ Ti
§ 100 bar ˜ 0.1 m3 1 bar ˜ 0.1 m3 ·
¨
¸ ˜ CP ˜ Tin CV ˜ Tref
¨
¸
R ˜Tf
R ˜ Ti
©
¹
which reduces to
Solutions to Chemical and Engineering Thermodynamics, 5th ed
100
CV ˜ T f CP ˜ Tin
Tf
Chapter 4
1
˜ CV ˜ Ti CP ˜ Tin
Ti
So that
Tf
100 ˜ CP ˜ Tin
489.86 K = 216.71o C
T
100 ˜ CV CV CP ˜ in
Ti
100 bar ˜ 0.1 m3
245.54 mol
3
-5 bar ˜ m
˜ 489.86 K
8.314 ˜10
mol ˜ K
241.51 mol
so that 'N 241.51 mol and the fill time is t
12.075 sec
20 mol/s
c) Final pressure, since volume and number of moles is constant, can be computer from
Pf ' Pf
Pf
100
˜Tf '
˜ 298.15 60.86 bar
so that Pf '
Tf ' Tf
Tf
489.86
b) Number of moles finally N f
4.44
c
1 bar
20°C
10 bar
e
d
Liquid, 85°C
10 bar = 1000 kPa =1 MPa
200°C
­1 bar, 20qC
L®
¯1 bar,85qC
ˆ
H
1
ˆ
H
3
ˆ
V ^10 bar, 200qC H
2
M
M
M
1
2
3
M.B.
E.B.
0
355.90
Sˆ 1
Sˆ
3
1.1343
2827.9
Sˆ 2
6.6940
83.96
0
M
3
0.2966
kg M1 M
2
s
M
1 M
1
2
1
ˆ
H
ˆ
ˆ
M
1 1 M 2 H 2 M3 H3
83.96 M
˜ 2827.9 1 ˜ 355.90
1 M
2
2
2827.9 83.96 355.90
83.96 M
2
M
2
M
1
355.90 83.96
0.0991
2827.9 83.96
1 0.0991 0.9009
S.B.
0
Sˆ M
Sˆ M
Sˆ S
M
1 1
2 2
3 3
gen
0
0.9091u 0.2966 0.0991u 6.6940 1u 1.1343 S gen
0
Solutions to Chemical and Engineering Thermodynamics, 5th ed
S gen
0.2013
Chapter 4
J
K ˜s
Reactor is I, initially evacuated tank is II. Mass balance is
4.45
N I N II , which when using the ideal gas law becomes
Ni
200 bar ˜ V I
673.15 K
P I V I P II V II
TI
T II
200
673.15
4 ·
§ 1
P I ¨ I II ¸
T
T
©
¹
or
§ V I 4V I ·
P I ¨¨ I II ¸¸
T ¹
©T
since P I
P II
The energy balance of the reactor and tank is
N I U I N II U II , which when using the ideal gas law and constant heat capacity becomes
NiU i
200 bar ˜ CV ˜ 673.15 K
673.15 K
or 200=P I 4 P I
P I CVT I 4 ˜ P I CVT II
TI
T II
200
and therefore P I
40 bar
5
The entropy balance on just the contents of the reactor that undergo a uniform expansion leads to
S
constant which leads to
R
T
I
§ P I · CP
Ti ¨¨
¸¸
© Pi ¹
8.314
§ 40 · 73.2
673.15 ¨
¸
© 200 ¹
560.7 K
Using this result in the mass balance gives
200
673.15
4 º
ª 1
II »
40 «
560.7
T ¼
¬
which gives T II
708.7 K
4.46
N
steam
1 bar, 250°C
M
steam
1 bar, 100°C
1 bar, liquid water
e
ˆ
H
2676.1
ˆ
O liquid 1 bar, 100qC O H
ˆ
N steam 1 bar, 250qC N H
419.04
kJ
Sˆ 7.3549
kg ˜ K
ˆS 1.3069
2974.3
Sˆ 8.0333
M steam 1 bar, 100qC M
M.B.
E.B.
M
M
M
0
1
2
3
H
ˆ M
H
ˆ M
H
ˆ
M
1
1
2
2
3
kJ
kg
M
M
2
3
3
0
M
1
1
Solutions to Chemical and Engineering Thermodynamics, 5th ed
M
2676.1 M
2974.3 M
419.04
M
2
3
2
3
Chapter 4
0
2974.3 2676.1 M
419.04 2676.1
M
2
3
M 2 2676.1 419.04
7.5689
2974.3 2676.1
M
3
Ÿ M
2
M
M
2
3
but
M
1 7.5689 M
3
3
1
0.1167
8.5689
Sˆ M
Sˆ M
Sˆ Sˆ
M
0
M
3
S.B.
1 1
2 2
3 3
1
M
2
1 0.1167
0.8833
gen
1u 7.3549 0.8833 u 8.0333 0.1167 u 1.3069 Sˆ gen
0.1066
Ŝgen
7.5689 M
3
0.1066
kJ ˆ
Sgen
K
0
0
kJ
Ÿ process is not possible
K
4.47
4.48
The maximum temperature difference will occur if the process is carried out reversibly.
In this case the energy balance is
1
1
1
H1 H 2 H 3 0 or H1
H 2 H 3 and since CP is constant T1
T2 T3
2
2
2
The entropy balance is
T
P
T
P
0 CP ln 1 R ln 1 CP ln 1 R ln 1
T2
P2
T3
P3
0
29.3ln
273.15 25
4
273.15 25
4
8.314 ln
29.3ln
8.314 ln
T2
T3
1.013
1.013
The solution to these equations using MATHCAD is
T2 298.15
T3 298.15
Given
298.15
§ 1 · ˜ ( T2 T3)
¨ ¸
© 2¹
29.3˜ ln §¨
298.15˜ 298.15 ·
©
T2˜ T3
¸
¹
8.314˜ 2˜ ln §¨
4
·
¸
© 1.013 ¹
z Find ( T2 T3)
z
§ 78.788 ·
¨
¸
© 517.512 ¹
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Chapter 4
T2=78.8 K and T3=517.5 K, a difference of more than 438 K. However, in fact a HilschRanque
vortex tube is very inefficient and very irreversible
4.49 In the folder Aspen for Textbook>Chapter 4>Problems
4.50 In the folder Aspen for Textbook>Chapter 4>Problems
4.51 In the folder Aspen for Textbook>Chapter 4>Problems
4.52 In the folder Aspen for Textbook>Chapter 4>Problems
4.53 In the folder Aspen for Textbook>Chapter 4>Problems
4.54 In the folder Aspen for Textbook>Chapter 4>Problems
4.55 In the folder Aspen for Textbook>Chapter 4>Problems
4.56 In the folder Aspen for Textbook>Chapter 4>Problems
4.57 In the folder Aspen for Textbook>Chapter 4>Problems
4.58 In the folder Aspen for Textbook>Chapter 4>Problems
4.59 Equation 4.6-14 is to be used to solve this problem
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Chapter 4
M ª Uˆ T1 , P1 PambVˆ T1 , P1 Tamb Sˆ T1 , P1 Uˆ T2 , P2 PambVˆ T2 , P2 Tamb Sˆ T2 , P2 º
¬
¼
a) So for part a, assuming the exiting stream is a vapor
WS ,max
ª Uˆ 800o C, 2 MPa (1 bar)Vˆ 800o C, 2 MPa 298.15 K Sˆ 800o C, 2 MPa
WS ,max M «
« Uˆ 298.15 K),1 bar (1 bar)Vˆ 298.15 K),1 bar 298.15 K Sˆ 298.15 K),1 bar
«¬
ª§
kJ
m3
kJ · º
- 298.15 K 8.1765
Ǭ 3657.0 +(1 bar)0.2467
¸»
kg
kg
kg K ¹ »
©
«
100 kg «
3
§
·»
«- ¨104.88 kJ +(1 bar)43.36 m - 298.15 K 8.5580 kJ ¸ »
kg
kg
kg K ¹ »¼
«¬ ©
ª§
kJ
m3
kJ
kJ · º
u100
3657.0
+(1
bar)0.2467
- 298.15 K 8.1765
Ǭ
¸»
3
kg
kg
bar m
kg K ¹ »
©
=100 kg ««
3
§
·»
«- ¨104.88 kJ +(1 bar)43.36 m u100 kJ 3 - 298.15 K 8.5580 kJ ¸ »
kg
kg
bar m
kg K ¹ »¼
«¬ ©
º
»
»
¼»
=100 kJ ª¬ 3657.0+24.7-2437.8 - 104.88+4336-2551.6 º¼
=100 kJ >1243.9-1889.3@ 64540 kJ
b) WS ,max
ª Uˆ 800o C, 2 MPa (1 bar)Vˆ 800o C, 2 MPa 298.15 K Sˆ 800o C, 2 MPa
M«
« Uˆ 673.15 K, 0.6 MPa (1 bar)Vˆ 673.15 K, 0.6 MPa 298.15 K Sˆ 673.15 K, 0.6 MPa
«¬
ª§
kJ
m3
kJ · º
- 298.15 K 8.1765
Ǭ 3657.0 +(1 bar)0.2467
¸ »
kg
kg
kg K ¹ »
©
«
100 kg «
3
§
·»
«- ¨ 2962.1 kJ +(1 bar)0.5137 m - 298.15 K 7.7079 kJ ¸ »
kg
kg
kg K ¹ »¼
«¬ ©
100 kJ ª¬ 3657.0+24.67-2437.8 - 2962.1+51.37-2298.1 º¼
100 kJ >1243.9-715.4@ 52850 kJ
c) The simplest way to calculate this is as
WS ,max
WS ,max part a WS ,max part b
64540 52850 kJ 11690 kJ
4.60 In the first printing, Problem 4.60 inadvertently duplicates Problem 4.59
4.61 In first printing, this problem duplicates parts a and b of Problem 4.59
4.62
º
»
»
¼»
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Chapter 4
The starting point is
Wˆ
S ,max
Hˆ T1 , P1 PambVˆ T1 , P1 Tamb Sˆ T1 , P1
Hˆ Tamb , Pamb PambVˆ Tamb , Pamb Tamb Sˆ Tamb , Pamb
o
For the 800 C and 2 MPa stream
WˆS ,max
Hˆ 800o C, 2 MPa (1 bar)Vˆ 800o C, 2 MPa 298.15 K Sˆ 800o C, 2 MPa
Hˆ 298.15 K),1 bar (1 bar)Vˆ 298.15 K),1 bar 298.15 K Sˆ 298.15 K),1 bar
ª§
kJ
m3
kJ · º
4150.3
+(1
bar)0.2467
- 298.15 K 8.1765
Ǭ
¸»
kg
kg
kg K ¹ »
©
«
=
« §
3
·»
«- ¨104.88 kJ +(1 bar)43.36 m - 298.15 K 8.5580 kJ ¸ »
kg
kg
kg K ¹ »¼
«¬ ©
For the 900o C and 1.4 MPa stream
WˆS ,max
Hˆ 900o C,1.4 MPa (1 bar)Vˆ 900o C,1.4 MPa 298.15 K Sˆ 900o C,1.4 MPa
Hˆ 298.15 K),1 bar (1 bar)Vˆ 298.15 K),1 bar 298.15 K Sˆ 298.15 K),1 bar
ª§
kJ
m3
kJ · º
4391.5
+(1
bar)0.3861
- 298.15 K 8.5556
Ǭ
¸»
kg
kg
kg K ¹ »
©
«
=
« §
3
·»
«- ¨104.88 kJ +(1 bar)43.36 m - 298.15 K 8.5580 kJ ¸ »
kg
kg
kg K ¹ ¼»
¬« ©
4.63
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Chapter 4
The starting point is
WS ,max
Hˆ T1 , P1 PambVˆ T1 , P1 Tamb Sˆ T1 , P1
Hˆ Tamb , Pamb PambVˆ Tamb , Pamb Tamb Sˆ Tamb , Pamb
o
For the 1000 C and 25 bar stream
WS ,max
H 1000o C, 25 bar (1 bar)V 1000o C, 25 bar 298.15 K S 800o C, 25 bar
H 298.15 K),1 bar (1 bar)V 298.15 K),1 bar 298.15 K S 298.15 K),1 bar
For the 900o C and 40 bar stream
WS ,max
H 900o C, 40 bar (1 bar)V 900o C, 40 bar 298.15 K S 800o C, 25 bar
Toevaluate which stream has the potential to do more work, calculate the difference in
available work
'WS ,max
WS ,max 1000o C, 25 bar WS ,max 900o C, 40 bar
= H 1000o C, 25 bar H 900o C, 40 bar 1 bar u V 1000o C, 25 bar V 900o C, 40 bar
298.15 K S 1000o C, 25 bar S 900o C, 40 bar
Now since the fluid is an ideal gas
§ 1273.15 K 1173.15 K ·
CP 1000o C 900o C 1 bar u R ¨
¸
40 bar ¹
© 25 bar
1273.15
25 ·
§
298.15 K ¨ CP ln
R ln ¸
1173.15
40 ¹
©
3
bar m
1273.15
25 ·
§
CP 100 1 bar u 8.314 u 105
R ln ¸
29.329 298.15 K ¨ CP ln
mol K
1173.15
40 ¹
©
'WS ,max
4.64
A U PambV Tamb S
§ wA ·
¨
¸
© wT ¹V
§ wA ·
¨
¸
© wV ¹T
§C ·
§ T ·
§ wU ·
§ wS ·
CV Tamb ¨ V ¸ CV ¨1 amb ¸
¨
¸ Tamb ¨
¸
T ¹
© wT ¹V
© wT ¹V
© T ¹
©
§ wU ·
§ wS ·
§ wS · § wP ·
§ wU ·
¨
¸ Pamb Tamb ¨
¸ ; but ¨
¸ ¨
¸ and ¨
¸
© wV ¹T
© wV ¹T
© wV ¹T © wT ¹V
© wV ¹T
§ wA ·
so ¨
¸
© wV ¹T
§ wP ·
§ wP ·
T¨
¸ P Pamb Tamb ¨
¸
© wT ¹V
© wT ¹V
§ wP ·
T¨
¸ P
© wT ¹V
§ wP ·
T Tamb ¨
¸ Pamb P
© wT ¹V
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Chapter 4
A U PambV Tamb S
§ wA ·
¨
¸
© wT ¹V
§ wA ·
¨
¸
© wV ¹T
§C ·
§ T ·
§ wU ·
§ wS ·
CV Tamb ¨ V ¸ CV ¨1 amb ¸
¨
¸ Tamb ¨
¸
T ¹
© wT ¹V
© wT ¹V
© T ¹
©
§ wU ·
§ wS ·
§ wS · § wP ·
§ wU ·
¨
¸ Pamb Tamb ¨
¸ ; but ¨
¸ ¨
¸ and ¨
¸
© wV ¹T
© wV ¹T
© wV ¹T © wT ¹V
© wV ¹T
§ wA ·
so ¨
¸
© wV ¹T
§ wP ·
§ wP ·
T¨
¸ P Pamb Tamb ¨
¸
© wT ¹V
© wT ¹V
§ wP ·
T¨
¸ P
© wT ¹V
§ wP ·
T Tamb ¨
¸ Pamb P
© wT ¹V
7KLV LV OLNH ,OOXVWUDWLRQ H[FHSW WKDW WKH 5DQNLQH UDWKHU WKDQ YDSRU
FRPSUHVVLRQ UHIULJHUDWLRQ F\FOH LV XVHG 2QO\ SURSHUWLHV RI SRLQW DQG SDWK
IURPoFKDQJHV
3RLQWLVHQWURSLF S N- NJ. H N- NJ q & T q & x IUDFWLRQYDSRU S 9 q & S / q & ˜ x x
x H / q & 9
H q & H
H
x x /
Q% H H W H H ,IWKHWXUELQHGULYHVFRPSUHVVRU
W
H H H H &23
9DSRUFRPSUHVVLRQF\FOHZLWKFKDQJHRI TORZ DQG TKLJK /RFDWLRQ
6WDWH
3DWK
T q& LVHQWKDOS
P FRQVW
LVHQWURSH
6DW¶GOLT
9/PL[
6DW¶GYDS
VXSHUKHDWHG
YDSRU
6DW¶GOLT
P N3D H N- NJ S N- NJ p
FRQVW
n
H
H
H
c
h c
&23
h
P
a
,QWHUSRODWLRQ
P
N3D q & q & q & S
S
S
f a
f
P
N3D q & q & S
S
DWq&DQG H
DWN3D S DWq&DQG H
DWN3D S Ÿ DWN3D T q&DQG H Q& H H W H H
H
H
&23
*HQHUDODQDO\VLVRIDKHDWHQJLQH
z
dS
dt
Q Q
T
Q TdS
,QWHJUDWLRQDURXQGDF\FOH
Q Q WQHW
WQHW Q Q DUHD a b a DUHD a b a
DUHD WQHW WQHW
DUHD K
QLQ
Q DUHD a b a
z
T
dS
Q dt
a
T S S
T
TdS
DUHD a b a T
f DUHD a b a 2ULJLQDOF\FOH,QFUHDVLQJHYDSRUDWRUTEXWQRWSUHVVXUH
2ULJLQDOF\FOH
,QFUHDVLQJHYDSRUDWRU TDQGP (LWKHUZD\PRUHZRUNLVREWDLQHGZLWKRQO\DVOLJKWLQFUHDVHLQKHDWUHMHFWHG
ŸDOPRVWDOOWKHDGGLWLRQDOKHDWLQSXWLVFRQYHUWHGWRZRUN
ŸHIILFLHQF\LPSURYHV
'HFUHDVLQJHYDSRUDWRUT
1RWHPD\JHWLQWRSKDVHUHJLRQ $JDLQHIILFLHQF\ZLOOLQFUHDVHGXHWRPRUHZRUNEHLQJSURGXFHGDQGOHVVKHDWUHMHFWHG
a c c bc a ZLWKRXWVXSHUKHDW
a b a ZLWKVXSHUKHDWLQJ
&OHDUO\JUHDWHUZRUNRXWSXWDQGKLJKHUHIILFLHQF\ZLWKVXSHUKHDW
a b a ZLWKRXWVXEFRROLQJ
a c c c b a c ZLWKVXEFRROLQJ
3UREDEO\LVJUHDWHUHIILFLHQF\ZLWKVXEFRROLQJEXWLVQRWDVHYLGHQWDVZLWKVXSHUKHDWLQJ
FDVH
LVHQWURSLF
T
P
S
H
V
ƒ&
N3D
03D
03D
ƒ&
DW P
03D a T H S T H S Ÿ T H N- NJ z
:RUNo
VdP
P NJ u 3D
N- NJ
K
P NJ u N3D
WQHW
WQHW
QLQ
u P NJ u - P QLQ K
Ÿ 7XUELQHLVRQO\EXWDGLDEDWLF
LGHDO
Ÿ WWXUELQH
DFW
WWXUELQH u Ÿ H
N- NJ 6RVWDWHŸ P H T q & H S T q & H S Ÿ T q & S N- NJ. K
P
T
H
S
N3D
03D
03D
ƒ&
N3D
a
,IWXUELQHLVRQO\HIILFLHQW
W LQSUHYLRXVFDVH
W Ÿ H K a u U /
N- NJ S /
N- NJ. U /
N- NJ S /
VDW/
T
H
N3D
VXSHUKHDWHG
YDSRU
VXSHUKHDWHG
YDSRU
S
V
03D
PN3D
03D
ƒ&
aƒ&
H H H H W H H W
K
Q QLQ
D $WVWHDG\VWDWH
dU QH Q M Q L VLQFHW dt
dS Q H Q M Q L
VLQFHSJHQ dt TH TM TL
§ Q
Q ·
Q M TM ¨ H L ¸ DQGIURPHQHUJ\EDODQFH
© TH TL ¹
§ TM · TM
QH ¨ ¸ QL
TL
© TH ¹
VRWKDW
Q
&23 L
Q
H
E TH
T
M
TH
TM
TL
TL TM TL
˜
TM TH TM
TL TL
TM TH
.TM
§ ·
TL ¨
¸
T
T
© M
H ¹
.DQGTL
§ ·
VRWKDW&23 ¨
¸ © ¹
D aVDWYDSRU
OLTXLGSURGXFHG
:RUNLQVWDJH N- NJ :RUNLQVWDJH N- NJ N- NJ N- NJ OLTXLG1 N- NJ 1 SURGXFHG .
M
M
M
c
H
M H
M H M
IUDFWLRQYDSRU
(QWKDOS\RIVWUHDPEHIRUHFRPSUHVVRU
6WDJH N- NJ1 6WDJH
DVEHIRUH H 03D. H 03D.
u u N- NJ
6WDJH 6WDJH DVEHIRUH N- NJ N- NJ/1 6\VWHP FRQWHQWVRIWKHWXUELQH7KLVLVDVWHDG\VWDWHDGLDEDWLFFRQVWDQWYROXPH
V\VWHP
dM
M
RU M
M
M
D 0DVVEDODQFH
dt
(QHUJ\EDODQFH
FRQVWDQW
dU
H M
H Q DGLDEDWLF W P dV YROXPH M
s
dt
dt
(QWURS\EDODQFH
dS
S M
S Q SJHQ
M
T
dt
E\SUREOHPVWDWHPHQW
7KXV
M
M
WS
c
h
S S 6WDWH T
P
q & EDU
6WDWH P
EDU
NJ K H H M
S
x ˜ x x x
Ÿ IUDFWLRQRIOLTXLG
Ws
S
0%
(%
6%
6WHDP
 o 7DEOHV
6WHDP
 o 7DEOHV
N
NJ.
NJ
N u K
NJ
NK
H N- NJ N- NJ S T # q & H | N- NJ N: a
f
E 6DPHH[LWSUHVVXUH P EDU DQGVWLOODGLDEDWLF
H H Ÿ W M
+HUHKRZHYHU
Ws
Ws 3DUWD
Ÿ
H N- NJ
P
EDU
6WHDP
 o 7DEOHV
c
h N-K c H h N-K T # .
S | N- NJ.
NN h NJK u NJ.
.˜K
H H RXWRIYDOYH
P
EDU
S
H LQWRYDOYH
N- NJ
T | q &
S | N- NJ.
)ORZDFURVVWXUELQHLVLVHQWURSLFDVLQSDUW D S
N- NJ.
P
EDU
Ws
6WHDP
 o 7DEOHV
NJ
Nu K
NJ
u q&
T # H | N- NJ
NK
N: D )RUHDFKVWDJHRIWKHFRPSUHVVRUDVVXPLQJVWHDG\VWDWHRSHUDWLRQDQG
UHYHUVLEOHDGLDEDWLFRSHUDWLRQZHKDYHIURPWKHPDVVHQHUJ\DQGHQWURS\
EDODQFHVUHVSHFWLYHO\
M
RUM
M
M
M
in
out
out
in
H M
H W RUW M
H H
M
in in
out out
out
in
DQG
S M
S RUS
S
M
in in
c
u 6WHDP
 o 7DEOHV
h
F )ORZ DFURVV YDOYH LV D -RXOH7KRPSVRQ LVHQWKDOSLF H[SDQVLRQ 6HH
,OOXVWUDWLRQ
7KXV H LQWRYDOYH H RXWRIYDOYH DQGWKHLQOHWFRQGLWLRQVWRWKHWXUELQHDUH
S S
M
SJHQ
7KXV
c
s
out out
out
in
6RWKURXJKHDFKFRPSUHVVRU EXWQRWLQWHUFRROHU VWDJHRQHIROORZVDOLQHRQ
FRQVWDQWHQWURS\LQ)LJ
7KHUHIRUHIRUILUVWFRPSUHVVRUVWDJHZHKDYH
H in T . P EDU N- NJDQGSin T . P EDU
N- NJ. P EDU N- NJDQGTout .
H out S N- NJ.
7KHUHIRUHWKHILUVWVWDJHZRUNSHUNJRIPHWKDQHIORZLQJWKURXJKWKHFRPSUHVVRULV
W ILUVWVWDJH N- NJ N- NJ
$IWHUFRROLQJWKHWHPSHUDWXUHRIWKHPHWKDQHVWUHDPLV.VRWKDWIRUWKH
VHFRQGFRPSUHVVRUVWDJHZHKDYH
H in T . P EDU
N- NJ. P
H S out
N- NJDQGSin T
EDU
. P
N- NJDQGTout
EDU
N- NJ.
.
7KHUHIRUHWKHVHFRQGVWDJHZRUNSHUNJRIPHWKDQHIORZLQJWKURXJKWKHFRPSUHVVRULV
W VHF RQGVWDJH N- NJ N- NJ
6LPLODUO\DIWHULQWHUFRROLQJWKHWKLUGVWDJHFRPSUHVVRUZRUNLVIRXQGIURP
H in T . P EDU N- NJDQGSin T . P EDU
H out S N- NJ. P EDU N- NJDQGTout .
N- NJ.
7KHUHIRUHWKHWKLUGVWDJHZRUNSHUNJRIPHWKDQHIORZLQJWKURXJKWKHFRPSUHVVRULV
W WKLUGVWDJH N- NJ N- NJ
&RQVHTXHQWO\WKHWRWDOFRPSUHVVRUZRUNWKURXJKDOOWKUHHVWDJHVLV
W N- NJ E 7KHOLTXHIDFWLRQSURFHVVLVD-RXOH7KRPVRQH[SDQVLRQDQGWKHUHIRUHRFFXUVDW
FRQVWDQWHQWKDOS\7KHHQWKDOS\RIWKHPHWKDQHOHDYLQJWKHFRROHUDWEDUDQG
.LVN-NJ$WEDUWKHHQWKDOS\RIWKHVDWXUDWHYDSRULVN-NJDQG
WKDWRIWKHOLTXLGLVN-NJ7KHUHIRUHIURPWKHHQHUJ\EDODQFHRQWKHWKURWWOLQJ
YDOYHDQGIODVKGUXPZHKDYH
H in H out RU
H . EDU
NNJ
x ˜ x H VDW GYDSRU EDU xH VDW GOLTXLG EDU NN
x ˜ NJ
NJ
ZKHUHx LVWKHIUDFWLRQRIYDSRUOHDYLQJWKHIODVKGUXPDQG x LVWKHIUDFWLRQRIWKHPHWKDQHWKDWKDVEHHQOLTXHILHG7KHUHIRUHIRUHDFK
NLORJUDPRIPHWKDQHWKDWHQWHUVWKHVLPSOHOLTXHIDFWLRQXQLWJUDPVRI
PHWKDQHDUHORVWDVYDSRUDQGRQO\JUDPVRI/1*DUHSURGXFHG)XUWKHU
VLQFHN-RIZRUNDUHUHTXLUHGLQWKHFRPSUHVVRUWRSURGXFHJUDPVRI
/1*DSSUR[LPDWHO\N-RIFRPSUHVVRUZRUNDUHUHTXLUHGIRUHDFKNJRI
/1*SURGXFHG
F
$VLQWKHLOOXVWUDWLRQZHFKRRVHWKHV\VWHPIRUZULWLQJEDODQFHHTXDWLRQVWREH
WKHVXEV\VWHPFRQVLVWLQJRIWKHKHDWH[FKDQJHUWKURWWOHYDOYHDQGIODVKGUXP
WKRXJKRWKHUFKRLFHVFRXOGEHPDGH 7KHPDVVDQGHQHUJ\EDODQFHVIRUWKLV
VXEV\VWHP VLQFHWKHUHDUHQRKHDWORVVHVWRWKHRXWVLGHRUDQ\ZRUNIORZV DUH
M
M
RUWDNLQJM
M
DQGOHWWLQJxEHWKHIUDFWLRQRIYDSRU
x x
H M
H
M H M
˜ H T . P EDU x ˜ H T . P EDU x ˜ H VDW GOLTXLG P
NNJ
x ˜ EDU
NN x ˜ NJ
NJ
7KHVROXWLRQWRWKLVHTXDWLRQLVx DVWKHIUDFWLRQRIYDSRUZKLFKLV
UHF\FOHGDQGDVWKHIUDFWLRQRIOLTXLG
7KHPDVVDQGHQHUJ\EDODQFHVIRUWKHPL[LQJRIWKHVWUHDPVLPPHGLDWHO\EHIRUH
WKHFRPSUHVVRUDUH
M
WKHQEDVLQJWKHFDOFXODWLRQRINJRIIORZLQWRWKHFRPSUHVVRU
M
M
M M
DQGM
+RZHYHU VLQFHERWKWKHUHF\FOHYDSRUDQGWKHLQOHWYDSRUDUHDW.DQGEDU WKHJDVOHDYLQJWKH
PL[LQJWHHPXVWDOVREHDWWKHVHFRQGLWLRQV VRWKDWWKHLQOHWFRQGLWLRQVWRWKHILUVWFRPSUHVVRUDUHWKH
VDPHDVLQWKHVLPSOHOLTXHIDFWLRQSURFHVV DQGH N $OVR DOORWKHUFRPSUHVVRUVWDJHV
NJ
RSHUDWHDVLQWKHVLPSOHOLTXHIDFWLRQSURFHVV
7KHUHIRUHWKHWRWDOFRPSUHVVRUZRUNSHUNJRIPHWKDQHSDVVHGWKURXJKWKH
FRPSUHVVRULV
W N- NJRIPHWKDQHWKURXJKWKHFRPSUHVVRU +RZHYHUHDFKNJRIPHWKDQHWKURXJKWKHFRPSUHVVRUUHVXOWVLQRQO\NJ
RI/1* WKHUHPDLQGHURIWKHPHWKDQHLVUHF\FOHG &RQVHTXHQWO\WKH
FRPSUHVVRUZRUNUHTXLUHGSHUNJRI/1*SURGXFHGLV N-NJ NJ N-NJRI/1*SURGXFHG7KLVLVWREHFRPSDUHGWRN-NJRI/1*
SURGXFHGLQWKHVLPSOHOLTXHIDFWLRQSURFHVV
DOVRDYDLODEOHDVD0DWKFDGZRUNVKHHW EDU 03Dƒ&
H N- NJ S N- NJ. S EDU
ƒ&
S ' Ÿ T qC H N- NJ W
N- PRO aLGHDOJDV N- PROf
$FWXDOZRUN
dU
H H Q P dV W
M
V
dt
dt
W
H H V M
c
N- NJ
h
)LQDOVWDWH P EDU H 'T
H EDU q & 'H
H EDU q & UV
W
H EDU q &
S EDU S EDU T
S EDU dS
S S S M
JHQ
dt
SJHQ
S S M
c
h
N- NJ.
N- NJ. LGHDOJDV 35 T q & P EDU H u S 1RZ P EDU S *XHVV T q & T
W
M
H
S
$FWXDOZRUN Ÿ H I
- PRO T
S
H
S )LUVWZULWHEDODQFHHTXDWLRQVIRUWKHFRQWHQWVRIWKHWXUELQH7KLVLVDVWHDG\
VWDWHFRQVWDQWYROXPHDGLDEDWLFV\VWHP
M
ŸM
M
0% M
H M
H W Ÿ W N H H M
(%
6%
s
s
a
S M
S Q
M
SJHQ T
0D[LPXPZRUNŸUHYHUVLEOHSURFHVVŸ SJHQ
f
VHH6HF Ÿ S S 7KXVWKHLQOHWDQGH[LWWXUELQHFRQGLWLRQVDUHFRQQHFWHGE\D
YHUWLFDOOLQHRQWKH0ROOLHUGLDJUDP
D H
- J Ws
H H - J - J M
E 7ZRVWDJHWXUELQH8VHEDODQFHHTXDWLRQVDERYHIRUHDFKVWDJH
6WDJH
P
H S
.
EDU
T
- J
P
S
- J.
6WDJH
T
H
S
EDU
q&
H T
E
- J
- J
q &
D
EDU
- J.

o
P
S
- J
q &
D
EDU
1RWH7KHUHLVQRZRUNGRQHLQUHKHDWLQJVWHS QRVKDIWZRUNDQGQRFKDQJH
RI UHKHDWHU ERXQGDULHV 2QO\ HQHUJ\ WUDQVIHU VWHS LV KHDWLQJ
- J Q M

o
- J
$IWHUUHKHDWLQJLQWKHUHDFWRU
P
H i
q&
E
IJ
K
FG
H
T
9HUWLFDOOLQH
H - J T q &
 o 6DWXUDWLRQT FRQQHFWLQJ.
P EDU
IRUP EDU
EDUDQGEDU
7RWDOZRUNRXWJ
Ws
M
- J a - J DGGLWLRQDOZRUNf F 6LPLODUWRSDUW E DERYHH[FHSWWKDWLQWHUPHGLDWHSUHVVXUHLVEDU
Q M
W M
s
- J
a - J DGGLWLRQDOZRUNf - J
3RLQW
1HZ
6WDWH 3DWK
6DW¶G/LT
,VHQWURSLF
,VREDULF
,VHQWURSLF
,VREDULF
3DWK
7 ƒ& 3 03D a
Ç
Ç
§ N- ·
6Ö ¨ ¸ © NJ ¹
È
È
Ö § N- · +
¨ ¸
© NJ ¹
1RWHWKDW3 03D 6Ö FRUUHVSRQGVWRD9/PL[WXUH/HW[
UHSUHVHQWWKHTXDOLW\RUWKHIUDFWLRQRIVWHDPWKDWLVYDSRU
6Ö [ ˜ [ ˜ [
&\FOHFDQQRWIXQFWLRQDVVSHFLILHFEHFDXVHWXUELQHZRXOGKDYHD9/PL[WXUH
/LTXLGGURSOHWVLQDKLJKVSHHGWXUELQHGHVWUR\VWXUELQH
%RLOHU
e
-7YDOYH
d
3XPS
f
7XUE
:3
&RQGHQVHU
c
g
:7
E
3RLQW
6WDWH 3DWK
3DWK
7 ƒ& 3 03D 6DW¶G/LT
,VHQWURSH
,VREDULF
,VHQWKDOSLF
,VHQWURSH
a
ƒ&
Ç
a
Ö § N- · +
¨ ¸
© NJ ¹
È
§ N- ·
6Ö ¨ ¸ © NJ ¹
È
Ç
8VH(QWKDOS\±(QWURS\GLDJUDP)LJDWRJHWDSSUR[LPDWHWHPSHUDWXUHDQGSUHVVXUH
YDOXHVDUHDSSUR[LPDWHDVWKH\ZHUHUHDGIURPWKDWILJXUH $OVRYHUWLFDOOLQH
LVHQWURSH WR03D N3DUHPDLQVLQYDSRUUHJLRQVRQROLTXLGLQWKH
WXUELQHV\VWHPFDQRSHUDWH
Ö N-NJDVEHIRUH
:
F
3
Ö
4 ± N-NJ
%
G
Ö ± ±N-NJ
:
7
Ö ± ±N-NJDVEHIRUH
4
&
ZRUNRXW KHDWLQ
FRPSDUHGWRIRURULJLQDOF\FOH (IILFLHQF\ Ș 0HWKDQH±03D.ĺ03D
)URP)LJPHWKDQHLVDOLTXLGDWWKHVHFRQGLWLRQV
Ö N-NJ 6Ö # N-NJ.
+
/LQHRI&RQVWDQW(QWURS\WR03Dĺ7ZRSKDVHUHJLRQDW03D
Ö
Ö
+
N-NJ +
N-NJ
/
9
6Ö N-NJ 6Ö # N-NJ ˜ .
/
9
([SORVLRQZRXOGEHDSURFHVVDWFRQVWDQWFQWURS\«
)LUVWGHWHUPLQHIUDFWLRQRIOLTXLGDIWHUH[SORVLRQ
/ ±/ ± /
/
OLTXLG DQG9 YDSRU (QWKDOS\RIPHWKDQHDIWHU([SORVLRQ
+ uu N-NJ
VRQHHGLQWHUQDOHQHUJLHVQRWHQWKDOSLHV
8I±8L :
Ö P NJ9
Ö
9
P NJ
9I 9
L
I /
P NJ 2QO\FRUUHFW8IRUILQDOYDSRU
NP
3D
NÖ
u u
u 8
03D u I 9
NJ
NJ
03D P 3D
NN NJ
NJ
NÖ
8
u u I
NJ
: NJ u NNJ
u N- 717 N-NJ
7KHSRVVLEOHH[SORVLRQKHUHZRXOGUHOHDVHWKHHQHUJ\RIDSSUR[LPDWHO\NJRI
717DELJEODVW $YHU\OLNHO\VFHQDULRLQDQDFWXDOH[SORVLRQLVWKDWWKHPHWKDQH
UHOHDVHZRXOGIRUPDWUDYHOLQJYDSRUFORXGZRXOGPL[ZLWKDLUDQGHYHQWXDOO\UHDFKDQ
LJQLWLRQVRXUFHUHVXOWLQJLQDVHFRQGDU\FKHPLFDOH[SORVLRQZKLFKZRXOGUHOHDVHPXFK
PRUHHQHUJ\ d
e
%RLOHU
:3
3XPS
&RQGHQVHU
f
c
4&
Ö
Ö
6WDWH 3
9
+
6Ö 3RLQW
7 ƒ& 3DWK
03D N-NJ PNJ N-NJ. 6DW¶G/LT
È
,VHQWURSH
&RPS/LT
a
a
Ç
,VREDULF
ƒ&
,UUHY
È
,VREDU
6DW¶G/LT
1HZWXUELQHSURGXFHV0:RISRZHUZLWKDVWHDPIORZRINJ6
:7
0%DURXQGWXUELQH
(%DURXQGWXUELQH
+
Ö
Ö
0
0 + 0
0
Ö
+
Ÿ 0
0
0: u :
NJ §
N- Ö ·
N¨ + ¸ u V ©
NJ
V
¹
N
NJ
V
u N
V
7KHUHIRUHVWHDPOHDYLQJWXUELQHLVDW3 03D
Ö N-NJ
DQG +
D
)URP6WHDP7DEOHV7 ƒ&DQG 6Ö N-NJ.
E
(QWURS\EDODQFH
6Ö 0
6Ö 6Ö
0
JHQ
,IWKHWXUELQHZDVLVHQWURSLF3 03DDQG 6Ö N-NJā.
6 JHQ
NJ
N V
NJ ˜ .
N
V˜.
8VLQJOLQHDULQWHUSRODWLRQ
7 7 ¯ ƒ&
DQGDQLQWHUSRODWLRQRQHQWKDOS\
Ö +
Ö N-NJ
+
6RIRUDQLVHQWKDOSLF UHYHUVLEOH WXUELQH
ǻ+ ± N-NJ
:KLOHIRUWKHUHDOWXUELQH
ǻ+ N-NJ
7XUELQH(IILFLHQF\ F
:RUNWRWXUELQH
NNJ
N
# u N: NJ
V
V
D )URPWKHLVREXWDQHWKHUPRG\QDPLFSURSHUWLHVILJXUHLQWKHSUREOHPZLWKWKH
Ö OHDYLQJFRQGHQVHU N- FRQGHQVHURSHUDWLQJDW.3|03D$OVR +
NJ
E ,VREXWDQHHQWHULQJWKHWXUELQHLVDW03DDQG.)URPWKHILJXUH
Ö N- DQG6Ö N- $OVRLVREXWDQHOHDYLQJWKHWXUELQHLVDW3 03D
+
NJ
NJ ˜ .
NÖ N- ZKLFKFRUUHVSRQGVWRDERXW.DQG +
NJ ˜ .
NJ
F )URPWKHHQHUJ\EDODQFHWKHZRUNSURGXFHGLQWKHWXUELQHLV
G8
:
N
Ö
Ö
0
LQ + LQ + RXW :VRWKDW GW
0 LQ
NJ
DQG 6Ö NNJ
V
6LQFH0: N: N-VWKHLVREXWDQHIORZUDWHLV
NV
NJ
G
H 3XPS QRKHDWIORZ P 3D
NN: '3 9 03D ˜ ˜
˜
˜
NJ 03D 3D ˜ P
NJ
N- u NJ N:
NJ
V
V
7XUELQH0: N-V
N- u NJ NJ &RQGHQVHU4 N-NJ 4
NJ
V
V
%RLOHU4 N-NJ 4
(IILFLHQF\
K
QHWZRUNRXW
KHDWLQDWERLOHU
NNJ
u NJ
V
NJ
V
RU 6HHVROXWLRQDV3UREOHPLQIROGHU$VSHQIRUWH[WERRN$OVRE\KDQG
D 7KLVV\VWHPLVRSHUDWLQJDWVWHDG\VWDWHDQGIRUPD[LPXPZRUNLV
DVVXPHGWRRSHUDWHLVHQWURSLFDOO\&RQVHTXHQWO\WKHUHOHYDQWEDODQFH
HTXDWLRQVDUH
HÖ T R &P 03D HÖ T "P 03D W N-NJ HÖ T
"P 03D
RUW HÖ T "P 03D N-NJDQG
SÖ T R &P 03D SÖ T "P 03D RUSÖ T "P 03D SÖ T R &P 03D N-NJ.
)URPWKH6WHDP7DEOHVLQ$SSHQGL[$,,,ZHKDYH
HÖ T
HÖ T
R &P
03D
R &P
03D
N-NJDQGSÖ T
N-NJDQGSÖ T
R &P
03D
N-NJ.
R &P
03D
N-NJ.
1RZE\OLQHDULQWHUSRODWLRQWRREWDLQDQHQWURS\RIN-NJ.ZH
REWDLQ
T
R &DQGHÖ T
VRWKDWW
R &P
N-NJ
03D
N-NJ
N-NJ
>1RWHWKDWXVLQJWKH$63(1SURFHVVVLPXODWRUZLWKWKH,$3:6WKHUPRG\QDPLFV
SDFNDJH GLVFXVVHGEHORZ ZHREWDLQT R&DQGW N:NJKU N-NJ8VLQJWKH$63(1VLPXODWRUWKHFDOFXODWLRQLVGRQHTXLFNO\ZLWKRXWWKH
QHHGIRUDQ\PDQXDOLQWHUSRODWLRQ7KHUHVXOWVRIXVLQJWKHVWHDPWDEOHVDQG$63(1DUH
LQVXFKJRRGDJUHHPHQWODUJHO\EHFDXVHRIWKHDYDLODELOLW\RIWDEOHVUDWKHUWKDQKDYLQJWR
UHDGSURSHUWLHVIURPJUDSKVDVZLOOEHWKHFDVHLQODWHULOOXVWUDWLRQVLQWKLVVHFWLRQ@
E ,QWKLVFDVHZHNQRZWKDWWKHZRUNSURGXFHGE\WKHWXUELQHLVRQO\
WKDWRIDQLVHQWURSLFWXUELQHRUN-NJ7KHUHIRUHIURPWKH
HQHUJ\EDODQFHZHKDYH
HÖ T R &P 03D HÖ T "P 03D W
RUHÖ T "P 03D N-NJ N-NJ
)URPWKHVWHDPWDEOHVZH
HÖ T R &P 03D
N-NJDQGHÖ T
%\LQWHUSRODWLRQZHILQGWKDW
DWT R &HÖ T R &P 03D
R &P 03D
N-NJ
N-NJ 6RZHVHHWKDWDVDUHVXOWRIWKHLQHIILFLHQF\RIWKHWXUELQHRQO\RI
WKHPD[LPXP LVHQWURSLF ZRUNLVREWDLQHGDQGWKHWHPSHUDWXUHRIWKH
H[LWLQJVWHDPLVPRUHWKDQR&KLJKHULQWHPSHUDWXUH
>8VLQJWKH$63(1SURFHVVVLPXODWRUWKHH[LWWHPSHUDWXUHLVFRPSXWHGWR
EHR&@
7KHPD[LPXPWHPSHUDWXUHGLIIHUHQFHZLOORFFXULIWKHSURFHVVLVFDUULHG
RXWUHYHUVLEO\,QWKLVFDVHWKHHQHUJ\EDODQFHLV
H H H RUH
H H DQGVLQFHC3 LVFRQVWDQWT
T T
7KHHQWURS\EDODQFHLV
T
P
T
P
C3 OQ R OQ C3 OQ R OQ T
P
T
P
OQ
OQ
OQ
OQ
T
T
7KHVROXWLRQWRWKHVHHTXDWLRQVXVLQJ0$7+&$'LV
7 7 *LYHQ
§ · ˜ 7 7
¨ ¸
© ¹
˜ OQ§¨
˜
·
©
7˜ 7
¸
¹
˜ ˜ OQ§¨
·
¸
© ¹
] )LQG 7 7
]
§ ·
¨
¸
© ¹ T .DQGT .DGLIIHUHQFHRIPRUHWKDQ.+RZHYHULQ
IDFWD+LOVFK5DQTXHYRUWH[WXEHLVYHU\LQHIILFLHQWDQGYHU\LUUHYHUVLEOH
6HHVROXWLRQDV3UREOHPLQIROGHU$VSHQIRUWH[WERRN
6HHVROXWLRQDV3UREOHPLQIROGHU$VSHQIRUWH[WERRN
6HHVROXWLRQDV3UREOHPLQIROGHU$VSHQIRUWH[WERRN
6HHVROXWLRQDV3UREOHPLQIROGHU$VSHQIRUWH[WERRN
6HHVROXWLRQDV3UREOHPLQIROGHU$VSHQIRUWH[WERRN
1RWHWKDWGXHWRDQHUURULQWKHILUVWSULQWLQJWKLVSUREOHPVWDWHPHQWZDVLGHQWLFDOWR
WKDWRI3UREOHP6RLQVXEVHTXHQWSULQWLQJVWKHLQOHWWHPSHUDWXUHZDVFKDQJHGWR
R&DQGWKHLQOHWSUHVVXUHWR03D6HH3UREOHPLQIROGHU$VSHQIRUWH[WERRN
3DUWD
3DUWE
6HHVROXWLRQDV3UREOHPLQIROGHU$VSHQIRUWH[WERRN
6HHVROXWLRQDV3UREOHPLQIROGHU$VSHQIRUWH[WERRN
6HHVROXWLRQDV3UREOHPLQIROGHU$VSHQIRUWH[WERRN
)URPWKHFORVHGV\VWHPHQHUJ\EDODQFH
'U
Q C9 'T VRC9
LGHDOJDVC3
Q 'T 6LQFHQ DQG'T DUHERWKSRVLWLYHC9 !$OVRIRUWKH
C9 R DQGPRUHJHQHUDOO\C3 t C9 ZHWKHQNQRZC3 !
7KHHQHUJ\EDODQFHIRUWKHVHFRQGH[SHULPHQWLV
P'V
'U C9 'T P'V RU'T C9
$VP'V DQGC9 DUHDOOSRVLWLYH'7WKHWHPSHUDWXUHFKDQJHPXVWEHQHJDWLYHVRWKH
WHPSHUDWXUHZLOOGHFUHDVH
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϱхWƌŽďůĞŵƐ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϱхWƌŽďůĞŵƐ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϱхWƌŽďůĞŵƐ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϱхWƌŽďůĞŵƐ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϱхWƌŽďůĞŵƐ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϱхWƌŽďůĞŵƐ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϱхWƌŽďůĞŵƐ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϱхWƌŽďůĞŵƐ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϱхWƌŽďůĞŵƐ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϱхWƌŽďůĞŵƐ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϱхWƌŽďůĞŵƐ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϱхWƌŽďůĞŵƐ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϱхWƌŽďůĞŵƐ
Solutions to Chemical and Engineering Thermodynamics, 5th ed
C
q &
F wT I F 'T I
H wP K H 'P K c
u u h3D P
H
H
u q & 3D
S
q & 03D
S
u q & 3D
awH wS f
awH wS f
q & 03D
a f a f wa H T f u waS Pf
a f a f wa H Pf waS T f waT H f wa P S f P
u
XQLWOHVV
wa P H f waT S f N
w S P
w H T
u
w H P
w S T
T
q &
F wT I | F 'T I
H wP K H 'P K c
u u h3D NS
P
S
D 6WDUWIURPHTQ
H T P H ,* T P
RT Z z LMN FH IK
V
T
V f
P
F I
H K
wP
RT
aT
wT V
V b V bV b
z LMN RST
V
RT Z V f
F
H
Iz
K
V
da
dV
dT V f V bV b
)URPLQWHJUDOWDEOHVZHKDYH
z
OP
Q
UV
W
R
da dt
RT
aT
dV V b V bV b
V b V bV b
T
RT Z a T
OP
Q
dP
P dV dT V
R
da dt
VR
V b V bV b
H T P H ,* T P
8VLQJWKH0ROOLHUGLDJUDP !!
dx
a cx bcx cc
bc a ccc
,Q RXU FDVH a c
bc
bc a ccc
b
OQ
a cx bc bc a ccc
a cx bc bc a ccc
b c
b IRU a ccc bc b VR a ccc bc
c h
˜ ˜ b b b DQG
Solutions to Chemical and Engineering Thermodynamics, 5th ed
H T P H ,* T P
aa Tda dT f LMOQ V b b OQ V b b OP b
MN V b b V b b PQ
aa Tda dT f OQ V d ib
RT Z V d ib
b
RT Z V f
V
RUILQDOO\
H T P H ,* T P
RT Z aTda dT af OQLM Z d iB OP b
MN Z d iB PQ
E 7KLVSDUWLVVLPLODUH[FHSWWKDWZHVWDUWIURPHTQ S T P S ,* T P
R OQ Z z LMNFH IK
z LMN
V
V f
R OQ Z OP
Q
dP
R
dv
dT V V
OP
Q
V
R
da dT
R
dV
V b V bV b V
V f
d i d i
da dT F Z d i B I
OQG
R OQ Z B J
b H Z d i B K
L F w P I POPdV 7KXV F w U I
6WDUW ZLWK HTQ dU C dT MT G
GH w T JK
MN H w T JK PQ
FG w U IJ C LMT FG w P IJ POPFG w V IJ DQG
HwTK
MN H w T K PQH w T K
FG w U IJ FG w U IJ LMT FG w P IJ POPFG w V IJ H w T K H w T K MN H w T K PQH w T K
V
V b
da dT V b
OQ
R OQ Z R OQ
V V f b V b
V
V f
C9 9
V
9
P
V
P
V
D ,GHDOJDV PV
T
P
V
RT FG w P IJ P Ÿ FG w U IJ FG w U IJ HwTK
HwTK HwTK
V
P
P
E YDQGHU:DDOVJDV
FG IJ
H K
wP
RT
a
wT V
V b V
P
$OVR dP
V
LM F I
MN GH JK
R
wP
Ÿ T
P
wT V
V b
RdT
RT
a
dV dV V b V b
V
OP a PQ V
V
Solutions to Chemical and Engineering Thermodynamics, 5th ed
L T a V b O
R V b
F w V IJ
ŸG
H w T K RT V b a V MN V b RV PQ
a
F w u IJ FG w u IJ
ŸG
H w T K H w T K V T V b a V b RV
P
P
V
aRV V b
RTV a V b F 7KH9LULDO(TXDWLRQRI6WDWH
PV
RT
RT
B RT
¦ i i V
i V
RU P
1RWH7KLVLVDWRWDO
FG w P IJ R ¦ B R ¦ RT F dB I m GHULYDWLYH
VLQFHB LV
HwTK V V
V H dT K
DIXQFWLRQRIRQO\WHPSHUDWXUH F w P IJ P ¦ RT dB
Ÿ TG
HwTK
V d OQ T
$OVRQHHG aw V w T f EXWWKLVLVKDUGHUWRHYDOXDWHDOWHUQDWLYHO\6LQFH
FG w V IJ FG w P IJ FG w T IJ Ÿ FG w V IJ aw P w T f H w T K H w V K H w PK
H w T K aw P w V f
i
i i V
i
i P
V
P
i
i V
i
i i B C
B
" ¦ ii V V
i V
T
V
P
T
FG w P IJ LVJLYHQDERYH
HwTK
V
FG w P IJ
H wV K
Ÿ
T
FG w V IJ
Hw TK
RT
i Bi RT
¦
V
V i i FG
H
i FG
H
T RT V ¦ i Bi RT V i P
i 8VLQJ
DQG
IJ
K
Bi RT
ZHJHW
i i V
P¦
FG w V IJ
Hw TK
RT
V
ia
d
V RT V ¦ Bi RT V i ¦ RT V i dBi d OQ T
d
ia
V P ¦ RT V i dBi d OQ T
P
a
f
T P ¦ iBi RT V
i f
fIJK Solutions to Chemical and Engineering Thermodynamics, 5th ed
FG w U IJ FG w U IJ
HwTK HwTK
P
LM
MN
ia
d
i dBi d OQ T
R dBi P ¦ RT V
¦ V i d OQ T
P ¦ iBi RT V i V
f OP PQ
D 6WDUWIURP
FG IJ OP Ÿ C LMV T FG w V IJ OP H KQ
Hw TK Q
PN
L F w V IJ OP T FG w aV T fIJ DQG C T FG w aV T fIJ EXW MV T G
P H wT K
N Hw TK Q H w T K
PC
F w aV T fIJ P C LQWHJUDWH F V I F V I
E G
dT z
H
K
H
K
H wT K T
T
T
T
P
LM
N
wV
V T
wT
C3
3
P
P
3
P
P
P
T P
3
T P
P
T P
T P
T P
3
a f V aT Pf TT T z PTC dT 7KXV V T P
P
3
T P
RT
a
V b V
FG w P IJ RT a H wV K V b V
FG w P IJ RT a H wV K V b V
T
T
DWWKHFULWLFDOSRLQW P o PC T o TC DQG V o VC FG w P IJ
H wV K
FG w P IJ
HwV K
Ÿ
T
'LYLGLQJ E\ Ÿ
a
$OVR
PV
RT
V b
C
a
V C RTC
a
VC b
aV bf
T
RTC
aV bf
C
a
V C
f
C
b RTC
b b bRTC
a
a
b b b
a
V C
f V Ÿ b V IURP V
a
DQG PC
V b RTV
Ÿ ZC
C
Ÿ
RTC
PCV C
RTC
VC
a
V C b RTCVC
VC RTC
a
b
a f
VC
V C RTC
V C RTC
VC
a f
Solutions to Chemical and Engineering Thermodynamics, 5th ed
dS
)RU
FG IJ dP >HTQ @
H K
C3
wV
dT T
wT
WKH
d
d S S ,*
P
LGHDO
C3
R
dT dP T
P
d S ,*
JDV
i LMN RP FGH ww VT IJK OPQdP DQG
7KXV
DW
FRQVWDQW
WHPSHUDWXUH
P
S T P S ,* T P S T P
z RST FGH IJK UVW
wV
R
wT
P
P
P +RZHYHU S T P S ,* T P
PV Z Tr Pr RT 7KXV
a
FG w V IJ
HwTK
RS a
T
FG a
H
w Z Tr Pr
RZ Tr Pr RT
wT
P
P
dP P
VLQFH DOO IOXLGV DUH LGHDO DW
f
S ,* T P
f
fIJ UV KW
P
DQG
FG IJ
H K
FG IJ
H K
L Z T FG w Z IJ OPdP Ÿ S T P S T P R z M
N P P Hw TK Q
L ZaT P f T F w Z I OPdP
R z M
G J
P H w T K PQ
MN P
R
wV
P
wT
P
RT w Z
R
Z Tr Pr P wT
P
ka
f p
,*
P
T P
TP P
Tr Pr
r
Tr Pr r
r
r
Pr
NRT
c
EDU P
N
h
NPRO
(QHUJ\EDODQFHFORVHGQRQIORZV\VWHP
z
'U
. u u EDU ˜ P NPRO.
+RZHYHUIRULGHDOJDV 'U
W
Q
z
PdV
Q PdV
Q W VLQFHTLVFRQVWDQW LVRWKHUPDO 7KXV
z
NRT
dV
V
V
NRT OQ V
NRT OQ
Q
u N
0
0
$OVRE\,GHDO*DV/DZDWIL[HGTDQGN
P
P
F I H K
NPRO u - PRO. u u OQ
r
D ,GHDOJDV
PV
r
r
P
$OVR
Solutions to Chemical and Engineering Thermodynamics, 5th ed
PV
Ÿ V
PV
V
P
P
P u
P E &RUUHVSRQGLQJVWDWHV
Tr
LQLWLDOVWDWH
ILQDOVWDWH
Z
H ,* H
TC
S ,* S
FDO
PRO.
FDO
PRO.
Pr
PV
ZRT
1XPEHURIPROHVRIJDV
)LQDOYROXPH V f
u u u u (QHUJ\EDODQFHRQJDV 'U Q W (QWURS\EDODQFHRQJDVSURFHVVHVLQJDVDUHUHYHUVLEOH 'S
N
Q
RU Q
T
'S
S f Si
PV
IURPDERYH RT
NPRO ZNRT
Pf
P
'S
Q
SJHQ Ÿ SJHQ
T
7KHUHIRUH
T 'S c
h N mcS S h cS S h cS S hr
P
R
UV
N S u OQ
u P
T
W
U
R
N S OQ
N - .
V W
T
N S f Si
f
,*
f
,*
f
,*
i
,*
i
i
f
i
u NPRO u Q
T 'S
W
'U Q
N U f Ui Q
0
N H f H i N Pf V f PV
i i Q
LM d H H i cH H h T dH H i Z RT Z RT OP Q T
T
MN
PQ
O
L u NPRO M
PQ 0 u Nu N TC
,*
f
f
,*
f
,*
i
,*
i
i
f
C
C
f
i
i
C
6LQFHSURFHVV
LVLVRWKHUPDO
u - 0
0-
0- F 3HQJ5RELQVRQ(26
P EDU DV WKH UHIHUHQFH VWDWH ZH REWDLQ
8VLQJ WKH SURJUDP 35 ZLWK T T q & P EDU Z V u P PRO H - PRO S - PRO. T q & P EDU Solutions to Chemical and Engineering Thermodynamics, 5th ed
u P PRO H
V
Z
- PRO S
- PRO N
Q
P
NPRO u P PRO
TN' S u u W
'U Q
V
V
0- H PV f H PV i Q
N H PV f H PV i Q
LM u u OP
Mu - EDU ˜ P P u u MMN u u u PPQ
0^1RWHWKDWNQDQGWDUHFORVHWRYDOXHVREWDLQHGIURPFRUUHVSRQGLQJVWDWHV`
FG w T IJ
H w PK
w T S
w P S
w T S w P T
˜
w P T w P S
w S T w P T
a f w S P w T P aw S w Pf
aw V w T f VDT C T
C
aw S w T f
S
T
P
P
3
3
DQG
a V faw V dPf
a V faw V dPf
NS
NT
w V S w P S
w V T w P T
S
T
FG w S IJ ˜ FG w T IJ
Hw TK Hw SK
w S V w T P
˜
w T V w S P
D
V
C9 T
˜
T C3
P
C9
C3
FG w H IJ w H T w H T ˜ w P T FG w H IJ FG w P IJ H w V K w V T w P T w V T H w P K H w V K
F w P IJ z H[FHSWDWWKHFULWLFDOSRLQW 6LQFH G
H wV K
T
w V S w P T
˜
w V T w P S
T
T
T
FG w H IJ
H wV K
FG w H IJ H w PK
F w S IJ w S P w S P ˜ w T P FG w S IJ ˜ FG w T IJ
E G
H w V K w V P w T P w V P H w T K H w V K
C TV
C
C F dT I
w SI
F
˜ ˜V G
J a V faw V w T f TVD Ÿ GH w V JK a D
T V H dV K
LI
T
T
P
P
3
3
P
P
3
P
P
Solutions to Chemical and Engineering Thermodynamics, 5th ed
D :HVWDUWE\XVLQJWKHPHWKRGRI-DFRELDQVWRUHGXFHWKHGHULYDWLYHV
FG w T IJ
HwV K
a f
w T H
w VH
H
w T H w T P w T V
˜
˜
w T P w T V w V H
w H T w P T w V T
w P T w H V w T V
FG w H IJ aw P w V f H w P K aw H w T f
T
T
aw H w V f
ad H w T f
V
T
V
1RZIURP7DEOHZHKDYHWKDW
FG w H IJ
H w PK
V T
T
FG w V IJ DQG FG w H IJ
Hw TK H w T K
P
DOWHUQDWLYHO\VLQFH H
a
V w P wV T T w P w T V
H
f
a
FG w T IJ
HwV K
a
FG w P IJ FG w V IJ
H wV K Hw TK
w T S
w VS
S
f
P
FG w S IJ FG w T IJ
HwV K H w SK
V
f a
f C V aw P w T f
9
V
FG w P IJ HwTK
V
w T S w V T
˜
w V T w V S
T
FG IJ
H K
w S T w T V
˜
w V T w S V
T wP
C9 w T V
E )RUWKHYDQGHU:DDOVIOXLG
FG w P IJ
HwTK
V
FG IJ
H K
R
wP
wV T
V b
a
RT
V b
V
7KXV
FG w T IJ
HwV K
V
a
f
w P wV T V T wV w T P
C9 V w P w T V
1RWH,KDYHXVHG
V
V
V
T
F dP I H dT K
P
FG w H IJ FG w U IJ FG w PV IJ
HwTK HwTK H wT K
C9 V
V
7KXV
FG w T IJ
HwV K
FG w V IJ OPFG w P IJ H w T K QH w T K
U PV V
LM
N
C3 V T
n
RTV V b a V RT V b
H
DIWHUVLPSOLILFDWLRQZHREWDLQ
C9 V R V b
s
Solutions to Chemical and Engineering Thermodynamics, 5th ed
FG w T IJ
HwV K
a V b RTV b
H
C& V b V R V b V DQG
FG w T IJ
HwV K
S
RT
C9 V b
7KHUH DUH D QXPEHU RI ZD\V WR VROYH WKLV SUREOHP 7KH PHWKRG , XVH LV D OLWWOH XQXVXDO EXW WKH
VLPSOHVW WKDW , NQRZ RI $W WKH FULWLFDO SRLQW DOO WKUHH URRWV RI V DUH HTXDO DQG HTXDO WR V C a
0DWKHPDWLFDOO\WKLVFDQEHH[SUHVVHGDV V V C
FRPSDUHWKLVZLWK
P
f
ZKLFKRQH[SDQVLRQEHFRPHV
V V CV V CV V C
RT
a
V b V V b b V b
RT
a
V b V bV b
ZKLFKPXOWLSO\LQJWKURXJKE\WKHGHQRPLQDWRUVFDQEHZULWWHQDV
F
H
V V b I FG
K H
I F
K H
RT
RTb ab
bRT a
b V b P
P
P
P
P
IJ K
&RPSDULQJWKHFRHIILFLHQWVRI V LQ(TQV DQG JLYHV TC PC V b V C bRTC a
PC
PC
V C V b RTC ab
b PC
PC
V C PCVC
RTC
ZC RU
PCb
RTC
V b RTC
PC
)URP(TQ PCb
RTC
)RUFRQYHQLHQFHOHW y
y
7KHQ
PCb
RTC
ZC RU ZC
)URP(TQ y
ZC Solutions to Chemical and Engineering Thermodynamics, 5th ed
FG P b IJ FG P b IJ aP
H RT K H RT K aRT f
C
C
C
C
aPC
Ÿ y y aPC
y y c
aRT f
C
C
C
C
C
C
C
C
RU
y y y 7KLVHTXDWLRQKDVWKHVROXWLRQ y
Ÿb
a
FG P b IJ FG bP IJ FG P b IJ FG aP IJ Z H RT K H RT K H RT K H a RT f K
y y y y ˜ c y y h h
)LQDOO\IURPHTQ C
RUH[SDQGLQJDQGUHDUUDQJLQJ
y a RT f
C
ZC
C
C
C
RTC
IURP(TQ PC
a RT f IURP(TQ C
PC
y
1RWHWKDWZHKDYHHTXDWHGaDQGbWR TC DQG PC RQO\DWWKHFULWLFDOSRLQW7KHUHIRUHWKHVHIXQFWLRQV
$OVR ZC
FRXOGKDYHRWKHUYDOXHVDZD\IURPWKHFULWLFDOSRLQW+RZHYHUDVZHKDYHHTXDWHGIXQFWLRQVRI V ZH KDYH DVVXPHG a DQG b ZRXOG RQO\ EH IXQFWLRQV RI T 7KHUHIRUH WR EH FRPSOHWHO\ JHQHUDO ZH
FRXOGKDYH
aRT f D FG T IJ
HT K P
RT F T I
b EG J
HT K
P
a
C
C
C
C
C
C
FG T IJ o DV T o DQG E FG T IJ o DV T o HT K
HT K
T
T
ZLWK D
,Q IDFW 3HQJ DQG 5RELQVRQ DQG RWKHUV KDYH VHW E DW DOO WHPSHUDWXUHV DQG DGMXVWHG D DV D
IXQFWLRQRIWHPSHUDWXUHWRJLYHWKHFRUUHFWYDSRUSUHVVXUH VHHFKDSWHU C
C
C
DOVRDYDLODEOHDVD0DWKFDGZRUNVKHHW C
Solutions to Chemical and Engineering Thermodynamics, 5th ed
N
Q
0%
(%
N N H N H Q
Ÿ
H H
T q & P EDU Z u P PRO V
- PRO H
- PRO - PRO. S
- PRO Q
N
EDUUHIHUHQFHVWDWHZHREWDLQ
H
Q
N
- PRO DOVRDYDLODEOHDVD0DWKFDGZRUNVKHHW 6LQFH
SURFHVV
LV
S . EDU
ST
H
DGLDEDWLF
DQG
UHYHUVLEOH
'S
- PRO DQG S
V
. P
EDU Z
u P PRO V
WS
W
N H N H RU S
N
H H
D
FG P a IJ V b
H VK
PV
RT
H
Q
WV P
N H N H OLP
PV
P o RT
V of
Po
V of
V of
. DQG P
. P
- PRO RT Ÿ
dV dt
- PRO V
a
V b RTV
FG V a IJ H
RTV K
PV I
RV a U
LL B OLP V F
H RT K OLP V STV b RTV VW RV V b a UV OLP RS bV a UV b a OLP V S
T V b RTV W T V b RT W RT
L
LH
u P PRO YROXPHRI
FRPSUHVVRU
FRQVWDQW
DGLDEDWLF
dU
(%
dt
S f - PRO. 6\VWHP FRQWHQWVRIFRPSUHVVRU
dN
0%
N N Ÿ N N dt
Si
. Ÿ Tf
RU
" EDU 8VLQJ WKH SURJUDP 35 ZLWK WKH T
%\ WULDO DQG HUURU NQRZLQJ P DQG S JXHVVLQJ T ZH REWDLQ T
Z
Ÿ N S
EDU UHIHUHQFH VWDWH ZH REWDLQ T
N N P
T q & P EDU
Z u P PRO dN
dt
dU
dt
$OVRQRZXVLQJWKHSURJUDP35ZLWK T
V
N
OLP
V of V b
V of
V of
S
EDU - PRO. Solutions to Chemical and Engineering Thermodynamics, 5th ed
LLL C
OLP V Po
V of
ŸC
FG PV B IJ OLP V RSbV b V b UV OLP b V b H RT V K
T V b W V b
V of
b E $WWKH%R\OHWHPSHUDWXUH OLP V
Po
a
TB
RTB
b
TB
V c RTc
RVc a
EXW a
Rb
Tc
V of
F PV I Ÿ B H RT K
V c RTc
b
Vc
(TQVD Tc D )URPWDEOH TB a . LH B TB
B . 7KHLQYHUVLRQWHPSHUDWXUHLVWKHWHPSHUDWXUHDWZKLFK
wT
wV
V T
C3
wP H
wT P
a f
FG IJ
H K
FG w T IJ
H w PK
F w V IJ
V TG
Hw TK
LM
N
LM
N
FG IJ OP
H KQ
OP R dB
Q P dT
P
RT
w
B
wT P P
P
dB
RT
RT
B
T
dT
P
P
dB
dT
dB
dt
3ORWXSBYVTREWDLQ dB dT HLWKHUJUDSKLFDOO\RUQXPHULFDOO\IURPWKHWDEXODUGDWD,ILQG
7KXV T LQY LVWKHWHPSHUDWXUHDWZKLFK B T
T LQY a . $OVR
dB dT GHFUHDVHV
ZLWK
LQFUHDVLQJ
WHPSHUDWXUH
LH
dB dT a FP PRO. DW.DQG FP PRO. DW.3UHVXPDEO\LWLVQHJDWLYH
DWHYHQKLJKHUWHPSHUDWXUH E *HQHUDOO\
FG w T IJ
H w PK
P
H
BT
RS
T
FG IJ UV RSB T dB UV H K W C T dT W
wV
V T
C3
wT
3
P
8VLQJWKHGDWDLQWKHWDEOHLWLVHDV\WRVKRZWKDWIRU T T LQY B T
dB
Ÿ P ! ZKLOHIRU
dT
dB
! Ÿ P dT
F 6LQFH)LJIRUQLWURJHQLVDQHPSORWLVHDVLHVWWRSURFHHGDVIROORZV
T ! T LQY B T
aw H w Pf
FG w T IJ
H w P K aw H w T f aw P w H f aw H w T f 6LQFH aw H w T f
C LV!DQGOHVVWKDQf>([FHSWDWDSKDVHWUDQVLWLRQ²VHH&KDSDQG
3UREOHP²KRZHYHU PKDVQRPHDQLQJLQWKHWZRSKDVHUHJLRQ@LI adT dPf LVWREH]HUR
WKHQ ad H dPf PXVWHTXDO]HUR7KDWLVDQLQYHUVLRQSRLQWRFFXUVZKHQLVRWKHUPVDUHSDUDOOHO
T
H
P
P
T
P
3
H
T
WROLQHVRIFRQVWDQWH YHUWLFDOOLQH 7KLVRFFXUVDWORZSUHVVXUHV LGHDOJDVUHJLRQ DQGDWKLJK
Solutions to Chemical and Engineering Thermodynamics, 5th ed
SUHVVXUHV QRQLGHDOJDVUHJLRQ 6HHIRUH[DPSOH T q & LVRWKHUPQHDU03D ZKLFK
LVRIIWKHILJXUH 7R LGHQWLI\ WKH LQYHUVLRQ WHPSHUDWXUHV RI QLWURJHQ ZH FDQ XVH )LJE D WHPSHUDWXUH
HQWURS\GLDJUDP)URPSDUWDRIWKLVSUREOHPZHQRWHWKDWDW T LQY T LQY
V
FG w V IJ Ÿ P T FG w P IJ Hw TK
HwTK
LQY
P
7KXV DW HDFK LQYHUVLRQ WHPSHUDWXUHV T LQY ZH FDQ ILQG D GHQVLW\ RU SUHVVXUH IRU ZKLFK WKLV
HTXDWLRQLVVDWLVILHG8QIRUWXQDWHO\LWLVGLIILFXOWWRUHDGWKHILJXUH
NJ V
P
P
NJ
u P NJ 8VLQJ)LJZHILQG P | EDU !
. DQG V
8VLQJWKHSURJUDP35ZLWK T
u P NJ ZHILQGE\WULDODQGHUURU
WKDW P EDU 8VLQJ WKH SURJUDP 35 ZH ILQG DW q& DQG EDU Z
Tr
:HILQG Z
d
Pr
i
H ,* H TC
- PRO S ,* S
T u T u T ' H ,*
' S ,*
z
z
C3 dT
- PRO
F
H
EDU
C3
dT R OQ
EDU
T
I - PRO.
K
7KXV
HT
q & EDU
H q & EDU ' H ,* TC
FG H H IJ
H T K
,*
C
- PRO. )URP$SSHQGL[$,,
C3
u P PRO H - PRO N- PRO DQG S - PRO. EDU
7RXVHWKHSULQFLSOHRIFRUUHVSRQGLQJVWDWHVZHZLOODVVXPHWKHVWDWHRI T q & DQG P LVDQLGHDOJDVVWDWH LHGRQ¶WQHHGFRUUHFWLRQVIRUQRQLGHDOLW\DWWKLVFRQGLWLRQ $Wq&DQG
EDUZHKDYH
V
ST
)LQDOO\ PV
q & EDU
ZRT V
- PRO
u u u q&EDU
- PRO. u P PRO Solutions to Chemical and Engineering Thermodynamics, 5th ed
(TXDWLRQRIVWDWH P V b
D
FG w P IJ
HwTK
V
RT FG IJ
H K
P
wV
wT
T
R
V b
P
C9 T
C3
FG w V IJ FG w P IJ
Hw TK HwTK
P
T
3
P
P
V
P
C9 T ZHPXVWKDYHWKDW
FG IJ
H K
w
wP
wTV wT V
FG w T IJ
H w PK
H
w
R
w T V V b
T
T
V
a f C T
Ÿ C9 T V
9
FRQVWDQW 3
6LQFH C3 LVLQGHSHQGHQWRIPLQWHJUDWLRQFDQEHGRQHHDVLO\
z
T
C3 T dT
a
f
b P P T
WRSURFHHGZHQHHGWRNQRZKRZ C p GHSHQGVRQT,I C p LVLQGHSHQGHQWRITZHKDYH
FG w P IJ
HwT K
9
3
P
FG w C IJ
H wV K
FG IJ OP L RT b RT O b H K Q C NM P
P QP
C
LM
N
wV
V T
C3
wT
3
P
E )LUVWFDVHLVFOHDUO\D-RXOH7KRPVRQH[SDQVLRQ Ÿ H
6LPLODUO\IRU C9 V T
FG w P IJ
HwT K
C9 R P
V
T
C9 T
FG w V IJ Hw T K
R
w FwV I
w
G J w T P Ÿ C T P C T
w T Hw TK
3
R P
˜
P T
P
V b
C3 T ZHPXVWKDYHWKDW
IRU C3 P T
FG w C IJ
HwPK
FG w V IJ
HwT K
7KXV
FG IJ
H K
V b
wP
DQG
wV T
T
R
P
T T
a
f
b
P P C3
(TQ DOVR KROGV LI C3 LV D IXQFWLRQ RI T EXW WKHQ LW LV WKH DYHUDJH KHDW FDSDFLW\ RYHU WKH
WHPSHUDWXUHLQWHUYDOZKLFKDSSHDUVLQ(TQ 7KHVHFRQGH[SDQVLRQLVDWFRQVWDQWHQWURS\ NH\ZRUGVDUHUHYHUVLEOHDQGDGLDEDWLF FG w T IJ
H w PK
S
aw S w Pf
aw S w T f
T
P
aw V w T f
C3 T
z
T
P
T R
dT
Ÿ C3
C3 P
T
T
z
P
R
dP
P
P
Solutions to Chemical and Engineering Thermodynamics, 5th ed
,I C3 LVLQGHSHQGHQWRITWKHQ
T
T
FG P IJ
HPK
VWDWHYDULDEOHFRQVWUDLQWV T
D ,GHDOJDVVROXWLRQ
(TQVRIVWDWH PV
)URP0%JHW
Pi
Ti
P f
Tf f
)URP(%JHW Ti
T f Ÿ Ti
T
T P
f
f
f
P
f
P V
V Ÿ U f
i
P
EDU f
C9T C3TR NRT U
Tf
q & P f
ZNRT RU PV
,*
,*
,*
,*
,*
,*
,*
u FG
H
Z Tr
u
Pf
Pf
RU f f
f f
Z T
Z T
u u Pr
Nf U f N f U f
u IJ K c N N hU c N N hU Ÿ U
f
f
ZKHUHZHKDYHXVHGWKHIDFWWKDW U f
U f U i
b g
f
DQG
f
U f 6LQFH T f
b g c
h
i
f
f
Tf DQG P f
c
h
U ,* T f U ,* T i H H ,* T f P f H H ,* Ti Pi
Z RT T f P f Z RT T i Pi
,*
T
)URPHQHUJ\EDODQFH
Ni U i
T U U
,*
u u Zi
ZRT T i U T d H H i PV PV
d
U T d H H i Z PV
U T d H H i Z RT
U
,*
U f ZKHUHZHKDYHXVHG
)URPWKHPDVVEDODQFH
Pi
Z i Ti
f
PV
C3 T E &RUUHVSRQGLQJVWDWHVVROXWLRQ
U T
Nf U f N f U f HQHUJ\EDODQFH Ni U i
PRUHFRPSOLFDWHGH[SUHVVLRQDULVHVLI C3
DOVRDYDLODEOHDVD0DWKFDGZRUNVKHHW *HQHUDO
PDVVEDODQFH Ni Nf N f R C3
Ui Pf %XW
Solutions to Chemical and Engineering Thermodynamics, 5th ed
c h
.h
c h C cT T h - PRO. cT U ,* T f U ,* T i
f
i
f
dH H i
u Z RT T i P i
u u ,*
T i Pi
Y
Ÿ
- PRO
- PRO
d
Tf ^ u ` u H H ,*
i
T f P f
Z RT T f P f
d
Tf H H ,*
Pf
TfZf
F w T IJ dV FG w T IJ dU Ÿ T T
dT G
HwV K HwU K
f
FG w T IJ
HwV K
V
U
aw U w V f
aw U w T f
T
Vi
a
f
T w P wT V P
C9
V
wT
dV wV U
a
C9V 1RZE\(TQ 9
T
V
9
V
EXW C9
9
V
C3 R
z
Vf
ŸT T
f
i
- PRO. z
a
VLQFHV i
C9V i
f
V a
dV
C9V Vi
z FGH IJK
Vf
i DORQJ
SDWKRI
FRQVWU
FG w C IJ T FG w P IJ IRUYG:HRV P RT a H wV K H w T K
V b V
FG w P IJ R FG w P IJ Ÿ C LVLQGHSHQGHQWRIYROXPH Ÿ C
H w T K V b H w T K
EXW
u ,IRXQGWKHVROXWLRQE\PDNLQJDJXHVVIRU T f XVLQJHTQ DQG)LJWRILQG P f E\WULDO
DQG HUURU 7KHQ JXHVVHG T f DQG FRPSXWHG P f ZHUH WHVWHG LQ HTQ 6ROXWLRQ IRXQG
T f a . q & P f a u 3D EDU F 7KHYDQGHU:DDOVJDV
f
:HNQRZWKDW U i U f DQG V i
V 7RHYDOXDWHWKHILQDOWHPSHUDWXUHZHVWDUWIURP
U
Z RT T f P f
T f P f
7REHVROYHGDORQJZLWK
i
Vf
a
dV
C9 V i V FG
H
a
i
f
C9 V
V
IJ
K
C9 Solutions to Chemical and Engineering Thermodynamics, 5th ed
7KXVWKHILUVWVWHSLVWRILQG V i F a I
GG P JJ dV bi RT
H dV i K
u P PRO V f
ŸVi
a
f c
.
ŸTf
RT
a
f
V b
Vf
d
u 3D
Pf
i c h
h . q &
u u u u f
Pf
u P PRO 3D ˜ P PRO
u - PRO. u u P PRO
T f Ti
i
i i
c
h
EDU
G +HUHZHZLOOXVHWKHSURJUDP358VLQJWKH.DQGEDUUHIHUHQFHVWDWHZHILQGWKDWDW
V i u P PRO H i - PRO DQG
WKH LQLWLDO FRQGLWLRQV Z - PRO. 7KHUHIRUH
Si
1RZVLQFH U f
V
V u P PRO :H PXVW E\ WULDODQGHUURU ILQG WKH WHPSHUDWXUH DQG SUHVVXUH RI WKH VWDWH KDYLQJ WKHVH
SURSHUWLHV , ILQG WKH IROORZLQJ DV WKH VROXWLRQ Tf . P a EDU IRU ZKLFK
Ui
f
u u u H i PiV i
Ui
- PRO - PRO DQG
i
V P PRO DQG U - PRO 7RVXPPDUL]HZHKDYHWKHIROORZLQJDQVZHUV
IRUWKHGLIIHUHQWSDUWVRIWKHSUREOHP
,GHDOJDV
&RUUHVSRQGLQJVWDWHV
YDQGHU:DDOV
3HQJ5RELQVRQ
Pf EDU
EDU
EDU
EDU
2QFHDJDLQWKHLGHDOJDVVROXWLRQLVVHULRXVO\LQHUURU
0DVVEDODQFH V\VWHP ERWKWDQNV Ni Nf N f Tf .
.
.
.
Nf U f N f U f HQHUJ\EDODQFH V\VWHP ERWKWDQNV Ni U i
HQWURS\EDODQFH V\VWHP SRUWLRQRILQLWLDOFRQWHQWVRIWDQNDOVRLQWKHUHILQDOO\ S i
$OVR P f
D ,GHDOJDVVROXWLRQREWDLQ
Pi
Pf
P f Ni
P f Pf
V
Nf
V i
Pi
Ti
P f Ÿ P f
V
DQG N f
V f
V
V f
P f Pf
IURPPDVVEDODQFHDQG
T f Tf
EDU
u 3D IURPHQHUJ\EDODQFH
S f Solutions to Chemical and Engineering Thermodynamics, 5th ed
T
f
Ti
FG P IJ
HPK
f
R C3
i
F I
H K
.
q & Ÿ T f
.
q &IURPHQWURS\EDODQFH
DQG
Tf
$OVR
Nf
Ni
Ÿ Tf
Ti T f
N f
Ni
f
i
u u RS UV u u u T Z T Z T W ,QLWLDO
H ,* H
- PRO. S ,* S
TC
0DVVEDODQFH
FRQGLWLRQV
Pi
Zi Ti
Pf
S f S i
Tr
Pr
- PRO. f f
f f
(QWURS\EDODQFH
dS S i dS
,* f
,*f
S ,*i
i dS S i f
f
T
P
R OQ
dS S i C OQ u ,* f
3
(QHUJ\EDODQFH
Ni U i
,* i
RU
FG N IJ HN K
E &RUUHVSRQGLQJ6WDWHV6ROXWLRQ
DQG
u ˜
u P f V RTi
˜
RT f PiV
RU
Nf U f N f U f EXW Ni
d
i
d
Nf N f Ÿ Nf U f U i N f U f U i
i ŸZ
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Pf V
P f V
f
f
i
f
i i
H f Pf V f H i PiV i
H
P
V
H
P
V
Zf RTf
Zf RT f
H f H f ,* H f ,* H i ,* H i H i ,* Zf RT f Zi RTi
f f
Z T
f f H f H f ,* H f ,* H i ,* H i H i ,* Zf RTf Zi RTi
Z T
nd
nd
nd
i d
i d
i d
is
nd
i d
i d
i d
i
is
s
i
s
6XEVWLWXWLQJLQWKHNQRZQYDOXHVJLYHV
cT h Z T s
ndH H i H
H
T
Z
T
c
h
i
nd
s
Z T
f ,*
f
Zf T f
f f
f
f ,*
f
f f
f
f f
(TQV QRZPXVWEHVROYHG2QHSRVVLEOHSURFHGXUHLV
L *XHVV P f LL 8VH(TQ WRILQG T f LLL 8VH(TQ WRILQG Tf LY 8VH(TQ WRJHWKHUZLWK T f DQG Tf WRVHHLIJXHVVHG P f LVFRUUHFW,IQRWJREDFNWR
VWHSL
$IWHU many LWHUDWLRQV , IRXQG WKH IROORZLQJ VROXWLRQ P f EDU T f .
Tf . Nf Ni N f Ni
F 3HQJ5RELQVRQHTXDWLRQRIVWDWH
+HUHZHXVHWKHHTXDWLRQV
P f
DQG Ni
V V i Nf
H PV S S i
Nf U f N f U f ZLWK U
Nf N f Ni
Ni U i
Pf
V V f N f
f
Pf V V f
V V f VLQFH V V YDOXHRI V FDQFHOVRXW
RISUREOHPVRDQ\FRQYHQLHQWYDOXHPD\EHXVHG 3URFHGXUH,XVHGWRVROYHSUREOHPZDVDV
IROORZV)URP35ZHNQRZV i Ÿ Ni DQG S i JLYHQLQLWLDOFRQGLWLRQV7KHQ
c
h
*XHVVYDOXHRI T ILQG P
8VH T f P f DQG V f WRJHW Nf WKHQ N f
)URP P f DQG V f ILQG WULDODQGHUURUZLWK35 Tf f
P WKDWVDWLVILHV S f
f
S i Ni Nf VR V f LVNQRZQ
6HH LI HTQ HQHUJ\ EDODQFH LV VDWLVILHG LI QRW JR EDFN WR VWHS $IWHU D QXPEHU RI
. Nf Ni LWHUDWLRQV , ILQG Pf EDU
T f . Tf N f Ni
f
Solutions to Chemical and Engineering Thermodynamics, 5th ed
6XPPDU\
Pf T f LGHDOJDV
SDUWD EDU
.
&RUUHVSRQGLQJ
VWDWHV SDUWE .
35(26
SDUWF .
.
.
.
Tf Nf Ni N f Ni &OHDUO\WKHLGHDOJDVDVVXPSWLRQLVVHULRXVO\LQHUURU
6\VWHP FRQWHQWVRIFRPSUHVVRU7KLVLVDVWHDG\VWDWHRSHQFRQVWDQWYROXPHV\VWHP
dN
PDVVEDODQFH
N N dt
dV dU
N H N H Q Ws P
HQHUJ\EDODQFH
dt
dt
Ÿ N H H Q Ws Q
Q
dS
N S N S SJHQ N S S HQWURS\EDODQFH
T
T
dt
7KXV
Q TN S S Q
Q T S S N
a
f
a
a
f
DQG
WS Q
N D &RUUHVSRQGLQJVWDWHVVROXWLRQ
Q
W Q
H H a f TndS S i dS S i dS S is
PU
R
T Sd S S i dS S i R OQ V
PW
T
T S S
f
a
,*
,*
,*
,*
,*
a Pr Pr Q
RT OQ
,*
T S S ,*
Pr S S Pr Tr Tr RSd
T
i
d
u u OQ ,*
1RZ Tr
f
i
7KXV
UV
W
- PRO
DQG
W Q
H H
H H ,* H ,* H ,*
VLQFH
T FRQVWDQW
H H ,* Solutions to Chemical and Engineering Thermodynamics, 5th ed
7KXV
W Q
TC
R|d H H i dH H i U| u S| T
V|
T
T
W
,*
C
W
C
- PRO
- PRO
Q E &ODXVLXVJDV
,*
P V b
FG IJ
H K
wV
RT
b wT
P
RT V
P
R
P
7KXV
z FGH ww SP IJK dP z FGH ww VT IJK dP R z P dP R OQ PP
- PRO
Q T'S RT OQF I H K
F w H IJ dP z LMV T FG w V IJ OPdP z L RT b RT OdP 'H z G
H w PK
P PQ
N H w T K Q MN P
z bdP ba P P f - PRO
P
P
'S
T
P
P
P
P
T
P
P
P
P
P
P
P
P
P
6R W Q - PRO DQG
W
Q - PRO F 3HQJ5RELQVRQHTXDWLRQRIVWDWH
8VLQJWKHSURJUDP35ZHILQG IRU T
V
q&EDU
u H
S
P
DQG P
P PRO -PRO
-PRO.
EDULGHDOJDVUHIHUHQFHVWDWH WKDW
q&EDU
u ±
1RWH IURP 35 WKH YDSRU SUHVVXUH RI 1+ DW q& LV EDU 7KHUHIRUH XVH YDSRU
VROXWLRQWR35HTXDWLRQ
7KHQ Q T S S - PRO DQG W
a
f
W Q
H H
Q - PRO - PRO Solutions to Chemical and Engineering Thermodynamics, 5th ed
DOVRDYDLODEOHDVD0DWKFDG &RQVLGHULQJ WKH JDV WKDW LV LQ WKH WDQN ILQDOO\ DV WKH V\VWHPWKLV LV D FORVHG V\VWHP XQGHUJRLQJ D
UHYHUVLEOHDGLDEDWLFH[SDQVLRQ7KHUHIRUH S i S f FG IJ dP EXW ZLWK P V b RT RU V RT b 7KHQ FG w V IJ R DOVR
H K
Hw TK P
P
FG w C IJ T FG w V IJ T w R Ÿ C LV LQGHSHQGHQW RI SUHVVXUH 7KHUHIRUH
H w P K HwT K
wT P
wV
C3
dT T
wT
D dS
3
3
T
C3
P
P
P
P
C3 7KXV
z
z
Tf
Pf
C3
dT R
dP T
P
T
P
'S
i
i
1RZWRILQGWKHLQLWLDODQGILQDOPRODUYROXPHVZHXVH
EDU
7KLVKDVWKHVROXWLRQ Pf
V
RT
b P
Vi
P PRO
Vf
P PRO
6RWKDW
Nf
Vi
Vf
Ni
RU E &RUUHVSRQGLQJVWDWHV
Tr
Pr
,*
f
,*
i
,*
i
i
Ÿ
S ,* S
Z
dS S i OQ P ,*
f
f
,*
S # DQG Z f
EDUIRUZKLFK
Pr
S
Nf
c P Z RT h P ˜ Z T u
a P Z RT f Z T P u RU Ni
,*
i
$VJLYHQE\HTQ DERYH:K\" *XHVVIRUILQDOVWDWH XVHPIURPSDUWD WKHQLWHUDWH)LQDOVROXWLRQLV Pf
,*
f
f
,QLWLDOVWDWH
dS S i dS S i d S S i S f Si
f
f
i
i
f
i
f
f
i i
f
i
F 3HQJ5RELQVRQHTXDWLRQRIVWDWH
8VHSURJUDP35ZLWKJLYHQKHDWFDSDFLW\FRQVWDQWVWRILQGDSUHVVXUHDW.ZKLFKKDVWKH
VDPH HQWURS\ DV WKH VWDWH T . P EDU %\ WULDODQGHUURU ZH ILQG WKDW
P EDU VRPHZKDW KLJKHU WKDQ WKH SUHYLRXV FDVHV $OVR V i u P PRO DQG
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Nf
u P PRO Ÿ
Vf
Ni
Vi
Vf
RU 7KHUHDUHWZRREYLRXVZD\VWRSURFHHG
UHWDLQ T DQG P DV WKH LQGHSHQGHQW YDULDEOHV VLQFH ZH KDYH D SURJUDP 35 WKDW FDOFXODWHV
V T P H T P DQG S T P :HFDQWKHQXVH
U T P H T P PV H T P ZRT ZKHUH Z Z T P G T P H T P TS T P DQG
A T P G PV U TS H PV TS H ZRT TS 1RZZHZLOOZULWH
T da dT a
X
H T P H ,* T RT Z b
DQG
T da dT
S T P S ,* T P R OQ Z B X
b
ZKHUHIRUFRQYHQLHQFH,KDYHXVHG
Z B
X OQ
Z B
a
f
a
7KHQZHILQG
U ,* T U T P
i OP
i PQ
LM d
MN d
G T P
a
f
T da dT a
X ZKHUH U ,*
b
G ,* T P RT Z OQ Z B dU
LM F w P I POPdV DQG d S C dT F w P I dV GH w T JK
T
MN GH w T JK PQ
9
C9dT T
V
S S ,*
z LMMNFGH IJK
T V
DQG
U U ,*
z LMMN FGH IJK
T V
T
T V f
OP
PQ
wP
R
dV wT V V
T V f
a
X
b
A,* T P RT OQ Z B WRJHW
a
X
b
7KXVZHFDQHLWKHUXVHHTQV WR DQGSUHYLRXVO\FDOFXODWHGYDOXHVZ H DQG S RUPRGLI\
35WRXVH(TQV LQVWHDGRI(TQV DQG 7KHVHFRQGDOWHUQDWLYHLVWRWDNHTDQG V DVWKHLQGHSHQGHQWYDULDEOHVDQGVWDUWIURP
A T P
V
H ,* RT DQG
f
OP
PQ
wP
P dV wT V
Solutions to Chemical and Engineering Thermodynamics, 5th ed
7KHQSXWLQWKH3HQJ5RELQVRQHTXDWLRQRIVWDWHDQGIURPEXLOGXSDSURFHGXUHWRFDOFXODWH S H U P A DQG G ZLWK T DQG V DV WKH LQGHSHQGHQW YDULDEOHV :H ZLOO QRW IROORZ WKLV
DOWHUQDWLYHIXUWKHU
:HZLOOGRWKHVHFDOFXODWLRQVXVLQJ
G H T S DQG A G PV H PV TS $V DQ H[DPSOH FRQVLGHU WKH T q & LVRWKHUP P
u A
EDU u P PRO u -
- PRO H - PRO
G - PRO
S - PRO. Ÿ
A - PRO
V P PRO
P EDU H - PRO
- PRO
G - PRO. Ÿ
S - PRO
A
V P PRO
P EDU H - PRO
- PRO
G S - PRO. Ÿ
- PRO
A V P PRO
- PRO
P EDU H G - PRO
Ÿ
S - PRO.
- PRO
A V P PRO
P EDU H - PRO
G - PRO
Ÿ
S - PRO.
A - PRO
V P PRO
- PRO
P EDU H G - PRO
- PRO.
Ÿ
S - PRO
A V P PRO
- PRO
P EDU H G - PRO
- PRO.
S Ÿ
A - PRO
V P PRO
P
- PRO G
EDU
EDU H
- PRO
S
- PRO.
V
P NPRO
Solutions to Chemical and Engineering Thermodynamics, 5th ed
6LPLODUO\ G DQG A DWRWKHUSRLQWVFRXOGEHFRPSXWHGWKRXJKWKLVZLOOQRWEHGRQHKHUH
D
U
V
wU
w
FG IJ V w V Ÿ FG w P IJ
HV K
Hw UK
V S
FG w P IJ
H wV K
S
a
f
V w S wV P
wS wPV
a
f
E\HTQ D 1RZ
FG IJ
FG IJ C FG w T IJ
H K
H K T H w PK
C
F w V IJ dP Ÿ FG w S IJ C FG w T IJ
dS
dT G
HwV K T HwV K
Hw TK
T
F w P IJ V C FG w T IJ ˜ T FG w P IJ V J FG w T IJ FG w P IJ
ŸG
HwV K H w T K
H w UK
T HwV K C H w T K
dS
wP
wS
C9
dT dV Ÿ
wT V
wP V
T
9
V
3
3
P
P
P
3
S
P
9
V
P
J V V
FG w P IJ
H wV K
E\HTQ D
FG w P IJ
H wV K
JV T
C3
IRUWKHLGHDOJDV C3
C9
)RUWKH&ODXVLXV*DV
J
J V E J
7KXV vS
C3
C9
ZLWK P V b
a
FG w T IJ FG w P IJ HwV K H w T K
P
V
C9 R
C9
C9 R Ÿ J
fa
f
C9 T w V w T P w P w T V
C9
RT R
C9
FG IJ FG w P IJ H K HwTK
T wV
C9 w T
P
V
T
Solutions to Chemical and Engineering Thermodynamics, 5th ed
FG w V IJ
Hw TK
P
J
FG IJ
H K
V b
wP
DQG
T
wT V
R
V b
7KXV
R
P
T V b R
˜
˜
C9 T V b
R
C9
7RVKRZWKDW Cv z Cv V ZHVWDUWIURPHTQ FG w P IJ EXW FG w P IJ R FG w P IJ IRULGHDOJDV
HwT K HwTK V HwT K
F w P IJ R FG w P IJ IRU&ODXVLXVJDV@
DQG G
H w T K V b H w T K
F w C IJ IRUWKHLGHDODQG&ODXVLXV*DVHV
ŸG
H wV K
FG w C IJ
H wV K
9
T
T
V
V
V
V
V
9
T
F vS LGHDOJDV
vS &ODXVLXVJDV
JV T P
V T
JPV
JRT T
R
V b V b
JV V
JRT
V b
V
vS LGHDOJDV
V b
DWVDPHTDQGV
3UHOLPLQDULHV
3UHVVXUH RXWZDUGIRUFHSHUXQLWDUHDH[HUWHGE\JDV
)RUFH WHQVLOHIRUFHH[HUWHGRQILEHU²DWPHFKDQLFDOHTXLOLEULXPILEHUH[HUWVDQHTXDODQGRSSRVLWH
LQZDUGIRUFH
Ÿ,QDOOWKHUPRG\QDPLFUHODWLRQVUHSODFHPE\ F A DQGVE\LADQGWKH\ZLOOEHDSSOLFDEOHWR
ILEHU
wS
wS
dV :HZLOOXVH S S T L dT ,QSDUWLFXODULQSODFHRI S S T V DQG dS
wT V
wV T
DQG dS
FG w S IJ dT FG w S IJ dL $OVR
H w T K H w LK
L
FG IJ
H K
T
FG w S IJ C Ÿ FG w S IJ C
Hw TK T Hw TK T
FG w S IJ FG w P IJ Ÿ FG w S IJ FG w F IJ H w T K H w T K H w LK H w T K
9
V
T
dS
FG IJ
H K
/
L
V
P
dU dV Ÿ dS
T
T
T
dU
T
L
F
dL
T
a f
Solutions to Chemical and Engineering Thermodynamics, 5th ed
FG w S IJ dT FG w S IJ dL C dT FG w S IJ dL DQG WKH DQDORJ RI WKH
H w T K H w LK
H w LK
T
/
D )URP WKH DERYH dS
L
T
V
a
f
a
/
L
a
f
f
a
z RST
a
f
Cv dT f UVW
a
f a f
&KRRVLQJWKHSDWK L T o L T o L T \LHOGV
T L
D ET
T L
T
z
f
dT J
za
TL
T L
f
L L dL
T
J
D OQ E T T L L T
a
f a
f
h a f mS c L T h S a L T fr kS a L T f S a L T fp
J
FT I
D OQG J E cT T h c L L h a L L f
HTK
c
G $UHYHUVLEOH VORZ DGLDEDWLFH[SDQVLRQ Ÿ S L f Tf S Li Ti
f
f
i
i
i
f
f
f
i
i
L
E dU TdS PdV Ÿ dU TdS FdL C/dT JT L L dL JT L L dL C/dT 1RWH7KLVLVDQDORJRILGHDOJDVH[SUHVVLRQ U U T RU dU
C/
F dS
dT J L L dL T
L T
C/
Ÿ S L T S L T
dT J L L dL T
L T
a
T
FG IJ H K
wF
A wT
FG IJ dL C dT J a L L fdL H K
T
wF
C/
dT wT
T
dS
S L T S L T
ZHJHW
T
FG w S IJ FG w P IJ Ÿ FG w S IJ
H w V K H w T K A H w LK
T
0D[ZHOOUHODWLRQ
1HHGWRVROYHWKLVWUDQVFHQGHQWDOHTXDWLRQWRILQG Tf H dU
FS
D
CLdT Ÿ
T
FG w S IJ
H w LK
FG w U IJ
H w LK
Ÿ FU
T
k a
T J L L
T
dU
TdS PdV GdN Ÿ
fp JTa L L f T FGH ww FT IJK L
FG w U IJ
HwSK
T
V N
Solutions to Chemical and Engineering Thermodynamics, 5th ed
FG w U IJ P DQG
H wV K
FG w U IJ G Hw NK
SN
S V
1RZHTXDWLQJPL[HGVHFRQGGHULYDWLYHV
V N
FG IJ
H K
VN
FG IJ
H K
F w GI
H wV K
S N
V N
FG IJ
H K
w
wU
wP
Ÿ
w V S N w N S V
w N S V
DOVRDYDLODEOHDVD0DWKFDGZRUNVKHHW D 7KHSURFHGXUHWKDWZLOOEHXVHGLVWRILUVWLGHQWLI\WKHWHPSHUDWXUHDWZKLFK P DQGWKHQ
VKRZWKDW
P DWODUJHUWHPSHUDWXUHVDQG P ! DWORZHUWHPSHUDWXUHV7KHVWDUWLQJSRLQWLVIURP6HF
P
D FG w T IJ
H w PK
H
V
TD C3
ZKHUHIURP,OOXVWUDWLRQIRUWKHYDQGHU:DDOVJDV
TV
a
V b DQG P
V b RV R|
S| a
T
V
C3
V V b a V b RTV f
a
a
f
f
V b V b a V b RTV
C3 V V b a V b RTV V RTb V b a V b V
C3 RTV V b a V b V U| V|
W
6LPSOLI\LQJ\LHOGV
S V
E F DQG G DUHGHULYHGLQVLPLODUIDVKLRQ
P
S V
FG w P IJ
HwSK
F w G IJ
G
HwSK
w
w S V N
w
wU
w N S V w V S N
FG IJ
H K
F w T IJ
ŸG
Hw NK
w
w N S V
FG IJ
H K
FG w U IJ
Hw NK
w
wU
wT
Ÿ
w S V N w V S N
w V SN
DQG
FG IJ
H K
FG w U IJ
HwSK
w
wU
w V S N w S V N
1RZIRUPWREH]HURHLWKHUWKHQXPHUDWRUPXVWEH]HURRUWKHGHQRPLQDWRULQILQLW\2QO\WKH
IRUPHULVSRVVLEOH7KXV
T LQY
a V b V b
˜
b
RV a V b WKHGHVLUHGH[SUHVVLRQ
RV b
Solutions to Chemical and Engineering Thermodynamics, 5th ed
WR GHWHUPLQH WKH VLJQ RI WKH -RXOH7KRPVRQ FRHIILFLHQW LQ WKH YLFLQW\ RI WKH LQYHUVLRQ
WHPSHUDWXUH ZH ZLOO UHSODFH T LQ HTQ E\ T LQY G ZKHUH G PD\ EH HLWKHU SRVLWLYH RU
QHJDWLYH7KHUHVXOWLV
P
V
Rb V b G
C3 RV V b G a V b bV ,W LV HDVLO\ VKRZQ WKDW WKH GHQRPLQDWRU LV DOZD\V SRVLWLYH 7KXV
PLVSURSRUWLRQDOWR G Ÿ LIT ! T LQY VR WKDW G ! P $OWHUQDWLYHO\ LI T T LQY G DQG P ! E 8VLQJ a
V RTC DQG b
C
T LQY
F I aV f
H K V
i
aV f
F V I aV f
T
˜
H K V V
V
f
C
r
r
V
TC C
r
r
C
r
LQY
r
TrLQY Vr ZKDWZHZDQWLV TrLQY DVDIXQFWLRQRI Pr 7KXVORRNDW
FG P IJ aV f T Ÿ P T H VK
V V
r
Vr r
a f
F ([SUHVVLRQDERYHJLYHV TrLQY
a
d
V V C ˜ V C RTC
RV V C T LQY
Ÿ
TC
VC
r
r
r
r
r
r
&KRRVH Vr DV LQGHSHQGHQW YDULDEOH XVH (TQ WR JHW TrLQY DQG XVH (TQ WR JHW Pr 5HVXOWVDUHWDEXODWHGDQGSORWWHGEHORZ
TrLQY T LQY . Pr P EDU T LQY q & ±
±
±
Solutions to Chemical and Engineering Thermodynamics, 5th ed
DOVRDYDLODEOHDVD0DWKFDGZRUNVKHHW Solutions to Chemical and Engineering Thermodynamics, 5th ed
5 7 % &
3 9 7
G
7 3 9 7 3 9 7 G7
&RQVHTXHQWO\WKHLQWHUQDOHQHUJ\RIWKLV
JDVLVDOVRQRWDIXQFWLRQRISUHVVXUHRU
YROXPHRQO\WHPSHUDWXUH+RZHYHULIWKH
YLULDOFRHIILFLHQWZHUHDIXQFWLRQRI
WHPSHUDWXUH ZKLFKLVWKHDFWXDOFDVH WKHQWKHLQWHUQDOHQHUJ\RIWKLVJDVZRXOG
EHDIXQFWLRQRIWHPSHUDWXUH
*XHVVHVIRU,QLWLDODQGILQDOYROXPHV
9L
9 9 9
57L
*LYHQ
3D 3 9L 7L
*LYHQ
3D 3 9I 7L 9I ILQG 9I
9I
3 9 7L G 9
:
9L
9I
3D
5 7L
3D
9L P PROH
9L ILQG 9L
9I P PROH
: PROH MRXOH
8 ILQDO 8 LQLWLDO VR4 :
4 PROH MRXOH
4 :
F 7KHYDQGHU:DDOVJDV
D 3DP
P
E PROH
PROH
57 D
3 9 7
9E 9
P
G
7 3 9 7 3 9 7 3D
G7
PROH 9
*XHVVHVIRU,QLWLDODQGILQDOYROXPHV
D9A ,QWKLVFDVHWKHLQWHUQDOHQHUJ\LVD
IXQFWLRQRIYROXPH RUSUHVVXUH
9L
57L
*LYHQ
3D 3 9L 7L
*LYHQ
3D 3 9I 7L 9I ILQG 9I
9I
:
'8
9L
9I D
'8 PROH MRXOH
G9
4 '8 :
3D
: PROH MRXOH
9
5 7L
9L P PROH
9L ILQG 9L
3 9 7L G 9
9L
9I
9I
3D
4 PROH MRXOH
P PROH
Solutions to Chemical and Engineering Thermodynamics, 5th ed
G 7KH3HQJ5RELQVRQIOXLG
.
Z E 5 E P PROH
3D
5 .
DF DOI
Z Z Z 3D
DOI D DF DOI
D P PROH
3D
D
5 7
3 9 7
9 E 9 9 E
E 9 E
P
G 3 9 7
3D 7
3
9
7
IORDW
G7
P
P
9 P
PROH 9 9 PROH
PROH
PROH
:KLFKVKRZVWKDWWKHLQWHUQDOHQHUJ\LVD
IXQFWLRQRIYROXPH RUSUHVVXUH
5 7L
5 7L
*XHVVHVIRU,QLWLDODQGILQDOYROXPHV
9L
9I
3D
3D
*LYHQ
3D 3 9L 7L
9L ILQG 9L
9L P PROH
*LYHQ
3D 3 9I 7L
9I ILQG 9I
9I P PROH
9I
:
3 9 7L G 9
: PROH
MRXOH
9L
7 .
9I
G
7
3 9 7
'8
3 9 7 G 9
'8
PROH
MRXOH
G7
9L
4 '8 :
4
PROH
MRXOH
)RUDQLVRWKHUPDOSURFHVVLQYROYLQJDIOXLGGHVFULEHGE\WKH5HGOLFK.ZRQJHTXDWLRQRI
VWDWHGHYHORSH[SUHVVLRQVIRUWKHFKDQJHVLQ
Solutions to Chemical and Engineering Thermodynamics, 5th ed
D LQWHUQDOHQHUJ\
E HQWKDOS\DQG
F HQWURS\
LQWHUPVRIWKHLQLWLDOWHPSHUDWXUHDQGWKHLQLWLDODQGILQDOYROXPHV
)RU\RXULQIRUPDWLRQWKH5HGOLFK.ZRQJHTXDWLRQRIVWDWHLV
RT
a
V b
T ˜V ˜ V b
P
DQG
z
dU
dx
x xc
F
H
I
K
x
OQ
c
xc
LMT F w P I POPdV
MN GH w T JK PQ
LMT ˜ R a fat RT a OPdV
N V b T V V b V b T V V b Q
V
adV
TV V b
a
f a f aT z V VdV b
V
U T V U T V V
a
LM
N
V V b
a
OQ
Tb V b V OP
Q
f a f U aT V f U aT V f P V PV
LV aV bf OP RT LM V V OP
a
OQM ˜
Tb N V aV bf Q
NV b V b Q
a L O
M
V
b
V
b PQ
T N
H T V H T V FG w P IJ dV LM R a OPdV
HwTK
NV b T V V b Q
V b
LV aV bf OP
a
OQM ˜
S aT V f S aT V f R OQ
V b T b N V aV bf Q
dS
V
Solutions to Chemical and Engineering Thermodynamics, 5th ed
a
f a f HaT V f TSaT V f HaT V f T SaT V f
LV aV bf OP RT LM V V OP
a
OQM ˜
Tb N V aV bf Q
NV b V b Q
O
a L M
P
T NV b V b Q
V b
LV aV bf OP
a
OQ M ˜
RT OQ
V b Tb N V aV bf Q
LV aV bf OP RT LM V V OP
a
OQ M ˜
Tb N V aV bf Q
NV b V b Q
a L M OP RT OQFGH VV bb IJK
T NV b V b Q
G T V G T V X
-RXOH7KRPVRQ([SDQVLRQ
P EDU T q & P EDU T " D ,GHDOJDVHQWKDOS\LVLQGHSHQGHQWRISUHVVXUH Ÿ T q & E YDQGHU:DDOVJDV
,*
H H Ÿ H H ,*
H ,*
H H ,* H
c
h c
h c
h
a fL
Ÿ RT a Z f MMNT FGH ww TP IJK POPPQdV C dT RT aZ f
a fL
MMNT FGH ww TP IJK POPPQdV RT
a F w PI
R
F w P JI P RT RT a
P
G
T G
J
V b V H w T K
V b H w T K
V b V b V
z
z
V T P
V f
z
T
V
p
T
V T P
V f
V
V
V
a
V
Solutions to Chemical and Engineering Thermodynamics, 5th ed
z
f z
a
f
RT Z a
RT Z a
PV RT a
a
z
z
z
V
V
T
a
f
V
T
a
f
dV
dV 9
C
dT
RT
Z
a
p
T
V fV
V fV
Ÿ
dV 9
C p dT RT Z T
V fV
FG IJ z ca bT cT dT hdT
HV V K
T
T
PV
RT
FG IJ aaT T f b cT T h
HV V K
c
d
cT T h cT T h PV RT
PV RT a
6ROYHGWRJHWKHUZLWKYG:(26
T . | q & TLQFUHDVHV" F 3HQJ5RELQVRQ(26
7KHUPRG\QDPLFSURSHUWLHVUHODWLYHWRDQLGHDOJDVDW.DQGEDU
H q & EDU u - PRO $IWHUVRPHWULDODQGHUURU
H q & EDU u - PRO
6RWKHVROXWLRQLVT R&
&ORVHHQRXJK
G 6WHDPWDEOHV
N- NJ
H q & EDU H T " EDU N- NJ
H T
H T
q & EDU
q & EDU
Ÿ T # q &
.
7KHVROXWLRQLVDYDLODEOHRQO\DVD0DWKFDGZRUNVKHHW
f T NV N Z N V T N
A N V T kT OQ Q N V T kT OQ
N
LM
N
NRT OQ f T NkT OQV NkT OQ Z
6WLUOLQJ¶VDSSUR[LPDWLRQ OQ N a
F N TI kT OQ N HV K
N OQ N N f OP
Q
Solutions to Chemical and Engineering Thermodynamics, 5th ed
F N TI NkT OQ N NkT
HV K
FG w A IJ P NkT NkT w OQ Za N V T f w a N V f
H wV K
waN V f
wV
V
NkT N kT w OQ Z a N V T f
P
waN V f
V
V
FG wA IJ S Nk OQ f T NkT d OQ f T Nk OQV k OQ Z F N TI
HV K
Hw TK
dT
w OQ Z a N V T f
NkT
Nk OQ N kT
A N V T
NkT OQ f T NkT OQV kT OQ Z
TN
T
T
T
V N
wT
S
F I
H K
F
H
d OQ f T
N
N
k OQ Z
Nk OQ
T
dT
V
V
w OQ Z N V T
kT
NkT
wT
N V
Nk OQ f T NkT
a
G PROHFXOH
N V
FG w A IJ
Hw NK
f
kT OQ f T kT OQ V kT
T V
a
f
I
K
a f a f
a f
w OQ Z N V T w N V
w NV
wN V
V
NkT OQ Z N V T kT OQ N kT kT
N ˜ G PROHFXOH
G
NkT OQ f T NkT OQ V NkT OQ N NkT OQ Z
a f
a f
NkT w OQ Z N V T
w NV
V
F N TI
HV K
$VDFKHFN
F N TI NkT OQ N
HV K
L NkT NkT w OQ Za N V T f OPV
NkT M
N V V waN V f Q
N
NkT OQ f T NkT OQV NkT OQ Z F T I NkT OQ N
HV K
A PV
G
NkT OQ f T NkT OQ V NkT OQ Z
T
NkT NkT a
f
a f
NkT w OQ z N V T
w NV
V
T
NkT OQ f T NkT OQ V NkT OQ Z
a f
a f
NkT w OQ Z N V T
V
w NV
F N TI NkT OQ N
HV K
Solutions to Chemical and Engineering Thermodynamics, 5th ed
ZKLFKFKHFNV
A
w
w T V N T
F I U
H K T
w
LM Nk OQ f T Nk OQV Nk OQ ZF N TI Nk OQ N Nk OP
HV K
wT
N
Q
LM Nk d OQ f T Nk F w OQ Za N V T fI OP
GH w T JK P
dT
NM
Q
V N
N V
FG
H
NkT C9
F dU I
H dT K
NkT
NV
NkT
fIJ
K
N V
d OQ f T
d OQ f T
NkT dT
dT FG w OQ Za N V T fIJ
H wT K
NkT N V
FG w OQ Za N V T fIJ
H wT K
H
U PV
NkT HWF
a
FG
H
d OQ f T
w OQ Z N V T
NkT dT
wT
a f
a f
fIJ
K
N V
NkT NkT w OQ Z N V T
V
V
w NV
T
d H UHV
a
d OQ f T
w OQ Z N V T
NkT dT
wT
U
d H d H ,*
LM
N
C3dT V T
FG w V IJ OPdP C dT
Hw TK Q
3
P
cC C hdT LMNV T FGH ww VT IJK OPQdP
cC C hdT LMNT FGH ww TP IJK POPQdV
3
3
P
dU UHV
9
V
dS
FG IJ
H K
wP
C9
dT dV Ÿ d S UHV
wT V
T
RUd S UHV
9
C9 C9
T
C9 C9
dT T
LMFG w V IJ V OPdP
NH w T K T Q
P
LMF w P I P OPdV
MNGH w T JK T PQ
V
N V
Solutions to Chemical and Engineering Thermodynamics, 5th ed
c
h d H Td S S dT
cC C hdT LMNV T FGH ww VT IJK OPQdP cC C hdT LMNT FGH ww VT IJK V OPQdP
dG UHV
d H UHV d TS UHV
3
UHV
UHV
UHV
S
3
3
P
P
S UHVdT
dG UHV
S UHVdT H T P H ,* T P
z LMMN FGH IJK
V
RT Z T
V f
P
OP
PQ
wP
P dV
wT V
RT
a
V b V
FG w P IJ R
H w T K V b
F w P IJ P RT RT a
TG
HwTK
V b V b V
V
a
V
V
H T P H ,* T P
RT Z z
V
a
dV
V fV
RT Z a
LM OP
NV f Q
RT a P
a
˜
RT Z P V RT
V
aP
RT Z ZRT
RT Z H T P H ,* T
aP
ZRT
H UHV
a f
d
RT Z U UHV
U T P U ,* T
H T P PV H ,* PV ,*
,*
PV PV
RT
RT
RT
aP
H UHV T P RT Z ZRT
H UHV T P RT
S UHV
S T P S ,* T P
OP
Q
z LMN
V
R
R
dV
V b V
V f
S UHV T P
R OQ
V
V b
R OQ
a f
H UHV T P RT Z
z LMMNFGH IJK
V
V f
R OQ
Z
B
ZB
i
OP
PQ
wP
R
dV
wT V V
V b
V
R OQ
V o f b
V of
Pb
RT
Solutions to Chemical and Engineering Thermodynamics, 5th ed
D 7KH6RDYH5HGOLFK.ZRQJHTXDWLRQRIVWDWHLV
aT
RT
V b V V b
a
P
f
5HZULWHWKLVLQWKHSRZHUVHULHVRIV
RT RT
aT
aTb
V V b b
V
P
P
P
P
1RWLFHWKDWWKHWKUHHURRWVRIYROXPHDWWKHFULWLFDOSRLQWDUHLGHQWLFDOVRZHFDQZULWH
FH
IK
bV V g
RU
C
V V C V V C V V C
$WFULWLFDOSRLQWWKHVHFRQGDQGIRXUWKHTXDWLRQVPXVWEHVDWLVILHGVLPXOWDQHRXVO\&RQVHTXHQWO\WKH
FRHIILFLHQWVRIHDFKSRZHURIVPXVWEHWKHVDPH7KXV
V C
RTC
PC
V C
b V C
a TC
b
PC
a f
a f
a TC
RTC
b
DQG
PC
PC
6ROYLQJWKHDERYHWKUHHHTXDWLRQVWRJHWKHUIRUa T b DQGVcZHJHW
E RTC
VC
PC
b
c hV
a f P bV
a TC
C
C
C
RTC
DQG
PC
a RT f
C
PC
$OVR
ZC
aT
a TC D T
a f
PCV C
RTC
PCV C RTC
RTC PC
a RT f D T
DOVRDYDLODEOHDVD0DWKFDGZRUNVKHHW RT
aT
P
V b V V b
C
PC
Solutions to Chemical and Engineering Thermodynamics, 5th ed
H T P H ,* T P
z LMMN FGH IJK
V
RT Z T
V f
OP
PQ
wP
P dV
wT V
FG w P IJ R da T
H w T K V b V V b dT
F w P IJ P RT T da T RT a
TG
HwTK
V b V V b dT
V b V V b
LMa T da T OP
V V b N
dT Q
V
V
6RWKHLQWHJUDOWREHGRQHLV
z
dV
V V b
V f
V
z
dV
V V b
V f
LM
N
LM
N
OP
Q
V
˜V b
OQ
V
b
V f
V
OP
Q
LM
N
OP
Q
V b
V b
OQ
OQ
b
V
b
V V f
6R
OP
Q
z
LMT FG w P IJ POPdV
MN H w T K PQ
a T ada dT f F V b I
OQG
RT Z H V JK
b
T ada dT f a L Z a Pb RT f O
OQ M
RT Z b
N Z PQ
T ada dT f a L Z B O
OQ M
RT Z b
N Z PQ
V
H T P H ,* T P
RT Z V f
LM
N
V b
OQ
b
V
S T P S ,* T P
z LMMNFGH IJK
V ZRT P
R OQ Z V f
FG w P IJ R R da R
H w T K V V b V V b dT V
V
V
OP
PQ
wP
R
dV
wT V V
Solutions to Chemical and Engineering Thermodynamics, 5th ed
z LMN
V ZRT P
OP
Q
R
da R
dV
V b V V b dT V
V f
V b
V
da V b
R OQ
OQ
V b V of
V V of dT b
V
R OQ
LM a f OP
N
Q
LM OP
N Q
V b da Z Pb RT
OQ
V
dT b
Z
ZB
da ZB
R OQ
OQ
Z
dT b
Z
R OQ
F
H
I
K
F Z B I da OQLM Z B OP
H Z K dT b N Z Q
da L Z B O
R OQ Z B OQ
dT b NM Z QP
S T P S ,* T P
G UHV
R OQ Z R OQ
H UHV TS UHV
RT Z 5HGOLFK.ZRQJ
P
aP
Z
RT OQ
zRT
ZB
RT
a
V b
TV V b
FG w P IJ R a fa
H w T K V b T V V b
F w P IJ P RT a fa RT a
TG
HwTK
V b
TV V b V b
TV V b
a
a fa
V
V
TV V b
TV V b
H T P H ,* T P
RT Z z
dV
a
b
V
V
T
V f
V
V
a
OQ
b T V b
Z
a
RT Z OQ
Z
bP
RT
b T
Z
a
OQ
RT Z b T Z B
RT Z a
f
Solutions to Chemical and Engineering Thermodynamics, 5th ed
U T P U ,* T P
a
Z
OQ
b T Z B
z LMMNFGH IJK OPPQ
z LMN a f
V
S T P S
,*
T P
V f
wP
R
dV
wT V V
V
OP
Q
a
R
R
dV
V b T V V b V
V f
a f
a
V
V
OQ
V b T b V b
Z
a
Z
OQ
R OQ
Z B T b Z B
R OQ
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Solutions to Chemical and Engineering Thermodynamics, 5th ed
&ULWLFDOSURSHUWLHVIRUFDUERQGLR[LGH
7F .
3F ˜ 3D
Z 6RDYH5HGOLFK.ZRQJ(TXDWLRQRI6WDWHDQG&RQVWDQWV
MRXOH
5 D 7 3D˜ P
RU
.˜ PRO
.˜ PRO
§ 5˜ 7F ·
¸
© 3F ¹
D ˜ ¨
ª ˜ Z ˜ Z ˜ § 7 · º
«
¨
¸»
7F ¹ ¼
¬
©
3 9 7 5˜ 7
9 E
E ˜
5˜ 7F
3F
D 7 D˜ D 7
D 7
9˜ 9 E
'DWDJLYHQLQWKHSUREOHP
7
.
9 5˜
7
3 ˜ 3D
3 ˜ 3D
9WRWDO P
,QLWLDOJXHVVQHHGHGIRUVROYHU
3
6ROYLQJIRUWKHLQLWLDOPRODUYROXPHDQGWKHQXPEHURIPROHVRIFDUERQGLR[LGH
3
*LYHQ
3 9 7
9PRODU )LQG 9
9WRWDO
1
1
9PRODU
u 9PRODU u 6ROYLQJIRUWKHILQDOPRODUYROXPHDQGWKHILQDOWRWDOYROXPH
5˜ 7
9
3
*LYHQ
3
3 9 7
9PRODU )LQG 9
9PRODU u L 9WRWDO
9WRWDO 9PRODU˜ 1
&DOFXODWLQJWKHDPRXQWRIZRUNGRQHWRFRPSUHVVWKHJDV
´
:RUN 1˜ µ
¶
9PRODU
3 9 7 G9
LL :RUN
u MRXOH
9PRODU
6LQFHWKHWHPSHUDWXUHLVFRQVWDQWWKHFKDQJHLQHQWKDOS\+ 73 + 73 LVMXVWHTXDO
WR+GHS 73 +GHS 73 § 3˜ 9
+GHS 7 3 5˜ 7˜ ¨
© 5˜ 7
¸· D 7 7˜
¹
E
G
D 7
G7
˜ OQ§¨
9
·
¸
© 9 E¹
+ 7 3
4 1˜ > + 7 3 + 7 3
3˜ 9PRODU 3˜ 9PRODU @ :RUN
4
u MRXOH
u + 7 3
+ 7 3 +GHS 7 3
u Solutions to Chemical and Engineering Thermodynamics, 5th ed
Solutions to Chemical and Engineering Thermodynamics, 5th ed
PRO EDU 3D
˜
3URSHUW\'DWD
7LQ.3LQEDU 5( ˜
7F ˜ . 3F ˜ EDU
5* ˜
PRO˜ .
EDU˜ P
PRO˜ .
RP NDS ˜ RP ˜ RP˜ RP
,QLWLDO&RQGLWLRQV
7L ˜ .
65.&RQVWDQWV E ˜
,QLWLDOWHPSHUDWXUH
MRXOH
3L ˜ EDU
5*˜ 7F
7 7L
DF ˜
3F
1RWHWKDWWKHVHDUHEHLQJGHILQHGDVD
ª
§
IXQFWLRQRIWHPSHUDWXUHVLQFHZHZLOOQHHGWR
DOI 7 ˜ « NDS˜ ¨ LQWHUDWHRQWHPSHUDWXUHODWHUWRREWDLQWKHILQDO
¬
©
VWDWHRIWKHV\VWHP
MRXOH
MRXOH
+HDWFDSDFLW\ &S ˜
&S ˜ ˜
PRO˜ .
PRO˜ .
FRQVWDQWV
MRXOH
&S ˜ ˜
*LYHQ 3L
9
5*˜ 7
9 E
5*˜ 7L
9
3L
D7
'(/6L
'(/6L . MRXOH
=L ¸»
D 7 DF ˜ DOI 7
7F ¹ ¼
'D 7 G
D 7
G7
MRXOH
˜
PRO˜ .
u P
9L )LQG 9
9˜ 9 E
9L
(QWURS\GHSDUWXUHDWWKH
LQLWLDOFRQGLWLRQV
3F
&S ˜ PRO˜ .
)LQGLQLWLDOPRODUYROXPHDQGQXPEHURIPROHV
6WDUWZLWKLQLWLDOJXHVVIRUYROXPHPAPRO
6ROYH65.(26IRULQLWLDOYROXPH
7 ·º
5* ˜ 7F
u P
ªOQª 9L E ˜ 3L º 'D 7 ˜ OQ§ 9L E ·º ˜ 5(
« «
»
¨
¸»
E ˜ 5* © 9L ¹¼
5*˜ 7¼
¬ ¬
3L˜ 9L
=L
5*˜ 7
E ˜ 3L ·
§ 7˜ G D 7 D 7 · §
¨
¸ ¨ =L 5*˜ 7 ¸
© G7
¹ ˜ OQ
'HO+L 5(˜ 7˜ =L ¨
¸
=L
E
©
¹
'HO+L
u MRXOH
)LQDOWHPSHUDWXUHLV.DQGILQDOSUHVVXUHLVXQNQRZQZLOOEHIRXQGE\HTXDWLQJWKHLQLWLDODQGILQDO
HQWURSLHV*XHVVILQDOWHPSHUDWXUHLVEDU
3I ˜ EDU
7 ˜ .
9
5*˜ 7
3I
9
u P
Solutions to Chemical and Engineering Thermodynamics, 5th ed
7HPSHUDWXUHSDUWRILGHDOJDVHQWURS\FKDQJH
WL '6LGHDO7 &S˜ OQ§¨
WI ·
WI &S
&S
¸ &S˜ WI WL ˜ WI WL ˜ WI WL
© WL ¹
'6LGHDO7
. MRXOH
1RWH7RXVHWKHJLYHQDQGILQGFRPPDQGVIRUYDULDEOHZLWKGLIIHUHQWGLPHQVLRQVVXFKDV3DQG9ZLOOKDYH
WRFRQYHUWWRGLPHQVLRQOHVVYDULDEOHVVRDVQRWWRKDYHDXQLWVFRQIOLFW'HILQH[ 9EDQG\ 33F
LQLWLDOJXHVV
*LYHQ
9
[
\ ˜ 3F
5*˜ ˜ .
[˜ E E
\ D ˜ .
[˜ E ˜ [˜ E E
\ ˜ 3F º 'D ˜ .
[ ·º
ª«OQª« [˜ E E ˜
˜ OQ§¨
¸
»
¸» ˜ 5( '(/6L
5*˜ ˜ .¼
E ˜ 5*
© 3L ¹ ¬ ¬
© [ ¹¼
'6LGHDO7 5(˜ OQ§¨
\ ˜ 3F ·
< )LQG [ \
<
§ ·
¨
¸
© ¹
9I < ˜ E
9I
u 3I < ˜ 3F
3I
u 3D
3I
[
E
EDU
P
)UDFWLRQPDVVUHPDLQLQJLQWDQN 9L
9I
Solutions to Chemical and Engineering Thermodynamics, 5th ed
DOVRDYDLODEOHDVD0DWKFDGZRUNVKHHW,QIDFWWKDWILOHFRQWDLQJUDSKVDQGRWKHULQIRUPDWLRQ w C3
w V
w C9
wP
WKDQZLWK
T
(DVLHUWRZRUNZLWK
T
wP T
wV T
w T P
w T V
FG
H
IJ
K
FG
H
IJ
K
f T
C9 V T C9 T
FG
H
C9 V T C9 V
6R
C9 V T
C9 T IJ
K
T
V f
a
V b
OQ
V V f
T
FG
H
z FGH ww TP IJK dV
a
F V b IJ
OQG
HV K
T
V
V
IJ
K
FG V b IJ
T b H V K
a
OQ
&OHDUO\ DV V o f LGHDO JDV ZH
UHFRYHU C9
3URFHGXUH
RI V ¶V
C9 &KRRVH FROOHFWLRQ
&DOFXODWH C9 C9 IRUJLYHQ V DQGT
&DOFXODWHPIURP5.(26JHW C9 C9 YVP
1H[WXVH
C3
C9 T
FG w V IJ FG w P IJ
H w PK H w T K
T
C9 T
V
aw P w T f aw P w V f
V
T
WRFRQYHUWIURP C9 WR C3 +DYHGRQHERWKSDUWVXVLQJ0$7+&$'6HHWKH0$7+&$')ILOH
Solutions to Chemical and Engineering Thermodynamics, 5th ed
DOVR DYDLODEOH DV D 0DWKFDG
ZRUNVKHHW RT
a
D
P
V b
TV V b
PV
B C
" Z RT
V V
PV
V
V
a
V
a
Z
RT V b RT TV V b V b RT V b
aV
b V RT b V
a V
b V RT b V
1RZH[SDQGLQJLQDSRZHUVHULHVLQ V
a
a V
Z b V b RT
V
RT
a
a
B b
B b
RT
RTB F
H
F I
H K
I
K
a a
Ÿ TB
. bR
bR
E 8VLQJWKH5HGOLFK.ZRQJSDUDPHWHUV
a TB
. bR
TB F I
H K
Solutions to Chemical and Engineering Thermodynamics, 5th ed
DOVR DYDLODEOH DV D 0DWKFDG
ZRUNVKHHW V
a
Z
V b RT V b
wZ
wV
V
wV
a
wV
w P T V b w P T V b w P T RT V b w P T
FG IJ
H K
P
FG IJ
H K
RT
a
V b
T V bV
FG w P IJ
H wV K
c
RT
a
V b
V b
T V bV
OLP
FG w P IJ
H wV K
d
T
Po
V of
OLP
FG w P IJ
H wV K
FG IJ
H K
h
P of
V of
FG IJ
H K
i
T
f
T
FG w Z IJ LM V a OPFG w V IJ
H w PK
NV b V b RT V b QH w P K
FG w Z IJ V b mV V b r ma RT V b r
H w P K m RT V b r {a T cV bV h } V b
F w Z IJ V b V V b b
OLP G
H w P K RT V b
RT
OLP
Po
V of
T
P of
V ob
T
FG w Z IJ V V a c RT V h
H w P K RT V a d TV i ˜ V
a d RT V i
aV
a
OLP
Po
V of
T
RT V T
RT V RT T
Solutions to Chemical and Engineering Thermodynamics, 5th ed
D 7KH5HGOLFK.ZRQJHTXDWLRQRIVWDWHLV
RT
a
V b
TV V b
P
ZKLFKZHUHZULWHDVIROORZV
PV
V
a
VRWKDW
RT V b RT V b
PV
V
a
b
a
DQG
RT
V b
RT V b V b RT V b
V
a
V
PV
V
b
V b RT V b
RT
F
H
I
K
PV I
/LP V F
H RT K /LP
Po
V
V of
F PV I /LP LMb V a V OP b a
H RT K
N V b RT V b Q RT
V of
BT
7RSURFHHGIXUWKHUZHQRZQHHGWRKDYHH[SUHVVLRQVIRUDDQGELQWHUPVRIWKHFULWLFDOSURSHUWLHV7R
REWDLQWKHVHZHSURFHHGDVLQ3UREOHPDQGUHZULWHWKLVLQWKHSRZHUVHULHVRIV
RT RT
ab
a
V V b V
b
P
P
TP
TP
1RWLFHWKDWWKHWKUHHURRWVRIYROXPHDWWKHFULWLFDOSRLQWDUHLGHQWLFDOVRZHFDQZULWH
F
H
I
K
bV V g
C
RU
V V C V V C V V C
$WFULWLFDOSRLQWWKHVHFRQGDQGIRXUWKHTXDWLRQVPXVWEHVDWLVILHGVLPXOWDQHRXVO\&RQVHTXHQWO\WKH
FRHIILFLHQWVRIHDFKSRZHURIVPXVWEHWKHVDPH7KXV
V C
V C
V C
RTC
PC
RT
a
b C b DQG
PC
TC PC
a
b
TC PC
6ROYLQJWKHDERYHWKUHHHTXDWLRQVWRJHWKHUIRUa T b DQGVcZHJHW
VC
RTC
PC
b
d iV
a
C
R TC
PC
RTC
DQG
PC
Solutions to Chemical and Engineering Thermodynamics, 5th ed
a
RTC
RTC
RT PC
PC T )RUQSHQWDQHTC .DQGPC EDU
7KHUHVXOWLQJYLULDOFRHIILFLHQWDVDIXQFWLRQRIWHPSHUDWXUHLVVKRZQEHORZ
6R B T
b
% 7
7
Solutions to Chemical and Engineering Thermodynamics, 5th ed
FG w T IJ V TD Ÿ DT H w PK C
FwV I
D
G J EXW FGH ww VT IJK FGH ww VP IJK FGH ww TP IJK E\WULSOHSURGXFWUXOH
V Hw TK
,QYHUVLRQWHPSHUDWXUH P
P
P
P
T
T
V
YG:(26 P
V
3
FG w V IJ
H w T K aw P w V f aw T w Pf
H
aw P w T f aw P w V f
V
T
FG IJ
H K
R
V b
wP
RT
a
V b V
wT V
FG w P IJ RT a H wV K V b V
R V b
F w V IJ
G
H w T K n RT V b s a V FwV I
D
G J n RTkVR Vb V s b pa V V Hw TK
T
P
P
TD
RT
V V b
TR
RT V V b
RT V b a V n
s
RT
a
V b
V
LM OP a NV V b V b Q V
TR
V b V
V V b bRT
a
V V b V
FG
H
a
V
IJ
K
a V b a
b
bR V bR
V
RT
a
DOVR P
V b V
&KRRVHV
&DOFXODWH TLQY IURP(TQ &DOFXODWHPIURP(TQ 6ROXWLRQGRQHZLWK0$7+&$' VHH0$7+&$'ZRUNVKHHW T
Solutions to Chemical and Engineering Thermodynamics, 5th ed
E 5.(26
RT
a
V b
TV V b
P
FG w P IJ
H wV K
FG w P IJ
HwTK
D
T
V
RT
a
a
V b TV V b
TV V b a f
a
R
V b T V V b
FG IJ
H K
wV
V wT
P
a
a
l
a f
q
R V b a T V V b
V RT V b a TV V b a TV V b m
r n
TD a
RT
V V b T V V b
a f
f
f
w P wT V
V w P wV T
s n
s
RT
a
a
V b
TV V b
TV V b a
LM OP
OP
LM
Q TV V b NV V b V Q
N
RT LV b V O
RTb
a
LM a f V b V OP
V b MN V V b PQ V V b
TV V b N V V b
Q
RT V b V V b
RTb
a
V b
V V b
TV V b V V b
RTb
a V b
V b TV V b T a V b V b V V b Rb
RS a L b O V b UV
T Rb MN V PQ V b W
T
8VLQJD0DWKFDGSURJUDPIRUWKH6RDYH5HGOLFK.ZRQJ(26ZHILQG
H q & EDU u - PRO UHODWLYHWRLGHDOJDVDW.DQGEDU 1RZE\WULDODQGHUURUXQWLOHQWKDOSLHVPDWFK
H q & EDU u H q & EDU u Ÿ T R & &ORVHHQRXJK
1RWHWKDWWKLVVROXWLRQLVRQO\YHU\VOLJKWO\GLIIHUHQWIURPWKDWREWDLQHGZLWKWKH3HQJ5RELQVRQ
HTXDWLRQ R&FRPSDUHGWRR&REWDLQHGKHUH 2QO\DYDLODEOHDVD0DWKFDGZRUNVKHHW
$YDLODEOHDVD0DWKFDGZRUNVKHHW
Solutions to Chemical and Engineering Thermodynamics, 5th ed
PRO EDU 3D
˜
3URSHUW\'DWD
7LQ.3LQEDU 5( ˜
7F ˜ .
MRXOH
5* ˜
PRO˜ .
3F ˜ EDU RP EDU ˜ P
PRO˜ .
&S ˜ 5(
NDS RP
˜
RP
˜ ˜ RP
,QLWLDO&RQGLWLRQVDQGWRWDOYROXPH9W
7L ˜ .
3HQJ5RELQVRQ&RQVWDQWV ,QLWLDOWHPSHUDWXUH
E ˜
5*˜ 7F
7 7L
1RWHWKDWWKHVHDUHEHLQJGHILQHGDVD
IXQFWLRQRIWHPSHUDWXUHVLQFHZHZLOOQHHGWR
LQWHUDWHRQWHPSHUDWXUHODWHUWRREWDLQWKHILQDO
VWDWHRIWKHV\VWHP
9
5*˜ 7
*LYHQ
3L
,QLWLDOPRODUYROXPHDQG
QXPEHURIPROHV
9L
u '(/6L . MRXOH
7˜
'HO+L 5(˜ 7˜ =L =L G
D 7 D 7
G7
˜ ˜ E
DF ˜
3F
5* ˜ 7F
3F
D 7 DF ˜ DOI 7
'D 7 6ROYH35(26IRULQLWLDOYROXPH
'(/6L
9W ˜ P
7 ·º
ª
§
DOI 7 ˜ « NDS˜ ¨ ¸»
7F ¹ ¼
¬
©
)LQGLQLWLDOPRODUYROXPHDQGQXPEHURIPROHV
6WDUWZLWKLQLWLDOJXHVVIRUYROXPHPAPRO
(QWURS\GHSDUWXUHDWWKH
LQLWLDOFRQGLWLRQV
3L ˜ EDU
9 E
5*˜ 7L
9
3L
u D 7
9˜ 9 E E ˜ 9 E
P
9W
1
9L
G
D 7
G7
1
P
9L )LQG 9
'D 7
3L º
ª ª
ª 9L ˜ E º º ˜ 5(
˜ OQ«
« OQ« 9L E ˜
»
»»
5*˜ 7¼ ˜ ˜ E ˜ 5* ¬ 9L ˜ E ¼ ¼
¬ ¬
3L˜ 9L
=L
5*˜ 7
ª =L ˜ E ˜ 3L º
«
5*˜ 7 »
˜ OQ«
»
« =L ˜ E˜ 3L »
5*˜ 7 ¼
¬
'HO+L
MRXOH
)LQDOSUHVVXUHLVEDUDQGILQDOWHPSHUDWXUHLVXQNQRZQZLOOEHIRXQGE\HTXDWLQJWKHLQLWLDODQGILQDO
HQWURSLHV*XHVVILQDOWHPSHUDWXUHLV.
3I ˜ EDU
7 ˜ .
9
5*˜ 7
3I
9
u P
1RWH7RXVHWKHJLYHQDQGILQGFRPPDQGIRUYDULDEOHVZLWKGLIIHUHQWGLPHQVLRQVVXFKDV7DQG9ZLOOKDYH
WRFRQYHUWWRGLPHQVLRQOHVVYDULDEOHVVRDVQRWWRJHWDXQLWVFRQIOLFW'HILQH[ 9E\ 77F
LQLWLDOJXHVV
[ \ Solutions to Chemical and Engineering Thermodynamics, 5th ed
LQLWLDOJXHVV
*LYHQ
[ \ 5*˜ \ ˜ 7F
3I
D \ ˜ 7F
[˜ E E
[˜ E ˜ [˜ E E E ˜ [˜ E E
3I º 'D \ ˜ 7F ˜ 5( ª [˜ E ˜ E º º
§ 3I · ª
ª
˜ OQ«
¸ 5(˜ OQ¨ ¸ « 5(˜ OQ« [˜ E E ˜
»
» » '(/6
5*˜ \ ˜ 7F¼
© 7L ¹
© 3L ¹ ¬
¬
˜ ˜ E ˜ 5*
¬ [˜ E ˜ E ¼ ¼
&S˜ OQ§¨
\ ˜ 7F ·
< )LQG [ \
9 < ˜E
§ ·
¨
¸
© ¹
<
9
u =I 3I ˜ 9
5*˜ 7
7I ˜
'HO+I 5(˜ 7I ˜ =I 'HO+I
P
=I
7I < ˜ 7F
7I
.
G
D 7I D 7I
G7I
˜ ˜ E
ª =I ˜ E˜ 3I º
«
5*˜ 7I »
˜ OQ«
»
« =I ˜ E ˜ 3I »
5*˜ 7I ¼
¬
MRXOH
8I8L +I=I 5 7I +L=L 5 7L +I+L=I 5 7I=L 5 7L
+I+I,* +I,* +L+L,* +L,*=I 5 7I=L 5 7L
'HO+I'HO+L&S 7I7L =I 5 7I=L 5 7L
'HO8 > 'HO+I 'HO+L &S˜ 7I 7L =I ˜ 5(˜ 7I =L˜ 5(˜ 7L@ ˜ 1
'HO8
u MRXOH
717HT 'HO8
˜
MRXOH
NJ
717HT
NJ
)LQDOWHPSHUDWXUH
Solutions to Chemical and Engineering Thermodynamics, 5th ed
:HVWDUWIURP
wP
d S C9dT dV wT V
6LQFHWKHHQWURS\DW.LVQRWDIXQFWLRQRIWHPSHUDWXUHLWIROORZVWKDWC9 $OVRVLQFHWKH
HQWURS\LVQRWDIXQFWLRQRIVSHFLILFYROXPHLWIROORZVWKDW
wP
wT V
F I
H K
F I
H K
+RZHYHU E\WKHWULSOHSURGXFWUXOH
F wP I F wV I FG wT IJ RU
H wT K H wP K H wV K
F wP I FG wP IJ F wV I H wT K H wV K H wT K
V
T
V
P
P
T
EXWIURPWKHWKHUPRG\QDPLFVWDELOLW\FRQGLWLRQ
FG wP IJ H wV K
T
ZKLFKLPSOLHVWKDW
F wV I
H wT K
F I
H K
wV
V wT
DQGD
P
P
5HZULWHWKH&ODXVLXVHTXDWLRQDV
RT
R
§ wV ·
V
b 7KHQ ¨
¸
P
T
T
w
©
¹P
§ wV ·
§ wV ·
ZKLFKPHDQVC3 LVLQGHSHQGHQWRISUHVVXUHDQGHTXDOWRC3 DQGV T ¨
¨¨ ¸¸
¸ b
© wT ¹ P
© wT ¹ P
D 7KHUHIRUH
'H
z
T
H T P H T P
z
P
T
E
RU
C3 T dT bdP
P
J
T T T T b P P
LVWKHOLQHRIFRQVWDQWHQWKDOS\
E
D T T 'S
S T P S T P
z FH IK
z
T
P
C3 T
wV
dT T
wT
T
P
z
z
T
P
C3 T
R
dT dP
T
P
T
P
RU
J
P
T
D OQ E T T T T R OQ P
T
dP
P
LVWKHOLQHRIFRQVWDQWHQWURS\
F )RUWKHIOXLGWRKDYHD-RXOH7KRPVRQLQYHUVLRQWHPSHUDWXUH
Solutions to Chemical and Engineering Thermodynamics, 5th ed
F wT I PXVWXQGHUJRDVLJQFKDQJH+RZHYHU
H wP K
LMV T F wV I OP RT b T R
wT
N H wT K Q P
P b
P F I
H wP K C
C
C
P
H
P
3
H
3
3
7KLVLVDOZD\VQHJDWLYHVRWKH&ODXVLXVGRHVQRWKDYHD-RXOH7KRPVRQLQYHUVLRQWHPSHUDWXUH
57
57
5
57
§ w9 ·
§ w9 · 57
E ¨
E
3
DQG&3 D E 7 J7 VR9
DQG97 ¨
E
¸
¸
9E
3
3
© w7 ¹3 3
© w7 ¹3 3
3 7
'+
3 7
3 7
ª
ª
§ w9 · º
§ w9 · º
&3 G7 «9 7 ¨
«9 7 ¨
¸ » G3 ¸ » G3
w7 ¹3 ¼
w7 ¹3 ¼
©
©
¬
¬
3 7
3 7
3 7
³
³
E 3 D 7 7 ³
E J
7 7 7 7 E 3 IRUFRQVWDQWHQWKDOS\
E
J
Ÿ E3 D7 7 7 IRUOLQHRIFRQVWDQWHQWKDOS\
3 7
'6
3 7
3 7
&3
§ w9 ·
§ w9 ·
G7 ¨
¸ G3 ¨
¸ G3
7
© w7 ¹3
© w7 ¹3
3 7
3 7
3 7
³
³
³
IRUFRQVWDQWHQWURS\
§3 ·
§7 ·
J
5OQ ¨ ¸ DOQ ¨ ¸ E 7 7 7 7
© 3 ¹
© 7 ¹
J
Ÿ E OQ 3 D OQ 7 E7 7 IRUOLQHRIFRQVWDQWHQWURS\
8VLQJWKH35(26SURJUDPIRUWKHLQLWLDOFRQGLWLRQVRIEDUDQGR&ZHKDYHIRUWKH
OLTXLGWKDW
9/ ¯PPRO+/ -PRODQG6/ - PRO. $WWKLVWHPSHUDWXUH
VLQFHWKHSURFHVVLVLVRWKHUPDO WKH9/(FRQGLWLRQVDUH3YDS EDUDQG
9/ ¯PPRO+/ -PRODQG6/ - PRO. DQG
99 ¯PPRO+9 -PRODQG69 - PRO. D 7KHUHIRUHWKHYROXPHFKDQJHLV
'9 ¯¯¯¯¯¯ ¯ P
E 7KHHQWURS\EDODQFHLV
G6 4
6 JHQ QRZVLQFH7LVFRQVWDQWDQGWKHIOXLGLVXQGHUJRLQJDXQLIRUPH[SDQVLRQ6 JHQ GW 7
N
VRWKDW'6 u u u u DQG
.
4 7'6 u N
)URP(T
9
§ w3 ·
&9 9 7 &9 9 f 7 7 ³ ¨ ¸ G9 9 f 7 © w7 ¹ 9
Solutions to Chemical and Engineering Thermodynamics, 5th ed
3
§ w3 ·
¨
¸
© w7 ¹ 9
&9 9 7
57
D7
9E 9 9E E 9E
§ w3 ·
5
GD 7 G7
G D 7 G7 ¨ ¸ 9E 9 9E E 9E
© w7 ¹9 9 9 E E 9 E
&9 7
9
G9
GD 7
G7 9³f 9 9 E E 9 E
EXW
G9
³ 9 9E E 9E
ª 9 E º
OQ «
»
E ¬« 9 E »¼
VRWKDW
&9 9 7
&9 7
)LQDOO\
ª 9 E º
GD 7
OQ «
»
G7 E «¬ 9 E »¼
5 7&
D 7 ZKHUHD 7
3&
D 7
VRWKDW
GD 7
G7
ª
§
7 ·º ª
º
˜ « N ¨ » DQG
¸¸ » ˜ « N ˜ ˜
¨
7& ¹ ¼» ¬«
7 ˜ 7& ¼»
©
¬«
ª
º
ª
§
7 ·º « N
» § N ·
˜ « N ¨ ¸¸ » ˜ « ˜ ¸¸
» ¨¨
¨
7& ¹ »¼ « «¬
©
» © ˜ 7 ˜ 7& ¹
¬ 7 7& ¼
­
½
ª
º
§
5 7& °° ª
7 ·º « N
» § N · °°
˜ ® ˜ « N ¨ ¸» ˜ « ˜
» ¨¨
¸¸ ¾
¨
3& ° «¬
7& ¸¹ »¼ « ©
» © ˜ 7 ˜ 7& ¹°
°¯
°¿
¬ 7 7& ¼
GD 7
G7 GD 7
G7 9 7 3 D E7 &3DQG8 73
§ w9 ·
¨
¸
© w7 ¹3
ª
§
7 ·º
« N ¨¨ ¸»
7& ¸¹ ¼»
©
¬«
G7 H3
§ w6 · § w8 ·
§ w6 ·
¨ ¸
¨
H Ÿ H E7
¸ ¨
¸
7
© w3 ¹7
© w8 ¹7 © w3 ¹7
%XWLWZDVFODLPHGWKDWDEFGDQGHDUHFRQVWDQWVZKLFKLVQRWWKHFDVHJLYHQWKH
LQWHUUHODWLRQVKLSEHWZHHQEDQGHJLYHQDERYH
E
D 5HZULWLQJWKHHTXDWLRQRIVWDWHDV
9
E
57 D3 § w9 ·
¨
¸
3
7 © w7 ¹3
§ w9 ·
5 D3
DQG ¨¨ ¸¸
3 7
© w7 ¹3
D3
3
&3 3 7
§ w9 ·
&3 3 7 7 ¨¨ ¸¸ G3
w7 ¹3
3 ©
³
7
3
&3 3 7 7
D3
³ 7 G3 & 3 7 7
3
3
D
3 3
DQG
&3 EDU7
&3 EDU7 EDU
P .
u
u EDUPRO
7 .
EDUP
7
PRO.
Solutions to Chemical and Engineering Thermodynamics, 5th ed
&3 EDU7 .
7KHUHIRUH &3 EDU7 .
PRO.
PRO.
6RWKDWWKHPHDQYDOXHRYHUWKHWHPSHUDWXUHUDQJHLV
EDU.
E
57 D3 § w9 ·
¨
¸
3
7 © w7 ¹ 3
EDU.
'+
EDU.
.
³
DQG
9
³
³
5 D3
§ w9 ·
DQG9 7 ¨
¸
3 7
© w7 ¹3
.
E
57 D3 57 D3
3
7
3
7
³
³
EDU.
.
D3
7 ³
EDU.
EDU.
E
EDU.
D3 º
D3 º
ª
ª
G7 « E 7 »G3 « E 7 »G3
¬
¼
¬
¼
EDU.
.
EDU.
§
§
D3 ·
D3 ·
u ¨¨ E3 ¨¨ E3 ¸¸
¸
7 ¹
7 ¸¹
©
©
EDU.
EDU.
'6
PRO.
ª
ª
§ w9 · º
§ w9 · º
& 3 G7 « 97 ¨
« 97 ¨
¸ »G3 ¸ »G3 © w7 ¹3 ¼
© w7 ¹3 ¼
EDU. ¬
.
EDU. ¬
E '+
PRO
EDU.
&3
§ w9 ·
§ w9 ·
G7 ¨
¸ G3 ¨
¸ G3
7
© w7 ¹ 3
© w7 ¹3
EDU.
.
EDU.
³
³
³
º
º
ª
ª
D u »
«
D u »
«
OQ 5 OQ u
5 OQ u
«
»
»
«
PRO.
7
7
¬
¼
¬
¼
Solutions to Chemical and Engineering Thermodynamics, 5th ed
D
]] 7RFDOFXODWHSRLQWVDWJHRPHWULFDOO\LQFUHDVLQJYDOXHVRIWKHPRODUYROXPH
DWWHQSRLQWVSHUGHFDGH
L L
XQLWVPAP
9 ]]
˜
L
YDQGHU:DDOVSDUDPHWHUV
D XQLWV
5 3D˜ P
XQLWV
E ˜ PRO
3 L
3 L
5˜ 9 E
D
9
L
5˜ 9 E
L
3 L
5˜ 9 E
L
D
9
L
3 L
XQLWV
3D˜ P
PRO˜ .
P
PRO
5˜ 9 E
L
D
9
L
D
L
9
L
3L 3L
3L
3L
ORJ 9L
Solutions to Chemical and Engineering Thermodynamics, 5th ed
E
7 5˜ 7
39 9 E
*LYHQ
D
5˜ 7
9LQLW,*
9
39
,GHDOJDVJXHVV
IRULQLWLDOYROXPH
9 9LQLW,*
9LQLW )LQG 9
9LQLW
,GHDOJDVJXHVV
IRULQLWLDOYROXPH
5˜ 7
9ILQ,*
9LQLW,* 9 9ILQ,*
*LYHQ
39
9ILQ )LQG 9
´
: µ
¶
9ILQ
u 9ILQ
3 9 G9
:
u 9ILQ,* u -
9LQLW
F
)RUWKHLGHDOJDV
´
:,* µ
µ
¶
9ILQ,*
9LQLW,*
5˜ 7
9
G9 :,*
u -
$OPRVWLGHQWLFDOWRWKHYG:UHVXOW
1RWHWKDWLIRQHZDQWHGWRGRWKHLQWHJUDOVDQDO\WLFDOO\LQ0DWKFDGLQVWHDGRIQXPHULFDOO\
WKHIROORZLQJZRXOGEHREWDLQHG
´
µ
µ
¶
57
9
´
µ
µ
µ
¶
´
µ
µ
µ
¶
G9 o 57˜ OQ 9
´
µ
µ
¶
YI
YL
57
9
G9 o OQ YI ˜ 57 OQ YL ˜ 57
DD
DD ·
§ 57
¨ 9 EE ¸ G9 o OQ 9 EE ˜ 57 9
9 ¹
©
YI
YL
OQ YI EE ˜ 57˜ YI DD
DD ·
OQ YL EE ˜ 57˜ YL DD
§ 57
¨ 9 EE ¸ G9 o
YI
YL
9 ¹
©
Solutions to Chemical and Engineering Thermodynamics, 5th ed
)RUWKHJDVLQWKHSXPSWKHSURFHVVLVDGLDEDWLFDQGUHYHUVLEOH XQLIRUPFRPSUHVVLRQ 7KHUHIRUHWKHSURFHVVLVLVHQWURSLF ZKHWKHURUQRWLWLVDQLGHDOJDV D )RUWKHLGHDOJDVFDVH
7
3
7I
' 6 &3 OQ I 5 OQ I OQ
OQ
Ÿ 7 .
7L
3L
39
1RZVLQFHIURPWKHLGHDOJDVODZ L L
7L
3I 9I
9
Ÿ I
7I
9L
3L 7I
u
7L 3I
7KHUHIRUH WKHSXPSKDQGOHZLOOEHSXVKHGGRZQ u u u FP
E 8VLQJWKH35(26SURJUDP
$W7 .DQG3 EDU9 ¯PPRODQG6 - PRO. 1RZDW3 EDUJXHVV7XQWLORQHREWDLQV6 - PRO. FDQXVHLGHDOJDVWHPSHUDWXUHDV
LQLWLDOJXHVV )LQGWKDW7 .IRUZKLFK9 ¯PPRO
§ u ·
7KHUHIRUH WKHSXPSKDQGOHZLOOEHSXVKHGGRZQ u ¨¨ ¸ FP u ¸¹
©
D %\WKHWULSOHSURGXFWUXOH
§ wT · § wU · § wV ·
¨
¸ ¨
¸ ¨
¸
© wV ¹U © wT ¹V © wU ¹T
§ wT ·
¨
¸
© wV ¹U
§ wT · § wU ·
¨
¸ ¨
¸ © wU ¹V © wV ¹T
DQGdU
Td SHJ PdV
§ wU ·
7KHUHIRUH ¨
¸
© wV ¹T
§ wU ·
¨
¸
© wV ¹T
§ wS ·
§ w S · § wP ·
T¨
¸ P DQGXVLQJ ¨
¸ ¨
¸ JLYHV
V
w
©
¹T
© wV ¹T © wT ¹V
§ wT ·
§ wP ·
T¨
¸ P DQG ¨
¸
T
w
©
¹V
© wU ¹V
C9
ª § wP ·
º
«T ¨
¸ P»
« © wT ¹V
¼»
VRWKDW[ ¬
C9
E ,GHDOJDV PV
§ wP ·
RT VRWKDW ¨
¸
© wT ¹V
R
V
P
VR[ T
F
P
RT B T RT § wP ·
¨
˜
¸
V
V
V
© wT ¹V
§ wP ·
T¨
¸ P
© wT ¹V
R B T R RT dB T
˜ ˜
V
V V V
dT
RT dB T
RT dB T
DQG
[
˜
˜
dT
V
V ˜ C9 dT
Solutions to Chemical and Engineering Thermodynamics, 5th ed
G
RT
aT
V b V V b b V b
P
da T
R
˜
V b V V b b V b
dT
§ wP ·
¨
¸
© wT ¹V
da T
a T
dT
DQG
V V b b V b
T
§ wP ·
T¨
¸ P
© wT ¹V
da T
dT
˜
V V b b V b ˜ C9
a T T
[
ZKHUHC9LVDIXQFWLRQRIWHPSHUDWXUHDQGVSHFLILFYROXPH
5HDFWRULV,LQLWLDOO\HYDFXDWHGWDQNLV,,7KHPDVVEDODQFHDVLQ3UREOHPLV Ni N I N II H[FHSWWKDWKHUHXQOLNHWKHUHKHUHZHXVHWKHYLULDOJDVHTXDWLRQRIVWDWH
7KHHQHUJ\EDODQFHRIWKHUHDFWRUDQGWDQNLV NiU i N I U I N II U II 7KHHQWURS\EDODQFHRQMXVWWKH
FRQWHQWVRIWKHUHDFWRUWKDWXQGHUJRDXQLIRUPH[SDQVLRQOHDGVWR
S I FRQVWDQW 7KHHTXDWLRQRIVWDWHLV
39
57
'6
%7
ZLWK% 7
9
u 9 f 7L
7I
L
L
P
&3
7
PRO
PRO.
9I 7I
&9
§ w3 ·
§ w3 ·
G7 ¨
¸ G9 ¨
¸ G9
w
w7 ¹9
7
7
©
¹
©
9
9 7
7
9 f 7
³
³
L
³
I
5OQ
ª 9I
7
º
u 5 « » OQ I
9L
7L
¬ 9I 9L ¼
VLQFH3
57 57% 7
9
9
§ w3 ·
¨
¸
© w7 ¹9
5 5 u u 9
9
57 57 u u 5 u 9
9
9
$OVRIRUODWHUUHIHUHQFH
9 f 7UHI
97
7
ª § w3 ·
º
ª § w3 ·
º
'8
«7 ¨
«7 ¨
¸ 3 » G9 &9 G7 ¸ 3 » G9
»¼
»¼
¬ © w7 ¹9
¬ © w7 ¹9
9UHI 7UHI «
7UHI
9 f 7 «
³
³
§ w3 ·
DQG7 ¨
¸ 3
© w7 ¹9
³
57 57 u u 57 57 u u 5 u 9
9
9
9
9
§ w3 ·
¨
¸
© w7 ¹9
VRWKDW
9 f 7UHI
8
³
9UHI 7UHI
5 u 9
97
7
G9 ³ & G7 ³
9
7UHI
5HVWRIWKHSUREOHPVROXWLRQLVLQ
0$7+&$'
9 f 7
5 u 9
G9
5 u 9
Solutions to Chemical and Engineering Thermodynamics, 5th ed
5 5* ˜ % 7 ˜ 7UHI 3UHI &Y 5
7
&DOFXODWHUHIHUHQFHYROXPH IRUODWHU
5*˜ 7UHI
9UHI 9UHI
3UHI
*LYHQ
3UHI˜ 9UHI
5*˜ 7UHI
% 7UHI
9UHI
9UHI )LQG 9UHI
3LQLW 9UHI
7LQLW 5*˜ 7LQLW
9LJ 9LJ
3LQLW
u 9LQLW 9LJ
*LYHQ
3LQLW˜ 9LQLW
5*˜ 7LQLW
% 7LQLW
9LQLW
9LQLW )LQG 9LQLW
1LQLW 9LQLW
1LQLW
9LQLW
&DOFXODWHGLQLWLDOPRODUYROXPH
u u &DOFXODWHLQLWLDOPRODULQWHUQDOHQHUJ\
8LQLW 5˜ ˜ ¨§
© 9UHI
·
¸ &Y˜ 7LQLW 7UHI
9LQLW ¹
8LQLW
u Solutions to Chemical and Engineering Thermodynamics, 5th ed
*XHVVILQDOSUHVVXUHDQGWHPSHUDWXUHLQWDQNXVHLGHDOJDVUHVXOWDVILUVWJXHVV7KH
UHVXOWEHORZLVWKHUHVXOWRIVHYHUDOLWHUDWLRQV
3I 7I 3I˜ 9I
% 7I
9I 5*˜
7I
9I u 3I
)LQGILQDOWHPSHUDWXUHDQGPRODUYROXPHLQUHDFWRUIRUWKLVILQDOSUHVVXUHJXHVV
*LYHQ
5*˜ 7I
9I
·
9I ·
§ § 7I ·
5 ˜ OQ§¨
¸ ˜ ˜ 5˜ ¨ 9I 9LQLW ¸ 5 ˜ OQ¨ 7LQLW ¸
© 9LQLW ¹
©
¹
©
¹
[ ILQG 7I 9I
§ ·
¨
¸
© u ¹
[
7I [
9I [
1I 1I
9I
&DOFXODWHILQDOPRODULQWHUQDOHQHUJ\RIJDVLQUHDFWRU
8I 5˜ ˜ ¨§
© 9UHI
·
¸ &Y˜ 7I 7UHI
9I ¹
8I
u &DOFXODWHILQDOQXPEHURIPROHVLQWDQNDQGPRODUYROXPHEDVHGRQJXHVV3I
1I 1LQLW 1I
1I
9I u 1I
9I
u 1RZXVHWKLVWRFDOFXODWHWHPSHUDWXUHLQWDQN
7I 7LQLW
*LYHQ
3I ˜ 9I
5*˜ 7I
% 7I
9I
7I ILQG 7I
7I
&DOFXODWHILQDOPRODULQWHUQDOHQHUJ\RIJDVLQWDQN
8I 5˜ ˜ ¨§
© 9UHI
·
¸ &Y˜ 7I 7UHI
9I ¹
8I
u ,VHQHUJ\EDODQFHVDWLVILHG"
'HO 1LQLW˜ 8LQLW 1I˜ 8I 1I˜ 8I
'HO
u 5HVXOWV
3ILVGHOLV
3ILVGHOLV
3ILVGHOLV
3ILVGHOLV
3ILVGHOLV
7I
7I
ZKXFKXVFORVHHQRXJK
1I
1I
u &RPSDUHZLWKLGHDOJDVUHVXOW
7I DQG7I 6RWKHUHVXOWVDUHQRWGUDVWLFDOO\GLIIHUHQWWKDQWKHLGHDOJDVUHVXOW
1RWHWKDWLQWKHQH[WSUREOHPZHVROYHWKLVXVLQJWKH35(26,QWKDWFDVHWKH
ILQDOSUHVVXUHLVEDU7I DQG7I ZKLFKDUHFORVHWRWKHUHVXOWV
REWDLQHGKHUH H[FHSWIRUWKHSUHVVXUH Solutions to Chemical and Engineering Thermodynamics, 5th ed
5HDFWRULV,LQLWLDOO\HYDFXDWHGWDQNLV,,7KHPDVVEDODQFHDVLQ3UREOHPLV Ni N I N II H[FHSWWKDWKHUHXQOLNHWKHUHZHFDQQRWXVHWKHLGHDOJDVHTXDWLRQRIVWDWH
7KHHQHUJ\EDODQFHRIWKHUHDFWRUDQGWDQNLV NiU i N I U I N II U II 7KHHQWURS\EDODQFHRQMXVWWKHFRQWHQWVRIWKHUHDFWRUWKDWXQGHUJRDXQLIRUPH[SDQVLRQOHDGVWR
S I FRQVWDQW 8VLQJWKH3HQJ5RELQVRQSURJUDPZLWKKHDWFDSDFLW\WKDWLVDIXQFWLRQRI
WHPSHUDWXUHZHREWDLQWKDW
P
V i ˜
VRWKDWNi PRO
PRO
H i VRWKDWU i H i PV
˜ ˜ ˜ i i
PRO
PRO
S i PRO ˜ .
7KHFDOFXODWLRQDOSURFHGXUHLVDVIROORZV*XHVVDILQDOSUHVVXUHILQGWKHWHPSHUDWXUHLQWKHUHDFWRU
IRUZKLFKWKHHQWURS\LVHTXDOWRSi8VLQJWKDWTDQGPFDOFXODWH XVLQJWKH35(26 WKHPRODU
YROXPHDQGPRODUHQWKDOS\IURPZKLFKWKHQXPEHURIPROHVLQWKHUHDFWRUDQGLWVLQWHUQDOHQHUJ\
FDQEHFRPSXWHG7KHXVHWKHPDVVEDODQFHWRFRPSXWHWKHQXPEHURIPROHVLQWDQN,,WKHPRODU
YROXPHLQWDQN,,DQGWKHNQRZQSUHVVXUHWRFDOFXODWHWKHWHPSHUDWXUHLQWDQN,,7KHQVHHLIWKH
HQHUJ\EDODQFHLVVDWLVILHG,IQRWUHSHDWWKHFDOFXODWLRQZLWKDQRWKHUJXHVVHGYDOXHIRUWKHILQDO
SUHVVXUH
6WDUWZLWKWKHLGHDOJDVVROXWLRQRISUREOHPRIP EDU)LQG
IRU S i WKHWHPSHUDWXUHLVT I .
PRO ˜ .
P
V I ˜
VRWKDWN I PRON II PRO
PRO
II
P
DQGV
˜
PRO
I
I
I
H
H P I V I ˜ ˜ ˜ VRWKDWU
PRO
PRO
%\WULDODQGHUURUZHILQGWKDWWKHFRQGLWLRQVLQWKHWDQN,,DUH
P
IRUV II ˜
WKHWHPSHUDWXUHLVT II .
PRO
H II VRWKDWU II H II P II V II ˜ ˜ ˜ PRO
PRO
DQGWKHHQHUJ\EDODQFHLV
"
˜ ˜ ˜ ˜ 6RWKDWWKHHQHUJ\EDODQFHLV127VDWLVILHG
7U\DQRWKHUJXHVV«P I EDU
IRU S i WKHWHPSHUDWXUHLVT I .
PRO ˜ .
VI
˜
P
VRWKDWN I
PRO
PRON II
˜ PRO
P
DQGV
˜
PRO
VRWKDWU I H I P I V I ˜ ˜ ˜
H I PRO
%\WULDODQGHUURUZHILQGWKDWWKHFRQGLWLRQVLQWKHWDQN,,DUH
II
PRO
Solutions to Chemical and Engineering Thermodynamics, 5th ed
IRUV II
˜
P
WKHWHPSHUDWXUHLVT II
PRO
VRWKDWU II
H
PRO
DQGWKHHQHUJ\EDODQFHLV
II
H
II
P V
II
II
.
˜ ˜ ˜
PRO
"
˜ ˜ ˜ ˜ ˜ :KLFKLVFORVHHQRXJKVRWKDWZHFDQFRQVLGHUWKHHQHUJ\EDODQFHWREHVDWLVILHG7KHILQDO
FRQGLWLRQVWKHQDUHDSUHVVXUHRIEDUTIRI.DQGTIIRI.
7KLVLVWREHFRPSDUHGZLWKWKHLGHDOJDVFRQVWDQWKHDWFDSDFLW\VROXWLRQRIDSUHVVXUHRIEDU
TIRI.DQGTIIRI.
7RFRPSDUHEHWWHUZLWKWKHSUHYLRXVLGHDOJDVVROXWLRQZHFRXOGUHSHDWWKHFDOFXODWLRQZLWKWKH
FRQVWDQWLGHDOJDVKHDWFDSDFLW\RI-PRO'RLQJWKLVZHILQGWKDWWKHILQDOSUHVVXUHLVDJDLQ
DERXWEDUTI .DQGTII .
§ w9 ·
§ w6 ·
¨
¨ ¸ HTQ ¸
© w7 ¹6
© w3 ¹ 9
DQGHTQ
&9
§ w3 ·
G6
G7 ¨
N7
¸ G9
7
© w7 ¹ 9
& 9 § w7 ·
§ w6 ·
¨ w3 ¸
7 ©¨ w3 ¹¸ 9
© ¹9
Ÿ
7ULSOHSURGXFW5XOH
w 9 w7 6
w 9 w7 3
§ w9 ·
9 ¨© w3 ¹¸7
&9 § w7 · § w7 ·
¨
¸ 7 ¨© w3 ¸¹ 9 © w 9 ¹3
§ w7 · § w 9 · § w3 ·
¨
¸ ¨
¸
¸ ¨
© w 9 ¹ 3 © w3 ¹7 © w7 ¹ 9
§ w7 ·
§ w3 · § w7 ·
6RWKDW ¨
¨
¸
¸ ¨
¸ 9
w
©
¹3
© w 9 ¹7 © w3 ¹ 9
w 9 w7 6
& § w7 · § w3 ·
&9 § w7 ·
D
9¨
¨
¸
7 © w3 ¸¹ 9 © w 9 ¹7
79N 7 ¨© w3 ¸¹9
w 9 w7 3
§ w9 ·
&9 ! 6LQFH ¨
VR N 7 ! E
¸ © w3 ¹7
Ÿ w 9 w7 6
DQG
w 9 w7 3 PXVWKDYHRSSRVLWHVLJQV &RQVLGHUWKHSURFHVVWREHFRQWLQXRXV QRWDEDWFKSLVWRQDQGF\OLQGHURSHUDWLRQ ,QWKLVFDVH
WKHHQHUJ\DQGHQWURS\EDODQFHVIRUWKHDGLDEDWLFSURFHVVDUH
G1
1
Ÿ 1
1
1
1
LQ
RXW
RXW
LQ
GW
G8
G6
1+
:
1+
16
16
6
LQ
RXW
LQ
RXW
JHQ
GW
GW
)RUPLQLPXPZRUN 6 JHQ VRWKDW6LQ 6RXW 8VLQJWKH35(26SURJUDPDWWKHLQLWLDOFRQGLWLRQVRI.DQGEDU
Solutions to Chemical and Engineering Thermodynamics, 5th ed
9/ ¯PPRO+/ -PRODQG6/ - PRO. 1RZDWEDUE\WULDODQGHUURU7 .DQG
9/ ¯PPRO+/ -PRODQG6/ - PRO. 7KHUHIRUH: +RXW±+LQ -PRO
Solutions to Chemical and Engineering Thermodynamics, 5th ed
D
)URP3HQJ5RELQVRQ(26
99 ˜
9/ ˜ 1L 9/
1I ˜ 1L
/ 1I
/ 9
+/ +9 1I
u JLYHQ
1I
= ILQG / 9
u 9 999
˜ 9/˜ /
/ =
1L
9 =
§ u ·
¨
¸
© ¹
=
)UDFWLRQYDSRUI
9
I
I
9 /
VRRIWKHSURSDQHUHPDLQLQJLQWKHWDQNLVYDSRU
E
0DVVEDODQFH
1RXW 1L 1I
1RXW
u (QHUJ\EDODQFH / +/9 +91RXW +91L +/4
4 1L˜ +/ /˜ +/ 9˜ +9 1RXW ˜ +9
4
u -
KHDWPXVWEHDGGHGWRYDSRUL]HWKHSURSDQH
,QWKLVSUREOHPPHWKDQHLVWRXQGHUJRDQLVHQWURSLFFRPSUHVVLRQIURPR&DQGVRPHXQNQRZQ
LQLWLDOSUHVVXUHWRR&DQG03D EDU7KHSUREOHPLVWRILQGWKHLQLWLDOSUHVVXUH8VLQJRQHRI
WKH35(26SURJUDPVDQGWKHLGHDOJDVKHDWFDSDFLW\IRUPHWKDQHLQWKH3523(57<SURJUDPZH
KDYHDWEDUDQGR& .WKDW
6 - PRO. DQG+ -PRO%\WULDODQGHUURUZHILQGWKDWDWEDUDQGR& .
6 - PRO. DQG+ -PRO7KHUHIRUHWKHORZHVWSUHVVXUHDOORZHGLVEDU$
ORZHUSUHVVXUHZRXOGUHVXOWLQWKHWHPSHUDWXUHIURPWKHRXWOHWWRWKHFRPSUHVVRUEHLQJDERYHWKH
VDIHW\OLPLW
7KHZRUNGRQHLQWKHFRPSUHVVRUIURPWKHHQHUJ\EDODQFHLV
G8
:
1+
+ RXW +LQ LQ 1+ RXW : RU GW
PRO
1
Solutions to Chemical and Engineering Thermodynamics, 5th ed
%\WKHPDVVDQGHQHUJ\EDODQFHVIRUD-RXOH7KRPVRQH[SDQVLRQZHKDYH
H LQ H RXW RU H EDU. H EDUT " 6LQFHWKHILQDOSUHVVXUHLVVRORZZHZLOODVVXPH
WKHH[LWLQJJDVLVLGHDO7KHUHIRUH H H ,*
EDU.
,*
H EDU.
T
H H ,*
EDU.
,*
,*
H EDU
T H EDU.
³
,*
H EDU
T T
C3 dT
.
³ T dT .
˜ T T 1RZ
V T P
H TP H
DQG
PV
Z
RT
§ wP ·
¨ wT ¸
©
¹V
,*
TP
B T
V
RT Z VRWKDWZ ª § wP ·
º
P »dV
«T ¨
¸
»¼
¬ © wT ¹V
V f «
³
B T
V
P
RT RTB T
V
V
R RB T
RT dB T
DQG
V
dT
V
V
§ wP ·
T¨
¸ P
© wT ¹V
RT RTB T
RT dB T
RT RTB T
V
dT
V
V
V
V
H T P H ,* T P
RTB T
V
V T P
³
V f
RT dB
dV
V dT
RTB T
V
RT dB T
dT
V
RT dB § ·
¨ ¸
dT © V f ¹
RT ª
dB º
B T T
«
V ¬
dT »¼
$OVR
dB T
T
dT
T
dB T
˜ B T T
dT
T
T
T
B T
dB T º
˜ FP
P
ª
u « B T T dT »
PRO
PRO
¬
¼T IURP
B T
PV
RT
V
2QHREWDLQVDW.DQGEDU
P §
P ·
V u ¨ 1RWHIRUWKHLGHDOJDVLHB V u ¸
PRO ¨©
PRO ¸¹
H T P H ,* T P
RT ª
dB º
B T T
V «¬
dT »¼
Solutions to Chemical and Engineering Thermodynamics, 5th ed
6RWKDWWKHHTXDWLRQWREHVROYHGLV ˜ T T 7KHVROXWLRQWRWKLVHTXDWLRQLVT .VRWKDWWKHWHPSHUDWXUHFKDQJHLV.
3 7 5˜
7
9
5 ˜ 7˜
%7
&Y 7 ˜ 7 9
5
G
37 o
5˜
9
G7
5 ˜ %7
G
%7
´
µ G 3 7 G9 o 5˜ OQ 9 5˜ % 7 5˜ 7˜ G7
µ G7
9
9
¶
%7 § · ˜ ¨
¸
7 ¹
©
'% 7 % 7 7˜
G
%7
G7
G
%7 o
G7
'% 5 ˜ 7˜
G
%7
G7
9
9
˜ 7
7KLVFRPELQDWLRQLVDFRQVWDQW
u % u 6ROYLQJIRUYROXPHV
573 5˜
º
ª
573 573 ˜ 573˜ % ¬
¼
9 9
u º
ª
573 573 ˜ 573˜ % ¬
¼
9 9
573 5˜
8VLQJ(T D
9 ·
¸
© 9 ¹
7HUP ˜ OQ§¨
7HUP 7HUP '% ˜ 9
'% ˜ 9
´
7HUP µ
µ
¶
&Y 7
7
7HUP
7HUP
7HUP
u 7HUP
G7
'HO6 7HUP 7HUP 7HUP 7HUP
'HO6
(QHUJ\DQGHQWURS\EDODQFHVDUH
PRO˜ .
Solutions to Chemical and Engineering Thermodynamics, 5th ed
G8
G9 G6 :DQG
6JHQ VLQFHV\VWHPLVFORVHGDQGDGLDEDWLF
3
GW
GW
GW
VLQFHSURFHVVLVGRQHVORZO\ZHFDQDVVXPHLWLVGRQHUHYHUVLEO\ 6JHQ 7KHUHIRUHWKHEDODQFHVEHFRPH
'8 :DQG6 FRQVWDQW ,QLWLDOFRQGLWLRQLV7 .DWZKLFKWKHSUHVVXUHZKLFKHTXDOVWKHYDSRUSUHVVXUHLV3 EDU8VLQJWKH35(26SURJUDPJLYHQWKDWRQO\OLTXLGLVSUHVHQWZHKDYHWKDW9/ ¯
PPRO+/ -PRODQG
6/ - PRO. 7KHSURFHGXUHQRZLVWRXVHWKH35(26SURJUDPJXHVVDWHPSHUDWXUH
FRPSXWHWKHYDSRUOLTXLGHTXLOLEULXPSURSHUWLHVDWWKDWWHPSHUDWXUHDQGVHHLIWKHHQWURS\DW
YDSRUDQGOLTXLGHTXDOVWKHHQWURS\RIWKHLQLWLDOVWDWH:HILQGWKDWDW7 .3YDS EDU
9/ ¯PPRO+/ -PRODQG6/ - PRO. DQG
99 ¯PPRO+9 -PRODQG69 - PRO. DQG6/9 696/ - PRO. 1RZWRFRPSXWHWKHZRUNZHQHHGWKHLQWHUQDOHQHUJLHVZKLFKDUHREWDLQHGIURP8 +±39IURP
/
/
9
ZKLFKZHREWDLQ 8LQLW
DQG
8 ILQDO
DQG8 ILQDO
PRO
J
J
9
/
/
'8 : 8 ILQ
8 ILQ
8 LQLW
u
u
- J
J
PRO
PRO
PRO
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Solutions to Chemical and Engineering Thermodynamics, 5th ed
&ULWLFDOSURSHUWLHVIRUFDUERQGLR[LGH
7F .
3F ˜ 3D
Z 6RDYH5HGOLFK.ZRQJ(TXDWLRQRI6WDWHDQG&RQVWDQWV
MRXOH
5 D 7 3D˜ P
RU
.˜ PRO
.˜ PRO
§ 5˜ 7F ·
¸
© 3F ¹
D ˜ ¨
7 ·º
§
ª
« ˜ Z ˜ Z ˜ ¨ ¸»
7F
¬
©
¹¼
3 9 7 5˜ 7
9 E
E ˜
5˜ 7F
3F
D 7 D˜ D 7
D 7
9˜ 9 E
'DWDJLYHQLQWKHSUREOHP
7
.
9 5˜
7
3 ˜ 3D
3 ˜ 3D
9WRWDO P
,QLWLDOJXHVVQHHGHGIRUVROYHU
3
6ROYLQJIRUWKHLQLWLDOPRODUYROXPHDQGWKHQXPEHURIPROHVRIFDUERQGLR[LGH
3
*LYHQ
3 9 7
9PRODU )LQG 9
9WRWDO
1
1
9PRODU
u 9PRODU u 6ROYLQJIRUWKHILQDOPRODUYROXPHDQGWKHILQDOWRWDOYROXPH
5˜ 7
9
3
*LYHQ
3
3 9 7
9PRODU )LQG 9
9PRODU u L 9WRWDO
9WRWDO 9PRODU˜ 1
&DOFXODWLQJWKHDPRXQWRIZRUNGRQHWRFRPSUHVVWKHJDV
´
:RUN 1˜ µ
¶
9PRODU
3 9 7 G9
LL :RUN
u MRXOH
9PRODU
6LQFHWKHWHPSHUDWXUHLVFRQVWDQWWKHFKDQJHLQHQWKDOS\+ 73 + 73 LVMXVWHTXDO
WR+GHS 73 +GHS 73 § 3˜ 9
+GHS 7 3 5˜ 7˜ ¨
© 5˜ 7
¸· D 7 7˜
¹
E
G
D 7
G7
˜ OQ§¨
9
·
¸
© 9 E¹
+ 7 3 +GHS 7 3
+ 7 3
4 1˜ > + 7 3 + 7 3
3˜ 9PRODU 3˜ 9PRODU @ :RUN
4
u MRXOH
u + 7 3
u Solutions to Chemical and Engineering Thermodynamics, 5th ed
Solutions to Chemical and Engineering Thermodynamics, 5th ed
PRO EDU 3D
˜
3URSHUW\'DWD
7LQ.3LQEDU 5( ˜
7F ˜ . 3F ˜ EDU
5* ˜
PRO˜ .
EDU˜ P
PRO˜ .
RP NDS ˜ RP ˜ RP˜ RP
,QLWLDO&RQGLWLRQV
7L ˜ .
65.&RQVWDQWV E ˜
,QLWLDOWHPSHUDWXUH
MRXOH
3L ˜ EDU
5*˜ 7F
7 7L
DF ˜
3F
1RWHWKDWWKHVHDUHEHLQJGHILQHGDVD
ª
§
IXQFWLRQRIWHPSHUDWXUHVLQFHZHZLOOQHHGWR
DOI 7 ˜ « NDS˜ ¨ LQWHUDWHRQWHPSHUDWXUHODWHUWRREWDLQWKHILQDO
¬
©
VWDWHRIWKHV\VWHP
MRXOH
MRXOH
+HDWFDSDFLW\ &S ˜
&S ˜ ˜
PRO˜ .
PRO˜ .
FRQVWDQWV
MRXOH
&S ˜ ˜
*LYHQ 3L
9
5*˜ 7
9 E
5*˜ 7L
9
3L
D7
'(/6L
'(/6L . MRXOH
=L ¸»
D 7 DF ˜ DOI 7
7F ¹ ¼
'D 7 G
D 7
G7
MRXOH
˜
PRO˜ .
u P
9L )LQG 9
9˜ 9 E
9L
(QWURS\GHSDUWXUHDWWKH
LQLWLDOFRQGLWLRQV
3F
&S ˜ PRO˜ .
)LQGLQLWLDOPRODUYROXPHDQGQXPEHURIPROHV
6WDUWZLWKLQLWLDOJXHVVIRUYROXPHPAPRO
6ROYH65.(26IRULQLWLDOYROXPH
7 ·º
5* ˜ 7F
u P
ªOQª 9L E ˜ 3L º 'D 7 ˜ OQ§ 9L E ·º ˜ 5(
« «
»
¨
¸»
E ˜ 5* © 9L ¹¼
5*˜ 7¼
¬ ¬
3L˜ 9L
=L
5*˜ 7
E ˜ 3L ·
· §
§ G
¨ 7˜ D 7 D 7 ¸ ¨ =L 5*˜ 7 ¸
© G7
¹ ˜ OQ
'HO+L 5(˜ 7˜ =L ¨
¸
=L
E
©
¹
'HO+L
u MRXOH
)LQDOWHPSHUDWXUHLV.DQGILQDOSUHVVXUHLVXQNQRZQZLOOEHIRXQGE\HTXDWLQJWKHLQLWLDODQGILQDO
HQWURSLHV*XHVVILQDOWHPSHUDWXUHLVEDU
3I ˜ EDU
7 ˜ .
9
5*˜ 7
3I
9
u P
Solutions to Chemical and Engineering Thermodynamics, 5th ed
7HPSHUDWXUHSDUWRILGHDOJDVHQWURS\FKDQJH
WL '6LGHDO7 &S˜ OQ§¨
WI ·
¸ &S˜ WI WL © WL ¹
WI &S
˜ WI
WL
&S
˜ WI
WL
'6LGHDO7
. MRXOH
1RWH7RXVHWKHJLYHQDQGILQGFRPPDQGVIRUYDULDEOHZLWKGLIIHUHQWGLPHQVLRQVVXFKDV3DQG9ZLOOKDYH
WRFRQYHUWWRGLPHQVLRQOHVVYDULDEOHVVRDVQRWWRKDYHDXQLWVFRQIOLFW'HILQH[ 9EDQG\ 33F
LQLWLDOJXHVV
*LYHQ
9
[
\ ˜ 3F
5*˜ ˜ .
[˜ E E
\ D ˜ .
[˜ E ˜ [˜ E E
\ ˜ 3F º 'D ˜ .
ª ª
§ [ ·º ˜ 5( '(/6L
¸ «OQ« [˜ E E ˜
» E˜ 5* ˜ OQ¨
¸»
5*˜ ˜ .¼
© 3L ¹ ¬ ¬
© [ ¹¼
'6LGHDO7 5(˜ OQ§¨
\ ˜ 3F ·
< )LQG [ \
<
§ ·
¨
¸
© ¹
9I < ˜ E
9I
u 3I < ˜ 3F
3I
u 3D
3I
[
E
EDU
P
)UDFWLRQPDVVUHPDLQLQJLQWDQN 9L
9I
Solutions to Chemical and Engineering Thermodynamics, 5th ed
DOVRDYDLODEOHDVD0DWKFDGZRUNVKHHW,QIDFWWKDWILOHFRQWDLQJUDSKVDQGRWKHULQIRUPDWLRQ w C3
w V
w C9
wP
WKDQZLWK
(DVLHUWRZRUNZLWK
T
T
wP T
wV T
w T P
w T V
FG
H
IJ
K
FG
H
IJ
K
f T
C9 V T C9 T
FG
H
C9 V T C9 V
6R
C9 V T
C9 T IJ
K
T
V f
a
V b
OQ
V V f
T
FG
H
z FGH ww TP IJK dV
a
F V b IJ
OQG
HV K
T
V
V
IJ
K
FG V b IJ
T b H V K
a
OQ
&OHDUO\ DV V o f LGHDO JDV ZH
UHFRYHU C9
3URFHGXUH
RI V ¶V
C9 &KRRVH FROOHFWLRQ
&DOFXODWH C9 C9 IRUJLYHQ V DQGT
&DOFXODWH PIURP5.(26JHW C9 C9 YVP
1H[WXVH
C3
C9 T
FG w V IJ FG w P IJ
H w PK H w T K
T
C9 T
V
aw P w T f aw P w V f
V
T
WRFRQYHUWIURP C9 WR C3 +DYHGRQHERWKSDUWVXVLQJ0$7+&$'6HHWKH0$7+&$')ILOH
Solutions to Chemical and Engineering Thermodynamics, 5th ed
DOVR DYDLODEOH DV D 0DWKFDG
ZRUNVKHHW RT
a
D
P
V b
TV V b
PV
B C
" Z RT
V V
PV
V
V
a
V
a
Z
RT V b RT TV V b V b RT V b
a V
b V RT b V
1RZH[SDQGLQJLQDSRZHUVHULHVLQ V
a V
a
Z b V b RT
RT
V
a
a
B b
B b
RT
RTB aV
b V RT b V
F
H
F I
H K
I
K
a a
Ÿ TB
. bR
bR
E 8VLQJWKH5HGOLFK.ZRQJSDUDPHWHUV
a TB
. bR
TB F I
H K
Solutions to Chemical and Engineering Thermodynamics, 5th ed
DOVR DYDLODEOH DV D 0DWKFDG
ZRUNVKHHW V
a
Z
V b RT V b
wZ
wV
V
wV
a
wV
w P T V b w P T V b w P T RT V b w P T
FG IJ
H K
P
FG IJ
H K
RT
a
V b
T V bV
FG w P IJ
H wV K
c
RT
a
V b
V b
T V bV
OLP
FG w P IJ
H wV K
d
T
Po
V of
OLP
FG w P IJ
H wV K
FG IJ
H K
h
P of
V of
FG IJ
H K
i
T
f
T
FG w Z IJ LM V a OPFG w V IJ
H w PK
NV b V b RT V b QH w P K
FG w Z IJ V b mV V b r ma RT V b r
H w P K m RT V b r {a T cV bV h } V b
F w Z IJ V b V V b b
OLP G
H w P K RT V b
RT
OLP
Po
V of
T
P of
V ob
T
FG w Z IJ V V a c RT V h
H w P K RT V a d TV i ˜ V
a d RT V i
aV
a
OLP
Po
V of
T
RT V T
RT V RT T
Solutions to Chemical and Engineering Thermodynamics, 5th ed
D 7KH5HGOLFK.ZRQJHTXDWLRQRIVWDWHLV
RT
a
V b
TV V b
P
ZKLFKZHUHZULWHDVIROORZV
PV
V
a
VRWKDW
RT V b RT V b
PV
V
a
b
a
DQG
RT
V b
RT V b V b RT V b
PV
V
a
V
b
V
RT
V b RT V b
F
H
I
K
PV I
/LP V F
H RT K /LP
Po
V
V of
F PV I /LP LMb V a V OP b a
H RT K
N V b RT V b Q RT
V of
BT
7RSURFHHGIXUWKHUZHQRZQHHGWRKDYHH[SUHVVLRQVIRUDDQGELQWHUPVRIWKHFULWLFDOSURSHUWLHV7R
REWDLQWKHVHZHSURFHHGDVLQ3UREOHPDQGUHZULWHWKLVLQWKHSRZHUVHULHVRIV
RT RT
ab
a
V V b V
b
P
P
TP
TP
1RWLFHWKDWWKHWKUHHURRWVRIYROXPHDWWKHFULWLFDOSRLQWDUHLGHQWLFDOVRZHFDQZULWH
I
K
F
H
bV V g
C
RU
V V C V V C V V C
$WFULWLFDOSRLQWWKHVHFRQGDQGIRXUWKHTXDWLRQVPXVWEHVDWLVILHGVLPXOWDQHRXVO\&RQVHTXHQWO\WKH
FRHIILFLHQWVRIHDFKSRZHURIVPXVWEHWKHVDPH7KXV
V C
RTC
PC
V C
b V C
a
b
TC PC
a
RTC
DQG
b
PC
TC PC
6ROYLQJWKHDERYHWKUHHHTXDWLRQVWRJHWKHUIRUa T b DQGVcZHJHW
VC
RTC
PC
b
d iV
a
C
R TC
PC
RTC
DQG
PC
Solutions to Chemical and Engineering Thermodynamics, 5th ed
a
RTC
RTC
RT PC
PC T )RUQSHQWDQHTC .DQGPC EDU
7KHUHVXOWLQJYLULDOFRHIILFLHQWDVDIXQFWLRQRIWHPSHUDWXUHLVVKRZQEHORZ
6R B T
b
% 7
7
Solutions to Chemical and Engineering Thermodynamics, 5th ed
FG w T IJ V TD Ÿ DT H w PK C
FwV I
D
G J EXW FGH ww VT IJK FGH ww VP IJK FGH ww TP IJK E\WULSOHSURGXFWUXOH
V Hw TK
,QYHUVLRQWHPSHUDWXUH P
P
P
H
3
T
V
FG w V IJ
H w T K aw P w V f aw T w Pf
P
T
V
YG:(26 P
aw P w T f aw P w V f
V
T
FG IJ
H K
R
V b
RT
a
wP
V b V
wT V
FG w P IJ RT a H wV K V b V
R V b
F w V IJ
G
H w T K n RT V b s a V FwV I
D
G J n RTkVR Vb V s b pa V V Hw TK
T
P
P
TD
RT
V V b
TR
RT V V b
RT V b a V n
s
RT
a
V b
V
LM OP a NV V b V b Q V
TR
V b V
V V b bRT
a
V V b V
FG
H
a
V
IJ
K
b
a V b a
bR V bR
V
RT
a
DOVR P
V b V
&KRRVHV
&DOFXODWH TLQY IURP(TQ &DOFXODWHPIURP(TQ 6ROXWLRQGRQHZLWK0$7+&$' VHH0$7+&$'ZRUNVKHHW T
Solutions to Chemical and Engineering Thermodynamics, 5th ed
E 5.(26
RT
a
V b
TV V b
P
FG w P IJ
H wV K
FG w P IJ
HwTK
D
T
V
RT
a
a
V b TV V b
TV V b a f
R
a
V b T V V b
FG IJ
H K
wV
V wT
P
a
a
l
a f
q
R V b a T V V b
V RT V b a TV V b a TV V b m
r n
TD RT
a
V V b T V V b
a f
f
f
w P wT V
V w P wV T
s n
s
RT
a
a
V b
TV V b
TV V b a
LM OP
OP
LM
Q TV V b NV V b V Q
N
RT LV b V O
RTb
a
LM a f V b V OP
V b MN V V b PQ V V b
TV V b N V V b
Q
RT V b V V b
V b
RTb
a
V V b
TV V b V V b
RTb
a V b
V b TV V b T a V b V b V V b Rb
RS a L b O V b UV
T Rb MN V PQ V b W
T
8VLQJD0DWKFDGSURJUDPIRUWKH6RDYH5HGOLFK.ZRQJ(26ZHILQG
H q & EDU u - PRO UHODWLYHWRLGHDOJDVDW.DQGEDU 1RZE\WULDODQGHUURUXQWLOHQWKDOSLHVPDWFK
H q & EDU u H q & EDU
u Ÿ T
R & &ORVHHQRXJK
1RWHWKDWWKLVVROXWLRQLVRQO\YHU\VOLJKWO\GLIIHUHQWIURPWKDWREWDLQHGZLWKWKH3HQJ5RELQVRQ
HTXDWLRQ R&FRPSDUHGWRR&REWDLQHGKHUH 2QO\DYDLODEOHDVD0DWKFDGZRUNVKHHW
$YDLODEOHDVD0DWKFDGZRUNVKHHW
Solutions to Chemical and Engineering Thermodynamics, 5th ed
PRO EDU 3D
˜
3URSHUW\'DWD
7LQ.3LQEDU 5( ˜
7F ˜ .
MRXOH
5* ˜
PRO˜ .
3F ˜ EDU RP EDU ˜ P
PRO˜ .
&S ˜ 5(
NDS RP
˜
RP
˜ ˜ RP
,QLWLDO&RQGLWLRQVDQGWRWDOYROXPH9W
7L ˜ .
3HQJ5RELQVRQ&RQVWDQWV ,QLWLDOWHPSHUDWXUH
E ˜
5*˜ 7F
7 7L
1RWHWKDWWKHVHDUHEHLQJGHILQHGDVD
IXQFWLRQRIWHPSHUDWXUHVLQFHZHZLOOQHHGWR
LQWHUDWHRQWHPSHUDWXUHODWHUWRREWDLQWKHILQDO
VWDWHRIWKHV\VWHP
9
5*˜ 7
*LYHQ
3L
,QLWLDOPRODUYROXPHDQG
QXPEHURIPROHV
9L
u '(/6L . MRXOH
7˜
'HO+L 5(˜ 7˜ =L =L G
D 7 D 7
G7
˜ ˜ E
DF ˜
3F
5* ˜ 7F
3F
D 7 DF ˜ DOI 7
'D 7 6ROYH35(26IRULQLWLDOYROXPH
'(/6L
9W ˜ P
7 ·º
ª
§
DOI 7 ˜ « NDS˜ ¨ ¸»
7F ¹ ¼
¬
©
)LQGLQLWLDOPRODUYROXPHDQGQXPEHURIPROHV
6WDUWZLWKLQLWLDOJXHVVIRUYROXPHPAPRO
(QWURS\GHSDUWXUHDWWKH
LQLWLDOFRQGLWLRQV
3L ˜ EDU
9 E
5*˜ 7L
9
3L
u D 7
9˜ 9 E E ˜ 9 E
P
9W
1
9L
G
D 7
G7
1
P
9L )LQG 9
'D 7
3L º
ª ª
ª 9L ˜ E º º ˜ 5(
˜ OQ«
« OQ« 9L E ˜ 5*˜ 7» »»
¼ ˜ ˜ E˜ 5* ¬ 9L ˜ E ¼ ¼
¬ ¬
3L˜ 9L
=L
5*˜ 7
ª =L ˜ E ˜ 3L º
«
5*˜ 7 »
˜ OQ«
»
« =L ˜ E˜ 3L »
5*˜ 7 ¼
¬
'HO+L
MRXOH
)LQDOSUHVVXUHLVEDUDQGILQDOWHPSHUDWXUHLVXQNQRZQZLOOEHIRXQGE\HTXDWLQJWKHLQLWLDODQGILQDO
HQWURSLHV*XHVVILQDOWHPSHUDWXUHLV.
3I ˜ EDU
7 ˜ .
9
5*˜ 7
3I
9
u P
1RWH7RXVHWKHJLYHQDQGILQGFRPPDQGIRUYDULDEOHVZLWKGLIIHUHQWGLPHQVLRQVVXFKDV7DQG9ZLOOKDYH
WRFRQYHUWWRGLPHQVLRQOHVVYDULDEOHVVRDVQRWWRJHWDXQLWVFRQIOLFW'HILQH[ 9E\ 77F
LQLWLDOJXHVV
[ \ Solutions to Chemical and Engineering Thermodynamics, 5th ed
LQLWLDOJXHVV
*LYHQ
[ \ 5*˜ \ ˜ 7F
3I
D \ ˜ 7F
[˜ E E
[˜ E ˜ [˜ E E E ˜ [˜ E E
3I º 'D \ ˜ 7F ˜ 5( ª [˜ E ˜ E º º
§ 3I · ª
ª
˜ OQ«
¸ 5(˜ OQ¨ ¸ « 5(˜ OQ« [˜ E E ˜
»
» » '(/6
5*˜ \ ˜ 7F¼
© 7L ¹
© 3L ¹ ¬
¬
˜ ˜ E ˜ 5*
¬ [˜ E ˜ E ¼ ¼
&S˜ OQ§¨
\ ˜ 7F ·
< )LQG [ \
9 < ˜E
§ ·
¨
¸
© ¹
<
9
u =I 3I ˜ 9
5*˜ 7
7I ˜
'HO+I 5(˜ 7I ˜ =I 'HO+I
P
=I
7I < ˜ 7F
7I
.
G
D 7I D 7I
G7I
˜ ˜ E
ª =I ˜ E˜ 3I º
«
5*˜ 7I »
˜ OQ«
»
« =I ˜ E ˜ 3I »
5*˜ 7I ¼
¬
MRXOH
8I8L +I=I 5 7I +L=L 5 7L +I+L=I 5 7I=L 5 7L
+I+I,* +I,* +L+L,* +L,*=I 5 7I=L 5 7L
'HO+I'HO+L&S 7I7L =I 5 7I=L 5 7L
'HO8 > 'HO+I 'HO+L &S˜ 7I 7L =I ˜ 5(˜ 7I =L˜ 5(˜ 7L@ ˜ 1
'HO8
u MRXOH
717HT 'HO8
˜
MRXOH
NJ
717HT
NJ
)LQDOWHPSHUDWXUH
Solutions to Chemical and Engineering Thermodynamics, 5th ed
:HVWDUWIURP
wP
dV d S C9dT wT V
6LQFHWKHHQWURS\DW.LVQRWDIXQFWLRQRIWHPSHUDWXUHLWIROORZVWKDWC9 $OVRVLQFHWKH
HQWURS\LVQRWDIXQFWLRQRIVSHFLILFYROXPHLWIROORZVWKDW
wP
wT V
F I
H K
F I
H K
+RZHYHU E\WKHWULSOHSURGXFWUXOH
F wP I F wV I FG wT IJ RU
H wT K H wP K H wV K
F wP I FG wP IJ F wV I H wT K H wV K H wT K
V
T
V
P
P
T
EXWIURPWKHWKHUPRG\QDPLFVWDELOLW\FRQGLWLRQ
FG wP IJ H wV K
T
ZKLFKLPSOLHVWKDW
F wV I
H wT K
F I
H K
wV
V wT
DQGD
P
P
5HZULWHWKH&ODXVLXVHTXDWLRQDV
RT
R
§ wV ·
V
b 7KHQ ¨
¸
P
T
T
w
©
¹P
§ wV ·
§ wV ·
ZKLFKPHDQVC3 LVLQGHSHQGHQWRISUHVVXUHDQGHTXDOWRC3 DQGV T ¨
¸ b
¨¨ wT ¸¸
© wT ¹ P
©
¹P
D 7KHUHIRUH
'H
z
T
H T P H T P
z
P
T
E
RU
C3 T dT bdP
P
J
T T T T b P P
LVWKHOLQHRIFRQVWDQWHQWKDOS\
E
D T T 'S
S T P S T P
z FH IK
z
T
P
wV
C3 T
dT wT
T
T
P
z
z
T
P
R
C3 T
dP
dT P
T
T
P
RU
T
P
J
D OQ E T T T T R OQ T
P
dP
P
LVWKHOLQHRIFRQVWDQWHQWURS\
G )RUWKHIOXLGWRKDYHD-RXOH7KRPVRQLQYHUVLRQWHPSHUDWXUH
Solutions to Chemical and Engineering Thermodynamics, 5th ed
F wT I PXVWXQGHUJRDVLJQFKDQJH+RZHYHU
H wP K
LMV T F wV I OP RT b T R
wT
N H wT K Q P
P b
P F I
H wP K C
C
C
P
H
P
3
H
3
3
7KLVLVDOZD\VQHJDWLYHVRWKH&ODXVLXVGRHVQRWKDYHD-RXOH7KRPVRQLQYHUVLRQWHPSHUDWXUH
57
57
5
57
§ w9 ·
§ w9 · 57
E ¨
E
3
DQG&3 D E 7 J7 VR9
DQG97 ¨
E
¸
¸
9E
3
3
© w7 ¹3 3
© w7 ¹3 3
3 7
'+
3 7
3 7
ª
ª
§ w9 · º
§ w9 · º
&3 G7 «9 7 ¨
«9 7 ¨
¸ » G3 ¸ » G3
w7 ¹3 ¼
w7 ¹3 ¼
©
©
¬
¬
3 7
3 7
3 7
³
³
E 3 D 7 7 ³
E J
7 7 7 7 E 3 IRUFRQVWDQWHQWKDOS\
E
J
Ÿ E3 D7 7 7 IRUOLQHRIFRQVWDQWHQWKDOS\
3 7
'6
3 7
3 7
&3
§ w9 ·
§ w9 ·
G7 ¨
¸ G3 ¨
¸ G3
7
© w7 ¹3
© w7 ¹3
3 7
3 7
3 7
³
³
³
IRUFRQVWDQWHQWURS\
§3 ·
§7 ·
J
5OQ ¨ ¸ DOQ ¨ ¸ E 7 7 7 7
© 3 ¹
© 7 ¹
J
Ÿ E OQ 3 D OQ 7 E7 7 IRUOLQHRIFRQVWDQWHQWURS\
8VLQJWKH35(26SURJUDPIRUWKHLQLWLDOFRQGLWLRQVRIEDUDQGR&ZHKDYHIRUWKH
OLTXLGWKDW
9/ ¯PPRO+/ -PRODQG6/ - PRO. $WWKLVWHPSHUDWXUH
VLQFHWKHSURFHVVLVLVRWKHUPDO WKH9/(FRQGLWLRQVDUH3YDS EDUDQG
9/ ¯PPRO+/ -PRODQG6/ - PRO. DQG
99 ¯PPRO+9 -PRODQG69 - PRO. D 7KHUHIRUHWKHYROXPHFKDQJHLV
'9 ¯¯¯¯¯¯ ¯ P
E 7KHHQWURS\EDODQFHLV
G6 4
6 JHQ QRZVLQFH7LVFRQVWDQWDQGWKHIOXLGLVXQGHUJRLQJDXQLIRUPH[SDQVLRQ6 JHQ GW 7
N
VRWKDW'6 u u u u DQG
.
4 7'6 u N
)URP(T
9
§ w3 ·
&9 9 7 &9 9 f 7 7 ³ ¨ ¸ G9 9 f 7 © w7 ¹ 9
Solutions to Chemical and Engineering Thermodynamics, 5th ed
3
§ w3 ·
¨ w7 ¸
©
¹9
&9 9 7
57
D7
9E 9 9E E 9E
§ w3 ·
5
GD 7 G7
G D 7 G7 ¨ ¸ 9E 9 9E E 9E
© w7 ¹9 9 9 E E 9 E
&9 7
9
G9
GD 7
G7 9³f 9 9 E E 9 E
EXW
G9
³ 9 9E E 9E
ª 9 E º
OQ «
»
E ¬« 9 E »¼
VRWKDW
&9 9 7
&9 7
)LQDOO\
ª 9 E º
GD 7
OQ «
»
G7 E «¬ 9 E »¼
5 7&
D 7 ZKHUHD 7
3&
D 7
VRWKDW
GD 7
G7
ª
§
7 ·º ª
º
˜ « N ¨ » DQG
¸¸ » ˜ « N ˜ ˜
¨
7& ¹ ¼» ¬«
7 ˜ 7& ¼»
©
¬«
ª
º
ª
§
7 ·º « N
» § N ·
˜ « N ¨ ¸» ˜ « ˜
¸
» ¨¨
¨
7& ¸¹ ¼» « ˜ 7 ˜ 7& ¸¹
©
¬«
7 » ©
7
& ¼
¬
GD 7
G7
ª
§
7 ·º
« N ¨¨ ¸»
7& ¸¹ ¼»
©
¬«
­
½
ª
º
§
5 7& °° ª
7 ·º « N
» § N · °°
˜ ® ˜ « N ¨ ¸» ˜ « ˜
¸¸ ¾
» ¨¨
¨
3& ° «¬
7& ¸¹ »¼ « G7 ©
» © ˜ 7 ˜ 7& ¹°
°¯
°¿
¬ 7 7& ¼
9 7 3 D E7 &3DQG8 73 G7 H3
GD 7
§ w9 ·
¨
¸
© w7 ¹ 3
§ w6 · § w8 ·
§ w6 ·
¨ ¸
¨
H Ÿ H E7
¸ ¨
¸
7
© w3 ¹7
© w8 ¹7 © w3 ¹7
%XWLWZDVFODLPHGWKDWDEFGDQGHDUHFRQVWDQWVZKLFKLVQRWWKHFDVHJLYHQWKH
LQWHUUHODWLRQVKLSEHWZHHQEDQGHJLYHQDERYH
E
D 5HZULWLQJWKHHTXDWLRQRIVWDWHDV
9
E
57 D3 § w9 ·
¨
¸
3
7 © w7 ¹3
§ w9 ·
5 D3
DQG ¨¨ ¸¸
3 7
© w7 ¹3
D3
3
&3 3 7
§ w9 ·
&3 3 7 7 ¨¨ ¸¸ G3
w7 ¹ 3
3 ©
³
7
3
&3 3 7 7
D3
³ 7 G3 & 3 7 7
3
3
D
3 3
DQG
&3 EDU7
&3 EDU7 EDU
P .
u
u EDUPRO
7 .
EDUP
7
PRO.
Solutions to Chemical and Engineering Thermodynamics, 5th ed
&3 EDU7 .
7KHUHIRUH &3 EDU7 .
PRO.
PRO.
6RWKDWWKHPHDQYDOXHRYHUWKHWHPSHUDWXUHUDQJHLV
EDU.
E
57 D3 § w9 ·
¨
¸
3
7 © w7 ¹3
EDU.
'+
EDU.
.
³
DQG
9
³
³
5 D3
§ w9 ·
DQG9 7 ¨
¸
3 7
© w7 ¹3
.
E
57 D3 57 D3
3
7
3
7
³
³
EDU.
.
D3
7 ³
EDU.
EDU.
E
EDU.
D3 º
D3 º
ª
ª
G7 « E 7 »G3 « E 7 »G3
¬
¼
¬
¼
EDU.
.
EDU.
§
§
D3 ·
D3 ·
¨¨ E3 u ¨¨ E3 ¸¸
¸
7 ¹
7 ¸¹
©
©
EDU.
EDU.
'6
PRO.
ª
ª
§ w9 · º
§ w9 · º
&3 G7 « 97 ¨
« 97 ¨
¸ »G3 ¸ »G3 © w7 ¹3 ¼
© w7 ¹3 ¼
EDU. ¬
.
EDU. ¬
E '+
PRO
EDU.
&3
§ w9 ·
§ w9 ·
G7 ¨
¸ G3 ¨
¸ G3
7
© w7 ¹3
© w7 ¹3
EDU.
.
EDU.
³
³
³
º
º
ª
ª
D u »
«
D u »
«
OQ 5 OQ u
5 OQ u
«
»
»
«
PRO.
7
7
¬
¼
¬
¼
Solutions to Chemical and Engineering Thermodynamics, 5th ed
D
]] 7RFDOFXODWHSRLQWVDWJHRPHWULFDOO\LQFUHDVLQJYDOXHVRIWKHPRODUYROXPH
DWWHQSRLQWVSHUGHFDGH
L L
XQLWVPAP
9 ]]
˜
L
YDQGHU:DDOVSDUDPHWHUV
D XQLWV
5 3D˜ P
XQLWV
E ˜ PRO
3 L
3 L
5˜ 9 E
D
9
L
5˜ 9 E
L
3 L
5˜ 9 E
L
D
9
L
3 L
XQLWV
3D˜ P
PRO˜ .
P
PRO
5˜ 9 E
L
D
9
L
D
L
9
L
3L 3L
3L
3L
ORJ 9L
Solutions to Chemical and Engineering Thermodynamics, 5th ed
E
7 5˜ 7
39 9 E
*LYHQ
D
5˜ 7
9LQLW,*
9
39
,GHDOJDVJXHVV
IRULQLWLDOYROXPH
9 9LQLW,*
9LQLW )LQG 9
9LQLW
,GHDOJDVJXHVV
IRULQLWLDOYROXPH
5˜ 7
9ILQ,*
9LQLW,* 9 9ILQ,*
*LYHQ
39
9ILQ )LQG 9
´
: µ
¶
9ILQ
u 9ILQ
3 9 G9
:
u 9ILQ,* u -
9LQLW
F
)RUWKHLGHDOJDV
´
:,* µ
µ
¶
9ILQ,*
9LQLW,*
5˜ 7
9
G9 :,*
u -
$OPRVWLGHQWLFDOWRWKHYG:UHVXOW
1RWHWKDWLIRQHZDQWHGWRGRWKHLQWHJUDOVDQDO\WLFDOO\LQ0DWKFDGLQVWHDGRIQXPHULFDOO\
WKHIROORZLQJZRXOGEHREWDLQHG
´
µ
µ
¶
57
9
´
µ
µ
µ
¶
´
µ
µ
µ
¶
G9 o 57˜ OQ 9
´
µ
µ
¶
YI
YL
57
9
G9 o OQ YI ˜ 57 OQ YL ˜ 57
DD
DD ·
§ 57
¨ 9 EE ¸ G9 o OQ 9 EE ˜ 57 9
9 ¹
©
YI
YL
OQ YI EE ˜ 57˜ YI DD
DD ·
OQ YL EE ˜ 57˜ YL DD
§ 57
¨ 9 EE ¸ G9 o
YI
YL
9 ¹
©
Solutions to Chemical and Engineering Thermodynamics, 5th ed
)RUWKHJDVLQWKHSXPSWKHSURFHVVLVDGLDEDWLFDQGUHYHUVLEOH XQLIRUPFRPSUHVVLRQ 7KHUHIRUHWKHSURFHVVLVLVHQWURSLF ZKHWKHURUQRWLWLVDQLGHDOJDV D )RUWKHLGHDOJDVFDVH
7
3
7I
Ÿ 7 .
' 6 &3 OQ I 5 OQ I OQ
OQ
7L
3L
39
1RZVLQFHIURPWKHLGHDOJDVODZ L L
7L
3I 9I
9
Ÿ I
7I
9L
3L 7I
u
7L 3I
7KHUHIRUH WKHSXPSKDQGOHZLOOEHSXVKHGGRZQ u u u FP
E 8VLQJWKH35(26SURJUDP
$W7 .DQG3 EDU9 ¯PPRODQG6 - PRO. 1RZDW3 EDUJXHVV7XQWLORQHREWDLQV6 - PRO. FDQXVHLGHDOJDVWHPSHUDWXUHDV
LQLWLDOJXHVV )LQGWKDW7 .IRUZKLFK9 ¯PPRO
§ u ·
7KHUHIRUH WKHSXPSKDQGOHZLOOEHSXVKHGGRZQ u ¨¨ ¸ FP u ¸¹
©
D %\WKHWULSOHSURGXFWUXOH
§ wT · § wU · § wV ·
¨
¸ ¨
¸ ¨
¸
© wV ¹U © wT ¹V © wU ¹T
§ wT ·
¨
¸
© wV ¹U
§ wT · § wU ·
¨
¸ ¨
¸ © wU ¹V © wV ¹T
DQGdU
Td SHJ PdV
§ wU ·
7KHUHIRUH ¨
¸
© wV ¹T
§ wU ·
¨
¸
© wV ¹T
§ wS ·
§ w S · § wP ·
T¨
¸ P DQGXVLQJ ¨
¸ ¨
¸ JLYHV
w
V
©
¹T
© wV ¹T © wT ¹V
§ wT ·
§ wP ·
T¨
¸ P DQG ¨
¸
w
T
©
¹V
© wU ¹V
C9
ª § wP ·
º
«T ¨
¸ P»
« © wT ¹V
¼»
VRWKDW[ ¬
C9
E ,GHDOJDV PV
§ wP ·
RT VRWKDW ¨
¸
© wT ¹V
R
V
P
VR[ T
F
P
RT B T RT § wP ·
˜
¨
¸
V
V
V
© wT ¹V
§ wP ·
T¨
¸ P
© wT ¹V
R B T R RT dB T
˜ ˜
V
V V V
dT
RT dB T
RT dB T
˜
˜
DQG
[
dT
V
V ˜ C9 dT
Solutions to Chemical and Engineering Thermodynamics, 5th ed
G
RT
aT
V b V V b b V b
P
da T
R
˜
V b V V b b V b
dT
§ wP ·
¨
¸
© wT ¹V
da T
a T
dT
DQG
V V b b V b
T
§ wP ·
T¨
¸ P
© wT ¹V
da T
dT
˜
V V b b V b ˜ C9
a T T
[
ZKHUHC9LVDIXQFWLRQRIWHPSHUDWXUHDQGVSHFLILFYROXPH
5HDFWRULV,LQLWLDOO\HYDFXDWHGWDQNLV,,7KHPDVVEDODQFHDVLQ3UREOHPLV Ni N I N II H[FHSWWKDWKHUHXQOLNHWKHUHKHUHZHXVHWKHYLULDOJDVHTXDWLRQRIVWDWH
7KHHQHUJ\EDODQFHRIWKHUHDFWRUDQGWDQNLV NiU i N I U I N II U II 7KHHQWURS\EDODQFHRQMXVWWKH
FRQWHQWVRIWKHUHDFWRUWKDWXQGHUJRDXQLIRUPH[SDQVLRQOHDGVWR
S I FRQVWDQW 7KHHTXDWLRQRIVWDWHLV
39
57
'6
%7
ZLWK% 7
9
u 9 f 7L
7I
L
L
P
& 3
7
PRO
PRO.
9I 7I
&9
§ w3 ·
§ w3 ·
G7 ¨
¸ G9 ¨
¸ G9
w
w7 ¹9
7
7
©
¹
©
9
9 7
7
9 f 7
³
³
L
³
I
5OQ
ª 9I
7
º
u 5 « » OQ I
9L
7L
¬ 9I 9L ¼
VLQFH3
57 57% 7
9
9
§ w3 ·
¨
¸
© w7 ¹9
5 5 u u 9
9
57 57 u u 5 u 9
9
9
$OVRIRUODWHUUHIHUHQFH
9 f 7UHI
97
7
ª § w3 ·
º
ª § w3 ·
º
'8
«7 ¨
«7 ¨
¸ 3 » G9 &9 G7 ¸ 3 » G9
»¼
»¼
¬ © w7 ¹9
¬ © w7 ¹9
9UHI 7UHI «
7UHI
9 f 7 «
³
³
§ w3 ·
DQG7 ¨
¸ 3
© w7 ¹ 9
³
57 57 u u 57 57 u u 5 u 9
9
9
9
9
§ w3 ·
¨
¸
© w7 ¹ 9
VRWKDW
9 f 7UHI
8
³
9UHI 7UHI
5 u 9
97
7
G9 ³ & G7 ³
9
7UHI
5HVWRIWKHSUREOHPVROXWLRQLVLQ
0$7+&$'
9 f 7
5 u 9
G9
5 u 9
Solutions to Chemical and Engineering Thermodynamics, 5th ed
5 5* ˜ % 7 ˜ 7UHI 3UHI &Y 5
7
&DOFXODWHUHIHUHQFHYROXPH IRUODWHU
5*˜ 7UHI
9UHI 9UHI
3UHI
*LYHQ
3UHI˜ 9UHI
5*˜ 7UHI
% 7UHI
9UHI
9UHI )LQG 9UHI
3LQLW 9UHI
7LQLW 5*˜ 7LQLW
9LJ 9LJ
3LQLW
u 9LQLW 9LJ
*LYHQ
3LQLW˜ 9LQLW
5*˜ 7LQLW
% 7LQLW
9LQLW
9LQLW )LQG 9LQLW
1LQLW 9LQLW
1LQLW
9LQLW
&DOFXODWHGLQLWLDOPRODUYROXPH
u u &DOFXODWHLQLWLDOPRODULQWHUQDOHQHUJ\
8LQLW 5˜ ˜ ¨§
© 9UHI
·
¸ &Y˜ 7LQLW 7UHI
9LQLW ¹
8LQLW
u Solutions to Chemical and Engineering Thermodynamics, 5th ed
*XHVVILQDOSUHVVXUHDQGWHPSHUDWXUHLQWDQNXVHLGHDOJDVUHVXOWDVILUVWJXHVV7KH
UHVXOWEHORZLVWKHUHVXOWRIVHYHUDOLWHUDWLRQV
3I 7I 3I˜ 9I
% 7I
9I 5*˜
7I
9I u 3I
)LQGILQDOWHPSHUDWXUHDQGPRODUYROXPHLQUHDFWRUIRUWKLVILQDOSUHVVXUHJXHVV
*LYHQ
5*˜ 7I
9I
·
9I ·
§ § 7I ·
5 ˜ OQ§¨
¸ ˜ ˜ 5˜ ¨ 9I 9LQLW ¸ 5 ˜ OQ¨ 7LQLW ¸
© 9LQLW ¹
©
¹
©
¹
[ ILQG 7I 9I
§ ·
¨
¸
© u ¹
[
7I [
9I [
1I 1I
9I
&DOFXODWHILQDOPRODULQWHUQDOHQHUJ\RIJDVLQUHDFWRU
8I 5˜ ˜ ¨§
© 9UHI
·
¸ &Y˜ 7I 7UHI
9I ¹
8I
u &DOFXODWHILQDOQXPEHURIPROHVLQWDQNDQGPRODUYROXPHEDVHGRQJXHVV3I
1I 1LQLW 1I
1I
9I u 1I
9I
u 1RZXVHWKLVWRFDOFXODWHWHPSHUDWXUHLQWDQN
7I 7LQLW
*LYHQ
3I ˜ 9I
5*˜ 7I
% 7I
9I
7I ILQG 7I
7I
&DOFXODWHILQDOPRODULQWHUQDOHQHUJ\RIJDVLQWDQN
8I 5˜ ˜ ¨§
© 9UHI
·
¸ &Y˜ 7I 7UHI
9I ¹
8I
u ,VHQHUJ\EDODQFHVDWLVILHG"
'HO 1LQLW˜ 8LQLW 1I˜ 8I 1I˜ 8I
'HO
u 5HVXOWV
3ILVGHOLV
3ILVGHOLV
3ILVGHOLV
3ILVGHOLV
3ILVGHOLV
7I
7I
ZKXFKXVFORVHHQRXJK
1I
1I
u &RPSDUHZLWKLGHDOJDVUHVXOW
7I DQG7I 6RWKHUHVXOWVDUHQRWGUDVWLFDOO\GLIIHUHQWWKDQWKHLGHDOJDVUHVXOW
1RWHWKDWLQWKHQH[WSUREOHPZHVROYHWKLVXVLQJWKH35(26,QWKDWFDVHWKH
ILQDOSUHVVXUHLVEDU7I DQG7I ZKLFKDUHFORVHWRWKHUHVXOWV
REWDLQHGKHUH H[FHSWIRUWKHSUHVVXUH Solutions to Chemical and Engineering Thermodynamics, 5th ed
5HDFWRULV,LQLWLDOO\HYDFXDWHGWDQNLV,,7KHPDVVEDODQFHDVLQ3UREOHPLV Ni N I N II H[FHSWWKDWKHUHXQOLNHWKHUHZHFDQQRWXVHWKHLGHDOJDVHTXDWLRQRIVWDWH
7KHHQHUJ\EDODQFHRIWKHUHDFWRUDQGWDQNLV NiU i N I U I N II U II 7KHHQWURS\EDODQFHRQMXVWWKHFRQWHQWVRIWKHUHDFWRUWKDWXQGHUJRDXQLIRUPH[SDQVLRQOHDGVWR
S I FRQVWDQW 8VLQJWKH3HQJ5RELQVRQSURJUDPZLWKKHDWFDSDFLW\WKDWLVDIXQFWLRQRI
WHPSHUDWXUHZHREWDLQWKDW
P
V i ˜
VRWKDWNi PRO
PRO
H i VRWKDWU i H i PV
˜ ˜ ˜ i i
PRO
PRO
S i PRO ˜ .
7KHFDOFXODWLRQDOSURFHGXUHLVDVIROORZV*XHVVDILQDOSUHVVXUHILQGWKHWHPSHUDWXUHLQWKHUHDFWRU
IRUZKLFKWKHHQWURS\LVHTXDOWRSi8VLQJWKDWTDQGPFDOFXODWH XVLQJWKH35(26 WKHPRODU
YROXPHDQGPRODUHQWKDOS\IURPZKLFKWKHQXPEHURIPROHVLQWKHUHDFWRUDQGLWVLQWHUQDOHQHUJ\
FDQEHFRPSXWHG7KHXVHWKHPDVVEDODQFHWRFRPSXWHWKHQXPEHURIPROHVLQWDQN,,WKHPRODU
YROXPHLQWDQN,,DQGWKHNQRZQSUHVVXUHWRFDOFXODWHWKHWHPSHUDWXUHLQWDQN,,7KHQVHHLIWKH
HQHUJ\EDODQFHLVVDWLVILHG,IQRWUHSHDWWKHFDOFXODWLRQZLWKDQRWKHUJXHVVHGYDOXHIRUWKHILQDO
SUHVVXUH
6WDUWZLWKWKHLGHDOJDVVROXWLRQRISUREOHPRIP EDU)LQG
IRU S i WKHWHPSHUDWXUHLVT I .
PRO ˜ .
P
V I ˜
VRWKDWN I PRON II PRO
PRO
II
P
DQGV
˜
PRO
I
I
I
H
H P I V I ˜ ˜ ˜ VRWKDWU
PRO
PRO
%\WULDODQGHUURUZHILQGWKDWWKHFRQGLWLRQVLQWKHWDQN,,DUH
P
IRUV II ˜
WKHWHPSHUDWXUHLVT II .
PRO
H II VRWKDWU II H II P II V II ˜ ˜ ˜ PRO
PRO
DQGWKHHQHUJ\EDODQFHLV
"
˜ ˜ ˜ ˜ 6RWKDWWKHHQHUJ\EDODQFHLV127VDWLVILHG
7U\DQRWKHUJXHVV«P I EDU
IRU S i WKHWHPSHUDWXUHLVT I .
PRO ˜ .
VI
˜
P
VRWKDWN I
PRO
PRON II
˜ PRO
P
DQGV
˜
PRO
H I VRWKDWU I H I P I V I ˜ ˜ ˜
PRO
%\WULDODQGHUURUZHILQGWKDWWKHFRQGLWLRQVLQWKHWDQN,,DUH
II
PRO
Solutions to Chemical and Engineering Thermodynamics, 5th ed
IRUV II
˜
P
WKHWHPSHUDWXUHLVT II
PRO
H
VRWKDWU II
PRO
DQGWKHHQHUJ\EDODQFHLV
II
H
II
P V
II
II
.
˜ ˜ ˜
PRO
"
˜ ˜ ˜ ˜ ˜ :KLFKLVFORVHHQRXJKVRWKDWZHFDQFRQVLGHUWKHHQHUJ\EDODQFHWREHVDWLVILHG7KHILQDO
FRQGLWLRQVWKHQDUHDSUHVVXUHRIEDUTIRI.DQGTIIRI.
7KLVLVWREHFRPSDUHGZLWKWKHLGHDOJDVFRQVWDQWKHDWFDSDFLW\VROXWLRQRIDSUHVVXUHRIEDU
TIRI.DQGTIIRI.
7RFRPSDUHEHWWHUZLWKWKHSUHYLRXVLGHDOJDVVROXWLRQZHFRXOGUHSHDWWKHFDOFXODWLRQZLWKWKH
FRQVWDQWLGHDOJDVKHDWFDSDFLW\RI-PRO'RLQJWKLVZHILQGWKDWWKHILQDOSUHVVXUHLVDJDLQ
DERXWEDUTI .DQGTII .
§ w9 ·
¨
¸
© w7 ¹6
§ w6 ·
¨ ¸ © w3 ¹9
HTQ DQGHTQ
&9
§ w9 ·
§ w3 ·
N7 ¨
G6
G7 ¨
¸ G9
¸
w
7
7
9
©
¹9
© w3 ¹7
& 9 § w7 ·
§ w6 ·
¨ w3 ¸
7 ¨© w3 ¸¹9
© ¹9
Ÿ
w 9 w7 6
& § w7 · § w7 ·
9¨
¨
¸ 7 © w3 ¸¹ 9 © w 9 ¹3
w 9 w7 3
7ULSOHSURGXFW5XOH
§ w7 · § w 9 · § w3 ·
¨
¸ ¨
¸
¸ ¨
© w 9 ¹3 © w3 ¹7 © w7 ¹9
§ w7 ·
§ w3 · § w7 ·
¨
6RWKDW ¨
¸
¸ ¨
¸ w
9
©
¹3
© w 9 ¹7 © w3 ¹9
w 9 w7 6
& § w7 · § w3 ·
&9 § w7 ·
9¨
D
¨
¸
w 9 w7 3
7 © w3 ¸¹ 9 © w 9 ¹7
79N 7 ¨© w3 ¸¹9
§ w9 ·
E
&9 ! 6LQFH ¨
VR N 7 ! ¸ © w3 ¹7
Ÿ w 9 w7 6
DQG
w 9 w7 3 PXVWKDYHRSSRVLWHVLJQV &RQVLGHUWKHSURFHVVWREHFRQWLQXRXV QRWDEDWFKSLVWRQDQGF\OLQGHURSHUDWLRQ ,QWKLVFDVH
WKHHQHUJ\DQGHQWURS\EDODQFHVIRUWKHDGLDEDWLFSURFHVVDUH
G1
1
Ÿ 1
1
1
1
LQ
RXW
RXW
LQ
GW
G8
G6
1+
:
1+
16
16
6
LQ
RXW
LQ
RXW
JHQ
GW
GW
)RUPLQLPXPZRUN 6 JHQ VRWKDW6LQ 6RXW 8VLQJWKH35(26SURJUDPDWWKHLQLWLDOFRQGLWLRQVRI.DQGEDU
Solutions to Chemical and Engineering Thermodynamics, 5th ed
9/ ¯PPRO+/ -PRODQG6/ - PRO. 1RZDWEDUE\WULDODQGHUURU7 .DQG
9/ ¯PPRO+/ -PRODQG6/ - PRO. 7KHUHIRUH: +RXW±+LQ -PRO
Solutions to Chemical and Engineering Thermodynamics, 5th ed
D
)URP3HQJ5RELQVRQ(26
99 ˜
9/ ˜ 1L 9/
1I ˜ 1L
/ 1I
/ 9
+/ +9 1I
u JLYHQ
1I
= ILQG / 9
u 9 999
˜ 9/˜ /
/ =
1L
9 =
§ u ·
¨
¸
© ¹
=
)UDFWLRQYDSRUI
9
I
I
9 /
VRRIWKHSURSDQHUHPDLQLQJLQWKHWDQNLVYDSRU
E
0DVVEDODQFH
1RXW 1L 1I
1RXW
u (QHUJ\EDODQFH / +/9 +91RXW +91L +/4
4 1L˜ +/ /˜ +/ 9˜ +9 1RXW ˜ +9
4
u -
KHDWPXVWEHDGGHGWRYDSRUL]HWKHSURSDQH
,QWKLVSUREOHPPHWKDQHLVWRXQGHUJRDQLVHQWURSLFFRPSUHVVLRQIURPR&DQGVRPHXQNQRZQ
LQLWLDOSUHVVXUHWRR&DQG03D EDU7KHSUREOHPLVWRILQGWKHLQLWLDOSUHVVXUH8VLQJRQHRI
WKH35(26SURJUDPVDQGWKHLGHDOJDVKHDWFDSDFLW\IRUPHWKDQHLQWKH3523(57<SURJUDPZH
KDYHDWEDUDQGR& .WKDW
6 - PRO. DQG+ -PRO%\WULDODQGHUURUZHILQGWKDWDWEDUDQGR& .
6 - PRO. DQG+ -PRO7KHUHIRUHWKHORZHVWSUHVVXUHDOORZHGLVEDU$
ORZHUSUHVVXUHZRXOGUHVXOWLQWKHWHPSHUDWXUHIURPWKHRXWOHWWRWKHFRPSUHVVRUEHLQJDERYHWKH
VDIHW\OLPLW
7KHZRUNGRQHLQWKHFRPSUHVVRUIURPWKHHQHUJ\EDODQFHLV
G8
:
1+
+ RXW +LQ LQ 1+ RXW : RU GW
PRO
1
Solutions to Chemical and Engineering Thermodynamics, 5th ed
%\WKHPDVVDQGHQHUJ\EDODQFHVIRUD-RXOH7KRPVRQH[SDQVLRQZHKDYH
H LQ H RXW RU H EDU. H EDUT " 6LQFHWKHILQDOSUHVVXUHLVVRORZZHZLOODVVXPH
WKHH[LWLQJJDVLVLGHDO7KHUHIRUH H H ,*
H H ,*
EDU.
EDU.
,*
H EDU.
T
,*
,*
H EDU
T H EDU.
³
,*
H EDU
T T
C3 dT
.
³ T dT .
T ˜ T 1RZ
V T P
H TP H
DQG
PV
Z
RT
§ wP ·
¨ wT ¸
©
¹V
,*
TP
B T
V
RT Z VRWKDWZ ª § wP ·
º
P »dV
«T ¨
¸
»¼
¬ © wT ¹V
V f «
³
B T
V
P
RT RTB T
V
V
R RB T
RT dB T
DQG
V
dT
V
V
§ wP ·
T¨
¸ P
© wT ¹V
RT RTB T
RT dB T
RT RTB T
V
dT
V
V
V
V
H T P H ,* T P
RTB T
V
V T P
³
V f
RT dB
dV
V dT
RTB T
V
RT dB T
dT
V
RT dB § ·
¨ ¸
dT © V f ¹
RT ª
dB º
B T T
«
V ¬
dT »¼
$OVR
dB T
T
dT
T
dB T
˜ B T T
dT
T
T
T
B T
dB T º
˜ FP
P
ª
u « B T T dT »
PRO
PRO
¬
¼T IURP
B T
PV
RT
V
2QHREWDLQVDW.DQGEDU
P §
P ·
V u ¨ 1RWHIRUWKHLGHDOJDVLHB V u ¸
PRO ¨©
PRO ¸¹
H T P H ,* T P
RT ª
dB º
B T T
V «¬
dT »¼
Solutions to Chemical and Engineering Thermodynamics, 5th ed
6RWKDWWKHHTXDWLRQWREHVROYHGLV ˜ T T 7KHVROXWLRQWRWKLVHTXDWLRQLVT .VRWKDWWKHWHPSHUDWXUHFKDQJHLV.
3 7 5˜
7
9
5 ˜ 7˜
%7
&Y 7 ˜ 7 9
5
G
37 o
5˜
9
G7
5 ˜ %7
G
%7
´
µ G 3 7 G9 o 5˜ OQ 9 5˜ % 7 5˜ 7˜ G7
µ G7
9
9
¶
%7 § · ˜ ¨
¸
7 ¹
©
'% 7 % 7 7˜
G
%7
G7
G
%7 o
G7
'% 5 ˜ 7˜
G
%7
G7
9
9
˜ 7
7KLVFRPELQDWLRQLVDFRQVWDQW
u % u 6ROYLQJIRUYROXPHV
573 5˜
º
ª
573 573 ˜ 573˜ % ¬
¼
9 9
u º
ª
573 573 ˜ 573˜ % ¬
¼
9 9
573 5˜
8VLQJ(T D
9 ·
¸
© 9 ¹
7HUP ˜ OQ§¨
7HUP 7HUP '% ˜ 9
'% ˜ 9
´
7HUP µ
µ
¶
&Y 7
7
7HUP
7HUP
7HUP
u 7HUP
G7
'HO6 7HUP 7HUP 7HUP 7HUP
'HO6
PRO˜ .
6HHDOVR3UREVDQGLQWKHIROGHU$VSHQIRU7H[WERRN!&KDSWHU!3UREOHPV
Solutions to Chemical and Engineering Thermodynamics, 5th ed
(QHUJ\DQGHQWURS\EDODQFHVDUH
G8
G9 G6 :DQG
6JHQ VLQFHV\VWHPLVFORVHGDQGDGLDEDWLF
3
GW
GW
GW
VLQFHSURFHVVLVGRQHVORZO\ZHFDQDVVXPHLWLVGRQHUHYHUVLEO\ 6JHQ 7KHUHIRUHWKHEDODQFHVEHFRPH
'8 :DQG6 FRQVWDQW ,QLWLDOFRQGLWLRQLV7 .DWZKLFKWKHSUHVVXUHZKLFKHTXDOVWKHYDSRUSUHVVXUHLV3 EDU8VLQJWKH35(26SURJUDPJLYHQWKDWRQO\OLTXLGLVSUHVHQWZHKDYHWKDW9/ ¯
PPRO+/ -PRODQG
6/ - PRO. 7KHSURFHGXUHQRZLVWRXVHWKH35(26SURJUDPJXHVVDWHPSHUDWXUH
FRPSXWHWKHYDSRUOLTXLGHTXLOLEULXPSURSHUWLHVDWWKDWWHPSHUDWXUHDQGVHHLIWKHHQWURS\DW
YDSRUDQGOLTXLGHTXDOVWKHHQWURS\RIWKHLQLWLDOVWDWH:HILQGWKDWDW7 .3YDS EDU
9/ ¯PPRO+/ -PRODQG6/ - PRO. DQG
99 ¯PPRO+9 -PRODQG69 - PRO. DQG6/9 696/ - PRO. 1RZWRFRPSXWHWKHZRUNZHQHHGWKHLQWHUQDOHQHUJLHVZKLFKDUHREWDLQHGIURP8 +±39IURP
/
/
9
8 ILQDO
DQG8 ILQDO
ZKLFKZHREWDLQ 8LQLW
DQG
PRO
J
J
9
/
/
'8 : 8 ILQ
8 ILQ
8 LQLW
u
u
- J
PRO J
PRO
PRO
3UHOLPLQDULHV
3UHVVXUH RXWZDUGIRUFHSHUXQLWDUHDH[HUWHGE\JDV
)RUFH WHQVLOHIRUFHH[HUWHGRQILEHU²DWPHFKDQLFDOHTXLOLEULXPILEHUH[HUWVDQHTXDODQGRSSRVLWH
LQZDUGIRUFH
Ÿ,QDOOWKHUPRG\QDPLFUHODWLRQVUHSODFH PE\ F A DQG VE\ LADQGWKH\ZLOOEHDSSOLFDEOHWR
ILEHU
wS
wS
dT dV :HZLOOXVH S S T L ,QSDUWLFXODULQSODFHRI S S T V DQG dS
wT V
wV T
a
DQG dS
FG IJ
H K
f
FG w SIJ dT FG w SIJ dL $OVR
H w T K H w LK
FG w SIJ C Ÿ FG w SIJ
Hw TK T Hw TK
FG w SIJ FG w PIJ Ÿ FG w SIJ
H w T K H w T K H w LK
L
V
T
)URP WKH DERYH dS
V
L
CL
T
FG w FIJ Hw TK
du
T
F
dL
T
T
P
du dV Ÿ dS
T
T
dS
a f
T
v
FG IJ
H K
L
FG w SIJ dT FG w SIJ dL C dT FG w SIJ dL DQG WKH DQDORJ RI WKH 0D[ZHOO
H w T K H w LK
H w LK
T
L
L
UHODWLRQ
T
FG w SIJ FG w PIJ Ÿ FG w SIJ
H w V K H w T K A H w LK
T
V
T
T
FG IJ
H K
wF
A wT L
Solutions to Chemical and Engineering Thermodynamics, 5th ed
ZHJHW
dS
FG IJ
H K
wF
CL
dT dL
wT L
T
a
)RUDUHYHUVLEOHSURFHVV
CL
dT
§ wF ·
dT ¨
¸ dL RU
T
dL
© wT ¹ L
T § wF ·
¨
¸ CL © wT ¹ L dS
dU
TdS PdV Ÿ dU TdS FdL CLdT JT L L dL JT L L dL
dS
a
a
f
f
CLdT 1RWH7KLVLVDQDORJRILGHDOJDVH[SUHVVLRQ U
CL
dT J L L dL T
U T RU dU
f
Ÿ SaL T f SaL T f
z RSTCT dT J aL L fdLUVW &KRRVLQJWKHSDWK aL T f o aL T f o aL T f \LHOGV
L T
L
L T
S L T S L T
T L
z
f
T L
D ET
T
za
T L
dT J
T L
f
L L dL
T
J
D OQ E T T L L T
a
f a
f
f
f
i
i
i
f
f
f
i
i
1HHGWRVROYHWKLVWUDQVFHQGHQWDOHTXDWLRQWRILQG Tf H dU
FG w U IJ Ÿ F H w LK
F w SI TkJ aL L fp JTaL L f TFG w FIJ TG J
H w LK
H w TK
CLdT Ÿ
U
T
FS
T
L
$WWKHFULWLFDOSRLQW
h a f mScL T h SaL T fr kSaL T f Sa L T fp
J
FT I
D OQ G J E cT T h cL L h aL L f
HT K
c
G $UHYHUVLEOH VORZ DGLDEDWLFH[SDQVLRQ Ÿ S Lf Tf S Li Ti
Cv dT a
a f a
CL
§ wF ·
dT ¨
¸ dL
T
© wT ¹ L
f
CL
dT J L L dL dS
T
Solutions to Chemical and Engineering Thermodynamics, 5th ed
6XEWUDFWLQJHTQ
IURP
JLYHV
8VLQJWKLVHTXDWLRQLQHTQ
JLYHV
7KHUHIRUHWKHIOXLGKDVDFULWLFDOSRLQW DQGWKHUHIRUHYDSRUOLTXLGHTXLOLEULD 7KH
FRPSUHVVLELOLW\IDFWRUDWWKHFULWLFDOSRLQWLV
DQG
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϲхWƌŽďůĞŵƐхWƌŽď͘ϲ͘ϳϭ
Solutions to Chemical and Engineering Thermodynamics, 5th ed
PV
RT
B PV
V B
V
RT
PV ,*
RT
VR
V V ,*
V UHV
P
V V ,*
RT
PV
RT RU PV r PB
V
P
RT RT
RT
PB
PB
VR
r r RT
RT
r RT § PB ·
¨¨ r ¸ ZKLFKLVDQH[SOLFLWIXQFWLRQRISUHVVXUH
P © RT ¸¹
,WLVFRQVLGHUEO\PRUHGLIILFXOWWRJHWU UHV DQGAUHV DVH[SOLFLWIXQFWLRQVRISUHVVXUH7KHHDV\
ZD\DURXQGWKDWSUREOHPLVWRREWDLQWKHVHUHVLGXDOIXQFWLRQVDVH[SOLFLWIXQFWLRQRIYROXPHDQG
WKHQDWHDFKSUHVVXUHDQGWHPSHUDWXUHVROYHWKHYLULDOHTXDWLRQIRUYROXPH6RWKHFDOFXODWLRQLV
WKHQDWZRVWHSSURFHVVUDWKHUWKDQDVLQJOHVWHSDVDERYH6RVWDUWZLWK
§ wU ·
¨
¸
© wV ¹T
§ wP ·
T¨
¸
© wT ¹V
§ wP ·
T¨
¸ P P
© wT ¹V
RT RTB § wP ·
¨
¸
V
V
© wT ¹V
R RB RT dB
DQG
V V V dT
RT RTB RT dB
§ wP ·
VRWKDWT ¨
¸ P
V
V
V dT
© wT ¹V
§ wP ·
T¨
¸ P
© wT ¹V
RT dB
U UHV T V
V dT
U T V U T V
RT RBT RT dB RT RBT
V
V
V dT V
V
RT dB
dB
dV RT
dV
³
³
V dT
dT V f V V f
V
f
V
RT dB
VRWRILQGU UHV DVDIXQFWLRQRISUHVVXUHDWHDFKSUHVVXUHILUVWVROYHWKH
V dT
YLULDOHTXDWLRQIRUYROXPHDQGVXEVWLWXWHWKDWYDOXHLQWKHHTXDWLRQDERYH1RZ
U UHV T V
§ wA ·
¨
¸
© wV ¹T
P
§ w A AIG
¨
¨
wV
©
AUHV T V
§ wA,* ·
RT RTB
DQG ¨
¸
V
V
© wV ¹T
·
RTB
¸
A AIG
¸
V
¹T
RT dB
V dT
P ,*
RT
VR
V
RT dB
dB
RT dB
RU
dV RT
dV ³
³
V dT
dT V f V
V dT
V f
V
V
D ,QLWLDOYROXPHLV
V LQLWLDO
V ILQ/
N ˜ Z ˜ R ˜T
P
N ˜ Z ˜ R ˜T
P
PRO ˜ . ˜ . ˜ EDUP / PRO. ˜ .
EDU
PRO ˜ . ˜ . ˜ EDUP / PRO. ˜ .
EDU
P
P
Solutions to Chemical and Engineering Thermodynamics, 5th ed
V ILQY
N ˜ Z ˜ R ˜T
P
PRO ˜ ˜ . ˜ EDUP / PRO. ˜ .
EDU
6RWKDWWKHFKDQJHLQYROXPHLV P
E 7KHHQHUJ\EDODQFHLV
dU
dV P
Q
dt
dT
6LQFHSUHVVXUHLVFRQVWDQWZHKDYH
dU
dV dU dPV dH
Q
P
RUQ ǻH dt
dT
dt
dT
dt
1RZ
V
ª § wP ·
º
H H IG RT Z ³ «T ¨
¸ P » dV
wT ¹V
V f¬ ©
¼
RT
a § wP ·
R
RT
RT
a
§ wP ·
DQGT ¨
¨
P
¸
¸ P
V b V © wT ¹V V b
V b V b V
© wT ¹V
V
H H IG
a
dV
V
V f
RT Z ³
RT Z a
V
RT Z P
a
V
aP
ZRT
8VLQJWKLVUHODWLRQZHKDYHWKDW
H H IG
. u -PRO
initial
H H
IG
H H IG
final ,l
final ,V
. u -PRO . u -PRO
7KHUHIRUHVLQFHWHPSHUDWXUHLVFRQVWDQWWKHDPRXQWRIKHDWWKDWPXVWEHDGGHGLV
Q ǻH . u u . u . u . u u . u
. u . u . u . u N
%\WKHPDVVDQGHQHUJ\EDODQFHVIRUD-RXOH7KRPVRQH[SDQVLRQZHKDYH
H LQ H RXW RU H EDU. H EDUT " 6LQFHWKHILQDOSUHVVXUHLVVRORZZHZLOODVVXPH
WKHH[LWLQJJDVLVLGHDO7KHUHIRUH H H ,*
H H ,*
EDU.
EDU.
,*
,*
H EDU
T H EDU.
,*
H EDU.
T
³
T
C3 dT
.
T ˜ T 1RZ
,*
H EDU
T ³ T dT .
Solutions to Chemical and Engineering Thermodynamics, 5th ed
H T P H ,* T P
DQG
PV
Z
RT
§ wP ·
¨ wT ¸
©
¹V
B T
V
V T P
ª § wP ·
V f
V
VRWKDWZ º
³ «¬«T ¨© wT ¸¹ P »¼»dV
RT Z B T
V
RT RTB T
V
V
P
R RB T
RT dB T
DQG
V
dT
V
V
§ wP ·
T¨
¸ P
© wT ¹V
H TP H
,*
RT RTB T
RT dB T
RT RTB T
V
dT
V
V
V
V
TP
RTB T
V
V T P
³
V f
RT dB
dV
V dT
RTB T
V
RT dB T
dT
V
RT dB § ·
¨ ¸
dT © V f ¹
RT ª
dB º
B T T
«
V ¬
dT »¼
$OVR
dB T
T
dT
T
dB T
˜ B T T
dT
T
T
T
B T
dB T º
˜ FP
P
ª
u « B T T dT »
PRO
PRO
¬
¼T IURP
B T
PV
RT
V
2QHREWDLQVDW.DQGEDU
P §
P ·
V u ¨ 1RWHIRUWKHLGHDOJDVLHB V u ¸
PRO ¨©
PRO ¸¹
H T P H ,* T P
RT ª
dB º
B T T
«
V ¬
dT »¼
6RWKDWWKHHTXDWLRQWREHVROYHGLV ˜ T T 7KHVROXWLRQWRWKLVHTXDWLRQLVT .VRWKDWWKHWHPSHUDWXUHFKDQJHLV.
Solutions to Chemical and Engineering Thermodynamics, 5th ed
dŚĞŐĂƐŝƐĚĞƐĐƌŝďĞĚďLJĂŶĂƵŐŵĞŶƚĞĚůĂƵƐŝƵƐĞƋƵĂƚŝŽŶ
^ŽůƵƚŝŽŶƚŽƚŚŝƐĞƋƵĂƚŝŽŶŝƐdϮсϰϬϰ͘ϰ<͘EŽǁƚŽĨŝŶĚƚŚĞǁŽƌŬĚŽŶĞ͙
hƐŝŶŐƚŚĞǀĂůƵĞƐŽĨdϭ͕dϮ͕WϭĂŶĚWϮǁĞŽďƚĂŝŶ
37(QYHORSH
9)5$& 3UHVVXUHEDU
ZϭϮ
7HPSHUDWXUH&
Solutions to Chemical and Engineering Thermodynamics, 5th ed
37(QYHORSH
9)5$& 3UHVVXUHEDU
ZϭϮϰ
7HPSHUDWXUH&
37(QYHORSH
9)5$& 3UHVVXUHEDU
Zϭϯϰ
7HPSHUDWXUH&
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϲхWƌŽďůĞŵƐхWƌŽď͘ϲ͘ϳϳ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϲхWƌŽďůĞŵƐхWƌŽď͘ϲ͘ϳϴ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϲхWƌŽďůĞŵƐхWƌŽď͘ϲ͘ϳϵ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϲхWƌŽďůĞŵƐхWƌŽď͘ϲ͘ϳ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϲхWƌŽďůĞŵƐхWƌŽď͘ϲ͘ϭϮ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϲхWƌŽďůĞŵƐхWƌŽď͘ϲ͘ϭϯ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϲхWƌŽďůĞŵƐхWƌŽď͘ϲ͘Ϯϵ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϲхWƌŽďůĞŵƐхWƌŽď͘ϲ͘ϯϭ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϲхWƌŽďůĞŵƐхWƌŽď͘ϲ͘ϯϮ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϲхWƌŽďůĞŵƐхWƌŽď͘ϲ͘ϲϱ
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
DOVRDYDLODEOHDVD0DWKFDGZRUNVKHHW q&
D G H TS DW P 03D DQG T G 9
G /
H 9 TS 9
H / TS /
E T q&
T .
. HTXDOZLWK
u - J
WKHDFFXUDF\ u - J
RIWDEOHV
H 9
TS 9
G /
|UV
|W
u - J
u u u u 1RWH$OO*LEEVIUHHHQHUJLHVDUHUHODWLYHWRWKHLQWHUQDOHQHUJ\DQGHQWURS\RIWKHOLTXLGSKDVH
EHLQJ]HURDWWKHWULSOHSRLQW6LQFH H / a U / DQG G / H / TS / ZHKDYHWKDW G / DW
WKHWULSOHSRLQW F T q& T .
H /
TS /
G 9
u - J
u u u u u 5(68/76
G T q&
V P NJ c
h
WR
T q&
3
3
h F 'H IJ H T 'T H T H :LOOFRPSXWH C IURP C a G
'T
H 'T K
c
V P NJ
P
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
T q&
a
f C3 N- NJ.
T q&
a
f
WR
C3 N- NJ. 7KHVHUHVXOWVDUHSORWWHGEHORZ
dU
dV
Q P
dt
dt
dS Q &ORVHGV\VWHPHQWURS\EDODQFH
SJHQ dt T
D 6\VWHPDWFRQVWDQWYROXPHDQGFRQVWDQWHQWURS\
&ORVHGV\VWHPHQHUJ\EDODQFH
dV
dt
Ÿ
dU
dt
DQG
Q DQG dS
dt
Q SJHQ Ÿ Q
T
TSJHQ Solutions to Chemical and Engineering Thermodynamics, 5th ed.
DQG
Ÿ
dU
d RU U
dt
1RZ
dP
dt
Ÿ P
FG w A IJ
HwT K
,
dA,
EXW dT ,
,
d
PV dt
d
U PV
dt
dH
dt
TSJHQ d dT , VM
FG w A IJ
H wV K
,
,
,
dV , T M
P, DQG
T M
FG w A IJ
Hw M K
FG w A IJ dM Hw M K
,
,
,
,
,
T V
G , T V
c
h
c
h
P , P ,, dV , G , G ,, dM , Ÿ P ,
P ,, DQGG , G ,, LL 6WDELOLW\DQDO\VLV
+HUHDJDLQZHIROORZDQDO\VLVLQ6HF²DQGILQG
d A
,
7KXVIROORZLQJWKHDQDO\VLVLQWKHWH[WZHREWDLQ
dA
FG w A IJ
H wV K
VLQFHWHPSHUDWXUHLVIL[HGDQG
AVV dV AVM dV dM AMM dM t 7KLVFDQEHUHZULWWHQDV
d A T dx T dx t a f
TSJHQ D 7KHFRQGLWLRQIRUHTXLOLEULXPDWFRQVWDQWTDQGVLVWKDWWKH+HOPKROW]IUHHHQHUJ\AVKDOOEHD
PLQLPXP
L (TXLOLEULXPDQDO\VLV IROORZLQJDQDO\VLVLQWH[W dV
Q P
dt
7KHUHIRUHHQWKDOS\LVDPLQLPXPDWHTXLOLEULXPDWFRQVWDQWSDQGP
d
PV 7KXV
dt
dU d
PV
dt dt
dV
dt
TSJHQ DQG
PLQLPXPDWHTXLOLEULXPDWFRQVWDQWVDQGS
dU
dt
TSJHQ T ! SJHQ t E 6\VWHPDWFRQVWDQWHQWURS\DQGSUHVVXUHDJDLQ Q
dU
dt
a f
ZKHUH
T
AVV T AMM AVV AVM
AVV
AMM AVM
AVV
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
DQG
FG w A IJ
H wV K
RU
FG w P IJ
H wV K
AVV
FG w P IJ
H wV K
FG w G IJ
Hw MK
T V
T V
T V
T V
FG
H
M
IJ
K
wP
V
w M T V
M
T V
OP
PQ
G T V
T V
FG w P IJ
Hw MK
FG
H
IJ
K
V
wP
V
w M T V
G DQG
T V
AMM VR
AMM FG IJ FG w P IJ
H K
Hw MK
FG w A IJ FG w G IJ Hw MK Hw MK
w wA
w M w V T M
LMFG w G IJ
MNH w M K
F w G IJ
ŸG
$OVR dG VdP SdT GdM
Hw MK
w M T V
AVM
t
AVV
1RZXVLQJ RQDPDVVEDVLV AMM w
FG w G IJ
Hw MK
d DVSUHYLRXVO\IRXQG
AVM
T M
AMM
FG IJ
H K
w
wP
P Ÿ t
w V T M
w V T M
T M
T
FG IJ
H K
w
wA
w V T M w V T M
T M
dM 7KXV T t DQG T t T
AVM
dM dx
AVV
dV dx
AVM
AVV
FG
H
a
a
IJ
K
f
f
w P w M T V
wP
V
w P w V T M
w M T V
FG w P IJ LMV FG w V IJ FG w P IJ OP
H w M K N H w PK H w M K Q
FG w P IJ LMV FG w V IJ OP E\WKHWULSOHSURGXFW
H w M K N H w M K Q UXOH (TQ T V
T M
T V
TP
T V
6LQFH
FG w V IJ
Hw MK
TP
A
V Ÿ AMM VM
AVV
T
E 7KH*LEEVIUHHHQHUJ\PXVWEHDPLQLPXPIRUDV\VWHPFRQVWUDLQHGDWFRQVWDQWTDQGP
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
L (TXLOLEULXPDQDO\VLV
FG w G IJ
HwT K
,
dG ,
,
7KXV
,
,,
FG w G IJ dM Hw M K
,
,
,
TP
G ,dM , ,
TP
DQG G ,
,
G ,, FG w G IJ
Hw M K
G MM dM dM ! TP
1RZ
FG w G IJ
Hw M K
FG w G IJ
Hw MK
TP
LMFG w G IJ
MNH w M K
(TQ
M
TP
Ÿ
G
TP
OP G G PQ M
7KXV G MM { DQGVWDELOLW\DQDO\VLVJLYHVQRXVHIXOLQIRUPDWLRQ
D $WFRQVWDQWMTDQGVAVKRXOGEHDPLQLPXP)RUDYDSRUOLTXLGPL[WXUHDWFRQVWDQWMT
DQGVZHKDYH
A/ A9 A
DQGDWHTXLOLEULXP dA
dA
dA / dA 9 7KXV
m P dV S dT G dM r m P dV S dT G dM r /
/
/
/
/
9
9
FRQVWDQW Ÿ dM / dM 9
V
FRQVWDQW Ÿ dV / dV 9
RUdV /
T
FRQVWDQW Ÿ dT / dT 9
9
c
h
c
RUdM /
9
9
9
dM 9
dV 9
h
P / P 9 dV / G / G 9 dM /
6LQFH dV / DQG dM / DUHLQGHSHQGHQWYDULDWLRQVZHKDYHWKDW
P/
/
M
Ÿ dA
EXW
T M
,
,
d G
,
dP , LL 6WDELOLW\DQDO\VLV
,
cG G hdM
dG
FG w G IJ
Hw P K
FG w G IJ dM
Hw M K
dG ,
P M
6LQFHTDQGPDUHIL[HG
dT , P 9 DQG G /
G 9 DOVR T / T 9 E\FRQVWUDLQWWKDWTLVFRQVWDQWDQGXQLIRUP
E $WFRQVWDQWMTDQGP G PLQLPXPRU dG RUHTXLOLEULXP
dG
mV dP S dT G dM r mV dP S dT G dM r /
/
/
/
/
/
9
9
9
9
9
9
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
DQGM
FRQVWDQW Ÿ dM /
dM 9
P
FRQVWDQW Ÿ dP /
dP 9
/
9
T FRQVWDQW Ÿ dT
dT
/
/
9
9
9
/
G dM G dM
G G dM /
c
Ÿ dG
/
RU G
h
9
G IRUYDSRUOLTXLGHTXLOLEULXPDWFRQVWDQWTDQGP
$OVRTDQGPDUHXQLIRUP²WKLVLVLPSOLHGE\FRQVWUDLQWV )URP6HFZHKDYH
C3
C9 T
FG w V IJ FG w P IJ
H w PK H w T K
T
FG w V IJ FG w P IJ H w T K H wV K
C9 T
V
P
T
a
f
,WLVWKHODVWIRUPRIWKHHTXDWLRQZKLFKLVXVHIXOKHUHQRZ T ! DQG w V w T P t +RZHYHU
FG w P IJ RS H w V K T DWFULWLFDOSRLQWRUOLPLWRIVWDELOLW\ T
7KXV C3 ! C9 LQJHQHUDOH[FHSWWKDW C3 C9 L DWWKHFULWLFDOSRLQWRUOLPLWRIVWDELOLW\RIDVLQJOHSKDVH
LL )RUWKHVXEVWDQFHVZLWK]HURYDOXHU RUYHU\VPDOOYDOXH RIWKHFRHIILFLHQWRIWKHUPDOH[SDQVLRQ
D V w V w T P VXFKDVOLTXLGVDQGVROLGVDZD\IURPWKHFULWLFDOSRLQW
a fa
f
6WDELOLW\FRQGLWLRQVIRUDIOXLGDUH
C9 ! DQG
FG w P IJ H wV K
T
IRUDILEHUWKHVHWUDQVODWHWR
C/ ! DQG
FG w F IJ ! H w LK
T
1RZ C/ D ET LI C/ ! IRUDOOTWKHQ C/ ! DW T Ÿ LPSOLHV D ! C/ ! DV T o f LPSOLHV E ! $OVR w F w L T JT ! VLQFH T ! WKLVLPSOLHV J ! a
f
P
dU dV T
T
wS
P
DQG
7KHVHUHODWLRQVWRJHWKHUZLWKWKHHTXDWLRQ
T
wV U T
dU
Td S PdV Ÿ d S
7KXV
FG w S IJ
HwUK
V
FG IJ
H K
S
S R D OQ
U
V
E OQ R UR
V
ZLOOEHXVHGWRGHULYHWKHUHTXLUHGHTXDWLRQ
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
>1RWH WKDW (TQ ZKLFK LV RI WKH IRUP S
VHQVHRI6HF@
D
FG w S IJ
HwUK
F w S IJ
E G
HwV K
V
U
FG U IJ HU K U
D
P
T
R
E
D
R
ŸU
T
U
VR ˜
V VR
E
FG w S IJ
HwU K
FG w S IJ
HwV K
w
D
wU V U
V
D
U
Ÿ D ! IRUIOXLG
DQG
FG IJ
H K
w S
w Uw V
D
FG w U IJ
HwTK
IURP
(TQ w
E
{ wU V V
C9 ! VLQFHDDQGEDUHSRVLWLYHFRQVWDQWV
V
DQG
E
w
wS
w U V wV U
7KXVWKHVWDELOLW\FULWHULD\LHOG
FG IJ
H K
FG IJ
H K
V w P
wP
!Ÿ
T wV T
wV T
7KXVIOXLGLVDOZD\VVWDEOHDQGGRHVQRWKDYHDILUVWRUGHUSKDVHWUDQVLWLRQ
a
f
$WOLPLWRIVWDELOLW\ w P w V T
IRUWKHYDQGHU:DDOVHTXDWLRQ
FG P a IJ V b
H VK
RT 6RWKDWDWOLPLWRIVWDELOLW\
FG w P IJ
H wV K
>1RWH D E ! E\SUREOHPVWDWHPHQW@
V
U
U
U
V
U|
|V WREH w
E
E
Ÿ E ! || VWDEOH
wV V
V
W
1RZ
FG w S IJ FG w S IJ FG w S IJ DQG FG w S IJ FG w S IJ H w Uw V K H w V K H w U K
HwV K
HwU K
ET 7KXV PV
V
U
DT >&OHDUO\ WKH IOXLG ZLWK DQ HTXDWLRQ RI VWDWH JLYHQ E\ LV DQ LGHDO JDV ZLWK FRQVWDQW KHDW
FDSDFLW\@
F 6WDELOLW\FULWHULD
S U V LV D IXQGDPHQWDO HTXDWLRQ RI VWDWH LQ WKH
a
V
RT
V b T
RT a
V b V
FG
H
RT
V b V b
RU
IJ FG P a IJ K V b H V K
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
7KXV P
a V b V RUXVLQJ
Vr Vr
VC
Ÿ Pr
PCVC DQG b
a
7RREWDLQWKHHQYHORSHZHFRPSXWH Pr IRUYDULRXVYDOXHVRI Vr Vr Pr a
f
LVWKHXSSHUOLPLWRIPHWDVWDELOLW\ LH Pr d DVZHOO
1RWLFHWKDWWKHFULWLFDOSRLQW Vr Pr
DVWKHOLPLWRIVLQJOHSKDVHVWDELOLW\
TDQGPZLOOEHWDNHQDVWKHLQGHSHQGHQWYDULDEOHVDWDVHFRQGRUGHUSKDVHWUDQVLWLRQ
7KHQ
DQG
G,
FG w G IJ HwTK
F w G IJ V ZKHUH V G
H w PK
G ,, S ,
S ,, VLQFH S
P
V,
,,
T
,
,,
,
,,
DQGRIFRXUVH T T DQG P
P ,
,,
)URP S S ZHKDYHWKDWDORQJWKHQGRUGHUSKDVHWUDQVLWLRQFXUYHWKDW d S ,
FG w S IJ dT FG w S IJ dP FG w S IJ dT FG w S IJ dP
H wT K H w PK H wT K H w P K F w V IJ dP C dT FG w V IJ dP
C
Ÿ
dT G
T
T
H wT K
H wT K
,
,
,,
P
T
P
,
,
3
,,
T
,,
,,
3
P
FG w P IJ
HwTK
6LPLODUO\HTXDWLQJ dV ,
od
DORQJ
WUDQVLWLRQ
FXUYH
7 wV , w T
i dw V w T i t
,,
P
P
dV ,, \LHOGV
,
,
P
,,
T
,,
P
T
7KXV
F dP I
H dT K
C3, C3,,
FG w V IJ dT FG w V IJ dP FG w V IJ dT FG w V IJ dP H wT K H w PK H wT K H w P K
P
7KXV
d S ,, RU
+RZHYHUVLQFH V ,
i dw V w T i t od
dw V w Pi dw V w Pi
wV , w T
DORQJ
WUDQVLWLRQ
FXUYH
,
,,
P
T
,,
P
D T
V ,, ZHFDQGLYLGHQXPHUDWRUDQGGHQRPLQDWRUE\ V DQGREWDLQ
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
FG w P IJ
HwTK
T
FG w P IJ
HwTK
H , H ,,
V , V ,,
DORQJ
WUDQVLWLRQ
FXUYH
+RZHYHUWKLVIRUPLVLQGHWHUPLQDWHIRUDQGRUGHUSKDVHWUDQVLWLRQ$SSO\LQJ/¶+RSLWDO¶VUXOHWR
HTQ WDNLQJGHULYDWLYHVRIQXPHUDWRUDQGGHQRPLQDWRUZLWKUHVSHFWWRTDWFRQVWDQWP
E 1RWH7KH&ODXVLXV&ODSH\URQHTXDWLRQLV
DORQJ
WUDQVLWLRQ
FXUYH
D , D ,,
N T, N T,,
T
FG w P IJ
HwTK
DORQJ
WUDQVLWLRQ
FXUYH
C3, C3,,
ZKLFKLVHTQ w V , w T w V ,, w T
d
i d
i
6LPLODUO\DSSO\LQJ/¶+RSLWDO¶VUXOHEXWQRZWDNLQJGHULYDWLYHVZLWKUHVSHFWWRPDWFRQVWDQWT
T
FG w P IJ
HwTK
cw H w Ph cw H w Ph
cw V w Ph cw V w Ph
V T cw V w T h V T cw V w T h
cw V w Ph cw V w Ph
,
DORQJ
WUDQVLWLRQ
FXUYH
,,
T
,
T
,
T
,
,,
EXW V ,
F w P IJ
Ÿ TG
HwTK
ZKLFKLVHTQ D D
F dP I
H dT K
DORQJ
WUDQVLWLRQ
FXUYH
dP
d OQ T
dP
DORQJ
dT WUDQVLWLRQ
FXUYH
dw V w T i dw V w T i T
dw V w Pi dw V w Pi
,
,,
P
,
DORQJ
WUDQVLWLRQ
FXUYH
P
,,
T
'H
dP
Ÿ T'V
d OQ T
H / H 6
V/ V6
T
u - NJ
P NJ
u - P u - P
RU T
E
T
V ,, VRWKDW
P
,,
T
,,
P
,
T
,,
u 3D Ÿ P P
T H[S
T
u OQ T
RS u a P P fUV T 3D
W
'H
RT
EXW 'V | V 9 a
P
T'V
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
' HP d OQ P
dT
RT dP
dT
,IZHDVVXPHWKDW 'H LVFRQVWDQWWKHQ
OQ
'H
RT P
P
FG
H
IJ
K
'H Ÿ T
R T T
c
m
u H[S u u .
hr
q & IUHH]LQJSRLQWHVVHQWLDOO\XQFKDQJHG
LM u OQ u OP
u Q
u u N T%
u 3D F 'HQYHU P
T)
LM R OQ P OP NT 'H P Q
.
q &
7KLVSUREOHPLQYROYHVWKHDSSOLFDWLRQRIWKH&ODXVLXV&ODSH\URQHTXDWLRQ:HZLOODVVXPHWKDWWKH
KHDWV RI IXVLRQ VXEOLPDWLRQ DQG YDSRUL]DWLRQ DUH DOO FRQVWDQW 7KXV ZH ZLOO XVH
'H P
LQ DOO FDVHV 1RZ ' YDS H ' VXE H ' IXV H 7R FDOFXODWH 'VXE H ZH
OQ P
R T T
ZLOOXVHWKHIROORZLQJVXEOLPDWLRQGDWD
u 3D 6WDWH7ULSOHSRLQW T q & . P q & . P u 3D 6WDWH T § u ·
' H§ ·
Ÿ OQ ¨¨
VXE ¨
¸
¸
¸
R © ¹
© u ¹
FG
H
IJ
K
u - PRO
Ÿ 'VXE H
' IXV H
u - PRO Ÿ ' YDS H
u
u - PRO
. 7RILQGWKHQRUPDOERLOLQJWHPSHUDWXUHZHDJDLQXVH&ODXVLXV&ODSH\URQ
DQG ' YDS H R
HTXDWLRQ
FG u I u F I 6WDWH 73
G
J
H T JK 6WDWH 1%3 H u K
OQ
Ÿ T
. q &
([SHULPHQWDOYDOXH q & GLIIHUHQFHGXHWRDVVXPSWLRQWKDW 'H YDS LVDFRQVWDQW
D $WHTXLOLEULXP P VDW LFH P VDW ZDWHU (TXDWLQJWKH OQ P VDW ¶VJLYHV
T
Ÿ T
T
q& Solutions to Chemical and Engineering Thermodynamics, 5th ed.
DQG
OQ P VDW LFH
E OQ P
A
B
d OQ P
DQG
dT
T
Ÿ P VDW
3D 'H
DOVR
RT d OQ P
dT
B
Ÿ 'H
T
B˜R 7KXV
'H
R LFHo YDSRU
DQG 'VXE H
u - PRO 'H
R ZDWHU o YDSRU
DQG ' YDS H
u - PRO ' IXV H
' H LFHo ZDWHU
$OVRDYDLODEOHDVD0DWKFDGZRUNVKHHW7KH0DWKFDGVROXWLRQLQFOXGHVJUDSKV D 8VHWKH&ODXVLXV&ODSH\URQHTXDWLRQ
T T
' OQ P YDS
'T
' YDS H
RT HLWKHUJUDSKLFDOO\RUDQDO\WLFDOO\ZHILQG
' YDS H a - PRO D 6WDUWZLWK(TQ f
EXW
dP
P
OQ
f
P
R
DQGJUDSKLFDOO\WDNLQJVORSH,ILQG ' YDS H a - PRO T
E 7KHYDSRUSUHVVXUHLVORZHQRXJKWKDWWKHLGHDOJDVDSSUR[LPDWLRQVKRXOGEHYDOLG²WKXV
OQ P P
d OQ P YDS
dT
' YDS H
3ORWWLQJ OQ P YV
u - PRO 'VXE H ' YDS H
P H[S
RS FV RT I dPUV Ÿ OQ f F PV I dP T RT z H P K W P z H RT K
d PV
dV
PV
V
z
Z
Z Z P
P
dZ dV
VR
Z
V
z FH
V
I
K
dZ
PV
dV
Z V f RT
V
Z OQ
z FGH
IJ
K
Z
P
dV V f RT V
V
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
RU
OQ
f
P
Z OQ Z E Z
BT
DQG P
V
OQ
f
P
z FGH
IJ
K
RT
P dV RT V f V
V
(TQ IJ
K
FG
H
RT
B
V
V
FG
H
IJ
K
z LMNFGH
B
B
OQ V
V
RT V f
z
V
B
OQ Z B
dV
V
V
V f
V
IJ OP
KQ
RT RT RT B
˜
dV
V
V
V V
ZB
OQ Z
V
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
F
YG:HRV
RT
a
Z YG:
V b V
P
z FGH
V
V f
z FGH
IJ
K
V
RT
P dV
V
V f
PV
RT
V
a
V b V RT
IJ
K
RT
RT
a
dV
V V b V
V
RT OQ
V
a
V b V V f
V
a
RT OQ
V b V
VR
f YG:
P
OQ
V
a
Z OQ Z
V b RTV
Pb
a RT P
˜
˜
Z OQ Z OQ Z OQ Z RT
RT PV RT
A
Z OQ Z B
Z
OQ
F
H
aP
Pb
DQG B
RT
RT
G 3HQJ5RELQVRQHTXDWLRQRIVWDWH6WDUWZLWK
ZKHUH A
z FGH RTV PIJK dV
z LMN
V
V
V f
V f
OP
Q
RT
RT
a
dV
V V b V V b b V b
LM d i OP
MN d i PQ
LV d ib OP
Z
a
OQM
RT OQ
Z B b NMV d ib QP
RT OQ
I
K
V b
V
a
OQ
V b b V b
>6HHVROXWLRQWR3UREOHPIRULQWHJUDO@7KHUHIRUH
OQ
f PR
P
Z OQ Z OQ
D f +OLT6
f +YDS
f YDS
6
d
d
i
i
V b
a
OQ
bRT V b
TC + 6
ZC + 6
i OP
i PQ a f
P f P ZKHUH WKH IXJDFLW\ FRHIILFLHQW f P ZLOO EH JRWWHQ IURP
FRUUHVSRQGLQJVWDWHV
PC + 6
LM d
MN d
V b
Z
a
OQ
Z B bRT V b
Z OQ Z B . Ÿ Tr
ZKLFKLVUHDVRQDEO\FORVHWR
EDU Ÿ Pr
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
f
f + S
P
E )RUDOLTXLGIURP(TQ )URP)LJ
f
u 3D DW WKH WHPSHUDWXUH RI LQWHUHVW ZH ZLOO DVVXPH WKDW f P VDW a z
P
V
dP
RT
YDS
P
P YDS H[S
f +S
z
P
V
dP
RT P YDS
h
YDS
VRWKDW
f+63D
u u u 5HSRUWHG
DOVRDYDLODEOHDVD0DWKFDGZRUNVKHHW D 7KHUHDUH DWOHDVW WZRZD\VWRVROYHWKLVSUREOHP2QHZD\LVWRVWDUWIURP
P
RT
f P H[S
V
dP RT P
RS
T
RU
z FH
P
z FH
I UV
K W
I
K
RT
dP
P
RT OQ
f
P
RT
P
u 03D ˜ P PRO. u .
P 03D u J PRO u NJ J
RT
LMV c P P h OP H[SL P O3D MN u u PQ
MN RT PQ
3UHVVXUH3D
c
V P P YDS
DQG
P
P YDS
$OVRZHZLOOFRQVLGHUWKHOLTXLGWREHLQFRPSUHVVLEOH7KXV
P
a f
6LQFH P YDS
F f I H[SLM z V dPOP H P K MN RT PQ
VDW
P YDS
u EDU V
P NJ
P
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
)URP6WHDP7DEOHV T
q & P NJ V
P03D
V RT P ±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
%\QXPHULFDOLQWHJUDWLRQRIWKLVGDWDZHILQGWKDW
f
a 03D ˜ P NJ
P
03D ˜ P NJ
f
Ÿ OQ
P 03D ˜ P NJ
RT OQ
VR f P DQG f 03D $VHFRQGZD\WRXVHWKHVWHDPWDEOHVLVWRDVVXPHWKDW
VWHDPDWq&DQG03DLVDQLGHDOJDV)URPWKHVWHDPWDEOHVDWWKHVHFRQGLWLRQVZHKDYH
H N- NJ S N- NJ. Ÿ G H TS u N- NJ
N- NPRO G q & 03D G ,* q & 03D
$OVR
G ,* T
q & 03D G ,* T
z
03D
q & 03D
V ,* dP
P 03D
G ,* T
q & 03D
N- NPRO OQ N- NPRO
z
03D
RT
dP
P
$OVRIURPVWHDPWDEOHV
GT
q & 03D
u u N- NPRO
f
P
FG G G IJ H[SF I
H
K
u RT
K
H
H[S
,*
f
u 03D
03D
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
a
E &RUUHVSRQGLQJVWDWHV TC
Tr
f
P
D
PV
B C
" Ÿ V
RT
V V
7KXV
PV
RT
03D
EDU
&RPPHQW7KHVWHDPWDEOHUHVXOWVDUHSUREDEO\WKHPRVWDFFXUDWHDQGWKHFRUUHVSRQGLQJVWDWHV
UHVXOWVWKHOHDVWDFFXUDWH1RWHWKDWZLWKWKHDYDLODELOLW\RIWKHFRPSXWHUSURJUDP35WKH35
HRVLVWKHHDVLHVWWRXVH7KHUHVXOWVZRXOGEHHYHQPRUHDFFXUDWHLIWKH3569HTXDWLRQZDV
XVHG
RT BRT CRT
" P
V P V P
B
RT P BRT V P CRT V P "
C
"
RT P BRT V P CRT V P "
a
B P RT
k
f
p CkRT P "p
B RT P "
PV
RT
a
E V
c
RT
B C B
P
hFH RTP IK " DQG V
RT
7KHUHIRUH
P
,*
hFH RTP IK " DQG
R LB cC B h P "OdPUV f
H[SS
PQ W
RT
P
T RT z MN
V V ,*
c
B C B
P
RU
f
C P RT " B V "
F P I F B P "I CF P I
H RT K H RT K H RT K BP
PI
"
cC B hF
H
RT
RT K
B
1RZNHHSLQJWHUPVRIRUGHUB B DQGCRQO\\LHOGV
03D
Ÿ f
f
Pr
f
F 8VLQJWKHSURJUDP35ZHILQG
03D w
)URPFRUUHVSRQGLQJVWDWHVFKDUW DFWXDOO\IURP7DEOHLQ+RXJHQ:DWVRQDQG5RJDW]9RO,,S
ZHKDYH
. PC
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
f
P
:HZLOO FRQVLGHU DQXPEHURI DOWHUQDWLYHV IRU XVLQJ WKHYLULDO FRHIILFLHQWGDWD 7KHILUVW LV WR
VWDUWZLWK(TQ D f
P
R| BP cC B h F P I "U| S| RT H RT K V|
W
T
H[S
H[S
RS FV RT I dPUV H[SRS dV V idPUV T RT z H P K W T RT z
W
P
P
zd
P
WKXV ZH QHHG WR HYDOXDWH WKH LQWHJUDO RT
i
V V ,* dP 6LQFH WKH WUXQFDWHG YLULDO HTXDWLRQ
,*
P RT V B V C V FDQ QRW HDVLO\ EH VROYHG IRU V DV D IXQFWLRQ RI T DQG P WKH
IROORZLQJSURFHGXUHZLOOEHXVHG
L &KRRVHYDOXHVRI V DQGFRPSXWH
P
IJ
K
FG
H
RT
B C
DQG P ,*
V
V V
RT
V
LL 3ORWPDQG P ,* DVDIXQFWLRQRI V LLL 8VHWKHVHWZRSORWVWRREWDLQ V ,* DQG V UHDOJDV DWWKHVDPHYDOXHRIPDOVRFRPSXWH
V T P V ,* T P LY )LQDOO\XVHDQXPHULFDORUJUDSKLFDOLQWHJUDWLRQVFKHPHWRJHW f P DVDIXQFWLRQRIP
6DPHUHSUHVHQWDWLYHYDOXHVRI V V ,* DUHJLYHQEHORZ
P 3D c h
dV V i
,*
P NPRO
P 3D c h
dV V i
,*
P NPRO
8VLQJWKHGDWDDQGSHUIRUPLQJWKHLQWHJUDWLRQZHREWDLQ
P 3D c
h
f P
FG w P IJ dV DQGZLWK
H wV K
RT F
P
G B C IJ WKDW FGH ww VP IJK FGH V V B VC IJK RT V H V V K
$QDOWHUQDWLYHLVWRQRWHWKDW dP
T
T
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
DQG
z
z
V P
P
V P o z
z
P
P
7KXV
EXW PV RT
f
P
H[S OQ
m
H[S B V C V B V C V r H[Sm V V r V V IRU V LQ P NPRO 7KH XVH RI WKLV HTXDWLRQ OHDGV WR UHVXOWV WKDW DUH VRPHZKDW PRUH DFFXUDWH WKDQ WKH JUDSKLFDO
LQWHJUDWLRQVFKHPH6WLOODQRWKHUSRVVLELOLW\LVWRXVHWKHUHVXOWVRISDUW D ZKLFK\LHOGV
m
r
H[S P u P IRUPLQEDU 7KHUHVXOWVRIXVLQJWKLVHTXDWLRQDUHOLVWHGEHORZ
)LQDOO\ ZH FDQ DOVR FRPSXWH f P XVLQJ FRUUHVSRQGLQJ VWDWHV )LJXUH )RU PHWK\O
IOXRULGH TC . DQG PC 03D QRWH ZC LV XQNQRZQ LV V C KDV QRW EHHQ
a f
DQGIRUHDFKSUHVVXUH Pr FDQEHFRPSXWHG
PHDVXUHG 7KXV Tr DQG f P IRXQGIURP)LJ
7KHUHVXOWVIRUHDFKRIWKHFDOFXODWLRQVDUHJLYHQEHORZ
P EDU HTQ f P HTQ f P HTQ f P &RUUHVSRQGLQJ
VWDWHV f P a f f @
B V C V DQG
f
P
RS RT B C UV RT H[SRS B C UV T PV V V W PV T V V W
f
P
P o B C
OQ PV OQ RT V
V
>1RWH OLP V P
V P
P o T
V P
z
c h RT V FGH ww VP IJK dV RT RTP dP
L F B C I
dP O
P
OLP M
J dV G
PP
MN a f H V V V K
Q
L
F IJ C FG IJ OP
OLP M OQ PV OQ PV a P f BG
H V P V a P fK H V P V a P fK Q
N
P
V V ,* dP
RT 1RWHWKDWDWORZSUHVVXUHDOOWKHUHVXOWVIRU f P DUHVLPLODU$WKLJKSUHVVXUHVKRZHYHUWKH
UHVXOWVGLIIHU(TXDWLRQ LVDSSUR[LPDWHDQGSUREDEO\WKHOHDVWDFFXUDWH(TXDWLRQ VKRXOG
EHWKHPRVWDFFXUDWHH[FHSWWKDWWKHUHLVDTXHVWLRQDVWRKRZDFFXUDWHLWLVWRXVHDQHTXDWLRQRI
VWDWHZLWKRQO\WKHVHFRQGDQGWKLUGYLULDOFRHIILFLHQWVIRUSUHVVXUHVDVKLJKDVEDU
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
D $VVXPH WKH YDSRU SKDVH LV LGHDO DQG WKDW ' YDS H LV DSSUR[LPDWHO\ FRQVWDQW RU DQ DYHUDJH
' YDS H FDQEHXVHG P
P
OQ
IJ
K
FG
H
' H YDS R
T T
§ · ' YDS H §
·
Ÿ OQ ¨
¸
¨
¸
R
©
¹
©
¹
' YDS H
u .
Ÿ
' YDS H u - PRO
R
E ' YDS H T
H VDWYDST H VDWOLTT ª H VDWYDST H ,* T º ª H VDWOLTT H ,* T º
¬
¼ ¬
¼
F Tr
T
TC
FG H H IJ
H T K
,*
C
FG H H IJ
H T K
,*
C
. > @
DW T
T
VDWOLT
Tr - PRO. u - PRO q & Tr
Z
q & Tr
Z
7KHDYHUDJHYDOXHRIWKHFRPSUHVVLELOLW\LV
Z
- PRO. DQG
G 7KH UHDVRQ IRU WKH GLVFUHSDQF\ LV SUREDEO\ QRW WKH LQDFFXUDF\ RI FRUUHVSRQGLQJ VWDWHV VLQFH
ZC ZKLFK LV FORVH WR EXW UDWKHU WKH DVVXPSWLRQ RI DQ LGHDO YDSRU SKDVH LQ WKH
&ODXVLXV&ODSH\URQHTXDWLRQ:HFRUUHFWIRUJDVSKDVHQRQLGHDOLW\EHORZ
VDWYDS
Tr ' YDS H T
ª§ H H ,* ·
º
§ H H ,* ·
»
TC Ǭ
¨
¸
¸
¸
¨
¸
«¨© TC
»
T
¹
©
¹
VDWYDS
VDWOLT
T
T
¬
¼
Ÿ ' YDS H T
:HQRZUHSODFHHTQZLWK
OQ
P
P
' YDS H § ·
¨ ¸ Ÿ ' YDS H
ZR © T T ¹
u u - PRO
u - PRO ZKLFKLVLQPXFKEHWWHUDJUHHPHQWZLWKWKHUHVXOWRISDUW F $EHWWHUZD\WRSURFHHGZRXOGEH
WR FRPSXWH WKH FRPSUHVVLELOLW\ DV D IXQFWLRQ RI WHPSHUDWXUH LH ILQG Z Z T P DQG WKHQ
LQWHJUDWH
' YDS H P
dP
dT Z T P RT Solutions to Chemical and Engineering Thermodynamics, 5th ed.
UDWKHUWKDQXVHDQDYHUDJHYDOXHRIZLH Z %DVLV YHVVHOYROXPH P FDQFHOVRXWRISUREOHP x IUDFWLRQRIYHVVHOILOOHGZLWKOLTXLGZDWHU
x
x
N WRWDOQXPEHURIPROHVRIZDWHU
SHU P RIYHVVHO V/
V9
7RWDOQXPEHURIPROHVVDPHDWDOOFRQGLWLRQV
xLVWKHVDPHDWLQLWLDOORDGLQJDQGDWFULWLDOSRLQW
6WDWH²ORZSUHVVXUH V 9 !! V / x
x
x
Ÿ N
| / /
9
V
V
V
x
x
6WDWH²&ULWLFDOSRLQW V 9 V / V C Ÿ N EXW
VC VC VC
x
V/
N N Ÿ /
RU x
VC
VC
V
D 8VLQJVWHDPWDEOHV
V / q & P NJ V C P NJ x
Ÿ,QLWLDOILOOVKRXOGFRQWDLQRIYROXPHZLWKOLTXLG ZKLFKZDVUHSRUWHGLQWKH&KHPLFDO
DQG(QJLQHHULQJ1HZVDUWLFOH E 3HQJ5RELQVRQHTXDWLRQRIVWDWH
V / q &
u P PRO DQGWKH35HTXDWLRQRIVWDWHSUHGLFWV ZC
VC
ZC RTC
PC
VROXWLRQWR3UREOHPE VR
u u 03D P PRO. u .
03D
u P PRO Ÿ x
RUDQLQLWLDOILOORIRIYROXPHZLWKOLTXLG
D 2QHWKHRU\IRUZK\LFHVNDWLQJLVSRVVLEOHLVEHFDXVHLFHPHOWVGXHWRWKHSUHVVXUHSXWXSRQLW
XQGHUWKHLFHVNDWHVDQGWKHQUHIUHH]HVZKHQVNDWHOHDYHVDQGWKHSUHVVXUHLVUHOHDVHG6NDWH
DFWXDOO\PRYHVRYHUDILOPRIZDWHURQWKHVKHHWRILFH7RILQGWKHORZHVWWHPSHUDWXUHZHXVH
WKH &ODSH\URQ HTXDWLRQ WR FDOFXODWH WKH FKDQJH LQ IUHH]LQJ SRLQW DV D UHVXOW RI WKH DSSOLHG
SUHVVXUH3URSHUWLHVRILFH
U J FF Ÿ V S FF J ' HÖ ' HÖ ' HÖ - J DWq& IXV
VXE
YDS
$SSHQGL[,,, Solutions to Chemical and Engineering Thermodynamics, 5th ed.
FG w P IJ
HwTK
RU
Ÿ 'T
- FF.
EDU .
FG w T IJ a . EDU Ÿ 'T . EDU 'P H w PK
'P
- J
. u FF J $VVXPHNJSHUVRQRQ FP VNDWHDUHD ZHOOVKDUSHQHG VDW
wP
'H
Ÿ
wT
T 'V
NJ FP u EDU NJ FP
q&
DVVXPLQJ VNDWH PDNHV FRPSOHWH FRQWDFW ZLWK LFH ,I WKH VXUIDFH LV LUUHJXODU DV LW LV PD\EH
FRQWDFWRQO\RYHURIDUHD,QWKLVFDVH 'T q & 0\REVHUYDWLRQLQ0LQQHVRWDZDV
WKDW LW ZDV SRVVLEOH WR VNDWH GRZQ WR a q & FRQWDFW DUHD"" 2I FRXUVH WKH
WKHUPRG\QDPLFPRGHOIRUWKLVSURFHVVPD\EHLQFRUUHFW2WKHUSRVVLELOLWLHVLQFOXGHWKHPHOWLQJ
RILFHDVDUHVXOWRIIULFWLRQRUE\KHDWWUDQVIHUIURPWKHVNDWHU¶VIRRWWRWKHLFH,EHOLHYHWKH
WKHUPRG\QDPLFWKHRU\WREHWKHDUHDVRQDEOHH[SODQDWLRQRIWKHSKHQRPHQD
E 6LQFH ' IXV HÖ ! DQG ' IXVVÖ ! IRU &2 DQG PRVW RWKHU PDWHULDOV IUHH]LQJ SRLQW ZLOO EH
HOHYDWHGQRWGHSUHVVHG/LTXLGILOPFDQQRWIRUPDQGLFHVNDWLQJLVLPSRVVLEOH
F 0RUH GLIILFXOW WR TXDQWLI\ 6LPLODU WR D IRU IUHH]LQJ SRLQW GHSUHVVLRQ ZKLFK RQ UHOHDVH RI
SUHVVXUHFDXVHVUHIUHH]LQJDQGIRUPDWLRQRIVQRZEDOOEXWLQWKLVFDVHWKHUHLVDOVRFRQVLGHUDEOH
KHDWWUDQVIHUIURPWKHKDQGVWRWKHVXUIDFHRIWKHVQRZEDOOWKDWFDXVHVPHOWLQJ
DOVRDYDLODEOHDVD0DWKFDGZRUNVKHHW )URP(T ' 7 ' 3 ' 9 7' +
'7 §
·
P
©
J ¹
˜
$FHWLFDFLG
˜
¸ ˜ ˜ .
MRXOH
MRXOH
MRXOH
'7 '7
.
'7
.
'7
.
J
§
·
P
©
J ¹
˜ EDU ˜ ¨ ˜ %LVPXWK
.
J
·
§
P
¨
¸ ˜ ˜ .
˜ EDU ˜ ˜
˜
J ¹
©
'7 ˜
'7
J
·
§
P
¨
¸ ˜ ˜ .
˜ EDU ˜ ˜
˜
J ¹
©
'7 ˜
7LQ
EDU ˜ 3D
˜ EDU ˜ ¨ ˜ :DWHU
˜
EDU
MRXOH
J
˜
¸ ˜ ˜ .
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
DOVRDYDLODEOHDVD0DWKFDGZRUNVKHHW $J O
7PD[ 7PLQ 7PD[ 7PLQ <L
L 7PLQ
7PD[ 7PLQ %H2 V
7PD[ 7PLQ <L
L 7PLQ
D
F
F
' +VXE-PRO
7.
<L
E
' + <L D E F
D
7.
E
' +P-PRO
<L
' + <L D E F
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
*H V
<L
0J V
<L
7PD[
7PD[ 7PLQ L
7PLQ
D
7PLQ
E
F
F
' +P-PRO
7.
<L
' + <L D E F
7PD[
7PD[ 7PLQ L
7PLQ
D
7PLQ
7.
E
'+P-PRO
<L
' + <L D E F
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
0J O
7PD[ 7PLQ 7PD[ 7PLQ <L
L 7PLQ
1D&O V
7PD[ 7PLQ 7PD[ 7PLQ <L
L 7PLQ
D
7.
E
F
'+P-PRO
<L
' + <L D E F
D
7.
E
'+P-PRO
<L
' + <L D E F
F
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
6L V
7PD[ 7PLQ D
E
F 7PD[ 7PLQ <L
L 7PLQ
7.
'+P-PRO
<L
' + <L D E F
7KHPHWDOWLQXQGHUJRHVDWUDQVLWLRQIURPDJUD\SKDVHWRDZKLWHSKDVHDW.DWDPELHQW
SUHVVXUH*LYHQWKDWWKHHQWKDOS\FKDQJHRIWKLVWUDQVLWLRQLVN-PROHDQGWKDW
WKHYROXPHFKDQJHRIWKLVWUDQVLWLRQLVFPPROHFRPSXWHWKHWHPSHUDWXUHDWZKLFK
WKLVWUDQVLWLRQRFFXUVDWEDU
)URP(T ' 7 ' 3 ' 9 7' +
§
P ·
©
PROH ¹
˜ EDU ˜ ¨ ˜ '7 EDU ˜ 3D
˜
˜
¸ ˜ ˜ .
PROH
)RUWKHVROLGOLTXLGWUDQVLWLRQ
'HI
'HI
wP
wP
Ÿ
'V
w T HT T'V
w OQ T HT
FG IJ
H K
FG w P IJ
H w OQ T K
IJ
K
FG
H
- FF
'7
MRXOH
u - P
u .
- J
FF J
- FF
EDU
u 3D HT
Ÿ P
Ÿ T 73
T H[S
T
P u OQ T
RS P P UV H[SRS P 3D UV T u 3D W
T u 3D W
73
73
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
)RUWKHVROLGYDSRUWUDQVLWLRQDVVXPLQJDQLGHDOYDSRUSKDVH
Ÿ
' H VXE
R
Ÿ OQ
P
P
73
LM P O
u OQ
P
T
P Q
N
T T
FG IJ
HT T K
a
f
OQ u u . DQG
FG IJ
H K
OQa P P f
' H VXE R
T T
P
P
OQ
T
73 OQ P 73 c
h
6ROYLQJ(TQV DQG VLPXOWDQHRXVO\JLYHV
P 73
EDU
N3D DQG T 73
. >7KH PHOWLQJ WHPSHUDWXUH RI EHQ]HQH a WULSOH SRLQW WHPSHUDWXUH
H[DFWO\ZLWKRXUSUHGLFWLRQ@
. ZKLFK DJUHHV
q&
)LUVWDW.OHWVUHODWHWKH*LEEVIUHHHQHUJ\DWDQ\SUHVVXUHWRWKHYDOXHJLYHQDWEDU
NJ
J PRO
P EDU u u u
NJ J
P
G dia P G dia P EDU EDUP
u .
u PRO.
NJ
J PRO
P EDU u u u
NJ J
P
G g P G g P EDU EDUP
u u .
PRO.
1RWHWKDW
P
NJ
J
P
V dia
u
u u NJ J
PRO
PRO
P
NJ
J
P
Vg
u
u u NJ J
PRO
PRO
7KHUHIRUH
P G g P G dia P EDU G g P EDU
G dia F
GG
GH
F
GG
GH
I
JJ
JK
F P EDU u P I
u G
PRO J
G
JJ
EDUP
GH u PRO.
.
u K
DWHTXLOLEULXPDW.ZHKDYH
P u u u EDU
P u 0EDU
I
JJ
JK
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
7RILQGWKHWUDQVLWLRQSUHVVXUHDWRWKHUWHPSHUDWXUHVZHXVHWKH&ODSH\URQHTXDWLRQ
wP
3DP
'S
PRO.
u
wT VDW 'V
P
u PRO
3D
EDU
u .
.
ZKLFKLQGLFDWHVWKDWIRUHYHU\GHJUHH.LQFUHDVHDERXW.ZHQHHGWRLQFUHDVHWKHSUHVVXUH
E\EDU+RZHYHUWKLVLVDVPDOOSHUFHQWDJHLQFUHDVHFRPSDUHGWRWKH03DSUHVVXUH
UHTXLUHGDW.6RWKHWUDQVLWLRQLVHVVHQWLDOO\ ZLWKLQHQJLQHHULQJDFFXUDF\ RQO\YHU\
ZHDNO\GHSHQGHQWRQSUHVVXUH
F I
H K
0DVVEDODQFH M f
Mi (QHUJ\EDODQFH U f
Ui (TXLOLEULXPFULWHULRQ G 6 G / G 9 DOVR S PD[LPXP
:HZLOODVVXPHWKHYDSRUSKDVHLVDQLGHDOJDV
3URSHUWLHVRIWKHWULSOHSRLQW DFRQYHQLHQWUHIHUHQFHVWDWH U OLTXLG T q & WULSOHSRLQW U VROLG T q & 'H VROLG o YDSRU
UHIHUHQFHVWDWH
- J
VLQFH'V # U YDSRU T
q &
'H YDS RT
- J u - PRO u
PRO
J
- J
(QHUJ\FRQWHQWRIRULJLQDOV\VWHP
U OLTXLG T q & - Jq & u q & - J
- Jq & u q & - J
U VROLG T q & - J - J C9 q &
U YDSRU T q & ZKHUHC9 a C3 R - J
q & $OVRLQLWLDOVSHFLILFYROXPHRIYDSRUSKDVH
F
H
V9
RT
P
3D ˜ P PRO. u .
3D
P PRO 5DWLRRIPDVVLQLWLDOO\SUHVHQWLQYDSRUWRPDVVLQOLTXLG
M9
M/
I
K
V/
V9
u P PRO
P PRO
u Ÿ1HJOLJLEOHPDVVRIV\VWHPLQLWLDOO\SUHVHQWLQJDVSKDVH
6LQFHWKHLQLWLDOV\VWHPLVDVXEFRROHGOLTXLGYDSRUWKHIROORZLQJSRVVLELOLWLHVH[LVWIRU
WKHHTXLOLEULXPVWDWH
AllWKHOLTXLGIUHH]HVDQGVRPHYDSRUUHPDLQV LHDVROLGYDSRUV\VWHPDWHTXLOLEULXP 7KH HQHUJ\ UHOHDVHG KHDW RI IXVLRQ ZRXOG WKHQ JR WR KHDW WKH V\VWHP²6LQFH -J DUH
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
UHOHDVHGDQG C3 a - Jq & WRRPXFKHQHUJ\LVUHOHDVHGIRURQO\WKHVROLGDQGYDSRUWREH
SUHVHQW
Some RI WKH OLTXLG YDSRUL]HV DQG VRPH IUHH]HV VR WKDW D VROLGOLTXLGYDSRU PL[WXUH LV
SUHVHQW 7KXV WKH V\VWHP LV DW WKH WULSOH SRLQW DW HTXLOLEULXP >7KH HQHUJ\ UHOHDVHG LQ
VROLGLILFDWLRQRIWKHZDWHUJRHVWRKHDWWKHV\VWHPXSWRq&WKHWULSOHSRLQWWHPSHUDWXUH@
:HZLOOFRQVLGHUWKLVVHFRQGSRVVLELOLW\KHUHDVDILUVWJXHVVWKHVPDOODPRXQWRIYDSRUZLOOQRW
EHLQFOXGHGLQWKHFDOFXODWLRQV
/HW
M
/
LQLWLDOPDVVRIOLTXLGLQWKHV\VWHP
M /U / q &
x M /U / q & xM /U 6 q &
- x x Ÿx
RQWKDWLVVROLG
IUDFWL
M9
M
N3D
c u P PROh u 3D
,I ZH QRZ LQFOXGH WKLV DPRXQW RI YDSRU LQ WKH HQHUJ\ EDODQFH LW PDNHV RQO\ DQ LQVLJQLILFDQW
FKDQJHLQWKHFRPSXWHGVROLGDQGOLTXLGIUDFWLRQVŸ:HZLOOQHJOHFWWKHSUHVHQFHRIWKHYDSRU
7R FRPSXWH WKH HQWURS\ FKDQJH ZH QRWLFH WKDW LI ZH VWDUWHG ZLWK JUDP RI OLTXLG WKH QHW
FKDQJHLQWKHV\VWHPZRXOGEH
JOLTXLG q &
G q &
o JVROL
u JYDSRU q & 3D
u YDSRU q & 3D
(QWURS\FKDQJHV
JOLTXLG ±q& oJOLTXLG q& 'S
u .
3D ˜ P PRO. u JOLTXLG q &
3D A6WHDPWDEOHV
$VDILUVWDSSUR[LPDWLRQDVVXPHWKDWWKHYDSRUVWLOORFFXSLHVDERXWRIWKHWRWDOYROXPH²
WKLV LV UHDVRQDEOH VLQFH ZH H[SHFW OLWWOH YROXPH FKDQJH RI WKH FRQGHQVHG SKDVH GXH HLWKHU WR
IUHH]LQJRUWKHUPDOH[SDQVLRQ
IUDFWLRQWKDWLVOLTXLG
x 1RZOHWVJREDFNDQGGHWHUPLQHWKHDPRXQWRIYDSRULQWKHV\VWHP
WULSOHSRLQWSUHVVXUH
(QHUJ\EDODQFH
IUDFWLRQRIOLTXLGWKDWVROLGLILHG
x
T
C3 OQ I
TL
- J. OQ
- J. JOLTXLG q& oJVROLG q& 'G
'H T'S
'S
'H
DQG'S
T
M
'H T
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
- J
u J
.
'S
- . u 'S
u JOLTXLG q & o u JYDSRU q & 'H YDS
u u u
u - . T
ŸYDSRUPDNHVDQHJOLJLEOHFRQWULEXWLRQLQFRPSXWLQJ 'S 'S - . IRUHDFKJUDPRIZDWHUSUHVHQW
*LEEV3KDVH5XOH (TQ ) P D 2QO\VROLGVFDQKDYHPDQ\DVSKDVHVSUHVHQW7KHVHFDQRQO\H[LVWDWDVLQJOHSRLQWRITDQG
PDVWKHUHDUHQRGHJUHHVRIIUHHGRP
,IRQO\VROLGSKDVHV ) Ÿ DVLQJOHGHJUHHRIIUHHGRP&DQIL[HLWKHUTRUP
,IRQO\RQHVROLGSKDVHSUHVHQWWKHQ ) ERWKTDQGPFDQYDU\LQGHSHQGHQWO\
E ,IDOLTXLGLVDOVRSUHVHQW
) TDQGPLQGHSHQGHQWO\
/LTXLGDORQH YDULDEOH
) &DQYDU\TRUPWKH
/LTXLGVROLG
RWKHUIROORZV
) 2QO\DVLQJOHSRLQW
/LTXLGVROLGV
F 6DPHDV E ZLWKYDSRUUHSODFLQJOLTXLGLQGLVFXVVLRQDERYH
G ,IOLTXLGDQGYDSRUDUHDOUHDG\SUHVHQWWKHQ P ŸGHJUHHRIIUHHGRPLIQRVROLGLVSUHVHQW LHHLWKHUTRUPFDQEHIL[HGQRWERWK ,IOLTXLGDQGYDSRUDQGVROLGDUHSUHVHQW P Ÿ ) 7ULSOHSRLQWLVDXQLTXHSRLQWRQSKDVHGLDJUDP
&ULWHULRQIRUHTXLOLEULXPDWFRQVWDQWTDQGVDVWKDW A PLQLPXP+RZHYHUIURP3UREOHP WKLVLPSOLHV G 9 G / QRW A9 A / 1RZ G A PV DQG P w A w V T 7KXVDWHTXLOLEULXP
a
G9
FG w A IJ V G A FG w A IJ V H wV K
H wV K
F w A IJ FG w A IJ P ŸG
H wV K H wV K
A9 9
9
/
$OVRDWHTXLOLEULXP P 9
9
/
/
9
/
T
Ÿ3UHVVXUHVDUHHTXDOZKHQGHULYDWLYHVRI A ZLWKUHVSHFWWR V DUHHTXDO7KHGHULYDWLYHRI A ZLWK
UHVSHFWWR V LVWKHWDQJHQWWRWKHFXUYHLQWKH A V SODQH7KXVERWKFXUYHVPXVWKDYHDFRPPRQ
WDQJHQWLIHTXLOLEULXPLVWRRFFXUDQGWKHVORSHRIWKLVWDQJHQWOLQHLVWKHQHJDWLYHRIWKHHTXLOLEULXP
SUHVVXUH1RZPXVWSURYHWKDWWKHSRLQWVRILQWHUVHFWLRQRIWKHWDQJHQWOLQHDQG A FXUYHDUHSRLQWV
DW ZKLFK G 9
G / )URP WKH ILJXUH ZH KDYH A9
ZKDWLVUHTXLUHGE\(TQ IRU G
/
/
T
T
/
/
T
f
9
a
fd
i
A/ w A w V T V 9 V / EXW WKLV LV H[DFWO\
/
G 6RZHKDYHLQGHHGLGHQWLILHGWKHHTXLOLEULXPVWDWH
A
VV
VL
V
§ wA· V
¸¸ V V L ¨¨
V
w
¹
©
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
D )URPWKHHTXLOLEULXPFULWHULDZHKDYH T 9
G9
G/ Ÿ
V / T
z
V/
Tr
Vr Vr
Pr
RS T UV TV aV f W
FG w P IJ T
H w V K V aV f
r
r
r
r
DQG
wP
dV
wV T
V
E )RUWKHYDQGHU:DDOVHTXDWLRQZHKDYH
P / DQG
z FGH IJK
V9
RU
VdP
V 9 T
T / P 9
r
r
r
r
7KXV
z FGH ww VP IJK dV z FGH V aVTVf IJK dV
Vr9
Vr/
Vr9
r
Vr
r
Vr/
r
V9
r r
r
r
Vr9
r
T
r OQ Vr Vr /
Vr
Vr V /
a
f a
r
f
r
DQG
OQ
cV h cV h cV h FG V IJ T V H V K
cV h
/
r
9
r
9
r
/
r
$QRWKHUHTXDWLRQDULVHVIURPWKHIDFWWKDW P 9
/
r r
P / Ÿ Pr9
Tr
Vr9 Vr9 RUVROYLQJIRUWKHUHGXFHGWHPSHUDWXUH
c h
Tr
{c
Pr/ VRWKDWXVLQJ(TQ Tr
Vr/ Vr/ c h
h c h}
h c
hr
Vr9 Vr/
9r9 9r/ mc
/
r
9
r
7KLVLVDQLQGHSHQGHQWHTXDWLRQEHWZHHQ Tr Vr9 DQG Vr/ 8VLQJ(TQ LQ(TQ JLYHV Solutions to Chemical and Engineering Thermodynamics, 5th ed.
R| cV h U| cV h cV h
OQS
|TcV h V|W
V V R U
V S
V V T V V W
/
r
9
r
9
r
9 /
r r
9
/
r
r
9
r
/
r
/
r
F 3URFHGXUHXVHGLQVROXWLRQ
L *XHVV RUFKRRVH DYDOXHRI Vr9 FRPSXWH Vr/ ZKLFKVDWLVILHV(TQ LL 8VH Vr9 DQG Vr/ VRREWDLQHGWRFRPSXWH Tr IURP(TQ LLL 8VH Vr DQG Tr DQG (TQ WR JHW Pr PrYDS >1RWH 7KLV FDOFXODWLRQ ZDV GRQH RQ D
GLJLWDOFRPSXWHU@5HVXOWVDUHDWHQGRISUREOHPVROXWLRQ
G 7KH&ODXVLXV&ODSH\URQHTXDWLRQFDQEHZULWWHQDV
dP
dT FRH[LVWDQFH
FXUYH
dPrYDS
dTr
' H YDS
T'V
6LQFH a
PCVC VC
' H YDS Tr 'V PC
' H YDS V C
Tr 'V a
b ' H YDS
a Tr 'V r
b 1RZ b a KDV XQLWV RI HQHUJ\ VR GHILQH D UHGXFHG KHDW RI
YDSRUL]DWLRQ 'HrYDS WREH b a ' H YDS 7KXV
dPrYDS
dT
7KXV 'HrYDS FDQ EH FRPSXWHG E\ WDNLQJ GHULYDWLYHV RI UHVXOWV RI SDUW F 7KLV ZDV GRQH
JUDSKLFDOO\ 7KHUHVXOWVDUHDVVKRZQEHORZ
Tr Tr 'HrYDS PrYDS V r9 V r/ ' H rYDS
PC dPrYDS
TC dTr
RU
'H
T'V
u u u u u u Tr 'Vr
u u u u u u Solutions to Chemical and Engineering Thermodynamics, 5th ed.
5HVXOWVDUHSORWWHGEHORZ
7KHUHGXFHGYDSRUSUHVVXUHDQGKHDWRIYDSRUL]DWLRQIRUWKHYDQGHU:DDOVIOXLG
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
D 5HVWULFWHGIRUPRI*LEEV3KDVH5XOH F P Ÿ P PXVWEHRUOHVV
ŸTXDWHUQDU\SRLQWFDQQRWH[LVWLQDFRPSRQHQWIOXLG
E SKDVHV Ÿ F GHJUHHRIIUHHGRP
7KXV LI DQ\ SURSHUW\ RI one RI WKH SKDVHV LV VSHFLILHG WKLV LV VXIILFLHQW WR IL[ DOO RI WKH
WKHUPRG\QDPLFSURSHUWLHVRIERWKSKDVHV+RZHYHULIRQO\WKHWRWDOPRODUYROXPHRIWKHWZR
SKDVHV\VWHP RUVRPHRWKHUWZRSKDVHSURSHUW\ LH V x ,V , T P x ,,V ,, T P WKLVLVQRW
VXIILFLHQWWRVROYHIRU x , V , DQG V ,, 7KDWLVPDQ\FKRLFHVRI TDQG PFDQ\LHOGWKHVDPH
YDOXHRIWKHWRWDOPRODUYROXPHE\YDU\LQJWKHPDVVGLVWULEXWLRQEHWZHHQWKHWZRSKDVHV
&RQVLGHU QRZ WKH VLWXDWLRQ LQ ZKLFK WKH WRWDO PRODU YROXPH DQG WRWDO PRODU HQWKDOS\ LV
VSHFLILHG,QWKLVFDVHZHKDYH
x , x ,,
V
x ,V , T P x ,,V ,, T P DQG H
7KH XQNQRZQV KHUH DUH x , x ,, DQG HLWKHU T RU P QRWH WKDW VLQFH WKH V\VWHP KDV RQO\ RQH
GHJUHHRIIUHHGRPHLWKHUTRU PEXWQRWERWKDUHLQGHSHQGHQWYDULDEOHV 7KXVJLYHQHTXDWLRQ
RIVWDWHLQIRUPDWLRQ UHODWLQJ V DQG H WR TDQG PDQGWKHWZRSKDVHFRH[LVWDQFHFXUYH WKH
HTXDWLRQVDERYHSURYLGHHTXDWLRQVWREHVROYHGIRUWKHXQNQRZQV
D dS
FG w S IJ dT FG w S IJ dP 7KXV
H w T K H w PK
FG w S IJ FG w S IJ FG w S IJ FG w P IJ
H w T K H w T K H w PK H w T K
P
T
VDW
FXUYH
EXW
FG w S IJ
Hw TK
7KXV
CVDW
T
FG w S IJ
Hw TK
FG IJ
H K
T
FXUYH
FG w V IJ
Hw TK
C3 DV
C3i D iV i
ZKHUHiGHQRWHVWKHSKDVH
E )RUWKHOLTXLG
' H YDS
'V YDS
VDW
FXUYH
' H YDS
T'V YDS
V D P
' H YDS
'V YDS
DQG
i
CVDW
VDW
FXUYH
T
C3
wP
T
w T VDW
FG w S IJ
H w PK
P
P
DQGE\WKH0D[ZHOOUHODWLRQV
x , H , T P x ,, H ,, T P
FG w V IJ ' H
H w T K 'V
C3 P
FG w V IJ ' H
H w T K 'V
i
C3i P
YDS
YDS
YDS
YDS
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
/
'V YDS !! V / DQG D i a Ÿ CVDW
a C3/ )RU
WKH
YDSRU
w V w T P a V T V T
a
a f afa
f
9
CVDW
ZH
ZLOO
T T DQG
a
a f a f aT T f XVH
f
C39 D 9V 9
C3 a H T H T
' H YDS
'V YDS
f
FDVH T ZLOOEHWDNHQDVWKHVDWXUDWLRQWHPSHUDWXUHDQG T WREHWKHQH[WKLJKHUWHPSHUDWXUHLQ
WKHVWHDPWDEOHV@
7KXVDWq& 03D F dV I
H dT K
P
N- NJ. P NJ. DWq& 03D C39 N- NJ. DQG
dV
P NJ. 7KXVDWq&
dT P
F I
H K
9
CVDW
N- NJ. P NJ. u
N- NJ
P NJ N- NJ.
DWq&
9
CVDW
N- NJ. P NJ. u
N- NJ
P NJ N- NJ.
D 0XOWLSO\RXWWKHWHUPVDQGWKLVLVHDVLO\SURYHG
w S
w
wT
E SUN
w Nw U w N U V T
T w N U V
FG IJ
H K
F I
H K
FG IJ
H K
w
wS
w U N V w N U V
EXW dS
w S
w Uw N
FG IJ
H K
G
P
wS
dU dV dN Ÿ
T
T
T
w N U V
F I FG w G IJ
H K
T HwU K
G FwTI
F w GI
G
J
G J
TN H w U K
NT H w U K
w
G
wU
T VN
N V
FG w S IJ H w Uw N K
G
VRWKDW
T
FG IJ
H K
G
F w GI F w T I
G J G J NT C
TN H w T K H w U K
V
DQG
> C3 DQG w V w T P ZLOOEHHYDOXDWHGXVLQJILQLWHGLIIHUHQFHVDERYHDQGVWHDPWDEOHV,QHDFK
C39
G wT
T w U V N
V
V
V
v
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
1RZ dG VdP SdT WKXV
FG w G IJ
HwTK
V
V
FG w P IJ S HwTK
V
DQG
SUN
SUN
F SVN
EXW G TS
U|
V|
W
R| F I
S| GH JK
T
V w P
G TS
NCv T w T V NT C9
H 7KXV
FG IJ
H K
wT
T w N U V
FG IJ
H K
wP
V
H
NC9T w T V NT C9
F I FG w P IJ P FG w T IJ H K T Hw NK T Hw NK
w wS
F w aG T fIJ FG w G IJ G FG w T IJ
G
H wV K
T H wV K
wV w N
T HwV K
F w GI
G J NTG FGH ww VT IJK NT FGH ww GT IJK FGH ww VT IJK NTG FGH ww VT IJK NT H w V K
FG w T IJ LMG T FG w G IJ OP
H w V K NT MN H w T K PQ
w wS
w N wV
w P
w N T U V
U V
U N
FG IJ
H K
G
wP
S V
wT V
C9 NT
NT C9
U V
U N
U N
U
U
U
U
U
U
U
EXW
dU
LM F w P I POPdV Ÿ F w T I
MN GH w T JK PQ GH w V JK
V
C9dT T
U
LM F I
MN GH JK
OP
PQ
wP
P T
C9
wT V
$OVR
FG w G IJ
HwTK
f w aG U f ˜ w aT V f
f w aT V f w aT U f
aw G w T f aw U w V f aw G w V f aw U w T f aw U w V f
FG w G IJ C aw G w V f H w T K aw U w V f
F w G IJ V FG w P IJ S DQG FG w G IJ V FG w P IJ dG VdP SdT G
HwTK HwTK
H wV K H wV K
U
a
a
w G U
w T U
V
T
T
V
T
9
T
V
T
V
V
T
7KXV
FG w G IJ
HwTK
V
U
FG w P IJ S C V aw P w V f HwTK
T aw P w T f P
9
V
T
V
T
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
GT
FG w G IJ H TS VT FG w P IJ T S C TV aw P w V f HwTK
HwTK
T aw P w T f P
|RST FG w P IJ P|UV|RSG T FG w G IJ |UV
|T H w T K |W|T H w T K |W
R|SH VT FG w P IJ U|VR|ST FG w P IJ PU|V C TV FG w P IJ H wV K
|T H w T K |W|T H w T K |W
9
U
V
V
T
V
U
9
V
FG IJ
H K
FG IJ
H K
U
V F w PI
|R
w P I |U|R F w P I
F
|
SH VT GH w T JK V|WS|TT GH w T JK PV|W NT GH w V JK
NC T |T
P wT
wP
T w N U V T w N U V
9
V
V
FG w S IJ w RS G UV FG w G IJ
Hw N K w N T T W T Hw NK
|RF w G I
|U G FG w T IJ GV S
G
J
TN |TH w N K
|W NT H w N K
T
G S NN
T
VRWKDWILQDOO\
SVN
V
HTQ
$
1RZ
U V
U V
U V
U V
U V
FG w G IJ V FG w P IJ SFG w T IJ G Hw NK
Hw NK Hw NK
V F w PI
S FwTI
G F wTI
ŸS
G
G
G J
J
J
TN H w N K
TN H w N K
T N Hw NK
V F w PI
H FwTI
V F w PI
HFwTI
G J T N GH w N JK T GH w N JK T GH w N JK
TN H w N K
dG VdP SdT GdN Ÿ
U V
NN
U V
U V
U V
U V
U V
U V
FG IJ
H K
V
H
wP
NC9T w T V NT C9
DQGIURPHTXDWLQJWKHWZRH[SUHVVLRQVIRU SVN FG IJ
H K
wP
T w N U V
U V
FG IJ
H K
U V
wT
T w N U V
U V
EXWIURPDERYH
FG IJ
H K
G wT
T w N U V
a
f V FG w P IJ V FG w P IJ NC T
NC H w T K
NT H w V K
H w P w T V
9
9
V
T
a
f
FG IJ
H K
3XWWLQJWKHVHH[SUHVVLRQVWRJHWKHU\LHOGV
S NN
FG IJ
H K
V w P w T V V w P
HV w P
H
NC9T w T V NC9T NC9
NT w V T
H ,WLVQRZVLPSOHDOJHEUDWRFRPELQHWKHH[SUHVVLRQVDERYHDQGWKRVHLQ6HFDQGVKRZWKDW
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
T
a
f
g
SUU S NN SUN
S S SUV SUN
UU VN
SUU
SUU SUU SVV SVN
b
LVH[DFWO\]HUR
0DWKFDGZRUNVKHHWVLQWKH0DWKFDG8WLOLWLHV'LUHFWRU\DUHDOVRDYDLODEOHWRGRWKHVHFDOFXODWLRQV 6WXGHQWVLQP\WKHUPRG\QDPLFVFRXUVHVKDYHSURGXFHGWKHUPRG\QDPLFVGLDJUDPVIRUPDQ\IOXLGV
XVLQJ WKH SURJUDP 35 DQG IROORZLQJ WKH PHWKRGV LQ LOOXVWUDWLRQV RI &KDSWHUV DQG 7KH
IROORZLQJ ILJXUHV DUH H[DPSOHV RI VRPH RI WKHVH GLDJUDPV ,W VKRXOG EH QRWHG WKDW DOO RI WKHVH
GLDJUDPVDUHLQTXDOLWDWLYHEXWQRWTXDQWLWDWLYHDJUHHPHQWZLWKWKHUPRG\QDPLFGLDJUDPVJHQHUDWHG
XVLQJ PRUH DFFXUDWH HTXDWLRQV RI VWDWH ,Q SDUWLFXODU OLTXLG GHQVLWLHV DUH QRW SUHGLFWHG YHU\
DFFXUDWHO\ IURP WKH 3HQJ5RELQVRQ HRV VR WKDW WKH ORFDWLRQ RI WKH WZRSKDVH GRPH LV VRPHZKDW
VKLIWHG DV DUH WKHRWKHU WKHUPRG\QDPLF SURSHUWLHV 'LDJUDPV IRURWKHU VXEVWDQFHVZLOO EHIRXQG LQ
WKH ILOH QDPHG ³2WKHU ILJV´ $OVR WKH 9,68$/ %$6,& FRPSXWHU SURJUDP ZLOO JHQHUDWH PDQ\ RI
WKHVHGLDJUDPVTXLFNO\DQGHDVLO\
7KHUPRG\QDPLFSURSHUWLHVRIQLWURJHQE\7RP3HWWL
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
3UHVVXUHYROXPHGLDJUDPIRUQLWURJHQ3HQJ5RELQVRQHRV
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
7KHUPRG\QDPLFSURSHUWLHVRIZDWHU VWHDP E\$OOHQ'RQQ
D 7KLVZRXOGEHDGLIILFXOWSUREOHPLILWZHUHQRWIRUWKHDYDLODELOLW\RIWKHSURJUDP358VLQJWKLV
SURJUDPWKHFULWLFDOSURSHUWLHVDQGWKHKHDWFDSDFLW\GDWDLQWKHWH[WDQGWKH T . P EDUUHIHUHQFHVWDWH ZKLFKFDQFHOVRXWRIWKHSUREOHP ZHILQGIRUHWK\OHQH
EDUDQG q & . H - PRO
S - PRO.
%\WULDODQGHUURUXVLQJJXHVVHGYDOXHVRITXQWLOZHREWDLQ P YDS EDU ZHREWDLQ
T . S 9 - PRO. V 9 u P PRO S/
- PRO. V /
u P PRO 1RZFRQVLGHULQJWKHIOXLGLQLWLDOO\LQWKHWDQNWKDWZLOOEHLQWKHWDQNILQDOO\DVWKHV\VWHPZH
KDYH
Ni
N f DQG S i
Sf E 1RZWKHUHFDQQRWEHRQO\YDSRULQWKHWDQN HQWURS\WRRKLJK RURQO\OLTXLG HQWURS\WRRORZ VRWKHUHPXVWEHWZRSKDVHPL[WXUH/HW x / PDVV RUPROH IUDFWLRQRIOLTXLG7KXV
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
c
h
x/ S / x/ S 9
h
x / x / x/
c
- PRO.
Si
x 9
7KXVZWRIIOXLGLQWDQNLVOLTXLGDQGLVYDSRU%DVHGRQPROHLQWDQNZHKDYH
V
u P PRO
u u u u YROXPHOLTXLG
u u u u YROXPHYDSRU
DOVRDYDLODEOHDVD0DWKFDGZRUNVKHHW FG w P IJ
HwTK
VDW
'H
T'V
FG w P IJ
HwTK
FG w OQ P IJ
H wT K
'H
ZRT VDW
F
H
w
OQ T
wT
T
T
T
I
K
T
T
T T
RU
'H
Z
ZR T - PRO DWq&
u u EXW
PV
RT
VDW
'H
ZRT IJ
K
7KXV
FG
H
'H
w OQ P
Ÿ
w T VDW
ZRT P
EXW
ZRT
P
$VVXPH V 9 !! V / Ÿ 'V a V 9
VR
B
V
Z
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
B
V
B
V RT P 7RILQGPXVH
7KHQ V
OQ P YDS
V
OQ P YDS EDU
u u u u P PRO DQG
B u P PRO :HVWDUWZLWK(TQ WKH&ODSH\URQHTXDWLRQ
w PVDW
'H
w T G , G ,, T'V
IJ
K
FG
H
)URP WKH SUREOHP VWDWHPHQW 'H
P
VDW
EDUDW T
q&
. %DVHGRQRWKHUK\GURFDUERQVZHFDQJXHVVWKDW
' IXV V a WR u P NJ :H ZLOO XVH WKLV DV DQ HVWLPDWH DQG GHWHUPLQH WKH HIIHFW RQ Tm $OVR WKH PROHFXODU ZHLJKW RI
KH[DGHFDQHLV7KXV
dP
d OQ T
N- PRO u - N-
G u P NJ u J PRO u NJ J
G
u G
N- PRO EXW QR GDWD RQ 'V LV JLYHQ $OVR
dP
T
G
u - P
>ZKHUHGLVRU@
- P u EDU ˜ P N- u N- -
d OQ T Ÿ P EDU
H[S
u T
.
G
LIG LIG uG O R
LM N PQ ST
OQ
G
EDU
6RWKHIUHH]LQJSRLQWLVUDLVHGEHWZHHQDQG.GHSHQGLQJRQWKH XQNQRZQ YDOXHRI ' IXV V DOVRDYDLODEOHDVD0DWKFDGZRUNVKHHW 7KLVLVDRQHFRPSRQHQWDGLDEDWLFIODVKSURFHVV,ZLOODVVXPHWKDWRQO\YDSRUOLTXLGDUHSUHVHQW
DQGWKHQVKRZWKDWWKLVLVLQGHHGWKHFDVH
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
7KHUHDUHWZRZD\VWRVROYHWKLVSUREOHP2QHLVWRFDOFXODWHDOOWKHWKHUPRG\QDPLFSURSHUWLHVDQG
WKHVHFRQGLVWRXVHWKHVWHDPWDEOHV%RWKPHWKRGVZLOOEHFRQVLGHUHGKHUH
&DOFXODWLQJDOOWKHUPRG\QDPLFSURSHUWLHVDQGDVVXPLQJWKHYDSRUSKDVHLVLGHDO
HQHUJ\EDODQFH U / T q &
x U / T xU 9 T F
H
P YDS
P 9 DQG G /
HTXLOLEULXPUHTXLUHPHQWV T / T 9 P /
$OVRXVLQJGDWDVXSSOLHGHDUOLHU
H[S G9 Ÿ P
P YDS I
K
T
DQGE\WKHLGHDOJDVODZ
P
x PRO u u EDU ˜ P PRO. T
u x N 9 RT
V9
YROXPHWDNHQXS
E\OLTXLG
F
H
x u u T
H[S T
u x $OVRZHKDYHIRUWKHLQWHUQDOHQHUJLHV
(TXDWLQJPDQG P YDS ZHKDYH
U / T
U / T
q &
UHIHUHQFHVWDWH
q &
q & u - Jq &
U / T
u T U 9 T
q &
U 9 T
T u 'H YDS RT
- J
DVVXPLQJCv
FRQVWDQW
u - J
VRWKDW
u T
I x ˜ K x x u u T x T u ,ILQGWKDWWKHVROXWLRQWRWKHVHHTXDWLRQVLV
T
. DQG P EDU x J 7KLVLVVRIDUDERYHWKHPHOWLQJSRLQWRIZDWHUWKDWWKHSUHVHQFHRIDQLFHSKDVHLVLPSRVVLEOH
8VLQJWKHVWHDPWDEOHV
HQHUJ\EDODQFH
˜ U / T
q &
u x U / T xU 9 T ERWKDWVDWXUDWLRQ
DOVR P
P VDW T DQG
V
P
x V / T xV 9 T Solutions to Chemical and Engineering Thermodynamics, 5th ed.
3URFHGXUH
L *XHVVTJHW P VDW T V / T V 9 T U / T DQG U 9 T IURPVWHDPWDEOHV
LL 6HHLI(TQV DQG DUHVDWLVILHGE\XVLQJ(TQ WRJHWxDQGWKHQVHHLQJLI(TQ LV
VDWLVILHG
)RUH[DPSOHJXHVV T q & u P J V / U / - J u P J V 9 U 9 - J P
N3D Ÿ x HTQ EDU DQG x
JUDPV E\LWHUDWLRQDQGLQWHUSRODWLRQ
T
x HTQ q & P
'LIIHUHQFH EHWZHHQ WKLV VROXWLRQ DQG WKH SUHYLRXV RQH LV GXH WR WKH LQDFFXUDFLHV RI WKH
DSSUR[LPDWHYDSRUSUHVVXUHHTXDWLRQLQ3DUWDQGWKHDVVXPSWLRQRIFRQVWDQWKHDWFDSDFLWLHV
$OOWKHPVGDWDIRUWKLVSUREOHPZDVREWDLQHGZLWKDVLPSOHEDVLFODQJXDJHSURJUDPZULWWHQIRUWKLV
SUREOHP &DOFXODWLRQV ZHUH GRQH IRU QEXWDQH DV D UHSUHVHQWDWLYH IOXLG 7KH YDQ GHU :DDOV ORRS
UHJLRQLVVKRZQRQWKHGLDJUDP:KDWLVLQWHUHVWLQJLVWKDWLQDGGLWLRQWRWKHYDQGHU:DDOVORRS
WKHUHLVPXFKVWUXFWXUHLQWKHP V SORW0XFKRILWRFFXUVLQWKHUHJLRQRI b ! V DQG V VRWKDW
LW KDV QR UHOHYDQFH WR RXU FDOFXODWLRQV ,Q WKH UHJLRQ V ! b WKHUH LV RQO\ WKH YDQ GHU :DDOV ORRS
EHKDYLRU DW ORZ UHGXFHG WHPSHUDWXUHV DQG WKH K\SHUEROLF EHKDYLRU PV RT DW YHU\ KLJK
WHPSHUDWXUHV 7KH PDLQ SRLQW LV WKDW WKH FXELF HTXDWLRQV ZH XVH H[KLELW TXLWH FRPSOLFDWHG P V EHKDYLRUEXWRQO\UHODWLYHO\VLPSOHEHKDYLRULQWKHUHJLRQRILQWHUHVWWRXVZKLFKLV V ! b Solutions to Chemical and Engineering Thermodynamics, 5th ed.
P V GLDJUDPIRUQEXWDQHFDOFXODWHGZLWKWKH3HQJ5RELQVRQHTXDWLRQRIVWDWHIRUUHDOL]DEOH V ! b DQG
SK\VLFDOO\XQUHDOL]DEOH V b UHJLRQV
/HW T( WKHHTXLOLEULXPWUDQVLWLRQWHPSHUDWXUHZKHQERWKVROLGSKDVHVDUHVWDEOH
dG VdP SdT $OVR dH TdS VdP VRDWFRQVWDQWSUHVVXUH
Cp
wH
wS
wT P T wT P T
FG IJ
H K
FG IJ
H K
ŸSKDVHZLWKKLJKHUKHDWFDSDFLW\ZLOOKDYHDKLJKHUHQWURS\VLQFHWKHHQWURS\RIERWKSKDVHVDUH
]HURDW.
7KHQDJDLQDWFRQVWDQWSUHVVXUH
FG w G IJ
HwTK
S P
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
6LQFH ERWK SKDVHV KDYH WKH VDPH *LEEV IUHH HQHUJ\ DW WKH WHPSHUDWXUH T WKLV LPSOLHV WKDW WKH
VXEVWDQFHZLWKWKHODUJHUHQWURS\ ZKLFKDULVHVIURPODUJHUKHDWFDSDFLW\ ZLOOKDYHWKHORZHU*LEEV
IUHHHQHUJ\DQGWKHUHIRUHEHWKHVWDEOHSKDVH
PV
RT
FG w P IJ
H wV K
BT
P
V
T
RT B T RT
V
V
a
PB
RT RT
r
RT
P P
B!
LM
N
a
OPLM
QN
PB
RT
r RT
P
B
RT B T RT
V
V
V
V
V
B T
!
B T ! B
T
!
V V
V
%DFNWRYLULDOHT
r PB RT
PV V B T V
P RT
RT
V
f
!
f
OP
Q
RT
IOXLGZLOOEHVWDEOH
P
r PB RT
V
,QIDFW B T ! LVVXIILFLHQWVLQFH B T V LQDOOFRQGLWLRQVZKHUHVHFRQGYLULDOFRHIILFLHQWLV
XVHG
V
RT
$SSUR[LPDWLRQ B T ! Ÿ B T a! IRUVWDELOLW\
P
(DV\ZD\ dV dS Q dU
Q P
SJHQ dt dt T
dt
6\VWHPRIFRQVWDQWHQWURS\ Q TSJHQ $OVRFRQVWDQWSUHVVXUH
d
dV
dU
PV
TSJHQ P
TSJHQ dt
dt
dt
dU d
d
dH
PV
U PV
TSJHQ d dt dt
dt
dt
Ÿ H PD[LPXPDWHTXLOLEULXP
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
dH
d H t Ÿ
dH
TdS VdP
FG w T IJ dS FG w T IJ dSdP FG w V IJ dPdS FG w V IJ dP
Hw SK
H w PK
Hw SK
H w PK
F w T IJ aw S f t Ÿ FG w T IJ t d H G
Hw SK
Hw SK
FG w T IJ T t Ÿ C ! Hw SK C
d H
P
S
P
P
P
3
3
P
0RUHWKHRUHWLFDOO\FRUUHFWZD\
(TXLOLEULXPFULWHULRQIRUDFORVHGV\VWHPDWFRQVWDQWHQWURS\DQGSUHVVXUH
dU
dV dS Q Q P
SJHQ dt
dt dt T
dS
Q T
TSJHQ
dt
dU
dS
dV
T
P
TSJHQ
dt
dt
dt
dV
d
dS
&RQVWDQWHQWURS\
PV FRQVWDQWSUHVVXUH P
dt
dt
dt
d
dU
d
U PV
TSJHQ Ÿ
PV TSJHQ dt
dt
dt
dH
dH
TSJHQ d Ÿ
d Ÿ H PLQLPXPVWDELOLW\ dt
dt
d H ! EXW dH TdS VdP Gi dN d H
cH H hcdS h c H H hcdN h
c H H hdS dN ! N N
N H cdS h N H dS dN N H cdN h ! N N
,
66
,
61
, ,,
66
,
,,
61
,,
,
,
,,
,
11
,
,
, ,
66
,
0DNLQJDWUDQVIRUPDWLRQRIYDULDEOHV
H
dx dS , 61 dN dx dN , H11
T
$VDFKHFN
T dx T dx
FG
H
66
,
,
,
, ,
11
11
61
NH66
H61
dN ,
H66
NH 66dS, NH 66 ˜
NH 11 dN , ,
61
c NH NH N H h NH66 T NH66 dS , , ,,
11
S
IJ NH NH N H dN K
NH
11
61
,
66
H 61 , ,
H
,
dS dN NH 66 ˜ 61
dN
H 66
H 66
NH 61
dN ,
H 66
ZKLFKLVFRUUHFWVR
66
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
T dx T dx t Ÿ T ! T ! FG w H IJ N w FG w H IJ N FG w T IJ ! H w S K w S H w S K Hw SK
C
F w V IJ FG w S IJ C Ÿ FG w S IJ NC dT G
EXW d S
Hw TK Hw TK T Hw TK T
T
T
NH66
N
P
P
3
P
3
3
T
ŸN
! N ! T ! Ÿ C3 ! NC3
6HFRQGFULWHULRQ
NH66 NH11 N H61
w H
H11
NH66
w N S P
P
P
IJ
K
FG
H
w
wN
w
FG IJ
FwTI
H K w N T GH w N JK
H
F w G IJ N aw T w N f
NH N
NG
Hw NK
H
aw T w S f
w
wH
w N S P w S P N
H61
P
SP
11
66
S P
S P
SP
SP
61
G
FG w G IJ
Hw NK
S P
"
P N
$YDLODEOHDVD0DWKFDGZRUNVKHHW
RT
&ODXVLXV(26 P
V b T
FG w P IJ H wV K
&RQGLWLRQIRUVWDELOLW\LV
)RUWKH&ODXVLXVHTXDWLRQ
wP
RT
6LQFH R ! T ! DQG V b ! wV T
V b T
FG IJ
H K
F w P IJ PXVWEHQHJDWLYHRU
7KHQ G
H wV K
FG w P IJ Ÿ 6LQJOHSKDVHLVVWDEOHDWDOOFRQGLWLRQV
H wV K
T
T
6HHVROXWLRQWR3UREOHP,IIOXLGLVXQVWDEOHWKHQDYDSRUOLTXLGSKDVHWUDQVLWLRQFDQRFFXU
5HGOLFK.ZRQJHTXDWLRQRIVWDWH
OQ
f
P
z FGH
IJ z FG
K
H
RT
V ZRT P
V f
z FGH RTV P dV
IJ
K
RT
P dV OQ Z Z V
V
V f
IJ
K
RT
RT
aT
dV
V V b V V b
z
V
RT OQ
V
V b
dV
RT OQ
a T
V of
V b V of
V V b
V f
RT OQ
V
V b
a OQ
V b
b
V
F I FG
H K H
IJ RT OQ Z a OQF Z B I
K
ZB b H Z K
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
F
H
I
K
Z
a
ZB
OQ
OQ Z Z Z B bRT
Z
a
ZB
Z OQ
OQ Z B bRT
Z
a
Z Pb RT
OQ
Z OQ Z B bRT
Z
aP
bP
A
B
RT
RT f
A
ZB
OQ
Z OQ Z B OQ
P
B
Z
8VLQJWKHVDPHDQDO\VLVIRUWKH6RDYH5HGOLFK.ZRQJHTXDWLRQRIVWDWHOHDGVWRWKHIROORZLQJ
Pb
aT
Z Pb RT
f
Z OQ Z OQ
OQ
RT
RTb
Z
P
aT
ZB
OQ
Z OQ Z B RTb
Z
OQ
f
P
OQ
F
H
F
H
F
H
I
K
F
H
I
K
I
K
LM a
N
LM OP
N Q
I
K
IL 3L H[S IRS L
fO
PQ
3
L
3D
$YDLODEOHDVD0DWKFDGZRUNVKHHW6HH0DWKFDGZRUNVKHHWIRUWKHJUDSKV
9
L
ORJ
E
I
L
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
5 7
D
3L 9 E
7 9L 9L E
L
9L
IRS L OQ
OQ =L
=L
9L E
I 3 H[S IRS
L
L
L
3
L
3D
I
L
=L
D
7
5 E
OQ
3L 9L
P PROH
9L
9L
ORJ
9
57
E
9
L
ORJ
E
9
L
E
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
D ,IHWKDQROLVDQLGHDOJDVWKH
f P, VRWKDWWKHIXJDFLW\RIHWKDQROLVN3D
E 6WDUWLQJIURPHTQZHKDYHWKDW
OQ I3
%9 OQ
Z =-
3 3D
˜
7
PRO ˜ .
5˜ 7
9
3˜ 9
*LYHQ
5˜ 7
=
3˜ 9
IVDW 3˜ H[S«ª
%
¬9
9
3
P
% ˜ PRO˜ .
˜
PRO
u P
%
9
=
5˜ 7
3D˜ P
5 ˜
9 )LQG 9
9
u P
˜ OQ = = »º
IVDW
¼
u 3D
$YDLODEOHDVD0DWKFDGZRUNVKHHW
7KHGHQVLW\RIHWKDQROLVJFFDW&ZKLFKZHZLOODOVRXVHDW&DQGLWVPROH
ZHLJKWLV7KHUHIRUHLWVOLTXLGPRODUYROXPHLV
NJ ·
§
¨ ˜ ˜
¸
PRO ¹
©
9O NJ
˜ ˜
D
N3D ˜ 3D
03D ˜ 3D
9O u P
P
I IVDW ˜ H[Sª«
¬
> ˜ 03D ˜ N3D ˜ 9O@ º
»
¼
5˜ 7
I
u 3D
E
99 S 9O˜ ª¬ ˜ ˜ N3D
˜ ˜ N3D S º¼
˜ 03D
§
·
¨ ˜´
¸
µ
99 S GS
I IVDW ˜ H[S
¨ 5˜ 7 ¶
¸
˜ N3D
©
¹
I
99 ˜ 03D
u P
u 3D
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
)8*$&,7<&$/&8/$7,2186,1*65.(48$7,21
5HDGLQSURSHUWLHVIRU3HQWDQH
7F 3F RP NDSSDFDOFXODWLRQ
NDS ˜ RP ˜ RP˜ RP
65.&RQVWDQWV
5 1RWHWKDWWKHVHDUHEHLQJGHILQHGDVD
IXQFWLRQRIWHPSHUDWXUH
&$ 7 3 ª
¬
5˜ 7F
D 7 ˜3
§
©
&% 7 3 DF ˜
3F
DOI 7 ˜ « NDS˜ ¨ 5˜ 7
= 7 3 E ˜
3˜ E
5˜ 7
7 ·º
5 ˜ 7F
3F
¸»
7F ¹ ¼
D 7 DF ˜ DOI 7
'D 7 G
D 7
G7
$ m &$ 7 3
% m &% 7 3
§ $ ˜ % ·
¨
¸
$ % %¸
9m ¨
¨
¸
¨
¸
©
¹
9HFWRURIFRHIILFLHQWVLQWKH65.HTXDWLRQ
LQWKHIRUP
$ % $%A% = =A=A
== m SRO\URRWV 9
6ROXWLRQWRWKHFXELF
IRU L  == m LI ,P == z L
L
6HWDQ\LPDJLQDU\URRWVWR]HUR
6RUWWKHURRWV
== m VRUW ==
·
© ¹
·
§
== == m == LI
© ¹
== m ==
LI § ==
6HWWKHYDOXHRIDQ\LPDJLQDU\URRWV
WRYDOXHRIWKHUHDOURRW
==
(QWHUWHPSHUDWXUH7DQGSUHVVXUH3
7 &
7 7
.
3 )XJDFLW\H[SUHVVLRQV>DFWXDOO\OQ I3 @IRUWKHOLTXLGIODQGYDSRUIY
IO 7 3 = 7 3
IY 7 3 = 7 3
OQ = 7 3
OQ = 7 3
&% 7 3
&% 7 3
&$ 7 3
&% 7 3
&$ 7 3
&% 7 3
§ = 7 3 &% 7 3 ·
¸
= 7 3
¨
¸
©
¹
˜ OQ¨
§ = 7 3 &% 7 3 ·
¸
= 7 3
¨
¸
©
¹
˜ OQ¨
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
)XJDFLW\)XJDFLW\FRHIILFLHQW
IXJO 3˜ H[S IO 7 3
IXJO
IO 7 3
IXJY 3˜ H[S IY 7 3
IXJY
IY 7 3
SKLO IXJO
3
SKLY IXJY
3
SKLO
SKLY
6800$5<2)5(68/76
7
.
9DSRUSUHVVXUHEDU
3
/,48,'9$325
&RPSUHVVLELOLW\
= 7 3
)XJDFLW\FRHIILFLHQW
SKLO
SKLY
)XJDFLW\EDU
IXJO
IXJY
= 7 3
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
5HDGLQSURSHUWLHVIRU%HQ]HQH
7F 3F RP NDSSDFDOFXODWLRQ
NDS ˜ RP ˜ RP˜ RP
65.&RQVWDQWV
5 1RWHWKDWWKHVHDUHEHLQJGHILQHGDVD
IXQFWLRQRIWHPSHUDWXUH
&$ 7 3 5˜ 7F
§
©
ª
¬
D 7 ˜3
&% 7 3 DF ˜
3F
DOI 7 ˜ « NDS˜ ¨ 5˜ 7
= 7 3 E ˜
3˜ E
5˜ 7
7 ·º
D 7 DF ˜ DOI 7
'D 7 G
D 7
G7
$ m &$ 7 3
§ $ ˜ % ·
¨
¸
$ % %¸
¨
9m
¨
¸
¨
¸
©
¹
9HFWRURIFRHIILFLHQWVLQWKH65.HTXDWLRQ
LQWKHIRUP
$ % $%A% = =A=A
== m SRO\URRWV 9
6ROXWLRQWRWKHFXELF
IRU L  LI ,P == z L
L
6HWDQ\LPDJLQDU\URRWVWR]HUR
6RUWWKHURRWV
== m VRUW ==
·
© ¹
·
§
== m == LI
== © ¹
== m ==
LI § ==
6HWWKHYDOXHRIDQ\LPDJLQDU\URRWV
WRYDOXHRIWKHUHDOURRW
==
(QWHUWHPSHUDWXUH7DQGSUHVVXUH3
7 &
7 7
.
3 )XJDFLW\H[SUHVVLRQV>DFWXDOO\OQ I3 @IRUWKHOLTXLGIODQGYDSRUIY
IO 7 3 = 7 3
IY 7 3 = 7 3
OQ = 7 3
OQ = 7 3
&% 7 3
&% 7 3
&$ 7 3
&% 7 3
&$ 7 3
&% 7 3
§ = 7 3 &% 7 3 ·¸
¨
¸
= 7 3
©
¹
˜ OQ¨
§ = 7 3 &% 7 3 ·¸
¨
¸
= 7 3
©
¹
˜ OQ¨
3F
¸»
7F ¹ ¼
% m &% 7 3
== m 5 ˜ 7F
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
)XJDFLW\)XJDFLW\FRHIILFLHQW
IXJO 3˜ H[S IO 7 3
IXJO
IO 7 3
IXJY 3˜ H[S IY 7 3
IXJY
IY 7 3
SKLO IXJO
3
SKLY IXJY
3
SKLO
SKLY
6800$5<2)5(68/76
7
.
9DSRUSUHVVXUHEDU
3
/,48,'9$325
&RPSUHVVLELOLW\
= 7 3
)XJDFLW\FRHIILFLHQW
SKLO
SKLY
)XJDFLW\EDU
IXJO
IXJY
= 7 3
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
= 7 3 $ &$ 7 3
% &% 7 3
$ %
9HFWRURIFRHIILFLHQWVLQWKH65.HTXDWLRQ
$ % %
LQWKHIRUP
9
$ % $%A% = =A=A
6ROXWLRQWRWKHFXELF
== SRO\URRWV 9
IRU L  ==L LI ,P ==L 6HWDQ\LPDJLQDU\URRWVWR]HUR
== VRUW ==
6RUWWKHURRWV
== == LI == 6HWWKHYDOXHRIDQ\LPDJLQDU\URRWV
WRYDOXHRIWKHUHDOURRW
== == LI == ==
(QWHUWHPSHUDWXUH7DQGSUHVVXUH3
&
7 7 7 .
3 )XJDFLW\H[SUHVVLRQV>DFWXDOO\OQ
I3 @IRUWKHOLTXLGIODQGYDSRUIY
&$ 7 3 = 7 3 &% 7 3
IO 7 3 = 7 3 OQ = 7 3 &% 7 3
OQ
&% 7 3
= 7 3 &$ 7 3 = 7 3 &% 7 3
IY 7 3
= 7 3 OQ = 7 3 &% 7 3
OQ
&% 7 3
= 7 3 Solutions to Chemical and Engineering Thermodynamics, 5th ed.
6ROXWLRQXVLQJ0DWKFDGZRUNVKHHW T R&
PYDS EDU
9
Z
Z / u H9
u H/
u S9
S/
T R&
Z 9 H9
u S 9 T R&
Z 9 PYDS EDU
Z / u H/
u S/
PYDS EDU
Z / H9
u H/
u S9
S/
T R&
Z 9 PYDS EDU
Z / H9
u H/
u S9
S/
T R&
Z 9 PYDS EDU
Z / H9
u H/
u S9
S/
T R&
Z 9 PYDS EDU
Z / H9
u H/
u S9
S/
T R&
Z 9 PYDS EDU
Z / H9
u H/
u S9
S/
7KH0DWKFDGZRUNVKHHWIRUWKLVILOHLVVKRZQEHORZ
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
9$32535(6685(&$/&8/$7,2186,1*65.(48$7,21
5HDGLQSURSHUWLHVIRUR[\JHQ
7F 3F RP +HDWFDSDFLW\FRQVWDQWV
&S &S ˜ &S ˜ &S ˜ 5HIHUHQFHVWDWHDQGNDSSDFDOFXODWLRQ
7UV 3UV NDS ˜ RP ˜ RP˜ RP
65.&RQVWDQWV
5 1RWHWKDWWKHVHDUHEHLQJGHILQHGDVD
IXQFWLRQRIWHPSHUDWXUHVLQFHZHZLOO
QHHGWRLQWHUDWHRQWHPSHUDWXUH
&$ 7 3 ª
¬
5 ˜ 7F
D 7 ˜3
&% 7 3 DF ˜
3F
§
©
DOI 7 ˜ « NDS˜ ¨ 5˜ 7
= 7 3 E ˜
7 ·º
3˜ E
3F
¸»
7F ¹ ¼
5˜ 7
5 ˜ 7F
D 7 DF ˜ DOI 7
'D 7 G
D 7
G7
$ m &$ 7 3
% m &% 7 3
§ $ ˜ % ·
¨
¸
$ % %¸
9m ¨
¨
¸
¨
¸
©
¹
9HFWRURIFRHIILFLHQWVLQWKH65.HTXDWLRQ
LQWKHIRUP
$ % $%A% = =A=A
== m SRO\URRWV 9
6ROXWLRQWRWKHFXELF
IRU L  == m L
LI ,P == z L
== m VRUW ==
·
© ¹
·
§
== m == LI
== © ¹
== m ==
LI § ==
6HWDQ\LPDJLQDU\URRWVWR]HUR
6RUWWKHURRWV
6HWWKHYDOXHRIDQ\LPDJLQDU\URRWV
WRYDOXHRIWKHUHDOURRW
==
(QWHUWHPSHUDWXUH7DQGSUHVVXUH3
'HSHQGLQJRQZKDWLVVSHFLILHGLQWKH*LYHQDQG)LQGVWDWHPHQWV
EHORZHLWKHU7RU3LVVSHFLILHGDQGWKHRWKHULVDQLQLWLDOJXHVV
ZKLFKPD\KDYHWREHDGMXVWHGDVWKHRWKHUYDULDEOHLVFKDQJHG
HVSHFLDOO\DVWKHFULWLFDOSRLQWLVDSSURDFKHG7KLVZRUNVKHHWZLOO
SUREDEO\QRWFRQYHUJHWRDVROXWLRQZKHQ7RU3DUHZLWKLQWR
RIWKHFULWLFDOYDOXHVXQOHVVDQH[WUHPHO\JRRGLQLWLDOJXHVVLV
SURYLGHG2QHZD\WRREWDLQDJRRGLQLWLDOLVWRVWDUWZHOOEHORZWKH
FULWLFDOUHJLRQDQGVWHSWRZDUGVLWXVLQJWKHUHVXOWRISUHYLRXVFDOFXODWLRQV
7 7 7
3 &
.
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
)XJDFLW\H[SUHVVLRQV>DFWXDOO\OQ I3 @IRUWKHOLTXLGIODQGYDSRUIY
OQ = 7 3
IO 7 3 = 7 3
IY 7 3 = 7 3
*LYHQ
IO 7 3 IY 7 3
3 )LQG 3
OQ = 7 3
3
&% 7 3
&% 7 3
˜ OQ¨
&% 7 3
§ = 7 3 &% 7 3 ¸·
= 7 3
¨
¸
©
¹
&$ 7 3
˜ OQ¨
&% 7 3
(TXDWHORJRIIXJDFLW\FRHIILFLHQWV
6ROYHHTXDOLW\RIIXJDFLWLHV
§ = 7 3 &% 7 3 ¸·
= 7 3
¨
¸
©
¹
&$ 7 3
)XJDFLW\)XJDFLW\FRHIILFLHQW
IXJO
IXJO 3˜ H[S IO 7 3
IXJO
IO 7 3
SKLO IXJY 3˜ H[S IY 7 3
IXJY
IY 7 3
SKLY 3
IXJY
3
SKLO
SKLY
5HVLGXDOHQWURS\IRUOLTXLG '(/6/ DQGYDSRU '(/69 SKDVHV
'(/6/
§ = 7 3 &% 7 3 · · §
¨ 5˜ OQ = 7 3 &% 7 3 'D 7 ˜ OQ¨
¸ ¸ ˜ E
= 7 3
¨
¨
¸¸
©
©
¹¹
'(/69
§ = 7 3 &% 7 3 · · §
¨ 5˜ OQ = 7 3 &% 7 3 'D 7 ˜ OQ¨
¸ ¸ ˜ E
¨
¨
¸¸
= 7 3
©
©
¹¹
5HVLGXDOHQWKDOS\IRUOLTXLG '(/+/ DQGYDSRU '(/+9 SKDVHV
'(/+/
= 7 3 &% 7 3 ·º
«ª5˜ 7˜ = 7 3 7˜ 'D 7 D 7 ˜ OQ§¨
¸» ˜ «
¨
¸»
= 7 3
E
¬
©
¹¼
'(/+9
ª«
§ = 7 3 &% 7 3 ¸·»º 7˜ 'D 7 D 7
5 ˜ 7˜ = 7 3 ˜ OQ¨
˜ = 7 3
«
¨
¸»
E
¬
©
¹¼
,GHDOJDVSURSHUWLHVFKDQJHVUHODWLYHWRWKHUHIHUHQFHVWDWH
'(/+,* &S ˜ 7 7UV '(/6,* &S ˜ OQ§¨
7 ·
&S ˜ 7 7UV
¸ &S ˜ 7 7UV © 7UV ¹
&S ˜ 7 7UV
&S ˜ 7 7UV
&S ˜ 7 7UV
&S ˜ 7 7UV
3 ·
¸
© 3UV ¹
5˜ ˜ OQ§¨
7RWDOHQWURS\DQGHQWKDOS\UHODWLYHWRLGHDOJDVUHIHUHQFHVWDWH
6/ '(/6,* '(/6/
69 '(/6,* '(/69 +/ '(/+,* '(/+/ +9 '(/+,* '(/+9
6800$5<2)5(68/76
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
7
.
9DSRUSUHVVXUHEDU
3
/,48,'9$325
&RPSUHVVLELOLW\
= 7 3
(QWKDOS\-PRO
+/
u +9
u (QWURS\-PRO.
6/
69
)XJDFLW\FRHIILFLHQW
SKLO
SKLY
)XJDFLW\EDU
IXJO
IXJY
u = 7 3
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
3XUHFRPSRQHQWSURSHUWLHVFDOFXODWLRQXVLQJWKH65.HTXDWLRQRIVWDWH
5HDGLQSURSHUWLHVIRUR[\JHQ
7F 3F RP +HDWFDSDFLW\FRQVWDQWV
&S &S ˜ &S ˜ &S ˜ 5HIHUHQFHVWDWHDQGNDSSDFDOFXODWLRQ
7UV 3UV NDS ˜ RP ˜ RP˜ RP
65.&RQVWDQWV
5 1RWHWKDWWKHVHDUHEHLQJGHILQHGDVD
IXQFWLRQRIWHPSHUDWXUHIRUFRQYHQLHQFH
&$ 7 3 ª
¬
5 ˜ 7F
D 7 ˜3
&% 7 3 DF ˜
3F
§
©
DOI 7 ˜ « NDS˜ ¨ 5˜ 7
= 7 3 E ˜
3˜ E
5˜ 7
7 ·º
5 ˜ 7F
3F
¸»
7F ¹ ¼
D 7 DF ˜ DOI 7
'D 7 G
D 7
G7
$ m &$ 7 3
% m &% 7 3
§ $ ˜ % ·
¨
¸
$ % %¸
9m ¨
¨
¸
¨
¸
©
¹
9HFWRURIFRHIILFLHQWVLQWKH65.HTXDWLRQ
LQWKHIRUP
$ % $%A% = =A=A
== m SRO\URRWV 9
6ROXWLRQWRWKHFXELF
IRU L  == m L
LI ,P == z L
== m VRUW ==
·
© ¹
·
§
== m == LI
== © ¹
== m ==
LI § ==
6HWDQ\LPDJLQDU\URRWVWR]HUR
6RUWWKHURRWV
6HWWKHYDOXHRIDQ\LPDJLQDU\URRWV
WRYDOXHRIWKHUHDOURRW
==
(QWHUWHPSHUDWXUH7DQGSUHVVXUH3
7 &
7 7
.
3 Solutions to Chemical and Engineering Thermodynamics, 5th ed.
)XJDFLW\H[SUHVVLRQV>DFWXDOO\OQ I3 @IRUWKHOLTXLGIODQGYDSRUIY
§ = 7 3 &% 7 3 ·¸
&$ 7 3
IO 7 3 = 7 3 OQ = 7 3 &% 7 3 ˜ OQ¨
&% 7 3
¨
¸
= 7 3
©
IY 7 3 = 7 3
OQ = 7 3
&% 7 3
¹
§ = 7 3 &% 7 3 ·
¸
= 7 3
¨
¸
©
¹
&$ 7 3
˜ OQ¨
&% 7 3
)XJDFLW\)XJDFLW\FRHIILFLHQW
IXJO
IXJO 3˜ H[S IO 7 3
IXJO
IO 7 3
SKLO IXJY 3˜ H[S IY 7 3
IXJY
IY 7 3
SKLY 3
IXJY
3
SKLO
SKLY
5HVLGXDOHQWURS\IRUOLTXLG '(/6/ DQGYDSRU '(/69 SKDVHV
'(/6/
§ = 7 3 &% 7 3 ¸· ·¸ §¨
'D 7
5˜ OQ = 7 3 &% 7 3 ˜ OQ¨
˜ = 7 3
E
¨
¨
¸¸
©
©
¹¹
'(/69
§¨
§ = 7 3 &% 7 3 ·¸ ·¸ 'D 7
5˜ OQ = 7 3 &% 7 3 ˜ OQ¨
˜ E
¨
¨
¸¸
= 7 3
©
©
¹¹
5HVLGXDOHQWKDOS\IRUOLTXLG '(/+/ DQGYDSRU '(/+9 SKDVHV
'(/+/
§ = 7 3 &% 7 3 ¸·º» ª«
7˜ 'D 7 D 7
5 ˜ 7˜ = 7 3 ˜ OQ¨
˜ E
«
¨
¸»
= 7 3
¬
©
¹¼
'(/+9
ª
§ = 7 3 &% 7 3 ·º «5˜ 7˜ = 7 3 7˜ 'D 7 D 7 ˜ OQ¨
¸» ˜ = 7 3
«
¨
¸»
E
¬
©
¹¼
,GHDOJDVSURSHUWLHVFKDQJHVUHODWLYHWRWKHUHIHUHQFHVWDWH
'(/+,* &S ˜ 7 7UV '(/6,* &S ˜ OQ§¨
7 ·
&S ˜ 7 7UV
¸ &S ˜ 7 7UV © 7UV ¹
&S ˜ 7 7UV
&S ˜ 7 7UV
&S ˜ 7 7UV
&S ˜ 7 7UV
3 ·
¸
© 3UV ¹
5˜ ˜ OQ§¨
7RWDOHQWURS\DQGHQWKDOS\UHODWLYHWRLGHDOJDVUHIHUHQFHVWDWH
6/ '(/6,* '(/6/
69 '(/6,* '(/69 +/ '(/+,* '(/+/ +9 '(/+,* '(/+9 Solutions to Chemical and Engineering Thermodynamics, 5th ed.
7
9 = 7 3 ˜ ˜ ˜
3
9
7
9 = 7 3 ˜ ˜ ˜
3
9
6800$5<2)5(68/76
7
.
9DSRUSUHVVXUHEDU
3
/,48,'9$325
&RPSUHVVLELOLW\
= 7 3
9ROXPHPANPRO
9
(QWKDOS\-PRO
+/
(QWURS\-PRO.
= 7 3
9
u +9
u 6/
69
)XJDFLW\FRHIILFLHQW
SKLO
SKLY
)XJDFLW\EDU
IXJO
IXJY
6RPHUHSUHVHQWDWLYHUHVXOWVDUHVKRZQEHORZ
T &
P EDU
Z
V
H
S
P EDU
Z
V
H
S
P EDU
Z
V
H
S
P EDU
Z
V
H
S
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
3XUHFRPSRQHQWSURSHUWLHVFDOFXODWLRQXVLQJWKH65.HTXDWLRQRIVWDWH
5HDGLQSURSHUWLHVIRU:DWHU
7F 3F RP +HDWFDSDFLW\FRQVWDQWV
&S &S ˜ &S ˜ &S ˜ 5HIHUHQFHVWDWHDQGNDSSDFDOFXODWLRQ
7UV 3UV NDS ˜ RP ˜ RP˜ RP
65.&RQVWDQWV
5 1RWHWKDWWKHVHDUHEHLQJGHILQHGDVD
IXQFWLRQRIWHPSHUDWXUHIRUFRQYHQLHQFH
&$ 7 3 ª
¬
5˜ 7F
D 7 ˜3
§
©
&% 7 3 DF ˜
3F
DOI 7 ˜ « NDS˜ ¨ 5˜ 7
= 7 3 E ˜
3˜ E
5˜ 7
7 ·º
5 ˜ 7F
3F
¸»
7F ¹ ¼
D 7 DF ˜ DOI 7
'D 7 G
D 7
G7
$ m &$ 7 3
% m &% 7 3
§ $ ˜ % ·
¨
¸
$ % %¸
¨
9m
¨
¸
¨
¸
©
¹
9HFWRURIFRHIILFLHQWVLQWKH65.HTXDWLRQ
LQWKHIRUP
$ % $%A% = =A=A
== m SRO\URRWV 9
6ROXWLRQWRWKHFXELF
IRU L  == m L
LI ,P == z L
6HWDQ\LPDJLQDU\URRWVWR]HUR
6RUWWKHURRWV
== m VRUW ==
·
© ¹
·
§
== == m == LI
© ¹
== m ==
LI § ==
6HWWKHYDOXHRIDQ\LPDJLQDU\URRWV
WRYDOXHRIWKHUHDOURRW
==
(QWHUWHPSHUDWXUH7DQGSUHVVXUH3
7 &
7 7
.
3 )XJDFLW\H[SUHVVLRQV>DFWXDOO\OQ I3 @IRUWKHOLTXLGIODQGYDSRUIY
IO 7 3 = 7 3
IY 7 3 = 7 3
OQ = 7 3
OQ = 7 3
&% 7 3
&% 7 3
&$ 7 3
&% 7 3
&$ 7 3
&% 7 3
§ = 7 3 &% 7 3 ·
¸
¨
¸
= 7 3
©
¹
˜ OQ¨
§ = 7 3 &% 7 3 ·
¸
¨
¸
= 7 3
©
¹
˜ OQ¨
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
)XJDFLW\)XJDFLW\FRHIILFLHQW
IXJO
SKLO IXJO 3˜ H[S IO 7 3
IXJO
IO 7 3
IXJY 3˜ H[S IY 7 3
IXJY
IY 7 3
u SKLY 3
IXJY
3
SKLO
SKLY
5HVLGXDOHQWURS\IRUOLTXLG '(/6/ DQGYDSRU '(/69 SKDVHV
'(/6/
§ = 7 3 &% 7 3 · · §
¨ 5˜ OQ = 7 3 &% 7 3 'D 7 ˜ OQ¨
¸ ¸ ˜ E
= 7 3
¨
¨
¸¸
©
©
¹¹
'(/69
§ = 7 3 &% 7 3 · · §
¨ 5˜ OQ = 7 3 &% 7 3 'D 7 ˜ OQ¨
¸ ¸ ˜ E
¨
¨
¸¸
= 7 3
©
©
¹¹
5HVLGXDOHQWKDOS\IRUOLTXLG '(/+/ DQGYDSRU '(/+9 SKDVHV
'(/+/
= 7 3 &% 7 3 ·º
«ª5˜ 7˜ = 7 3 7˜ 'D 7 D 7 ˜ OQ§¨
¸» ˜ «
¨
¸»
E
= 7 3
¬
©
¹¼
'(/+9
= 7 3 &% 7 3 ·º
«ª5˜ 7˜ = 7 3 7˜ 'D 7 D 7 ˜ OQ§¨
¸» ˜ «
¨
¸»
= 7 3
E
¬
©
¹¼
,GHDOJDVSURSHUWLHVFKDQJHVUHODWLYHWRWKHUHIHUHQFHVWDWH
'(/+,* &S ˜ 7 7UV '(/6,* &S ˜ OQ§¨
7 ·
&S ˜ 7 7UV
¸ &S ˜ 7 7UV &S ˜ 7 7UV
© 7UV ¹
&S ˜ 7 7UV
&S ˜ 7 7UV
&S ˜ 7 7UV
3 ·
¸
© 3UV ¹
5˜ ˜ OQ§¨
7RWDOHQWURS\DQGHQWKDOS\UHODWLYHWRLGHDOJDVUHIHUHQFHVWDWH
6/ '(/6,* '(/6/
69 '(/6,* '(/69 +/ '(/+,* '(/+/ +9 '(/+,* '(/+9
6800$5<2)5(68/76
7
.
9DSRUSUHVVXUHEDU
3
/,48,'9$325
&RPSUHVVLELOLW\
= 7 3
u = 7 3
(QWKDOS\-PRO
+/
u +9
u (QWURS\-PRO.
6/
69
)XJDFLW\FRHIILFLHQW
SKLO
SKLY
)XJDFLW\EDU
IXJO
IXJY
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
,6(17+$/3,&3(1*52%,1621(48$7,212)67$7(&$/&8/$7,21
7F 3F &S &S ˜ RP NDS RP
˜
RP
˜ ˜ RP
&S ˜ 5 E ˜
3HQJ5RELQVRQ&RQVWDQWV
,QSXWLQLWLDOWHPSHUDWXUHDQGSUHVVXUHRIFDOFXODWLRQ
,QSXWILQDOSUHVVXUH
§
©
ª
¬
7 ·º
7 7L
3 3L
D 7 DF ˜ DOI 7
&$ 7 3 D 7 ˜3
5˜ 7
G
'D 7 D 7
G7
= 7 3 3F
7L .
¸»
7F ¹ ¼
5˜ 7F
DF ˜
5 ˜ 7F
3F
EDU
3L EDU
3I ,QLWLDOVWDWHFDOFXODWLRQV
DOI 7 ˜ « NDS˜ ¨ &S ˜ &% 7 3 3˜ E
5˜ 7
$ m &$ 7 3
% m &% 7 3
«ª $ ˜ % % % »º
«
»
9 m « $ ˜ % ˜ % »
«
»
%
«
»
¬
¼
9HFWRURIFRHIILFLHQWVLQWKH35HTXDWLRQ
LQWKHIRUP
$ %%A%A $ %A % = % =A=A
6ROXWLRQWRWKHFXELF
== m SRO\URRWV 9
IRU L  == m L
LI ,P == z 6HWDQ\LPDJLQDU\URRWVWR]HUR
6RUWWKHURRWV
L
== m VRUW ==
·
© ¹
·
§
== m == LI
== © ¹
== m ==
LI § ==
==
&DOFXODWHLQLWDOSURSHUWLHV
&DOFXODWHLQLWLDOPRODUYROXPH
DQGHQWKDOS\DQGHQWURS\
GHSDUWXUH
6HWWKHYDOXHRIDQ\LPDJLQDU\URRWV
WRYDOXHRIWKHUHDOURRW
=I 7 3 = 7 3
9/ = 7 3 ˜ 5˜ 7
3
˜ = 7 3
§ ·
¨ ¸
¨
¸
© ¹
'(/+LQ
ª
ª = 7 3 ˜ &% 7 3 º º « 5˜ 7˜ = 7 3 7˜ 'D 7 D 7 ˜ OQ«
» » ˜ «
« = 7 3 ˜ &% 7 3 » »
˜ ˜ E
¬
¬
¼¼
'(/6LQ ª
ª = 7 3 ˜ &% 7 3 º º « 5˜ OQ = 7 3 &% 7 3 'D 7 ˜ OQ«
» » ˜ «
˜ ˜ E « = 7 3 ˜ &% 7 3 » »
¬
¬
¼¼
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
'(/+LQ u '(/6LQ *XHVVIRUILQDOVWDWH
7 ˜ 7L
3 3I
)XJDFLW\H[SUHVVLRQV>DFWXDOO\OQ I3 @IRUWKHOLTXLGIODQGYDSRUIY
IO 7 3 = 7 3
OQ = 7 3
IY 7 3 = 7 3
*LYHQ
IO 7 3 IY 7 3
OQ = 7 3
&% 7 3
&% 7 3
ª = 7 3 ˜ &% 7 3 º
»
« = 7 3 ˜ &% 7 3 »
¬
¼
&$ 7 3
˜ OQ«
˜ ˜ &% 7 3
ª = 7 3 ˜ &% 7 3 º
»
« = 7 3 ˜ &% 7 3 »
¬
¼
&$ 7 3
˜ OQ«
˜ ˜ &% 7 3
7 )LQG 7
7
5HVLGXDOHQWURS\IRUOLTXLG '(/6/ DQGYDSRU '(/69 SKDVHV
'(/6/ 7 3 ª = 7 3 ˜ &% 7 3 º º ª
« 5˜ OQ = 7 3 &% 7 3 'D 7 ˜ OQ«
» » ˜ «
˜ ˜ E « = 7 3 ˜ &% 7 3 » »
¬
¬
¼¼
'(/697 3 ª
ª = 7 3 ˜ &% 7 3 º º « 5˜ OQ = 7 3 &% 7 3 'D 7 ˜ OQ«
» » ˜ «
˜ ˜ E « = 7 3 ˜ &% 7 3 » »
¬
¬
¼¼
5HVLGXDOHQWKDOS\IRUOLTXLG '(/+/ DQGYDSRU '(/+9 SKDVHV
'(/+/ 7 3 ª
ª = 7 3 ˜ &% 7 3 º º « 5˜ 7˜ = 7 3 7˜ 'D 7 D 7 ˜ OQ«
» » ˜ «
« = 7 3 ˜ &% 7 3 » »
˜ ˜ E
¬
¬
¼¼
'(/+97 3 ª
ª = 7 3 ˜ &% 7 3 º º « 5˜ 7˜ = 7 3 7˜ 'D 7 D 7 ˜ OQ«
» » ˜ «
« = 7 3 ˜ &% 7 3 » »
˜ ˜ E
¬
¬
¼¼
,GHDOJDVSURSHUWLHVFKDQJHVUHODWLYHWRWKHLQLWLDOVWDWH
'(/+,*7 3 &S ˜ 7 7L &S ˜ 7 7L
7
'(/6,*7 3 &S ˜ OQ§¨ ·¸ &S ˜ 7 7L © 7L ¹
)LQGYDSRUOLTXLGVSOLW
&S ˜ 7 7L
&S ˜ 7 7L
&S ˜ 7 7L
&S ˜ 7 7L
5˜ ˜ OQ§¨
3·
¸
© 3L ¹
[ *LYHQ
[˜ '(/+97 3 [ ˜ '(/+/ 7 3 '(/+,*7 3
[ )LQG [
[
'(/+LQ
)UDFWLRQYDSRU
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
+9 '(/+97 3 '(/+,*7 3
69 '(/697 3 '(/6,*7 3
+/ '(/+/ 7 3 '(/+,*7 3
6/ '(/6/ 7 3 '(/6,*7 3
G+ [˜ +9 [ ˜ +/ '(/+LQ
G6 [˜ 69 [ ˜ 6/ '(/6LQ
6800$5<2)5(68/76
7HPSHUDWXUH.
)(('/,48,'9$325
7L 7 7 3UHVVXUHEDU
3L
3
3
9DSRUOLTXLGVSOLW
[
&RPSUHVVLELOLW\
= 7L 3L
(QWKDOS\-PRO
UHODWLYHWRIHHG
+/
u +9
(QWURS\-PRO.
UHODWLYHWRIHHG
6/
69
= 7 3
u (QWKDOS\FKDQJH
-PRO
(QWURS\FKDQJH
-PRO.
G+
G6
= 7 3
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
,6(17523,&3(1*52%,1621(48$7,212)67$7(&$/&8/$7,21
7F 3F &S &S ˜ RP NDS RP
˜
RP
˜ ˜ RP
&S ˜ 5 E ˜
3HQJ5RELQVRQ&RQVWDQWV
,QSXWLQLWLDOWHPSHUDWXUHDQGSUHVVXUHRIFDOFXODWLRQ
,QSXWILQDOSUHVVXUH
ª
¬
3I 7 ·º
§
©
7 7L
3 3L
D 7 DF ˜ DOI 7
&$ 7 3 D 7 ˜3
5˜ 7
G
'D 7 D 7
G7
= 7 3 3F
7L .
¸»
7F ¹ ¼
5˜ 7F
DF ˜
5 ˜ 7F
3F
EDU
3L EDU
,QLWLDOVWDWHFDOFXODWLRQV
DOI 7 ˜ « NDS˜ ¨ &S ˜ &% 7 3 3˜ E
5˜ 7
$ m &$ 7 3
% m &% 7 3
ª« $ ˜ % % % º»
«
»
9 m « $ ˜ % ˜ % »
«
»
%
«
»
¬
¼
9HFWRURIFRHIILFLHQWVLQWKH35HTXDWLRQ
LQWKHIRUP
$ %%A%A $ %A % = % =A=A
6ROXWLRQWRWKHFXELF
== m SRO\URRWV 9
IRU L  == m L
LI ,P == z 6HWDQ\LPDJLQDU\URRWVWR]HUR
6RUWWKHURRWV
L
== m VRUW ==
·
© ¹
== m == LI § == ·
© ¹
== m ==
LI § ==
==
&DOFXODWHLQLWDOSURSHUWLHV
&DOFXODWHLQLWLDOPRODUYROXPH
DQGHQWKDOS\DQGHQWURS\
GHSDUWXUH
6HWWKHYDOXHRIDQ\LPDJLQDU\URRWV
WRYDOXHRIWKHUHDOURRW
=I 7 3 = 7 3
9/ = 7 3 ˜ 5˜ 7
3
˜ = 7 3
§ ·
¨ ¸
¨
¸
© ¹
'(/+LQ
ª
ª = 7 3 ˜ &% 7 3 º º « 5˜ 7˜ = 7 3 7˜ 'D 7 D 7 ˜ OQ«
» » ˜ «
« = 7 3 ˜ &% 7 3 » »
˜ ˜ E
¬
¬
¼¼
'(/6LQ ª = 7 3 ˜ &% 7 3 º º ª
« 5˜ OQ = 7 3 &% 7 3 'D 7 ˜ OQ«
» » ˜ «
˜ ˜ E « = 7 3 ˜ &% 7 3 » »
¬
¬
¼¼
'(/+LQ u '(/6LQ Solutions to Chemical and Engineering Thermodynamics, 5th ed.
*XHVVIRUILQDOVWDWH
7 ˜ 7L
3 3I
)XJDFLW\H[SUHVVLRQV>DFWXDOO\OQ I3 @IRUWKHOLTXLGIODQGYDSRUIY
IO 7 3 = 7 3
IY 7 3 = 7 3
*LYHQ
IO 7 3 IY 7 3
OQ = 7 3
OQ = 7 3
&% 7 3
&% 7 3
&$ 7 3
˜ ˜ &% 7 3
ª = 7 3 ˜ &% 7 3 º
»
« = 7 3 ˜ &% 7 3 »
¬
¼
˜ OQ«
ª = 7 3 ˜ &% 7 3 º
»
« = 7 3 ˜ &% 7 3 »
¬
¼
&$ 7 3
˜ OQ«
˜ ˜ &% 7 3
7 )LQG 7
7
5HVLGXDOHQWURS\IRUOLTXLG '(/6/ DQGYDSRU '(/69 SKDVHV
'(/6/ 7 3 ª
ª = 7 3 ˜ &% 7 3 º º « 5˜ OQ = 7 3 &% 7 3 'D 7 ˜ OQ«
» » ˜ «
˜ ˜ E « = 7 3 ˜ &% 7 3 » »
¬
¬
¼¼
'(/697 3 ª
ª = 7 3 ˜ &% 7 3 º º « 5˜ OQ = 7 3 &% 7 3 'D 7 ˜ OQ«
» » ˜ «
˜ ˜ E « = 7 3 ˜ &% 7 3 » »
¬
¬
¼¼
5HVLGXDOHQWKDOS\IRUOLTXLG '(/+/ DQGYDSRU '(/+9 SKDVHV
'(/+/ 7 3 ª
ª = 7 3 ˜ &% 7 3 º º « 5˜ 7˜ = 7 3 7˜ 'D 7 D 7 ˜ OQ«
» » ˜ «
« = 7 3 ˜ &% 7 3 » »
˜ ˜ E
¬
¬
¼¼
'(/+97 3 ª = 7 3 ˜ &% 7 3 º º ª
« 5˜ 7˜ = 7 3 7˜ 'D 7 D 7 ˜ OQ«
» » ˜ «
« = 7 3 ˜ &% 7 3 » »
˜ ˜ E
¬
¬
¼¼
,GHDOJDVSURSHUWLHVFKDQJHVUHODWLYHWRWKHLQLWLDOVWDWH
'(/+,*7 3 &S ˜ 7 7L '(/6,*7 3 &S ˜ OQ§¨
7·
&S ˜ 7 7L
¸ &S ˜ 7 7L © 7L ¹
&S ˜ 7 7L
)LQGYDSRUOLTXLGVSOLW
&S ˜ 7 7L
&S ˜ 7 7L
&S ˜ 7 7L
3·
¸
© 3L ¹
5˜ ˜ OQ§¨
[ *LYHQ
[˜ '(/697 3 [ ˜ '(/6/ 7 3 '(/6,*7 3
[ )LQG [
[
'(/6LQ
)UDFWLRQYDSRU
+9 '(/+97 3 '(/+,*7 3
69 '(/697 3 '(/6,*7 3
+/ '(/+/ 7 3 '(/+,*7 3
6/ '(/6/ 7 3 '(/6,*7 3
G+ [˜ +9 [ ˜ +/ '(/+LQ
G6 [˜ 69 [ ˜ 6/ '(/6LQ
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
6800$5<2)5(68/76
7HPSHUDWXUH.
)(('/,48,'9$325
7L 7 7 3UHVVXUHEDU
3L
3
3
9DSRUOLTXLGVSOLW
[
&RPSUHVVLELOLW\
= 7L 3L
(QWKDOS\-PRO
UHODWLYHWRIHHG
+/
u +9
(QWURS\-PRO.
UHODWLYHWRIHHG
6/
69
= 7 3
u (QWKDOS\FKDQJH
-PRO
G+
(QWURS\FKDQJH
-PRO.
G6
= 7 3
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
,6(17+$/3,&65.(48$7,212)67$7(&$/&8/$7,21
7F 3F &S &S ˜ RP NDS ˜ RP ˜ RP˜ RP
&S ˜ 5 E ˜
65.&RQVWDQWV
,QSXWLQLWLDOWHPSHUDWXUHDQGSUHVVXUHRIFDOFXODWLRQ
,QSXWILQDOSUHVVXUH
3I ,QLWLDOVWDWHFDOFXODWLRQV
ª
¬
7 ·º
§
©
DOI 7 ˜ « NDS˜ ¨ DF ˜
3F
7L .
7 7L
3 3L
D 7 DF ˜ DOI 7
&$ 7 3 D 7 ˜3
5˜ 7
G
'D 7 D 7
G7
= 7 3 5˜ 7F
5 ˜ 7F
3F
EDU
3L EDU
¸»
7F ¹ ¼
&S ˜ &% 7 3 3˜ E
5˜ 7
$ m &$ 7 3
% m &% 7 3
§ $ ˜ % ·
¨
¸
$ % %¸
9m ¨
¨
¸
¨
¸
©
¹
9HFWRURIFRHIILFLHQWVLQWKH65.HTXDWLRQ
LQWKHIRUP
$ % $%A% ==A=A
== m SRO\URRWV 9
6ROXWLRQWRWKHFXE LF
IRU L  == m L
LI ,P == z L
6HWDQ\LPDJLQDU\URRWVWR]HUR
6RUWWKHURRWV
== m VRUW ==
·
© ¹
== m == LI § == ·
© ¹
== m ==
LI § ==
==
&DOFXODWHLQLWDOSURSHUWLHV
&DOFXODWHLQLWLDOPRODUYROXPH
DQGHQWKDOS\DQGHQWURS\
GHSDUWXUH
6HWWKHYDOXHRIDQ\LPDJLQDU\URRWV
WRYDOXHRIWKHUHDOURRW
=I 7 3 = 7 3
9/ = 7 3 ˜ 5˜ 7
3
˜ = 7 3
'(/+LQ
ª«
§ = 7 3 &% 7 3 ¸·º» 7˜ 'D 7 D 7
5 ˜ 7˜ = 7 3 ˜ OQ¨
˜ E
= 7 3
«
¨
¸»
¬
©
¹¼
'(/6LQ = 7 3 &% 7 3 · ·
¨§ 5˜ OQ = 7 3 &% 7 3 'D 7 ˜ OQ§¨
¸ ¸ ˜ E
= 7 3
¨
¨
¸¸
©
©
¹¹
'(/+LQ u '(/6LQ § ·
¨ ¸
¨
¸
© ¹
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
*XHVVIRUILQDOVWDWH
7 ˜ 7L
3 3I
)XJDFLW\H[SUHVVLRQV>DFWXDOO\OQ I3 @IRUWKHOLTXLGIODQGYDSRUIY
OQ = 7 3
IO 7 3 = 7 3
IY 7 3 = 7 3
*LYHQ
IO 7 3 IY 7 3
OQ = 7 3
&% 7 3
&% 7 3
&$ 7 3
&% 7 3
&$ 7 3
&% 7 3
§ = 7 3 &% 7 3 ·
¸
= 7 3
¨
¸
©
¹
˜ OQ¨
§ = 7 3 &% 7 3 ¸·
= 7 3
¨
¸
©
¹
˜ OQ¨
7 )LQG 7
7
5HVLGXDOHQWURS\IRUOLTXLG '(/6/ DQGYDSRU '(/69 SKDVHV
'(/6/ 7 3 § = 7 3 &% 7 3 ¸· ¸· §¨
'D 7
˜ OQ¨
˜ 5 ˜ OQ = 7 3 &% 7 3 E
= 7 3
¨
¨
¸¸
©
©
¹¹
'(/697 3 § = 7 3 &% 7 3 · · §¨
'D 7
¸ ¸ ˜ ˜ OQ¨
5˜ OQ = 7 3 &% 7 3 E
= 7 3
¨
¨
¸¸
©
©
¹¹
5HVLGXDOHQWKDOS\IRUOLTXLG '(/+/ DQGYDSRU '(/+9 SKDVHV
'(/+/ 7 3 = 7 3 &% 7 3 ·º
«ª5˜ 7˜ = 7 3 7˜ 'D 7 D 7 ˜ OQ§¨
¸» ˜ = 7 3
«
¨
¸»
E
¬
©
¹¼
'(/+97 3 § = 7 3 &% 7 3 ¸·»º ª«
7˜ 'D 7 D 7
˜ OQ¨
˜ 5 ˜ 7˜ = 7 3 = 7 3
«
¨
¸»
E
¬
©
¹¼
,GHDOJDVSURSHUWLHVFKDQJHVUHODWLYHWRWKHLQLWLDOVWDWH
'(/+,*7 3 &S ˜ 7 7L &S ˜ 7 7L
7
'(/6,*7 3 &S ˜ OQ§¨ ¸· &S ˜ 7 7L © 7L ¹
)LQGYDSRUOLTXLGVSOLW
&S ˜ 7 7L
&S ˜ 7 7L
&S ˜ 7 7L
&S ˜ 7 7L
3·
¸
© 3L ¹
5˜ ˜ OQ§¨
[ *LYHQ
[˜ '(/+97 3 [ ˜ '(/+/ 7 3 '(/+,*7 3
[ )LQG [
[
'(/+LQ
)UDFWLRQYDSRU
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
+9 '(/+97 3 '(/+,*7 3
69 '(/697 3 '(/6,*7 3
+/ '(/+/ 7 3 '(/+,*7 3
6/ '(/6/ 7 3 '(/6,*7 3
G+ [˜ +9 [ ˜ +/ '(/+LQ
G6 [˜ 69 [ ˜ 6/ '(/6LQ
6800$5<2)5(68/76
7HPSHUDWXUH.
)(('/,48,'9$325
7L 7 7 3UHVVXUHEDU
3L
3
3
9DSRUOLTXLGVSOLW
[
&RPSUHVVLELOLW\
= 7L 3L
(QWKDOS\-PRO
UHODWLYHWRIHHG
+/
u +9
(QWURS\-PRO.
UHODWLYHWRIHHG
6/
69
= 7 3
= 7 3
(QWKDOS\FKDQJH
-PRO
G+
(QWURS\FKDQJH
-PRO.
G6
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
,6(17523,&65.(48$7,212)67$7(&$/&8/$7,21
7F 3F &S &S ˜ RP NDS ˜ RP ˜ RP˜ RP
&S ˜ 5 E ˜
65.&RQVWDQWV
,QSXWLQLWLDOWHPSHUDWXUHDQGSUHVVXUHRIFDOFXODWLRQ
,QSXWILQDOSUHVVXUH
3I ,QLWLDOVWDWHFDOFXODWLRQV
ª
¬
7 ·º
§
©
DOI 7 ˜ « NDS˜ ¨ DF ˜
3F
7L .
7 7L
3 3L
D 7 DF ˜ DOI 7
&$ 7 3 D 7 ˜3
5˜ 7
G
'D 7 D 7
G7
= 7 3 5˜ 7F
5 ˜ 7F
3F
EDU
3L EDU
¸»
7F ¹ ¼
&S ˜ &% 7 3 3˜ E
5˜ 7
$ m &$ 7 3
% m &% 7 3
§ $ ˜ % ·
¨
¸
$ % %¸
9m ¨
¨
¸
¨
¸
©
¹
9HFWRURIFRHIILFLHQWVLQWKH65.HTXDWLRQ
LQWKHIRUP
$ % $%A% ==A=A
== m SRO\URRWV 9
6ROXWLRQWRWKHFXE LF
IRU L  == m L
LI ,P == z L
6HWDQ\LPDJLQDU\URRWVWR]HUR
6RUWWKHURRWV
== m VRUW ==
·
© ¹
== m == LI § == ·
© ¹
== m ==
LI § ==
==
&DOFXODWHLQLWDOSURSHUWLHV
6HWWKHYDOXHRIDQ\LPDJLQDU\URRWV
WRYDOXHRIWKHUHDOURRW
=I 7 3 = 7 3
&DOFXODWHLQLWLDOPRODUYROXPH
DQGHQWKDOS\DQGHQWURS\
GHSDUWXUH
9/ = 7 3 ˜ 5˜ 7
3
˜ = 7 3
'(/+LQ
ª
§ = 7 3 &% 7 3 ·º «5˜ 7˜ = 7 3 7˜ 'D 7 D 7 ˜ OQ¨
¸» ˜ «
¨
¸»
= 7 3
E
¬
©
¹¼
'(/6LQ = 7 3 &% 7 3 · ·
¨§ 5˜ OQ = 7 3 &% 7 3 'D 7 ˜ OQ§¨
¸ ¸ ˜ ¨
¨
¸¸
= 7 3
E
©
©
¹¹
'(/+LQ u *XHVVIRUILQDOVWDWH
§ ·
¨ ¸
¨
¸
© ¹
'(/6LQ 7 ˜ 7L
3 3I
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
)XJDFLW\H[SUHVVLRQV>DFWXDOO\OQ I3 @IRUWKHOLTXLGIODQGYDSRUIY
OQ = 7 3
IO 7 3 = 7 3
IY 7 3 = 7 3
*LYHQ
IO 7 3 IY 7 3
OQ = 7 3
&% 7 3
&% 7 3
&$ 7 3
&% 7 3
&$ 7 3
&% 7 3
§ = 7 3 &% 7 3 ·
¸
¨
¸
= 7 3
©
¹
˜ OQ¨
§ = 7 3 &% 7 3 ·
¸
= 7 3
¨
¸
©
¹
˜ OQ¨
7 )LQG 7
7
5HVLGXDOHQWURS\IRUOLTXLG '(/6/ DQGYDSRU '(/69 SKDVHV
'(/6/ 7 3 § = 7 3 &% 7 3 · · §¨
'D 7
¸ ¸ ˜ ˜ OQ¨
5 ˜ OQ = 7 3 &% 7 3 = 7 3
E
¨
¨
¸¸
©
©
¹¹
'(/697 3 § = 7 3 &% 7 3 ·¸ ·¸ §¨
'D 7
5˜ OQ = 7 3 &% 7 3 ˜ OQ¨
˜ = 7 3
E
¨
¨
¸¸
©
©
¹¹
5HVLGXDOHQWKDOS\IRUOLTXLG '(/+/ DQGYDSRU '(/+9 SKDVHV
'(/+/ 7 3 ª
§ = 7 3 &% 7 3 ·º «5˜ 7˜ = 7 3 7˜ 'D 7 D 7 ˜ OQ¨
¸» ˜ = 7 3
E
«
¨
¸»
¬
©
¹¼
'(/+97 3 ª
§ = 7 3 &% 7 3 ·º «5˜ 7˜ = 7 3 7˜ 'D 7 D 7 ˜ OQ¨
¸» ˜ = 7 3
E
«
¨
¸»
¬
©
¹¼
,GHDOJDVSURSHUWLHVFKDQJHVUHODWLYHWRWKHLQLWLDOVWDWH
'(/+,*7 3 &S ˜ 7 7L &S ˜ 7 7L
7
'(/6,*7 3 &S ˜ OQ§¨ ¸· &S ˜ 7 7L © 7L ¹
)LQGYDSRUOLTXLGVSOLW
&S ˜ 7 7L
&S ˜ 7 7L
&S ˜ 7 7L
&S ˜ 7 7L
5˜ ˜ OQ§¨
3·
¸
© 3L ¹
[ *LYHQ
[˜ '(/697 3 [ ˜ '(/6/ 7 3 '(/6,*7 3
[ )LQG [
[
'(/6LQ
)UDFWLRQYDSRU
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
+9 '(/+97 3 '(/+,*7 3
69 '(/697 3 '(/6,*7 3
+/ '(/+/ 7 3 '(/+,*7 3
6/ '(/6/ 7 3 '(/6,*7 3
G+ [˜ +9 [ ˜ +/ '(/+LQ
G6 [˜ 69 [ ˜ 6/ '(/6LQ
6800$5<2)5(68/76
7HPSHUDWXUH.
)(('/,48,'9$325
7L 7 7 3UHVVXUHEDU
3L
3
3
9DSRUOLTXLGVSOLW
[
&RPSUHVVLELOLW\
= 7L 3L
(QWKDOS\-PRO
UHODWLYHWRIHHG
+/
u +9
(QWURS\-PRO.
UHODWLYHWRIHHG
6/
69
= 7 3
= 7 3
(QWKDOS\FKDQJH
-PRO
G+
(QWURS\FKDQJH
-PRO.
G6
u 7KLVSUREOHPZDVVROYHGXVLQJWKHDWWDFKHG0DWKFDGZRUNVKHHW7KHUHVXOWVDUH
T R& PYDSZLWKD(T)
PYDSZLWKD PLQN3D Solutions to Chemical and Engineering Thermodynamics, 5th ed.
$VFDQEHVHHQWKH65.HTXDWLRQLVRIFRPSDUDEOHDFFXUDF\WRWKH35HTXDWLRQ,QERWKFDVHVLI
WKHDSDUDPHWHULVVHWWRRQHWKHUHVXOWVDUHQRWYHU\JRRGLQGHHGTXLWHEDGDWORZWHPSHUDWXUHV
7KH0DWKFDGZRUNVKHHWXVHGLQVROYLQJWKLVSUREOHPLVJLYHQEHORZ
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
9$32535(6685(&$/&8/$7,2186,1*65.(48$7,21
5HDGLQSURSHUWLHVIRU:DWHU
7F 3F RP NDSSDFDOFXODWLRQ
NDS ˜ RP ˜ RP˜ RP
65.&RQVWDQWV
5 1RWHWKDWWKHVHDUHEHLQJGHILQHGDVD
IXQFWLRQRIWHPSHUDWXUHVLQFHZHZLOO
QHHGWRLQWHUDWHRQWHPSHUDWXUH
&$ 7 3 ª
¬
5˜ 7F
D 7 ˜3
&% 7 3 DF ˜
3F
§
©
DOI 7 ˜ « NDS˜ ¨ 5˜ 7
= 7 3 E ˜
7 ·º
¸»
7F ¹ ¼
D 7 DF ˜ DOI 7
3˜ E
5˜ 7
$ m &$ 7 3
§ $ ˜ % ·
¨
¸
$ % %¸
9m ¨
¨
¸
¨
¸
©
¹
9HFWRURIFRHIILFLHQWVLQWKH65.HTXDWLRQ
LQWKHIRUP
$ % $%A% = =A=A
== m SRO\URRWV 9
6ROXWLRQWRWKHFXELF
IRU L  L
LI ,P == z L
== m VRUW ==
·
©
¹
== m == LI § == ·
©
¹
== m ==
LI § ==
6HWDQ\LPDJLQDU\URRWVWR]HUR
6RUWWKHURRWV
6HWWKHYDOXHRIDQ\LPDJLQDU\URRWV
WRYDOXHRIWKHUHDOURRW
==
(QWHUWHPSHUDWXUH7DQGSUHVVXUH3
'HSHQGLQJRQZKDWLVVSHFLILHGLQWKH*LYHQDQG)LQGVWDWHPHQWV
EHORZHLWKHU7RU3LVVSHFLILHGDQGWKHRWKHULVDQLQLWLDOJXHVV
ZKLFKPD\KDYHWREHDGMXVWHGDVWKHRWKHUYDULDEOHLVFKDQJHG
HVSHFLDOO\DVWKHFULWLFDOSRLQWLVDSSURDFKHG7KLVZRUNVKHHWZLOO
SUREDEO\QRWFRQYHUJHWRDVROXWLRQZKHQ7RU3DUHZLWKLQWR
RIWKHFULWLFDOYDOXHVXQOHVVDQH[WUHPHO\JRRGLQLWLDOJXHVVLV
SURYLGHG2QHZD\WRREWDLQDJRRGLQLWLDOLVWRVWDUWZHOOEHORZWKH
FULWLFDOUHJLRQDQGVWHSWRZDUGVLWXVLQJWKHUHVXOWRISUHYLRXVFDOFXODWLRQV
3F
% m &% 7 3
== m 5 ˜ 7F
7 3 Solutions to Chemical and Engineering Thermodynamics, 5th ed.
)XJDFLW\H[SUHVVLRQV>DFWXDOO\OQ I3 @IRUWKHOLTXLGIODQGYDSRUIY
IO 7 3 = 7 3
IY 7 3 = 7 3
*LYHQ
OQ = 7 3
OQ = 7 3
IO 7 3 IY 7 3
3 )LQG 3
3
&% 7 3
&% 7 3
§ = 7 3 &% 7 3 ¸·
&$ 7 3
˜ OQ¨
¨
©
&% 7 3
&$ 7 3
&% 7 3
= 7 3
¸
¹
§ = 7 3 &% 7 3 ¸·
˜ OQ¨
¨
©
= 7 3
¸
¹
(TXDWHORJRIIXJDFLW\FRHIILFLHQWV
6ROYHHTXDOLW\RIIXJDFLWLHV
)XJDFLW\)XJDFLW\FRHIILFLHQW
IXJO 3˜ H[S IO 7 3
IXJO
IO 7 3
IXJY 3˜ H[S IY 7 3
IXJY
IY 7 3
SKLO IXJO
3
6800$5<2)5(68/76
7
.
9DSRUSUHVVXUHEDU
3
/,48,'9$325
&RPSUHVVLELOLW\
)XJDFLW\EDU
= 7 3
IXJO
= 7 3
IXJY
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
DOVRDYDLODEOHDVD0DWKFDGZRUNVKHHW 7KHVROXWLRQLVWKDWWKHILQDOWHPSHUDWXUHLV.DQGWKHILQDOSUHVVXUHLVEDU
8VLQJ65.(26ZLWKWKHDSSUR[LPDWHWZRFRQVWDQWKHDWFDSDFLW\H[SUHVVLRQ
3URSHUW\'DWD
7LQ.3LQEDU 7F 3F 5 RP 7L 3HQJ5RELQVRQ&RQVWDQWV
E ˜
3L 5˜ 7F
7 7L
7 ·º
(QWURS\GHSDUWXUHDWWKH
LQLWLDOFRQGLWLRQV
'(/6L
1RZFRQVLGHUILQDOVWDWH
(QWURS\GHSDUWXUH
DWILQDOFRQGLWLRQV
3I 7 '(/6 7 7 )LQG 7
3L
1L
9L
u ª5˜ OQª 9L E ˜ 3L º 'D 7 ˜ OQ§ 9L E ·º ˜ « «
»
¨ 9L ¸»
5˜ 7¼
E
¬ ¬
©
¹¼
1I
)LQDOSUHVVXUHZLOOFKDQJHLQFRXUVH
RIVROYLQJIRUWKHILQDOWHPSHUDWXUH
7\SHRXWVROXWLRQ
9W
1I 1L ˜ 9I 7\SHRXWILQDOQXPEHURI
PROHVDQGVSHFLILFYROXPH
*LYHQ
1L G
D 7
G7
5˜ 7L
9L )LQG 9
> 9˜ 9 E @
9L u 3F
'D 7 D 7
,QLWLDOPRODUYROXPHDQG
QXPEHURIPROHV
6ROYHIRUILQDO
WHPSHUDWXUHXVLQJ
6 ILQDO 6 LQLWLDO 5˜ 7
9 E
D 7 DF ˜ DOI 7
9
3L
5 ˜ 7F
¸»
7F ¹ ¼
)LQGLQLWLDOPRODUYROXPHDQGQXPEHURIPROHV
6WDUWZLWKLQLWLDOJXHVVIRUYROXPHPAPRO
*LYHQ
9W DF ˜
3F
1RWHWKDWWKHVHDUHEHLQJGHILQHGDVD
§
ª
IXQFWLRQRIWHPSHUDWXUHVLQFHZHZLOOQHHGWR
DOI 7 ˜ « NDS˜ ¨ ¬
©
LQWHUDWHRQWHPSHUDWXUHODWHUWRREWDLQWKHILQDO
VWDWHRIWKHV\VWHP
6ROYH35(26IRULQLWLDOYROXPH
&S NDS ˜ RP ˜ RP˜ RP
,QLWLDO&RQGLWLRQV 9W WRWDOYROXPHPA ,QLWLDOWHPSHUDWXUH
&S 5˜ 7
9 E
9 9I
1I
9I
u D 7
9˜ 9 E
3I 7 º 'D 7
9 E ·º ª ª
˜ OQ§¨
«5˜ OQ« 9 E ˜
»
¸» ˜ 5˜ 7 ¼
E
¬ ¬
© 9 ¹¼
7·
§ 3I 7 ·
¸ ˜ 7 7L 5˜ ˜ OQ¨ 3L ¸ '(/6 7 '(/6
© 7L ¹
©
¹
˜ OQ§¨
7 3I 7
D $WDJLYHQWHPSHUDWXUHWKHVWDELOLW\OLPLWRIDIOXLGLVGHWHUPLQHGE\WKHIROORZLQJFULWHULRQ
1RWHWKDWWKLVOHDGVWRWKHVSLQRGDOFXUYH Solutions to Chemical and Engineering Thermodynamics, 5th ed.
FG wP IJ
H wV K
T
)RUWKHJLYHQ(26WKHVWDELOLW\OLPLWRIDIOXLGXQGHUJRLQJDSUHVVXUHFKDQJHDWFRQVWDQW
WHPSHUDWXUHLV
wP
RT
BRT
CRT
wV T
V
V
V
FG IJ
H K
RU V BV C
,Q RUGHU WR KDYH D SKDVH WUDQVLWLRQ WKHUH PXVW EH WZR GLVWLQFW VWDELOLW\ OLPLWV LH WKH DERYH
TXDGUDWLFHTXDWLRQPXVWKDYHWZRGLIIHUHQWURRWVRIV7KHUHIRUH
B u u C ! RU B ! C E $FFRUGLQJWR,OOXVWUDWLRQ
wP
dU C9dT T
P dV wT V
LM F I
NH K
OP
Q
LM F wP I POP N H wT K Q
%XWIRUWKHJLYHQ(26 T
V
7KHUHIRUH
'U
dU
z z
6LQFH
FG wC IJ
H wV K
z
T
FG w P IJ
H wT K
9
7KHUHIRUH CY
DQG
'U
C9 V T dT T
T
CY T
a bT dT
V
C9
a bT a T T T
%HFDXVHBDQGCDUHQRWIXQFWLRQVRIT b T T 7KHLQWHUQDOHQHUJ\FKDQJHLVWKHVDPHIRUDQLGHDOJDV
F $FFRUGLQJWR(TQ
C9
wP
dS
dT dV VRWKDW
T
wT V
F I
H K
wP
FG RT BRT CRT IJ
TF I
H
K
T
w
HV V V K
P
T
w
FG IJ H wV K
C
a bT
a bT
V
9
S
)RUDQLGHDOJDV
FG wT IJ
H wV K
S
P
a bT
FG RT IJ
HV K
a bT
RT
V a bT
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
D 6LQFH
' )o ' 9 ! DQG' )o ' + ! DQGWKDWIRUDQHTXLEULXPWUDQVLWLRQ LH* )
*'
WKH
' )o ' +
§ w* ·
! VRWKDW6' ! 6) )XUWKHUVLQFH ¨
6
¸
7
© w7 ¹3
VORSHRIWKH*LEEVHQHUJ\YHUVXVWHPSHUDWXUHPXVWEHJUHDWHUIRUWKHGHQDWXUHGRIXQIROGHGSURWHLQ
WKDQIRUWKHIROGHGSURWHLQ7KLVLVVKRZQEHORZ
LWIROORZVWKDW' )o' 6
*
)
'
7 ,I77 WKHQ*)!*'VRWKDWWKHIROGHGVWDWHRUQDWXUDOVWDWHLVWKHVWDEOHHTXLOLEULXPVWDWH,I7
!7 WKHQ*)*'VRWKDWWKHXQIROGHGVWDWHLVWKHQWKHVWDEOHHTXLOLEULXPVWDWH6R7 LVWKH
³PHOWLQJWHPSHUDWXUH´RIWKHSURWHLQ
§ G3 · ' )o' +
E 6LQFHE\WKH&ODSH\URQHTXDWLRQ ¨
! LWIROORZVWKDWDVWKHSUHVVXUHRQWKH
¸
© G7 ¹ 7' )o ' 9
SURWHLQLVLQFUHDVHV7 ZLOOLQFUHDVH7KDWLVWKHSURWHLQZLOOXQIROGHGRUGHQDWXUHDWDKLJKHU
WHPSHUDWXUHDVWKHSUHVVXUHLVLQFUHDVHG
7KH9,68$/%DVLFSURJUDP3HQJ5RELQVRQ(TXDWLRQRI6WDWHLVXVHGIRUWKH3HQJ5RELQVRQ
SUHGLFWLRQVWKHFRUUHODWLRQLQWKH9LVXDO%DVLF3URSHUW\SURJUDPZLOOEHXVHGIRUWKHµOLWHUDWXUHGDWD´DQG
WKH0$7+&$'ZRUNVKHHW359$330&'LVXVHGIRUWKHD 7 FDOFXODWLRQVLQFHLWLVWKHHDVLHVWWR
PRGLI\7KHUHVXOWVDUHDVIROORZV
7 . 3 EDU OLWHUDWXUH
3 EDU 35D 7 3 EDU 35D 7 DQGRIFRXUVHDW7F ERWK35FDOFXODWLRQVJLYH3 EDU 3FVLQFHWKH(26SDUDPHWHUVKDYHEHHQ
ILWWHGWRJLYHWKHFRUUHFWFULWLFDOSRLQWFRQGLWLRQV VLQFH D 7& Name
: methane
Formula: CH4
Molecular weight (g/mol) = 16.043
Normal boiling point (K) = 111.6
Critical temperature (K) = 190.4
Critical pressure (bar)
= 46.0
Critical volume (cm3/mol) = 99.2
Critical compressibility factor = 0.288
Pitzer's acentric factor = 0.011
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
Isobaric heat capacity of the ideal gas (J/mol.K)
Cp = A + B*T + C*T^2 + D*T^3
A = 1.925e+01
B = 5.213e-02
C = 1.197e-05
D = -1.132e-08
-----------------------------------------------------------------** Vapor-Liquid Equilibrium **
Temperature
(K)
= 111.6
Vapor pressure (bar) = 1.0249
Vapor
Compressibility factor
0.96673
Specific volume (m^3/mol)
8.75155e-03
Density (g/cm^3)
1.83316e-03
Fugacity (bar)
0.99183
Fugacity coeff
0.96771
Enthalpy departure (J/mol)
-73.98
Enthalpy ideal
(J/mol)
-5661.75
Enthalpy
(J/mol)
-5735.73
Entropy departure (J/mol K)
-0.39
Entropy ideal
(J/mol K)
-29.21
Entropy
(J/mol K)
-29.60
** Vapor-Liquid Equilibrium **
T (K)
115.0
120.0
125.0
130.0
135.0
140.0
145.0
150.0
155.0
160.0
165.0
170.0
175.0
180.0
185.0
190.0
P (bar)
1.343
1.943
2.726
3.723
4.969
6.496
8.342
10.539
13.128
16.143
19.625
23.612
28.146
33.270
39.030
45.478
Liquid
0.00373
3.37236e-05
4.75720e-01
0.99186
0.96774
-8266.44
-5661.75
-13928.18
-73.80
-29.21
-103.01
Molar volume (m^3/mol)
Density (g/cm^3)
Vapor
Liquid
Vapor
Liquid
6.82605e-03 3.41685e-05
2.35026e-03 4.69525e-01
4.85847e-03 3.48771e-05
3.30207e-03 4.59986e-01
3.54925e-03 3.56597e-05
4.52011e-03 4.49891e-01
2.65044e-03 3.65291e-05
6.05295e-03 4.39184e-01
2.01609e-03 3.75015e-05
7.95749e-03 4.27797e-01
1.55770e-03 3.85977e-05
1.02992e-02 4.15647e-01
1.21889e-03 3.98450e-05
1.31620e-02 4.02636e-01
9.63678e-04 4.12800e-05
1.66477e-02 3.88639e-01
7.67798e-04 4.29535e-05
2.08948e-02 3.73497e-01
6.15040e-04 4.49387e-05
2.60845e-02 3.56998e-01
4.93846e-04 4.73457e-05
3.24858e-02 3.38848e-01
3.96190e-04 5.03529e-05
4.04932e-02 3.18611e-01
3.15943e-04 5.42758e-05
5.07781e-02 2.95583e-01
2.48265e-04 5.97618e-05
6.46204e-02 2.68449e-01
1.88258e-04 6.85514e-05
8.52179e-02 2.34029e-01
1.18996e-04 9.18577e-05
1.34820e-01 1.74651e-01
9ROXPHWHPSHUDWXUHFRH[LVWHQFHFXUYHRIPHWKDQH
9DSRU3UHVVXUHRIPHWKDQH
9 PAPRO
9DSRU
/LTXLG
7HPSHUDWXUH . Solutions to Chemical and Engineering Thermodynamics, 5th ed.
0RGLILHG0$7+&$'ZRUNVKHHWIRU3HQJ5RELQVRQYDSRUSUHVVXUHFDOFXODWLRQZLWKD 7 7F 3F RP 3HQJ5RELQVRQ&RQVWDQWV
D 7 DF
NDS RP
˜
RP
˜ ˜ RP
5 E ˜
D 7 ˜3
3˜ E
&$ 7 3 5˜ 7
= 7 3 &% 7 3 5 ˜ 7F
3F
DF ˜
'D 7 5˜ 7
5 ˜ 7F
3F
G
D 7
G7
$ m &$ 7 3
% m &% 7 3
ª« $ ˜ % % % º»
«
»
9 m « $ ˜ % ˜ % »
«
»
%
«
»
¬
¼
9HFWRURIFRHIILFLHQWVLQWKH35HTXDWLRQ
LQWKHIRUP
$ %%A%A $ %A % = % =A=A
6ROXWLRQWRWKHFXELF
== m SRO\URRWV 9
IRU L  == m L
LI ,P == z 6HWDQ\LPDJLQDU\URRWVWR]HUR
6RUWWKHURRWV
L
== m VRUW ==
·
© ¹
== m == LI § == ·
© ¹
== m ==
LI § ==
6HWWKHYDOXHRIDQ\LPDJLQDU\URRWV
WRYDOXHRIWKHUHDOURRW
==
7 .
3 EDU
)XJDFLW\H[SUHVVLRQV>DFWXDOO\OQ I3 @IRUWKHOLTXLGIODQGYDSRUIY
IO 7 3 = 7 3
IY 7 3 = 7 3
*LYHQ
IO 7 3 IY 7 3
OQ = 7 3
3 )LQG 3
OQ = 7 3
3
&% 7 3
&% 7 3
&$ 7 3
˜ ˜ &% 7 3
&$ 7 3
˜ ˜ &% 7 3
ª = 7 3 ˜ &% 7 3 º
»
« = 7 3 ˜ &% 7 3 »
¬
¼
˜ OQ«
ª = 7 3 ˜ &% 7 3 º
»
« = 7 3 ˜ &% 7 3 »
¬
¼
˜ OQ«
(TXDWHORJRIIXJDFLW\FRHIILFLHQWV
6ROYHHTXDOLW\RIIXJDFLWLHV
)XJDFLW\)XJDFLW\FRHIILFLHQW
IXJO
IXJO 3˜ H[S IO 7 3
IXJO IO 7 3
SKLO SKLO 3
IXJY 3˜ H[S IY 7 3
IXJY
IY 7 3
SKLY IXJY
SKLY
3
6800$5<2)5(68/76
7
.
9DSRUSUHVVXUHEDU
3
/,48,'9$325
&RPSUHVVLELOLW\
= 7 3
)XJDFLW\FRHIILFLHQW
SKLO
SKLY
)XJDFLW\EDU
IXJO
IXJY
= 7 3
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
7KH9,68$/%DVLFSURJUDP3HQJ5RELQVRQ(TXDWLRQRI6WDWHLVXVHGIRUWKH3HQJ5RELQVRQ
SUHGLFWLRQVWKHFRUUHODWLRQLQWKH9LVXDO%DVLF3URSHUW\SURJUDPZLOOEHXVHGIRUWKHµOLWHUDWXUHGDWD´DQG
WKH0$7+&$'ZRUNVKHHW359$330&'LVXVHGIRUWKHD 7 FDOFXODWLRQVLQFHLWLVWKHHDVLHVWWR
PRGLI\7KHUHVXOWVDUHDVIROORZV
7 . 3 EDU OLWHUDWXUH
3 EDU 35D 7 3 EDU 35D 7 DQGRIFRXUVHDW7F ERWK35FDOFXODWLRQVJLYH3 EDU 3FVLQFHWKH(26SDUDPHWHUVKDYHEHHQ
ILWWHGWRJLYHWKHFRUUHFWFULWLFDOSRLQWFRQGLWLRQV VLQFH D 7& 7KH9,68$/%DVLFSURJUDP3HQJ5RELQVRQ(TXDWLRQRI6WDWHLVXVHGIRUWKH3HQJ5RELQVRQ
SUHGLFWLRQVWKHFRUUHODWLRQLQWKH9LVXDO%DVLF3URSHUW\SURJUDPZLOOEHXVHGIRUWKHµOLWHUDWXUHGDWD´DQG
WKH0$7+&$'ZRUNVKHHW359$330&'LVXVHGIRUWKHD 7 FDOFXODWLRQVLQFHLWLVWKHHDVLHVWWR
PRGLI\7KHUHVXOWVDUHDVIROORZV
7 . 3 EDU OLWHUDWXUH
3 EDU 35D 7 3 EDU 35D 7 DQGRIFRXUVHDW7F ERWK35FDOFXODWLRQVJLYH3 EDU 3FVLQFHWKH(26SDUDPHWHUVKDYHEHHQ
ILWWHGWRJLYHWKHFRUUHFWFULWLFDOSRLQWFRQGLWLRQV VLQFH D 7& Solutions to Chemical and Engineering Thermodynamics, 5th ed.
:LOOXVHWKH35(26
D ILQGWKDWI EDU 03D
E WRFRUUHFWIRULQFUHDVHLQSUHVVXUH
L ZHFDQXVH35(26DQGILQGI EDU
RULL XVH3R\QWLQJ&RUUHFWLRQ
ª 9 / EDU º
I 03D I 03D H[S «
»
5 ˜7
¬
¼
EXW
ª J
º
EDU u PFF
PRO
» H[S > @ u
H[S «
J
EDU P
«¬ FF u PRO . u . »¼
I 03D H[S î 6RWKHWZRUHVXOWVDUHFORVHEXWQRWH[DFWO\WKHVDPH3HUKDSVEHFDXVHRILQDFFXUDF\LQOLTXLG
GHQVLW\RUWKDWWKHOLTXLGGHQVLW\FKDQJHVZLWKSUHVVXUH
&ODXVLXV&RDSH\URQ(TDSSOLHGWRVXEOLPDWLRQ
D
' +§ ·
3VXE 7
OQ VXE VXE ¨ ¸
3 7
5 © 7 7 ¹
E
OQ
' VXE + § ·
5 ¨© ¸¹
' VXE +
u 5
' VXE +
u 5
- PRO
(VWLPDWH6XEOLPDWLRQSUHVVXUHDW.
OQ
3VXE .
·
§ ¨
¸
© ¹
3VXE u N3D
6LQFHWKHSUHVVXUHLVEHORZDWPRVSKHULFZHFDQDVVXPHWKDW
IVROLG .N3D N3D EDU
1RZQHHGWRFRUUHFWWREDU
ª EDU u u P º
PRO
»
IVROLG .EDU EDU([S «
EDU P
«
»
<
.
u
PRO
N
¬
¼
î EDU
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
D $SORWRIWKHORJDULWKPRIWKHVXEOLPDWLRQSUHVVXUHLVVKRZQEHORZ:HVHHWKDWWKHSORWLVYHU\FORVHWROLQHDU6LQFH
' +
OQ 3VXE VXE & ZHILQGWKDW
5 ˜7
'VXE +
.RUWKDW'VXE + N- PRO 5
OQ3VXEYHUVXV7
OQ 3L
§·
¨ ¸
© 7L ¹
E (VWLPDWH6XEOLPDWLRQSUHVVXUHDW.
3VXE .
·
§ ¨
OQ
¸
© ¹
VXE
3
u N3D
6LQFHWKHSUHVVXUHLVEHORZDWPRVSKHULFZHFDQDVVXPHWKDW
IVROLG .N3D N3D EDU
1RZQHHGWRFRUUHFWWREDU
ª EDU u u P º
PRO
»
IVROLG .EDU EDUH[S «
EDU P
u PRO . < .
«¬
»¼
î EDU
§ w3 ·
)URPWKHFRQGLWLRQRIPHFKDQLFDOVWDELOLW\WKHFRQGLWLRQIRUDVWDEOHHTXLOLEULXPVWDWHLV ¨
¸ © w9 ¹7
,IRQHIROORZVWKHLVRWKHUP7 '9 9 9 ! ZKLOH'3 3 3 ! 7KHUHIRUH
§ '3 ·
§ w3 ·
¨
¸
¨
¸ ! &RQVHTXHQWO\WKHSUHVVXUHHQWKDOS\GLDJUDPIRUWKLVV\VWHPLVQRW
9
'
©
¹7 © w9 ¹7
FRQVLVWHQWZLWKWKHUPRG\QDPLFVWDELOLW\
D 6WDUWLQJIURP 3
5 ˜7
D
ZHREWDLQ
9E
7 ˜ 9
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
§ w3 ·
¨
¸
© w9 ¹7&
5 ˜ 7&
9E
§ w3 ·
D
DQG ¨¨ ¸¸
7& ˜ 9
© w9 ¹7&
˜ 5 ˜ 7&
9E
D
7& ˜ 9 5 ˜ 7& 9& 6XEVWLWXWLQJ
5 ˜ 7&
VRWKDWWKH
WKLVH[SUHVVLRQLQWRWKHHTXDWLRQRIVWDWHDWWKHFULWLFDOSRLQWZHREWDLQ 3&
9&
FULWLFDOFRPSUHVVLELOLW\LVDVIRUWKHYDQGHU:DDOVHTXDWLRQ8VLQJWKLVODVWUHODWLRQWR
HOLPLQDWH9&ZHREWDLQ
5 ˜ 7&
5 7& D
DQGE
3&
˜ 3&
E 6WDUWLQJIURP
6ROYLQJWKHVHWZRHTXDWLRQVVLPXOWDQHRXVO\ZHJHW 9&
E DQG D
97
&9 97
§ w3 ·
¨ ¸
© w7 ¹9
&9 97
&9 7 7
§ w3 ·
¨¨ ¸¸ G9 DQGXVLQJWKDW
w7 ¹9
9 f©
³
§ w3 ·
5
D
DQG ¨¨ ¸¸
9E 7 9
© w7 ¹9
&9 7 7
97
³
˜ D
˜ 9
7 9 f
G9
D
VRWKDW
7 9 &9 7 D
7 ˜9
7KLVH[SUHVVLRQGHVQRWGLYHUJHDWWKHFULWLFDOSRLQWVRWKHKHDWFDSDFLW\LVILQLWHDWWKHFULWLFDO
SRLQW
6WDUWZLWKWKHWULSOHSURGXFWUXOH IRUVLPSOLFLW\LQW\SLQJQHJOHFWLQJWKHXQGHUEDUVWKDWLQGLFDWH
PRODUSURSHUWLHV § w6 ·
¨
¸
§ w9 · § w6 · § w7 ·
§ w9 ·
© w7 ¹9
Ÿ ¨
¨
¸ ¨
¸ ¨
¸
¸ w6
§
·
© w7 ¹6 © w9 ¹7 © w6 ¹9
© w7 ¹6
¨
¸
© w9 ¹7
&9
&9
§ w3 ·
§ w6 ·
§ w6 ·
§ w3 ·
6WDUWLQJIURPG6
G7 ¨
DQG ¨
¸ G9JLYHV ¨
¸
¸
¨
¸ w
w
w
7
7
7
7
9
©
¹9
©
¹9
©
¹7 © w7 ¹9
§ w9 ·
VRWKDW ¨
¸
© w7 ¹6
§ w6 ·
¨
¸
© w7 ¹9
§ w6 ·
¨
¸
© w9 ¹7
§ w9 · § w3 · § w7 ·
¨
¸ ¨
¸ ¨
¸
© w7 ¹3 © w9 ¹7 © w3 ¹9
&9 § w7 ·
¨
¸
7 © w3 ¹9
§ w9 ·
Ÿ ¨
¸
© w7 ¹3
7KHUHIRUH
§ w9 ·
¨
¸
© w7 ¹6 &9 § w3 · § w7 ·
¨
¸ ¨
¸ DOVRN7
7 © w9 ¹7 © w3 ¹9
§ w9 ·
¨
¸
© w7 ¹3
E &OHDUO\
§ w3 · § w7 ·
¨
¸ ¨
¸
© w9 ¹7 © w3 ¹9
§ w9 ·
¨
¸
§ w9 ·
© w3 ¹6
¨
¸ VRWKDW w9
9 © w3 ¹7
§
·
¨
¸
© w3 ¹3
&9 § w7 ·
¨
¸ 79N7 © w3 ¹9
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
§ w7 ·
¨
¸ ! 7!DQG9 ! $OVRZHKDYHVKRZQWKDWIRUDVWDEOHV\VWHP&9 t DQGN7 t © w3 ¹9
§ w9 ·
¨
¸
§ w9 ·
§ w9 ·
© w7 ¹6
RU ¨
7KHUHIRUH
¸ DQG ¨
¸ PXVWKDYHRSSRVLWHVLJQV
§ w9 ·
© w7 ¹6
© w7 ¹3
¨
¸
© w7 ¹3
F 7KHHQWURS\EDODQFHIRUWKHILUVWPHDVXUHPHQWLV
G6 VLQFHWKHUHLVQRKHDWIORZDQGWKHSURFHVVLVUHYHUVLEOH6RWKDWFDQEHGHWHUPLQHGIURPWKH
§ w9 ·
PHDVXUHPHQWLV ¨
¸ ,QWKDWPHDVXUHPHQW7KDVLQFUHDVHGZKHQ9LVGHFUHDVHG JDVLVFRPSUHVVHG
© w7 ¹6
§ w9 ·
VRWKDW ¨
¸ ,QWKHVHFRQGPHDVXUHPHQWDVSUHVVXUHLVFRQVWDQWWKHUHODWLRQEHWZHHQ7DQG9LV
© w7 ¹6
§ w9 ·
§ w9 ·
§ w9 ·
¨
¸ 6LQFHZHKDYHVKRZQLQSDUWEWKDW ¨
¸ KDVRSSRVLWHVLJQIURP ¨
¸ ZKLFKLVQHJDWLYH
© w7 ¹3
© w7 ¹3
© w7 ¹6
§ w9 ·
LWIROORZVWKDW ¨
¸ ! VRWKDWWKHWHPSHUDWXUHLQFUHDVHVDVWKHYROXPHLQFUHDVHV
© w7 ¹3
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
*O\FLQH
§ ·
¨
¸
¸
¨
GDWD ¨ ¸
¨ ¸
©
¹
; GDWD
;
¢²
< GDWD
¢²
1XPEHURIGDWDSRLQWV
< OQ <
;
Q URZV GDWD
E LQWHUFHSW ; <
E
6ORSH
E VORSH ; <
E
§
E
© 7 ¹
'IXV+ 5˜ E
Q 5HJUHVVLRQ6WDWLVWLFV
,QWHUFHSW
3 7 H[S¨ E 5 ·
¸
3 3 'IXV+
u UHDVRQDEO\JRRGILW
u Solutions to Chemical and Engineering Thermodynamics, 5th ed.
$ODQLQH
§¨ ·¸
¸
GDWD ¨
¨ ¸
¨ ¸
©
¹
; GDWD
;
5 ¢²
< GDWD
¢²
1XPEHURIGDWDSRLQWV
< OQ <
;
Q URZV GDWD
5HJUHVVLRQ6WDWLVWLFV
,QWHUFHSW
E LQWHUFHSW ; <
E
6ORSH
E VORSH ; <
E
§
3 7 H[S¨ E ©
'IXV+ 5˜ E
Q E
·
¸
7 ¹
3 3 'IXV+
u UHDVRQDEO\JRRGILW
u 7KHERLOLQJWHPSHUDWXUHRIZDWHULVR&ZKHQLWVYDSRUSUHVVXUHLVEDU6RWKHERLOLQJWHPSHUDWXUHRID
VPDOO
OLTXLGZDWHUGURSLVJLYHQE\
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
f Z T EDU
§ V ·
P YDS T H[S ¨
V¸
© rRT ¹
§
FP ·
u
u
u
¨
¸
P
PRO ¸
P YDS T H[S ¨
¨ r u - T ¸
¨
¸
PRO ˜ .
©
¹
§
·
P
u
u
u
u
FP
¨
¸
P
FP ¸
P YDS T H[S ¨
¨
¸
r u T
¨
¸
.
©
¹
§ u u u ·
§ u ·
YDS
H[S
P
T
P YDS T H[S ¨
¨¨
¸¸ EDU
¨ r FP u u T . ¸¸
u
FP
.
r
T
©
¹
©
¹
6RWKHSURFHGXUHQRZLVWRSURFHHGLQRQHRIWZRZD\V)LUVWZRXOGEHWRFKRRVHDYDOXHRI
r DQGILQGWKHYDOXHRIT WKDWVDWLVILHVWKLVHTXDWLRQ7KHVHFRQGDQGVRPHZKDWVLPSOHUZD\LV
WRFKRRVHDYDOXHRIT DQGILQGWKHYDOXHRIr,QIDFWWKHHTXDWLRQDERYHFDQEHPDGHLQWRD
OLQHDUHTXDWLRQIRUUDVIROORZV
u r FP u T .
OQ
RUr FP
YDS
P EDU
u
T . u OQ YDS
P EDU
)RUH[DPSOHDWR &P YDS EDUVR
u
u u
r FP
u
OQ
u OQ
7KDWLVIRUWKHERLOLQJWHPSHUDWXUHRIDZDWHUGURSOHWWREHUHGXFHGR &WR.
WKHEXEEOHUDGLXVVKRXOGEH u FPRU u P
6LPLODUFDOFXODWLRQVFDQEHGRQHDWRWKHUWHPSHUDWXUH
ƚƚŚĞĐƌŝƚŝĐĂůƉŽŝŶƚ
^ƵďƚƌĂĐƚŝŶŐĞƋŶ͘;ΎͿĨƌŽŵ;ΎΎͿŐŝǀĞƐ
hƐŝŶŐƚŚŝƐĞƋƵĂƚŝŽŶŝŶĞƋŶ;ΎͿŐŝǀĞƐ
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
dŚĞƌĞĨŽƌĞ͕ƚŚĞĨůƵŝĚŚĂƐĂĐƌŝƚŝĐĂůƉŽŝŶƚ;ĂŶĚƚŚĞƌĞĨŽƌĞǀĂƉŽƌͲůŝƋƵŝĚĞƋƵŝůŝďƌŝĂͿ͘dŚĞ
ĐŽŵƉƌĞƐƐŝďŝůŝƚLJĨĂĐƚŽƌĂƚƚŚĞĐƌŝƚŝĐĂůƉŽŝŶƚŝƐ
ĂŶĚ
9DSRUSUHVVXUHRIUHIULJHUDQWV
3/EDU
3/YV7HPSHUDWXUH
/,48,'5
/,48,'5$
/,48,'5
/,48,'5
7(03 35(6 5
&
EDU
EDU
7HPSHUDWXUH&
5$ 5
EDU
EDU
5
EDU
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϳхWƌŽďůĞŵƐхWƌŽď͘ϳ͘ϭ /ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϳхWƌŽďůĞŵƐхWƌŽď͘ϳ͘ϯϯ /ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϳхWƌŽďůĞŵƐхWƌŽď͘ϳ͘ϯϰ )URPHTQZHKDYHIRUWKHYDQGHU:DDOVHTXDWLRQWKDW
f
A
Z OQ Z B P
Z
:HVHHLQ)LJ WKDWLVIRUFRUUHVSRQGLQJVWDWHVQRWWKHYDQGHU:DDOVIOXLGEXWLV
UHSUHVHQWDWLYHRIDOOIOXLGV WKDWDVWKHSUHVVXUHLQFUHDVHVDWFRQVWDQWWHPSHUDWXUHWKHFRPSUHVVLELOLW\
IDFWRU=ILUVWGHFUHDVHVDQGWKHQDWKLJKHUSUHVVXUHVVRLWVEHKDYLRULVQRWPRQRWRQLF6LQFH=
DSSHDUVLQDYHU\QRQOLQHDUPDQQHULQWKHIXJDFLW\HTXDWLRQDERYHLWLVWREHH[SHFWHGWKDWWKH
IXJDFLW\LVDQRQOLQHDUIXQFWLRQRISUHVVXUHDWFRQVWDQWWHPSHUDWXUH
OQ
^ĞĞƐŽůƵƚŝŽŶƚŽWƌŽďůĞŵϳ͘ϳϰ
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
D %\(TQ FG w H IJ
Hw N K
i
EXW Gi
Hi TSi 7KXV
FG w H IJ
Hw N K
i
E 6LQFH U
f
F w U IJ
dU G
HwSK
a
U S V N dS VN
SN
i
i
+RZHYHUZHDOVRKDYH U
(TXDWLQJ DQG VKRZVWKDW Gi
A
A T V N
Ÿ dA
FG w A IJ
HwTK
dT FG w U IJ
Hw N K
V N
FG w A IJ
H wV K
TN
dA
SdT PdV ¦
i
TdS PdV ¦ Gi dNi 1H[WZHVWDUWIURP
S V N j zi
dV ¦
i
FG w A IJ
Hw N K
i T V N j zi
dNi
RU
dNi
dH PdV VdP DQGE\(TQ i
S V N j zi
i
S V N j zi
VdP TdS ¦ Gi dNi PdV VdP
dU
i
i
H PV dU
Hi TSi P S N j zi
FG w U IJ dV ¦ FG w U IJ
H wV K
Hw N K
F w U IJ dN
TdS PdV ¦ G
Hw N K
Gi P S N j zi
FG w A IJ
Hw N K
i T V N j zi
dNi +RZHYHUZHDOVRKDYHWKDW A U TS dA
dU TdS SdT
TdS PdV ¦ Gi dNi TdS SdT Solutions to Chemical and Engineering Thermodynamics, 5e
SdT PdV ¦ Gi dNi dA
i T P N j zi DQG
¦ T i dNi ¦ Ni dT i FG w T IJ
Hw TK
dT V N
FG w T IJ
HwV K
TN
dV ¦
FG w T IJ
Hw TK
dT V N
FG w T IJ
HwV K
TN
LM F w T I
MN GH w N JK
dV ¦ T i i T V N j zi
i
i
i
OPdN N dT PQ ¦
FG w T IJ
Hw N K
FG w A IJ
Hw N K
i
i
i
JHQHUDOHTXDWLRQ )RU T
A T i
Ai DQG
¦ Ni dAi T V
i T V N j zi
Gi 7KXV T i i T V N j zi
P¦ Vi dNi T V VSHFLILFHTXDWLRQIRU T
FG w T IJ
Hw N K
i T V N j zi
Ai Gi
A
E )ROORZLQJWKHDQDO\VLVDERYHZHDOVRJHW
FG w T IJ
HwUK
dU V N
FG w T IJ
HwV K
U N
LM F w T I
MN GH w N JK
dV ¦ T i i
i
i
OPdN N dT PQ ¦
G
F w S IJ
ZKLFKLVHDVLO\GHULYHG\LHOGV
1RZFKRRVLQJ T S DQGXVLQJWKDW G
Hw N K
T
i U V N j zi
OPdN N dT PQ ¦
DQGDWFRQVWDQWUDQGV
L F wT I
N
¦ MMT GH w N JK
i
i
i
dNi OPdN N dT PQ ¦
L F wT I
MN
¦ MT i GH w N JK
i T V N
i T V N j zi
$WFRQVWDQWTDQGV
FG w T IJ
Hw N K
6XEWUDFWLQJ IURP \LHOGV
+RZHYHUZHDOVRKDYHWKDW
dT
aw T w N f
dT
i T V N j zi
¦ NiT i ZKHUH T i
D *HQHUDO T
FG w A IJ
Hw N K
Gi
&RPSDULQJ DQG \LHOGV
Chapter 6
RU
i
i
U V N j zi
i
i U V N j zi
PVi DQG
Solutions to Chemical and Engineering Thermodynamics, 5e
T ¦ Ni dSi U V
F )ROORZLQJDVLPLODUDQDO\VLVWRWKRVHDERYHZHREWDLQ
¦ Hi dNi U V FG w T IJ
Hw SK
dS V N
FG w T IJ
HwV K
SN
LM F w T I
MN GH w N JK
dV ¦ T i i
ZKLFKDWFRQVWDQWVDQGSUHGXFHVWR
L F wT I
MN
)LQDOO\XVLQJ T U DQG aw U w N f
¦ Ni dUi S V ¦ l PVi TSi qdNi S V ¦ MT i GH w N JK
i
,
,
i
i
Gi \LHOGV
C
C
i i ¦ Ni, Si, ¦ Ni,, Si,, ,
,
,,
V N
,
i U V N j zi
U N
,,
,,
,
,
,
V N
,,
,,
dNi,
,,
,,
,,
i U V N j zi
U N
dNi,,
6LQFH U U , U ,, FRQVWDQW dU ,, dU , 6LQFH V V , V ,, FRQVWDQW dV ,, dV , DQGVLQFH Ni Ni, Ni,, FRQVWDQW dNi,, dNi, $OVR
FG w S IJ
HwU K
i
FG w S IJ dU FG w S IJ dV ¦ FG w S IJ
HwU K
HwV K
Hw N K
F w S IJ dU FG w S IJ dV ¦ FG w S IJ
G
Hw N K
HwV K
HwU K
FG
H
FG IJ
H K
V N
IJ
K
P
wS
DQG
w Ni U V N
T
wS
w V UN
T
j zi
Gi
T
VHHSUHYLRXVSUREOHP i
EXW
S , S ,,
S
dS
i
PD[LPXPDWHTXLOLEULXP
D $WFRQVWDQWUDQGV S
i
OPdN N dT PQ ¦
i S V N j zi
S V N j zi
S V N j zi
OPdN N dT PQ ¦
7KXV
dS
Ÿ T,
F I dU FG P P IJ dV ¦ FG G G IJ dN
HT T K HT T K
HT T K ,
T ,, P ,
,
,,
,
,,
P ,, DQGGi ,
Gi ,,
,,
,
,
,
,
i
,,
,,
,
i
Solutions to Chemical and Engineering Thermodynamics, 5e
IRUHTXLOLEULXPLQDFORVHGV\VWHPDWFRQVWDQWUDQGV
E )RUDFORVHGV\VWHPDWFRQVWDQWSDQGV U KDVDQH[WUHPXP7KXV
dU
FG w U IJ dS FG w U IJ dV ¦ FG w U IJ
HwS K
H wV K
Hw N K
F w U IJ dS FG w U IJ dV ¦ FG w U IJ
G
Hw N K
H wV K
HwS K
,
,
EXWSVDQG N j j
,
%XW Ni
dA V T
,,
i U V N j zi
i
dNi,,
" C DUHFRQVWDQW7KXV
,
,,
T P
,
,
,,
,
i
,
,,
P DQGGi
,
,
i
,,
,
i
Gi
,,
PdV SdT ¦ Gi dNi RU dA V T
Ni Q i X 7KXV dNi
b¦ Q G gdX RU FGH ww XA IJK
i
i
dS U V
)URP(TQ ¦ Gi dNi Q i dX DQG
¦ Q i Gi
V T
i
PD[LPXP RU dS U V
)URP (TQ
P
dU dV ¦ Gi dNi WKXV
T
T
T
¦ Gi dNi RU dS U V
T
¦ GiQ i dX T
b
g
DQG
wS
w X U V
SN
E )RU D FORVHG V\VWHP DW FRQVWDQW U DQG V S
dS
,,
,,
V N
,,
dA
,,
S V N j zi
,,
D )RUDFORVHGV\VWHPDWFRQVWDQWTDQGVALVDPLQLPXPDWHTXLOLEULXPWKXV dA V T
i
,,
dNi,
IRUHTXLOLEULXPLQDFORVHGV\VWHPDWFRQVWDQWSDQGV
SN
,
i
i
ŸT
,
V N
,
,
cT T hdS c P P hdV ¦ cG G hdN
dU
,
,
,,
,,
Chapter 6
¦ Q i Gi
T i
/HW mi PROHFXODUZHLJKWRIVSHFLHV i0XOWLSO\LQJ(TQ D E\ mi DQGVXPPLQJRYHUDOOVSHFLHV i\LHOGVIRUD closed
V\VWHP
¦ mi Ni
WRWDOPDVVLQV\VWHP
¦ mi Ni X ¦ Q i mi WRWDOPDVVLQ
V\VWHPLQLWLDOO\
+RZHYHUVLQFHWKHWRWDOPDVVLVDFRQVHUYHGTXDQWLW\
¦ mi Ni ¦ mi Ni Ÿ X ¦ Q i mi
ZKHUHXFDQWDNHRQDQ\YDOXH
&RQVHTXHQWO\LIWKLVHTXDWLRQLVWREHVDWLVILHGIRUDOOYDOXHVRIXWKHQ ¦ Q i mi
Solutions to Chemical and Engineering Thermodynamics, 5e
6LPLODUO\LQWKHPXOWLUHDFWLRQFDVHVWDUWLQJIURP Ni
C
C
M
C
M
i i i j i j j ¦ mi Ni ¦ mi Ni o ¦ mi ¦ Q ij X j Ÿ ¦ mi ¦ Q ij X j
C
M
Ni ¦ Q ij X j ZHJHW
M
C
j i ¦ X j ¦ Q ij mi 6LQFHWKH X j ¶VDUHQRWLQJHQHUDOHTXDOWR]HURZHKDYH
C
¦ Q ij mi
¦ Q ij mi
i
ZHKDYH
a
f
a
f
V V 'V PL[ x
w 'V PL[
w x T P
V 'V PL[ x
w 'V PL[
w x T P
DQG
+ 2 RU + 2 + 2
)URP(TQV ZHKDYH
a f
a f
F I H K
,QSDUWLFXODUIRUWKHUHDFWLRQ + 2
i V
1RZVLQFHTPDQGXDUHWKHLQGHSHQGHQWYDULDEOHVZHKDYHWKDW
dV T P
VLQFHSXUHFRPSRQHQWYROXPHLVDIXQFWLRQRI
TDQG PRQO\
w 'V PL[
dV T P d 'V PL[ T P d x
w x
T P
a
a
LMN a
f
a
f
f OP
Q
a
f
f
w 'V PL[
w 'V PL[
w 'V PL[
w x
dx
dx x
w x T P
w x T P w x
w x
T P
a
6LPLODUO\
f
w 'V PL[
wx
dxVLQFH w x
w x
T P
x
dV T P
x
a
f
a
f
w 'V PL[
dx w x
TP
7KXV
¦ xiDVi T P
xx
a
f
w 'V PL[
w 'V PL[
dx { dx
x
x
w x
w x
TP
TP
Solutions to Chemical and Engineering Thermodynamics, 5e
Chapter 6
7KXVV DQGV JLYHQE\HTXDWLRQV DQG LGHQWLFDOO\VDWLVI\WKH*LEEV'XKHPHTXDWLRQ ¦ xi dT i T P
$VLPLODUDUJXPHQWDSSOLHVIRUWKHSDUWLDOPRODUHQWKDOSLHVRI(TQ DOVRDYDLODEOHDVD0DWKFDGZRUNVKHHW 7KHVWXGHQWVFDQVROYHWKLVSUREOHPE\GUDZLQJWDQJHQWOLQHVWRWKH 'V PL[ FXUYHV3RODNDQG/XVPRRWKHGWKHLUGDWDXVLQJWKH
5HGKLFK.LVWHUHTXDWLRQ VHH(TQ D 7KDWLVWKH\ILWWHGWKHLUGDWDWR
1RZ
a
w 'V PL[
w x
a
n
j f a x f¦ C a x f
7KXV
DQG
a
j PL[
j
j a'V f x w a'wVx f x k A x Bp PL[
PL[
¦ C j a xf j DQG B ¦ C j j a xf j A
n
n
j j 7DNLQJVSHFLHVWREHPHWK\OIRUPDWH3RODNDQG/XIRXQG
C
C
C
C
PHWK\OIRUPDWH 0HWKDQRO PHWK\OIRUPDWH (WKDQRO
>XQLWVDUHFFPROPXOWLSO\E\ WRJHW P NPRO @
,KDYHXVHGWKHHTXDWLRQVDERYHDQGWKHFRQVWDQWVJLYHQWRILQG V V DQG V V VLQFHWKLVOHDGVWRPRUHDFFXUDWHUHVXOWV
WKDQWKHJUDSKLFDOPHWKRG
7KHUHVXOWVDUHWDEXODWHGDQGSORWWHGEHORZ
0HWK\OIRUPDWH0HWKDQRO
'V PL[ 'V PL[ a
V0
± ± ± ± ±
f
V V 7KXV V0)
x0) ±
±
±
±
FF PRO V V ± ± ± ± ±
± ± ± ± ± ±
V V x0) aFF PROf V V j ZKHUH
PL[
V V f
x x ¦ C j x j f x a x f¦ C j a x f
w a'V f
V V a'V f x
a x f k A x Bp wx
j
x ¦ C j x
a
f
xx ¦ C j x x j 'V PL[
± ± ± ±
± ± ± ± ±
b
g
V V FFPRORU P NPRO b
g
V V Solutions to Chemical and Engineering Thermodynamics, 5e
0HWK\OIRUPDWH(WKDQRO
x0) 'V PL[ FF PRO a
V V V V 'V PL[ 7KXV V0)
V(
a
V V x0) FF PRO V V f
f
b
g
V V FFPRO0XOWLSO\E\ IRU P NPRO b
g
V V 7KLVSUREOHPLVVLPLODUWRWKHODVWRQHDQGZLOOEHWUHDWHGLQDVLPLODUIDVKLRQ)HQE\DQG5XHQNUDLUHUJDVDJLYHWKHLUGDWDLQ
WKHIRUP
Solutions to Chemical and Engineering Thermodynamics, 5e
a
'H PL[ - PRO
f x a x f¦ C a x f j
j j ZKHUHFRPSRQHQWLVWKHIOXRUREHQ]HQH7KHFRQVWDQWVJLYHQLQWKHDIRUHPHQWLRQHGUHIHUHQFHDQG)HQE\DQG6FRWWJ. Phys.
Chem DUHJLYHQEHORZ
6\VWHP
C C C C ±
& + & )&O ±
& + & )%U ±
±
& + & ), ±
& + & ) & + &)+ ,IZHUHSODFH x ZLWK x LQ(TQ ZHUHJDLQWKHHTXDWLRQRIWKHSUHYLRXVLOOXVWUDWLRQH[FHSWIRUDIDFWRURI j LQWKH
VXPDQGWKHFRUUHVSRQGLQJSODFHVLQWKHRWKHUHTXDWLRQV
x& + 'H PL[ ±
±
±
±
±
±
±
±
±
bH Hg
& + ±
±
±
±
±
±
±
±
±
±
bH Hg
& )&O
±
±
±
±
±
±
±
±
±
±
x& )&O >1RWH-PRO@
x& + Chapter 6
n
& + & )%U 'H PL[ H H H H & + & )%U
±
± ± ± ± ± ± ± ± ± ± ± ± ±
±
± ±
±
± ±
± ± ±
± ± ±
± ± ± b
gb
g
& + & ), x& ) x 'H PL[ H H H H
& + & ),
±
± ± ±
± ± ±
± ± ±
± ± ±
± ± ±
± ± ±
±
± ±
±
± ±
±
± ±
±
b
gb
g
Solutions to Chemical and Engineering Thermodynamics, 5e
x& + & + &)+ x& ) x 'H PL[ H H H H 'H PL[ H H H H
& + & )
& + & )+
±
± ± ± ±
±
± ± ± ±
±
± ± ± ± ± ± ± ±
±
±
± ±
±
±
± ±
±
±
±
±
±
±
±
±
±
n
n
n
1RWH&KDQJHVLQVLJQLQFROXPQ
& + & ) b
F )RUPDWLRQUHDFWLRQV
& 2 o &2
& 2 o &2
+ o + & + o &+ + 2 o + 2
8VH 2
gb
g
+ WRHOLPLQDWH2DQG+IURPWKHVHWVRWKDW
&2 & DQG +
& &2 & o &2
a
a
&2 o &2 &
f
& + o &+ Ÿ & + o &+ f
+ &2 & o + 2
+ &2 o + 2 &
7KXVZHKDYHIRXQG aVHW WKHUHLVQRXQLTXHVHW RIWKUHHLQGHSHQGHQWUHDFWLRQVDPRQJWKHVL[VSHFLHV&RQVHTXHQWO\
C M P VROLGFDUERQJDVSKDVH F C M P GHJUHHVRIIUHHGRP$VDFKHFN
c
RIXQNQRZQV T 6 P6 T 9 P 9 x&2 x&2 x+ x&+ b
D *LEEV3KDVH5XOH F C M P P C M Ÿ F GHJUHHVRIIUHHGRP
7KXVFDQIL[WZRYDULDEOHVXVXDOO\IURPDPRQJTPxDQGy
E P C DQG M Ÿ F GHJUHHVRIIUHHGRP
7KXVZHFDQIL[YDULDEOHVIRUH[DPSOHTPDQG x+ g
gb
1RWH x+ 2
x&2 x&2 x+ x&+ 5HODWLRQVDPRQJWKHXQNQRZQV T 6
RIWKHIRUP ¦ Q ij Gi T 9 P6
h
P 9 QRSKDVHHTXLOLEULXPUHODWLRQVEXWFKHPLFDOHTXLOLEULXPUHODWLRQV
Solutions to Chemical and Engineering Thermodynamics, 5e
XQNQRZQV HTQV XQVSHFLILHGXQNQRZQVRU
GHJUHHVRIIUHHGRP
Chapter 6
D ,QJHQHUDOIRUDELQDU\WZRSKDVHPL[WXUH C M P F C M P GHJUHHVRIIUHHGRP
+RZHYHUIRUDQD]HRWURSHWKHUHLVWKHDGGLWLRQDOUHVWULFWLRQ x y ZKLFKHOLPLQDWHVRQHGHJUHHRIIUHHGRP7KXVWKHUH
LVRQO\GHJUHHRIIUHHGRPIRUDELQDU\D]HRWURSLFV\VWHP
E ,QRVPRWLFHTXLOLEULXP P , z P ,, VLQFHWKHPHPEUDQHLVFDSDEOHRIVXSSRUWLQJDSUHVVXUHGLIIHUHQFHDQG G, z G,, ZKHUH
a
f
LVWKHVSHFLHVZKLFKGRHVQRWSDVVWKURXJKWKHPHPEUDQH7KHUHIRUHWKHLQGHSHQGHQWXQNQRZQVDUH T , P , x, T ,, P ,, DQG x,, >1RWH x, DQG x,, DUH QRW LQGHSHQGHQW XQNQRZQV VLQFH x,
HTXLOLEULXPUHODWLRQVEHWZHHQWKHVHVL[XQNQRZQVYL] T ,
T ,, DQG G,
x, DQG x,,
x,, @ 7KHUH DUH two
G,, &RQVHTXHQWO\WKHUHDUHIRXUGHJUHHVRI
IUHHGRP}WKDWLVDVZHVKDOOVHHLQ6HFLIT P , P ,, DQG x, DUHVSHFLILHG x,, ZLOOEHIL[HG
F &DVH, M C P Ÿ F &DVH,, M C P Ÿ F D *LEEV3KDVH5XOH F C M P C M Ÿ F P P GHJUHHVRIIUHHGRP
7KHUHIRUHDPD[LPXPRISKDVHVFDQH[LVWDWHTXLOLEULXP IRUH[DPSOHDVROLGWZROLTXLGVDQGDYDSRURUWZRVROLGVD
OLTXLGDQGDYDSRUHWF E *LEEV3KDVH5XOH F C M P C M Ÿ F P P GHJUHHVRIIUHHGRP
7KHUHIRUHDPD[LPXPRIHSKDVHVFDQH[LVWDWHTXLOLEULXP IRUH[DPSOHDWZROLTXLGVDQGDYDSRURUDVROLGDOLTXLGDQG
DYDSRUHWF dNi
N i N i U[Q D dt
dV
dU
N i Hi Q WV P
¦
dt dt
Q
dS
N i Si SJHQ
¦
T
dt
dS
T ¦ N i Si TSJHQ Q
T
dt
dV
dS
dU
¦ N i Hi T dt T ¦ N i Si TSJHQ P dt
dt
dS
dV
dU
N i Hi TSi TSJHQ
P
T
¦
dt
dt
dt
dN
dX
dS
dV
dU
P
T
¦ N i Pi TSJHQ ¦ dt i Q i dt P i TSJHQ dt
dt
dt
*HQHUDOH[SUHVVLRQ
1RZ
6\VWHPLVRQO\SHUPHDEOHWRVSHFLHV
dS
dN
dX
dV
dU
P TSJHQ d P
T
Q
dt
dt
dt
dt
dt
:KHQTDQGPFRQVWDQW
d
d
N Q X P d U PV TS dt
dt
b
g
F
H
F
H
I
K
a
f
I
K
Solutions to Chemical and Engineering Thermodynamics, 5e
a
a
f
f
a
a
f
d
G N Q X P d dt
Ÿ G N Q X P PLQLPXPDWHTXLOLEULXP
E :KHQTDQGVDUHFRQVWDQW
d
d
N Q X P d U TS dt
dt
Ÿ A N Q X P PLQLPXPDWHTXLOLEULXP
D
f
1 o 1 2 o 2
2 o 1 2
1 2 o 12 1 2 o 1 2
1 1 2 o 12
1 2 o 1 2
1 2 o 1 2
1 2 o 12
1 2 o 12
1 2 o 1 2
1 2 o 1 2
Ÿ LQGHSHQGHQWUHDFWLRQV
E F C M P F GHJUHHVRIIUHHGRP
F GHJUHHRIIUHHGRPXVHGLQ 21 UDWLRŸGHJUHVRIIUHHGRP
0DVVEDODQFH M M M f 0ROHFXODUZHLJKW+ 2 J PRO (QHUJ\EDODQFH MU M U ,QHDFKFDVHWKHV\VWHPLV MNJ RIVROXWLRQ M NJ RIVROXWLRQ
6LQFH Q Ws DGLDEDWLFPL[LQJ )RUOLTXLGV U { H 7KXVZHKDYH
MH M H H f
M M H H ZKHQ M M H f
D 5HDGIURP)LJXUH
H u - NJ
H
u - NJ
c
M f U f h
u u - NJ 7KXV H f
7RILQGWKHFRPSRVLWLRQVRDVXOIXULFDFLGEDODQFH
U M U M U f M f Ÿ U f
U U VLQFH M M ZKHUH U i ZHLJKWSHUFHQW RIiWKIORZVWUHDP
7KXV U f
ZW VXOIXULFDFLG)URP)LJXUH
c
h
a
f
ZW+ 62
Ÿ Tf a q & H U u - NJ
Solutions to Chemical and Engineering Thermodynamics, 5e
E +HUH H Chapter 6
u - NJ u ZW 8VLQJ
H u - NJ Ÿ H f
U
ZW Ÿ U f
u - NJ DQG
U ZW )LJXUH Tf a q & 1RWLFH WKDW WKHUH LV D EDODQFH EHWZHHQ WKH HQHUJ\ UHOHDVHG LQ PL[LQJ 'H PL[ DQG WKH HQHUJ\ DEVRUEHG LQ KHDWLQJ WKH
PL[WXUH C 'T ,QFDVH D 'H LVYHU\ODUJHDQG T ! T RU T ZKLOHLQFDVH E 'H LVVPDOOHUVRWKDW T a T 3
PL[
D 0: + 2 J PRO 0: + 62
J+ 62
PL[
f
1RWH:KHQWKHVHDUHPL[HGDVROXWLRQFRQWDLQLQJ
PRO + 2 PRODFLGLVIRUPHG 'H V IRUVXFKDVROXWLRQLV±-PRODFLG7KXV
f - 1HJDWLYHVLJQPHDQVWKDWKHDWLVUHOHDVHG E $GGLQJ DQRWKHU JUDPV RI ZDWHU SURGXFHV D VROXWLRQ ZKLFK FRQWDLQV PRO + 2 PRO DFLG )URP WKH JUDSK
'H V - PRODFLG +RZHYHU±-PRORIDFLGZHUHUHOHDVHGLQSUHSDULQJWKHILUVWVROXWLRQVRWKDWRQO\±
-PRODFLGRU-DUHUHOHDVHGRQWKLVIXUWKHUGLOXWLRQ
F ZW + 62 Ÿ
PROHV+ 2 PROHVDFLG IRUZKLFK 'H V - PRODFLG DQG
PRODFLG
'HV - PRODFLG u
- 1RWH (QWKDOS\ RI :7 VROXWLRQ LV ± - UHODWLYH WR SXUH FRPSRQHQWV DW WKH VDPH WHPSHUDWXUH 6LPLODUO\
ZW+ 62 Ÿ PRO+ 2 PRODFLG 'H V a - PRODFLG DQG
u )LQDOVROXWLRQ JUDPV JUDPVDFLG PRO
JUDPVZDWHU PRO Ÿ PRO+ 2 PRODFLG 6RWKDW
'HV
- PRODFLG u
- PRODFLG
'HV
PRO
a
PRO
PRODFLG u - PRO DFLG
WRWDOKHDWUHOHDVHG J PRO J+ 2
f
-
'HV
7KXVHQWKDOS\FKDQJHRQPL[LQJ 'HPL[ LV
'HPL[ - 7KXV - N- PXVWEHremovedWRNHHSVROXWLRQLVRWKHUPDO
N
G )RUPROHRIVROXWH N H PL[ H N H ˜ ' H V DUJXPHQWRI 'H V DQGIRU N PROHVRIVROXWHDQG N N
PROHVRIVROYHQW
N
HPL[ N N H PL[ N H N H N' H V N
1RZ
w 'HV
N
w HPL[
w N N
H ' H V N
˜
H
N
w N T P
w N N T P
w N T P
a
a
FG
H
RU
FG IJ
H K
f
f
IJ
K
FG IJ
H K
FG IJ
H K
a f
a
a f
f
Solutions to Chemical and Engineering Thermodynamics, 5e
H ZWDFLG Ÿ
TP
TP
V
TP
PRO + 2 PRODFLG - PRO DQGIURPWKHDFFRPSDQ\LQJJUDSK
'H V a
PL[
a
f
w 'H V N N
w N N DW N N V
FG N IJ N LM w ' H a N N f OP VLQFH w a N N f N H N K N N waN N f Q
wN
N
F w H IJ ZHREWDLQ
6LPLODUO\VWDUWLQJIURP H G
H wN K
w 'H aN N f
H H
wa N N f
'HV
H H VRWKDW H H H H f
- PRO - PRO DQG
- PRO 7RJHWSDUWLDOPRODUSURSHUWLHVLWLVHDVLHVWWRILUVWFRQYHUWDOOGDWDLQSUREOHPWRPROHIUDFWLRQVDQGSURSHUWLHVSHUPROH
ZW &&O x&&O ZW&&O ZW&&O a
ZKHUH 0:&&O C3 PROHPL[WXUH
f a
f
0:& + C3 JUDPVPL[WXUH u 0:RIPL[WXUH
c
h
x&&O u C3 u x&&O u DOVRFRPSXWH ¦ xi C3i ZKHUH C3i
KHDWFDSDFLW\RISXUHVSHFLHViDQG 'C3 PL[
JLYHQEHORZ
C3 PL[WXUH ¦ xi C3i 5HVXOWVDUH
Solutions to Chemical and Engineering Thermodynamics, 5e
:W &&O x&&O C3 - PRO. f
'C3PL[ - PRO. ±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
C3& + x&&O ±
±
±
±
x&&O 3
3 &&O
C3&&O bC C g
3
3 & +
bC C g
3
3 &&O
bC C g
3
3 & +
C3& + C3&&O a
¦ xi C3i 8VLQJWKHVHGDWDDQGWKHJUDSKLFDOSURFHGXUHLQWURGXFHGLQ6HFZHREWDLQWKHIROORZLQJUHVXOWV
bC C g
Chapter 6
±
±
±
±
±
Solutions to Chemical and Engineering Thermodynamics, 5e
$QDOWHUQDWHVROXWLRQWRWKLVSUREOHPIROORZV
$OWHUQDWH6ROXWLRQWR3UREOHP
,QVWHDGRIXVLQJ(TXDWLRQV DDQGE DQG 'C3 PL[ GDWD(TXDWLRQV DDQGE DQGWKHKHDWFDSDFLW\GDWDIRUWKH
PL[WXUHFDQEHXVHG6LQFH(TXDWLRQV DDQGE DUHYHU\VLPLODUWR(TXDWLRQV DDQGE >RIZKLFK(TXDWLRQV
DDQGE DQG DDQGE DUHVSHFLDOFDVHV@LWIROORZVWKDWWKHJUDSKLFDOFRQVWUXFWLRQGLVFXVVHGLQ6HFFDQEH
Solutions to Chemical and Engineering Thermodynamics, 5e
Chapter 6
XVHG 7KH GLIIHUHQFH KRZHYHU LV WKDW WKH WDQJHQWV WR WKH C3 PL[ YV PROH IUDFWLRQ FXUYH ZLOO JLYH C3&&O DQG C3 &+ b
g
b
g
GLUHFWO\UDWKHUWKDQ C3 C3 &&O DQG C3 C3 & + DVEHIRUH$QLOOXVWUDWLYHJUDSKDQGWKHQXPHULFDOUHVXOWVREWDLQHG
XVLQJDPXFKODUJHUJUDSKDUHJLYHQEHORZ
x&&O C3&&O C3 &+ x&&O C3&&O C3 &+ 1RWHWKDWWKHVHUHVXOWVGLIIHUIURPSUHYLRXVUHVXOWVE\VPDOODPRXQWV3UHYLRXVUHVXOWVDUHSUREDEO\PRUHDFFXUDWHVLQFHWKHFXUYDWXUH
RI 'C3 PL[ YV x&&O LVJUHDWHUWKDQWKDWRI C3 PL[ YV x&&O VRWDQJHQWVDUHIRXQGZLWKJUHDWHUDFFXUDF\
/HW
x
y
UV
W
OEVRI:7VROXWLRQ XVHGWRPDNHOE
OEVRISXUHDFLG
RI:7VROXWLRQ
D 7RWDOPDVVEDODQFH x y 6SHFLHVPDVVEDODQFHRQDFLG x y Ÿ x x RU x NJ:7VROXWLRQ y
E )URP)LJXUH
NJSXUHDFLG
Solutions to Chemical and Engineering Thermodynamics, 5e
H ZW q & u - NJ
u - NJ
H ZW q & H ZW q & u - NJ
H ZW ERLOLQJSRLQW H ZW q & a - NJ
8VLQJWKHFKDQJHRYHUDWLPHLQWHUYDOIRUPRIWKHHQHUJ\EDODQFHHTXDWLRQFRQVLGHULQJWKHLQLWLDOVWDWHWREHWZROEVRI
VHSDUDWHGZWDQGSXUHDFLGVROXWLRQVDQGWKHILQDOVWDWHWREHOERIPL[HGVROXWLRQDQGQHJOHFWLQJWKHGLIIHUHQFH
EHWZHHQ H DQG U IRUWKHVHOLTXLGV\LHOGV
c
h
H ZW Tf H ZW q & H ZW q &
DW Tf
Q
b
g
b
u u u u u u - NJ ILQDOVROXWLRQ
DW Tf
Q
Q
q & ERLOLQJSRLQW
b
g
q & g b
u u g u - NJ ILQDOVROXWLRQ 6XSSRVHWKHUHZDVHQRXJKLQIRUPDWLRQDYDLODEOHRQ 'T PL[ ZKHUH TLVDQ\H[WHQVLYHWKHUPRG\QDPLFSURSHUW\RIDPL[WXUHDV
DIXQFWLRQRIWKHWKUHHPROHQXPEHUV N N DQG N WKDWWKHGDWDFRXOGEHILWWHGWRDSRO\QRPLDOH[SUHVVLRQLQ x x DQG
¦ Ni 7KH SDUWLDO PRODU SURSHUWLHV FRXOG WKHQ EH REWDLQHG E\
x RU HTXLYDOHQWO\ LQ N N DQG N ZKHUH N
i
GLIIHUHQWLDWLRQRIWKHSRO\QRPLDOH[SUHVVLRQIRU 'T PL[ 7KDWLVVLQFH
T
¦ Ni T i 'T PL[ a N N Nf
i NT
Ti
a
f
w 'T PL[
Ti w N i T P N j zi
wT
w N i T P N j zi
VRWKDW
Ti Ti
a
f
w 'T PL[
w N i T P N j zi
$OWHUQDWHO\JUDSKLFDOPHWKRGVFRXOGEHGHYHORSHGIRUILQGLQJ T i T i DORQJSDWKVZKHUH Ni LVYDULHGDQGRWKHUPROHQXPEHUV
DUHIL[HG LH xi LVYDULHGZKLOHWKHPROHUDWLRVRIWKHRWKHUVSHFLHVLQWKHPL[WXUHDUHIL[HG 6LQFHLWLVXQOLNHO\WKDWHQRXJKLQIRUPDWLRQZLOOEHDYDLODEOHIRUDQ\PL[LQJSURSHUW\WRREWDLQ 'T PL[ DVDQH[SOLFLWIXQFWLRQRI
PROHIUDFWLRQVRUVSHFLHVPROHQXPEHUVIRUWHUQDU\TXDWHUQDU\HWFPL[WXUHVLWLVQRWVXUSULVLQJWKDWWKHUHLVOLWWOHLQIRUPDWLRQ
RQSDUWLDOPRODUSURSHUWLHVLQVXFKV\VWHPV
Solutions to Chemical and Engineering Thermodynamics, 5e
Chapter 6
D 7KH*LEEV'XKHPHTXDWLRQLV (TXDWLRQE x
w H
w H
x
w x T P
w x T P
w H
w x T P
w x
w x
1RZ
xb
bx DQG
w H
w x T P
b x VRWKDW
x
E OLP T i
xi o
w H
w H
x
w x T P
w x T P
b b xx
T i Ÿ a
H ZKHUH H DQG H DUHWKHSXUHFRPSRQHQWPRODUHQWKDOSLHV7KXV
H DQG a
H
a
f
H bx H
IRUDOO x Ÿ b
b
b
H bx DQG
' PL[ H
x H H x H H ' PL[ H
x x bx x
bx x
xbx xbx
1RWH6RUU\DERXWVHWRIGDWDEHLQJJLYHQLQDOFRKROZWDQGRWKHULQZDWHUPROHEXWWKLVLVWKHZD\WKHGDWDDSSHDUHGLQ
WKH,QWHUQDWLRQDO&ULWLFDO7DEOHV
D )LUVWZLOOFRQYHUWWKHGDWDWRPROHIUDFWLRQV
NJ$ u NJ$ 0:$
ZW$
x$
NJ$ NJ:
NJ$ 0:$ NJ: 0::
ZW$
Ÿ x$
ZW$ ZW$ 0:$ 0::
$OVR V PL[ 0: U PL[ ZKHUH U PL[ PL[WXUH GHQVLW\ DQG 0: LV WKH PROH IUDFWLRQ DYHUDJHG PROHFXODU ZHLJKW RI
PL[WXUH LH 0:
$OVR V $
V:
¦ xi 0:i 0:$ U ZWDOFRKRO DQG
0:: U DOFRKRO Solutions to Chemical and Engineering Thermodynamics, 5e
ZW
DOFRKRO
x$ 0: V PL[ ¦ xiV i ' PL[ V FFPRO >PXOWLSO\E\ IRU
P NPRO @
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
b
g
b
g
7KH 'V PL[ GDWDDUHSORWWHGDQGWKHJUDSKLFDOSURFHGXUHRI6HFXVHGWRILQG V$ V $ DQG V: V : 5HVXOWVDUH
JLYHQLQWKHIROORZLQJWDEOH
1H[WQRWHWKDW
Solutions to Chemical and Engineering Thermodynamics, 5e
§ SHUPROH ·
' PL[ H ¨
¸
© PL[WXUH ¹
HYROYHG' PL[ H
LVQHJDWLYH
b
g
b
g
2QFH ' PL[ H LV FRPSXWHG JUDSKLFDO SURFHGXUH LV XVHG WR JHW H$ H $ DQG H: H : 7DEOH EHORZ JLYHV
bV V g bV V g b H H g DQG b H H g DVDIXQFWLRQRIWKHZDWHUPROHIUDFWLRQ
$
Chapter 6
+HDWHYROYHG
0ROHIUDFWLRQ
u
u
SHUPROHHWKDQRO
RIHWKDQRO
6LQFHKHDWLV
x: $
:
V: V : FFPRO
±
±
±
±
±
±
±
±
±
±
:
$
V$ V $ ±
±
±
±
±
±
±
±
±
±
:
$
' PL[ H ±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
n
:
H: H : N-PRO
±
±
±
±
±
±
±
±
±
±
±
±
H$ H $ ±
±
±
±
±
"
n
Solutions to Chemical and Engineering Thermodynamics, 5e
c
h
:H ZDQW WR HYDOXDWH H$9 T P y H$/ T P x ZKHUH x$ DQG y$ GHQRWH WKH OLTXLG DQG YDSRU DOFRKRO PROH IUDFWLRQV
UHVSHFWLYHO\ DQG WKH VXSHUVFULSWV 9 DQG / GHVLJQDWH WKH YDSRU DQG OLTXLG SKDVHV 7R DQ H[FHOOHQW DSSUR[LPDWLRQ DW WKH
WHPSHUDWXUHVKHUH H$9 T P y H $9 T P 7RSURFHHGIXUWKHUZHXVH
c
9
$
ZKHUH
/
H9
$ TP H $ TP
- PRO 6HH3UREOHPD ' YDS H SXUH
HWKDQRO
DQG H$/ T P x H $/ T P ZDVFRPSXWHGLQ3UREOHP7KXV
' YDS H $ PROH
$
/
H $
T P º
¼
- PRO ª H $/ T P x$
¬
a - PRO
' YDS H $ PROH
- PRO
' YDS H $ PROH
a - PRO
$
$
H $9 T P H $/ T P H $/ T P H$/ T P x H T P H T P x
h
/
$
d
i
7RHYDOXDWH ' YDS H $ oPROH FDUHPXVWEHWDNHQVLQFH H$/ H $/ EHFRPHVYHU\ODUJH DQGQHJDWLYH LQWKLVOLPLW7RDYRLG
$
VHULRXV HUURUV ZH ZLOO ILW WKH ORZ DOFRKRO KLJK ZDWHU PROH IUDFWLRQ GDWD ZLWK D SRO\QRPLDO LQ PROH IUDFWLRQ DQG HYDOXDWH
H$ H $ DQDO\WLFDOO\,XVHG
b
g
' PL[ H
7KHQ
H$ H $
b
DQG OLP H$ H $
x: o
x$ o x$ x: A B x$ x:
FG w N' H IJ
H wN K
PL[
$
T P N :
6HH(TQ D c
h
x: Ax: B x: x:
g A B )LWWLQJ ' H GDWD DW x
PL[
DQG , ILQG WKDW A
:
N- PRO DQG N-PRO7KXV
H$ H $ x o N- PRO 'H $YDS
N- PRO $
7KXV
x$ o &OHDUO\ DW PRGHUDWH DQG KLJK HWKDQRO FRQFHQWUDWLRQV DQ HWKDQRO PROHFXOH PXVW LQWHUDFW ZLWK WKH ZDWHUHWKDQRO PL[WXUH WR
DOPRVWWKHVDPHH[WHQWDVDQHWKDQROPROHFXOHLQWHUDFWVZLWKSXUHHWKDQRO7KXVLWVKHDWRIYDSRUL]DWLRQIURPVROXWLRQLVDERXW
WKHVDPHDVIURPSXUHHWKDQRO+RZHYHUDWYHU\ORZDOFRKROFRQFHQWUDWLRQVWKHUHLVDGUDPDWLFFKDQJH3UHVXPDEO\ZDWHU
QRZDOPRVWSXUH IRUPVDPRUHRUGHUHGVWUXFWXUH SUREDEO\DVDUHVXOWRIK\GURJHQERQGLQJ 7KXVLWVHHPVUHDVRQDEOHWKDW
DWYHU\ORZHWKDQROFRQFHQWUDWLRQVHDFKHWKDQROPROHFXOHLVPRUHLQYROYHGLQK\GURJHQERQGLQJ ZLWKZDWHUPROHFXOHV WKDQLW
LVLQSXUHHWKDQRO ZLWKHWKDQROPROHFXOHV Solutions to Chemical and Engineering Thermodynamics, 5e
i D OLP V T P x
E V
x o
a
n
f
xb xb xx ¦ ai x x i V T P x
w
w N
a
f V T P b b OLP V T P xi
V T P
x o
NV T P N LM Nx V Nx V a Nx fa Nx f ¦ a a Nx Nx f OP
wN N
N
N
Q
w L
a N N f OP kNx
N V N V N N ¦a
M
wN N
aN N f Q
a aN N f
a ia N N f
N N ¦
V N ¦
aN N f
aN N f
aN N f
N N ¦a i aN N f
V x ¦ a a x x f x x ¦ a ia x x f
x x ¦ a i ax x f
V x ¦ a a x x f x x ix i x a x x f
w
n
n
i
n
i
i i n
i
a
i i
i i
i
i
Ni
p
i i i
i
i
n
n
i i
i
n
i
i i
i i
i
i n
Chapter 6
&RQVHTXHQWO\WKHKHDWRIYDSRUL]DWLRQRIHWKDQROIURPGLOXWHDOFRKROVROXWLRQVLVJUHDWHU WKDWLVPRUHHQHUJ\LVQHHGHG WKDQ
IRUWKHSXUHVXEVWDQFH
i i i
i i
f
V V x ¦ ai x x i i x i 6LPLODUO\E\WDNLQJGHULYDWLYHZLWKUHVSHFWWR N ZHREWDLQ
7KHUHIRUH
a
a
VH[
V V x ¦ ai x x i i x f V
VH[ T P x o a
f
x ¦ ai x x i i x F VH[ T P x o a
V V n
VH[
i DQG
f
V x ¦ ai x x i i x V
H[f
a
n
n
i i ¦ ai i ¦ ai i a a a a " f V
H[f
f
n
¦ ai i i n
¦ ai i a a a a " 7KHVHGHULYDWLRQVDUHWKHVDPHDVLQ6HF)RUH[DPSOHVWDUWLQJZLWKWKHFORVHGV\VWHPHTXDWLRQV
dS Q dN
dU
dV
Q WV P
DQG
SJHQ dt
dt
dt T
dt
ZLWK SJHQ t U ¦ NiUi DQG S ¦ Ni Si :HKDYHILUVWN RUM FRQVWDQW
Solutions to Chemical and Engineering Thermodynamics, 5e
1RZIRUDFRQVWDQWYROXPHDGLDEDWLFV\VWHPZLWKQRVKDIWZRUNZHKDYH
dU
dt
DQG
dS
dt
FRQVWDQW
SJHQ t Ÿ S FDQRQO\LQFUHDVHDWFRQVWDQWNVDQGU
ŸS
6LPLODUO\ZLWK WV
PD[LPXPDWHTXLOLEULXPDWFRQVWDQWNVU
IRUDQLVRWKHUPDOV\VWHP FRQVWDQWNTDQGV DWFRQVWDQWYROXPHZHKDYH
dU
dt
Ÿ FRQVWDQW DOVR V
Q SJHQ RU Q
T
dS
Q DQG
dt
T
dS
TSJHQ dt
VR
dU
dS
d U TS
dA
T
TSJHQ Ÿ
TSJHQ
dt
dt
dt
dt
Ÿ A PLQLPXPDWHTXLOLEULXPDWFRQVWDQWN T DQGV 6LPLODUO\RQHFDQVKRZWKDW G PLQLPXPDWFRQVWDQWNTDQGP
M M x x x N w NM
wM
Mi
MN
w Ni T P N
dNi T P N
a
f
FG
H
j zi
EXW
Fw MI
FG w M IJ
Hw N K
¦ GH w x JK
i T P N j zi
j zi
FG w x IJ
Hw N K
j
T P N jzk
i T P N j zi
1RZ
j
j
IJ
K
xj
Nj
N
w
w Ni
xj
R N
FG N IJ |S N
wN H N K | N
TN N
w
j
i
j
i
xj
LIj z i
N
xi
LIj i
N
7KHUHIRUH
Solutions to Chemical and Engineering Thermodynamics, 5e
FG w M IJ
Hw N K
i T P N j zi
Chapter 6
Fw MI
x Fw MI
F x I
G
J
GH w x JK
H NK
N Hwx K
Fw MI
x
Fw MI
¦G
J
G
J
NHwx K
N
Hwx K
F w M IJ ¦ x FG w M IJ
ŸM
M G
Hwx K
Hwx K
¦
j
j zi
i
T P N k z j
j
i
T P N k zi
j
i
j zi
T P x j zi
i
i
j
T P xk z j
j
T P x j zi
j
T P xk z j
6HH6HFWLRQ 6LQFHWKHVWDEOHHTXLOLEULXPVWDWHRIWKHPL[WXUHIRUVRPHUDQJHRIFRPSRVLWLRQVLVDVWZROLTXLGSKDVHVUDWKHUWKDQDVLQJOH
KRPRJHQHRXVSKDVHWKH*LEEVIUHHHQHUJ\RIWKHKRPRJHQHRXVPL[WXUHPXVWEHJUHDWHUWKDQDOLQHDUFRPELQDWLRQRIWKH*LEEV
IUHHHQHUJLHVRIPL[WXUHVRIFRPSRVLWLRQVRQHLWKHUVLGHRIWKHDFWXDOPL[WXUHFRPSRVLWLRQ7KDWLVWKH*LEEVIUHHHQHUJ\RI
WKHPL[WXUHPXVWEHDVLQGLFDWHGLQ)LJXUHEHORZ
1RZWKH*LEEVIUHHHQHUJ\RIWKHPL[WXUHLV
G N, G, N , G, N,,G,, N ,,G,, ZLWKWKHUHVWULFWLRQVWKDW
N, N,, FRQVWDQW NR DQG N , N ,,
N R 7KXVZHKDYHWKDW
FRQVWDQW
G
c
h
$WHTXLOLEULXP
FG w G IJ
Hw N K
Ÿ G,
,
T P N ,
c
h
N,G, N , G, NR N, G,, N R N ,, G,, G,, ZKHUHLQHYDOXDWLQJHDFKRIWKHVHGHULYDWLRQVZHKDYHXVHGWKH*LEEV'XKHPHTXDWLRQ DQG
FG w G IJ
Hw N K
Ÿ G,
,
T P N,
a
G,, f
+RZHYHU G, w G w N LVDWDQJHQWWRWKH GYHUVXV N FXUYH7KXVWKHHTXLOLEULXPUHTXLUHPHQWWKDW G, G,, UHTXLUHV
WKDW WKH FRPSRVLWLRQV RI WKH WZR HTXLOLEULXPSKDVHVDUHDWWKHLQWHUVHFWLRQVRIHDFKOREHRIWKH GYHUVXV N FXUYHZLWKWKH
VLQJOHWDQJHQWOLQHWRERWKFXUYHV6HH)LJ>1RWH6HH3UREOHP@
7KHOLPLWRIVWDELOLW\RIDVLQJOHSKDVHLVIRXQGIURPWKHFRQGLWLRQWKDW d G +HUHWKDWLPSOLHV
w G
1RZ G NG N G w N T P N
FG
H
IJ
K
FG w G IJ
Hw N K
T P N G N
FG w G IJ
Hw N K
T P N N
FG w G IJ
Hw N K
T P N G 6LQFHWKHODVWWZRWHUPVVXPWR]HURE\WKH*LEEV'XKHPHTXDWLRQ7KXV
w G
w G
w N T P N
w N T P N
FG
H
IJ
K
FG
H
IJ
K
7KXV WKH OLPLWV RI VWDELOLW\ RI D VLQJOH SKDVH DUH WKH LQIOHFWLRQ SRLQWV RQ WKH G YV N RU G YV x FXUYH DQG WKH ORFDO
PD[LPXPDQGPLQLPXPRQWKH G YV N RU G YV x FXUYHV 6HH)LJXUH 7KHUHJLRQEHWZHHQHDFKFRH[LVWLQJSKDVH
FRPSRVLWLRQDQGWKHDGMDFHQWLQIOHFWLRQSRLQWUHSUHVHQWVDPHWDVWDEOHUHJLRQ7KLVLVLOOXVWUDWHGLQ)LJXUH
Solutions to Chemical and Engineering Thermodynamics, 5e
a
f
x a x f a x f a x f EXW ' PL[ H
D
x B H B x B H ($ H
B
B
B
H xB H B xB H ($ xB xB ª xB xB º ¬
¼
)URP(TXDWLRQ E w ' PL[ H
' PL[ H xB
H ($ H ($ DOVR
w xB
' PL[ H
1RZ
w ' PL[ H
w xB
HB H B xB ª xB xB º
¬
¼
º
xB xB ª xB
¬
¼
b H H g x a x f a x f x a x f a x f
($
B
($
B
DQG
B
B
B
B
b H H g a x f a x f a x f x a x f a x f
B
B
B
B
E $W xB
B
B
b H H g FH IK FH IK FH IK - PRO
I F I
I
F
F
b H H g H K H K H K - PRO
($
($
B
B
B
7KHUHIRUH
w ' PL[ H
w xB
' PL[ H xB
B
B
F 0L[LQJSURFHVV
6\VWHP PROHRIHDFKLQLWLDOPL[WXUH PROHVWRWDO 'LIIHUHQFHIRUPRIPDVVEDODQFH N f N i N i PRO %DODQFHRQEHQ]HQH VSHFLHVEDODQFH x Bi Ni x Bi N i
xB f N f
xB f
(QHUJ\EDODQFH N f U f
)RUOLTXLGVROXWLRQV U | H &RQVHTXHQWO\
REYLRXVO\
Ni U i N i U i Q Solutions to Chemical and Engineering Thermodynamics, 5e
a
H f H i H i
Q
H xB
Chapter 6
f a
H xB
f a
H x B
f
EXW
' PL[ H
H PL[ ª¬ xB H B xB H ($ º¼
Ÿ H PL[
' PL[ H
xB H B xB H ($ SRO\QRPLDOJLYHQ
p
p
LQSUREOHPVWDWHPHQW
SXUHFRPSRQHQW
HQWKDOSLHV
ª¬ ' PL[ H xB
H B H ($ º¼
ª¬ ' PL[ H xB
H B H ($ º¼
ª¬ ' PL[ H xB
H B H ($ º¼
' PL[ H xB
' PL[ H xB
Q
Q
' PL[ H xB
u -
RS N- UV u PROHIUDFWLRQ QRS N- UV T PROQ SURSDQRO W RIQ SURSDQRO T PRORIVROXWLRQ W
7KXVZHREWDLQWKHIROORZLQJ
0ROH + 2 QN-PROSURSDQRO
' PL[ H N-PROVROXWLRQ
0ROH + 2 QN-PROSURSDQRO
' PL[ H N-PROVROXWLRQ
0ROH + 2 Q
' PL[ H 0ROH + 2 Q
' PL[ H ± ± ± ±
± ± ± ±
± ± ± ±
± ± ± ±
' PL[ H LVSORWWHGEHORZDQG H H DQG H H DUHHYDOXDWHGXVLQJWKHJUDSKLFDOSURFHGXUHRI6HF
Solutions to Chemical and Engineering Thermodynamics, 5e
'DVKHGOLQHVDUHVDPSOHWDQJHQWOLQHV>1RWH5HVXOWVLQWDEOHDQGILJXUHEHORZJRWWHQXVLQJDPXFKELJJHUJUDSK@
x+ 2 HQ3 H Q3 H+ 2 H + 2 N-PRO
±
±
±
±
±
±
D
' PL[ H
x x ª¬ A B x x º¼
cx x h A Ba x f x x ª¬ A B x º¼ Solutions to Chemical and Engineering Thermodynamics, 5e
Chapter 6
§ w PL[ ' H ·
¨
¸
© w x ¹
x x ¬ª A B x x ¼º Bx x § w ' PL[ H ·
' PL[ H x ¨
¸
© w x ¹
x x ª¬ A B x x º¼
x x x A x x x x x B Bx x x ª¬ A B x º¼
DW x
A B H H § w ' PL[ H ·
' PL[ H x ¨
¸
© w x ¹
H H A B § w ' PL[ H ·
' PL[ H x ¨
¸
© w x ¹
x x ª¬ A B x x º¼
x x x » A B
x ª¬ A B x º¼
DW x
I OP A B KQ - PRO
H H - PRO
w
wA
wB
E C3 i C3i
Hi H i wT
wT
wT
Ÿ H H b
g
w
H H
wT C3 C3 b
g FGH ww TA ww TB IJK Ÿ C3 C3 - PRO.
- PRO.
N N N QRFWDQROPDVVEDODQFH x N x N x N F 2YHUDOOPDVVEDODQFH x
$OVR SUREOHPLQIRUPDWLRQ N N %DVLVRIFDOFXODWLRQ N
Ÿ N
N
H H DW. A u u B F
H
LM
N
A B H H x x ¼º Bx x
(QHUJ\EDODQFH u u N H N H N H Q Solutions to Chemical and Engineering Thermodynamics, 5e
H Ÿ Q
' PL[ H ' PL[ H ' PL[ H ' PL[ H u ¬ª ¼º
' PL[ H u ª¬ º¼ - PRO
' PL[ H u ª¬ º¼
Q
Q
N 3 N $ N PL[ Ÿ N PL[
N 3 N $ Ÿ N PL[
7KHHQHUJ\EDODQFHLV
N 3 H 3 N $ H $ N PL[ H PL[ Q N 3 H 3 N $ H $ N PL[ x3 H 3 x$ H $ ' H PL[ Q
a
a
f a
f
a
˜ H 3 H 3 ˜ H $ H $ ˜ ' H PL[ x3
VR
Q ˜ ' H PL[ x3
f
NPRO
PLQ f
Q f
1RZIURPWKHWDEOH
' H PL[ x3 - PRODQG
a
' H ax
PL[
- PROVROXWLRQ a
u u u - PRO
7KHPDVVEDODQFHRIWKHDFHWLFDFLGS\ULGLQHVWUHDPVDWVWHDG\VWDWHLV - PRO
- PROVROXWLRQ
Ÿ Q Ÿ +HDWPXVWEHDGGHG
x x ª¬ x x º¼
' PL[ H
' PL[ H x H x H N ' PL[ H N ' PL[ H N N ' PL[ H H x H x H ' PL[ H
3
f
f - PRO
%\LQWHUSRODWLRQ
a
' H PL[ x3
f
| - PRO
Q
NPRO
N- PRO
˜ ˜
˜
PLQ
PRO NPRO
N
PLQ
1HJDWLYHVLJQPHDQVWKDWKHDWPXVWEHUHPRYHG RUFRROLQJVXSSOLHG WRNHHSWKHSURFHVVDWDFRQVWDQWWHPSHUDWXUH6LQFH
HWK\OHQHJO\FROKDVDYDOXHRIC3 N-NJ.)URPDQHQHUJ\EDODQFHZHKDYHWKDW
Solutions to Chemical and Engineering Thermodynamics, 5e
N
u . u M
NJ ˜ .
7KHUHIRUH
M
N
PLQ
N
u .
NJ ˜ .
Chapter 6
N
PLQ
NJHWK\OHQHJO\FRO
PLQ
DOVRDYDLODEOHDVDQ0DWKFDGZRUNVKHHW [ +
[ +
[ +
[ +
[ +
[ [ +
[ +
[ +
[ +
[ [ [
+
[ +
[ [
+
[
+
[
+
[ +
[ [ [
+
[
+
[
[
+
[ +
[ [
+
[
+
[
+
[ + [ L +
L
[
L
2QHFRQVWDQW0DUJXOHVILW
[ [
6 OLQILW [ + I
6
I [
7ZRFRQVWDQW0DUJXOHVILW
[ [
I [
66 OLQILW [ + I
66
[ [ [ 7KUHHFRQVWDQW0DUJXOHVILW
[ [
I [
[ [ [ 66 OLQILW [ + I
66
[ [ [ +
+
+
+
++
++
+
Solutions to Chemical and Engineering Thermodynamics, 5e
Solutions to Chemical and Engineering Thermodynamics, 5e
Chapter 6
Solutions to Chemical and Engineering Thermodynamics, 5e
++ [L
' + [L
' + [L
Solutions to Chemical and Engineering Thermodynamics, 5e
Chapter 6
6WDUWLQJE\ZULWLQJWKHHTXDWLRQIRUWKHIRUPDWLRQRIHDFKRIWKHVL[FRPSRXQGVSUHVHQWIURPWKHLUHOHPHQWV
&+ &+
2 2
&2 &2
&2 &2
+ +
+2 +2
1RZXVLQJHTQ WRHOLPLQDWHWKHR[\JHQDWRPDQGHTQ WRHOLPLQDWHWKHK\GURJHQDWRP:HREWDLQ
&+ &+
&2 &2
&2 &2
+2 +2
7KXVIURPWKH'HQELJKPHWKRGZHILQGWKHUHDUHIRXULQGHSHQGHQWUHDFWLRQV2QHVXFKVHWLVOLVWHGDERYH
D 1 g + g
' U[Q H
u N- PRO ' U[Q G
u N- PRO E &+ g & + g &+ g ' U[Q H N- PRO ' U[Q G N- PRO F &D&2 s &D2 s &2 g ' U[Q H N- PRO ' U[Q G N- PRO G &2 g + g &+ g &2 g + 2 g ' U[Q H
u u u N- PRO ' U[Q G
u u u N- PRO 1+ g %XFNPDVWHUIXOOHUHQH& %) 2 &2IRUZKLFK
'FRPEH N-PRO N-PROV&
*UDSKLWH&2 &2IRUZKLFK'FRPEH u N-PROV&
)RUWKHVHUHDFWLRQVVLFHRQO\FDUERQFDUERQGLR[LGHDQGR[\JHQDUHLQYROYHG'IH 'FRPEH
6XEWUDFWLQJWKHILUVWFKHPLFDOUHDFWLRQDERYHIURPWKHVHFRQG\LHOGV
&!& %) !± N-PRO& %) Solutions to Chemical and Engineering Thermodynamics, 5e
DOVRDYDLODEOHDVDQ0DWKFDGZRUNVKHHW [ + [ +
+
[ +
[
[[ +
[
+
+
[
+
[ [ +
[
+
+ [ +
[
+
[ ++
+ [
[ [
+
[
+
[
+
[
+
[ + [ +
[
+
[
+
[[ +
+
[ [
++
[
[
[ [
++
+
+
[
+
[
+
[
+
[
+
[
+
[
+
[
+
L
+
L
[
L
2QHFRQVWDQW0DUJXOHVILW
I [
[ [
66
++ [
66 [ [
G++ [
G
++ [
G[
3+ [
++ [
OLQILW [ + I
[ G++ [
66
3+ [
++ [
[ G++ [
7ZRFRQVWDQW0DUJXOHVILW
[ [
I [
66 OLQILW [ + I
66
[ [ [ [ [ 66 66 [ G
++ [
G++ [
++ [
G[
3+ [
++ [
[ G++ [
3+ [
++ [ [ G++ [ Solutions to Chemical and Engineering Thermodynamics, 5e
7KUHHFRQVWDQW0DUJXOHVILW
[ [
I [ [ [ [ 66 OLQILW [ + I
66
[ [ [
++ [ 66 [ [
66 [ [ [ 66 [ [ [ G
G++ [
++ [
3+ [
++ [
[ G++ [
3+ [
++ [
G[
Chapter 6
[ G++ [
Solutions to Chemical and Engineering Thermodynamics, 5e
)RXUFRQVWDQW0DUJXOHVILW
[ [
[ [ [ I [
66 OLQILW [ + I
[ [ [ [ [ [ ++ [
[ [ 66 66 [ 66 [ 66 [ G
G++
++ [
[
3+ [
++ [
[ G++ [
3+ [
G[
++ [
[ G++ [
Solutions to Chemical and Engineering Thermodynamics, 5e
Chapter 6
Solutions to Chemical and Engineering Thermodynamics, 5e
++ [
L
3+ [L
3+ [L
Solutions to Chemical and Engineering Thermodynamics, 5e
$VVXPHUHDFWRURSHUDWHVLQVWHDG\VWDWH7KHQWKHPDVVEDODQFHLV
N i LQ N i RXW Q i X
RU
Ni LQ
Chapter 6
N i RXW Q i X
DQGWKHHQHUJ\EDODQFHLV
N i LQ H i LQ N i RXW H i RXW Q
¦
¦
i
i
RU
¦ N H ¦ N H
¦ N Q X H ¦ N H
¦ N b H
H g 'H T X
Q
i LQ
i RXW
i LQ
i
i RXW
i
i RXW
i
i RXW
i LQ
i
i RXW
i
i RXW
i RXW
i LQ
U[Q
LQ
i
8VLQJD0DWKFDGZRUNVKHHWWKHKHDWVRIIRUPDWLRQDQGKHDWFDSDFLWLHVLQWKHDSSHQGLFHVRIWKHWH[WERRNZHILQGWKDWN-
PXVWEHVXSSOLHGSHUPROHRI1HQWHULQJUHDFWRU6HH0DWKFDGZRUNVKHHWIRUWKLVSUREOHP
DOVRDYDLODEOHDVDQ0DWKFDGZRUNVKHHW 1RPHQFODWXUH
[
+ [ + + HQWKDOS\DW&
[ +
[ +
+ HQWKDOS\DW&
++DQG++DUH
[ +
[ +
FRUUHODWHGHQWKDOSLHV
[
+
[ +
' + GLIIHUHQFH
EHWZHHQSDUWLDOPRODU
[
+
[
+
DQGSXUHFRPSRQHQW
[
+
[ + HQWKDOSLHVRIVSHFLHV
DW&
[
+
[
+
HWF
[
+
[ + [
+
[
+
[
+ [ + [
+ [ + [ + [ + [ + [ + L Solutions to Chemical and Engineering Thermodynamics, 5e
7KUHHFRQVWDQW0DUJXOHVILW
[ [
[ [ [ I [
[ [ [ V OLQILW [ + I
V
V OLQILW [ + I
V
[
++
V [ [ V [ [ [ V [ [ [ ++
V [ [ V [ [ [ V [ [ [ [
Solutions to Chemical and Engineering Thermodynamics, 5e
G++ [
G
++ [
G[
G++ [
G
++ [
G[
' + [
++ [
[ G++ [
' + [
++ [
[ G++ [
' + [
++ [
[ G++ [
' + [
++ [
[ G++ [
Chapter 6
Solutions to Chemical and Engineering Thermodynamics, 5e
DOVRDYDLODEOHDVDQ0DWKFDGZRUNVKHHW [ +
[ [ +
[ [ +
[ [ +
[ L +
L
' +L
' +L
+[[L
+
[
+
+
[
+
+
[
+
+
+L
[L [L
+[[
L
[L
[
L
' +L
[L
+[[L
' +L
[
L
Solutions to Chemical and Engineering Thermodynamics, 5e
Chapter 6
D &+ 2 1 o &2 + 2 1 ' U[Q H ' I H &2 ' I H + 2 ' I H & +
' U[QU
N-
N- u u
' U[Q H 'NRT
N- N u u C3 RIPL[WXUH u C9 RIPL[WXUH C3 NR
A
u N
- PRO. - PRO. RI
PROHV
' U[QU
u C9
E\LGHDOJDVODZ
N f Tf
N f Tf
NT
Pf Pi
PV NRT Ÿ i i
Ni Ti
Pi
Pf
EDU
Pf EDU
u
E $GLDEDWLFH[SDQVLRQ
C3 SHUPROH
TLQ TILQDO
FG P IJ
HPK
R C3
F I
H K
T
. dU
dS
SJHQ W dt
dt
W C9 'T u - PRORIRFWDQH 7KLVLVOLNH&DUQRWF\FOHZLWKDYDU\LQJXSSHUT
dU
dT dS Q
C3 dT
SJHQ
T
Q W C9
dt
T dt
dt dt T/
dU
dS T/
W RU
dt
dt
'U T/ 'S W
T
W C9 'T T/ ˜ C3 OQ /
T+
F
T
u
. T
W
W6
dU
dt
u OQ
u - PRORIRFWDQH
dT
C dT dT
C dT
C9
T/ 3
W Ÿ W C9
T/ 3
dt
T dt
T dt
dt
T/
C9 T/ T+ T/C3 OQ
T+
a
f
¦ N i LQ HiLQ ¦ N i RXW Hi RXW Q W
q & Solutions to Chemical and Engineering Thermodynamics, 5e
Q
dS
SJHQ
N i LQ Si LQ ¦ N i RXW Si RXW ¦
TDPE
dt
IRUPD[LPXPZRUN Q TDPE ¦ N i LQ Si LQ ¦ N i RXW Si RXW
W
$EVROXWHPD[LPXPZRUN TLQ
,GHDOJDV Hi
¦ N i LQ b Hi LQ TDPE SiLQ g ¦ N iRXW b Hi RXW TDPE Si RXW g W ¦ N i LQ b Hi LQ TDPE SiLQ g ¦ N iRXW b Hi RXW TDPE Si RXW g
¦ N i LQ b Hi LQ TDPE SiLQ g ¦ b N iLQ XQ i gb HiRXW TDPE SiRXW g
H i Gi LQ
TRXW
TDPE S i R OQ xi Gi RXW Si
¦ N G RT ¦ N OQ x ¦ N G RT ¦ N OQ x X ¦ Q bG RT OQ x g
W
i LQ
i LQ
DPE
i LQ
i
i LQ
i LQ
i LQ
i RXW
i
i
W
RTDPE
RTDPE
¦N
¦N
i
i RXW
DPE
i
xi LQ
i LQ OQ
i LQ OQ
i LQ
i
i
DPE
xi RXW
xi LQ
xi RXW
X
¦Q G RT ¦ OQ xQ
i
DPE
i
X ' U[Q G RTDPE
L
i RXW
¦ OQ x
QL
i RXW
&LVWKHQXPEHURIFRPSRQHQWV DQG0LVWKHQXPEHURISKDVHV
7KHQWKHXQNQRZQVDUH
Ni. QXPEHURIPROHVRIVSHFLHViLQSKDVH.
& u 3XQNQRZQV
.
P SUHVVXUHLQSKDVH. .
3XQNQRZQV
T WHPSHUDWXUHLQSKDVH. 3XQNQRZQV
7RWDOQXPEHURIXQNQRZQVLV& u 3 3 3
7KHQUHVWULFWLRQVDUHWKDW
TLVWKHVDPHLQDOOSKDVHV LH T ,
3 u & T ,,
T ,,, P ,,
P ,,, 3 UHVWULFWLRQV
PLVWKHVDPHLQDOOSKDVHV LH P ,
3 UHVWULFWLRQV
.
Gi PXVWEHWKHVDPHIRUVSHFLHViLQDOOSKDVHV
& u 3 UHVWULFWLRQV
,QDGGLWLRQZHKDYHWKHVWRLFKLRPHWULFUHODWLRQIRUHDFKVSHFLHVWKDW
.
Ni
¦ NiN N M
Ni R ¦ Q ij X j ZKLFKSURYLGHVDQDGGLWLRQ&UHVWULFWLRQV
j 7KHUHIRUHWKHQXPEHURIGHJUHHVRIIUHHGRP)DUH
) & ˜ 3 & ˜ 3 & & & LQGHSHQGHQWRIWKHQXPEHURIFRPSRQHQWV SKDVHVRULQGHSHQGHQWFKHPLFDOUHDFWLRQV
7KHUHIRUH'XKHP VWKHRUHPLVYDOLG
Solutions to Chemical and Engineering Thermodynamics, 5e
Chapter 6
7KH*LEEVHQHUJ\RIDPL[WXUHFRQVLVWLQJRIN PROHVRIVSHFLHVDQGN PROHVRIVSHFLHVLV
R
R
N G RT OQ x N G RT OQ x
G
§ R
§ R
N ·
N ·
N ¨ G RT OQ
¸ N ¨ G RT OQ
¸
N N ¹
N N ¹
©
©
6RWKHFKDQJHLQ*LEEVHQHUJ\EHWZHHQDVWDWH,DQGDVWDWH,, ZLWKRXWFKHPLFDOUHDFWLRQ LV
R
R
R
R
N,, G RT OQ x,, N ,, G RT OQ x,, N, G RT OQ x, N , G RT OQ x,
'G
6LQFHWKHUHLVQRFKHPLFDOUHDFWLRQN,, N, DQGN ,, N , VRWKLVHTXDWLRQUHGXFHVWR
N,, RT OQ x,, N ,, RT OQ x,, N, RT OQ x, N , RT OQ x,
'G
ĂƐĞĚŽŶϭŵŽůĞŽĨĨĞĞĚƚŚĂƚŝƐϬ͘ϳϵŵŽůĞƐŽĨŶŝƚƌŽŐĞŶĂŶĚϬ͘ϮϭŵŽůĞƐŽĨŽdžLJŐĞŶŐŽŝŶŐƚŽŽŶĞƐƚƌĞĂŵŽĨϬ͘ϳϵŵŽůĞƐŽĨ
ƉƵƌĞŶŝƚƌŽŐĞŶĂŶĚϬ͘ϮϭŵŽůĞƐŽĨƉƵƌĞŽdžLJŐĞŶ͕ǁĞŚĂǀĞ
'G
RT OQ RT OQ RT OQ RT OQ RT OQ RT OQ RT
PRODLU
NPRODLU
6LQFHW 'G WKHZRUNUHTXLUHGIRUWKLVLVRWKHUPDOFRQVWDQWVHSDUDWLRQLVN-SHUPROHRILQFRPLQJDLU ͘
EĞĞĚƚŽƐƚĂƌƚǁŝƚŚƚŚĞŵĂƐƐďĂůĂŶĐĞďĂƐĞĚŽŶĂϭϬϬϬŵŽůĞƐŽĨƚŽƚĂůŚĞdžĂĨůƵŽƌŝĚĞƐƚƌĞĂŵ͘dŚŝƐĐŽŶƚĂŝŶƐϳ͘ϮŵŽůĞƐhϮϯϱ
ĂŶĚϵϵϮ͘ϴϬĨhϮϯϴ͘dŚĞŐŽĂůŝƐƚŽƌĞĐŽǀĞƌϮϱйŽĨƚŚĞŝŶŝƚŝĂůhϮϯϱĂƐĂƉƵƌĞƐƚƌĞĂŵ͘^ŽƚŚĞĞdžŝƚƐƚƌĞĂŵƐǁŝůůďĞĂƉƵƌĞ
ƐƚƌĞĂŵŽĨϭ͘ϴŵŽůĞƐŽĨhϮϯϱĂŶĚĂƐĞĐŽŶĚƐƚƌĞĂŵƚŚĂƚĐŽŶƚĂŝŶƐϱ͘ϰŵŽůĞƐŽĨhϮϯϴĂŶĚϵϵϮ͘ϴŵŽůĞƐŽĨhϮϯϴ͘dŚĞƌĞĨŽƌĞ͕
ŽŶĞĞdžŝƚƐƚƌĞĂŵŽĨϵϵϴ͘ϮŵŽůĞƐŚĂƐĂƚŚĞŵŽůĞĨƌĂĐƚŝŽŶŽĨhϮϯϱŝŶŽŶĞĞdžŝƚƐƚƌĞĂŵŝƐϱ͘ϰͬϵϵϴ͘ϮсϬ͘ϬϬϱϰϭŵŽůĞĨƌĂĐƚŝŽŶŽĨ
hϮϯϱĂŶĚ;ϵϵϴ͘ϮͲϱ͘ϰͿͬϵϵϴ͘ϮсϬ͘ϵϵϰϱϵŵŽůĞĨƌĂĐƚŝŽŶŽĨhϮϯϴ͕ĂŶĚƚŚĞƐĞĐŽŶĚƐƚƌĞĂŵŽĨϱ͘ϰŵŽůĞƐŝƐ;ĞƐƐĞŶƚŝĂůůLJͿƉƵƌĞ
hϮϯϱ͘^Ž
ĂƐĞĚŽŶƚŚĞƐŽůƵƚŝŽŶƚŽƚŚĞƉƌĞǀŝŽƵƐƉƌŽďůĞŵǁĞŚĂǀĞ
ª RT OQ OQ RT OQº
«
»
¬ RT OQ RT OQ ¼
RT ª¬ º¼
'G
RT > @
RT
6LQFHW
'G PRO
7KHZRUNUHTXLUHGWRSURFHVVPROHVRIWKHJDVHRXVWRSURGXFHWKHGHVLUHG8VWUHDPLVN-
Solutions to Chemical and Engineering Thermodynamics, 5e
6WDUWIURPHTQ C9 V T
C9 V T T
z
V T
V f T
FG w P IJ dV H wT K
V
1RZIURP
P
RT
a
§ wP ·
¨
¸
V V T © wT ¹V
§ w P ·
a T DQG ¨ ¸
V © wT ¹V
a T V ZHKDYH
a T V
7KLVHTXDWLRQJLYHYWKHFRQVWDQWYROXPHKHDWFDSDFLW\LQWHUPVRIWKHPRODUYROXPH7RGHWHUPLQHKRZWKHKHDW
FDFLW\YDULHVZLWKSUHVVXUHWKHIROORZLQJSURFHGXUHZRXOGEHXVHG)RUHDFKSUHVVXUHFKRVHQVROYHWKH
HTXDWLRQRIVWDWHYRUYROXPH7KHQXVHWKDWYROXPHLQWKHHTXDWLRQDERYHWRGHWHUPLQHWKHGLIIHUHQFHEHWZHHQ
WKHDFWXDODQGLGHDOJDVFRQVWDQWYROXPHKHDWFDSDFLW\
CV V T CV V
f T
D PROHVRI PROHVRI WRWDOPROHV x x 9ROXPHRIDQLGHDOPL[WXUH VLG xVxV FPPRO
E $FWXDOPL[WXUHYROXPH VLGxx> xx @ FPPRO
F
' PL[V
x x ª¬ x x º¼
x x ª¬ x x º¼ x x ª¬ x º¼
§ w' PL[V ·
¨
¸ x ¬ª x ¼º x x
© wx ¹
' PL[V
' PL[V
x x ª¬ x x º¼
§ w' PL[V ·
¨
¸
© wx ¹
x x ª¬ x º¼
x ¬ª x ¼º x x
§ w' V ·
V V x ¨ PL[ ¸ V x ª¬ x ª¬ x º¼ x x º¼
© wx ¹
§ w' V ·
V V x ¨ PL[ ¸ V x x ª¬ x º¼ x x
© wx ¹
6RV FFPRODQGV FFPRO
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
a
i
i
f ¦ N U T P Ÿ U T P x U T P U T P N N !
f ¦ N RT Ÿ V T P x V T P
PV T P N N !
a
i
i
i
i
$OVR Si T P x
S i T P R OQ xi ' PL[ U
i
i
¦ x ª¬U T P x U T P º¼ ¦ x >@ ' V ¦ x ª¬V T P x V T P º¼ ¦ x > @ i
i
i
i
i
PL[
i
i
i
i
i
i
' PL[ H
' PL[ S
i
' PL[
i
i
i
i
' PL[ U ' PL[ V
¦ x ¬ª S T P x S T P ¼º ¦ x > R OQ x @ R¦ x OQ x G ¦ x ª¬G T P x G T P º¼ ¦ x ª¬ H TS H T S º¼
¦ x H H T¦ x S S
' H T ' S RT ¦ x OQ x
i
i
i
PL[
i
i
i
i
i
PL[
i
i
i
i
i
i
i
i
i
i
6LPLODUO\
' PL[ A ' PL[ U T ' PL[ S
7KHSLFWXUHRIWKHSURFHVVKHUHLVDVIROORZV
0L[WXUH ¦ Ni 0ROHVRIJDVDW TDQGV T V P RT
¦ x OQ x i
i
B
T V P
0L[NHHSLQJ
o T V Pf
 
#
TDQGVIL[HG
C T V PC
D
/HW
Pi LQLWLDOSUHVVXUHRIVSHFLHVi SUHVVXUHLQXQPL[HGVWDWH P ILQDOSUHVVXUHRIPL[HGJDV
Pi xi P SDUWLDOSUHVVXUHRIVSHFLHViLQILQDOVWDWH
ZDQWWRVKRZWKDW Pi
P
Pi Ni RT V DQG
NRT V
Pi b¦ N gRT V i
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
pLQLWLDOSUHVVXUHRISXUHi
Pi xi P xi NRT V Ni RT V Pi nSDUWLDOSUHVVXUHRIVSHFLHVi
Q.E.D.
E 6LQFH WKH LQWHUQDO HQHUJ\ RI DQ LGHDO JDV LV LQGHSHQGHQW RI WRWDO SUHVVXUH LW IROORZV WKDW
Ui IGM T x U iIG T IRUPL[LQJDWFRQVWDQWWRWDOSUHVVXUHRUFRQVWDQWSDUWLDOSUHVVXUH7KXV
' PL[ U ,*0
1H[W
UV RT RTx RTx
W P Px P V a P Tf
RT
x V aT P f
x
x
N
P
RS
T
wV
¦ N j RT
w Ni T P
P
Vi ,*0 T P x
,*
i
i
i
i
,*0
¦ x ^T T P x T T P ` DQG ' T
i
,*
i
i
a f
&DQQRZGHILQHWZR ' PL[ T
i
xiV i,* T Pi 7KXV Vi ,*0 T P x
' PL[ T ,*0
i
i
i
i
i
i
PL[
i
¦ x ^T T P x T T P ` ,*0
i
i
i
' PL[ T ,*0 ZDV FRPSXWHG LQ 6HFWLRQ DQG ZLOO QRW EH FRQVLGHUHG KHUH :H ZLOO EH
FRQFHUQHGZLWK ' PL[ T ,*0 ' PL[ V ,*0
¦ x ^V
i
,*0
i
T P x V i T Pi
V
i
V
i
i
i
i
i
i
i
i
C
¦
V
xi N i C
¦N
i
i C
1RWH ¦ C i ZKHUH C QXPEHURIFRPSRQHQWV
)RUHQWKDOS\ZHKDYH
Hi,*0 T P x
7KXV
Ui,*0 T P x PVi ,*0 T P x
U i,* T P
' PL[ H ,*0
i
i
V C
V
C
N
i
¦ x x N ¦ x x x N
` ¦ x ^x V T P V T P `
DQG
,*
T P
¦x H
,*0
i
PVi
i
a f
U i,* T P PxiV i,* T Pi
H i,* T P
T P x H i,* T P
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
' PL[ H ,*0
C
¦ x H T P x H T P
¦ x ^U T P PV T P U T P PV T P ` ¦ x ^ PV T P PV T P ` ¦ x ^RT RT ` i
,*0
i
i
,*
i
,*
i i
,*
i
i
i
i
i
i
i
i
i
i
i
i
7RFRPSXWH ' PL[ S ,*0 ZHXVHWKHVDPHVRUWRIDUJXPHQWDVLQ6HFWLRQEXWQRWLQJKHUHWKDW
WKH YROXPH RFFXSLHG E\ HDFK JDV LQ WKH LQLWLDO DQG ILQDO VWDWHV DUH WKH VDPH 7KHUHIRUH
Si,*0 T P x S i,* T Pi VLQFH T DQG V RI HDFK VSHFLHV LV XQFKDQJHG VHH HTQ a f
7KHUHIRUH
' PL[ S ,*0
7KXV ' PL[ A,*0
Gi,*0 T P x
)LQDOO\ G
,*
i
i
A PV Ÿ a f
a f
Ai,*0 T P x PVi ,*0 T P x
Ai,* T Pi PxiV i,* T Pi
,*
Ai,* T Pi PV
T Pi
i i
i
a f
a f G aT P f
,*
i
6RWKDW
' PL[ G ,*0
Ui,*0 T P x TSi,*0 T P x
a f A aT P f
` i
U i,* T T S i,* T Pi
T P x S i,* T Pi
,*0
i
A U TS Ÿ Ai,*0 T P x
i
)RUWKH+HOPKROW]IUHHHQHUJ\ZHQRWHWKDW
¦ x ^S
*HQHUDOO\PL[LQJDWFRQVWDQWTDQGPDQGPL[LQJDWFRQVWDQWTDQGVDUHTXLWHdifferent
+RZHYHUIRUWKHideal gasZHKDYH
PVi Ni RT SXUHIOXLGV DQG PV ¦ Ni RT PL[WXUHV 7KXVIRUWKHSXUHIOXLGV VDPHTDQGP NRT
N RT
DQG V
V
P
P
NRT N RT
RT
Ÿ V V
N N V
P
P
P
6RIRUWKHLGHDOJDVWKHPL[LQJSURFHVVGHVFULEHGLQSUREOHPVWDWHPHQWLVDOVRDPL[LQJSURFHVVDW
FRQVWDQWTDQGPDQG7DEOHDSSOLHVKHUHDOVR
a
f
a
f
:HKDYHWKHIROORZLQJSURSHUWLHVIRUDPL[WXUHIRUPL[LQJDWFRQVWDQWTDQGP
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
¦ Ni U i T P
¦ N iV i T P
¦ Ni S i T P R¦ Ni OQ xi U T P x
V T P x
S T P x
DQG S i
:HNQRZ dU
Vi
V i
Td S PdV o
SXUHFRPSRQHQWi
wS
o Ui
C9i
wU V T
Ui
wS
wU V
wS
T wV U
R
w S wU wV
˜
˜
DQG
wV U w S V wU S
Vi
wU
wS V
T
wV
wU S
P
o
¦ Ni C9i T
C9i T
6R U i
RT
P
wU
w Ni T P N
Ui
C9i T Vi
wV
w N i T P N j zi
wS
Ÿ
wV U
P
w
w Ni
j zi
P
DQG S i
T
S i C9i OQ
C9i T wS
wV U
R OQ
S i U i V i DUHDWVRPHUHIHUHQFHVWDWH
ŸVi
Ui
U i
D )LQG Vi U i S i DQG Gi LQWHUPVRI S i U i V i C9i RTDQG P 1HHG U i V i S i C9i OQ
w
¦ NiV i
wN
i
T˜
RT
P
Vi
wS
wV U
R
Vi
P
T
Ui Vi D 6WDUWZLWKHTQ
OQ
fi
xi P
P
RT
a
V b V
OQ I i
wP
wNi V
z
LM
MN
FG IJ OP
H K PQ
V ZRT P RT
dP
dV OQ Z N
RT V f
V
wNi V
NRT
N a
V Nb V
¦ N j aij
RT
NRT
bi V ¦ Nibi V ¦ Nibi V
b
g
a f
RT
NRTbi ¦ N j aij
V Nb V b V
N
wP
wNi V
NRT
¦ ¦ Ni N j aij
V ¦ Nibi
V
¦ x j aij
RT
RTbi
V
V
V b
Ui
U i
R OQ
Vi
V i
IRU
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
f
OQ i
xi P
OQ I i
LM
z MM
N
ZRT P
RT
RT
RTbi
RT V f V V b V b LM
N
OP
Q
V
RTbi
RT OQ
V b V f
V b V f
RT
f
OQ i
xi P
Ÿ OQ I i
OQ
F
H
Bi
OQ Z B ZB
j
RTV
fi
P
a
a
f RTV
Z OQ Z Bi i
i
PL[WXUH
ii
i
SXUHIOXLGi
RS
|T
LMF ¦ x A I Z OPUV
JK Q|W
NGH
H[Sk Z OQaZ B f a A f Z p
H[S Bi Z B OQ Z B j ij
j
i
PL[WXUH
SXUHIOXLG
ii
Ji 1RWHWKDWWKHFRPSUHVVLELOLWLHVLQSXUHIOXLGDQGPL[WXUHZLOOJHQHUDOO\EHGLIIHUHQWDWWKHVDPHT
DQGP
$VDSUHOLPLQDU\QRWHWKDWIURP(TQV DQG H T P H ,* T P
DQG
S T P S ,* T P
T
wP
P dV wT V
z LMMNFGH IJK
V ZRT P
R OQ Z OP
PQ
z LMMN FGH IJK
V ZRT P
RT Z V f
V f
ij
j
i
VR
fi
xi f i
I
K
¦ x j aij
R|
¦ x A U
|V
B
x P H[SS
OQ Z B Z
|T Z B
|W
A
P H[SRS Z OQa Z B f UV
ZW
T
fi
Pbi
RT
OQ Z ZKHUHBi
j
RTV
xi f iJ i Ÿ
fi
V f
j
OQ Z
V
DQGE\GHILQLWLRQRIWKHDFWLYLW\FRHIILFLHQW
fi
ZRT P
E )RUDSXUHYDQGHU:DDOVIOXLG (TQ V
OP
PPdV OQ Z
Q
j
¦ x j aij
Z
Bi
OQ
ZB ZB
j
¦ x j aij
ZRT P
ZRT P
¦ x j aij
OP
PQ
wP
R
dV wT V V
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
RT
a
VR
V b V
YGZ(26 P
FG w P IJ R T FG w P IJ P RT RT a
H w T K V b H w T K
V b V b V
F w P IJ R R R G
H w T K V V b V
V
a
V
a
V
RT Z V
V
z
V ZRT P
Ÿ H T P H ,* T P
RT Z V f
DQG
R OQ Z V f
RTA
Z
ZRT P
R OQ Z B
1RZRQWRVROXWLRQRISUREOHP
RT
D V H[ V PL[ xi V i
Z PL[ xi Z i
' PL[ V P
ZPL[ FRPSUHVVLELOLW\RIPL[WXUHDWTDQGP
Zi FRPSUHVVLELOLW\RISXUHIOXLGiDWTDQGP
:LOO OHDYH DQVZHU WR WKLV SDUW LQ WKLV IRUP VLQFH WKH DQDO\WLF H[SUHVVLRQ IRU Zi DQG ZPL[ VROXWLRQ WR FXELF LV PHVV\ 7KRXJK LW FDQ EH DQDO\WLFDOO\ DQG V\PEROLFDOO\ ZLWK D FRPSXWHU
DOJHEUDSURJUDPVXFKDV0DWKFDG0DWKHPDWLFD0DSOHHWF RT
RTAPL[
RTAi
xi H i
xi RT Zi E H H[ H PL[ ZPL[ ZPL[
ZPL[
i
¦
¦
L
¦ MN a
¦
PL[
i i
PL[
i
H[
OP
Q
f
FG x A IJ RTA
HZK Z F x A A IJ
RT bZ ¦ x Z g RT G ¦
H Z Z K
U
a H PV f ¦ x a H PV f
b H ¦ x H g PbV ¦ x V g
F x A A IJ RTbZ ¦ x Z g RT bZ ¦ x Z g RT G ¦
H Z Z K
F x A A IJ
RT G ¦
H Z Z K
g
RT ZPL[ ¦ xi Zi ¦ RT
PL[
OP
Q
R
R
dV
V b V
V b
R OQ Z R OQ
V V f
b
RT Z z LMN
V ZRT P
S T P S ,* T P
a
dV
V
i i
PL[
i
PL[
i i
PL[
PL[
PL[
i
PL[
i
i
i
PL[
i
i
i i
PL[
i
PL[
i i
i i
PL[
i
PL[
i
PL[
i i
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
F S PL[ ¦ xi Si R¦ xi OQ xi
S ex
a
a f
f
R OQaZ B f R¦ x OQa Z B f OQ x
R OQaZ B f R x OQ x a Z B f
Z B
R OQ
3a Z B f
P OQ ZPL[ BPL[ R X i OQ Zi Bi R¦ xi OQ xi
PL[
PL[
PL[
PL[
PL[
PL[
xi
i
i
i
G G ex
U
ex
i
a
FG x A A IJ
H Z Z K
g
i i
PL[
i
PL[
f
FG x A A IJ RT OQ Z B
H Z Z K
3aZ B f
RT ¦
T S ex
OQ I i
OQ
i i
PL[
PL[
i
PL[
i
i
LM
MN
z
FG IJ
H K
fi
yi P
V ZRT P RT
wP
N
RT V f
V
wNi T V N
BPL[
V
PL[
xi
i
j zi
OPdV OQ Z PQ
EXW
PV
RT
P
¦ ¦ yi y j Bij
V
¦ Ni RT RT ¦ ¦ Ni N j Bij
RT BPL[ RT
V
V
wP
wNi
RT
V
wP
N
wNi
NRT
V
V
V
¦ N B RT
j ij
j
V
N
¦ N B RT
j ij
j
V
¦
RT x j Bij
RT
V
V
Ÿ
ª
f
OQ i
yi P
RT
V
i
3 Zi Bi xi
i
i
b
D 6WDUWIURPHTQ i
RT ZPL[ ¦ xi Zi RT ¦
i
ZPL[ BPL[
i
A
i
H ex TS ex
RT OQ
ex
i
1RWHDOVRWKDW
³
V ZRT P « RT
V f
¦ x B OQ Z
j ij
RT
RT
«
V
«V
¬«
HTQ
¦ x B º»
j ij
j
V
»dV OQ Z
»
¼»
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
PV
RT
B
PV RU
V
RT
V B
PV V B
RT
PV
r PB RT
RU
RT
P RT
ŸV
r PB RT d
Z
i
DV P o Z o LGHDOJDVOLPLW VRRQO\VLJQDOORZHG
PB RT d
i
Z
1RWHWKDWDWORZSUHVVXUHVZHFDQREWDLQDVLPSOHUH[SUHVVLRQ
$WORZSUHVVXUHV
PV
RT
BPL[
B P
| PL[ V
RT
7KHQ
m
$OVR
V
a N N f RTP N N mN B N B N N B r NV
w N
V
zc
f
f
f
V V,*0 dP
RT P
f
Ÿ y P
¦ N j Bij
RT
j
¦ ¦ Ni N j Bij N N
P
N N i
a
a
a
V V,*0
f
y P
RT
¦ ¦ yi y j Bij ¦ y j Bij
P
RT
yB y B y B y y B y B
P
RT
y y B y y B y B
P
RT
y y B y B y B
P
RT
RT
y y B y B y B P
P
OQ
w
V
r
RT
RT
BPL[
y B y y B y B
P
P
RT
RT
H[
y
y
B
V
V V IG
BPL[ ¦ ¦ i j ij
P
P
V
a
a
h
f
f
y y B y B y B
P U
H[SRS y a y f B y B y B
V
RT W
T
P
RT
7KLVLVDQDOWHUQDWHVRPHDSSUR[LPDWHH[SUHVVLRQWKDWZHZLOOXVHLQZKDWIROORZV$OVRIRU
WKHSXUHFRPSRQHQWZHKDYH
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
f
P
f
y f
l a
f
a
H[S y y B y B y B P RT
J
k
H[S B P RT
RS b
g
T
P O
H[SLM y a B B B f
PQ J
RT
N
p
H[S y y B y B y B
UV
W
ZKHUH G B B B RT
E 5HSHDWLQJWKHDUJXPHQWIRUDWHUQDU\PL[WXUH
a
f
P y G y y G G G yG RT
OQ J P
RT
fq
G y P
RU OQ J LM B P OP N RT Q
1RWH WKDW WKHVH H[SUHVVLRQV DUH VOLJKWO\ HDVLHU WR XVH WKHQ WKH IXOO H[SUHVVLRQV VLQFH ZH GRQ¶W
KDYHWRVROYHIRU V RUZ ILUVW
H[S
%\VLPSOHJHQHUDOL]DWLRQ
c
h
P
¦ ¦ yi y j G i G ij G ij 1RWHG ii
RT i j
OQ J i z j z
iz j
a
fp
NN R
G
NG
S A BFGH NN NN IJK UVW
N N T
FG w G IJ
N
RS A BFG N N IJ UV
G
H w N K N N T H N N KW
N N
RS A B a N N f UV
aN N f T N N W
N N R B
Ba N N f U
S
N N TN N
a N N f VW
x k A Ba x x fp x x k A Ba x x fp x x Ba x x f
D G H[
k
xx A B x x H[
H[
H[
H[
TP
RT OQ J RTa xqxq
Ÿ G H[
xq xq
$IWHUVRPHDOJHEUD
E G H[
1RZE\UHSHDWLQJWKHFDOFXODWLRQRUE\XVLQJWKHV\PPHWU\RI G H[ DQGUHSODFLQJBE\ B DQG
LQWHUFKDQJLQJWKHVXEVFULSWVDQGZHREWDLQ
GH[
Ax Bx Bx Bx Bx
A B x Bx 7KXV GH[ RT OQ J A B x Bx N G H[
RTaqq N N Nq N q
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
GH[
d
w N G H[
6LPLODUO\ OQ J F
G H[
RT
RTaq
a
a
a
aq
D
a
f
xq xq E
f
xq xq T P N GH[
RT
Ÿ OQ J i
w N
f
Dx Ex f Ex Dx a zz azz az z ZKHUH
xi qi
¦ x jq j
zi
DQG
j
FG N q IJ
H ¦ K ¦ ¦ a N N q q WKXV
N G H[
G H[
RT
k k
kj
k
k
GiH[ w
G H[
RT w Ni
T P N
OQ J i
k
j k
j
j
qi ¦ akj qk q j xk x j
j k
FG ¦ x q IJ
H
K
j zi
¦ aij qi q j x j
¦ xk q k
k
k k
D qE VLPLODUO\ q
D qa
,QWHUFKDQJLQJLQGLFHVDQG
)LQDOO\LQWHUFKDQJLQJLQGLFHVDQGLQWKHRULJLQDOHTXDWLRQ\LHOGV
nx E aD E f x E aD E f s x x aD E faD E f E E E aE D f kx x aD E f x aD E fp
DOVRDYDLODEOHDVD0DWKFDGZRUNVKHHW 8VLQJ(TQV \LHOGV D
D nx E aD E f x D aE D f s x x aD E faE D f E D D aE D f kx x aD E f x aE D fp
OQ J qE D Ÿ q
qa
HWF)RUWKHFDVH i
OQ J qa E nx D aE D f x D aE D f s x x aE D faE D f D D D aD E f kx x aE D f x aE D fp
OQ J k
1RZVHWWLQJ
j
b
RT
LM a a OP DQG E b LM a a OP )URP6HFWLRQ
RT N b
b Q
Nb b Q
ZHKDYHWKUHHGLIIHUHQWH[SUHVVLRQVUHODWLQJWKHaDQGbSDUDPHWHUVWRWKHFULWLFDOSURSHUWLHV
R TC
RTC
DQG b
(TQ D a
PC
PC
a
PCV C DQG b
VC
(TQ E Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
VC
V C RTC
(TQ D DQG b
6LQFH ZC LV QRW HTXDO WR IRU WKH IOXLGV XQGHU FRQVLGHUDWLRQ HDFK VHW RI UHODWLRQV ZLOO JLYH D
a
GLIIHUHQWSDLURIYDOXHVIRU aDQG b*HQHUDOO\VHWLVXVHGVLQFH V C LVNQRZQZLWKOHVVDFFXUDF\
WKDQ PC DQG TC $OOWKUHHVHWVRISDUDPHWHUVZLOOEHFRQVLGHUHGKHUH
VC
0: U C %HQ]HQH
WUXPHWK\OSHQWDQH
P NJ P NJ c h
c P h NPRO u P 3D NPRO 6HW
a
b
6HW
D
E
a
u P 3D NPRO b
6HW
D
E
a
P NPRO u P 3D NPRO b g
b
P NPRO u P 3D NPRO c h
c h
u P 3D NPRO D
E
6HW ([DPSOH
)LWWLQJWKHYDQ/DDUHTXDWLRQ
6HW c h
c P h NPRO D
E
([DPSOH
D
5HJXODU6ROXWLRQ7KHRU\
E
P NPRO c h
u P 3D NPRO P NPRO a
f
V
G G RT
V
D V
7KHVHWVRIUHVXOWVDUHSORWWHGEHORZ
1XPEHUVLQFLUFOHVGHQRWHSDUDPHWHUVHWVXVHG
3DUDPHWHUVHW ILWWHGWRWKHH[SHULPHQWDOGDWDVKRXOGEHWKHPRVWDFFXUDWH3DUDPHWHUVHW REWDLQHGXVLQJ V C DQG TC GDWDVKRXOGEHUHDVRQDEO\JRRGDOVR
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
%
L 2QHFRQVWDQW0DUJXOHVHTXDWLRQ
RT OQ J Ax RT OQ J x A 7KXV
J x
Ax A A x RT OQ J RT OQ
J x RU
A x
A x
x ˜ H[S
DQG J J H[S
RT
mV M
RT
a
a
RT OQ J 7KXV
a f D x E x RT OQ J a x f D E
J ax f
RT OQ
D c x h E c x h J a x f
DQG
J
LM D c x h E c x h OP RT
MN
PQ
L D c x h E c x h OP x ˜ H[SM
mM
RT
MN
PQ
J
h
LM c
MN
LL 7ZRFRQVWDQW0DUJXOHVHTXDWLRQ
RT OQ J x
c
f
h OP
PQ
LM c
MN
a f
f
H[S
V
h OP PQ
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
LLL YDQ/DDUHTXDWLRQ
a f aDxD Ex f OQ J ax f D D
D OQ J a x f OQ J a x f OQ J a x f
aD x E x f
OQ J x
RUXSRQUHDUUDQJHPHQW
DQG
7KXV
V
xV Ÿ DV x o xV xV V
Y 81,48$&0RGHO
LM
c h
MM
N
OQ J a x o f OLP aOQ J f
OP
PP
Q
T jW ij
z
T
I
OQ qi OQ i li i ¦ x j l j qi OQ ¦ T jW ij ¦
xi xi
Ii
j ¦ T k W kj j
Ii
OQ J i
i
i
k
i
xi o 1RZFRQVLGHU xi o WKHQ T j o DQG T i o a
f qr
x
a
f rq
I
W O
F
F
L
r
z
q rI
r
OQ J
OQG J q OQG ˜ J l l q M OQcW h P
W Q
Hr K Hq r K r N
J
F I r IJ z q OQFG T ˜ q r IJ I ¦ x l r l OQ
OQ J
OQG
H x r K HI q r K x
r
J a x o f
O
LM
T W
W P
q M OQ ¦ T W ¦
OQ W W P
¦T W
PQ
MN
Ii
i
I
ri
i
xr xr xi
xi ri
˜
xr xr xi
i
i
i
j
a f RT OQ J ax f RT OQ J a x f V cI haG G f LV cI haG G f OP DQG
J
H[S M
RT
PQ
MN
LV cI haG G f OP x ˜ J
H[S M
mM
RT
PQ
MN
RT OQ J x
H[S
VLQFH I a f V I aG G f DQG RT OQ J ax f V aG G f RT OQ J x
LY 5HJXODU6ROXWLRQ7KHRU\
J
LM D x aE x D x f OP N aE x D x f Q
L D x aE x D x f OP x ˜ H[S M
mM
N aE x D x f Q
J
i
j
j
i
xi qi xq xq
xi ri xr xr
ij
i
i
j
i
ij
jj
j
i j
ri
T
i
rj
Ii
i
j i
i
i
i j
i
i
i
j j
i i
ij
j ij
i
ij
j ij
j
k kj
j
k
i
j
jj
i
j
i
j
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
J
x ˜ J ZKHUH J LVDVJLYHQDERYH
mV M
$QLGHDOJDVFRQVWUDLQHGWRUHPDLQDWFRQVWDQWYROXPHDQGTLVDOVRDV\VWHPDWFRQVWDQWLQWHUQDO
HQHUJ\ DQG YROXPH VLQFH U LV RQO\ D IXQFWLRQ RI WHPSHUDWXUH IRU WKH LGHDO JDV &RQVHTXHQWO\ DW
HTXLOLEULXPWKHHQWURS\VKRXOGEHDPD[LPXP
6XSSRVH WKHUH ZHUH N PROHFXOHV DQG N ODWWLFH VLWHV )RU DQ\ GLVWULEXWLRQ RI WKH N PROHFXOHV
DPRQJ WKH N ODWWLFH VLWHV WKHUH ZLOO EH N ODWWLFH VLWHV ZLWK PROHFXOHV DQG N N N HPSW\
ODWWLFHVLWHV7KXVZHFDQFRQVLGHUWKH³ODWWLFHJDV´WREHDPL[WXUHRI N PROHFXOHVDQG N KROHV
DQG WKH HQWURS\ RI YDULRXV FRQILJXUDWLRQV RI WKLV ELQDU\ V\VWHP FDQ EH FRPSXWHG )ROORZLQJ WKH
DQDO\VLV RI $SSHQGL[ LW LV FOHDU WKDW WKH UDQGRP PL[WXUH RU XQLIRUP GLVWULEXWLRQ RI JDV
PROHFXOHVLVWKHVWDWHRIPD[LPXPHQWURS\$FRPSOHWHO\RUGHUHGVWDWH IRUH[DPSOHWKHILUVW N ODWWLFH VLWHV ILOOHG DQG WKH QH[W N N N ODWWLFH VLWHV HPSW\ LV DQ HVSHFLDOO\ ORZ HQWURS\
FRQILJXUDWLRQ
7KH SULQFLSOH RI FRUUHVSRQGLQJ VWDWHV DQG WKH SVHXGRFULWLFDO FRQVWDQW FRQFHSW ZLOO EH XVHG ILUVW
WKHQWKH3HQJ5RELQVRQHTXDWLRQRIVWDWH SURJUDP35 Pr
D 2 T& . P& EDU Tr
f
f
EDU Ÿ f P
)URP)LJXUH
P
P
8VLQJWKH35(26DQGWKHSURJUDP35 f 2 EDU F I
H K
E 1 T&
f1
. P&
EDU Tr
EDU 8VLQJWKH35HRV f 1 Pr
f
# P
EDU F /HZLV5DQGDOO5XOH
G
f 2 &RUUHVSRQGLQJVWDWHV
u EDU f1 u EDU
8VLQJWKHSURJUDP35ZHILQG f 2 35HRV
u EDU u EDU EDUDQG f 1 6800$5<
f 1 /HZLV5DQGDOO
EDU
ZLWKFRUUHVSRQGLQJVWDWHV
ZLWK3HQJ5RELQVRQHRV
3HQJ5RELQVRQHRVGLUHFWO\
SURJUDP35 EDU f 2 EDU
7KLVSUREOHPZDVVROYHGXVLQJWKHRULJLQDO81,)$&PRGHO7KHUHVXOWVPD\EHVOLJKWO\GLIIHUHQWLQ
WKH9,68$/%$6,&81,)$&SURJUDPLVXVHGDVWKDWXVHVWKHXSGDWHGPRGLILHG81,)$&PRGHO
7RSUHVHQWWKHH[WHQWRIQRQLGHDOLW\WZRPHDVXUHVZLOOEHXVHG2QHLVWKHLQILQLWHGLOXWLRQDFWLYLW\
FRHIILFLHQWVDQGWKHRWKHULV G H[ PD[ WKDWLVWKHPD[LPXPYDOXHRIWKHH[FHVV*LEEVIUHHHQHUJ\
7KHUHVXOWVDSSHDUEHORZIRUWKHFDVHRI T q & Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
ZDWHU
HWKDQRO
EHQ]HQH
J f HWKDQRO
EHQ]HQH
WROXHQH
EHQ]HQH
WROXHQH
WROXHQH
J f G H[ -PRO ±
7KHVHUHVXOWVZHUHREWDLQHGWUHDWLQJWROXHQHDV$&+JURXSV $&&+ JURXS
$QDOWHUQDWLYHLVWRFRQVLGHUWROXHQHWREH$&+JURXSV$&JURXSDQG &+ JURXS:HGRWKLV
MXVW WR GHPRQVWUDWH WKDW WKHUH FDQ EH D QXPEHU RI SRVVLEOH JURXS DVVLJQPHQWV HDFK RI ZKLFK ZLOO
UHVXOWLQVRPHZKDWGLIIHUHQWDFWLYLW\FRHIILFLHQWV
ZDWHU
HWKDQRO
EHQ]HQH
J f WROXHQH
WROXHQH
WROXHQH
J f G H[ -PRO :H VHH IURP WKH UHVXOWV LQGHSHQGHQW RI ZKLFK JURXS DVVLJQPHQW LV XVHG IRU WROXHQH WKDW WKH
EHQ]HQHWROXHQH PL[WXUH ZKLFK FRQWDLQV FKHPLFDOO\ VLPLODU VSHFLHV LV YLUWXDOO\ DQ LGHDO VROXWLRQ
7KHZDWHUWROXHQHDQGZDWHUEHQ]HQHPL[WXUHVFRQVLVWRIYHU\GLVVLPLODUVSHFLHVDQGWKHUHIRUHWKH
PL[WXUHV DUH YHU\ QRQLGHDO (WKDQRO FRQWDLQV D K\GURFDUERQ HQG DQG D SRODU 2+ HQG
&RQVHTXHQWO\ LW LV DOPRVW HTXDOO\ FRPSDWLEOH RU LQFRPSDWLEOH ZLWK ERWK ZDWHU DQG K\GURFDUERQ
VROYHQWVDQGIRUPVRQO\PRGHUDWHO\QRQLGHDOPL[WXUHVZLWKERWKWKLVEHKDYLRULVSUHGLFWHGDERYH
5HJXODU VROXWLRQ WKHRU\ VKRXOG QRW EH XVHG ZLWK K\GURJHQERQGHG VROYHQWV VXFK DV ZDWHU DQG
HWKDQRO +RZHYHU PHUHO\ IRU GHPRQVWUDWLRQ ZH ZLOO XVH UHJXODU VROXWLRQ WKHRU\ IRU WKHVH
FRPSRXQGV 7KH ³+DQGERRN RI &KHPLVWU\ DQG 3K\VLFV´ UHSRUWV G (W2+ DQG G + 2 $OVR V + 2
FF PRO DQG V (W2+
FF PRO ,QUHJXODUVROXWLRQWKHRU\
RT OQ J i
VR
a
f
V iI j G G VR J if
H[S
LMV aG G f OP N RT Q
i
J f J f :DWHU
HWKDQRO
G (W2+ EHQ]HQH
G EHQ] WROXHQH
G WRO (WKDQRO EHQ]HQH
WROXHQH
%HQ]HQH WROXHQH
6LQFH WKH VROXELOLW\ SDUDPHWHUV RI DOO WKH FRPSRQHQWV DUH VLPLODU UHJXODU VROXWLRQ WKHRU\ SUHGLFWV
HVVHQWLDOO\ LGHDO VROXWLRQ EHKDYLRU HYHQ WKRXJK IRU H[DPSOH WKH ZDWHUDURPDWLF K\GURFDUERQ
PL[WXUHVDUHKLJKO\QRQLGHDO7KLVLVDQH[DPSOHRIKRZEDGWKHUHJXODUVROXWLRQWKHRU\SUHGLFWLRQV
FDQEHZKHQXVHGIRUPL[WXUHVIRUZKLFKLWLVQRWDSSURSULDWH
7KLVH[DPSOHVKRXOGVHUYHDVDZDUQLQJDERXWWKHLPSURSHUXVHRIWKHUPRG\QDPLFPRGHOV
7KHDFWLYLW\FRHIILFLHQWLVGHILQHGDV
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
I L 7 3 [
[ L J L 7 3 [ I L 7 3 RUJ L 7 3 [
I L 7 3 [
[ L J L IL 7 3
7R JHW WKH DFWLYLW\ FRHIILFLHQW XVH (T IRU I L 7 3 [ DQG (T IRU IL 7 3 +RZHYHUUHPHPEHUWKDWWKH3HQJ5RELQVRQHTXDWLRQRIVWDWHSDUDPHWHUVDUHGLIIHUHQWIRUWKHSXUH
FRPSRQHQWDQGIRUWKHPL[WXUH$OVRWKHUHIRUHWKHOLTXLGYROXPH UHDOO\FRPSUHVVLELOLW\= ZLOO
EHGLIIHUHQWIRUWKHSXUHFRPSRQHQWDQGIRUWKHPL[WXUH
T, P
xPROHVSXUH
T, P
3URFHVV
xx
T, P
xPROHVSXUH
6\VWHP PROHRILQLWLDOPL[WXUH
V\VWHPLVFORVHGLVRWKHUPDODQGLVREDULF
0DVVEDODQFH x x dV dU
(QHUJ\EDODQFH
Q P
WV dt
dt
dS Q (QWURS\EDODQFH
SJHQ dt T
7KXV
dV
dU Q P
WV
dT
dt
dS
TSJHQ
Q T
dt
dV
dS
dU
Ÿ WV
T
TSJHQ P
dt
dt
dt
D 6LQFHERWKPDQGTDUHFRQVWDQWZHFDQZULWH
dU d
d
WV
PV TS TSJHQ
dt dt
dt
dG
d
TSJHQ
U PV TS TSJHQ
dt
dt
dG
&OHDUO\IRU WV WREHDPLQLPXP SJHQ DQG WVPLQ
dt
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
WVPLQ
FG SHUPROHRI IJ G G x G x G x G x G
H LQLWLDOPL[WXUHK
x bG G g x bG G g
f
WVPLQ
T
f i T P x
fi T P
T
xi 7KHUHIRUH
dS
ZKLFK RFFXUV ZKHQ WV
dt
l
Q PD[
T x S T P S T P x x S T P S T P x
WVPLQ )ROORZLQJ VDPH
q
F 8VLQJWKHLGHDOJDVRULGHDOPL[WXUHDVVXPSWLRQVLQFHLVRPHUVFDQEHH[SHFWHGWRIRUPLGHDO
PL[WXUHVDQGWKHUHVXOWDERYH
WVPLQ RT x OQ x x OQ x OQ OQ u . u PRO ˜ .
u . u OQ PRO ˜ .
PRORIIHHG
LM
N
G
a
w NG
w N T P N f x x A Ca x x f
L a N N f Ca N N f OP
x N MC
N aN N f aN N f Q
x a x f A Ca x x f x x Ca x x f Ca x x f x A Ca x x f Cx x a x x f x x
x A Ca x x f Cx x a x x f RT OQ J
f OP
f Q
a
N N C N N A
N N N N N G
a
x A C x x
G
w NG
w N T P N
f x x A Ca x x f
L a N N f Ca N N f OP
x x a N N fM
N aN N f aN N f Q
x a x f A Ca x x f x x Ca x x f x x
x A Ca x x f Cx x a x x f RT OQ J
a
x A C x x
RT x OQ x x OQ x t VRZRUNPXVWEHDGGHG
dS
TSJHQ WKXV Q PD[
dt
DQDO\VLVDVDERYHOHDGVWR
>1RWH Q
E 1RZIRUHLWKHULGHDOPL[WXUHVRU/HZLV5DQGDOOPL[WXUHV
f T P
f T P
x RT OQ f T P x
f T P x
x RT OQ
i
RU
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
RT OQ J RT OQ J a f
a f
x A Ca x x f Cx a x x f
x A C x x Cx x x
D L 2QHFRQVWDQW0DUJXOHVHTXDWLRQ
a
f
A x RT
RT OQ J a
LL 7ZRFRQVWDQW0DUJXOHVHTXDWLRQV
a
RT OQ J w OQ J w x
D
a
f a f o DVx o E a x f Dx
D E x
LY 5HJXODU6ROXWLRQ7KHRU\H[SUHVVLRQ
567KDVWKHVDPHIRUPDVWKHYDQ/DDU(TXDWLRQVRWKDWSURRIIROORZVIURP LLL DERYH
E 6WDUWLQJIURPWKH*LEEV'XKHP(TXDWLRQ(TQ F w OQ J I
H K
C
¦ xi G w x i J
j
TP
ZHREWDLQ
Dx Ex i 7KXV
w OQ J w x
LLL YDQ/DDU(TXDWLRQ
f a f a f
D a x f E a x f s o DVx o n
RT
OQ J f
D x E x a
A x f
w OQ J A x
RT
w x
RU
A x
w OQ J OLP
OLP
x o w x
x o
RT
w OQ J w x
x
$OWHUQDWLYHO\VLQFH dx
x
FG w OQ J IJ
H wx K
TP
x
FG w OQ J IJ
H wx K
TP
dx ZHKDYH
FG w OQ J IJ
H wx K
TP
x
FG w OQ J IJ
H wx K
TP
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
1RZ OLP x
DQG OLP
x o
x o
FG w OQ J IJ
H wx K
Ÿ OLP x
x o
FG w OQ J IJ
H wx K
TP
ZKLFKDOVRLPSOLHVWKDW OLP x
x o OLP xi
xi o FG w OQ J IJ
H wx K
OLP
i
FG w OQ J IJ
H wx K
TP
RUPRUHJHQHUDOO\
TP
FG w OQ J IJ
H wx K
i
i
TP
OQ J GHSDUWV ZLWK
Ÿ ]HURVORSH IURPLWV i
xi o
x o
7KXVZHKDYH
Ÿ OLP x
FG w OQ J IJ
H wx K
TP
TP
YDOXHRIDWxi
DQG
F w OQ J IJ
OLP x G
H wx K
i
xi o i
i
TP
OQ J ULVHVOHVVUDSLGO\
Ÿ WKDQ DVxi o 7KXV xi
xi OQ J i LVERXQGHG
/HW M PRODOLW\RIVDOWLQVROXWLRQ
L )RU.&O z z M . M M&O M I
¦ zi Mi u M u M M z z LL )RU &U&O I
u M u M
M M M M M
z z LLL )RU &U 62 I
u M u M
M M M M M
UV
W
a f
m
UV
W
r
m
r
1RZWKH'HE\H+FNHOH[SUHVVLRQLV OQ J r D z z I DQG(TXDWLRQ D z z I
PRO OQ J r
z z I D DWq&
OLW
I
F I
H K
L .&O
M
OQ J r H[SHULPHQW
'HE\H+FNHO
OQ J r
M (TQ OQ J r
M
M
M
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
LL &U&O M
OQ J r OQ J r
LLL &U 62 OQ J r OQ J r
u u u u u a f
M
M OQ J r
M u u u u u u u OQ J r
M
M
M
M
M M
7KXV WKH 'HE\H+FNHO DQG H[WHQGHG 'HE\H+FNHO ZLWK D IL[HG YDOXH RI WKH GSDUDPHWHU WKHRULHVDUHQRWYHU\DFFXUDWH+RZHYHULIWKHGSDUDPHWHULVDGMXVWHGPXFKEHWWHUDJUHHPHQW
ZLWKH[SHULPHQWDOGDWDFDQEHDFKLHYHG7KLVLVOHIWWRWKHVWXGHQWWRSURYH
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
DOVRDYDLODEOHDVD0DWKFDGZRUNVKHHW 7KH*LEEV'XKHPHTXDWLRQZULWWHQLQWHUPVRIPRODOLWLHVDQGXVLQJWKHPHDQLRQLFDFWLYLW\
FRHIILFLHQWLVDVIROORZV Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
M6dG6 M (dG (
EXW
G(
c
G RT OQa x J f
G o( RT OQ J Qr M Q M Q o
G6
6
6RWKDW
M6dG 6 M ( dG (
6 6
a f
c
h M d OQa x J f M d OQcJ M h M d OQa x J f M Qd OQdJ M i
6
h
M6d OQ x6J 6 M ( d OQ J Qr M Q M Q ZKHUH6LVVROYHQWDQG(LVHOHFWURO\WH
6 6
Q
(
Q
r
6
r
)RU+&O
M M +&O M M +&O M r
Q
Q
M +&O
˜ M +&O
M6
M(
M6 M ( x6
M6d OQ J 6 x6 M +&O d OQ J r M +&O
a
a
f
a
f
d OQ J 6 x6
M +&Od OQ J r M +&O
M + 2
d OQ J 6 x6
M +&Od OQ J r M +&O
' OQ J 6 x6
M +&O
' OQ J r M +&O
a
a
M +&Od OQ J r M +&O
M + 2
f
f
a f
a
f
M +&Od OQ J r
M +&Od OQ M +&O
M +&Od OQ J r
M +&Od OQ M +&O
d OQ J 6 d OQ
M +&O
d OQ J 6 d OQ x6
FG
H
f 7KLV FDQ QRZ EH XVHG DV D EDVLV IRU QXPHULFDO LQWHJUDWLRQ ZLWK WKH DFWLYLW\ FRHIILFLHQW H[SUHVVLRQ
IURP,OOXVWUDWLRQ2USURFHHGLQJIXUWKHU
d OQ J 6 x6
r
M +&O
r
M6
(
7KLVLVWKH*LEEV'XKHPHTXDWLRQIRUWKHVROXWHHOHFWURO\WHV\VWHP
6 6
)URP,OOXVWUDWLRQ
IJ
K
a f
a
f
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
M +&O
M +&O M +&O
OQ J r
FG H M
d OQ J r
d OQ J 6 IJ
K
dM +&O M +&O +&O M +&O
dM +&O
M +&O
IJ
K
FG
H
M +&O
dM +&O dM +&O
M +&O M +&O M +&O
d OQ J 6
FG M
H +&O
IJ
K
M +&O
M +&O
M +&O
dM +&O
˜ M +&O
˜ M +&O
7KLVFDQRQO\EHVROYHGE\QXPHULFDOLQWHJUDWLRQ 6HH0$7+&$'ILOHIRUWKLVSUREOHP JDPPD 0
0
D 7KH WZRFRQVWDQW 5HGOLFK.LVWHU H[SDQVLRQ ZKLFK OHDGV WR WKH WZRFRQVWDQW 0DUJXOHV
HTXDWLRQLV
G H[
7KXV
G H[
xx
G H[
fp
a
f
A B x ZKLFKLVDOLQHDUIXQFWLRQRIx
7KHIRUPRIWKH:RKO(TXDWLRQZKLFKOHDGVWRWKHYDQ/DDU(TXDWLRQLV
a
k
xx A B x x ZKLFKFDQEHUHDUUDQJHGWR
RTa xqxq
xq xq
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
xx
G H[
a
f
xq x q
RTaqq
ZKLFK LV DOVR D OLQHDU IXQFWLRQ RI x (TXDWLRQV DQG SURYLGH WKH MXVWLILFDWLRQ IRU WKH
SURFHGXUH
E 7KH ILJXUH EHORZ LV WKH UHTXLUHG SORW &OHDUO\ QHLWKHU HTXDWLRQ LV DQ DFFXUDWH ILW RI WKH GDWD
>7KHFRQVWDQW:RKO RUYDQ/DDU HTXDWLRQSORWRIWKHGDWDLHWKHIRUPRI(T LVFORVHVW
WREHLQJOLQHDUDQGWKHUHIRUHVKRXOGEHWKHEHWWHURIWKHWZRFRQVWDQWILWVRIWKHGDWD7KHGDWD
FDQ KRZHYHU EH ILW TXLWH ZHOO ZLWK D FRQVWDQW 5HGOLFK.LVWHU H[SDQVLRQ²6HH ,OOXVWUDWLRQ
@
([SUHVVLRQIRU G H[ LQWKLVSUREOHPLVWKHVDPHDVWKDWRI(T ,IZHUHFRJQL]HWKDWADQGB
a
LQ(T LVUHSODFHGE\ARTDQGBRTKHUH$OVRVLQFH x$U
PHWKDQHDQGVSHFLHVLVDUJRQ
f cx
&+ x$U
h VSHFLHVLV
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
D
7KHUHIRUH
DQG
$W x
OQ J OQ J a
G H[ x$U
a
a
f
f
a
f AT
G ax
DQG
f ART
R
A
$U
. u
. . u
u
f
T
'T G H[
R
'
FG G IJ HRK
G H[
RT
'
±
±
±
±
h OP
PQ
H H[
RT H[
FG G IJ H RT K
H[
1H[WXVLQJ
a
f
w 'G RT
wT
P
7KXV
' G H[ RT
'T
LM c
MN
H[
w 'G RT
' H
DQG
RT
wT
H[
1RZUHSODFLQJGLIIHUHQWLDOVZLWKILQLWHGLIIHUHQFHV
$U
G H[ x$U DWT .
R
G H[ x$U DWT .
R
G H[ x$U DWT .
R
fp DW x
7KXV
J &+ J $U fk
a
ART BRT x BRTx RTx$U x$U A B x$U
RT OQ J A B
A B
a
E G H[
ART BRT x BRTx x
RT OQ J u RT P
' H PL[
RT ' PL[ H ZKHUH T
DYHUDJHWHPSHUDWXUHRYHU 'T RS
T
UV - PRO W
Ÿ 'H PL[ # u . u
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
F )URP
FG w G IJ
Hw TK
S ZHREWDLQ
P
S
H[
d
R' G H[ R
FG IJ
H K
w G H[
wT R P
i - PRO. u RS UV
T W
'T
- PRO.
$OVR
' PL[ S
R
¦ x OQ x S
i
H[
u u OQ i
- PRO.
:HVWDUWZLWK
OQ
zc
dP
P
RT i T P N j zi
RT
P
dZ dV
V
V
P
dZ
Z
P
h
(TQ D
OP
PQ
RT
dP P
P
d PV dV
V
V
P
dV V
$OVRE\WULSOHSURGXFWUXOH
FG w V IJ
Hw N K
i T P N j zi
FG w P IJ
H wV K
i
P T N j zi
˜
F w V IJ
ŸV G
Hw N K
i
LMF w V I
z MNGH w N JK
Vi Vi ,*0 dP
RT fi
xi P
1RZ
S H[
7KHUHIRUH
R
FG w N IJ
HwPK
F w P IJ
G
Hw N K
˜
i
TN j
T V N j zi
i T V N j zi
FG w V IJ
H w PK
TN j
6R
OQ
fi
xi P
z FGH IJK FGH IJK z LMN
z FGH IJK
z
z
wP
w Ni T V N
RT
P
wV
P
dP dZ dV
w
V
P TN
P Z
j zi
j
wP
RT V f w Ni T V N
V
j zi
z LMMN FGH IJK
LM RT N F w P I
z
RT
MN V GH w N JK
wP
RT
N
w Ni T V N
RT V f V
V
V f
Z
Z j zi
V
dV dZ
V
Z
V f
P V
NdV i T V N j zi
OP
Q
OPdV OQ Z
Z PQ
OPdV OQ Z
PQ
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
LM
N
OP
Q
LM ¦ x / OP
G
NG
¦ N OQ M
RT
RT
MN N PPQ
G
w FG I
Ÿ
G J
RT w N H RT K
LM / ¦ N / dN OP LM ¦ x / OP
N
¦N
OQ M
dN P
N
¦ N / MM N
PQ
MN N PPQ
N
x ¦x /
L
O
x/
G
OQ J
OQM ¦ x / P ¦
RT
N
Q ¦x / ¦ ¦x /
L
O x/
OQM ¦ x / P ¦
N
Q ¦x /
G H[
RT
C
¦ xi OQ ¦ x j /ij
i j
H[
H[
j
C
i H[
i
H[
ij
j
j
T P N j zi
ij
j
C
j
i i
i
j
ij
j
ij
j
H[
C
C
j j
ij
i j
j
C
C
j ij
i ij
i
j
ij
j
i j
ij
j
i
i
j
C
i
i
j
ij
j
:KLFKLVWKHDQVZHUWRSDUWE7RREWDLQWKHUHVXOWRISDUWDZHUHVWULFWiDQGjWRWKHYDOXHVDQG
DQGQRWHWKDW /ii 7KXV
x
x / OQ J OQ x x / x x / x/ x
EXW
x
x x / x
x /
x x /
x x /
x x /
VRWKDW
/
/ OQ J OQ x x / x
x x / x/ x
LM
N
OP
Q
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
$OOWKHFDOFXODWLRQVIRUWKLVSUREOHPZHUHGRQHXVLQJWKHSURJUDP9/08ZLWKWKHELQDU\LQWHUDFWLRQ
SDUDPHWHU IRU &2 n& HTXDO WR DV JLYHQ LQ 7DEOH 7KH UHVXOWV DUH RQO\ JLYHQ LQ
JUDSKLFDOIRUPKHUH
T = 377.6 K
kij = 0.13
f CO
VXSHUFULWLFDO
2
YDSRU
OLTXLG
YDSRU
f nC
4
x nC
4
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
T =300 K
kij = 0.13
OLTXLG
f CO
2
OLTXLG
YDSRU
YDSRU
f nC
4
OLTXLG
x nC
4
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
a
G H[
axx x x
f
f ALM N N N N OP aN N f
N aN N f Q
L N N N c N N N N h OP
AM
a N N f PQ
MN a N N f
a
AN N N N N G H[
w N G H[
w N N T P
A xx x x x xx
Ax x x x xx
Ax x Ax xx x xx
a
f RT OQ J
L N N N c N N N N h OP
AM
a N N f PQ
MN a N N f
Ax x
H[
w NG
w N
N T P
a
Ax x x xx x
a
Ax x
a
Ax x
f
c
h
f
c
h
h
h
c
f
h
a
c
f
c
c
a
7KLV LV MXVW D FKHFN VLQFH E\ V\PPHWU\ RI RULJLQDO HTXDWLRQ l JLYHV PLQXV VLJQ 7KHUHIRUH
l RQ OQ J PXVWJLYHPLQXVVLJQ 'RHVWKLVH[SUHVVLRQVDWLVI\WKH*LEEV'XKHP(TXDWLRQ"
d OQ J d OQ J d OQ J d OQ J RU x
x
x
x
dx
dx
dx
dx
A d
d OQ J A d x x
x x
RT dx
dx
RT dx
A
x x
RT
A d
d OQ J A d
x x
x x
RT dx
dx
RT dx
A
x x
RT
*LEEV'XKHP(TXDWLRQ
A
A
x x x
x
x x
RT
RT
A
x x xx x x x x
RT
A A
x x x x xx xx xx x x
RT
RT
A x x RT
Ÿ6DWLVILHV*LEEV'XKHP(TXDWLRQ
a
f RT OQ J
f
Ax x x x xx Ax x
Ax x xx xx
a
Ax x A x xx x x xx
f
Ax x x x xx c
h
h
a
f
7RFKHFNWKHXWLOLW\RIWKHVHPRGHOVZHZLOOXVHWKH*LEEV'XKHPHTXDWLRQLQWKHIRUP
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
x
FG w OQ J IJ
H wx K
OQ J Ax
A x OQ J Bx
FG w OQ J IJ
H wx K
TP
TP
TP
a
f
*LYHV
x
FG w OQ J IJ
H wx K
)RUWKHPRGHO
x
FG w OQ J IJ
H wx K
TP
x
a
xx B A
7KH RQO\ ZD\ WKLV HTXDWLRQ FDQ EH VDWLVILHG LV LI A
YLRODWHG
)RUWKHPRGHO OQ J Axn OQ J Axn x
FG w OQ J IJ
H wx K
TP
x
FG w OQ J IJ
H wx K
a
TP
f
x ˜ ˜ A x x B ˜ x
B LI QRW WKH *LEEV'XKHP HTXDWLRQ LV
f
x ˜ n ˜ A x n x ˜ n ˜ Axn nAxx xn xn 7KH RQO\ ZD\ WKH *LEEV'XKHP HTXDWLRQ FDQ EH VDWLVILHG IRU DOO YDOXHV RI x DQG x ZLWK
x x LVLI n LQZKLFKFDVHWKHWHUPLQEUDFNHWVLVDOZD\V]HUR
)RUWKHPRGHO OQ J x
FG w OQ J IJ
H wx K
TP
Axn OQ J x
FG w OQ J IJ
H wx K
Bxn :HKDYH
a
TP
f
x ˜ n ˜ A x n x ˜ n ˜ Bxn nxx Axn Bxn )RUWKLVHTXDWLRQWREHVDWLVILHGWKHWHUPLQWKHEUDFNHWVPXVWEH]HUR7KLVFDQRQO\EHLQ n
DQG A B Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
:HZLOOZULWHWKH)ORU\+XJJLQVH[SUHVVLRQDV
G H[
Fxxm
F x mx II x mx
RT UHV
a
f
G H[
RT FRPE
N OQ
G H[
RT UHV
N G H[
RT UHV
I
N OQ
x
I
x
x
I
x mx
ZLWKI N mN H[
H[
G
RT FRPE
a
F I I I FG
w N GH RT
H[
w
I
I
x
N OQ
Fm I I I f xmI
I
x
FRPE
a
f FI a I f FI
I OQ I N w OQ I N w OQ I
JK x w N x w N x
x mx
x
N OQ
N
N mN DQG
w
w N
I
x
w
w N
OQ
LM
N
I
x
m
x mx
OQ
I
x
OP
fQ
N mN N
w
I
N
OQ N
N mN w N x
N mN a
x I Nm
N mN LM m OP N w OQ I
N Nm N mN Q w N x
x I
m
7KHUHIRUH
OQ J FG I w FG I
JK w N GH RT JK
w N GH RT
I
I
OQ a x I f F x I FI
H mK
x
H[
w
OQ
UHV
H[
F G I FN m FN N m
N mN
w N GH RT JK N mN
w
UHV
6LPLODUO\
FN N m
F G I FN m FN N m
w N GH RT JK N mN a N mN f
w
mx
x mx
$OVR
I
x
H[
FRPE
I I
m
xI UHV
OQ
I
x
F
H
I I
K
FI m
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
F G I w L N OQ I N OQ I O
JK w N MN x
x PQ
w N GH RT
H[
w
FRQI
OQ
I
w
x w I
N
N x
I w N x I w N x
x
I
x mI OQ J OQ
I
OQ
I
OQ
I
x
x
x
a
x I f a
f
x mI x I FI I mI xI OQ
I
x
m I FI m I FI DOVRDYDLODEOHDVD0DWKFDGZRUNVKHHW L [L L
IPDL [L H[S [L
IP
[L H[S [L L
[
IPD
+PD H[S +P H[S IP
+PD
+P Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
6KRZ A(26 P o f
VWDUW
P
C
LM a
Nb
PL[
PL[
¦ xi
OP
Q
ai
bi
wA
o A V A V of
wV T
z
V
PdV f
V o f LVFRQYHQLHQWVLQFHZHKDYHLGHDOJDVDQGLGHDOJDVPL[WXUHV 35
RT
a
P
m SXUHFRPSRQHQW V b V V b b V b
z
z
V
A V A V of
V
RT
a
dV
V
b
V
V
b
b V b
f
f
RT OQ V b OQ f a
dV
V bV b
f
z
V
1HHGWRLQWHJUDWH
)URP3UREOHPZHKDYHWKDW
V
V b
OQ
dV
b V b
f V bV b
Ÿ A V A V of
F d
GH d
z
RT OQ V b OQ f 6RIRUSXUHFRPSRQHQW
i IJ
iK
F d
GH d
i IJ
iK
V b
a
OQ
b V b
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
p ab OQFG VV bb dd iiIJ H
K
%\H[DFWDQDORJ\IRUPL[WXUH
F V b d iI a
RT kOQaV b f OQ fp A A
OQG
b
H V b d iJK
ka
f
RT OQ V bi OQ f Ai V Ai V of
i
i
i
V
PL[
PL[
PL[
V of
i
PL[
PL[
PL[
RT ^OQ V b OQ f` A V A ,*0
RT OQ V PL[ bPL[ OQ f
ka
DQG
H[
A,*0
AH[
AH[
E\
F
GH
f
p
AH[
dA A i A A
,*0
,0
,*0
iIJ
iK
d
d
iIJ
iK
A A,0 LH
F
GH
V PL[ bPL[ aPL[
OQ
bPL[
V PL[ bPL[ GHILQLWLRQ
d
d
V b a
OQ
b V b A V A ,*
A
PL[
1RZZKHQ V o f JHWLGHDOJDVRULGHDOJDVPL[WXUH
PL[
A,0 AH[ A
VR
DQG
¦ xi Ai,* RT ¦ xi OQ xi A,0 ¦ xi Ai RT ¦ xi OQ xi A A,*0 ¦ xi Ai RT ¦ xi OQ xi ¦ xi Ai,* RT ¦ xi OQ xi
d A A,*0 i ¦ xi d Ai Ai,* i
WKLVLVZK\ZHIRXQG' AIRUSXUHFRPSRQHQW
6RSXWWLQJLQUHVXOWVIRU A V A f
z
V
PdV IRUSXUHiDQGPL[WXUH
f
iIJ
iK R|
F V b d iI U|
a
OQG
¦ x S RT OQaV b f RT OQ f JV
b H V b d i K |W
|T
1HHGWRFROOHFWVRPHWHUPV
O
L
RT MOQaV
b f OQ f ¦ x OQaV b f OQ f P
A
Q
N
F
F
b d i I
V b d i I
V
a
a
OQG
x
OQ
b
H V b d iJK ¦ b GH V b d iJK
AH[
a
f
RT OQ V PL[ bPL[ OQ f F
GH
d
d
V PL[ bPL[ aPL[
OQ
bPL[
V PL[ bPL[ i
i
i
i
H[
PL[
PL[
i
i
i
i
i
i
i
PL[
PL[
PL[
i
i
i
PL[
PL[
i
i
i
i
i
i
PL[
i
1RZOHW P o f ZKLFKLVWKHVDPHDV V i o bi DQG V PL[ o bPL[ a
f
a
f
RT>OQ bPL[ bPL[ OQ f ¦ xi >OQ bi bi OQ f@
Fc
GH c
h I
h JK
Fc
GH c
hI
h JK bPL[
bi
ai
aPL[
¦ xi
OQ
OQ
bPL[
bPL[
bi
bi
1RZ DQG FDQFHO OQ ¦ xi OQ DQG DQG FDQFHO OQ f ¦ xi OQ f
6R
LH f ˜ f
KDYH
A A,*0 VR
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
FG
H
IJ
K
FG
H
ai
aPL[
¦ xi
OQ
OQ
bPL[
bi
AH[
P R
P of
˜
FG IJ ˜ L a ¦ x a O
b PQ
H K MN b
L a ¦ x a OP
A
M
bQ
Nb
PL[
OQ
i
i
PL[
H[
P R
P of
PL[
PL[
FG IJ C IRUP R
H K IJ
K
i
i
i
i
ZKHUH OQ
1RZIRUYDQGHU:DDOVŸVDPHSURFHVVWKRXJKVROXWLRQLVEULHIHU
V RT
V a
wA
P
Ÿ A V A V of dV dV f
f
V b
wV T
V
3XUHFRPSRQHQW
a
,*
AV A
RT OQ V i bi RT OQ f i
Vi
z
a
z
f
DQGIRUWKHPL[WXUH
AV A
a
,*0
f
a0
V0
RT OQ V 0 b0 RT OQ f d A A i ¦ x d A A i m VDPHDVDERYH
DQG AH[ { A A,0
,*0
,*
i
i
z
V
PdV IRUSXUHiDQGIRUWKHPL[WXUH
3XWWLQJLQUHVXOWVIRU A V A f
AH[
a
f
f
RT OQ V M bM RT OQ f RS
T
a
aM
VM
f
¦ xi RT OQ V i bi RT OQ f a
a
f
ai
Vi
UV
W
f
RT OQ V M bM ¦ xi OQ V i bi OQ f ¦ xi OQ f 7DNHOLPLW P o f V i o bi V M o bM )LUVWWHUPVFDQFHO
a
a
AH[
M ¦ xi i 9':
b
bi
P of
M
OP
Q
LM
N
6WDUWLQJIURP
aij
a
bM M ¦ ¦ xi x j bij { QDQG
RT
RT
i j
FG
H
G
ex
RU
L a ¦ x a OP
C M
bQ
Nb
IJ
K
i
M
i
M
i
i
a M G ex
a
¦ xi i { DRTDQGbM
bM
bi
C
i
6XEVWLWXWLQJZHWKHQREWDLQ
aM DRT
a
aM
¦ xi i
Vi
VM
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
aM
a
M
DRT RT
DQG
QVRWKDW
aM
Q
RT
bM
Q
aM
RT
D
Q
D
Q
D
D
Q
D
(TXDWLRQ LVHDVLO\GHULYHGLVJHQHULFDQGDSSOLHVWRDQ\PL[LQJUXOH7KLVZLOOEHXVHGDV
WKHVWDUWLQJSRLQW:LWKWKH:RQJ6DQGOHUPL[LQJUXOH
1RWHWKDWGHULYDWLYHVPXVWEHWDNHQZLWKUHVSHFWWRPROHQXPEHUV
7KHUHIRUH
§
aij ·
¦¦ x x ¨© b RT ¸¹QHHGVWREHLQWKHIRUPRI
Q
i j
i
ij
j
§
aij ·
¦¦ N N ¨© b RT ¸¹
N Q
i
i
j
ij
j
6LPLODUO\
i
ai
G ex
QHHGVWREHLQWKHIRUPRI
bi RT C RT
¦
Ni
¦
D
ND
xi
i
ai
NG ex
bi RT C RT
)URPWKLVVWDUWLQJSRLQWHTQV DQG DUHHDVLO\GHULYHG
6WDUWLQJIURPHTQ OQ
RT
fk
xk P
z
V ZRT P
V f
LM RT N F wP I
MN V GH wN JK
k
T V N j z k
OPdV OQ Z PQ
7KH6RDYH5HGOLFK.ZRQJHTXDWLRQRIVWDWHLV
RT
aT
NRT
N a T
P
V b V V b V Nb V V Nb
ZLWK
Nb ¦ Nibi DQGN a ¦ ¦ Ni N j aij i
i
j
1RZWDNLQJWKHGHULYDWLYHZHREWDLQ
¦ xi aik
RT
RTbk
wP
abk
i
N
V b V b
wN k T V N
V V b V V b FG IJ
H K
jzk
FG IJ
H K
wP
RT
N
wN k
V
DQGWKHQ
T V N j z k
¦ xi aik
RT
RT
RTbk
abk
i
V V b V b
V V b V V b Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
OPdV OQ Z
PQ
LM
R| ¦ x a ab U| ZRT P OP ZRT P
RTb
RT OQ
S
V OQ
ZRT P b ZRT P b |
b
M
b | ZRT P b P
M
PP OQ Z
W
T
RT M
MM ab
PP
N b ZRT P b
Q
OQ
fk
xk P
RT
z LMMN
V ZRT P
V f
FG IJ
H K
wP
RT
N
wN k
V
T V N j z k
i ik
k
k
i
k
1RZXVLQJB=Pb/RTDQGA = Pa/(RT)ZHREWDLQ
f
OQ k
xk P
Z
Bk
A
OQ
Z B Z B B
R| ¦ x A B U| Z A B
S| A B V| OQ Z B B Z B OQ Z W
T
i ik
i
k
k
+RZHYHUWKH6RDYH5HGOLFK.ZRQJHTXDWLRQRIVWDWHFDQEHUHZULWWHQDVIROORZV
RT
a
P
V b V V b
V
a
Z
A
PV
Z
V b RT V b
ZB ZB
RT
B
A
Z ZB ZB
8VLQJWKLVH[SUHVVLRQLQWKHQGDQGWKWHUPVRQWKHULJKWKDQGVLGHRIWKHIXJDFLW\H[SUHVVLRQ
\LHOGVWKHGHVLUHGUHVXOW
¦ xi Aik
fk
Z
A
B
Z
B
i
OQ
OQ Z B
k OQ
k Z xk P
Z B B
A
B
Z B B
R|
S|
T
U|
V|
W
1RWHWKDWLQWKLVGHULYDWLRQZHKDYHXVHGWKHIROORZLQJ
dx
x
OQ
x x b
b
x b
dx
x
OQ
x b
b x b
b
x x b
z
z
z
F
H
dx
x b
OQ x b
F
H
I
K
I
K
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
L [ L
˜
L
936 97 P
936
[ ˜ 97
L
SKL7 L
SKL36 SKL7
L
[ ˜ 97 [ ˜ 936
L
L
97˜ H[S«ª ¨§ ¬©
J7L 7 97
·
º
¸ ˜ SKL36L F ˜ SKL36L »
¼
P¹
J36L [ ˜ 97 [ ˜ 936
L
L
*UDSKRIDFWLYLW\FRHIILFLHQWVYHUVXVPROHIUDFWLRQ
F L
936˜ H[Sª P ˜ SKL7 F ˜ SKL7
¬
º
¼
L
L
*UDSKRIDFWLYLW\FRHIILFLHQWVYHUVXVYROXPHIUDFWLRQR
WROXHQH
J7 L
J7 L
J36L
L
[ ˜ 97 [ ˜ 936
J36L
L
[L
SKL7 L
*UDSKRIDFWLYLW\FRHIILFLHQWVYHUVXVORJRIWKHYROXPH
IUDFWLRQRIWROXHQH
J7 L
J36L
L
ZW7 ORJ SKL7 L
˜ [
L
˜ [ ˜ [
L
L
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
*UDSKRIDFWLYLW\FRHIILFLHQWVYHUVXVZHLJKW
IUDFWLRQRIWROXHQH
*UDSKRIDFWLYLW\FRHIILFLHQWVYHUVXVORJRIWKHZHLJKW
IUDFWLRQRIWROXHQH
J7 L
J36L
J7 L
J36L
ZW7 L
ORJ ZW7 L
7KHKHDWRIPL[LQJLVFRPSXWHGDVIROORZV
ª ˜ ˜ P˜ [ ˜ [ º
L
L ¼
¬
+ L
[ P˜ [
L
L
+L
+L
[L
SKL7 L
+L
ZW7 L
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
F &OHDUO\
a
FRT x mx II Ax xRm
a
f
H H[
fa
Ax RT
x
x mx
T
x mx
fa
mx
x mx
f
x mx
G H[
RT
I
x OQ
x
x OQ
I
x
a
f
Ax
x mx I I T
F wF N Ax a x mx fI I I I
KJ
I
GH T
F
OQ J
OQ I G
JJ
w
x H mK
N
GH
K
F wF N Ax ax mx fI I I I
KJ
A w L Nx a x mx f x mx O A w L
N N mN
OP
GG H T
P
M
M
J
wN
T wN N
ax mx f Q T wN N a N mN fa N N f Q
GH
JK
OP
AL
N mN
N N mN
N N mN
M
T N a N mN fa N N f a N mN f a N N f a N mN fa N N f Q
IJ
K
FG
H
wN G H[
RT wN T P N
I
T P N T P N I
K
F
H
OQ J FG
H
IJ
K
wN G H[
RT wN T P N
AxI I x
T
F wF N Ax a x mx fI I I I
KJ
GH T
m I G
OQ
JJ
wN
x
GH
K
I
AxI I x
T
A
xI xI I xxI T
Ax I
I
OQ I I x
T
x
m
OQ J T P N
F wF N Ax ax mx fI I I I
OP
KJ
A w L Nx a x mx f x mx O A w L
N Nm
GG H T
M
P
M
J
T wN N
wN
ax mx f Q T wN N a N mN fa N N f Q
GH
JK
OP AL
N Nm
N Nm
M N mN
T N a N mN fa N N f a N mN f a N N f a N mN fa N N f Q
T P N
A
xI xI xxI T
I
Ax I
OQ m I I x
x
T
OQ J AxI I x
T
AxI I x
T
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
3DUWVGDQGH
:LWKQHZDFWLYLW\FRHIILFLHQWPRGHO
ª
97˜ H[S« §¨ ¬©
J7L ·
¸ ˜ SKL36L P¹
ª¬˜ [L ˜ SKL36Lº¼ ˜ SKL36L [L º
»
¼
[ ˜ 97 [ ˜ 936
L
L
ª
936˜ H[S« P ˜ SKL7 L
¬
J36L ª¬˜ [L ˜ SKL36Lº¼ ˜ SKL7 L [L º
»
¼
[ ˜ 97 [ ˜ 936
L
L
*UDSKRIDFWLYLW\FRHIILFLHQWVYHUVXVPROHIUDFWLRQ
*UDSKRIDFWLYLW\FRHIILFLHQWVYHUVXVYROXPHIUDFWLRQR
WROXHQH
J7 L
J7 L
J36L
J36L
[L
SKL7 L
*UDSKRIDFWLYLW\FRHIILFLHQWVYHUVXVORJRIWKHYROXPH
IUDFWLRQRIWROXHQH
7KLVLVQRWSK\VLFDOO\
FRUUHFWEHKDYLRU
J7 L
J36L
ORJ SKL7 L
*UDSKRIDFWLYLW\FRHIILFLHQWVYHUVXVZHLJKW
IUDFWLRQRIWROXHQH
*UDSKRIDFWLYLW\FRHIILFLHQWVYHUVXVORJRIWKHZHLJKW
IUDFWLRQRIWROXHQH
J7 L
J36L
J7 L
J36L
ZW7 L
ORJ ZW7 L
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
D [\
1,1
0%RQHDFKFRPSRQHQW
1
[ 1
,1
287
1
287
[ 1
,1
1
[1
,1
287
1
287
[ 1
,1
(% SHUPROHRIIHHG (QWURS\EDODQFH
1RZ 6 JHQ
ª [ + 76 [ + 76 º ª [ + 76 [ + 76 º :
¬
¼ ¬
¼
,VRPHUVFDQEHH[SHFWHGWRIRUPDQLGHDOVROXWLRQ7KHUHIRUH
57 > [ OQ [ [ OQ [ @ :
57 > [ OQ [ [ OQ [ @
RU :
1287[ 1287[ 4
[+ [ + [ + [ + :
4
6 JHQ 7
IRUPLQLPXPZRUN&RPELQLQJWKHHQHUJ\DQGHQWURS\EDODQFHV
[6 [6 [ 6 [ 6 E
:
u . > OQ OQ @ - PRO PRO ˜ .
D *LEEV'XKHPHTXDWLRQ
1 § w* ·
§ w* ·
§ w* ·
IRUDELQDU\PL[WXUH[ ¨
¦ ¨¨ L ¸¸
¸ [ ¨
¸
L w1
M ¹ 731
© w[ ¹73
© w[ ¹73
©
Nz M
D L 7 3 [
D L IL q
[L3
*L
w*
w[
[ L 3 H[S 57
* L * L,*0
IL 7 3q [ q
IL
q
`
q
­ * * L,*0 ½
D L IL q
,*0
H[S ® L
DQG
*
*
57
OQ
¾
L
L
[L3
¯ 57 ¿
R
* L 57 OQ [ L 57 OQ D L 57 OQ I 57 OQ [ L 57 OQ 3
w OQ D
57
57
w[
[
§ w* ·
§ w* ·
[ ¨ ¸ [ ¨
¸
[
w
© ¹73
© w[ ¹73
EXWDOVR
^
IL 7 3 [
w OQ D
w OQ D [
[
w[
w[
w* w[
w OQ D 57
57
w[
[
57 57 [
w OQ D
w OQ D 57 57[ w[
w[
Ÿ GOQD
[
G OQ D [
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
[
G OQ [ [
G OQ [ RU G OQ [
Ÿ
[ G[ G[ [
[ [ [
G OQ [ [
D
G OQ §¨ L ·¸
© [L ¹
E
DL
IL
q
G[
G[ [
[
VRVXEWUDFWLQJWKLVIURPHDFKVLGH
[
D
G OQ §¨ ·¸
[
© [ ¹
4('
IL
7q 3q [ q
§ w OQ D L ·
¨ w7 ¸
©
¹ 3[
w OQ IL
w7
*L *L
w ­°
®OQ [ L OQ 3 w7 °
57
¯
§*
·
§ * ,*0
·
w¨ L
w¨ L
¸
¸
57
57
©
¹ ©
¹
w7
w7
G[
G[ [
[
,*0
½
°
¾
°¿
+L
+
G
L OQ [ qL 57
57
G7
^
`
2 § w OQ D L ·
¨ w7 ¸
©
¹ 3[
+L +L
57 1
[2 ƒ&EDU
$LU
EDUƒ&
1 [2 ƒ&EDU
1RPRYLQJSDUWVŸ: 0DVVEDODQFH EDVLVPROHDLU 1
2
1
1
DLU 1
1RWHWKDWDPDVVEDODQFHRQQLWURJHQZRXOGEHUHGXQGDQWZLWKWKHWZRPDVVEDODQFHVDERYH 1
1
1
1
1
(QHUJ\EDODQFH
x+DLU±î+±+ 4
(QWURS\EDODQFH
4
x6DLU±î6±6 6JHQ 7
±6DLU±6±6± 76 4
JHQ
&RPELQHGZLWKWKHHQHUJ\EDODQFH
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
*DLU±*±*± 76 JHQ RU
76 JHQ *DLU±*±*•
1RZVLQFHSUHVVXUHLVORZDVVXPHDQLGHDOJDVPL[WXUH
76 JHQ *2*157OQ57OQ
±î*2±î57OQ
±î*1±î57OQ
±î*2±î57OQ
±î*1±î57OQ
ª
º
57 OQ «
u u u u »
¬ ¼
u ª
º
57 OQ «
»
¬ u u u ¼
57 OQ 57
6LQFH5!DQG7!6JHQ”ZKLFKLVQRWDOORZHG'HYLFHFDQQRWZRUN
$GLDEDWLFDOO\DQGUHYHUVLEO\Ÿ6,1 6287
:LOOKDYHWRGRWKLVE\WULDODQGHUURUXVLQJ35(26SURJUDP
D $W.DQGEDU
+ -PRO
6 ±-PRO
3I EDU 7 . 6 ±-PRO.
+ -PRO
(QHUJ\EDODQFH
:
+ 1
+ :
1
+ + 1
- PRO
E DQGF 1HHG(QWKDOS\ DQG(QWURS\ FDOFXODWLRQVLQ3HQJ5RELQVRQPL[WXUHSURJUDP
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
)LOHLV+&O
L , L ˜ L
GHO ª ª ˜ ˜ , º º
«¬
L
¼»
'+ H[S«
»
L
¬
¼
ª ª ˜ ˜ , º º
«¬
L
¼»
('+ H[S«
»
L
«
,
»
L
¬
¼
º»
«ª ª¬ ˜ ˜ ,L º¼
(('+ H[S«
GHO ˜ , »
L
L
«
,
»
L
¬
¼
§ ·
¨ ¸
¨
¸
¨ ¸
¨ ¸
¨
¸
¨ ¸
0 ¨ ¸
¨ ¸
¨
¸
¨ ¸
¨ ¸
¨ ¸
¨
¸
© ¹
<(
'+
('+
(('+
0,
§ ·
¨ ¸
¨
¸
¨ ¸
¨ ¸
¨
¸
¨ ¸
<( ¨ ¸
¨ ¸
¨
¸
¨ ¸
¨ ¸
¨ ¸
¨
¸
© ¹
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
)LOHLV&D&O
, L ˜ L L
GHO ª« ª ˜ ˜ , º º»
L
¬
¼
('+ H[S«
»
L
«
,
»
L
¬
¼
º
º
ªª
« ¬ ˜ ˜ ,L ¼
»
(('+ H[S«
GHO ˜ , »
L
L
«
,
»
L
¬
¼
ª« ª ˜ ˜ , º º»
L
¬
¼
'+ H[S«
»
L
¬
¼
§ ·
¨ ¸
¨
¸
¨ ¸
¨ ¸
¨ ¸
¨
¸
0 ¨ ¸
¨ ¸
¨
¸
¨ ¸
¨ ¸
¨ ¸
¨
¸
© ¹
§ ·
¨ ¸
¨
¸
¨ ¸
¨ ¸
¨ ¸
¨
¸
<( ¨ ¸
¨ ¸
¨
¸
¨ ¸
¨ ¸
¨ ¸
¨
¸
© ¹
§ ·
¨ ¸
¨
¸
¨ ¸
¨ ¸
¨
¸
¨ ¸
,6 ¨ ¸
¨ ¸
¨
¸
¨ ¸
¨ ¸
¨ ¸
¨
¸
© ¹
<(
'+
('+
(('+
,6 ,
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
)LOHLV=Q62
, L ˜ L L
GHO ª« ª ˜ ˜ , º º»
L
¬
¼
('+ H[S«
»
L
«
,
»
L
¬
¼
º
º
ªª
« ¬ ˜ ˜ ,L ¼
»
(('+ H[S«
GHO ˜ , »
L
L
«
,
»
L
¬
¼
ª« ª ˜ ˜ , º º»
L
¬
¼
'+ H[S«
»
L
¬
¼
§ ·
¨ ¸
¨
¸
¨ ¸
¨ ¸
¨ ¸
¨
¸
0 ¨ ¸
¨ ¸
¨
¸
¨ ¸
¨ ¸
¨ ¸
¨
¸
© ¹
§ ·
¨ ¸
¨
¸
¨ ¸
¨ ¸
¨ ¸
¨
¸
<( ¨ ¸
¨ ¸
¨
¸
¨ ¸
¨ ¸
¨ ¸
¨
¸
© ¹
§ ·
¨ ¸
¨
¸
¨ ¸
¨ ¸
¨
¸
¨ ¸
,6 ¨ ¸
¨ ¸
¨
¸
¨ ¸
¨ ¸
¨ ¸
¨
¸
© ¹
<(
'+
('+
(('+
,6 ,
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
:DWHULQ+&O
·
3: H[S§¨ ¸
©
3: u 3:
3:
§ ·
¨
¸
¹
3 ©
Z
PK Z 3
PZ Z
PZ Z ˜ PK Z
[KR
0 PK Z ˜
3
JDP Z
PZ Z [KR JDP
[KR˜ 3:
Z
0
0
ZHLJKWSHUFHQW
§ ·
¨
¸
¹
3 ©
PK Z Z
PZ Z ˜ PK Z
Z
PK Z
[KR 0 PK Z ˜
Z
ZHLJKWSHUFHQW
§ ·
¨
¸
¹
3 ©
PK Z JDP
[KR˜ 3:
Z
[KR
3
JDP Z 3
PZ Z PZ Z
[KR E
3:
ZHLJKWSHUFHQW
Z Z ¹
3
PZ Z Z
PZ Z
PZ Z ˜ PK Z
0 PK Z ˜
[KR
Z
0
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
3
JDP JDP
[KR˜ 3:
ZHLJKWSHUFHQW
Z ·
§
¨ ¸
¹
3 ©
Z
PK Z PZ Z PZ Z
[KR u Z
[KR
PZ Z ˜ PK Z
3
0 PK Z ˜
3
JDP [KR˜ 3:
JDP
0
Z
&RPSDUHZLWKSUHGLFWLRQVEDVHGRQ3UREOHPD
]S ]P , 0 ,
VLJ \
PV \ EHW ˜ ,
GHO DOI § · ˜ ª \ ˜ OQ \ § ·º
¨ \ ¸»
¨ ¸ «¬
©
¹¼
©\ ¹
VLJ \ VLJ \
] PV˜ QX ˜ 0 ˜ ª« DOI˜ ]S˜ ]P˜ ,
¬
]
[Z 0 ,
VLJ \
§¨ GHO˜
©
, ·º
¸»
¹¼
D H[S ]
˜ 0
0
, QX EHW \ EHW ˜ ,
JDP [Z
VLJ \ D
[Z
JDP
§ · ˜ ª \ ˜ OQ \ § ·º
¨ \ ¸»
¨ ¸ «¬
©
¹¼
©\ ¹
VLJ \
] PV˜ QX ˜ 0 ˜ ª« DOI˜ ]S˜ ]P˜ ,
¬
D
§¨ GHO˜
©
, ·º
¸»
¹¼
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
]
D H[S ]
[Z 0
˜ 0
, \ EHW ˜ ,
0 ,
VLJ \
JDP [Z
D
[Z
JDP
VLJ \
] PV˜ QX ˜ 0 ˜ ª« DOI˜ ]S˜ ]P˜ ,
§¨ GHO˜
˜ 0
[Z
, 0 ,
JDP \ EHW ˜ ,
0
JDP
D
[Z
VLJ \ D
[Z § · ˜ ª \ ˜ OQ \ § ·º
¨ \ ¸»
¨ ¸ «¬
©
¹¼
©\ ¹
VLJ \
] PV˜ QX ˜ 0 ˜ ª« DOI˜ ]S˜ ]P˜ ,
¬
0
¸»
©
[Z ]
, ·º
¹¼
D H[S ]
VLJ \
D
§ · ˜ ª \ ˜ OQ \ § ·º
¨ \ ¸»
¨ ¸ «¬
©
¹¼
©\ ¹
VLJ \ ¬
]
§¨ GHO˜
©
, ·º
¸»
¹¼
D H[S ]
˜ 0
JDP [Z
D
[Z
JDP
D
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
D
4
¦ [ L 4L 7 3 ¦ ¦ D LM [ L D [ [ [ L 1 1 1
1 1 1 D D D D D 1 1 1 1 1 1 1 1 1 1 1 1 14
¦ [ L 4L D
L D D 1 1
1 1 1 1 D D
1 1 1 1 1 1 1 1 1 w14
w1 731 1
4
4 D [ D [ D D [ D D [ [ D D [ D D [ [ D [ D D [ [ D [ D [ [ D [ [ [ 4 D [ [ D D [ [ D D [ [ D [ D D [ [ D [ D [ [ [
ĬREWDLQHGE\LQWHUFKDQJLQJLQGLFHVDQG
ĬREWDLQHGE\LQWHUFKDQJLQJLQGLFHVDQG
E 4f 4 [ o 4 D D [ D D [ D [ D D [ [ D [ D [ [ ,QILQLWHGLOXWLRQKHDWRIVROXWLRQ ±FDJUDP
'P +
/
'I + 'I +
9L
N&DO PRO
N &DO PRO
0:XUHD
&DO PRO
3
57
D
9E 9 9E E 9E
157
D1 9 1E 9 9 1E 1E 9 1E
157
D1 9 1E 9 1E9 E 1 D E
6WDUWLQJIURP
6
§ w9 ·
¨
¸
© w1L ¹731 MzL
§ w3 ·
¨
¸
© w1L ¹791 MzL
§ w3 ·
¨
¸
© w9 ¹71 M
E\WKHWULSOHSURGXFWUXOH ,QWDNLQJGHULYDWLYHVQHHGWRUHPHPEHUWKDW
1E 1E 1 E DQG1 D D1 D 11 D 1 Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
§ w3 ·
¨
¸
© w1 ¹791 § wD ·
§ wE ·
¨
¨
¸
¸
© w1L ¹791 © w1L ¹791
§ wD ·
¨
¸
© w1 ¹791 157 E
57
9 1E
9 1E
§ wE ·
¨
¸
© w1 ¹791 1 DE 9 1E
1D 1 D
9 E19 E 1 9 E19 E 1 DQG
§ w3 ·
¨
¸
© w9 ¹71 M
§ wD ·
¨
¸
© w9 ¹71 M
§ wE ·
¨
¸
© w9 ¹71 M
57 91 E E
9 1E
§ wD ·
§ wE ·
¨
¨
¸
¸
© w9 ¹71 M © w9 ¹71 M
157
9 1E
D1 ˜ 9 E1
9 E19 E 1 1RZVXEVWLWXWLQJHTXDWLRQVWR EDFNLQWRHTQ JLYHVWKHGHVLUHGUHVXOW
7KH SURFHGXUH LV WKH VDPH DV LQ 3UREOHP H[FHSW WKDW WKH GHULYDWLYHV DUH PXFK PRUH
FRPSOLFDWHG
7KHVWDUWLQJSRLQWLIWKHGHILQLWLRQRIWKHDFWLYLW\FRHIILFLHQWLQWHUPVRIWKHSXUHFRPSRQHQWDQG
PL[WXUHIXJDFLWLHV
IL [7 3
J L [7 3
ZKHUH
[ L IL 7 3
IL [7 3
ª %L
º
« % [ = [7 3 OQ = [7 3 % [
»
«
»
«
»
­
½
[
$
[ L 3 H[S «
M LM
»
°
°
% ° °­ % [ °½»
« $ [ ®° M
L ¾ OQ ®
¾ «
% [ ° °¯ % [ °¿»
% [ ° $ [
«
»
¯°
¿°
¬
¼
¦
ª
­° %L ½°º
$L
3 H[S « = 7 3 OQ = 7 3 %L OQ ®
¾»
%L ¯° %L ¿°¼»
¬«
:KHUH$DQG%DUHWKHGLPHQVLRQOHVVIRUPVRIDDQGEDQGDUHREWDLQHGIURPWKHYDQGHU:DDOVRQHIOXLGPL[LQJUXOHV1
WKDWIRUWKHSXUHFRPSRQHQWIXJDFLW\RIVSHFLHVLWKHFRPSUHVVLELOLW\WKDWLVXVHGLVWKDWREWDLQHGIURPWKHHTXDWLRQRI
ZLWKWKHSXUHFRPSRQHQWSDUDPHWHUVZKLOHLQWKHPL[WXUHFDVHWKHDDQGE RU$DQG% SDUDPHWHUVDUHIRUWKHPL[WXUH
GHSHQGRQFRPSRVLWLRQ
IL 7 3
6LPLODUWR3UREOHPH[FHSWWKDWWKHDDQGE RU$DQG% SDUDPHWHUVDUHREWDLQHGXVLQJWKH
:RQJ6DQGOHUPL[LQJUXOHV
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
(T LVOQ J 7 J 7
+H[5 77 7KHUHIRUHZHKDYHWKHIROORZLQJ
HTXDWLRQ+H[ 5 OQ J 7 J 7 77 ZKLFKLVWKHHTXDWLRQZHZLOOXVH
OQ§¨
+ ˜
¸
© ¹
+ ˜
·
·
¸
© ¹
u +
u OQ§¨
+
8VHDYHUDJHIRUWKHYDOXHDWWKHPLGUDQJHWHPSHUDWXUHRI&
+H[ + +
+H[
u 8VLQJWKLVDYHUDJHYDOXHWRH[WUDSRODWHWR&
·º
ª +H[˜ §
¨
¸
«
¹ »
©
J H[S«
»
¬
¼
J
ZKLFKLVWKHH[WUDSRODWHGYDOXHWR&
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
7KHUHJXODUVROXWLRQPRGHOLV57OQJ 9 I A GG ADQG57OQJ 9 I A GG A
ZLWKI L [L 9L9PL[LQZKLFK9PL[ [ 9[ 9
7KHLQILQLWHGLOXWLRQDFWLYLW\FRHIILFLHQWVWKHQDUH
J LQI H[S 9 GG A57 DQGJ LQI H[S 9 GG A57
7KHUHJXODUVROXWLRQSDUDPHWHUVDW&DUHJLYHQLQWKHWH[WERRNDVIROORZV
EHQ]HQH9 FPAPRODQG G FDOPRO A
7039 FPAPRODQG G FDOPRO A
:HZLOODVVXPHWKHVHSDUDPHWHUVFDQEHXVHGDW&DOVR
ª
JLQI H[S« ˜
¬
ª
º
»
˜ ¼
JLQI
º
»
˜ ¼
JLQI
JLQI H[S« ˜
¬
,WLVDOVRRILQWHUHVWWRFRPSXWHWKHDFWLYLW\FRHIILFLHQWVRYHUWKHZKROH
FRQFHQWUDWLRQUDQJH
L IL [ ˜
[ ˜ L
L
9 [ ˜ [ ˜ L
L
IL IL
L 9
L
ª ˜ I ˜ º
L
»
JL H[S«
¬ ˜ ¼
ª ˜ I ˜ º
L
»
˜
¼
¬
JL H[S«
J
J
L
[
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
7KLVLVDPRUHFRPSOHWHVROXWLRQWKDQDVNHGIRULQWKHSUREOHPVWDWHPHQWEXWWKLVDOVRFRQWDLQVWKH
VROXWLRQWR3UREOHP7KHVHUHVXOWVDUHIURPWKHPRGLILHG81,)$&PRGHODQGZHUHFRPSXWHGZLWK
WKH9LVXDO%DVLF0RGLILHG81,)$&SURJUDP
-------------COMPONENT DATA
-------------Molecular Subgroups
---------------------------------------------------------------------Comp
Name
Number
Description
Frequency
---- -------------------------------- -----------------------1
benzene
10
ACH
6
2
2,2,4-trimethylpentane
1
2
3
4
CH3
CH2
CH
C
EHQ]HQHDQGWULPHWK\OSHQWDQHDW.
[RIEHQ]HQH
EHQ]HQH
WULPHWK\OSHQWDQH
---------------------------------------------------x(1)
GAM(1)
GAM(2)
GEX(J/MOL)
---------------------------------------------------0.0000
1.3634
1.0000
0.00
0.0250
1.3522
1.0001
20.86
0.0500
1.3411
1.0004
41.14
0.0750
1.3299
1.0010
60.83
0.1000
1.3186
1.0018
79.89
0.1250
1.3074
1.0029
98.30
0.1500
1.2961
1.0043
116.04
0.1750
1.2848
1.0060
133.09
0.2000
1.2735
1.0080
149.41
0.2250
1.2622
1.0105
164.97
0.2500
1.2510
1.0133
179.74
0.2750
1.2397
1.0166
193.70
0.3000
1.2285
1.0203
206.80
0.3250
1.2173
1.0246
219.01
0.3500
1.2061
1.0294
230.30
0.3750
1.1950
1.0348
240.62
0.4000
1.1840
1.0409
249.93
0.4250
1.1731
1.0477
258.18
0.4500
1.1623
1.0553
265.33
0.4750
1.1515
1.0637
271.34
0.5000
1.1409
1.0731
276.14
0.5250
1.1305
1.0836
279.67
0.5500
1.1202
1.0952
281.90
0.5750
1.1101
1.1080
282.73
0.6000
1.1002
1.1222
282.13
0.6250
1.0905
1.1380
280.00
0.6500
1.0811
1.1555
276.28
5
1
1
1
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
0.6750
0.7000
0.7250
0.7500
0.7750
0.8000
0.8250
0.8500
0.8750
0.9000
0.9250
0.9500
0.9750
1.0000
1.0719
1.0631
1.0547
1.0466
1.0389
1.0318
1.0252
1.0191
1.0138
1.0091
1.0053
1.0025
1.0006
1.0000
1.1749
1.1965
1.2205
1.2471
1.2768
1.3100
1.3471
1.3888
1.4356
1.4884
1.5482
1.6161
1.6936
1.7825
6RWKHLQILQLWHGLOXWLRQDFWLYLW\FRHIILFLHQWVDUH J f%
270.89
263.74
254.73
243.79
230.78
215.62
198.17
178.32
155.91
130.81
102.85
71.85
37.64
0.00
f
DQGJ 703
EHQ]HQHDQGWULPHWK\OSHQWDQHDW.
[RIEHQ]HQH
---------------------------------------------------x(1)
GAM(1)
GAM(2)
GEX(J/MOL)
---------------------------------------------------0.0000
1.3634
1.0000
0.00
0.0250
1.3522
1.0001
20.86
0.0500
1.3411
1.0004
41.14
0.0750
1.3299
1.0010
60.83
0.1000
1.3186
1.0018
79.89
0.1250
1.3074
1.0029
98.30
0.1500
1.2961
1.0043
116.04
0.1750
1.2848
1.0060
133.09
0.2000
1.2735
1.0080
149.41
0.2250
1.2622
1.0105
164.97
0.2500
1.2510
1.0133
179.74
0.2750
1.2397
1.0166
193.70
0.3000
1.2285
1.0203
206.80
0.3250
1.2173
1.0246
219.01
0.3500
1.2061
1.0294
230.30
0.3750
1.1950
1.0348
240.62
0.4000
1.1840
1.0409
249.93
0.4250
1.1731
1.0477
258.18
0.4500
1.1623
1.0553
265.33
0.4750
1.1515
1.0637
271.34
0.5000
1.1409
1.0731
276.14
0.5250
1.1305
1.0836
279.67
0.5500
1.1202
1.0952
281.90
0.5750
1.1101
1.1080
282.73
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
0.6000
0.6250
0.6500
0.6750
0.7000
0.7250
0.7500
0.7750
0.8000
0.8250
0.8500
0.8750
0.9000
0.9250
0.9500
0.9750
1.0000
1.1002
1.0905
1.0811
1.0719
1.0631
1.0547
1.0466
1.0389
1.0318
1.0252
1.0191
1.0138
1.0091
1.0053
1.0025
1.0006
1.0000
1.1222
1.1380
1.1555
1.1749
1.1965
1.2205
1.2471
1.2768
1.3100
1.3471
1.3888
1.4356
1.4884
1.5482
1.6161
1.6936
1.7825
6RWKHLQILQLWHGLOXWLRQDFWLYLW\FRHIILFLHQWVDUH J f%
282.13
280.00
276.28
270.89
263.74
254.73
243.79
230.78
215.62
198.17
178.32
155.91
130.81
102.85
71.85
37.64
0.00
DQGJ f703
EHQ]HQHDQGWULPHWK\OSHQWDQHDW.
[\RIEHQ]HQH
3[
x(1)
0.0000
0.0250
0.0500
0.0750
0.1000
0.1250
0.1500
0.1750
0.2000
0.2250
0.2500
0.2750
0.3000
0.3250
0.3500
0.3750
0.4000
0.4250
0.4500
0.4750
y(1)
0.0000
0.0597
0.1144
0.1647
0.2112
0.2542
0.2942
0.3315
0.3664
0.3990
0.4297
0.4585
0.4858
0.5115
0.5360
0.5592
0.5813
0.6024
0.6226
0.6420
3\
y(2)
1.0000
0.9403
0.8856
0.8353
0.7888
0.7458
0.7058
0.6685
0.6336
0.6010
0.5703
0.5415
0.5142
0.4885
0.4640
0.4408
0.4187
0.3976
0.3774
0.3580
P(bar)
0.238
0.246
0.255
0.263
0.272
0.280
0.287
0.295
0.302
0.310
0.317
0.323
0.330
0.336
0.343
0.349
0.354
0.360
0.366
0.371
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
0.5000
0.5250
0.5500
0.5750
0.6000
0.6250
0.6500
0.6750
0.7000
0.7250
0.7500
0.7750
0.8000
0.8250
0.8500
0.8750
0.9000
0.9250
0.9500
0.9750
1.0000
0.6606
0.6785
0.6959
0.7128
0.7291
0.7451
0.7608
0.7762
0.7915
0.8066
0.8217
0.8369
0.8522
0.8679
0.8839
0.9005
0.9178
0.9361
0.9557
0.9768
1.0000
0.3394
0.3215
0.3041
0.2872
0.2709
0.2549
0.2392
0.2238
0.2085
0.1934
0.1783
0.1631
0.1478
0.1321
0.1161
0.0995
0.0822
0.0639
0.0443
0.0232
0.0000
0.376
0.381
0.385
0.390
0.394
0.398
0.402
0.406
0.409
0.412
0.416
0.419
0.421
0.424
0.426
0.429
0.431
0.432
0.434
0.435
0.435
7KLVLVDPRUHFRPSOHWHVROXWLRQWKDQDVNHGIRULQWKHSUREOHPVWDWHPHQWEXWWKLVDOVRFRQWDLQVWKH
VROXWLRQWR3UREOHP7KHVHUHVXOWVDUHIURPWKHPRGLILHG81,)$&PRGHODQGZHUHFRPSXWHGZLWK
WKH9LVXDO%DVLF0RGLILHG81,)$&SURJUDP
-------------COMPONENT DATA
-------------Molecular Subgroups
---------------------------------------------------------------------Comp
Name
Number
Description
Frequency
---- -------------------------------- -----------------------1
acetone
1
CH3
1
21
CH3CO
1
2
water
19
H2O
1
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
DFHWRQHDQGZDWHUDW.
[RIDFHWRQH
DFHWRQH
ZDWHU
---------------------------------------------------x(1)
GAM(1)
GAM(2)
GEX(J/MOL)
---------------------------------------------------0.0000
8.0381
1.0000
0.00
0.0250
6.8594
1.0020
124.06
0.0500
5.9421
1.0076
238.53
0.0750
5.2142
1.0164
344.08
0.1000
4.6264
1.0281
441.26
0.1250
4.1449
1.0425
530.56
0.1500
3.7451
1.0595
612.38
0.1750
3.4095
1.0789
687.09
0.2000
3.1248
1.1009
755.01
0.2250
2.8810
1.1252
816.42
0.2500
2.6707
1.1521
871.55
0.2750
2.4878
1.1816
920.63
0.3000
2.3278
1.2137
963.85
0.3250
2.1868
1.2486
1001.36
0.3500
2.0621
1.2865
1033.32
0.3750
1.9512
1.3276
1059.84
0.4000
1.8520
1.3721
1081.05
0.4250
1.7631
1.4204
1097.02
0.4500
1.6829
1.4727
1107.85
0.4750
1.6106
1.5295
1113.57
0.5000
1.5450
1.5912
1114.26
0.5250
1.4853
1.6584
1109.93
0.5500
1.4311
1.7317
1100.61
0.5750
1.3815
1.8120
1086.30
0.6000
1.3362
1.9001
1067.00
0.6250
1.2948
1.9971
1042.68
0.6500
1.2569
2.1043
1013.31
0.6750
1.2221
2.2233
978.82
0.7000
1.1903
2.3561
939.15
0.7250
1.1613
2.5050
894.20
0.7500
1.1348
2.6729
843.87
0.7750
1.1107
2.8634
788.02
0.8000
1.0890
3.0811
726.50
0.8250
1.0695
3.3318
659.12
0.8500
1.0522
3.6230
585.66
0.8750
1.0373
3.9644
505.87
0.9000
1.0246
4.3689
419.46
0.9250
1.0143
4.8539
326.09
0.9500
1.0066
5.4429
225.38
0.9750
1.0017
6.1689
116.86
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
1.0000
1.0000
7.0786
0.00
6RWKHLQILQLWHGLOXWLRQDFWLYLW\FRHIILFLHQWVDUH J f$
DQGJ f:
DFHWRQHDQGZDWHUDW.
[RIDFHWRQH
DFHWRQHDQGZDWHUDW.
[\RIDFHWRQH
3[
3\
x(1)
0.0000
0.0250
0.0500
0.0750
0.1000
0.1250
0.1500
0.1750
0.2000
0.2250
0.2500
0.2750
0.3000
0.3250
y(1)
0.0000
0.6310
0.7514
0.8020
0.8296
0.8469
0.8587
0.8672
0.8736
0.8786
0.8827
0.8861
0.8890
0.8915
y(2)
1.0000
0.3690
0.2486
0.1980
0.1704
0.1531
0.1413
0.1328
0.1264
0.1214
0.1173
0.1139
0.1110
0.1085
P(bar)
0.031
0.083
0.121
0.149
0.171
0.187
0.200
0.211
0.219
0.226
0.232
0.236
0.240
0.244
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
0.3500
0.3750
0.4000
0.4250
0.4500
0.4750
0.5000
0.5250
0.5500
0.5750
0.6000
0.6250
0.6500
0.6750
0.7000
0.7250
0.7500
0.7750
0.8000
0.8250
0.8500
0.8750
0.9000
0.9250
0.9500
0.9750
1.0000
0.8937
0.8957
0.8976
0.8994
0.9011
0.9027
0.9044
0.9060
0.9077
0.9095
0.9113
0.9132
0.9153
0.9175
0.9199
0.9225
0.9254
0.9286
0.9323
0.9365
0.9413
0.9469
0.9536
0.9617
0.9716
0.9840
1.0000
0.1063
0.1043
0.1024
0.1006
0.0989
0.0973
0.0956
0.0940
0.0923
0.0905
0.0887
0.0868
0.0847
0.0825
0.0801
0.0775
0.0746
0.0714
0.0677
0.0635
0.0587
0.0531
0.0464
0.0383
0.0284
0.0160
0.0000
0.247
0.250
0.253
0.255
0.257
0.259
0.262
0.263
0.265
0.267
0.269
0.271
0.273
0.275
0.277
0.279
0.282
0.284
0.286
0.288
0.291
0.293
0.296
0.299
0.301
0.304
0.306
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
:HVWDUWE\QRWLQJWKDW
* H[ [ [ D[[ D[ [
D[ D[ DQGWKDW1* H[ [ [ 7KHUHIRUH
§ w1* H[ ·
¨¨
¸¸
© w1 ¹731 D1[[ D
11 1 1 ª 1
w
11
11 º
» D [ [[ D D«
w1 731 1 1 « 1 1 1 1 »¼
¬
D[ [
D[ +RZHYHU
§ w* H[ ·
¨¨
¸¸
© w[ ¹73
DQG
w
w
D[[ D[ D[
w[ 73
w[ 73
§ w* H[ ·
¨¨
¸¸
© w[ ¹73[ w
D[[ D[ w[ 73
D [ D [ § wOQJ ·
D
+RZHYHU IURP WKH *LEEV
D [ LWIROORZVWKDW ¨
¸
57
© w[ ¹73
'XKHPHTXDWLRQ (T § wOQJ ·
§ wOQJ ·
§ wOQJ ·
[ § wOQJ ·
[D
VRWKDW
[ ¨
[ ¨
¨
¸
¸ VRWKDW ¨
¸
¸
w
w
w
w
[
[
[
[
[
[
¹73
¹73
¹73
©
¹73
57
©
©
©
6LQFH 57 OQ J
D[ § wOQJ ·
[D
D
¨
Ÿ OQJ > [ OQ [ @ FRQV WDQ W ¸
w
[
[
57
57
©
¹73
&OHDUO\WKHUHLVSUREOHPZLWKWKLVH[SUHVVLRQ)RUH[DPSOHDVDUHVXOWRIWKHORJDULWKPWHUPZKHQ
[ LV XQLW\ WKH SXUH FRPSRQHQW OLPLW WKH DFWLYLW\ FRHIILFLHQW LV LQILQLWH QR XQLW\ $OVR DW DQ\
FRPSRVLWLRQRWKHUWKDQWKHSXUHFRPSRQHQWOLPLWWKHUHLVDORJDULWKPRIDQHJDWLYHQXPEHUZKLFK
LVXQDFFHSWDEOH
E·
§
)URP 57 OQ J $[ ZHLPPHGLDWHO\KDYH* H[ $[[ ¨ D ¸ [[ 7¹
©
E·
§
DQG * [* [ * [57 OQ [ [ 57 OQ [ ¨ D ¸ [[ 7¹
©
w * H[ 7
w7
3
w §D E ·
¨ ¸ [[ w7 © 7 7 ¹
§ D E ·
¨ ¸ [[ 7 ¹
©7
E ·
E ·
§
§
¨ D ¸ [[ DQG+ [+ [ + ¨ D ¸ [[ 7 ¹
7 ¹
©
©
E ·
E·
ª
º
§
§
[+ [ + ¨ D ¸ [[ [* [ * 57[ OQ [ 57[ OQ [ ¨ D ¸ [[ »
7 ¹
7¹
+ * «¬
©
©
¼
6
7
7
E
[6 [ 6 5[ OQ [ 57[ OQ [ [[ 7
)LQDOO\ 9H[ T[ Ÿ 9 H[ T[[ VRWKDW9 [9 [ 9 T[[ 6RWKDW + H[
+ H[
7
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
'G 'H T 'S
6LQFHSURFHVVLVLVRWKHUPDODQGWKHJDVHVDUHLGHDO'H
'G
T 'S
7KHUHIRUH'G
RT OQ OQ OQ OQ T 'S
RT OQ u u -PRO -PRO
7KLVLVWKHPLQLPXPDPRXQWRIZRUNQHHGHGSHUPRORIWKHLQLWLDOPL[WXUH
M
LQ
DLU
PROH2
PROH1 0DVVEDODQFH2
[ 1
± 1
[ 1
± 1
1
1
2YHUDOO $OVRVLQFHKDOIWKHR[\JHQLVWREHUHFRYHUHGLQHQULFKHGVWUHDP
1
˜ Ÿ 1
1
DQG
±x [ ± [
u [
PROH IUDFWLRQ (QHUJ\EDODQFH LGHDOJDVPL[WXUH (QWURS\EDODQFH
:
< + ,1 + + 4
"[PRO2
N
PRO2
< 6,1 6 6 4
6 JHQ
7
0XOWLSO\LQJE\7DQGVXEWUDFWLQJWKHHQWURS\EDODQFHIURPWKHHQHUJ\EDODQFH
* ,1 * * 76 JHQ :ZKHUH6
JHQ
IRUPLQLPXPZRUN
: * ,1 * * 57 OQ OQ ±57>OQOQ@
±57>OQOQ@
57>±@ ±57
: î57 -PROLQOHWDLU
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
D $WHTXLOLEULXPf i 9
fi / RUOQfi 9
OQ fi / %\WKHSUREOHPVWDWHPHQWWKHOLTXLGPL[WXUHLVLGHDO
VRfi /
xi Pi YDS DQGOQfi 9 LVJLYHQE\HTQ DV
xi Pi YDS
yi PIi9 DQGK i
yi
xi
Pi YDS
PIi9
E ,IWKHOLTXLGPL[WXUHLVQRQLGHDOWKHQfi /
xiJ i Pi YDS
yi PIi9 RUK i
yi
xi
fi 9
Ii9 7KHUHIRUH
yi P
xiJ i Pi YDS VR
J i Pi YDS
PIi9
7KH VWDUWLQJ SRLQW LI WKH GHILQLWLRQ RI WKH DFWLYLW\ FRHIILFLHQW LQ WHUPV RI WKH SXUH FRPSRQHQW DQG
PL[WXUHIXJDFLWLHV
IL [7 3
J L [7 3
ZKHUH
[ L IL 73
IL [7 3
ª %L
º
« % [ = [7 3 OQ = [7 3 % [
»
«
»
«
»
­
½
[
$
[ L 3 H[S «
M LM
»
°
°
­
½
% ° ° % [ °»
« $ [ °® M
L ¾ OQ ®
¾ «
% [ ° °¯ % [ °¿»
% [ ° $ [
«
»
¯°
¿°
¬
¼
¦
ª
­° %L ½°º
$L
3 H[S « = 7 3 OQ = 7 3 %L OQ ®
¾»
%L ¯° %L ¿°¼»
¬«
ZKHUH$DQG%DUHWKHGLPHQVLRQOHVVIRUPVRIDDQGEDQGDUHREWDLQHGIURPWKHYDQGHU:DDOV
RQHIOXLGPL[LQJUXOHV1RWHWKDWIRUWKHSXUHFRPSRQHQWIXJDFLW\RIVSHFLHVLWKHFRPSUHVVLELOLW\
WKDWLVXVHGLVWKDWREWDLQHGIURPWKHHTXDWLRQRIVWDWHZLWKWKHSXUHFRPSRQHQWSDUDPHWHUV
ZKLOHLQWKHPL[WXUHFDVHWKHDDQGE RU$DQG% SDUDPHWHUVDUHIRUWKHPL[WXUHDQGGHSHQGRQ
FRPSRVLWLRQ
IL 7 3
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
A x A x
o G H[
A A
x x RT
G H[
§ wNG H[ ·
¨
¸
© wN ¹ N
OQ J x x RTA A
DQGNG H[
A x A x
N RTA A
N N RTA A
A N A N A N A N § wNG H[ ·
¨
¸
RT © wN ¹ N
N N RTA A
A N A N x RTA A
x x RTA A
A x A x
A x A x
x A A u A x A x x x A A
A x A x
x A A A x x A A A x x x A A
x x A A x A A x x A A
A x A x
A x A x
x x A A A x A A
A x A x
x A A x A x A
A x A x
6LQFHG H[ LVV\PPHWULFLQWKHLQGLFHV WKDWLVXQFKDQJHGLIWKHLQGLFHVDUHLQWHUFKDQJHG WKHQ
E\LQWHUFKDQJLQJWKHLQGLFHVLQWKHHTXDWLRQIRUWKHDFWLYLW\FRHIILFLHQW
OQ J x A A x A x A
A x A x
RT OQ J G H[
E
Ax RT OQ J f
Ax x
RTx x
A RT
RT OQ J f
Solutions to Chemical and Engineering Thermodynamics,5th ed
Chapter 9
E D 7KHDFWLYLW\FRHIILFLHQWVRIPHWKDQRODUHFDOFXODWHGIURP
YDS
xPHWK J PHWK xPHWK PPHWK
xPHWK
P xPHWK 6LQFHWKH>%0,1@>2FW6@LVLQYRODWLOHDQGQRWSUHVHQWLQWKHYDSRUSKDVH7KHUHIRUH
J PHWK xPHWK
P xPHWK
YDS
xPHWK u PPHWK
xPHWK
7KHUHVXOWVDUHVKRZQEHORZ
[PHWK JPHWK
Ϭ͘ϮϮϮϮ Ϭ͘ϰϯϮϯϱϲ
Ϭ͘ϮϲϮϵ Ϭ͘ϰϰϲϳϴϮ
Ϭ͘ϰϲϮϵ Ϭ͘ϰϰϮϱϮϱ
Ϭ͘ϲϭϱϲ Ϭ͘ϱϲϰϰϲϴ
Ϭ͘ϲϱϰϵ Ϭ͘ϲϭϭϵϲϳ
Ϭ͘ϳϲϯϴ Ϭ͘ϳϳϬϳϬϲ
Ϭ͘ϴϳϵϮ Ϭ͘ϵϮϭϳϳ
Ϭ͘ϵϮϯϱ Ϭ͘ϵϴϳϰϮϭ
Ϭ͘ϵϯϵϮ ϭ͘ϬϬϭϲϮϭ
P xPHWK
xPHWK u Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
Ϭ͘ϵϳϱϰ ϭ͘ϬϬϬϮϵϯ
Ϭ͘ϵϴϳϰ Ϭ͘ϵϵϰϰϳϳ
ϭ
ϭ
7KHVHGDWDDUHHDVLO\ILWZLWKWKHHTXDWLRQ
OQ J PHWK
xPHWK
)URPZKLFKLWIROORZVLPPHGLDWHO\E\V\PPHWU\RUIURPWKH*LEEV'XKHPHTXDWLRQWKDW
OQ J >%0,1@>2FW6@
x>%0,1@>2FW6@
xPHWK
7KHUHIRUH
[PHWKJ>%0,1@>2FW6@
Ϭ͘ϮϮϮϮ Ϭ͘ϵϭϵϮϴ
Ϭ͘ϮϲϮϵ Ϭ͘ϴϴϴϴϱϱ
Ϭ͘ϰϲϮϵ Ϭ͘ϲϵϰϬϬϴ
Ϭ͘ϲϭϱϲ Ϭ͘ϱϮϰϭϯϯ
Ϭ͘ϲϱϰϵ Ϭ͘ϰϴϭϯϲϳ
Ϭ͘ϳϲϯϴ Ϭ͘ϯϲϵϵϭϭ
Ϭ͘ϴϳϵϮ Ϭ͘Ϯϲϳϳϰϵ
Ϭ͘ϵϮϯϱ Ϭ͘ϮϯϯϲϳϮ
Ϭ͘ϵϯϵϮ Ϭ͘ϮϮϮϯϬϴ
Ϭ͘ϵϳϱϰ Ϭ͘ϭϵϳϱϯϱ
Ϭ͘ϵϴϳϰ Ϭ͘ϭϴϵϳϲϭ
ϭ Ϭ͘ϭϴϭϴϯϭ
7KH*LEEVHQHUJ\RIDPL[WXUHFRQVLVWLQJRIN PROHVRIVSHFLHVDQGN PROHVRIVSHFLHVLV
G
R
R
N G RT OQ x N G RT OQ x
§ R
§ R
N ·
N ·
N ¨ G RT OQ
¸ N ¨ G RT OQ
¸
N N ¹
N N ¹
©
©
6RWKHFKDQJHLQ*LEEVHQHUJ\EHWZHHQDVWDWH,DQGDVWDWH,, ZLWKRXWFKHPLFDOUHDFWLRQ LV
'G
R
R
R
R
N,, G RT OQ x,, N ,, G RT OQ x,, N, G RT OQ x, N , G RT OQ x,
6LQFHWKHUHLVQRFKHPLFDOUHDFWLRQN,, N, DQGN ,, N , VRWKLVHTXDWLRQUHGXFHVWR
'G
N,, RT OQ x,, N ,, RT OQ x,, N, RT OQ x, N , RT OQ x,
ĂƐĞĚŽŶϭŵŽůĞŽĨĨĞĞĚƚŚĂƚŝƐϬ͘ϳϵŵŽůĞƐŽĨŶŝƚƌŽŐĞŶĂŶĚϬ͘ϮϭŵŽůĞƐŽĨŽdžLJŐĞŶŐŽŝŶŐƚŽŽŶĞ
ƐƚƌĞĂŵŽĨϬ͘ϳϵŵŽůĞƐŽĨƉƵƌĞŶŝƚƌŽŐĞŶĂŶĚϬ͘ϮϭŵŽůĞƐŽĨƉƵƌĞŽdžLJŐĞŶ͕ǁĞŚĂǀĞ
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
'G
RT OQ RT OQ RT OQ RT OQ RT OQ RT OQ RT
PRODLU
NPRODLU
6LQFHW 'G WKHZRUNUHTXLUHGIRUWKLVLVRWKHUPDOFRQVWDQWVHSDUDWLRQLVN-SHUPROHRILQFRPLQJDLU dŽĂǀŽŝĚĂůŽƚŽĨĂůŐĞďƌĂ͕ƵƐĞƐƉĞŶĂŶĚƚŚĞWĞŶŐͲZŽďŝŶƐŽŶĞƋƵĂƚŝŽŶ͘dŚĞƌĞƐƵůƚŝƐŐŝǀĞŶŝŶ
WƌŽďůĞŵϵ͘ϲϯ͘
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϵхWƌŽďůĞŵƐхWƌŽďϵ͘ϲϯ
dŚĞŵĂŝŶƌĞƐƵůƚƐĂƌĞ͗
,ĞƌĞ,/'DyŝƐƚŚĞŝĚĞĂůŐĂƐĞŶƚŚĂůƉLJĂŶĚ,DyŝƐƚŚĞƌĞĂůŐĂƐĞŶƚŚĂůƉLJ͘^ĂŵĞŶŽƚĂƚŝŽŶĨŽƌ
ĞŶƚƌŽƉLJ͘
&DOFXODWHGXVLQJDQ([FHOVSUHDGVKHHW
V
§
·
¨
V ¸
'S R x OQ I x OQ I ¨ OQ
OQ
V
V ¸
¸
¨ V
V ¹
©
1RWHWKDW )RUDPL[WXUHRIHTXDOVL]HPROHFXOHVVRWKDWV V 'S R
·
§
¨ OQ
OQ
¸ OQ ¹
© RU'S R
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϵхWƌŽďůĞŵƐхWƌŽďϵ͘ϲϱ
Chapter 9
Solutions to Chemical and Engineering Thermodynamics,5th ed
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϵхWƌŽďůĞŵƐхWƌŽďϵ͘ϲϲ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϵхWƌŽďůĞŵƐхWƌŽďϵ͘ϵ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϵхWƌŽďůĞŵƐхWƌŽďϵ͘ϭϯ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϵхWƌŽďůĞŵƐхWƌŽďϵ͘ϮϮ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϵхWƌŽďůĞŵƐхWƌŽďϵ͘ϰϵ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϵхWƌŽďůĞŵƐхWƌŽďϵ͘ϱϬ
S
C
ca a d E
T
d
a
c ,, 5
Chap e 13
d
13
N e ha a
f he
be
h cha e ca be
ed e a e ea
h
g a . The f
he DOS BASIC CHEMEQ
he VISUAL BASIC Che ca
E
b
g a ha a e he ca c a
f he che ca e
b
c
a a
a
e e a e e ea . The ec d a e a
g
g a , ch a Ma hcad,
MATLAB, MATHEMATICA, e c. f
g he
ea a geb a c e a
( ) h ch
e . I ad ab e ha
de
h
e b h. [I ha e ed Ma hcad f
a
f he
be
e
ed he e.] 13.1
F
E a
13.1-18
Ÿ
Ne
g e
Ka
25qC
G
'
25qC
Ka T
RT
7.1566 a d K a T
h 'a
g E . 13.1-23b
Ka T 500 K
. K
Ka T 29815
'a
R
17, 740
8.314 u 298.15
25qC 7.7967 u10 4
16.736 J
K; '
'
'
'
500
.
29815
1
' H 29815
. 'a u 29815
. 5001 29815
. 1
R
16.736
500
.
8.314 29815
1
1
1
.
55,480 16.736 u 29815
.
8.314
500 29815
10407
.
8.2228 9.2635
Ka T 29815
. 9.2635 7.1566 9.2635 2107
.
F
H
Ka T
500 K
Ÿ Ka T
500 K
8.225
I
K
0
S
C
ca a d E
T
Ma Ba a ce Tab e
S ec e
I
a
c ,5
Chap e 13
d
O
.
X 09436
0.0290
IPOH
1
1 X
P
0
X
X 1 X
0.4855
H2
T a
0
X
X 1 X
0.4855
Ka 1 X 2 ; 1 Ka X 2
aIPOH
a
8225
.
ŸX
f
PCO Pa
X
X2
.
1 X 2
b
g
a
f
Ka 1 Ka ;
c
e ed.
CaCO3 CO
Ka
T qC
Ka
f
09436
.
Ÿ 94.36% f a c h
: CaC2O4
Reac
o ca c a ed af e X a
f d be
.
ba 1 ba
X 2 P 1013
1 X 1 X
X2
Ka
1 X
a
aH2 aP
Ka
Pd
1 X
1 X
N
Ka
13.2
d
aCaCO 3 aCO
aCO
aC C 2 O 4
PCO
375
388
403
410
416
418
1.09
4.00
17.86
33.33
78.25
91.18
0.0109 0.0400 0.1786 0.3333 0.7825 0.9181
PCO 100
4.5282 32.315 1.7356 1.1112 0.2581 0.1052
Ka
648.15 661.15 676.15 683.15 689.15 691.15
TK
RT
Ka
J/
CaC2 O 4
'
G
24.401 17.763
9.757
6.3113 1.4788 0.6351
eac ed
N
a d
'
G RT
T
'
S
Ka
T
'
H
RT 2
(1)
' H ' G
T
(2)
e f
e 0.106.
Ka . T. I fa
a ea ab
a gh
P
he da a f
Ka
0103
.
. Th , ' H # 0.103RT 2 , h ch f
.e.,
f
E . (1). O ce
T
' H
e a a ed, E . (2) ca be ed ge ' S . The e
a e g e be
:
S
d
a
c ,5
Chap e 13
C
ca a d E
T
d
T qC
375qC
403qC
418qC
'
H
358.7
390.4
409.9
J/
CaC2O4 eac ed
'
S
0.5159
0.5630
0.5984
J/
K CaC2O4 eac ed
0
Ka
2
4
6
640
650
660
670
Te
13.3
Reac
(1)
ea
680
690
700
e (K)
:
(2)
M e ba a ce ab e
S ec e
I a
C
CO2
CO
0
O2
0
C CO2
2CO
2CO
2C O2
F a
1
a1 X f a1 X X f
2a X X f a1 X X f
X a1 X X f
1 X1
1
2 X1 2 X 2
1
X2
1
2
2
2
1
1
2
2
1 X1 X 2
a
(a)
P
1 ba
F
a T
Ka ,1
(a
(b)
e
1 ba
ce P
he
ga
CHEMEQ e f d ha Ka,1 T
19
2000 K Ka ,2
2
aCO
aCaCO2
a
f
Ÿ Ka ,2
0, X 2
39050
0 a d Ka ,1
39050
a f
fa f
4 X1 X 2 2
4 X1 2
#
Ÿ X12
1 X 1 1 X1 X 2
1 X1 1 X1
a
d e ec
A 1000 K
2.445 u 10
2000 K
fa
f a
h ch a a ge e
g CHEMEQ
b
c
a )
Ka,1
4 Ka ,1
1
S
C
ca a d E
T
d
Ka ,1 T
1000 K
1835
.
Ka , 2 T
1000 K
3984
.
u 1023
a
0561
.
f he ga ea
S ec e
Chap e 13
d
1835
.
4 1835
.
X1
Th , he c
c ,5
g he g a h e bed
2000 K
2.594 u 10 5
1.0
0
CO2
CO
O2
1000 K
0.283
0.717
0
13.4
e
Ka
°­ ' G °½
®
¾
RT °¿
°¯
ad a d
;
ag a h e
d
g
nP
RdV
S
T
c
e
i P 1 ba UV
RT
W
12 g
3.51 g cc
3.4188 cc
3.4188 u106
3
Vg a h
12 g
2.25 g cc
5.3333 cc
5.3333 u106
3
3.4188 5.3333 P 1 cc-ba
Ÿ P 1 ba
2866 J
1.9145 cc
P 14971 ba .
f
P 14971 ba ;
f
RT
e
' G
RT
ad a
! Ka a d g a h e
ag a h
ab e ha e
ad a
Ka a d d a
ag a h
ca ab e f e e g 14971 ba
de a h ghe e e a e!)
F c
e e ce,
e eac
S ec e ba a ce ab e
2866
J
R
1497.0 J cc 14970 ba
P ! 14971 ba ;
Ÿ Need a h d a c
d . (A , h d c
g c ec
a
ed
e be
d
V g
d
Vda
0.3147
13.
e
­V
½
P 1 ba ¾
®
RT
¯
¿
e
T , P 1 ba
kV P 1 ba RT
e nV P 1 ba RT
he e
da
T, P
he e a
e
ad a d
ag a h e
Th
2866
­
½
®
¾ 0.3147
¯ 8.314 u 298.15 ¿
e
a N2 O2
2NO2
d
ab e ha e
c
e
e c
ead
S
C
ca a d E
T
S ec e
d
I
a
c ,5
a
Chap e 13
d
F a
O2
1
1 X
1 X 4.762
N2
NO
079
. 021
.
3762
.
0
3762
.
X
3762
.
X 4.762
2X
2 X 4.762
4.762
P
1 ba
S ce P 1013
.
ba ; a
2X 2
1 X 3762
.
X
2
NO
Ka
N 2 O2
h ch ha he
.
.
2.381 19073
15048
Ka
4 Ka 1
U g he
g a CHEMEQ a d
( h ch ag ee
h F g e 13.1-2)
X
13.
a
f
da a ba e he f
g e
a e b a ed
T qC
Ka
X
1500
1600
1800
2000
0.9795 u 104
0.00954
01924
.
u 103
0.0133
0.5861 u 103
0.0231
01455
.
u 102
0.0361
T qC
2100
2200
2400
2500
0.007487
Ka
X
0.2154 u 10
0.0438
T qC
2600
2800
2900
3000
Ka
X
0.009539
001450
.
0.1086
0.01732
0.02028
0.118
0.1269
2
0.3077 u 10
0.0520
0.0893
(a)
F
A
e d
0.5718 u 10
0.0701
2
'f H
4322.5
1382.8
241.8
Na 2SO4
H 2O
'f G
3642.3 J/
1265.2
228.6
Ÿ'
H T
25qC
4322.5 1382.8 10 228.6
'
G T
25qC
91.1 J
Ka T
25qC
91,100
8.314 u 298.15
Ÿ
Ka T
25qC
0.0796
A.IV
Na 2SO 4 ˜ 10H 2O
N
2
9.139 u 1015
36.751
521.7 J
S
C
ca a d E
T
d
a
aNa 2SO 4 ˜ a10
H 2O
A 15qC. S ce 15qC
H
he '
e
Ka T
15qC
Ka T
25qC
15qC
PH2 O 15qC
13.
Ka
ea 25qC
e
0.0253 ba
c
ec Ka f
e
ea
e
'
H § 1
1 ·
¨
¸
R © 288.15 298.15 ¹
25qC e
Ka T
1.221u102 ba
7.3438
1.358 u1019
7.304
0.01221 ba
e a da a (Ba e a d La
g, J.A.C.S. 42, 419 (1920))
PH 2 O 0q C 0.003693 ba
e
(a
aCS2
0.9139 u 1015
H 2O
, .e.
Ka T
E
10
2.503 u 102 ba
Ÿ PH 2 O 25q C
g
Chap e 13
d
F 1 ba I
GH P JK
aNa 2SO 4 ˜10H 2 O
Ka
(b)
c ,5
PH 2 O 15q C
0.01228 ba
PH 2 O 25q C
0.0256 ba
a a ab e a a Ma hcad
CS 2
aCaS2
S2
. S ce aC
hee )
d), a d P 1 ba
1(
a da d a e
e
e.
S ec e ba a ce ab e:
S ec e
I
C
S2
a
1
1 X
X
0
CS2
F a
X 750q C
.
a d X 1000q C
0894
c
f
13.
0869
.
.
f .
X (1 X )
X
Ka (1 Ka )
Ÿ
1
g CHEMEQ I f d Ka 750q C
U
e
1 X
X
Ka
8.478 a d Ka 1000q C
X
CS2
he e ce age e
a f
(a)
Ba NO 3 2
I
1
1
2
2
2
2
2
C
2 CBa 1 CNO3 1 CAg 1 CC
2
2
1
4CBa CNO3 CAg CC
2
¦
m
m
b
6.607 . The ef e
CAg
CC
CAgC ; CBa
CBaaNO3 f2 ; CNO3
2CBaa NO3 f2
b
S
C
ca a d E
T
d
a
c ,5
Chap e 13
d
1
4CBaaNO3 f2 2CBaa NO 3 f2 2CAgC
2
n
I
I
CAgC
CBaa NO3 f2
3CBaa NO3 f2 CAgC
K , he e
I
2
CAgC
K
2.111 u 10 4
7.064 u 10 4
44.02 u 10 4
56.0 u 10 4
(
/ e)
6.4639 u 10 4
213259
.
u 104
132.21 u 104
168147
.
u 104
2.542 u 10 2
4.618 u 10 2
11498
.
u 10 2
12.967 u 102
22.4873
22.4420
22.2827
22.2594
a f
(b)
U
01309
.
u 104
01339
.
u 104
01450
.
u 104
01467
.
u 104
(
/ e)
La NO 3 2
g
a a a
ha ab
e
CAgC
CLaa NO3 f2
ed I
6CLaaNO3 f3 CAgC .
I
I
K , he e
2
CAgC
K
1438
.
u 104
5.780 u 104
16.6 u 10 4
28.07 u 10 4
(
/ e)
01317
.
u 104
01367
.
u 104
01432
.
u 104
01477
.
u 104
(
/ e)
8.7597 u 104
34.8167 u 104
99.7432 u 104
168.568 u 104
/
I
fa
E ce f he
a a gh
e.
ge
h gh
2.9597 u 102
5.9006 u 102
9.9872 u 102
12.983 u 102
c
e
1
22.4751
22.4006
22.3077
22.2458
2
e g h (AgC
KNO3 ), a
he da a
S
C
13.
ca a d E
d
a
c ,5
1
2
I
1
¦ M #2 2C
2
2
Ba b
f BaSO4
K
CBa CSO 2
S . N e
T qC
D Tab e 9.6
S
5
10
15
20
25
1.140
1..149
1.158
1.167
1.178
15.6 u 10 6
16.7
18.3
19.8
21.6
he e S
a
Chap e 13
d
Ba SO4 , e ha e
BaSO4
F
T
4
22 CSO 4CBaSO4
4
4S
e/ e
e ha e eg ec ed he d ffe e ce be ee M
a d C. Th
(
N
Ka
Ka 1
K
Kq
a d
a Q Q 1
Jr
K
K q
e
7.9 u 103
8.173
8.556
8.900
9.295
e 2
e 12
f a
Kq
a d K
Q Q K q Ka 1
j
D I
Jr
r
4
Ba SO4
2
N e ha Ka
he e
BaSO4
5
10
15
20
25
G
T
R
b
c
a
K a . Th
Kq
2.261 u 1010
2.587
3.094
3.607
4.275
22.2100
22.0754
21.8964
21.7430
21.5731
f
2
a
Ba a , dea 1
Kq
a
K q16D I
d16D S i S e d16D S i .
K e
'
a Q Q Q Q K q2 ˜ 2 u 2 D I
a d
f
Q Q .
K q Q Ba Q SO Kq
2 S
I
2.43 u 1010
2.789
3.349
3.920
4.666
a
/ e)
a Q Q S2
K
f
he eac
a
f
a SO4 a , dea 1
a
f
e ha e
G
'
T
R
Ka
184.654
183.535
182.047
180.771
179.359
J
K
F
H
I
K
H
' G
'
20490
21234
21991
22760
23543
(J/
)
51,362
51,968
52,457
52,993
53,476
(J/
)
110.99
108.54
105.73
103.13
100.40
(J/
)
'
S
S
A
C
,f
ca a d E
' GT
he
T
e e a
e ha e
a
d
e e
w GT
wT
F a
H
T
c ,5
ea f c
w ' G T
wT
H
a
2
f T a ca be ee b
G T
' '
'H 'G
Ÿ'
T
'S
w '
T 2
H
Ka
1
01343
.
u 1012 0.4806 u 1014
2
0.3332 u 1012 0.3851 u 1014
A e ege
J/
f C 3H 8
3
7
aH 2 aCO
a d Ka,2
(b)
Ka,1
aC 3H 8 aH3 2 O
P
1 ba
S ec e ba a ce ab e
a
S ec e
I
C3H8
1
10
CO2
0
age.
a e b a ed
3CO 7H 2
3CO2 10H 2
' H
1000 K
537.260
432.380
' G
1000 K
213.030
220.590
' H
1100 K
538.140
436.440
he e,
ce P
3
a10
H 2 aCO 2
aC 3 H 8 aH6 2 O
10 3X1 6 X 2
1 X1 X 2
a1 X X f ¦
a
f
3X1
3X1 ¦
1
0
7 X1 10 X 2
a
¦ 11 6 X1 X 2
2
10 3 X1 6 X2 ¦
3X2 ¦
3X2
a7 X 10X f ¦
1
f
2
Th
Ka,1
a
f
a1 X X fa10 3 X 6 X f a11 6 X 6 X f
27 X13 7 X 1 10 X 2 7
1
2
1
2
3
1
2
' G
1100 K
288.11
286.09
1 ba
.
0
H2
G
g e
O
H 2O
CO
T 2 0.26485
e
2 C3H 8 6H 2O
1100 K
K2
H '
T
'
S
1 C3H 8 3H 2O
Ka
G T
wT
B h 'H a d 'S a e g e
he ab e
13.10 (a a a ab e a a Ma hcad
hee )
(a)
U g he
g a CHEMEQ, he f
1000 K
g he da a.
0.26485 J
'T
Ÿ'
Chap e 13
d
6
S
C
ca a d E
T
d
a
c ,5
Chap e 13
d
a
f
27 X 23 7 X 1 10 X 2 10
Ka,2
a1 X X fa10 3 X 6 X f a11 6 X 6 X f
1
2
1
6
2
1
2
c
a
6
A
a
I
a
f
X 23 7 X1 10 X 2 3
Ka ,2
X1 X 2
e f he e a ge
e ca a e
1. U g h a
a
e ge
f he e
a
f
a1 X X fa7 3 X f 17 ; K
27 X13 7 3 X 2 7
Ka,1
1
2
f
X13 10 3 X1 6 X 2 3
Ka ,1
2
3
, ee
a
f
a1 X X fa7 3X f 17
27 X 23 7 3 X 2 10
a,2
6
b
1
2
2
6
ec
6
a d
a
f Ÿ FG K IJ
HK K
X a7 3 X f
X13 7 3 X 2 3
Ka ,1
3
2
Ka ,2
N e ha h
Ka,2 . A
, he
2
3
a e a
13
a ,2
a
e
a
a
X1 7 3 X 2
X 2 7 3X2
ad a c e
a
f a1 X fa7 3 X f
f X a7 3 X f
2
2
f
X1 1 X 2 .
a 1000 K: X1
(c)
a ,1
a 1100 K: X1
Th
e ba
.
0527
; X2
0603
. ; X2
S ec e
C3H8
0473
.
0397
.
1000 K
0
1100 K
0
H 2O
CO
CO2
0.328
0.342
0.093
0.083
0.106
0.070
H2
0.495
0.482
a a ab e a a Ma hcad
hee )
13.11
(a
Reac
: SO2 1
O2
2
SO3
2
2
X 2 g e Ka,1 a d
S
C
ca a d E
T
d
a
c ,5
Chap e 13
d
S ec e ba a ce ab e
S ec e
I
N2
F a
1
1 X
1
2
1 0.79
u
2 0.21
1
1 X
2
SO2
O2
a
1.88
188
.
0
SO3
X
338
. P
1 ba
S ce P 1013
.
ba ; a
Ka
be
ga
aSO 3
aSO 2 aO1 22
T qC
X
14
1016
.
u 10
08625
.
u 109
01012
.
u 107
01265
.
u 105
05903
.
u 103
0.6188 u 102
01104
.
u 102
0.2847 u 101
0.9566
0.3951
0.1863
0.09969
0.05862
0.03722
0.02518
1
1
0.9998
0.9967
0.9750
0.8935
0.7090
0.4569
0.2467
0.1252
0.0655
0.0366
0.0220
0.0141
0.0096
. X 12
X 3.38 05
1013
. 1 2 1 X 3 2 05
. 12
SO 3
12
SO 2 O 2
c
a f
be
a
0
100
200
300
400
500
600
700
800
900
1000
1100
1200
1300
1400
1
X
2
1013
.
The che ca e
b
CHEMEQ a d he he
Ka
1 X
338
. 05
. X
1 2 1 X
338
. 05
. X
188
.
338
. 05
. X
X
338
. 05
. X
h
ed
be
a ca c a ed
g Ma hcad. The e
SO 2
O2
6148
.
u 105
0.001139
0.008635
0.0363
0.0962
0.1723
0.2313
0.2637
0.2792
0.2866
0.2903
0.2923
0.2934
3.074 u 105
0.000569
0.0004317
0.0182
0.0481
0.0862
0.1157
0.1319
0.1396
0.1433
0.1451
0.1462
0.1467
0
0
0
0
SO3
0.3472
0.3472
0.3472
0.3459
0.3371
0.3046
0.2343
0.1450
0.0757
0.0377
0.0196
0.0109
0.00652
0.00418
0.00285
a
g he
ea
S
C
ca a d E
T
d
a
c ,5
Chap e 13
d
1
1
X
YSO2
0.5
YO2
YSO3
0
13.12
0
0
0
500
Reac
: C2H 4 C 2
1000
T
C2 H4C 2
d a 50qC
e e , C2H 4C 2 a
e h ca
e da a
H
S
S ec e
N
a b
g
C2H4C 2
83.47qC
C2 H4
88.63qC
34.6qC
C2
1500
3
1.4 10
2 ha e eac
!
P a 50q C
0.288 ba
TC
PC ba
122.5 ba
14.87 ba
9.2qC
50.36
S ce e h e e ab e c ca e e a e,
d- ha e a - e
e
ha e be e
a ed f e a e d he a de
b
ca c a
.H e e,
ce e eed
a
de a e e a a
(f
T 9.2q C
T 50q C ), e
d
a e a a
f he a - e
e da a, a d
e Sha
c ea
. U g a e
ee a
he Ha
C
a P
, e f d P a 50q C 122.5
ba f e h e e.
N e: T be c
e ,a a
e
e da a f h
b e ha e bee a e
f
he Ha
C
a P
. The da a d ffe ,
a ca e b r20%
f
T C
a E
Ha
. I be e e he a e a be
e acc a e.
S ec e ba a ce ab e
S ec e
I a
F a
1
1 X
C2 H4
C2 H 4V C2 H 4 L
A
C2
1
1 X
C2H4C 2
0
X
ce P
1 ba , a
,
N e: A
a d
C2
C2 H4
b
P
1 ba
f
g e
ee
he
C 2 H 4 C 2V C 2 H 4 C 2 L
f
ha
C 2V C 2 L
ec e
C2H4C 2
h e a da d a e
ee
he a
d ha e. S ce he a da d a e
a a
.
ha e,
f C2 H4 a d
S
C
ca a d E
T
d
a d C2H 4C 2
C2 aea
e a
ea
f he f
a
c ,5
a a
d, h
aC2 H 4 C 2
Ka
Chap e 13
d
d e
1
1 2 ˜1 2
aC2 H 4 aC 2
a e
4
h ch c ea ca
be e! The ef e,
b a he c ec
e
c
de he
b
ha a
ec e a be e e
I he ab e ab e, L a d V a e he a
be f
e
he
ha e , e ec e . L V 1
L
Pha e e
b
: 1V
J Pa
Ÿ P
be
Che
ca e
: ( a da d a e: C2 H4 a d C 2
b
d)
Ka T
g e ,e e a , c
de a
C2 H 4 C 2
C2 H4C 2 J C2 H 4C 2 0.284 ;
g
c
hc
Th , he
,
a
G
e , e e ec a
c ea ha he
I
he
J
ae
ac a
y
de , J y
1,
y
C2
J C 2 14.68
C 2 H 4C 2 J C2 H 4 C 2
c eff c e
1,
C2 H 4 C 2
be f he de 10 ]
he a de
b
1,
ha
g e
he e
c
e
C2 H 4
0,
C2
be f ea
C2 H 4
ab e
0
C2
a
e
a
,
he
ha e e e a
d ha e,
0.
J y y he e
Nq
¦N q
ce he e a
e
dea .
N
N NBq
u A
N A N B N B2
Nq
N A NBq
a d JB
N A N B N B2
de he che
h
C2
N
¦N
e f ac
ŸJ A
c
J C2H 4 121 ;
13
G RT
e f ac
ŸJ
N
C2 H 4
he ac
h
be
d ha e. The eac
G RT
a
C2 H 4
.
c
C2 H 4 C 2
13.13
e
h
, he e a e
e f ac
P gg g he e a e bac
1. Ÿ N a
ha e!
b
e
h
g bac
che ca e
b
ea
J
J
u
14
.
68
121
11
. u 1023
C 2 H 4 C2 H 4
C2 C2
[ac a
e
ha
; C2H 4C 2
a
C2 H 4 C 2
c
S ce, f h
e ( e ha , a , 10),
¦
h
ha e !
da d a
g da a A e d ce II a d IV, a d E . 9.1-23b e b a
11
. u 1023 a h ge
be . Ÿ C 2 H 4 C 2 C 2 u 11
. u 1023
C2 H 4C 2 J C 2 H 4 C 2
Ÿ Reac
Va N
a
C 2 H 4C 2 J C2 H 4 C 2
Ka
N
50q C
b
ca e
b
:
NB
NBq
N
N A NBq
u
N q N A N B N B2
N A NBq
N A N B N B2
ef d
S
C
ca a d E
T
I a
be
N
d
e
a
c ,5
F a
be
e
Chap e 13
d
M e f ac
NA
N A N Bq N B2
A
NA
NA
B
N Bq
N Bq 2 X
N Bq 2 N B2
N A N Bq N B2
N B2
X
B2
N A N Bq N B2
N A N Bq X
N A N Bq N B2
S
g h e
a
aB2
q
B
q
B
A
a 2
A
q 2
B
A
2
N A N Bq
a 2
A
4Ka 1 , a d
2 1
B2
c N N h cN N h 4K c N h
2 N B2
a dG
2
q
B
g e
N B2
he e
c
h
cN 2N h
N B2 N A N Bq N B2
aB2
Ka
B
e ba
c
B
c
B G
2 12
a B a
a 2
2
B 2
h
2 12
a B a
2
B 2
2 G
h
cN N h 2 G ; N N cN N h 2 G
2
N N N
a N N qfFH 2G IK a d N N N N Nq 2G
A
q
B
A
N B2
B
B2
B
A
q
B
A
q
B
A
B
A
B
B
B2
Th
N A N Bq
N A N B N B2
JA
JB
c
N B N A N Bq
A
B
13.14
(a
a a ab e a a Ma hcad
Reac
B
G
a d
h
cN N N h
MNL1 2 G OPQ 2G 2G c 2
N Bq
2
B2
B
2
A
A B
h
2 12
B
A
hee ) 1
C2 H5OH O2 CH 3CHO H 2O
2
C2H5OH CH3CHO H2
1
:
2
S
C
ca a d E
T
d
a
c ,5
Chap e 13
d
S ec e ba a ce ab e:
S ec e
I
1
O2
0.75
C2H5OH
O
1 X1 X2
075
. 05
. X1
0.79
u 0.75 2.8214
0.21
0
N2
H 2O
CH3CHO
2.8214
X1
X1 X2
0
0
H2
X2
6
U
g he
ga
13
Ka,1
7.228 u 10
Ka,2
6.643
W h he e a e
eh
ac h
N2
0556
. ,
13.1
C3H8
(a
CHEMEQ e b a
ae b c eac
f he e
ed
H2O
1
X1 X 2
2
4.571 f
0197
.
,
g ea
fa
ed!
b
c
a
eac
CH 3CHO
].
a a ab e a a Ma hcad
C3H6 H2
C3H8 C2 H4 CH4
The ef e,
0197
.
a d
H2
X1 1 a d X2 # 0 [a
0 , O2 0.049 ,
C3H 5OH
e ba
0.
hee )
1
2
S ec e ba a ce ab e
S ec e
I
a
F a
C3H8
1
1 X1 X2
C3H6
0
X1
H2
0
X1
C2H4
0
X2
CH4
0
X2
6
I ge e a , e ha e a
1 X1 X2
P
, h
1 ba
1 X1 X 2
1 X1 X 2
X1
1 X1 X 2
X1
1 X1 X 2
X2
1 X1 X 2
X2
1 X1 X 2
S
C
ca a d E
aC 3H 6 aH 2
Ka,1
S
T
d
a
Chap e 13
d
X
F PI P
H 1 ba K 1 ba a1 X X fa1 X X f
2
1
C 3H 6 H 2
aC3H 8
c ,5
1
C 3H 8
2
1
a
aC2 H 4 aCH 4
Ka ,2
aC3H8
X 22
,
X 12
Ka ,2
A
Ka ,1
C
Ÿ X1
Re
(b)
a
1
d
C
P
e ca e: P 1 ba
i
a d X2
ab e be
2
1
1
d
Ka ,2
.
e ca e: A
e ga
N T
NRT Ÿ P
N T
dea
LMa1 X X f T OP ba
29815
. Q
N
1
f
RS K K
T
R
.
29815
K S K K
T
T
Ÿ X1
.
29815
Ka1,12
T
12
a ,1
12 2
a ,2
Ÿ X2
12
a ,2
12
a ,1
12 2
a ,2
ab e be
i
2
Ka1,12 Ka1,22
1 1 D X1 T
.
29815
. X 12
1 1 D X 1 T 29815
Ÿ Ka ,1
1 1 D X1 1 1 D X1
a eg e
2
1
2
1
1
a
Re
f
. Th
Ka ,1
1
Ka ,1
PV
Ka ,2
2
1
2
Ka1,12 Ka1,22
a
fa
a P 1 ba f X
a P 1 ba f X
1
1
1
1
X
X
D
D
a
fa
f 1 1 D X
e
a eg e
a
X2
X1
def e D
Ka,1
(a)
X 22
1 X1 X 2 1 X1 X 2
P
1 ba
2
X 12T
. 1 1 D X1
29815
UV
W
U
4T
K K V
.
29815
W
4T
Ka1,12 Ka1,22
.
29815
12
a ,1
12
a ,2
.
TK
Ka,1
1000
2.907
1200
38.88
1400
246.0
1500
512.6
1600
972.4
1800
2809
2000
6511
Ka,2
534.3
2581
7754
11950
17350
31900
50870
2
S
C
ca a d E
T
d
a
c ,5
Chap e 13
d
Pa a
0
0
0.034
0.055
0
0.076
0
0.086
0
0.096
0
0.114
0
0.132
0.465
0.445
0.424
0.414
0.404
0.386
0.368
C 3H 8
C3 H 6
H2
C2 H 4
CH 4
Pa b
C 3H 8
C3 H 6
H2
C2 H 4
CH 4
0.003
0.001
0.000
0.000
0.000
0.000
0.000
0.034
0.055
0.076
0.086
0.096
0.114
0.132
0.464
0.445
0.424
0.414
0.404
0.386
0.368
P ba
6.69
8.045
A
a,a e
b
c
a d Ma hcad.
13.1 The eac
g c e 1- h
9.389 10.061 10.732
12.074
a
eec
ed
g he
ga
ha e R g c e 6- h
13.416
CHEMEQ
ha e
F
he da a
he
b e , he f a c ce a
f g c e 1- h ha e 0.001 M
a db a
e a ba a ce e ha e ha he f a c ce a
f g c e 6- h ha e
= 0.019 M,
ha he a a e (
ea ed) e
b
c
a
M G6 0.019
K'
19
M G1 0.001
The G bb e e g cha ge f he
ce
J
'
'G RT K 8.314 ˜ 298.15 ˜ (19) 7299
13.1
S
e
ba a ce ab e :
e a
h1
S ec e
I
e f h d ge a d
a
ge . The
ec e
1 3X
1 2X
X
1 2X
2X
1 2X
1 3X
N2
X
0
NH3
f
F a
1
H2
e
2X
¦ 1 2 X
a d
Ka
cP
1 ba
NH 3
h
2
2
NH 3
3
N2 H2
c P 1 ba hc P 1 ba h
P I
4 X 1 2 X
ŸK F
H 1 ba K X 1 3 X
N2
2
a
3
H2
2
2
3
F 1 ba I
H PK
2
(1)
S
C
ca a d E
T
d
c ,5
Chap e 13
d
Pc
N e: We a e a
gP
e
gh ha
N
e a
h
X cha ge
h ,
e
a c
a T a d P. S a g f
E . (1) e b a
X
I
K
X I 12 X 1 2 X F
XI
4X 1 2 X F
1
K X 1 3 X H 3 K
X 1 3X H
2
2
N2
3
a f
de e
d Ta
1 1 G X 2 2 2 X G
e he g
a
1
1
1 G X 1 X G 2
G
F
H
a d
G
2 1 X
.
FG
H
I
K
e NH3
0,
05
. , X
e e , G 0 , .e., e
N2
N2
be f
ha NH3 dec
added ha
eeded f
2
e N2
e , a d N2
N2
05
. , he
ed.
N e: If,
ead f N 2 add
a c
a
e
e, he
a
e,
ha a a e
e f e he
ec e e e
e
e f N 2 c ea ed, he , f
Ka
IJ G
K 2 1 X
f G . If G ! 0 , .e.,
he a e a he g
d ced.
If, h
a
a a
1 X
G
G
1
1
1
1
1 X
1 X
1 X
2 1 X
f X
a he
a f
e
f N 2 ca e
e eed
N
e e e a
ed f
e,
.
added ha
add
a a
f X
N
Th , he g
.
a2 X f 4 2 2 X G 1 1 X G 9 1 3 X
S ce X d 1 3 , he de
N
4
be e a ,
a e ha , he a e f 1/3 a d .
X
.
1 . Le
05
Ÿ
1 G he e G a be e he +
1 2 X
X
ea
2
2
1 X 2 1 2X
2 X 4 1 2 X 1 X 9 1 3X
X
Ÿ
F
a e eeded!
a he de a e
F
H
2
2
he e X
ec
X 8X 2 1 2 X
X
8X 1 2 X
3
3 1 2
X 1 3X
X 1 3X
0
c
a
cP
NH 3
1 ba
h
2
c P 1 ba hc P 1 ba h
H2
H2
3
ge
a added a
affec ed, a d he
S
C
a
a
d c
ea da c
13.1
ca a d E
(a
T
d
a
c ea ha he eac
. Th
a
a
a
e
e
h ca e.
a a ab e a a Ma hcad
Reac
C4H8
S ec e ba a ce ab e
I
da a g
he d ec
c
be ee eac
d
a
1
F a
C4H10
1
1 X1
C4H8
0
X1 X2
C4H6
0
X2
H2
0
X1 X2
e
f c ea ed
a c
a
hee )
1 X1
1 X1 X 2
X1 X 2
1 X1 X 2
X2
1 X1 X 2
X1 X 2
1 X1 X 2
1 X1 X2
6
S ce
Chap e 13
d
C4H10 C4H8 H2
C4H6 H2
2
:
S ec e
c ,5
e
P
1 ba
1 ba , a
. Th
a X X fa X X f
a1 X fa1 X X f
Ÿ K c1 X X X X h X X
C4 H8 H 2
Ka,1
1
2
C 4 H 10
a, 1
1
1
2
1
2
1
2
1
2
2
1
2
2
2
(1)
a d
C4 H 6 H 2
Ka,2
c
a
C4 H 8
a
Ÿ Ka,2 X1 X12 X 2 X 22
13.1
U
g CHEMEQ (f
T
Ka,1
e
b
c
f
X 2 X1 X 2
X1 X 2 1 X1 X 2
a
fa
h X X X
1
2
f
2
2
(2)
) a d Ma hcad (f
)I ba
Ka,2
X1
X2
C4 H10
C4 H8
C4 H6
H2
900
1000
0.9731 0.1191
5.814 0.5575
0.724
0.951
0.147
0.464
0.147
0.020
0.308
0.202
0.079
0.192
0.466
0.586
We
c
de each f he eac
e aae .
S
C
2Ag +
'
1
O 2 o Ag 2 O; K1
2
G
,1
e
K1
ca a d E
T
aAg2O
2
0.5
Ag O 2
a a
d
a
c ,5
1
aO0.52
Chap e 13
d
1
ce he ac
0.5
O2
1
J
'Gf,Ag 2O 2'Gf,Ag 'Gf,O2
'Gf,Ag 2O 9.33
2
§ ' ,1G ·
9330
§
·
f h
¨
¸ e ¨
¸ 43.15;
RT ¹
© 8.314 * 298.15 ¹
©
1
0.000538 he e
b
43.152
f
ge
he a
0.21. S
ge ca ce a
The e eac
e
c
de
aAg2O aH2
aH 2
2Ag + H 2 O o Ag 2O + H 2 ; K 2
2
aAg aH 2O aH 2O
he
'
K2
O2
,2
G
e
Th e
'Gf,Ag 2O 'Gf,Ag 2'Gf,Ag 'Gf,H2O
§ ' ,2G ·
¨
¸ e
RT ¹
©
b
c
a
e f ac
a
H2
h
d
.
eac
eea e
f O2. H
e e , he
e f ac
f each
d
e
b
,
a e.
ce he ac
.
H2O
9.33 228.59
219260
§
·
¨
¸ e
© 8.314* 298.15 ¹
a ha h eac
f each
88.453
219.26
J
3.849 u10 39
cc
a d a
h he
e.
S
C
ca a d E
The h d eac
T
e
c
0.05
0.01
H2
H 2S
'
,3
§ ' ,3G ·
¨
¸ e
RT ¹
©
The ef e
0.05
0.01
H 2S
H 2S
aAg 2S aH2
aH 2
H2
a a
aH2S
H 2S
2
Ag H 2S
ce he ac
5 ba ed
31.80 33.6 1.8
1800
§
·
¨
¸ e
© 8.314* 298.15 ¹
a c ce
a
117.225
H2
,a d
f each
d
.
J
0.726
0.726 ba ed
e
b
.
H 2S
0.2 ba ed
a c ce
a
H 2S
,a d
1.377 ba ed
e
b
ea
.
H2
H2
Ba ed
he a
The a
ab e, he d ec
d c g H 2S a he ha
e e e.
F
he da a
,4
f he eac
, f
h g he
e.
a
ee
f
he eac
f each
d
cc
he
aAg2S aO2
1
1
a a
aSO2
SO 2
a
0.03
2
Ag SO2
b e , SO 2 c ce
'Gf,Ag2S 'Gf,O2 2'Gf,Ag 'Gf,SO2
G
ce he ac
,
ha
31.80 299.91
3 u10 8
SO 2
268.11
§ ' ,4G ·
268110
§
·
¨
¸ e ¨
¸ e
RT ¹
© 8.314* 298.15 ¹
©
The ef e, ba ed
he e
a a e f he e
108.16
1.063 u10 47
b
a , h
c
S
b e
he a h g f
he f
eac
c
e.
b e
h g f
13.20
U
a e e , Tab e 3.4 a d A
K4
g
eac
2Ag + SO 2 o Ag 2S + O 2 ; K 4
'
Chap e 13
d
'Gf,Ag2S 'Gf,H 2 2'Gf,Ag 'Gf,H2S
G
H2
c ,5
5
e
K3
a
de
2Ag + H 2S o Ag 2S + H 2 ; K 2
S
d
e
g he da a
he
be
Ka,1 750 K
aCaOS O 2 aCO 2
aCaCO 3 aS O 2
he a
1481
.
aCO 2
c
e d
e.
A.IV, I f d
CO 2 P
1 ba
eac
,
(1)
J
.
S
C
ca a d E
ce he ac
fa
T
d
a
c ,5
he
d ae
.
aS3 O 2 aFe 3O 4 aCO
Ka,2 750 K
Chap e 13
d
0.0277
3
a FeO
˜S O 2 aCO 2
aCO
aCO 2
CO
,
(2)
CO 2
a d
aFe 3O 4 aS3 O 2
Ka,3 750 K
12
3
aFeOS
O 2 aO 2
Ÿ
F
e
CO 2
,f
O2 P
F 1 ba I
GH P JK
1
aO1 22
12
O2
(3)
.
1242
u 1028
1 ba
. (2) e ha e
CO
A
08973
u 1014
.
e
0.0277 , h e f
ec
c
CO 2
148 ba , h e f
. (1), PCO2
CO
c be a
he
be, he
# 104
a a
he c
e
e
be ee 75 a d 105 ba . F a , f
E . (3), e c c de he e
O2
he a
he e, c
a ed a ace f
ec
c c be a
.
C c
? S e ha a b g
!
Ca c a
a d da a a e
a a e ag ee e , b a e ce a
a a e ag ee e . C
de he ce a
a he ea e e , he a
he c
de
d b ed a ea ab e e, a d ca
be e ec ed.
13.21 (a)
The c
he e '
.
G
G
The ha e e
F
ha
d
be
e
he eac
H TS , a d f
f
che
ca e
K aV
e
§ ' G V ·
¨¨
¸¸
RT
©
¹
K aL
e
§ ' G L ·
¨¨
¸¸
© RT
¹
b
V
aEB
aSV aHV
L
aEB
aSL aHL
; a
;
ha e
(1)
d ha e
(2)
he a da d a e G bb f ee e e g cha ge
b
ae e
e
e e
eac
ha e
ae
L
EB
V
EB ,
L
S
'
GV
83.0 J
g , e e a ,
d e he ' a H
V
S a d
L
H
V
H
(3)
.
u 1014 . Th
, a d K V 3482
c
e
he ga ha e. N
T' a S
e ha e e a J/
.
S
C
A
,
ce, f
ca a d E
T
d
a
c ,5
Chap e 13
d
he ab e ha e, G LH 2 ! G V
H 2 . The ef e,
h d ge , he a
L
ee
V
be f ab
he a e e a d g a ' G . C e e
, he
ha ' G
d ha e che ca e
b
c
a
a
be a ge, a d he h d ge a
eac
e e a g
c
e
he
d ha e.
Ÿ
e f ac
f
e e
be e
a
b h ha e . The
b e he
ed ce
de e
g he
b
f he e ce h d ge
he
d e h be e e,
a d de e
g he a
f e h be e e he a . Th , he e a
be
ed
ae
L
V
L
EB
EB Ÿ EBJ EB EB
EB P
a d
L
V
L
H
H Ÿ HJ H H
HP
He e e ha e a
ed ha he a
ha e dea .
A af
g e , e
a
e ha e
e h d ge
d
ed
he
d
g eg a
he
ha e. Th , J EB 1 , I EB 1 , a d,
e
a
V LH G EB G H
RT
JH
f
2
2
31
. cc
u 8.8 3.25 ca cc u 4.184 J / ca
K u 29815
8.314 J
. K
01612
.
Ÿ J H 1175
.
Ne
e ha e e
a e he f gac
f h d ge
he
d ha e. A b
a
ceed
e Sha
c ea
, Sec. 11.1. H e e , h d ge a
ed
de e
g h c ea
,a dPa
a
aga
ef
gh ga e
ch a
h d ge a d he
; ce e e e a da a a e
a a ab e, e ha e
e ch ce b
. K a d PC 12.97 ba
e h c ea
. N e, h e e , ha f h d ge , TC 332
,
c
ha T
T
ea
898
. , h ch
T TC
898
. (a e
a d
H
U
g he a
1 T , e f d ha PEBa
J
a
), he
a
L
H H
e
e da a f e h
1273
.
Pa a 25qC.
EB
J EB PEBa
EB
P
.
a d H 0996
[S ce he ga ha e
eed
eae a
].
0.049
H
H
EB
#0
A a e a e ca c a
e e f h d ge a e g e
S
c
ca
0.951 a d
e e
0.049 , a d
be e e,
EB
c
PC
5188
. ba
3 ba
e ha he ga - ha e
3 ba
1.175 u 51.88 ba
P
Ÿ
e ba
1013
.
ba , 25q C 4 u 12.97 ba
g e
e c ec
f h e
e
H
ee
L
e
L
[N e he he P
eg g b e.]
A af
g e ,
The ef e,
ff he ca e f F g. 11.1-1. If
EB
ed
0.004
a T
ae h
1.013 ba
a
e
4,
e
a
e h d ge .
1 H
0951
. .
he f
f
0.951 u 1 u 1273
.
# 0.004
3 u 100 Pa
a
e h d ge , a a
0.996
h
a
Pa
ed, he e
25q C
a d P 3 ba
# 0.0
e he Pe g-R b
e a
f a e. The
Tab e 6.6-1. The a e f e h be e e a e
S
S
C
ca a d E
T
d
a
c ,5
Chap e 13
d
TC = 617.2 K, PC = 36 ba , Z=0.302, a d TB=409.3 K. The e a e
b a
e ac
a a e e f h d ge
h he c
e
Tab e 9.4-1,
e
a
e ha
a e
e . U g he
he a f a h ca c a
he
g a VLMU e b a he
f
g e
0.9952
0.0018
H
H
a T 25q C
0.9982 a d EB 0.0048
EB
a d P 3 ba
S # 0.0
S # 0
Th
a be a
e acc a e ca c a
ha
g eg a
he
h ch
e
ed a e a a
f he P a
-Sha c e a
.. H e e , he e
ba ed
he a
ha
= 0. I
d be be e
ha e
e e e e a da a ge a
be e e
ae f h aa ee.
(b)
A 150qC a d 3 ba .
U g he da a
he P b e
a e e , A e d ce A.II a d A.IV e f d
Ka T 150q C 31
. u 108 . S aga
e ca
e
e ha a he
e e
he a
a d
d ha e
c
e ed e h be e e.
A af
a
a
( ea
), e
a
e ha he
d ha e
e e a
e e h be e e. Th
e ba
. 3.25 88
. 2
31
JH e
.
112
.
.
1987
u 42315
He e aga , e f d, e a a g F g. 11.1-1, ha
3
L
5188
. ba Ÿ H
0.052
H
5188
. u 112
.
.
0948
. .
a d EB 1 0052
RS
T
N
,h
e e , PEBa
1303
.
ba
H
1303
.
u 0.948
0.412
3
a he e a
, e ba
T 150q C
.
0.029
0586
H
EB
0.971
Ÿ
N
a
g he e a e f
Aga
ha
UV
W
EB
EB
0.414 3 ba
ST
0.0
H
0.0016
H
.
05034
EB
0.9984
EB
0.4966 3 ba
0
ST
g he Pe g-R b
= 0, e b a he f
e
a
g
f a e, he
ga
T
VLMU a d he
150q C
0
0.0
ST
ST
I b h a a a d b e ee ha he e
f he e a
f a e ca c a
ae
a a e, b
a a e ag ee e
h he P a
-Sha c e a
.
H e e , he a e ed c
ch h ghe
b e f h d ge
he
d ha e. The
e a
f a e ca c a
ch ea e
d (g e he a a ab
f he
ga
VLMU). If
e e e e a da a e e a a ab e f h d ge
b
e h be e e
(
he a
a c ), he a e f
c d be ad ed
e d ce ha da a. The
e
d ha e
e c f de ce
he e a
f a e ed c
f he
b e he e. If
ch e e e a da a e e a a ab e,
c ea h
e
d ad
he P a
Sha c e a
a ch ha ch da a.
S
C
ca a d E
13.22 (a)
T
d
E e g ba a ce
U
c ,5
a f ed
0
dV
Q P
d
a
. 0
f ga
FG w U IJ FG w U IJ FG w T IJ
H w K HwTK Hw K
Ÿ Q
W
Chap e 13
d
V
V
V
FwU I
FG IJ GH w T JK .
Hw K
Q
wT
CV
b
a
V
V
e
ea
U T
¦NU ¦N U ,
U
N
e
a d
a f
ce
e
e c
e ed
he e.
A
he e TR
ec
e e
efe e ce e
e
e
a
e he ga
dea a
N ,0 Q X
N
he
a d
T
U TR CV T
ea
e.
TR
FG w U IJ FG w IJ ¦ N U
H w T K Hw TK
Fw X I
¦ Q U GH w T JK ¦ N C
CV ,eff
V
¦Q U
FwU I
V
V
'U
V,
V
he e 'U
Fw N I
¦ GH w T JK U ¦ N GH w T JK
T
V
FG w X IJ ¦ N C T
H wT K
V,
V
e a e e g cha ge
eac
.
S ec e ba a ce ab e
S ec e
I
O
N2O4
1
1 X
NO2
0
2X
1 X
6
Ka
he
1 X
1 X
2X
1 X
c
c
2
aNO
2
aN 2O 4
B
he dea ga a
a c
d
PV
NO2 P 1 ba
h
NRT Ÿ
P
NT
2
N 2 O 4 P 1 ba
2
NO2 P
h
N 2 O 4 1 ba
P0
N 0T0
a
f
bc
0 de
4 X 2 P 1 ba
1 X 1 X
he e he
.
ba 1 X T
FG N IJ FG T IJ 1013
H N KH T K
300
ŸP
P0
Ÿ Ka
4 X 1 X T ˜ 1013
.
1 X 1 X T0
X2
1 X
T0 Ka
4 ˜ 1013
. ˜T
0
2
0
D
4 ˜ 1013
. ˜ X 2T
1 X T0
X 2 D 1 X
0
e
S
C
ca a d E
T
d
D
b
V
Ka 1
T
T
T
¦Q
1Ÿ '
F
he d d a c
che ca eac
e
e
. Th
'
U
RT 2
0Ÿ X
D
U
'
g e he c
; he a e
e ha
e'
¦Q
(2)
1
«X «
1 4 D
¬
º
» (3)
»
¼
'
H RT
U
2 ª
H RT
'
(
e f ac
he e ha ce e
U de e de ce
RT 2
)
e gh ed hea ca ac
f
f he hea ca ac d e
he
ce ha a
fe eg
ab bed a he eac
e
b
h f , a d a ec d ' U de e de ce,
de e
e he e e
fa hf
he e
b
h e e a e.
(b)
U g CHEMEQ a d he da a
A e d A.II (f CP* ),
de e
E
ed a each e
ea
ea
. (1) a d (3) X a d CV,eff a
T (K)
300
350
400
450
500
550
600
700
D
g
h CV f
0.044275 0.1891
1.0016
0.6177
10.0613 0.9163
58.883
0.9835
236.25
0.9958
721.36
0.9986
1796.25 0.9994
7220.4 0.99986
ce h
Ka
b h NO2 a d N2O4 . The , f
e . The e a e ab a ed a d
X
(1)
. A
1 X CV,N 2O4 XCV,NO2 Ÿ CV,eff
Chap e 13
d
¦Q U ¦Q H RT
U
'
c ,5
RS 1 4 1UV
2T
D
W
LMD RS 1 4 1UVOP LM X 1 OP D
T N2 T
D
W Q N 1 a4 D f Q T
X DX D
F XI
H TK
a
ed be
.
P (ba )
CV,NO2
CV,N2O4
CV,eff
1.205
1.912
2.588
3.014
3.370
3.712
4.051
4.727
37.11
38.94
40.65
42.22
43.67
45.00
46.23
48.36
78.83
84.47
89.55
94.06
98.01
101.39
104.20
108.12
J/
K
410.5
546.9
195.9
69.11
49.23
46.49
46.71
48.45
a
S
C
ca a d E
T
d
a
c ,5
Chap e 13
d
13.23
a) U g he De b gh e h d, e ha e
5C + 12H = C5
(1)
5C + 12H = C5
(2)
5C + 12H = e C5
(3)
S ce
C H
e e , b ac E . (1) f
E . (2 a d 3)
ba
C5 = C5
a d
C5 = e C5
The e a e he
de e de eac
.I h
e , he e a e 3 c
e ( he h ee
e a e ), C=3, e ha e, P=1 a d
de e de eac
M=2. The ef e he
be
f deg ee f f eed
a e F=C-P-M+2=3-1-2+2=2 deg ee f f eed .
b) F h
b e , ce T a d P a e ec f ed, he
be
e -def ed a d h
be a
e
. The e
b
ea
ha
ha e be
ed a e:
a C5
a e C5
C5 P /1 ba
C5
e C5 P /1 ba
e C5
.
K a,1
; a d K a,2
a C5
a C5
C5 P /1 ba
C5
C5 P /1 ba
C5
T ca c a e he e
b
c
a , e e he V
f
g a e Ka,1=0.18240 a d Ka,2=0.06737.
a BASIC
ga ,
e ba
he
S
C
ca a d E
K1 0.18240
T
d
a
c ,5
Chap e 13
d
K2 0.06737
c5 ( X1 X2) 1 X1 X2
c5( X1 X2) X1
X1 0.1
e c5 ( X1 X2) X2
X2 0.1
G e
X1
K1
K2
1 X1 X2
f d( X1 X2)
X1 0
X2 § 0.146·
¨
¸
© 0.054¹
1
c5 ( X1 X2)
0.8
c) P e
N
e ha
0.
13.24
The eac
¦
e
b
c
Ka
)F
X2
1 X1 X2
c5( X1 X2)
effec
he e
PCH 4
a CH 4 ˜ a 5.75
H 2O
ac
e c5 ( X1 X2)
b
c ce
RT
A 283 K, K a
'G f
Ka
a
f he
˜
H 2O ˜ J H 2O
f
ce he
h ch he
5.75
1 ba
f he
be , ae
he
1 ba
. The ef e K a
. A 278 K, K a
PCH 4
a
8.314 ˜ 278 ˜ (0.0238) 8640
1 ba
6.8 MPa ˜
e
1.0
ha
'G f
0.054
CH4 g 5.75H2O( ) R CH 4 ˜ 5.75H2O( )
a
ah d a e
h
0.146
1 ba
0.1 MPa
ee ,a d
1 ba
1 ba
4.2 MPa ˜
0.1 MPa
e
0.0238
.
0.0147 a d
8.314 ˜ 838 ˜ (0.0147) 9929
b) Need c e 10 % CH3OH
e gh f a e
be 18, a d e ha
J
d
J
ae
e f ac
. Ta g he
ec a
be 32, a d f 100 g f
ha 10
%
90
18
0.9412 . A
e ha , he a e
e f ac
e ha he e ha
H 2O
90 10
18 32
+ ae
e ca be de c bed b he Va Laa
de
h aa ee g e
Tab e
9.5-1 ( h ch a e a d a 25 C, b
e
a
e a e e e a e de e de ),
D a d E=0.46 he
e ha
c
e 1. Th ead
a ac
c eff c e
f a e f 1.002. (We c d c
de he e f he UNIFAC
de
ead, h e e ,
S
C
ca a d E
T
he a e ac
c eff c e
he e
b
ea
ah d a e
Ka
PCH 4
a CH 4 ˜ a 5.75
H 2O
˜
1 ba
d
c
a
e
,
1.0
H 2O ˜ J H 2O
c ,5
e d ffe e ce
PCH4
5.75
Chap e 13
d
1 ba
d be ee .) S
1.0
˜ 0.9412 ˜1.002
ha
1.0 ba
PCH 4 ˜ 0.714
5.75
0.1 MPa
0.1 MPa
ha a 278 K PCH4
5.885 MPa
K a ˜ 0.714
0.0238 ˜ 0.714
0.1 MPa
9.528 MPa
a d a 283 K PCH4
0.0147 ˜ 0.714
1
1
H 2 g I 2 g a d I2 g I2 .
13.2
The
eac
a e HI g
2
2
U g he da a
he C
a E
Ha
e ha e
PCH4
,1 G
'
1.95 ca =8.159 J
°­ ' ,1G °½
®
¾
¯° RT
¿°
e
K a ,1
e
K a ,2
S ec e
I
a
HI
1
H2
0
I2
0
6
N e: a da d a e
e
e
2 Ka , 1 1
Th a a
ed) e ha e
e
e (
2478.3
2
e
2
aHI
aI 2
aI 2 g
e a de he a
c
d e
b e , ha , a
cha ged.
eac
1
.
a
1 X
1 X
1
X
2
1
X
2
1
he C
1
X
2
1
X
2
a1 2f X a P 1 a f
1 X aP 1 a f
2 Ka , 1
g
a1H 2 a1I 2
2
F a
12 2
. u 102
Ÿ 372
3.72 u 10
­° ' ,2 G ½°
®
¾
¯° RT
¿°
S d ec a
f e
c a
he e
e a h ch he f
b f
d a ea
The ef e, he f
e
c
e he a
c
4.63 ca =-19.37 J
,2 G
'
1 X P
1a
1 2 XP
1a
1 2 XP
1a
a f
a f
a E
1a
Ha
X
2 1 X
Ka,1
0.069248 ( de e de
f
e
e!)
gh ha
ha e c
ec
dea
.
a
ca be
S
C
ca a d E
T
d
H2
h ch
cc
.
N
I2
he c
c
d
0.03462 ;
HI
he f
1
aI 2
I2
2478.3
1
I2 P 1 a
Th , f P ! 0.01181 ba
2NaHCO3
ec
a
ha e a d N H 2 O
P
CO2
f
0.01181 ba .
cc .
H2O
UV
W
b
aNaHCO 3 1
aNaCO 3 1
Ÿ Ka
aH 2 OaCO 2
PH 2 O PCO 2
˜
1 ba 1 ba
he e N
a
aNaCO 3 aCO 2 aH 2 O
aNaHCO 2
N
P
N
N CO2 Ÿ PH 2 O
PCO2
The ef e
d I2
f
1a
Ka , 2 I 2
ŸP
d I2
f
3
P
a
F PI
H1 a K
0.01166 a
a
NaCO3
Ka
b
ec
:
1a
0.03462 u 2478.3
P
093075
.
he
de he ec d eac
Ÿ Ka , 2
Chap e 13
c ,5
f he a
aI 2
13.2 (a)
a
e
d
ha e, N
ga
e
f
ga
1
P.
2
Ka
LM a1 2f P OP
N 1 ba Q
2
a d
LM a1 2f u 0.826 Pa OP 1706
N 100 Pa Q . u 10
a1 2f u 166.97 Pa O 0.697
K 110q C LM
N 100 Pa PQ
2
Ka 30q C
5
2
a
Ka 30q C
10.979;
0.3610
Ka 110q C
N
K a T2
K a T1
K a T2 K a T1
'
R
H§ 1 1·
¨ ¸
© T2 T1 ¹
(1)
S
C
ca a d E
T
d
Th a
e ha ' H
ed da a
he
be
h he
a
c ,5
de e de
ae e
Chap e 13
d
f T, he
a
e ca
a e
Ÿ ' H 128.2 J
(b)
Ka T
Ka
G
g bac
E
. (1) ab e e ha e
§ ' H§1
1 ··
5
¨
¨ ¸ ¸ 1.706 u10 e
R © T 303.15 ¹ ¹
©
15420
15420
10.9788 508659
39.8871 .
T
T
Ka T
30 C e
Ÿ Ka (T )
ŸT
13.2 Reac
U
10
. ba Ÿ PH 2O
PCO2
(c)
LM a1 2f u 2.0 OP
N 1 Q
2
g he
ga
P
PCO2
1 ba .
CH4
CHEMEQ e ha e Ka T
S ec e
I
a (ga )
1000 K
H2
1
1 2 X
CH4
0
X
6
1 X
aCH 4
aCaH2 2
aCH 4
aH2 2
009838
.
F a (ga )
C
Ka
2.0 ba
1
3866
. K 11345
. qC f
C 2H2
10
. ba
§
1 ··
§1
¨ 15420 ¨ ¸¸
© T 303.15 ¹ ¹
©
1 2 X
1 X
X
1 X
F
H
XP
1 X 2 1 ba
˜
P
1 X 1 ba 1 2 X 2
I
K
2
X 1 X
X 1 X 1 ba
1 2 X 2 P
1 2 X 2
h e a
a e X 0.0769 a d X 0.9231 .
a a e f he e
b
c
a , he X 0.0769
0.917 a d CH 4 0.083
e
H2
01004
.
c
The
W h ch a
ec
e. Th
Th
( bab
h e e e a a d ca c a
a e
)e e
a e a he e
b
c
. The ef e, he eac
ce
he
d a ca
ed,
a
a fe
ed. C e e
c ea
e
ba
eb
g he h d ge f
ha e
effec
he
13.2
Reac
(a a a ab e a a Ma hcad
hee )
he
a
he
g he
ce .
S
C
ca a d E
CO H 2
C 2H 2O
CO2 C
CO2 2H 2
2CO
CO H 2O
CO2 H 2
c
, e
e eac
CO H 2
C H 2O
CO2 2H 2
CO2 C
C 2H 2O
2CO
CO2 H 2
CO H 2O
N
eed
Sa b
de
g
f
he
CO
CO
E
2H O
C 2O
H 2O
CO2
eac
ce
H2
H
2H
e O
d
C H 2O
F
N
T
4
che
C 2O
CO2
ce
e e e
ca eac
CO
H 2O
aeO
Chap e 13
d
CO
Ÿ H2 O
ee
e
c ,5
1, 2 a d 4
de e de
ae
1
1
CO2 C
2
2
a
a
c
1
CO2
2
1
H 2 CO2
2
1
C CO
2
1
C H 2O
2
C CO 2
UV Th e e f
C 2H O W de e de eac
C
2H 2 CO 2
.
ge
ee .
2CO
2
.
Add he e
2H 2 C 2CO 2
We
2CO C 2H 2O Ÿ H 2 CO2
CO H 2O
e
H 2 CO 2
2H 2 CO 2
UV a he
C 2H O W de e de eac
CO H 2 O
2
.
S
C
S ec e
ca a d E
I
T
a
d
1 X1 X 2
H2
1
1 X1 2 X 2
CO
0
X1
H 2O
0
X1 2 X 2
C
0
aa
Ka,2
aCO aH 2 O
f
2
2
fa
1
2
1
2
CHEMEQ, I f d he f
2
1
2
g
Ka,2
758.6
48.43
5.950
1.137
0.2974
1
03665
.
u 10
0.1110
0.2493
0.4596
0.7387
X2 0
de
Ka,1
a
f
2
2
Ka,1
600
700
800
900
1000
N ca b
f
a X 2 X f a2 X f 1 ba
a1 X 2 X f a1 X X f P
1 a H2 2 O
C
a
a
aCO 2 aH 2
T K
de
1
1
X 1 X1 2 X 2
1 X1 X 2 1 X1 2 X 2
a H2 2 aCO 2
ga
g he e e
2
1
X1 2 X 2
¦
0
2
1
ga ha e)
¦ 2 X2
Ka,1
S
LM1 X X OP P
N ¦ Q 1 ba
LM1 X 2 X OP P
N ¦ Q 1 ba
LM X OP P
N ¦ Q 1 ba
LM X 2 X OP P
N ¦ Q 1 ba
1 X1 X 2
¦
1 X1 2 X 2
¦
X1
¦
aX f
(
(a)
Chap e 13
d
a
1
g he
c ,5
F a
CO 2
U
a
X 12
a1 X f
1
X 12 ˜ 2 1 ba
1 X1 3 P
a d Ka,2
2
a
f
,If d
TK
600
700
800
900
1000
P ba
1151
. u 104
6111
. u 103
0.126
1.357
9.237
If he
.
e
ef
ag e
e
ea
e
ab
e he
e
e ca c a ed, ca b
S
C
(b)
ca a d E
E ac
T
30% f ca b
Ka,1
a
TK
X1
P ba
a
de
d
a
c ,5
ed X 2
03
.
f
Chap e 13
d
f
a 0.6f 17. 1 ba
a0.7 X fa0.4 X f P
600
700
800
900
1000
0.0157
0.0410
0.0750
0.1104
0.1427
0.0084
0.170
1.972
15.266
85.419
X 1 X 1 0.6
a d Ka,2
0.7 X 1 0.4 X 1
fa
2
1
1
1
2
13.2
The eac
he e g ee
c ce ed ab
T ( ) + S O 2( ) = T O 2( ) + S ( )
Th
e
ae
he f
eac
he
be
ae e
eac
. The ef e
J
J
J
' G
674 (644)
30
30, 000
a d he e
Ka
C
e
b
c
a
f
h
he ec
d
eac
§ ' G ·
§ 30000 ·
¨¨ ¸¸ e ¨
¸ 17.02
R ˜T ¹
© 8.314 ˜1273 ¹
©
e e
, a he e g ee fea , he a
e a e c ac
h c d
de.
e
be affec ed b h gh
13.30
a)
R 8.314
T 80 273.15
1203.828 ·
§
¨ 4.01816
¸
T
53.229 ¹
P a B( T) 10©
1209.299 ·
§
¨ 3.98022
¸
T 49.623 ¹
P a C( T) 10©
B 0.2
P( T) P a C ( T)
0.991
2
ª
(1 ) º
« 100˜
»
R˜ T ¼
¬
ga C( T) e
2
ª
( ) º
« 100˜
»
R˜ T ¼
¬
˜ ga B( T) ˜ P a B ( T) ( 1 ) ˜ ga C( T) ˜ P a C( T)
P( 0.2 353.15)
1
ba
( ˜ ga B( T) ˜ P a B( T) )
P( T)
So b=0.206 and nC=0.794
) The
1.01
C 1 B
ga B( T) e
( T) P a B ( T)
a
ba a ce ab e :
( 0.2 353.15)
0.206
S
C
ca a d E
T
a
0.794
0.206
0
f a
0.794-X
0.206+X
3X
S ec e
C6
be e e
h d ge
a
S
d
a
c ,5
Chap e 13
d
e f ac
(0.794-X)/(1+3X)
(0.206-X)/(1+3X)
3X/(1+3X)
1+3X
ha he e
b
ea
a 3H a B
aC
Ka
3
3
§ H P · § BP ·
¨
¸ ¨
¸
© 1 ba ¹ © 1 ba ¹
§ CP ·
¨
¸
© 1 ba ¹
§ 3X · § 0.206 X ·
¨
¸ ¨
¸
© 1 3X ¹ © 1 3X ¹
§ 0.794 X ·
¨
¸
© 1 3X ¹
From the Vis al Basic chemical eq ilibri m program
Ka 0.064608
X 0.1
3
1
1 º
· ª
§
¨ 3˜ X˜
¸ ˜ «( 0.206 X) ˜
»
1 3˜ X ¹ ¬
1 3˜ X¼
©
G e
Ka
( 0.794 X) ˜
X f d( X)
cC H
X
( 0.799 X)
1
1 3˜ X
0.248
cC 0.316
1 3˜ X
( 3˜ X)
cB ( 0.201 X)
1 3˜ X
cB
0.257
H 0.427
1 3˜ X
c)
Since h drogen is so far abo e its critical point, onl a negligible amo nt of h drogen ill
appear in the liq id phase,.
PP 1
g e
B˜ ga B( B T) ˜ P a B ( T)
0.257˜ PP
cC˜ ga C( B T) ˜ P a C ( T)
0.32˜ PP
B cC
1
f d( B cC PP)
§ 0.44 ·
¨ 0.56 ¸
¨
¸
© 1.747 ¹
So the de point press re is 1.77 bar, and the liq id that forms has a mole fraction
of ben ene of 0.448, a mole fraction of c clohe ane of 0.552, and (b ass mption), a
negligible amo nt of h drogen.
S
C
ca a d E
T
d
13.31 The che ca eac
CH3-CHOH-CH3 = CH3-CO-CH3 + H2
A
g e a
h
e ace e, he
(g e he h gh e e a e a d
e
S ec e
ace e
h d ge
6
B X = .564,
Ka
aace aH
a P
0.8043 c ,5
Chap e 13
d
a ba a ce ab e
e)
ha
ec e a
a
a
1-X
(1-X)/(1+X)
(1-X)u95.9/(1+X) )u100
X
X/(1+X)
Xu95.9/(1+X) )u100
X
X/(1+X)
Xu95.9/(1+X) )u100
1+X
aI-P=0.2673, a d aace = aH =0.3458. The ef e
1
0
0
0.3458 u 0.3458
0.2673
'
a
G
;
RT
'
G
0.4474
§ ' G ·
¨¨ ¸
RT ¸¹
©
e
8.314 u 452.2 u 0.8043 3023.8
J
13.32 The eac
ae
C6H6 + H2 = 1,3-c c he ad e e
C6H6 + 2H2 = c c he e e
C6H6 + 3H2 = c c he a e
The G bb f ee e e g
ff
J
' f G (be e e) 124.5
' f G (c c he a e)
26.9
a
da a eeded
e h
' f G (c c he e e) 106.9
J
be
J
' f G (1, 3 c c he d e e) 178.97
J
The G bb f ee e e g f f
a
f 1,3-c c he ad e e
a a ab e
IV, Pe
The Che ca E g ee Ha db
he Ha db
f Che
Ph c . The a e a f d
g da a
he WWW e
h :// ebb
. .g /che
. Th Web e c a
he Na
a I
S a da d a d Tech
g (NIST) che
da a b
. The a e f d
ea e
J
' f H (1,3 c c he d e e, 298.15 K) 71.41
S (1,3 c c he d e e, 298.15 K) 197.3
S (C,g a h e, 298.15 K)
S (H 2 , 298.15 K) 130.68
5.88
J
J
A
e d
a d
e f
h Web
J
˜K
˜K
˜K
N e ha he e e
e ae
h e ec
he e
c
e a d 0 K. A , he e
cha ge f eac
The ef e
e
a
e f he
a 0K
e f a
e
eac
.
S
C
ca a d E
T
d
a
c ,5
Chap e 13
d
' f S (1,3 c c he d e e, 298.15 K)
S (1, 3 c c he d e e, 298.15 K) 6 ˜ S (C, 298.15 K) 4 ˜ S (H 2 , 298.15 K)
197.3 6 ˜ 5.88 4 ˜130.68
360.75
' f G (1,3 c c he d e e, 298.15 K)
J
˜K
' f H T 'f S
178967
J
178.97
71410 298.15 ˜ 360.75
J
The a ba a ce ab e a
g a he ga c a e e e
he
d ha e, a d
ha he h d ge
ee
g ea e ce
ee
a a e
e f ed a 1 ba .
A , ce a he ga c a e
a, e
a
e he f
a dea
d
e, a d ha he e
h d ge
he
d ha e. Ma ba a ce ab e f
d
ha e:
S ec e
I
O
Be e e
1,3-c c he
c c he e e
c c he a e
T a
1
0
0
0
1-X1-X2-X3
X1
X2
X3
1
The e
ea
b
a1,3 c c
Ka,1
abe aH 2
e
Ka,2
( 21973
. )
Ka,3
X2
1 X1 X 2 X 3
(7.1002)
1212.2
ac cha e
X3
1 X1 X 2 X 3
abe aH3 2
e
(39.374)
1-X1-X2-X3
X1
X2
X3
1-X1-X2-X3
X1
X2
X3
e
F 178970 124500 I
H 8.314 u 29815
K
.
e
F 106900 124500 I
H 8.314 u 29815
K
.
e
F 26900 124500 I
H 8.314 u 29815
K
.
2.866 u 1010
ac che e
abe aH2 2
e
ae
X1
1 X1 X 2 X 3
ac
12587
u 1017
.
B e a
g he a e f he e
b
c
a
he e e a
, e ee ha X3 ab
e a
f he e e a
he h d, e ha e
10
X 1 2.866 u 10
2.27 u 1027 | X 1
X 3 12587
.
u 1017
a d b a g he a
f he ec d f he e e a
X2
1212.2
9.63 u 1015 | X 2
17
X 3 12587
u 10
.
,
e d ec b a
. The b a g he a
he h d
Th
gge
ha X3 1, X2
f he de
10-17, a d X1
f de
-27
10 . Th he be e e
eac
f
e e a a c c he a e.
13.33
(a
a a ab e a a Ma hcad
hee ).
g a
f
f he f
S
C
ca a d E
T
G
2400 e
R
8.31451 aA
A J A
AJ A
aC
A J C
1
Ka
e
K
e
T
Chap e 13
d
298.15 K
V
4 e
1
Ka
2.633
a D 0.5 10 Pa a C
A 1
R T
c ,5
e
a D( P )
G
a
e
aB
A J C
d
P
5
10 Pa
Part (a):
A
(initial g ess)
0.5
5
G e
Ka
C
1
A
0.16
a B a A
F d
A
A 1
A
A
C
0.84
Part (b):
Recogni ing that the partial molar Gibb's e cess is in the form of the one constant Marg les
e pression ields:
2
0.3 C
J A e
G e
J C e
1
A
0.132
2
5
a D 0.5 10 Pa a C
Ka
C
0.3 A
a B a A
A
C
0.868
A e
A e
0.3 A
0.3 1
2
A
2
A
F d
A
S
C
ca a d E
T
d
a
c ,5
Chap e 13
d
Part (c):
Mass Balance Table:
Species
A
B
C
D
In
1
2
0
0
O t
1-X
2-X
X
X
Ass ming that D is an ideal gas:
P=(n*R*T)/V
P( X )
e X R T a hee
13.3
(a a a ab e a Da Ma hcad
C( X). 1 )
V
F e he S ea TabKae
P a = 12.349
Pa
G
F
CHEMEQ Ka = 171.2 a B a A ( 1 X 1 )
HNH3 (f
be
a e e ) = 384.5 Pa/
e f ac
a
NA
1
X
A
1
X
NA
0.575
A
0.575
V
(initial g ess)
X 0.5
13.34 Need
X
The 0.425
X R T
a
e
e
P ( Pa)
NB
0.10
0.25
0.50
1.00
2.00
4.00N B
6.00
8.00
10.00
12.00
14.00
16.00
18.00
18.40
18.50
19.00
20.00
25.00
30.00
35.00
40.00
50.00
2
X
1.575
X
F d( X)
X NC X
0.0899
0
0.2017 C X
0
0.3467
0
0.5478
0
0.7868
0
1.0322 N C 0.425 0
1.1670
0
1.2559 C 0.425
0
1.3204
0
1.3701
0
1.4099
0
1.4429
0
1.4707
0
1.4757
2.02u10-05
1.4797
0.0114
1.4996
0.0680
1.5339
0.1589
1.6437
0.3989
1.7063
0.5068
1.7488
0.5713
1.7806
0.6162
1.8267
0.6792
ND
PD
X
X R T e
V
ND
0.425
PD
5
2.633 10 Pa
S
C
ca a d E
T
d
60.00
70.00
80.00
90.00
100.00
104.50
a
c ,5
Chap e 13
d
1.8602
1.8869
1.9093
1.9291
1.9472
1.9449
0.7261
0.7661
0.8030
0.8388
0.8745
0.8908
2.5
X and f
2.0
1.5
X, molar e tent of reaction
f, fraction liq id
1.0
0.5
0.0
0
20
40
60
80
100
HCONH 3 R NH 3 CO , f
h ch he
Press re, kPa
13.34
ab e
The che
S ec e
HCONH2
NH3
CO
ca eac
a
0
0
f a
1
X
X
T a
A
, P
NRT
V
a NH3 ˜ a CO
Ka
a HCONH3
=
1-X
X2
ba ˜ 3 1
˜ 8.314 u105
u
˜ K 25
1 X
1 X ˜ 8.314 u 105 ˜ T
25 u 10 3
ba
2
RT
§ X ·
¨
¸ ˜ 1 X
V
© 1+X ¹
1
X
§
·
¨
¸
© 1+X ¹
u
103
3
uT
ba a ce
e f ac
(1-X)/(1+X)
X/(1+X)
X/(1+X)
1+X
1 X RT
1 X P
V
§ NH3 ˜ P · § CO ˜ P ·
¨
¸˜¨
¸
© 1 ba ¹ © 1 ba ¹
§ HCONH3 ˜ P ·
¨
¸
© 1 ba
¹
a
X 2 RT
1 X V
S
X2
1 X
C
ca a d E
',
he e ' =
T
d
' X'; X 2 'X ' 0, a d X
U
g he che
N
ca e
b
Ka
7.8184u105
4.5987u105
2.8598u105
1.8660u105
1.2695u105
8.9571u104
c) I
a
PV
RT
be
c
e
HCONH2
fa
ba ˜
u 10-3
3
A d he
T(K)
P(ba )
e
CO
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
1.022
e a
u 298.15 K
ge a d 0.807
ce he e
b
e
c
f
a
ge .
3.022 u 8.315 u 105 u T
25 u 10 3
ha
480
4.82
500
5.02
NH3=0.331
CO=0.331
O2=0.071
N2=0.267
e
P
NRT
V
400
4.02
P(ba )
2.66
2.794
2.927
3.06
3.193
3.326
3
˜K
The ef e, he e
be 0.215
e f
The c
a a e ea e (
eac
g
c
e
)
1.0
e NH3
1.0
e CO
0.215
e O2
0.807
e N2
NH3
0
0
0
0
0
0
1.013 ba u 25
8.314 u 10 5
Chap e 13
d
' ' 2 4'
2
ga
c
e K a, e b a
a
X
1
1
1
1
1
1
f
c ,5
K a u 25
8.314 u 102
X2
T(K)
400
420
440
460
480
500
a
420
4.22
440
4.42
460
4.62
a ge ha he
S
C
ca a d E
T
d
a
c ,5
Chap e 13
d
13.3
(a a a ab e a a Ma hcad
hee ).
F
he S ea Tab e
P a = 12.349 Pa
F
CHEMEQ Ka = 171.2
HNH3 (f
be
a e e ) = 384.5 Pa/
e f ac
The
a a
e
e
P ( Pa)
X
0.10
0.0899
0
0.25
0.2017
0
0.50
0.3467
0
1.00
0.5478
0
2.00
0.7868
0
4.00
1.0322
0
6.00
1.1670
0
8.00
1.2559
0
10.00
1.3204
0
12.00
1.3701
0
14.00
1.4099
0
16.00
1.4429
0
18.00
1.4707
0
18.40
1.4757
2.02u10-05
18.50
1.4797
0.0114
19.00
1.4996
0.0680
20.00
1.5339
0.1589
25.00
1.6437
0.3989
30.00
1.7063
0.5068
35.00
1.7488
0.5713
40.00
1.7806
0.6162
50.00
1.8267
0.6792
60.00
1.8602
0.7261
70.00
1.8869
0.7661
80.00
1.9093
0.8030
90.00
1.9291
0.8388
100.00
1.9472
0.8745
104.50
1.9449
0.8908
2.5
X and f
2.0
1.5
X, molar e tent of reaction
f, fraction liq id
1.0
0.5
0.0
0
20
40
60
Press re, kPa
80
100
S
C
ca a d E
T
d
a
c ,5
Chap e 13
d
13.3 a) The eac
2H 2S CH 4 R CS2 4H 2
F
he he che ca e
b
c
a
g a , e ha e he a e be
e
b
c
a (A ,
a e ace, I h
e f he a e ca c a ed
T(K)
Ka
X
773.15
863.15
923.15
1013.15
1073.15
4.7915u10-7
2.9923u10-5
3.0602u10-4
6.0722u10-3
3.3960u10-2
0.023
0.052
0.080
0.137
0.183
b)
S ec e
H2S
CH4
CS2
H2
1
1
0
0
a
CH4
CS2
H2
0.466
0.426
0.388
0.319
0.268
0.477
0.451
0.426
0.379
0.345
0.011
0.025
0.037
0.060
0.077
0.045
0.099
0.149
0.242
0.310
f a
1-2X
1-X
X
4X
T a
(
e f ac
)
(1-2X)/(2+2X)
(1-X)/(2+2X)
X/(2+2X)
4X/(2+2X)
2+2X
A 1 ba
Ka
H2S
f he
a b)
4
4X
X
˜
2 2X 2 2X 4
44 X 5
2
64X5
2
1 2X 1 X 2 2X
§ 1 2X · 1 X
¨
¸
2
2X
2
2X
©
¹
The
f X a d he
e f ac
a eg e
c) A 10 ba (a
g he a
ha e
dea )
2
1 2X
2
he ab e ab
1 X 2 2X
2
e.
4
Ka
5
4X
X
§ 10 ba ·
˜
¨
¸
2 2X 2 2X 4 © 1 ba ¹
2
3
§ 1 2X · 1 X § 10 ba ·
¨
¸
¨
¸
© 2 2X ¹ 2 2X © 1 ba ¹
The e
ae
he ab e be
T(K)
X
773.15
863.15
923.15
1013.15
1073.15
9.611u10-3
0.021
0.033
0.059
0.082
100 ˜ 64 ˜ X5
1 2X
2
1 X 2 2X
H2S
CH4
0.486
0.469
0.452
0.416
0.386
0.490
0.479
0.468
0.444
0.424
S he eac
e ed b he h ghe
LeCha e e
c e a d he eac
ch
he a
ha e).
2
CS2
e
4.76u10-3
0.010
0.016
0.028
0.038
e
H2
0.019
0.042
0.065
0.112
0.151
e, a
d be e ec ed f
( ha c ea e he
be f
e
S
C
ca a d E
T
13.3 (a
a a ab e a a Ma hcad
d
a
Chap e 13
c ,5
d
P2
10 Pa
hee ).
Gi en:
298.15 K
T1
5
10 Pa
P1
e
' G C3H8
24300
' H C3H8
104700 e
e
e
650 K
T2
' G CH4
50500 ' H CH4
74500
e
e
e
e
8.31451 R
6
' G C2H4
68500 ' H C2H4
52500
e
K
e
e
(at 298.15 K)
e
e
Mass Balance Table:
Species
In
O t
C3H8
CH4
C2H4
Total
1
0
0
1-X
X
X
1+X
(1-X)/(1+X)
X/(1+X)
X/(1+X)
Calc lation of mole fractions and acti ities:
C3H8 ( X )
1
X
1
X
CH4( X )
P
aC3H8( X P )
C3H8( X ) aC2H4( X P )
C2H4( X ) X
1
CH4( X ) aCH4( X P )
5
10 Pa
C2H4( X )
X
10 Pa
P
5
10 Pa
' G CH4
' G C3H8
'G
4
4.23 10
e
'H
' H C2H4
' H CH4
' H C3H8
'H
4
8.27 10
e
'G
e
Ka 298.15
R T1
Part (a):
X
G e
Ka 298.15
C3H8( Xa )
1
10
4
3.885 10
1
e
1
e
8
(initial g ess)
aC2H4( X P1 ) aCH4( X P1 )
aC3H8( X P1 )
C2H4( Xa )
1.971 10
4
Xa
F d( X)
CH4( Xa )
1.971 10
Xa
1.971 10
Xb
0.854
4
Part (b):
From eq ation 9.1-22b:
Ka 650
X
Ka 298.15 e
.5
'H
R
1
T2
1
Ka 650
T1
2.704
(initial g ess)
G e
Ka 650
C3H8( Xb )
0.079
X
P
' G C2H4
Ka 298.15
X
1
5
'G
aC2H4( X P1 ) aCH4( X P1 )
aC3H8( X P1 )
C2H4( Xb )
0.461
Xb
CH4( Xb )
F d( X)
0.461
e
4
S
C
ca a d E
T
d
a
c ,5
Chap e 13
d
Part (c):
X
(initial g ess)
.5
G e
aC2H4( X P2 ) aCH4( X P2 )
Ka 650
C3H8( Xc )
Xc
aC3H8( X P2 )
0.369
C2H4( Xc )
13.3
(a
R
8.31451 aA
A J A
AJ A
aC
A J C
1
Ka
e
0.316
CH4( Xc )
a a ab e a a Ma hcad
e
K
e
T
V
4 e
P
5
10 Pa
Ka
2.633
a D 0.5 10 Pa a C
A 1
RT
298.15 K
1
a D( P )
G
0.461
0.316
e
aB
A J C
Xc
hee ).
2400 G
e
F d( X)
Part (a):
A
(initial g ess)
0.5
5
G e
Ka
C
1
A
0.16
a B a A
A
A 1
F d
A
A
0.84
C
Part (b):
Recogni ing that the partial molar Gibb's e cess is in the form of the one constant Marg les
e pression ields:
J A e
2
0.3 C
J C e
0.3 A
5
G e
a D 0.5 10 Pa a C
Ka
C
1
A
0.132
a B a A
A
C
0.868
A e
2
A e
0.3 A
0.3 1
2
A
2
A
F d
A
S
C
ca a d E
T
d
a
c ,5
Chap e 13
d
Part (c):
Mass Balance Table:
Species
A
B
C
D
In
1
2
0
0
O t
1-X
2-X
X
X
Ass ming that D is an ideal gas:
P=(n*R*T)/V
X
P( X )
X R T
V
(initial g ess)
0.5
G e
Ka
X
0.425
NA
1
X
A
1
X
NA
0.575
A
0.575
aD
X R T V
a Ba A ( 1
NB
NB
2
X
1.575
e a C( X 1 )
X
X 1)
F d( X)
NC
X
ND
X
C
X
PD
NC
0.425
ND
0.425
C
0.425
PD
5
2.633 10 Pa
X R T e
V
S
C
13.3
(a
ca a d E
T
d
a
a a ab e a a Ma hcad
c ,5
Chap e 13
d
hee ).
Witho t dissociation:
a h drogen gas is eq al to the press re of h drogen gas:
Ass
ming
it ofh molec
A
_Adthe acti
bed_W
K1lar
H2
A
_Ad
K1 P H2
bed_W h
With dissociation:
A
_Ad
bed_W h K1 a H2
1
2
K3 a H
Using the eq ilibri m constant for the reaction H2 = 2H, the acti it of atomic h drogen can be
sol ed for in terms of the acti it of molec lar h drogen:
2
G e
K2
aH
F d aH
a H2
Using the positi e root for the acti it of h drogen ields:
A
_Ad
bed_W h K1 a H2
1
2
K3 K2 a H2
Ass ming the acti it of molec lar h drogen gas is eq al to the press re of h drogen gas:
A
_Ad
bed_W h K1 P H2
1
2
K3 K2 P H2
If the amo nt adsorbed aries linearl
ith the partial press re of molec lar h drogen then no
dissociation is occ rring. If the amo nt adsorbed aries as the sq are root of the partial press re,
then dissociation is occ rring.
13.40
F
he da a
he
be
G
'
H T'
S
'
'
The e
b
c
a
§ ' G ·
K e ¨
¨
¸¸
© R ˜ 298.15 ¹
The e
b
c
a
e
ae h
be
ae e
H
'
e ha e ha
G T'
S
148
a 25 C
e
19600
§
·
¨
¸
© 8.314 ˜ 298.15 ¹
a af c
f e
2716
ea
e
b a ed f
E .13.1-22b; he
S
C
ca a d E
T
d
a
c ,5
Chap e 13
d
Eq ilibri m constant
5
1 10
4
1 10
3
1 10
K
100
10
260
280
f
he eac
The
a
ba a ce ab e f
S ec e
M
e
D e
I a
M
0
Ka
aD
2
aM
b
h
h ch ca be
2
eac
F a
M(1-2D)
DM
ea
DM
1 2D
320
2A U A2
Th
The e
300
T
D
M
2
1 2D
2
M
e a
1 ·
§
4D 2 ¨ 4 ¸ 1 0
M ˜K ¹
©
a d ha he
2
D
The
1 ·
1 ·
§
§
¨4
¸ ¨4
¸ 16
˜
˜K ¹
M
K
M
©
¹
©
8
f
he f ac
d
e
ed a a f
c
f e
ea
e
g e be
:
S
C
13.41
(a
ca a d E
H N2 N2 e a
aN2 ga
P N2
H N2
A P N2
ge
Process #2:
aN e a
Chap e 13
d
hee ).
here
A
2 Ka 1
H N2
N2(gas) =2*N(metal)
2
H N N e a
aN2 ga
2
P N2
Ka 2 P N2
N ea
%
c ,5
Ka 1 P N2
N2 e a
Ka 2
a
N2(gas) = N2(metal)
aN2 e a
%
d
a a ab e a a Ma hcad
Process #1:
Ka 1
T
HN
ge
B P N2
here
B
Ka 2
HN
The empirical e pression gi en in the problem is s pported b process #2.
13.42 a) F
he che
ca e
b
a CO 2
P
1 ba
CaCO3 R CaO CO2
Ka
1.1395
a CaO a CO 2
a CaCO3
The ef e, he e
b
a d e f ha a d
ba e e a a ed
N
e
e
c a ed. The
c
a
ga
a 1150 K f
he eac
ce a CaO =a CaCO3 =1 a he a e
e
d .
he c
de
1.1395 ba f a CaCO3
ee ,
e f CO2 ha
d be
he ga ha e f 1.1395
1.1395 ba u 10 u10-3 3
0.119
e
3
5 ba ˜
8.314 u10
˜1150 K
˜K
ha 0.801
e f
d CaCO3 e a a 1150 K.
PV
RT
b) If a 100 e e e
ed a 1150 K, 1.19
e f CO2 c d be acc
da ed a a
e
e f 1.1395 ba , h ch
e ha a he ca c
ca b a e
d ha e d
c a ed
d ce 1
e f CO2 a d he e
e
d be
S
C
NRT
V
P
ca a d E
1
T
u 8.314 u 10 5
d
3
ba ˜
10 u10-3
3
˜K
a
c ,5
˜1150 K
Chap e 13
d
0.965 ba
2H2Se g R 2H2 g Se2
The a ba a ce ab e
S ec e
a
f a
H2Se
1
1-2X
1-2X
H2
0
2X
2X
Se2
0
(X)
(X)
13.43 The eac
T a
S
ha K a 1 4X 4X 2
F
each
X
1000
1050
1100
1150
1200
1250
0.239
0.271
0.299
0.324
0.346
0.364
13.44
K a,1
S ec e
N2
O2
N 2O
NO
NO2
a
0.79
0.21
0
0
0
S
e
f Se2 f
ed e
a 2H2Se
f Se2
f
a N 2O
; K a,2
0.5
2
0
e H2Se ha eac = X/2
a N 2 ˜ a O2
a 2NO
; a d K a,3
a N 2 ˜ a O2
f a
0.79-X-Y-0.5Z
0.21-0.5X-Y-Z
X
2Y
Z
4Y 2
0.79-X-Y-0.5Z ˜ 0.21-0.5X-Y-Z
c
1 2X
ed.
a
f
he che
a NO2
0.5
a N2 ˜ a O2
e f ac
(0.79-X-Y-0.5Z)/(1-0.5X-0.5Z)
(0.21-0.5X-Y-Z)/ (1-0.5X-0.5Z)
X/ (1-0.5X-0.5Z)
2Y/ (1-0.5X-0.5Z)
Z/ (1-0.5X-0.5Z)
1
1
1
1
a d K a,3
ca e
b
1
X
§
·§ 1-0.5X-0.5Z · 2 § 1 · 2
¨
¸¨
¸ ¨
¸
© 0.79-X-Y-0.5Z ¹© 0.21-0.5X-Y-Z ¹ © 1.013 ¹
0.79-X-Y-0.5Z § 0.21-0.5X-Y-Z · 2 § 1.013 · 2
˜¨
¸ ˜¨
¸
1-0.5X-0.5Z © 1-0.5X-0.5Z ¹ © 1 ¹
a , e ba
b
2
2X ˜1
0.119
0.135
0.150
0.162
0.173
0.182
X
(1-0.5X-0.5Z)
K a,2
E
e
a 2H2 ˜ a Se2
4X 2 Ÿ 4 K a 1 X 2 4K a X K a
e f H2Se ha eac ,
T(K)
K a,1
Ka
1
1
Z
§ 1-0.5X-0.5Z · 2 § 1 · 2
¨
¸ ¨
¸
0.21-0.5X-Y-Z © 0.79-X-Y-0.5Z ¹ © 1.013 ¹
c
a
ga
S
C
ca a d E
T
d
a
c ,5
Chap e 13
d
T(K)
1000
1200
1400
1600
1800
2000
Ka,1
6.7142u10-9
3.6675u10-8
1.2688u10-7
3.3069u10-7
7.1595u10-7
1.3675u10-6
Ka,2
7.5363u10-9
2.8345u10-7
3.7886u10-6
2.6496u10-5
1.2000u10-4
4.0052u10-4
Ka,3
1.0131u10-5
1.9428u10-5
3.1098u10-5
4.4434u10-5
5.8845u10-5
7.3931u10-5
T(K)
1000
1200
1400
1600
1800
2000
X
1.719u10-9
1.336u10-8
4.605u10-8
1.200u10-7
2.588u10-7
4.909u10-7
Y
1.986u10-5
1.084u10-4
1.002u10-4
1.044u10-3
2.216u10-3
4.026u10-3
Z
1.902u10-6
3.648u10-6
5.839u10-6
8.300u10-6
1.092u10-5
1.359u10-5
T(K)
1000
0.79
0.21
1.719u10-9
3.972u10-5
0.79
0.21
1.336u10-8
2.168u10-4
0.79
0.21
4.605u10-8
2.004u10-4
0.789
0.209
1.200u10-7
2.089u10-3
0.788
0.208
2.588u10-7
4.432u10-3
0.786
0.206
4.909u10-7
8.052u10-3
1200
1400
1600
1800
2000
N2
1.902u10-6
3.648u10
-6
5.839u10
-6
8.300u10
-6
1.092u10
-5
1.359u10
-5
N e ha a he e
he a
e
(
de
O2
N2O
f
ge (f e e
) a ge.
efe ed
NO
a NOX) a e a
NO2
a
S
C
13.4
(a
ca a d E
T
d
a a ab e a a Ma hcad
a
c ,5
Chap e 13
d
hee ).
Gi en:
'GAgC 108700˜
'GAg 77110˜
e
'GT C 186020˜
R 8.31451˜
e
e
'GT 32450˜
e
e
K˜
e
'GC 131170˜
e
e
e
e
e
T 298.15K
˜
e
Part (a):
'G
AgC 'GAg 'GC 'GAgC
'G
§ 'G AgC ·
¨
¸
R˜ T
©
¹
KAgC
KAgC e
AgC
4
-1
5.464 u 10
e
e
10
2.676 u 10
The sol bilit prod ct gi en in ill stration 9.3-2 is 1.607E-10. This e perimental al e is of the
same order of magnit de as the theoretical al e calc lated abo e.
Part (b):
'G
T C 'GT 'GC 'GT C
'G
§ 'G T C ·
¨
¸
R˜ T
©
¹
KT C
KT C e
TC
4
2.24 u 10
-1
e
e
4
1.19 u 10
The sol bilit prod ct gi en in ill stration 9.3-2 is 1.116E-2. This e perimental al e is t o orders
of magnit de greater than the theoretical al e calc lated abo e.
C2H4 H2O R C2H5OH
13.4 The eac
N e ha e h e e ha a b
g
f 169.3 K a d a c ca e e a e f 282.4 K.
The ef e, e h e e a ga a 298.15 K, h e a e a d e ha
ae
d.I a ,I
a
e ha a e a d e ha
f
a dea
, ha he e
ae
e ha
he a , a d
eh e e
he
d. The
c ec f he e a
b
ea
.
a C2H5OH
C2 H5OH J C2 H5OH
Ka
1 ba
a C 2 H 4 ˜ a H 2O
˜ H 2 O J H 2O
1.013 ba
a , a 298.15 K, f
he P e
ga
a
a
PH O 0.03171 ba a d PC H OH 0.07889 ba
2
The
S ,a
2
a
ba a ce ab e
S ec e
Wa e
C2H5OH
T a
d
g dea
1
0
5
a
beha
,
f a
1-X
X
1
,
1-X
X
d
e f ac
S
C
ca a d E
Ka
C 2H5OH
1 ba
˜ H 2O
1.013 ba
S he
a e
aef
0.9578 a d
C2H5OH
T
22.89
he
H 2O
d
a
c ,5
Chap e 13
d
X
22.89
, a d X=
1 ba
22.89+1.013
˜ 1 X
1.013 ba
d
e f ac
ae
0.0422.
0.9578
H e e, e
ha e ha
+ ae
e ae
dea . U g he V a Ba c
UNIFAC
ga
e
a e he ac
c eff c e g e
JC2H5OH ( 0.9578) 1.0011 a d J H2O (
0.0422) 2.4802 . The ef e,
Ka
X ˜1.0011
22.89
,
1 ba
˜ 1 X ˜ 2.4802
1.013 ba
22.89
a d X=
0.9824
0.0176
C2 H5OH a d H 2O
22.89+0.4089
Reca c a g he ac
c eff c e
h he UNIFAC
ga g e
JC2H5OH (
0.9824) 1.0002 a d J H2O (
0.0176) 2.5557
a d eca c a
g he e
b
X=0.9830
C2H5OH a d
H 2O
h ch
dea
ea
c
c
g e
0.0170 .
ee
gh
he e
e
ha I
: C2H5OH 0.9578 a d H2O 0.0422.
:
C2 H5OH =0.9830
a d
c
a
e
e a e:
0.0170
H 2O
N
eed c ec f e ha
a d ae
he a
PC 2 H5OH 0.9830 u 1.0002 u 0.07889 0.0776 ba
PH 2O
e a e aga . S f
ha e. The
a a
0.017 u 2.5557 u 0.03171 0.00138 ba
S
he a a e
e f e h e e 1 - 0.0776 - 0.00138 = 0.9210 ba a d
ac
0.9210
a C2 H 4
0.9092 ( 1/1.013 a ha
fa bee a
ed).
1.013
S eca c a g he e
b
(
g he a e f he e
b
c
a
f
a ea
) e ha e
X ˜1.0002
X
K a 22.89
0.4360
,
1 ba
1 X
˜ 1 X ˜ 2.5557 u 0.9092
1.013 ba
22.89
a d X=
0.9811 C2H5OH a d H 2O 0.0189.
22.89+0.44
A h e
e d ffe e f
he e
ea
,I
e a e aga .
13.4
F
he che ca e
K.
The a ba a ce ab e
S ec e
a
f a
SO2
1
1-X
O2
0.5
0.5-0.5X
SO3
0
X
T a
(1.5-0.5X)
b
c
a
ca c a
(
e f ac
)
(1-X)/(1.5-0.5X)
(0.5-0.5X)/(1.5-0.5X)
X/(1.5-0.5X)
he
g a , Ka = 4.5905 a 1000
S
C
ca a d E
T
d
a
c ,5
Chap e 13
d
X
P
ISO3
1.5
0.5X
1.013
Ka
0.5
0.5
a SO2 a O
1 X
P
§ 0.5 0.5X P
·
2
ISO2 ˜ ¨
IO 2 ¸
1.5 0.5X 1.013
© 1.5 0.5X 1.013
¹
he e he I e
a e f gac c eff c e . The e a
be
ed f
XISO3
1
4.5905
0.5
1 X ISO2 § 0.5 0.5X P
·
I
¨
O ¸
© 1.5 0.5X 1.013 2 ¹
a)
A P = 1.013 ba , e ca a
e ha he a e f a f gac
,
ha
a SO3
4.5905
X § 1.5 0.5X ·
¨
¸
1 X © 0.5 0.5X ¹
0.304,
SO2
0.5
Ÿ X
0.152 a d
O2
a
aa dba e
c eff c e
ae
0.642
SO3 =0.544.
b)
A P=101.3 ba , he
dea ga ha e beha
( ha , he f gac c eff c e
ca
be eg ec ed. The
ced e I
e
f
e he f gac c eff c e
ca c a e he e
b
c
, e he e c
he Pe g-R b
e a
f ae
ca c a e he f gac c eff c e , eca c a e he e
b
c
h e f gac c eff c e , a d c
e ea g
c e ge ce ach e ed.
·
X § 1.5 0.5X
¨¨
¸¸
1 X © 100 ˜ 0.5 0.5X ¹
4.5905
0.089,
SO2
0.045 a d
O2
0.5
Ÿ X 0.907
SO3 =0.866.
A
g ha he b a
e ac
a a e e f he e
f gac c eff c e a e
ISO2 1.0056, IO2 1.0405 a d ISO3 =1.0090. The ef e
KI
ISO3
1.0090
0.5
ISO2 IO
2
The
X 0.908,
1.0056 u 1.0405
he
SO2
S he cha ge f
e e a e be g
0.088,
O2
he dea ga
h gh, ha
0.5
ga c ac d a e
, he
0.984
0.048 a d
ha e ca e
SO3 =0.867.
h ea
e
g.
a , a ge
a a e
f he
)
,
h
S
C
13.4
ca a d E
(a
T
d
a a ab e a a Ma hcad
a
c ,5
Chap e 13
d
hee ).
Pa a
'G
R T
( Ka )
57.33
d
d T R T
0.17677
'H
R
R T
2
57.33 ' H
R T
2
R T
2
57.33 'H
J
e
Part (b):
'G
( T)
Ka( T )
e
e
57330 'G
176.77 e
e T
eK
R
In
1
0
1
O t
1-X
2*X
1+X
8.31451 K
e
e
( T)
R T
Mass Balance Table:
Species
N2O4
NO2
Total
Calc lation of mole fractions and acti ities:
N2O4( X )
a N2O4( X P )
KaX( X P )
1
X
1
X
NO2( X )
N2O4( X )
a NO2( X P )
2
a N2O4( X P )
P
5
10 Pa
a NO2( X P )
2X
1
X
NO2( X )
P
5
10 Pa
S
C
ca a d E
T
Pa b
X 0.7
(initial g ess)
G e
Ka( T ) KaX( X P )
Xb 0.1
4
X 323.15 K 10 Pa
d
a
c ,5
Chap e 13
d
X( T P )
Xb 1
F d( X)
5
X 323.15 K 10 Pa
Xb 10
6
X 323.15 K 10 Pa
NO2 Xb 0.1
0.91
NO2 Xb 1
0.605
NO2 Xb 10
0.261
N2O4 Xb 0.1
0.09
N2O4 Xb 1
0.395
N2O4 Xb 10
0.739
Part (c):
Xc 0.1
4
X 473.15 K 10 Pa
NO2 Xc 0.1
13.4
N2O4 Xc 0.1
1
1.246 10
4
Xc 1
5
X 473.15 K 10 Pa
NO2 Xc 1
0.999
N2O4 Xc 1
1.243 10
3
Xc 10
6
X 473.15 K 10 Pa
NO2 Xc 10
0.988
N2O4 Xc 10
0.012
S
C
ca a d E
T
d
a
c ,5
Chap e 13
d
13.49
Gi en:
Ta
e
58620 'H
138.2 'S
e
298.15 K
Tb
373.15 K
K
e
R
e
4
10 Pa
P1
P2
8.31451 5
10 Pa
K
P3
e
e
6
10 Pa
Mass Balance Table:
Species
In
O t
M
D
Total
2
0
2-2*X
X
2-X
(2-2*X)/(2-X)
X/(2-X)
Acti ities, Eq ilibri m Constant, and Eq ilibri m E pression:
a M( X P )
'G
( T)
KaX( X P )
X
0.999
2
2
2X X
P
a D( X P )
5
10 Pa
T ' S
'H
Ka( T )
X P
5
2 X 10 Pa
e
'G
( T)
R T
a D( X P )
a M( X P )
2
(initial g ess for sol er)
G e
KaX( X P ) Ka( T )
Part (a):
Deg eeOfD e a
( Ta P1 )
0.953
Deg eeOfD e a
( Ta P2 )
0.985
Deg eeOfD e a
( Ta P3 )
0.995
X 1
Deg eeOfD e a
( T P)
F d( X)
S
C
13. 0
(a
ca a d E
T
d
a
a a ab e a a Ma hcad
c ,5
Chap e 13
d
hee ).
Gi en (All Units are SI):
T1
267
T2
255
6
2 10
P1 CH4
Acti ities:
ah d ae
1
a H2O
1
6
1.5 10
P2 CH4
R
8.31451
P CH4
a CH4 P CH4
5
10
Part (a):
ah d a e
Ka P CH4
5.75 a H2O
'G
a CH4 P CH4
'G
T1 P1 CH4
3
6.65 10
'G
T2 P2 CH4
3
5.742 10
R T T P CH4
Ka P CH4
Part (b):
3
'H
10
G e
a
'G
F d 'H
'H
(initial g esses)
'S
10
T1 P1 CH4
'H
'S
T1 ' S
'H
4
1.357 10
'S
a
'G
0
T2 P2 CH4
'S
a
1
75.737
Part (c):
Ka 273
P273 CH4
G e
P273 CH4
'H
Ka P1 CH4 e
6
10
R
1
1
273
T1
Ka 273
0.044
(initial g ess)
Ka P273 CH4
6
2.288 10
Ka 273
P273 CH4
F d P273 CH4
'H
T2 ' S
S
C
ca a d E
13. 1 The eac
The
T
d
a
c ,5
Chap e 13
d
ae
ea
a
B1 œ B
ba a ce
ec e A
a
NA
be
e
O ea
a
f
a
e
a d
A1 B1 œ A B
¦ N ¦¦ N
A
f A1
ba a ce
NB
A e
A1 œ A
F
I
¦ GH N ¦ N JK
AB
A
AB
ec e B
be
f B1
f
¦ N ¦¦ N
B
F
I
¦ GH N ¦ N JK
AB
B
AB
b
0;
GA GA1
GB GB1
c
0 ; GA B G A1 GB1
h 0;
a d
G
A e
b
G
a
¦ N A G A ¦ N B GB ¦ ¦ N A B G A B
a
h e ec
0
each e e
f eac
.
¦ GA dNA ¦ GB dNB ¦ ¦ GA B dNA B
dGtotal
¦ N A dGA ¦ N B dGB ¦ ¦ N A B dGA B
0
0 b the GibbsD hem eq ation
A
,
G
g he e
a
0
0
b
e
a
¦ G A1 N A ¦ GB1 N B ¦ ¦ b G A1 GB1g N A B
G A1 ¦ N A GB1 ¦ N B ¦ G A1¦ N A B ¦ GB1 ¦ N A B
LM F N ¦ N I OP G LM¦ e N ¦ N jOP
JK Q N
N GH
Q
G A1 ¦
A
AB
B1
GA . A
FG w G IJ
H wN K
AB
G A1 N A GB1 N B
FG w G IJ
H wN K
a
A
N BT , P
{ GA
a
B
N A, T, P
{ GB
GB
S
C
ca a d E
T
G A1
The ef e
A
d
eg a
c ,5
Chap e 13
d
GA
, e ha e ha , b def
Th ,
a
GB1
a d
FG w G IJ
H w PK
g be ee a
RT
T
FG w IJ
H wP K
a e , e ha e
GB
(1)
(2)
T
a f a f RT aaTT,, PP ff
2
G T , P2 G T , P1
N
g E
a f ed a a T a d P
. (3)
h E
e ha
A T, P
A e a e
e ha E . (1)
. (1), a d ec g
A1 T , P
A1 P
b
Ÿ
A P
a
0
a d
Po0
A1 P
0
A1 P
ha
13. 2
A P
ee
be
B1 T , P
e
e ha
A P
be
A1
. (1)
P 0 a d he
A P
0
g ha E
B T, P
e c d eg a e E . (2) be ee
be a f ed a a T a d P. Th
A1 P
(3)
1
ePa d
0
(LeCha e e
c
.
A1 A1 œ A2
A1 A2 œ A3
#
A1 A œ A 1
e c.
T a
f A1
N0
NT
NT
T
¦
he e N T
a
¦N
be
f
e
a1
d¦ i
¦
¦
N
a
h ch
N1 2 N 2 3 N 3 " A "
N
¦ N
N0
NT
Ÿ
e
a
¦¦
e ha
¦
1
aa
1
¦
a d
¦¦
a1
2
e
e)
S
(a)
C
a
ca a d E
a1
T
FG N IJ a d
HN K
2
0
1
T
A
K 1
d
a
c ,5
Chap e 13
d
FG N IJ
HN K
K
0
T
a
f
a P 1 ba f
a 1
I 1
1 P 1 ba
a a1
I I1
1
I 1 1 1 ba
I I1 1 P
2
The
I I1P
I 1
F
he
e
e
1K
a
S
f he
1K
1
e ha
I I1P
I 1
de e de
I 1
I I1P
1
1
1K
de
D
a d he
2
1D K
2
1 2D K
3
4
3
4
1 DK
a D Kf
1
h
2
ha h
deed he ca e. The
a D Kf D K
2
1
1
a f
2
3
1 D K
a
a f
#
aD Kf
1
1
#
e c.
NT
The
N0 ¦
A
N
f
he
N0 ¦ KD 1 1 .
¦
1
e e
f ge
f
1
1 T
¦T
0
¦ KD 1 1
1
e c
e ha e
f
¦ KD 1 1
1
f
1
¦ a KD 1f 1
1
f
1
¦ a KD 1f
0
1
1 KD 1
1
(*)
S
C
ca a d E
T
d
a
c ,5
Chap e 13
d
ha
1
1 KD 1
1
1 ; KD
1 1 KD
1
1
A
FG ¦ T IJ
K
TH
f
F 1I
T H1T K
f
¦ T 1
0
0
f
1
1 T 2
¦ T 1
1
ha
f
f
¦ KD 1 1
1
1
¦ a KD 1f 1
1
1
1
Ÿ
F
1
NT
N0
1
he a de Waa e
0
1
2
1
FG
IJ
H 1 KD K
2
FG N IJ
HN K
2
1
1
E . (*) ab e
a d
1
a
FG N IJ
HN K
0
T
1 1; a
a1
0
T
1
1
12
2
1 a1
f a e e ha e
RT
a
2
V
V
P
¦ a KD 1f 1
NT
N0
a1 KD f
f
f
1
NRT
N 2a
VN
V
a d
I
B
ZB
B1
ZB
ZB h ch he e bec
I
2¦ a
RTV
e
I
ha
ZB 2¦
a1
B1
2a
ZB ¦
RTV
ZB
RTV
B
2 a1 N 0
ZB ZB
RTV N T
S
C
ca a d E
T
I I1
I 1
d
a
I 1
de e de
RT
V
2a1 N 0
RTV N T
ZB
1
ZB
RT
V 1 N0 NT
a
1
ZB
1
RT P
V
N T RT
N TV N 0 1
f
h ch
de e de
f he de
f he de . We ca
e
a
D
a a
N T RT
NTV N0 1
1 a dD
1 1 DK
B1
1 1 1
B
1
I I1
I 1
PI I1
Chap e 13
d
I I1 I 1
1
(b)
c ,5
ed. N
ha
RT
V N0 NT
a
1
1 1
2
1 1
2
1 1
1
1 1
1
1
aV RTK f r aV RTK f 4 V
1
1
RT
a
2
V
V
2
RT
a
1 21
V 1
V
1
2
2
1
1
21
ha
P
a
RT
a
12
V 1 V
a de Waa e a
.
1
f a e f he
h ch de e d
aV f r dV 4 V i aV f aV f 1
1
2
(***)
E a
(**) a d (***) a e he e h ch f
he e a
a c a g a de Waa f d. N ce ha
e f V e eed 1
ge c b c.
V ; he ef e, he e a
N e ha f he f d
-a c a g, he K 0
h
he
1
21
(**)
A
h ch
1 1
1
(c)
P
RT
V 1
f
FG1 RTK IJ 1 Ÿ aV RTK f V H V K
aV RTK f V 0
aV RTK f V 0
1
ha D
e
1
21
1
S
C
ca a d E
T
d
a
c ,5
Chap e 13
d
13. 3 The de c
f HF c a
g
e
de c bed
he a c e C ec
f Pha e E
b
Da a f Se a a
Tech
g b W a Sch e I . E
C
.P
D . D . (1980), 19, 432 439. B a ca ef e a
a
f he de
a d he da a, he
ed ha HF a c a e
he a
ha e acc d g he eac
.
2HF œ HF 2
6HF œ HF 6
a d,
e he e
8HF œ HF 8
e a ge f 195 240 K, he e
ea
K2
2
2
1
K6
6
6
1
K8
8
8
1
b
c
a
ae
LM 6429.4 24.1456OP
N T
Q
21100.9
O
e LM
N T 69.7292PQ
25224.5
e LM
83.4689OP
N T
Q
e
1
f a
,a d
he f gac
f ec e . Ne , Sch e
ed a (
e b Ta
a d W a [C
. E . S (1978), 33, 651]
P f he
I
e , d e , he a e a d
ha he f gac
c eff c e
ca e aea a
a e e a , a d ca be ca c a ed f
he f gac
f
e HF.
Th a e a ed he eed
ec f he
ec a a a e e f he a c a
c
e e .
U g h
de , Sch e b a ed e g d ag ee e f he a c a
fac (de
)
f
e HF a d a HF-F e
e
g a
e e a
f a e ch a Pe gR b
( h ch
be
ed e a e
ce che ca e
b
e
ed
he ha e e
b
ca c a
).
A a e a e,
ead f
g he Ta -W a a
f e a
f f gac
c eff c e
e he
de
P b e 9.52
he e K ha
e
ab e) a g
a d he
ea HF a a che
HF 8 . S
a
, HF +
e :
-a
ca
gc
-a
ca
a
2
ca
eac
gc
a1 a d
1
e
e
h HF, HF 2 , HF 6 a d
d be ea ed a a f e c
e
e + HF, HF 2 , HF 6 a d HF 8 . I each f he e ca e
he c
f he HF c
e cha ge a he e
b
cha ged. The
be
h h
a
ha he a a a e e f
he a c a
c
e e bec e
ea
ca
a ge.
F
e a
e, a8 64 u a1 . C e e
,
c
ee
he e ca -c ec
de f he HF a c a g
e e
, h gh he
de
ea e
bab a fac
f e g ee g ca c a
.
S
C
13. 4
Sa
af c
13.
ca a d E
KW
KW
gf
§ K T2
¨¨
© K T1
a
a ge f e
fa
C
H
1)
c ,5
Chap e 13
d
7.085
25
7.0
30
6.915
40
6.77
50
6.63
60
6.51
g ha '
1
KW
2
' H ( J/
e, a d ha
20
º
H
e ha e ha
H
)
59.22
58.79
56.89
58.83
52.70
54.24
49.46
K
W

o H OH
BOH o B OH a d H 2 O m

ªM - º
¬ OH ¼ W
ª¬ M H+ º¼
W
ª¬ MOH- º¼
W
` ª¬M
º J 2r
OH- ¼ W
M H+
^M
h
ha
º
BOH ª
¬ MOH- ¼ W
` ª¬M
º
OH- ¼ W
J r2
ha
M H
E
ea ed
ea
a
H§ 1
1 ·
¸ a da
¨
R © T2 T1 ¹
7.27
OH- ¼ BOH
M OH a
'
10
ªM º
¬ OH ¼ W
13.
H
7.47
de he eac
S
·
¸¸
¹
d
T( C)
0
M H M OH J r2 b
^ª¬M
T
a
e
M BOH M 2BOH 4K W / J r2
M 2BOH 4K W / J r2 M BOH
2
2
ª
º
¬ M OH ¼ W M BOH
KW
M 2BOH 4K W / J r2 M BOH
2K W
M OH J 2r
M 2BOH 4K W / J r2 M BOH J r2
(15.1-10)
e aced b K a,W
,
e aced b MH J r . B
g M H J r , e
cha ged.
1
2
ce aga
ha e H
|
KW
M BOH J 2r
M 2H J 2r
M HA, M 1
he e e MH
ha
h he def
1
1
K a,HA g M HA,
2
2
ha
,
e
. (15.1-
S
C
ca a d E
13.
The eac
T
d
a
c ,5
Chap e 13
d
ae
K a ,HA
K
a ,W

o H A a d H 2O m

o H OH HA m

M H J r M A J r
h K a,HA
M HA
b
The ef e
M H J r M OH J r
a d K a,W
M HA, M H a d M A K a,HA
ha
M HA ˜ M 1
M 1
2
M H
M 2H J r2
M HA, M H J r ˜ M 1
he e e MH a
ea ed
e
½°
Ka,HA ­°
4M HA,
1¾ a d
® 1
2 °¯
Ka,HA
°¿
e . 15.1-10
cha ged.
M H J r
13.
Wea ba e +
f ae
g ac d, eg ec
e aced b MH J r .
,
g H g M H J r
g H+ a d OH-
d ced f
he
a
K
BOH
ZZZZ
X B OH HA o H A a d BOH YZZZZ
h K a,BOH
MBOH
M B MOH
M BOH M 1
U g D be he
MBOH, D; MB+
(
a
f BOH d
c a ed,
D; a d MOH- D MHA
ce each
e f OH- f
ed b
+
fH f
ed f
he d
ca
D MHA ˜ D
K a,BOH
MBOH, D ˜ M 1
he d
ca
f he ac d f
( h ch
e e
e cha ge f BOH
(ab e I
a
e ca
ed a e
e a
h HA.) S
K a,BOH M HA D
K a,BOH M HA
2
15.1-7) e ce
e
f
e
he
4K a,BOH ˜ M BOH,
2
M OH D M HA
K a,BOH M HA K a,BOH M HA
M H
KW
M OH The ef e
4K a,BOH ˜ M BOH,
2K W
K a,BOH M HA
2
4K a,BOH ˜ M BOH, K a,BOH M HA
Add g e ec
e
dea
M B J r M OH J r
K a,BOH
K a,BOH
Ÿ
M BOH M 1
J r2
ha e e
2
2
a d
13.
f BOH eac
ha
a e .) S ha
he e e ace K a,BOH
M B M OH M BOH M 1
h K a,BOH / J r2
S
D
C
ca a d E
§K
·
¨¨ a,BOH
M HA ¸¸ 2
© Jr
¹
M OH T
d
13. 0
2
§ K a,BOH
·
K
M HA ¸¸ 4 a,BOH
˜ M BOH,
¨¨
2
J 2r
© Jr
¹
2
KW
M OH J 2r
2K W
2
§K
·
§K
·
K
M HA ¸¸
J 2r ¨¨ a,BOH
M HA ¸¸ 4 a,BOH
˜ M BOH, ¨¨ a,BOH
2
2
2
Jr
© Jr
¹
© Jr
¹
Add g e ec
e
K a,1M HA
M A
; J r M A 2
M H J 2r
ST S
ST S
S
13. 1 The e
Chap e 13
d
2
a d
M H
c ,5
§ K a,BOH
·
K
M HA ¸¸ 4 a,BOH
˜ M BOH,
¨¨
2
J 2r
© Jr
¹
2
§K
·
¨¨ a,BOH
M HA ¸¸ 2
© Jr
¹
D M HA
a
K a,1M HA
M H J 2r
dea
K a,2 M A J r
M H J r
K a,1K a,2 M HA
M H
2 3
Jr
ha M A2
K a,1Ka,2 M HA
MH J 3r
ª
º
K
K K
... « a,1 a,1 a,22 ...» M HA
«M »N
M H
¬« H
¼» S
ª K
º
K1 K 2
10 1 10
«
.... « H ....»»
2 3
H 2
M H J r2
10
J
r
M H J r
10
Jr
«¬
»¼
K a,1
b
K a,1K a,2
ea
ae
H ˜ NH3 CH 2 COO
NH3 CH 2COO
H ˜ NH 2CH 2COO
K1 =
; K2 =
; KD =
NH3 CH2COOH
NH3 CH 2 COO
NH 2CH 2COOH
The ef e
S
C
ca a d E
T
d
a
c ,5
Chap e 13
d
NH 3 CH 2COO K D ˜ NH 2CH 2COOH=NH 2CH 2COOH ˜10 K D
NH 3 CH 2COOH
H ˜ NH 3 CH 2COO K1
H ˜ NH 2 CH 2 COOH ˜10 KD
K1
NH 2 CH 2 COO
K 2 ˜ NH3 CH 2 COO
H
NH 2 CH 2COOH ˜10 H K 2 K D
NH 2 CH 2 COOH
NH 2 CH 2 COOH
NH 2 CH 2 COOH NH CH 2 COO NH 3 CH 2COOH NH 2CH 2COO 3
1
1 10
KD
10
K1 H K D
10 H K2 KD
10 K D
NH 3 CH 2 COO
NH 3 CH 2 COOH
NH 2 CH 2 COO A
H=3
NH 2 CH 2 COOH
A
1 10 K D 10 K1 H K D 10 H K 2 K D
10 K1 H KD
1 10 KD 10 K1 H K D 10 H K2 KD
10 H K2 KD
1 10 KD 10 K1 H KD 10 H K2 KD
| 0;
NH 3 CH 2COO 0.818;
NH3 CH 2 COOH
;
0.182;
NH 2 CH 2 COO |0
H=10
NH 2 CH 2 COOH
NH 2CH 2COOH ˜10 K1 H K D
| 0;
NH 3 CH 2COO 0424;
NH 3 CH 2COOH
| 0;
NH 2CH 2 COO | 0.576
S
C
ca a d E
13. 2
T
d
a
c ,5
Chap e 13
d
a) The eac
NH 3 + 2O 2 + (7.5238N 2 ) o HNO 3 + H 2O+ (7.5238N 2 )
The
e gh a e HNO 3 =63.01; H 2O= 18.01; NH 3 =17.03; H 2O=18.01
ec a
S 100 g f a 60
ae,
HNO3
=
The
e
%
f HNO 3 c
he f
g
g
a
e f ac
0.9522
e
he
d c
0.9522
=0.0979, H 2 O =0.1285 a d
0.9522 1.25 7.5238
he feed a e
e f ac
N2
f HNO3 a d 1.25
c d g
S
f
ge :
0.773576
1
=0.0950 O2 =0.1900 a d N 2 0.71493
1 2 7.5238
; 'Gf (H 2O, ) 237.1 J/
; 'Gf (HNO 3 , )
'Gf (NH3 ,g) 16.5 J/
NH3
e
=
'G = 111.3 237.1 ( 16.5)
Th
he
a
a
S
effec
he
f
d c
331.9 J/
ha c
e
111.3 J/
a eac ed,
fa
d be b a ed
h
c
de
g
effec .
e NH 3 eac ed
'GP = -RT (1 0.0979+1 0.1285+7.7238 0.7736)=1.710 J/
S
effec
he feed
'Gf = -RT (1 0.0950+2 0.1900+7.7238 0.71493)=2.714 J/
The
S
e a 'G f
he
a
ce ha
50 g=64.7
he
a
ce ='G
f
'GP 'Gf
ha ca be
effec a e eg g b e c
=0.446 MPa
A 50
g=64.7
a =0.446 MPa, I
331.9 1.7 2.7
d ced
a ed
a 25 C H =104.89 J/ g a d S =0.3674 J/ g-K
332.9 J/
he f ee e e g
T 147.9 C, H =2743.9 J/ g a d S =6.8565 J/ g-K
S ce
1
= 2000 b =4410 g,
J/ g
332.9 J/
NH 3
NH 3 eac ed. A
f eac
.
G=104.89-0.3674 u 298.15=-4.65
e 0.45 MPa
'G f ea = -143.03-(-4.65)=-138.4 J/ g
NH 3
ce a e ca be ead f
ea
ab e
G =2743.9-6.8586 u (147.9+273.15)=-143
S
C
ca a d E
13. 3 See f
13. 4
g
T
d
a
c ,5
c ce
a
Chap e 13
d
be
S
gf
he h d ge
a
he
ec e
b
gf
a de
a
g he c
ce
a
f
e ge
1
§
·2
K K a1
ª¬ H º¼ ¨10 14 PCO2 ¸
KH
©
¹
h ch ead
PCO2
0
H=7
PCO2
2.7 u 104
H=4.233
PCO2
3.85 u 104
H=4.156
PCO2
5.5 u 104
H=4.079
13.
Pa a Ma
bala ce able
S b a ce
I
CH3OH
F a
M e f ac
1
1-X
(1-X)/(1+X)
CH2O
0
X
X/(1+X)
H2
0
X
X/(1+X)
T a
1
1+X
Ka
aCH 2O aH 2
CH 2 O H 2
aCH3OH
Ka 1 X
X
a
2
P / 1 ba
X2
1 X 1 X
CH 3OH
2
X2
;
1 X 2
(1 K a ) X 2 ;
X ; Ka
Ka
1 Ka
' G ( J/
) ' H ( J/
CH 2O
-102.5
-108.6
H2
0
0
CH 3OH
-162.0
-200.7
Reac
59.5
92.1
K 298.15 K
e
§ ' G·
¨
¸ e
© RT ¹
)
59000
§
·
¨
¸ e ( 24.0034); K
© 8.314 u 298.15 ¹
3.7625 u1011
S
C
ca a d E
Pa c Ma
T
bala ce able
S b a ce
CH3OH
CH2O
H2
N2
T a
Ka
aCH2 O aH2
I a
1
0
0
0.5
1.5
CH 2 O H 2
aCH3OH
P / 1 ba
CH3OH
K a 1.5 0.5 X X
2 2
S
B
e
a
c ,5
Chap e 13
d
F a
1-X
X
X
0.5
1.5+X
M e f ac
(1-X)/(1.5+X)
X/(1.5+X)
X/(1.5+X)
0.5/(1.5+X)
X2
1.5 X 1 X
X2
;
1.5 0.5 X X 2
X 2 ; 1.5K a 0.5 K a X K a X 2
(1 K a ) X 2 0.5 K a 1.5 K a
13.
13.
d
X 2;
0
he a e a P b e 13.39
, h
he a e a
be
h
ge e ac g h d ge .
13.38,
ee he
ha
be .
S
C
ca a d E
a) Reac
ac
fBa dDa e
§ 'G ·
¨
¸
© RT ¹
Th ha he
a
c ,5
Chap e 13
d
A
he e a
0.5 CJ C
J
A
c) U e N A
S ec e
§ P ·
aC ¨ D ¸
© 1 ba ¹
aA
aC aD
aA aB
ha e a e
0.5 C
0.5 1 A
A
A
ed
e ee
he
=0.166 a d C =0.834.
he
be
d ha e a a
f
e
e
be c
each
e f A eac ed. D
M e
a
NA
NA N
B
NB
NB N
C
NA N
D
N
S
he e
b
K
aC aD
aA
NA N
f A added
a
a . N e ha he
a h
g he c
e f a
A
be
, e ha e he f
g
e f ac
Pa a
be
uN
/ NA
ed
u R uT / V
NA N
, h e a
be
ed f
>Cha
>Cha
>Cha
>Cha
>Cha
>Cha
>Cha
>Cha
e 13>P
e 13>P
e 13>P
e 13>P
e 13>P
e 13>P
e 13>P
e 13>P
N .
13.
13.
h
h
h
h
h
h
h
h
be
be
be
be
be
be
be
be
f
f
f
f
f
f
f
f
de A
de A
de A
de A
de A
de A
de A
de A
e
e
e
e
e
e
e
e
f
f
f
f
f
f
f
f
Te
Te
Te
Te
Te
Te
Te
Te
b
b
b
b
b
b
b
b
e
/ NA
N
ea
f
e f he eac
N
e
be
dC
NA N
ce
a
e f
d
/ NA
A N A , R, T a d V a e a
13. S
13. 0 S
13. 1 S
13. 2 S
13. 3 S
13. 4 S
13. S
13. S
e, he
=0.195 a d C =0.805
be
Th ha he
d ced f
d a d ga e
0.5 C e (0.3(1 C ) 2 )
(1 C ) e (0.3 C2 )
A A
e
g he
. The ef e
2.058
K
K
d
A( )+B( ) o C( )+D(g). S a
e
b) N
T
be
be
be
be
be
be
be
be
13.69
13.70
13.71
13.72
13.73
13.74
13.75
13.76
u R uT / V
e
S
13.
C
ca a d E
T
d
a
c ,5
d
Chap e 13
S
C
13.
M A M H
K1
b
M A
K1M A MA
M H
def e
F
ca a d E
c ,5
Chap e 13
d
M A, M A M A M A 2 M A3
H K1
10
a d def e M
def e
=
M A-
K 2M A
M H
M / M A,
K2
M A, M A M A M A 2 M A3
M H
;
M A
2
K2
M H
10
2
1 M A + M A- M A2- M A3- =
MA
M A,
a
M A, M A M A M A 2 M A3
eac
2
M A M H
M A
d
1 M A + M A- M A 2- M A3-
1 M A+
K2
MA
K1
M H
1
T
H K 2
2 1 M A+
2 1M A+
F
eac
3a d4
M A 2 M H K3
; M A 2
M A
M A 2- =
K 3M A 3M A MH
3 M A- =
3
2 1 M A+
4 M A 2- =
4
3
10
H K 3
M A
a d
M A3h
N
10
3
1 M A+ =
H K3
2 1M A+
a d
10
4
H K4
1 M A+ M A- M A2- M A3-
1+ 1 M A +
1
1 2+
1
2
3+
1 2
3
4
M A+ a d
1
M A+
1 1 1 2 + 1 2 3 + 1 2
The cha ge he c
ed f
ch a g e M A + M A- 2M A2- 3 M A3= M A+ 1
2 M A+ 2 1
2
3
4
3 M A+ 3 1 2
1 1 2 2 1 2 3 3 1 2 3 4
1 1 1 2 + 1 2 3 + 1 2 3 4
3
4 M A+
S
C
ca a d E
1 2.20
h
1 d
2 8.75
0 1 140
R1 10
T
Chap e 13
d
4 13.40
h
h 10
h
R2 10
c ,5
3 9.81
h 0.1˜
1
a
2
R3 10
h
3
R4 10
h
4
1
1 R1 R1 ˜ R2 R1 ˜ R2 ˜ R3 R1 ˜ R2 ˜ R3 ˜ R4
1 R1 ˜ R2 ˜ 1
2 R1 ˜ R2 ˜ R3 ˜ 1
3 R4 ˜ 2
0 1
1 1 2 3
cha ge 1 1 2˜ 2 3˜ 3
1
2
1
0
1
0
cha ge
0.5
2
2
3
4
0
0
5
10
h
15
0
5
10
h
15
S
C
ca a d E
T
d
a
c ,5
Chap e 13
d
13.80
part a
1 2.63
RP 10
cha ge
2 14.30
1 h
RN 10
1
1 RN
cha ge
cha ge cha ge
0 1 150
h
h 0.1˜ 1.0
2
RP
1 RP
cha ge
1
e ec c cha ge
0
1
e ec c
0
5
10
15
(
1
2)
2
8.465
20
h
ª 1 10 1 h 10 h 2º
¬
¼
S 16˜
(
1
7
)
(
7
2
)
ª¬ 1 10
º¼
10
3
1 10
S
100
10
5
10
h
15
10
h
2
S
84.249
S
C
ca a d E
T
d
a
c ,5
Chap e 13
d
13.81
1 2.55
0 1 70
R1 10
2 h
2 4.37
h
h 0.1˜
1
R2 10
h 10
h
2
1
1 R1 ˜ 2
1 R1 R1 ˜ R2
0 R1 ˜ R2 ˜ 2
cha ge 1
2
1
0.5
0
0
0
2
4
6
8
h
2
cha ge
1
0
0
2
4
h
6
8
2˜ 2 1
S
C
ca a d E
T
d
a
c ,5
Chap e 13
d
13.82
1 2.55
2 4.37
0 1 140
R1 10
2 h
3 13.03
h
h 0.1˜
1
h 10
h
R2 10
2
R3 10
1
1 R1 ˜ R2 ˜ R3 ˜ 2
cha ge 2˜ 2 1
2
1
0
0.5
1
0
0
5
10
15
h
2
1
cha ge
0
1
0
3
1 R1 ˜ 2
1 R1 R1 ˜ R2 R1 ˜ R2 ˜ R3
0 R1 ˜ R2 ˜ 2
h
5
10
h
15
1 1
S
C
ca a d E
T
d
a
c ,5
Chap e 13
d
13. 3
1 1.0
0 1 120
R1 10
2 2 2.1
3 8.02
h
h 0.1˜
h
1
R2 10
4 8.71
h 10
h
2
h
R3 10
3
R4 10
1
1 R1 R1 ˜ R2 R1 ˜ R2 ˜ R3 R1 ˜ R2 ˜ R3 ˜ R4
1 R1 ˜ 2
1 R1 ˜ R2 ˜ R3 ˜ 2
2 R1 ˜ R2 ˜ R3 ˜ R4 ˜ 2
0 R1 ˜ R2 ˜ 2
cha ge 2˜ 2 1 1 2˜ 2
1
2
1
0
0.5
1
2
0
0
2
4
6
h
8
10
12
h
4
S
13. 4
C
ca a d E
T
a) C
de he f
g eac
K a1
AH 2 
o AH H f
h ch he e
K a1
c ,5
ea
ae
M A MH
M AH
2
The a e age e cha ge
he
ab e
ec e = M AH M A .
2
The ef e, a he
e ec c
H
M AH M H M A M H ˜
M AH M AH
e e K a1 ˜ K a2
Chap e 13
d
K
a d K a2
M AH a
a2
a d AH 
o A H
b
M AH M H d
M AH = M A ,
2
e ec c
b
e .. F
K
M AH
2
=
M AH M H 2
M AH
2
M H
M AH 2
, 2 H= K a1 K a 2
b) He e he e a e
M A M H
K a2
M A M H
2
ha a he
ha
c
H=
K a1 K a 2
2
de
K
K
a1
a2
a3
AH3 
o AH 2 H , AH 2 
o AH H , a d AH 
o A H
f
h ch he e
K a1
b
M AH M H 2
M AH
ea
ae
M AH M H , K a2
M AH 3
he
ab e
3
e ec c
H
M AH M H M AH M 2
H
˜
M AH M AH e e K a1 ˜ K a2
M A
ec e = 2M AH M AH M A .
The ef e, a he
3
M AH 2
M AH
K a3
a d
2 M H
M AH
2
The a e age e cha ge
a
M A MH
a d K a3
M H
K a2
3
K a1 ˜ K a2
+
. S
M H
Ka 2
2M AH M AH ,
3
e ec c
N e ha d ffe e , b
h ch cha ge ec e
The ec d
b
e
e
ha
3
M AH
+
M H
M AH
K a1 ˜ K a2
3
M AH M H ,
3
+K a1 ˜ M H
2
= M H
3§
K ·
¨ 2 a1 ¸
¨
M H ¸¹
©
3 g M H
K a1 K a 2 K a3 g 2 10 H K a1
ae ,e a
d be b a ed de e d g
a ed
g he cha ge e a
c d
.
c
de
3 H
2
M AH
2
M AH
3
K a1 ˜ K a2 ˜ K a3 =2 M H §
K ·
K a1 K a 2 K a3 g ¨ 2 a1 ¸
¨
M H ¸¹
©
2M AH M H
K a3
2
M AH 2
§
K ·
g K a1 g K a2 g K a3 g ¨ 2 a1 ¸
¨
M H ¸¹
©
ha
2
M AH M H 2
ha a he
2
S
C
ca a d E
T
d
K
a
c ,5
Chap e 13
d
K
K
a1
a2
a3
AH3 
o AH 2 H , AH 2 
o AH H , a d AH 
o A H
f
h ch he e
b
M AH M H
K a1
2
M AH ea
, K a2
M AH M H 3
The a e age e cha ge
The ef e, a he
ha
H
K a3
ae
M AH
he
MA
M AH M H M M 3
- AH H
2M AH 2M AH K a3
a d
ha a he
M H
M AH
3
1
M AH M AH ,
3
2
M AH M H M 3
- H
2M AH
2
2
M AH M H 2
M AH ha
3
M H
M AH
K a1 ˜ K a2
3
,
M H
M H
3 1
1
K a1 ˜ K a2 ˜ K a3 = M H - K a1 ˜ K a 2 ˜ M H =
2
2
2
-
2
§
·
1 K ˜K
g K a1 g K a2 g K a3 g ¨ a1 a22 ¸
¨2
¸
¨
2 M H ¸
©
¹
§
·
1 K ˜K
K a1 K a 2 K a3 g ¨ a1 a22 ¸
¨2
¸
¨
2 M H ¸
©
¹
2
M AH e ec c
3
2K a1 ˜ K a2
M A M H
ec e = M AH M AH 2M A .
M AH M H M M 2
˜ AH H
M AH M AH
3
S
2
ab e
e ec c
e e K a1 ˜ K a2
a d K a3
3 §
·
¨1 K a1 ˜ K a2 ¸
2 ¸
¨
¨
M H ¸
©
¹
3 g M H
§1
·
K a1 K a 2 K a3 g ¨ 1 102 H K a1 K a 2 ¸
2
©
¹
Aga d ffe e , b e
ae ,e a
d be b a ed de e d g
h ch cha ge ec e
e
a ed
g he cha ge e a
c d
.
3 H
S
C
ca a d E
T
d
a
c ,5
Chap e 13
d
13.
1 2.24
RP 10
cha ge
2 9.04
1 h
RN 10
1
cha ge
1 RN
cha ge cha ge
0 1 140
h
h 0.1˜ 1.0
2
RP
1 RP
cha ge
1
e ec c cha ge
0
1
e ec c
0
5
10
h
15
(
1
2
5.64
2)
S
13.
F
C
ca a d E
h
eac
T
d
a
c ,5
e ha e
'G
2.303 H
K 2.303 H RT
M
Chap e 13
d
˜ M NADH
D2
M ac ˜ M NAD
1 D
2
he e D he f ac
a c
e
f ac a e
a e. F
gI
a
15.5-2 e ha e
D
R
2.303˜ H / 2
2.303˜ H / 2
D
K ˜e
R
he e R
K ˜e
1 D
1 R
The e
ae
ed
he Ma hcad
hee be
.
T de e
e he H a h ch he e e
f eac
50%, e
e ha he D
= 0.5, R
e a 1,
ha
§ 1 ·
2.303˜ H / 2
R 1
K ˜e
Ÿ ¨
¸ 2.303 ˜ H / 2 Ÿ K 2.303 ˜ H
© K¹
' G
' G
2.303 ˜ H a d H 50% eac
RT
2.303RT
g 25900
e
0 1 139
R 0.5
˜e
g
§
·
¨
¸
© 8.314˜ 298.15¹
5
2.899u 10
h 0.1˜ 0.1
h ·
§
¨ 2.303˜ ¸
2 ¹
©
D R
1R
for 50% reaction
1
PH D
g
2.3038.314
˜
˜ 298.15
0.5
PH
0
0
5
10
h
15
4.537
S
C
ca a d E
T
d
a
c ,5
Chap e 13
d
13.87
g 15100
e
0 1 139
R D 0.5
˜e
g ·
§
¨ 8.314˜ 298.15
¸
©
¹
442.133
h 0.1˜ 0.1
h·
§
¨ 2.303˜ ¸
2¹
©
R
1 R
1
0.98
D
0.96
0.94
0
5
10
15
h
13.88
g 50200
e
0 1 139
R D 0.5
˜e
g
§
·
¨ 8.314˜ 298.15
¸
©
¹
9
1.603 u 10
h 0.1˜ 0.1
h·
§
¨ 2.303˜ ¸
2¹
©
R
1 R
1
D
0.5
0
0
5
10
h
PH g
2.303˜ 8.314˜ 298.15
PH
8.794
15
S
13.
C
ca a d E
T
d
a
c ,5
d
Chap e 13
S
h
be
f de A e f
Te b
>Cha e 13>P b e
13.89
13. 0 S
h
be
f de A e f
Te b
>Cha e 13>P b e
13.90
13. 1 S
h
be
f de A e f
Te b
>Cha e 13>P b e
13.91
13. 2 S
h
be
f de A e f
Te b
>Cha e 13>P b e
13.92
13. 3 S
h
be
f de A e f
Te b
>Cha e 13>P b e
13.93
13. 4 S
h
be
f de A e f
Te b
>Cha e 13>P b e
13.94
13.
S
h
be
f de A e f
Te b
>Cha e 13>P b e
13.95
13.
S
h
be
f de A e f
Te b
>Cha e 13>P b e
13.96
13.
S
h
be
f de A e f
Te b
>Cha e 13>P b e
13.97
13.
S
h
be
f de A e f
Te b
>Cha e 13>P b e
13.98
13.
S
h
be
f de A e f
Te b
>Cha e 13>P b e
13.99
13.100 S
h
be
f de A e f
Te b
>Cha e 13>P b e 13.100
13.101 S
h
be
f de A e f
Te b
>Cha e 13>P b e 13.101
13.102 S
h
be
f de A e f
Te b
>Cha e 13>P b e 13.102
13.103 S
h
be
f de A e f
Te b
>Cha e 13>P b e 13.103
13.104 S
h
be
f de A e f
Te b
>Cha e 13>P b e 13.104
13.10 S
h
be
f de A e f
Te b
>Cha e 13>P b e 13.105
13.10 S
h
be
f de A e f
Te b
>Cha e 13>P b e 13.106
13.10 S
h
be
f de A e f
Te b
>Cha e 13>P b e 13.107
13.10 S
h
be
f de A e f
Te b
>Cha e 13>P b e 13.108
13.10 S
h
be
f de A e f
Te b
>Cha e 13>P b e 13.109
13.110 S
h
be
f de A e f
Te b
>Cha e 13>P b e 13.110
13.111 S
h
be
f de A e f
Te b
>Cha e 13>P b e 13.111
13.112 S
h
be
f de A e f
Te b
>Cha e 13>P b e 13.112
13.113 S
h
be
f de A e f
Te b
>Cha e 13>P b e 13.113
13.114 S
h
be
f de A e f
Te b
>Cha e 13>P b e 13.114
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Chapter 14
7KLV SUREOHP FDQ EH VROYHG JUDSKLFDOO\ DV VKRZQ KHUH RU DQDO\WLFDOO\ DV VHHQ LQ WKH 0DWKFDG
ZRUNVKHHW
5HDFWLRQ + 2 + 2 8VLQJWKHSURJUDP&+(0(4WKHHTXLOLEULXPFRQVWDQWFDQEHFRPSXWHGDWHDFKWHPSHUDWXUH7KH
yi P
EDU ai
yi u UHVXOWVDUHJLYHQRQWKHQH[WSDJH$OVRVLQFH P EDU
D 6WRLFKLRPHWULFDPRXQWRISXUHR[\JHQ
6SHFLHV
,Q
2XW
yi X
X + X
X
X 2 X
X
X
+ 2 X
6
X
a+ 2
X
X
X Ÿ Ka
˜
˜
X X X a+ a2 X X X RU
Ka X X X 7KLVZLOOEHVROYHGXVLQJ0DWKFDG
)URPWKHHQHUJ\EDODQFHZHREWDLQWKHIROORZLQJ
X (QJ 3DUWF X (QJ 3DUWE T . X (QJ 3DUWD Solutions to Chemical and Engineering Thermodynamics, 5th ed
DQGIURP(TQ E ZHJHW
z
THG
C
X
a f 'H
¦ Ni LQ
i i C
a f
ZKHUH ¦ Ni LQ C3i
a f
U[Q THG
C32 DQG TLQ
. E H[FHVVR[\JHQ
6SHFLHV
,Q
2XW
+ X 2 + 2 ¦ a Ni fLQ C3i
C
i X
X
Ÿ Ka X X X X C3+ C3 i dT
TLQ
X
yi X
X
X
X
X
X
DQGIRUHQHUJ\EDODQFH
C31 C32 F H[FHVVR[\JHQLQDLU
6SHFLHV
,Q
2XW
+ X 2 X
1 u + 2 X
X
Ÿ Ka X X X X yi X
X
X
X
X
X
X
Chapter 14
Solutions to Chemical and Engineering Thermodynamics, 5th ed
¦ a Ni fLQ C3i
C
i C3 + C32 C3 1 )URP WKH LQWHUVHFWLRQV RI WKH HTXLOLEULXP DQG HQHUJ\ EDODQFH FXUYHV ZH REWDLQ WKH IROORZLQJ
VROXWLRQV>FXUYHVRQIROORZLQJSDJH@ RUGLUHFWO\E\VROYLQJWKHHTXDWLRQVXVLQJ0$7+&$' D TDG . y+ X
y2 y+ 2
E TDG
. y+ Chapter 14
DQG
X
y2 y+ 2
F TDG
. y+ a X # y2 y+ 2
y1 7KLVLVDYHU\GLIILFXOWSUREOHP,XVHGWKH1$6$FKHPLFDOHTXLOLEULXPSURJUDPUHIHUHQFHGLQ6HF
LQWKHVROXWLRQRIWKLVSUREOHP,ZLOOGHVFULEHKHUHKRZWKLVSUREOHPFRXOGEHVROYHGZLWKRXW
WKLVSURJUDP
)LUVWZHQHHGWRLGHQWLI\WKHLQGHSHQGHQWFKHPLFDOUHDFWLRQVDPRQJWKHFRPSRQHQWV6WDUWLQJIURP
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Chapter 14
& + +&&+
2 2
DQGILUVWXVLQJ
& 2 &2
+ +&&+ &
+ 2 + 2
WRHOLPLQDWH+ DQGWKHQ
2
& 2 &2
2
+ + WRHOLPLQDWH2
\LHOGV
& 2 &2
+&&+ 2 + 2 & & 2 &2
+&&+ + &
a f &2 LV YHU\ ODUJH RYHU WKH ZKROH
WHPSHUDWXUHUDQJH LH K 2c h DW . DQG 2c h DW . 6LQFH 2 LV SUHVHQW LQ
H[FHVVWKLVLPSOLHVWKDWWKHLUZLOOEHQRVROLGFDUERQSUHVHQW7KXVZHZLOOHOLPLQDWH&XVLQJWKH
UHDFWLRQHTXDWLRQ & &2 a f2 Ÿ )URP )LJ ZH KDYH WKDW Ka IRU WKH UHDFWLRQ & 2
a
+&&+ 2 + 2 &2 &2 2 &2 +&&+ 2 + &2 7KHWKUHHHTXDWLRQVDERYHIRUPDVHWRILQGHSHQGHQWUHDFWLRQVWKDWFDQEHXVHGIRUWKHGHVFULSWLRQRI
WKLVUHDFWLRQV\VWHP,QIDFWVLQFHHTQV DQG DUHERWKH[SHFWHGWRJRWRFRPSOHWLRQ,XVHGWKH
IROORZLQJUHDFWLRQVHTXHQFHIRUWKHGHVFULSWLRQRIWKLVV\VWHP
2
&2 2
+ 2
+&&+ + 2 &2
U[Q
&2
U[Q + 2
U[Q
7KHILUVWVWHSLQWKHQXPHULFDOVROXWLRQRIWKLVSUREOHPLVWKHFDOFXODWLRQRIWKHHTXLOLEULXPFRQVWDQWV
DQGKHDWVRIUHDFWLRQIRUWKHUHDFWLRQVDERYH8VLQJWKHSURJUDP&+(0(4,REWDLQHG
T
' U[Q H Ka ±
±
±
±
±
' U[Q H Ka N-
u u u u u N-
' U[Q H Ka N-
Solutions to Chemical and Engineering Thermodynamics, 5th ed
T
' U[Q H Ka ' U[Q H Ka N-
u u u u u N-
' U[Q H Ka ±
±
±
±
±
+&&+ 2
+ 2 &2
2
+ 2
&2
&2 + 2
6SHFLHV ,QLWLDOO\
X X &2 X + 2 X + X &2
X b g
Ÿ Ka ¦
X
(TXLOLEULXP
X
2 U|
| rHDFWLRQVDUHWKH
"V
|| LQYHUVHVRI
" rHDFWLRQVDQGDERYH
|W
1RWH7KHVH
X
$IWHU U[Q JRHVWRFRPSOHWLRQ
+&&+
N-
&OHDUO\ ZLWK VXFK D ODUJH YDOXH RI WKH HTXLOLEULXP FRQVWDQW UHDFWLRQ PXVW JR HVVHQWLDOO\ WR
FRPSOHWLRQ,ZLOODVVXPHLWGRHV7KXVWKHUHDFWLRQVWRLFKLRPHWU\LV
Chapter 14
yi F
H
X X a
f
a&2a2 y&2 y2 a&2 y&2 X X
X
¦
X
¦
X
¦
X
¦
a f a f a f
a X f a f a f X a f X
X X X I ¦
K
ZKHUH IRU VLPSOLFLW\ , KDYH DVVXPHG WKDW WKH VWDQGDUG VWDWH DQG DWPRVSKHULF SUHVVXUHV ZHUH WKH
VDPH DQG
bK g
a a+ a2 y+ y2 a+ 2
y+ 2
a f a f a f
a X f a f a f X a f X
X X X ZKHUHWKHHTXLOLEULXPFRQVWDQWV Ka DQG Ka DUHWKHRQHVZKRVHQXPHULFDOYDOXHVDUHJLYHQLQWKH
WDEOHDERYH
Solutions to Chemical and Engineering Thermodynamics, 5th ed
,QVWHDG RI VROYLQJ WKHVH QRQOLQHDU DOJHEUDLF HTXDWLRQV , XVHG WKH 1$6$ *LEEV IUHH HQHUJ\
PLQLPL]DWLRQSURJUDPWRILQGWKHHTXLOLEULXPPROHIUDFWLRQV6LQFHWKLVSDFNDJHXVHVDGLIIHUHQWVHW
RIWKHUPRG\QDPLFGDWDWKHFRPSXWHGPROHIUDFWLRQVGRQRWDJUHHZLWKHTQV DQG DQGWKHWDEOH
RIHTXLOLEULXPFRQVWDQWVJLYHQDERYH7KHUHVXOWVDUH
T . y&2 y&2 y+ y+ 2 y2 X X ' U[Q H N- z
T$'
C3dT N- ± ± ± ±
TLQ
,QWKHWDEOHDERYHWKHPROHIUDFWLRQVZHUHFRPSXWHGXVLQJWKH1$6$SURJUDP X DQG X ZHUH
WKHQFRPSXWHGIURPWKHPROHIUDFWLRQVXVLQJ
y+ Chapter 14
NR
DQG' I G62
PRO
X X $OVRIURP3HUU\ V+DQGERRN
N' I G R$J 62 PRO
a
ZKHUH ¦
LQWHJUDO
z
X ¦ X ¦ DQG y&2
' I G R$J f
NPRO
T$'
C3dT ZHUHFRPSXWHGXVLQJ(TQ D DQGWKH ' U[Q H GDWDLQ$SSHQGL[$,9DQG
TLQ
C3 GDWDLQ7DEOH$OVR
' U[Q H T X ' U[Q H T X ' U[Q H T ' U[Q H T
DQG
z FHC
T$'
3 +&&+ TLQ
I
K
C32 dT
z
T$'
C3 dT TLQ
)LQDOO\IURPHTQ D ZHKDYHDWWKHDGLDEDWLFUHDFWLRQ IODPH WHPSHUDWXUHWKDW
RULQWKHQRWDWLRQKHUH
T$'
C
¦
i DQG WKH
Ni LQ
³
TLQ
M
C3i dT ¦' H T
j U[Q
j
$'
Xj Solutions to Chemical and Engineering Thermodynamics, 5th ed
Chapter 14
T$'
³ C dT ' H T
3
3ORWWLQJXSWKHUHVXOWVLQWKHSUHYLRXVWDEOHLH
$'
z
T$'
C3dT YVTDQG ' U[Q H YVTOHDGVWRWKHVROXWLRQ
TLQ
U[Q
TLQ
T$'
y&2 y+ .
X
X
y&2 y+ 2 DQG
y2 &RPPHQW7KHVROXWLRQDERYHFRQVLGHUHGRQO\ 2 + 2 &2 &2 + DQG+&&+DVSRVVLEOH
UHDFWLRQ VSHFLHV $W WKH KLJK WHPSHUDWXUHV LQYROYHG KHUH RWKHU UHDFWLRQV DQG RWKHU VSHFLHV DUH
SRVVLEOH 7KLV LV REYLRXV LQ WKH UHVXOWV EHORZ 7KH GDVKHG OLQHV UHVXOW IURP WKH FKHPLFDO
HTXLOLEULXPSURJUDPRI1$6$ZLWKRQO\WKHVSHFLHVPHQWLRQHGDERYHDVDOORZHGVSHFLHVDQGWKH
VROLGOLQHVUHVXOWIURPWKH&KHPLFDO(TXLOLEULXP3URJUDPZLWKDOOVSHFLHVDOORZHG
Solutions to Chemical and Engineering Thermodynamics, 5th ed
1RWHKRZGLIIHUHQWWKHWZRVROXWLRQVDUH
7KHDFWXDO XQUHVWULFWHG VROXWLRQKDV2DQG+ QRWLRQVEXWDWRPV DVLPSRUWDQWUHDFWLRQSURGXFWV
EXW WKHVH VSHFLHV KDYH QRW HYHQ EHHQ FRQVLGHUHG LQ WKH DGLDEDWLF UHDFWLRQ WHPSHUDWXUH FDOFXODWLRQ
7KHFRQFOXVLRQLVWKDWUHVWULFWLQJRYHUFRQVLGHUDWLRQWRMXVWWKHVSHFLHVLQWKHSUREOHPVWDWHPHQWLV
XQMXVWLILHG
DOVRDYDLODEOHDVD0DWKFDGZRUNVKHHW CH
CH
CH
CH
CHCH
8VLQJWKHSURJUDP&+(0(4ZHREWDLQWKDWDW. Ka DQG ' U[Q H
IRUWKHUHDFWLRQ
& + g & + g & +& + g 7RILQGWKHH[WHQWRIUHDFWLRQZHXVH
6SHFLHV
,QLWLDO
)LQDO
& + X & + X & +&+ X
X
Ka
a&+&+
a&+ ˜ a&+
1RZ P
yi X
X
X
X
X
X
Ÿ Ka
ai F X IF P I H X K H EDU K
F X IF P I H X K H EDU K
F X IF P I H X K H EDU K
a
a
f
f
f
˜ X ˜ X
˜ X X
˜ X
˜ X EDU ZKLFKKDVWKHVROXWLRQ X DQG P +HDWZKLFKPXVWEHremovedWRNHHSUHDFWRULVRWKHUPDOLV
u - UHPRYHG )URP(TQV DQG ZHKDYHWKDW
N-PRO
X X P EDU
P EDU
X X
X X
X P EDU
7RWDORIPROHV
X
PLQLWLDO u
EDU u
,QLWLDO RIPROHV
a
Chapter 14
nFE R RT OQ K a ' U[Q G R &RQVHTXHQWO\E\PHDVXULQJWKH]HURFXUUHQWFHOOSRWHQWLDOZHREWDLQWKHVWDQGDUGVWDWH*LEEVIUHH
HQHUJ\FKDQJHRQUHDFWLRQ LIDOOWKHLRQVDUHLQWKHLUVWDQGDUGVWDWHV 1RZLIZHFRQWLQXHIXUWKHU
DQGPHDVXUHKRZWKH]HURFXUUHQWVWDQGDUGVWDWHFHOOSRWHQWLDOYDULHVDVDIXQFWLRQRIWHPSHUDWXUH
ZHKDYH
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Chapter 14
§ w' G R ·
§ wE R ·
¨ U[Q
{ ' U[Q S R
nF ¨¨
¸¸
¨ wT ¸¸
w
T
©
¹P
©
¹P
&RQVHTXHQWO\E\NQRZLQJWKH]HURFXUUHQWVWDQGDUGVWDWHFHOOSRWHQWLDO DQGLWVWHPSHUDWXUHGHULYDWLYHZHFDQFDOFXODWH' U[Q H R IURP
' U[Q G R
' U[Q H R T ' U[Q S R RU' U[Q H R
' U[Q G R T ' U[Q S R
6LPLODUO\VWDUWLQJIURP nFE ' U[Q G DQGWKHPHDVXUHG]HURFXUUHQWSRWHQWLDOVZHFDQFDOFXODWH
WKHHQWKDOS\DQGHQWURS\FKDQJHVIRUWKHUHDFWLRQZKHQWKHLRQVDUHQRWLQWKHLUVWDQGDUGVWDWHV
7KHSURFHVVLV
5 u 10 mol/kg N
5 u 10 mol/kg
N, IN
5 u 10 mol/kg
N
N, OUT
, OUT
5 u 10 mol/kg
, IN
$VVXPHDOORWKHUFRPSRQHQWFRQFHQWUDWLRQVDUHXQFKDQJHGVLQFHWKHJOXFRVHFRQFHQWUDWLRQLVVR
ORZ7KHPDVVEDODQFHLV
N LQ N LQ N RXW N RXW
,IGLVWKHUDWHRIJOXFRVHWUDQVSRUWHG WKHQ
N * RXW
N * LQ G
b g
b N g
b g
b N g G
* RXW
* LQ
7KHHQHUJ\EDODQFHLV
¦ N L H L
¦ N L H L
c
h
LQ
c
h c¦ N H h
L
L
LQ
RXW
c
h
¦ N L H L
RXW
Q W DQGWKHHQWURS\EDODQFHLV
Q SJHQ LQ
RXW
RXW
T
7KHNLGQH\RSHUDWHVUHYHUVLEO\DQGPLQLPXPZRUNLPSOLHV SJHQ c¦ N S h c¦ N S h c¦ N S h
L
L
L L LQ
L
c
¦ N L S L
L
h
6XEWUDFWLQJTWLPHVWKHHQWURS\EDODQFHIRUPWKHHQHUJ\EDODQFHJLYHV
¦ N L G L
¦ N L G L
¦ N L G L
¦ N L G L
W c
h
LQ
RU
W
c
h
c
LQ
h
c
RXW
h
RXW
c¦ N G h c¦ N G h c¦ N G h c¦ N G h
c N G h
c N G h c N G h
c N G h
eb N g G jcG h
c N G h c N G h
eb N g G jcG h
L
L
*
*
*
RXW
*
* LQ
L
L
RXW
*
*
RXW
L
L
RXW
*
LQ
*
* LQ
RXW
L
LQ
*
RXW
*
L
LQ
LQ
*
*
LQ
*
*
LQ
1RZVLQFHWKHFRQFHQWUDWLRQVDUHYHU\ORZDQGKDYHQRWFKDQJHGVLJQLILFDQWO\
G*
G*
G*
DQG G *
c h
LQ
c h
7KHUHIRUH
W
c h
RXW
LQ
c h
RXW
c h c h G G * G *
VLQFHIURPWKHSUHYLRXVHTXDWLRQZHFDQHOLPLQDWHWKHVXEVFULSWVLQDQGRXW 7KHQ
f*
W
G
RT
OQ
* G*
f
G
c h c h
Fc h I
GH c h JK
* Solutions to Chemical and Engineering Thermodynamics, 5th ed
Chapter 14
1RZDVVXPLQJLGHDOVROXWLRQV RUWKDWWKHDFWLYLW\FRHIILFLHQWVRIJOXFRVHLQEORRGDQGXULQHDUHWKH
VDPH f*
C* x
W
RT OQ
RT OQ * RT
OQ
G
C
x
f
Fc h I
GH c h JK
* FG a f IJ
Ha f K
* FG a f IJ
Ha f K
* ZKHUHZHKDYH DVVXPH WKDWERWKEORRG DQGXULQHEHLQJPRVWO\ ZDWHUKDYH DERXW WKH VDPH PRODU
FRQFHQWUDWLRQ7KHUHIRUH
FG aC f IJ RT OQFG u IJ RT OQ H u K
H aC f K
W
G
RT OQ
. u OQ u PRO.
* * 1RWHWKDWERG\WHPSHUDWXUHLVR) .
PRO
N
PRO
Solutions to Chemical and Engineering Thermodynamics, 5th ed
7 .
˜
0 ˜
D ˜ §¨
PROH ·
PROH
0 ˜
OLWHU
Chapter 14
PROH
OLWHU
5 ˜
¸
© OLWHU ¹
MRXOH
.˜ PROH
8VLQJHTQDQGWKH'HE\H+XFNHOOLPLWLQJODZRIHTQ
OQ J
D ˜ ˜ ˜ 0 &X62
J 0 &X62 H[S D ˜ ˜ 0 &X62
§ J 0 ·
§ 0 ·
¸ ˜ 5˜ 7˜ OQ¨
¸
© 0 ¹
© J 0 ¹
'*PROHV ˜ 5˜ 7˜ OQ¨
7KHFKDQJHLV*LEEV(QHUJ\FDOFXODWHGDERYHLVIRUWZRPROHVRIHOHFWURQV Q 7KHQXPEHURIPROHVRIHOHFWURQVLQWKLVSUREOHPLVFDOFXODWHGEHORZ
Q ˜ §¨
©
'* ˜ PROH PROH
˜
·
Q
u PROH
§ Q · ˜ '*
¨ ¸ PROHV
© ¹
:PD[ '*
¸
¹
:PD[
IRUDSURFHVVDWFRQVWDQWWHPSHUDWXUHDQGSUHVVXUH
MRXOH
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Chapter 14
DOVRDYDLODEOHDVD0DWKFDGZRUNVKHHW 5[Q&+ 2 +2 &2
*LYHQ
MRXOH
MRXOH
MRXOH
' +I 1 ' +I 2 ' +I &+
PROH
PROH
PROH
MRXOH
MRXOH
' +I +2
' +I &2
PROH
PROH
MRXOH
MRXOH
MRXOH
MRXOH
&S
7
7 7 7 &2
PROH .
PROH .
PROH .
PROH .
MRXOH
MRXOH
MRXOH
MRXOH
&S
7
7 7 7 1
PROH .
PROH .
PROH .
PROH .
MRXOH
MRXOH
MRXOH
MRXOH
7 7 7 &S +2 7
PROH .
PROH .
PROH .
PROH .
0DVV%DODQFH7DEOH
6SHFLHV
,Q
2XW
&+
2
1
&2
+2
7RWDO
1LQ 2
PROH
1 1 PROH
1LQ &+ PROH
1RXW +2 PROH
1RXW &2 PROH
(QHUJ\%DODQFHDW6WHDG\6WDWH
N
ZKHUH+LLVWKHSDUWLDOPRODUHQWKDOS\RIVSHFLHVL
1 L + L 4
3G 9 :
L Solutions to Chemical and Engineering Thermodynamics, 5th ed
Chapter 14
$VVXPLQJQRKHDWIORZQRFKDQJHLQYROXPHRIWKHHQJLQHDQGDQLGHDOJDVPL[WXUH\LHOGV
N
ZKHUHLV+LLVWKHPRODUHQWKDOS\RIVSHFLHVL
:
1 L + L
L $W&
.
+ 1 ' +I 1
&S 1 7 G 7
.
.
+
'
+I
&S +2 7 G 7
+2
+2
.
.
&S &2 7 G 7
+ &2 ' +I &2
.
: 1RXW &2 + &2 1RXW +2 + +2 1 1 + 1 ' +I 1 1LQ &+ ' +I &+ 1LQ 2 ' +I 2
MRXOH
:
7KLVZRUNREWDLQHGLVSHUPROHRIQRFWDQH
DOVRDYDLODEOHDVD0DWKFDGZRUNVKHHW $OOLQ6,XQLWV
5[Q&+ 2 &2 +2
' +I &2
' +I +2
' *I +2
' *I &2
+HDWFDSDFLW\
&S &+ 7
&S 2 7
7
7
7
7
5
' +I 2
' +I 1
' +I &+
' *I 2
' *I 1
' *I &+
7
&S1 7 ˜ ˜7
&S+2 7 ˜ &S&2 7 ˜ ˜ 7 ˜ ˜ 7 ˜ ˜7 Chapter 14
Solutions to Chemical and Engineering Thermodynamics, 5th ed
˜7
7
'HILQLQJ]DVWKHPHWKDQHWRDLUUDWLR
0DVV%DODQFH7DEOH
6SHFLHV
&+
2
1
&2
+2
7RWDO
,Q
]
]
2XW
] ;
;
;
;
]
'+U[Q '+I&2 ˜ '+I+2 '+I&+ ˜ '+I2
'+U[Q
u ,I]WKHQPHWKDQHLVWKHOLPLWLQJUHDFWDQWDQG; ]
,I]!WKHQR[\JHQLVWKHOLPLWLQJUHDFWDQWDQG; ; ] LI ] ˜ ] &SRXW ] 7 ˜ ] ; ] ˜ &S&+ 7 ˜ ; ] ˜ &S2 7
&SRXW ] 7 ˜ &S1 7 ; ] ˜ &S&2 7 ˜ ; ] ˜ &S+2 7
&SRXW ] 7 &SRXW ] 7 &SRXW ] 7
8VLQJHTXDWLRQE
LQLWLDOJXHVV
7RXW §´
7DG ] URRW ¨ µ
¨¶
7RXW
© ] ·
&SRXW ] 7 G7 '+U[Q˜ ; ] 7RXW ¸
¸
¹
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Chapter 14
7 DG ] ]
7KHVROXWLRQLVDSSUR[LPDWHEHFDXVHWKHUDQJHIRUWKHKHDWFDSDFLW\RIPHWKDQHXVHGLVRQO\
YDOLGEHWZHHQ.DQG.
)URPHTQ
§ 7 7 ·
¸
© 7 ¹
IRUD&DUQRWF\FOH
:
4˜ ¨
G:
G4˜ ¨
:
´ 7 7
µ
G4
7
µ
¶7
§ 7 7 ·
¸
© 7 ¹
7
G4
&SRXW ˜ G7 7
´ DG
: &DUQRW ] µ
µ
¶
]
7 ˜ &SRXW ] 7 G7
7
: &DUQRW ]
'LYLGLQJE\JLYHVWKHZRUNSHUPROHRIDLU
7KHZRUNREWDLQHGE\WKH&DUQRWF\FOHLVSORWWHGYHUVXV]DWWKHHQGRI3UREOHP :REWDLQHG &DUQRW ] Solutions to Chemical and Engineering Thermodynamics, 5th ed
Chapter 14
Chapter 14
Solutions to Chemical and Engineering Thermodynamics, 5th ed
1RXW 1 1RXW &2 ] ; ]
1RXW &+ ] ˜ ] ; ]
1RXW +2 ] ˜ ; ]
1LQ&+ ] ˜ ]
1LQ2 IURPWKHPDVVEDODQFHWDEOH
1RXW 2 ] ˜ ; ]
1 ] ˜ ] 1LQ1 7 3DUWLDO0RODU*LEEV)UHH(QHUJ\
§ 1RXW 1 ·
¸
© 1] ¹
*RXW1 ] '*I1 5˜ 7˜ OQ¨
§ 1RXW 2 ] ·
¸
© 1]
¹
*RXW2 ] '*I2 5˜ 7˜ OQ¨
§ 1RXW +2 ] ·
¸
1]
©
¹
*RXW+2 ] '*I+2 5˜ 7˜ OQ¨
§ 1RXW &2 ] ·
¸
1]
©
¹
*RXW&2 ] '*I&2 5˜ 7˜ OQ¨
§ 1LQ1 ·
¸
© 1] ¹
§ 1RXW &+ ] ·
¸
1]
©
¹
*LQ1 ] '*I1 5˜ 7˜ OQ¨
§ 1LQ&+ ] ·
¸
© 1]
¹
*LQ2 ] '*I2 5˜ 7˜ OQ¨
*RXW&+ ] '*I&+ 5˜ 7˜ OQ¨
*LQ&+ ] '*I&+ 5˜ 7˜ OQ¨
§ 1LQ2 ·
¸
© 1] ¹
)URPHTQE
2XW ] 1RXW 1˜ *RXW1 ] 1RXW 2 ] ˜ *RXW2 ] 1RXW &+ ] ˜ *RXW&+ ]
2XW ] 1RXW +2 ] ˜ *RXW+2 ] 1RXW &2 ] ˜ *RXW&2 ]
,Q ] 1LQ1˜ *LQ1 ] 1LQ2˜ *LQ2 ] 1LQ&+ ] ˜ *LQ&+ ]
: )XHO&HOO ] 2XW ] 2XW ] ,Q ]
:REWDLQHG )XHO&HOO ] : )XHO&HOO ]
7KHZRUNREWDLQHGKDVXQLWVRIMRXOHVSHUPROHRIDLU
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Chapter 14
Solutions to Chemical and Engineering Thermodynamics, 5th ed
D )URP7DEOHZHKDYHWKDW
NNR
DQG' I G62
' I G R$J PRO
PRO
$OVRIURP3HUU\ V+DQGERRN
N' I G R$J 62 PRO
1RZFRQVLGHUWKHUHDFWLRQ
$J62 $J62²
7KHFKHPLFDOHTXLOLEULXPUHODWLRQIRUWKLVUHDFWLRQLV
u Ka H[S H[S u u I
K
F
H
H[S F
H
a$J
a
62 u I
K
Chapter 14
KsR
a$J62 1RZIURPHTQ DVVXPLQJWKHVLPSOH'HE\H+FNHOHTXDWLRQ
OQ Ks OQ KsR Q Q z z D
¦ zi Mi
M $J M62 OQ KsR u D
M $J M62 OQ KsR u D
M VRWKDW
%XWM62 $J
a
f
e
j
e
j
e
j
M $J M62 OQ Ks OQ KsR D
OQ M $J M $J OQ
PRODO
7KLVKDVWKHVROXWLRQ
PRO
PRO
M $J u u DQGM62 OLWHU
OLWHU
7KHUHIRUH Ks
e
j
PRO
PRO I
F u PRO I u u u F
H
K
H
OLWHU
OLWHU
OLWHU K
E 1RWHWKDWWKHUHLVDHUURULQWKHSUREOHPVWDWHPHQWRIWKHILUVWSULQWLQJRIWKHWH[W7KHVROXWLRQ
VKRXOGEH0&X62DQGVDWXUDWHGZLWK$J627KH7KHKDOIFHOOUHDFWLRQVDUH
$J e o $J V IRUZKLFKE R YROWV
&X V o &X eIRUZKLFKE R
YROWV
7KHUHIRUHIRUWKHUHDFWLRQ
$J &X V o $J V &X E R
1H[WZHKDYHIURPHTQ WKDW
YROWV
Solutions to Chemical and Engineering Thermodynamics, 5th ed
c h E RT OQ ca h
E
F
ea j
e j
F c M h aJ f I
G OQ
GH e M j OQ aJ f JJK
F aJ f IJ
OQ
G OQ
GH e M j aJ f JK
a RT
E OQ &X F
a &X R
o
$J $J
&X r &X $J r &X r &X $J r &X c
Chapter 14
a f h
e a f j
M &X J r &X RT
E OQ
F
M J r $J R
$J
7RSURFHHGIXUWKHUZHKDYHWRFRPSXWHWKHVROXELOLW\RI$J62LQWKH0&X62VROXWLRQ)RU
WKLVDQGDOOWKHFDOFXODWLRQVWKDWIROORZZHZLOOXVHWKHIDFWWKDWVLQFHWKH&X62FRQFHQWUDWLRQLVVR
PXFK KLJKHU WKDQ WKDW RI $J62 ZH ZLOO QHJOHFW WKH FRQWULEXWLRQ RI $J62 WR WKH WRWDO VROXWLRQ
LRQLFVWUHQJWK$OVREHFDXVHRIWKHKLJKLRQLFVWUHQJWKRIWKHVROXWLRQZHZLOOXVH(TQ WR
FRPSXWHWKHPHDQLRQLFDFWLYLW\FRHIILFLHQWDVIROORZV
z z I
I
I
u ZKHUHI
u 7KHUHIRUH
OQ J r
h c
a f
OQaJ f
OQ J r $J 62
r $J 62 u u u u 6RQRZZHKDYH
OQ Ks OQ KsR Q Q a
f
OQ KsR VR
Ks
u eM j M
$J PRODO
62 e M j $J PRODO DQG
M $J E
u u PRODO
F
I
u u J
GH c
u h
K
OQ
YROW
6LQFHWKLVLVSRVLWLYHLWLVWKHSRWHQWLDOWKDWLVSURGXFHGE\WKHFHOO UDWKHUWKDQPXVWEHDSSOLHG IRU
PHWDOOLFVLOYHUWRIRUP
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Chapter 14
MRXOH
5 *LYHQ
.DDQG+U[QZHUHFDOFXODWHGRQ&+(0(4
. PROH
MRXOH
.D +U[Q
3 3D
7 .
1L PROH
PROH
0DVV%DODQFH7DEOH
6SHFLHV
,Q
2XW
\
&+
;
; ;
&+
;
; ;
&+&+
;
; ;
;
;
;
D&+ ;
D&+ ;
D&+&+ ;
;
;
;
D&+&+ ;
.D; ;
D&+ ; D&+ ;
LQLWLDOJXHVV
; *LYHQ
.D; ; .D
; )LQG ;
; ;; ; PROH
$VVXPLQJWKHFRQWHQWVRIWKHUHDFWRUEHKDYHDVDQLGHDOJDV
1L 5 7
1I 5 7
9L
1I
; PROH
9I
1I PROH
3
3
$QHQHUJ\EDODQFHRQWKHUHDFWRU\LHOGV
9I
3G 9
4 ;; +U[Q
9L
4
MRXOH +HDWPXVWEHUHPRYHGEHFDXVH4LVQHJDWLYH
DOVRDYDLODEOHDVD0DWKFDGZRUNVKHHW 5HDFWLRQ 1 2 12 6SHFLHVEDODQFHWDEOH
6SHFLHV ,QLWLDO
6LQFH P
EDU ai
yi P
EDU
X 2 X 12
6
X a12
a1 a2
X X yi X
X
X
yi 7KXV
Ka
RU
Ka )LQDO
1 Chapter 14
Solutions to Chemical and Engineering Thermodynamics, 5th ed
X
ŸX
X
Ka
Ka
1RZWKHHQHUJ\EDODQFHIRUWKHDGLDEDWLFUHDFWRU(TQ E LV
¦ a N f z C dT ' H aT f X TRXW
3 i
i LQ
U[Q
RXW
TLQ
RU
zm
TRXW
X
TLQ
r
C3 1 C32 dT
a f
' H U[Q TRXW
8VLQJ WKH SURJUDP &+(0(4 DQG WKH GDWD LQ $SSHQGL[ ,, WRJHWKHU ZLWK HTQV DQG WKH
IROORZLQJUHVXOWVDUHREWDLQHG
T . Ka 'H U[Q N-PRO X HT X HQHUJ\ Solutions to Chemical and Engineering Thermodynamics, 5th ed
7KHVROXWLRQLV T a . DQG X
VRWKDW y12
Chapter 14
y1 y2 6LQFHWKLVLVDFRPEXVWLRQUHDFWLRQWKHUHDFWLRQFDQEHDVVXPHGWRJRWRFRPSOHWLRQ
5[Q
&+ ˜ 2 ˜ ˜ 1
˜ &2 ˜ +2 ˜ ˜ 1
*LYHQ
'*I&+ ˜
'*I2 ˜
'*I1 ˜
PROH
MRXOH
'*I&2 ˜
PROH
MRXOH
'*I+2 ˜
PROH
0: &+ ˜ ˜
1, 7KHYDOXHVIRUWKH*LEEVIUHHHQHUJ\RIIRUPDWLRQDUHJLYHQ
DWRQHEDU7KHGLIIHUHQFHLQ*LEEVIUHHHQHUJ\EHWZHHQ
RQHEDUDQGRQHDWPRVSKHUHZLOOEHLJQRUHGEHFDXVHLWLV
LQVLJQLILFDQWLQWKLVFDOFXODWLRQ
MRXOH
JP
PROH
˜ ˜
˜
MRXOH
7 .
˜
PROH
MRXOH
5 ˜
PROH
MRXOH
.˜ PROH
JP
PROH
1) ˜
7RWDOQXPEHURIPROHV
§
§ ·
§ ˜ · ·
¨
¨
¸
¨ ¸ ¸
· ˜ OQ¨
˜ '*I2 ˜ ˜ '*I1 5˜ 7˜ ¨ OQ§¨
˜ ˜ OQ¨
*, '*I&+ ¸
¸¸
¸
© © 1, ¹ © 1, ¹ © 1, ¹ ¹
§ ˜ · ·
§
¨ § ·
¨ ¸ ¸
·
˜ ˜ '*I1 5˜ 7˜ ¨ ˜ OQ¨
˜ ˜ OQ¨
*) ˜ '*I&2 ˜ '*I+2 ˜ OQ§¨
¸¸
¸
¸
© © 1) ¹
© 1) ¹ © 1) ¹ ¹
*,
*)
u PROH MRXOH
1&+ u PROH MRXOH
)URPHTQ
: PRODU : 1&+˜ : PRODU
˜ JP
0: &+
*) *, 5˜ 7˜ 1) 1,
:
u MRXOH
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Chapter 14
:HFDQVWDUWZLWKWKHHQHUJ\DQGHQWURS\EDODQFHVIRUDFORVHGFRQVWDQWYROXPHV\VWHPWKDWLVVWDWLRQDU\
dU
dS Q Q WS DQG
SJHQ IRUPD[LPXPZRUNSJHQ VRWKDWFRPELQLQJWKHWZRHTXDWLRQV
dt
dt T
dU TdS dA WS PD[
dt
dt
dt
1RZLQWHJUDWLQJRYHUWLPH
AILQDO ALQLWLDO
WPD[
C
C
L L ¦ NL AL ILQDO ¦ NL AL LQLWDO
6ROXWLRQWRWKLVSUREOHPLQIROGHU$VSHQIRU7H[WERRN!&KDSWHU!3UREOHP
6ROXWLRQWRWKLVSUREOHPLQIROGHU$VSHQIRU7H[WERRN!&KDSWHU!3UREOHP
6ROXWLRQWRWKLVSUREOHPLQIROGHU$VSHQIRU7H[WERRN!&KDSWHU!3UREOHP
6ROXWLRQWRWKLVSUREOHPLQIROGHU$VSHQIRU7H[WERRN!&KDSWHU!3UREOHP
6ROXWLRQWRWKLVSUREOHPLQIROGHU$VSHQIRU7H[WERRN!&KDSWHU!3UREOHP
6ROXWLRQWRWKLVSUREOHPLQIROGHU$VSHQIRU7H[WERRN!&KDSWHU!3UREOHP
6ROXWLRQWRWKLVSUREOHPLQIROGHU$VSHQIRU7H[WERRN!&KDSWHU!3UREOHP
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Chapter 14
7KHFKHPLFDOUHDFWLRQLV& + 2 o &2 + 2
6LQFHH[FHVVDLULVXVHGDQGH[WUDPROHVRIDLUDUHXVHGIRUDWRWDORIPROHVRI
R[\JHQ$OVRVLQFHDLULVXVHGWKHQXPEHURIPROHVRIDFFRPSDQ\QLWURJHQLV u :HZLOODVVXPHWKDWWKHZDWHULVDOLTXLGDWDPELHQWFRQGLWLRQVIRUPD[LPXPZRUNRXWSXW
6SHFLHV,QLWLDO)LQDOPROHIUDFWLRQ'GI
PROHVPROHVLQOHWJDVH[LWJDV
2FWDQH H[WUDSRODWLRQIURPWDEOHYDOXHV
R[\JHQ
QLWURJHQ
+ 2
&2 )RUWKHUHDFWLRQ
'GU[Q u 'GI+ u 'GI&2 'GI&
u u N-PROHRFWDQHUHDFWHG
)RUWKHVROXWLRQWHUP JDVHRXVFRPSRQHQWVRQO\
'GLQ RT 1 2 LQ u OQ x2 LQ 1 1 LQ u OQ x1 LQ RT u OQ u OQ u -PRORFWDQHUHDFWHG N'GRXW RT 1 2 RXW u OQ x2 RXW 1 1 RXW u OQ x1 RXW 1 + 2RXW u OQ x+ 2RXW 1 &2 RXW u OQ x&2 RXW
RT OQ OQ OQ OQ u -PRORFWDQHUHDFWHG N'GWRWDO 'GU[Q 'GRXW 'GLQ N-PRORFWDQHUHDFWHG
1RWHWKDWWKH*LEEVHQHUJ\RIVROXWLRQWHUPVKDGDQHWFRQWULEXWLRQRIRQO\N-RIDWRWDO
RIN-RURQO\
W6PD[ N-PRORFWDQHUHDFWHG PLQXVVLJQLQGLFDWHZRUNRXWSXW
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Chapter 14
7KHFKHPLFDOUHDFWLRQLV& + 2 o &2 + 2DVLQWKHSUHYLRXVSUREOHP
6LQFHH[FHVVDLULVXVHGDQGDWRWDORIPROHVRIDLUDUHXVHG$OVRVLQFHDLULVXVHGWKH
QXPEHURIPROHVRIDFFRPSDQ\LQJQLWURJHQLV u :HZLOODVVXPHWKDWWKHZDWHULVDOLTXLGDWDPELHQWFRQGLWLRQVIRUPD[LPXPZRUNRXWSXW
6SHFLHV,QLWLDO)LQDOPROHIUDFWLRQ'GI
PROHVPROHVLQOHWJDVH[LWJDV
2FWDQH H[WUDSRODWLRQIURPWDEOHYDOXHV
R[\JHQ
QLWURJHQ
+ 2
&2 )RUWKHUHDFWLRQ
'GU[Q u 'GI+ u 'GI&2 'GI&
u u N-PRORFWDQHUHDFWHG
)RUWKHVROXWLRQWHUP JDVHRXVFRPSRQHQWVRQO\
'GLQ RT 1 2 LQ u OQ x2 LQ 1 1 LQ u OQ x1 LQ
RT u OQ u OQ u -PRORFWDQHUHDFWHG
N-
'GRXW RT 1 2 RXW u OQ x2 RXW 1 1 RXW u OQ x1 RXW 1 + 2RXW u OQ x+ 2RXW 1 &2 RXW u OQ x&2 RXW
RT OQ OQ OQ OQ N-PRORFWDQHUHDFWHG
'GWRWDO
'GU[Q 'GRXW 'GLQ
N-PROHRFWDQHUHDFWHG
1RWHWKDWWKH*LEEVHQHUJ\RIVROXWLRQWHUPVKDGDQHWFRQWULEXWLRQRIRQO\N-RIDWRWDO
RIN-RURQO\
W6PD[ N-PRORFWDQHUHDFWHG PLQXVVLJQLQGLFDWHZRUNRXWSXW
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Chapter 14
7KHFKHPLFDOUHDFWLRQLV& + 2 o &2 + 2DVLQWKHSUHYLRXVSUREOHP
+HUHXQOLNHWKHSUHYLRXVSUREOHPVR[\JHQQRWDLUXVXVHG:HZLOODVVXPHWKDWWKHZDWHU
LVDOLTXLGDWDPELHQWFRQGLWLRQVIRUPD[LPXPZRUNRXWSXW6RWKHPDVVEDODQFHLV
6SHFLHV,QLWLDO)LQDOPROHIUDFWLRQ'GI ' AI
PROHVPROHVLQOHWJDVH[LWJDVN-PRO
2FWDQH
R[\JHQ
+ 2
&2 7KH*LEEVHQHUJ\RIIRUPDWLRQIRURFWDQHLVJRWWHQE\H[WUDSRODWLRQIURPWDEOHYDOXHVIRURWKHU
K\GURFDUERQV$OVRVLQFHDWWKHDPELHQWFRQGLWLRQVWKHJDVHVDUHLGHDO
'G I ' AI PVJM
)RUWKHUHDFWLRQ
' AI RT RU' AI
'AU[Q u 'GI+ u 'GI&2 'GI&
'G I RT
'G I u u N-PRORFWDQHUHDFWHG
)RUWKHVROXWLRQWHUP JDVHRXVFRPSRQHQWVRQO\
'ALQ 'ARXW RT 1 + 2RXW u OQ x+ 2RXW 1 &2 RXW u OQ x&2 RXW
RT OQ OQ N-PRORFWDQHUHDFWHG
'AWRWDO 'AU[Q 'ARXW N-PROHRFWDQHUHDFWHG
1RWHWKDWWKH*LEEVHQHUJ\RIVROXWLRQWHUPVKDGDQHWFRQWULEXWLRQRIRQO\N-RIDWRWDO
RIN-RURQO\RIWKHWRWDO
W6PD[ N-PRORFWDQHUHDFWHG PLQXVVLJQLQGLFDWHZRUNRXWSXW
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Chapter 14
7KHFKHPLFDOUHDFWLRQLV& + 2 o &2 + 2DVLQWKHSUHYLRXVSUREOHP
:HZLOODVVXPHWKDWWKHZDWHULVDOLTXLGDWDPELHQWFRQGLWLRQVIRUPD[LPXPZRUNRXWSXW6RWKH
PDVVEDODQFHLV
6SHFLHV,QLWLDO)LQDOPROHIUDFWLRQ'GI ' AI
PROHVPROHVLQOHWJDVH[LWJDVN-PRO
2FWDQH
R[\JHQ
+ 2
&2 7KH*LEEVHQHUJ\RIIRUPDWLRQIRURFWDQHLVJRWWHQE\H[WUDSRODWLRQIURPWDEOHYDOXHVIRURWKHU
K\GURFDUERQV$OVRVLQFHDWWKHDPELHQWFRQGLWLRQVWKHJDVHVDUHLGHDO
'G I ' AI PVJM ' AI RT RU' AI 'G I RT 'G I )RUWKHUHDFWLRQ
'AU[Q u 'GI+ u 'GI&2 'GI&
u u N-PRORFWDQHUHDFWHG
)RUWKHVROXWLRQWHUP JDVHRXVFRPSRQHQWVRQO\
'ALQ 'ARXW RT 1 + 2RXW u OQ x+ 2RXW 1 &2 RXW u OQ x&2 RXW 1 2 RXW u OQ x2 RXW
RT OQ OQ OQ N-PRORFWDQHUHDFWHG
'AWRWDO 'AU[Q 'ARXW N-PROHRFWDQHUHDFWHG
1RWHWKDWWKH*LEEVHQHUJ\RIVROXWLRQWHUPVKDGDQHWFRQWULEXWLRQRIRQO\N-RIDWRWDO
RIN-RUOHVVWKDQ
W6PD[ N-PRORFWDQHUHDFWHG PLQXVVLJQLQGLFDWHZRUNRXWSXW
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Chapter 14
6LQFHWKHUHDFWLRQLVXQFKDQJHGIURPWKDWLQWKHLOOXVWUDWLRQWKHWVPD[ QHJOHFWLQJWKHVROLWLRQ
WHUPVLVXQFKDQJHG6RLWLVRQO\WKHN L OQ yL WHUPVWKDWFKDQJH
6LQFHH[FHVVDLULVXVHGPROHVRIR[\JHQDUHQHHGHG
R[\JHQ$OVRVLQFHDLULVXVHGWKHQXPEHURIPROHVRIDFFRPSDQ\QLWURJHQLV u :HZLOODVVXPHWKDWWKHZDWHULVDOLTXLGDWDPELHQWFRQGLWLRQVIRUPD[LPXPZRUNRXWSXW
6SHFLHV,QLWLDO)LQDOPROHIUDFWLRQV
PROHVPROHVLQOHWJDVH[LWJDV
0HWKDQH
R[\JHQ
QLWURJHQ
+ 2
&2 )RUWKHVROXWLRQWHUP JDVHRXVFRPSRQHQWVRQO\DVVXPLQJZDWHULVDYDSRU
'GLQ RT N &+ LQ u OQ x&+ LQ N 2 LQ u OQ x2 LQ N 1 LQ u OQ x1 LQ
RT u OQ u OQ u OQ RT
N-PROPHWKDQHUHDFWHG
'GRXW RT N 2 RXW u OQ x2 RXW N 1 RXW u OQ x1 RXW N + 2RXW u OQ x+ 2RXW N &2 RXW u OQ x&2 RXW
RT OQ OQ OQ OQ RT
N-PROPHWKDQHUHDFWHG
'GWRWDO 'GU[Q 'GRXW 'GLQ N-PROPHWKDQHUHDFWHG
1RWHWKDWWKH*LEEVHQHUJ\RIVROXWLRQWHUPVKDGDQHWFRQWULEXWLRQRIRQO\N-RIDWRWDO
RINW6PD[ N-PROPHWKDQHUHDFWHG PLQXVVLJQLQGLFDWHZRUNRXWSXW
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Chapter 14
7KHUPRG\QDPLFSURSHUWLHVIURP6WHDP7DEOHV
/LTXLGZDWHUDWEDUDQGR& HÖ N-NJ
6WHDPDWR&DQGEDU 03D HÖ N-NJ SÖ N- NJ. R
6DWXUDWHGZDWHUDW &
P N3D EDU
R
HÖ N-NJ
/LTXLGZDWHUDW & )LUVWFDOFXODWHWKHHQHUJ\SURYLGHGSHUNJRIVWHDPE\FRPEXVWLQJFRDO
' HÖ N-NJWKHUHIRUHWKHHQHUJ\SURYLGHGE\WKHFRDORQO\
RIZKLFKLVWUDQVIHUUHGWRWKHVWHDPLV N-
3DUW D $VWKHWXUELQHLQDGLDEDWLFDQGUHYHUVLEOHDQGRSHUDWLQJLQVWHDG\VWDWHZHKDYH
IURPWKHPDVVDQGHQHUJ\EDODQFHVWKDW
W
WÖ
HÖ RXW HÖ LQ DQGSÖLQ SÖRXW M
7KHRXWOHWFRQGLWLRQVIURPWKHWXUELQH EHIRUHWKHFRQGHQVHU DUHP N3D EDU
DQG SÖ N- NJ. ,QWHUSRODWLQJLQWKHVWHDPWDEOHVZHILQGWKDWTaR&DQG
HÖ RXW N-NJ7KHUHIRUHWKHZRUNREWDLQHG DVHOHFWULFDOHQHUJ\ LV
NW
WÖ
HÖ RXW HÖ LQ NJ
M
u DQGWKHRYHUDOOHIILFLHQF\LV
3DUW E $VLQSDUW D WKHWXUELQHLQDGLDEDWLFDQGUHYHUVLEOHDQGRSHUDWLQJLQVWHDG\VWDWH
ZHKDYHIURPWKHPDVVDQGHQHUJ\EDODQFHVWKDW
W
WÖ
HÖ RXW HÖ LQ DQGSÖLQ SÖRXW M
7KHRXWOHWFRQGLWLRQVIURPWKHWXUELQHKHUHDUHP EDU 03D
DQG SÖ N- NJ. ,QWHUSRODWLQJLQWKHVWHDPWDEOHVZHILQGWKDWTaR&DQG
HÖ RXW N-NJ7KHUHIRUHWKHZRUNREWDLQHG DVHOHFWULFDOHQHUJ\ LV
W
NWÖ
HÖ RXW HÖ LQ NJ
M
ZKLFKLVOHVVWKDQLQSDUW D +RZHYHUDOVRUHFRYHUHGDVXVHIXOKHDWLV QÖ 4 N-NJ6RWKDW
WRWDOHQHUJ\UHFRYHUHG DVDFRPELQDWLRQRIHOHFWULFDOZRUNDQGXVHIXOKHDWLQJIRUKHDWLQJ
RUDVSURFHVVVWHDP LV N-NJRIVWHDPDQGWKHHIILFLHQF\LV
u Solutions to Chemical and Engineering Thermodynamics, 5h ed
Chapter 15
Solutions to Chemical and Engineering Thermodynamics, 5h ed
Chapter 15
Solutions to Chemical and Engineering Thermodynamics, 5h ed
Chapter 15
Solutions to Chemical and Engineering Thermodynamics, 5h ed
Chapter 15
7KHHTXLOLEULXPUHODWLRQLVK
aXQI
aI
X
RUX
K
K
K
'GXQI
§ 'GXQI ·
§
·
H[S ¨ ¸ H[S ¨ ¸ IRU'GXQI LQFDOPROQRWNFDO7KHUHVXOWVDUH
© RT ¹
© ˜ ¹
0 *GQ K u X u ZKHUHK
0 *GQ K X 0 XUHD K u X u 0 XUHD K X 6LQFHWKHUHDUHRQO\WZRSRLQWVLWLVQRWXVHIXOWRSORW7KHLPSRUWDQWREVHUYDWLRQLVWKDWDV
WKHFRQFHQWUDWLRQRIGHQDWXUDWLRQLQFUHDVHVWKHIUDFWLRQXQIROGHGRUGHQDWXUHGLQFUHDVHV
Solutions to Chemical and Engineering Thermodynamics, 5h ed
Chapter 15
' XQI H ' XQI G
T
1D&OTXQI TXQI ' XQI H ' XQI G ' XQI S
' XQI G
' XQI H T ' XQI S VR' XQI S
0 R& . NFDOPRONFDOPRONFDOPRO ˜ .
7KRXJK,KDYHQRWGRQHVRWRSXWHUURUERXQGVRQWKHHQWURS\UHVXOWVDGGWKHDEVROXWHYDOXHV
RIXQFHUWDLQWLHVLQ' XQI H DQG' XQI G DQGGLYLGHE\WKHDEVROXWHWHPSHUDWXUH6RIRUH[DPSOH
IRUWKH0FDVH r ' XQI S
. D+$
0 + 0 $ J r
0 +$ 0 J +$
. D+$
S. D+$
r 6R
0 +$ DQG67
rNFDOPRO ˜ .
0 +$ 0 $ 6R 0 $ VRWKDW0 $
0 + 67 6R J r
§ 0 Jr ·
XVLQJS+ ORJ ¨ +
¸
6R 0 J +$
© 0 ¹
§ 6 6R J r ·
S+ ORJ ¨ 7
˜
¸
J +$ ¹
© 6R
67 6R J r
˜
6R
J +$
S+ S. D +$
DQG67 6R
6R
J +$ S+S. D +$
Jr
VRWKDW
ª J
S+ S. D +$ º
6R « +$ »
Jr
¬
¼
ª J
º
S. D +$
»
67 S+ 6R « +$
J r S+ «¬
»¼
DQGILQDOO\
67 S+
67 S+
67 S+
ª J
º
S+ S. D +$
« +$ »
J r S+
«¬
»¼
ª J
º
S. D +$
« +$
»
J r S+ «¬
»¼
67 6R Solutions to Chemical and Engineering Thermodynamics, 5h ed
Chapter 15
*OX FRV H & +2 Ÿ &+ 2
(WKDQRO & +2+ Ÿ &+2
VRRQD&PROEDVLVWKHVWRLFKLRPHWU\LV
&+ 2 <1 6 1+ <2 62 <: 6+ 2 o <% 6&2 1 <3 6&+2 <& 6&2 7KH&EDODQFHLV
<% 6 <3 6 <& 6 <3 6 VRWKDW<36 7KH1EDODQFHLV
<1 6 <%6 u 7KH+EDODQFHLV
<1 6 <: 6 <% 6 <3 6
u <: 6 u u <: 6 u u u VRWKDW<: 6 7KDWLVZDWHULVSURGXFHGLQWKLVELRFKHPLFDOSURFHVV
2QD&PROEDVLVWKHVWRLFKLRPHWU\LV
&+2+ <1 6 1+ <2 62 <: 6+ 2 o <% 6&2 1 <& 6&2 7KH&EDODQFHLV <% 6 <& 6 7KH1EDODQFHLV <1 6 <% 6 7KH+EDODQFHLV <1 6 <: 6 <% 6 7KH2EDODQFHLV <2 6 <: 6 o <% 6 <& 6 2QD&PROHEDVLVWKHPROHFXODUZHLJKWRIPHWKDQROLVu DQGWKHPROHFXODUZHLJKWRIWKH\HDVWLV
7KHUHIRUHNJRIPHWKDQRO PROHV &PROHVRI
PHWKDQROJ\HDVW &PROHVRI\HDVW7KHUHIRUH<%6 DQG
<& 6 <% 6 <1 6
u <: 6 <%6 <1 6 u u DQG<: 6
<2 6
<% 6 <& 6 <: 6
u u DQG<2 6 2QD&PROEDVLVWKHVWRLFKLRPHWU\LV
&+ 2 <1 6 1+ <: 6 + 2 o <% 6&2 1 <3 6&+2 <& 6&2 7KH1EDODQFHLV <1 6 <% 6 u 7KH+EDODQFHLV
<1 6 <: 6 <% 6 <3 6
u <: 6 u <3 6
7KH2EDODQFHLV <: 6 o u <3 6 <& 6 7KH&EDODQFHLV <% 6 <3 6 <& 6 <3 6 <& 6 7KHVROXWLRQWRWKHVHHTXDWLRQVLV
<36 <:6 <&6 <16 PROHVRI1+XVHGSHU&PROHRIJOXFRVHFRQVXPHGDQG
<&6 PROHVRI&2SURGXFHGSHU&PROHRIJOXFRVHFRQVXPHG
Solutions to Chemical and Engineering Thermodynamics, 5h ed
Chapter 15
2QD&PROEDVLVWKHVWRLFKLRPHWU\LV
&+ 2 <1 6 1+ <: 6 + 2 o <% 6&2 1 <3 6&+2 <& 6&2 7KH&EDODQFHLV <% 6 <3 6 <& 6 <3 6 <& 6 7KH1EDODQFHLV <1 6 <% 6 7KH+EDODQFHLV <1 6 <: 6 <% 6 <3 6 7KH2EDODQFHLV <: 6 o u <% 6 <3 6 <& 6 <2 6 6LQFHWKHELRPDVVLVWKHVDPHDVLQ,OOXVWUDWLRQ [% 7KHUHIRUH
<2 6
<4 6
[6 [ 1 <1 6 [%<% 6 [3 <3 6
u u <% 6 u <% 6 <3 6
u <% 6 u <3 6 u <% 6 u u <1 6 u <% 6 u <3 6
7KHUHVXOWVDUHDVIROORZV
*LYHQ <%6
<&6
<36
<16
<:6 <26
$0DWKFDG:RUNVKHHWWRGRWKHVHFDOFXODWLRQVIROORZV
6JHQ
\E \F FDUERQEDODQFH
\S \E \F
QLWURJHQEDODQFH
\Q ˜ \E
K\GURJHQEDODQFH
\Z \Q
\Z
˜ \E ˜ \S ˜ \F \Z
\T ˜ \E \S ˜ \Q ˜ \V ˜ \E ˜ \S ˜ \Q
R[\JHQEDODQFH
\R \S
\Q ˜ \E ˜ \S ˜ \R
\T
\V
<46
6JHQ
PHDVXUHG
<46
Solutions to Chemical and Engineering Thermodynamics, 5h ed
Chapter 15
1LWULFDFLGDVWKHQLWURJHQVRXUFH7KHVWRLFKLRPHWU\RQD&PROHEDVLVLV
VXEVWUDWH <1 6+12 <2 62 <: 6+ 2 o <% 6&2 1 <& 6&2 8VLQJWKHIRUPXODRIHTQ WKHJHQHUDOL]HGGHJUHHRIUHGXFWLRQRIQLWULFDFLGLV
[ 1 u ZKLFKLVXQUHDOLVWLFDOO\KLJK7KLVLVEHFDXVHHTQZDVGHYHORSHGIRURUJDQLFFKHPLFDOV
QRWLQRUJDQLFVVXFKDVQLWULFDFLG7KHUHLVQRLQIRUPDWLRQRQWKHKHDWRIFRPEXVWLRQRIQLWULFDFLGDQGLWLVQRW
HYLGHQWWKDWLWLVFRPEXVWLEOH(VWLPDWLQJWKHKHDWRIFRPEXVWLRQIURPWKHKHDWVRIIRUPDWLRQ WRSURGXFHZDWHU VXJJHVWVWKDWWKHKHDWRIFRPEXVWLRQRIQLWULFDFLGVKRXOGEHFORVHWR]HUR7KHUHIRUHZHZLOODVVXPHWKDW [ 1 7KH1EDODQFHLV <1 6 <% 6 7KH&EDODQFHLV <% 6 <& 6 1RWHWKDWVLQFHWKHFKHPLFDOFRPSRVLWLRQRIWKHVXEVWUDWHLVXQNQRZQ ZHZLOORQO\KDYHYDOXHVIRULWVJHQHUDOL]HG
GHJUHHRIUHGXFWLRQZHFDQQRWGRDULJRURXVK\GURJHQRUR[\JHQEDODQFH
[6 <% 6 [6
7KH2EDODQFHLV <2 6
<% 6 7KHVHFRQGODZ RUDYDLODELOLW\ UHVWULFWLRQLV
[6 [ 1 <1 6 t [%<% 6 [3 <3 6 Ÿ [6 t <%6 $OVR <4 6 [ % <% 6 [ 3 <3 6 [6 [ 1 <1 6
[ % <% 6 [6
<% 6 [6 6RWKDWWKHILQDOHTXDWLRQVDUH
[6 <2 6
[
<% 6 <1 6 <% 6
<% 6 d 6 DQG<46 <% 6 [6 7KHVHUHVXOWVDUHSORWWHGRQWKHJUDSKEHORZ
Solutions to Chemical and Engineering Thermodynamics, 5h ed
1D2+ R 1D 2+ DQG3 2+ =
3 = = ˜ 2+ ,QWKHSURWHLQIUHHFRPSDUWPHQW , 0 ,1D ,
0 2+
,QWKHSURWHLQFRQWDLQLQJFRPSDUWPHQW ,, =0 ,,3 = 0 ,,1D RU0 ,,1D Chapter 15
0 2+ 0 2+ =0 3,,=
)RUVLPSOLFLW\,ZLOOOHDYHWKHDQG±VLJQVRIIWKHLRQV 1RZE\HTQ ,
0 ,1D 0 2+
0 ,1D
DQGWKHUHIRUH0 ,,1D
0 ,,1D
,,
0 ,,1D 0 2+
,,
0 2+
=0 ,,3 ˜ 0 ,,1D
6RWKDW0 ,,1D
0 ,1D
,
0 ,1D 0 2+
0 ,1D
=0 3,, 0 ,,1D =0 3,,
=0 3,,
0 ,1D
§ =0 ,,3 ·
=0 3,,
,
¨¨
¸¸ 0 1D © ¹
,,
DQG0 2+
0 ,,1D =0 ,,3
§ =0 ,,3 ·
=0 ,,3
,
¨¨ ¸¸ 0 1D ©
¹
D )URP7DEOH""""
[* VRWKDW' F + * u u N-PRO FRPSDUHGWRWKHH[SYDOXHRI )UXFWRVHKDVWKHVDPHFKHPLFDOIRUPXODDVJOXFRVHVRXVLQJWKHJHQHUDOL]HGGHJUHHRIUHGXFWLRQ
DSSUR[LPDWLRQ ' F + ) N-PRO FRPSDUHGWRWKHH[SYDOXHRI E (YHQWKRXJKWKHUHZLOOEHVPDOOHUURUV,ZLOOXVHWKHJHQHUDOL]HGGHJUHHRIUHGXFWLRQDSSUR[LPDWLRQIRU
WKHDQDO\VLVKHUH$VVXPLQJQRQLWURJHQVRXUFHDQGQRSURGXFWVRWKHUWKDQEXWDQHGLROZHKDYHIURP
WKHVHFRQGODZ DYDLODELOLW\ DQDO\VLV
[
[6 t [3 <3 6 Ÿ <3 6 d 6
DVLQWKHLOOXVWUDWLRQ [3 7KHUHIRUH&PROHVRIEXWDQHGLROLVWKHPD[LPXPWKDWFDQEHSURGXFHGSHU&PROHRIJOXFRVHRU
IUXFWRVH+RZHYHUIUXFWRVHDQGJOXFRVHKDYHFDUERQDWRPVVRWKDWu &PROHVRI
EXWDQHGLROLVWKHPD[LPXPWKDWFDQEHSURGXFHGSHUPROHRIJOXFRVHRUIUXFWRVH%XWEXWDQHGLROKDV
FDUERQDWRPVZKLFKLPSOLHVWKDWDWPRVWPROHVRIEXWDQHGLROFDQEHSURGXFHGSHUPROHRI
JOXFRVHRUIUXFWRVH)LQDOO\WKHPROHFXODUZHLJKWRIEXWDQHGLROLVDQGWKHPROHFXODUZHLJKWRI
JOXFRVHDQGIUXFWRVHLVuuu VRWKDW4 JUDPVRIEXWDQHGLROSHU
JUDPVRIIUXFWRVHRUJOXFRVHRUJRIEXWDQHGLROSHUNJRIIUXFWRVHRUJOXFRVHLVWKHPRVWWKDWFDQEH
SURGXFHG
/LJQLQKDVDUHSHDWLQJXQLWRI&+2ZKLFKRQD&PROHEDVLV&+2,WVJHQHUDOL]HGGHJUHHRI
UHGXFWLRQLV [ u u DQGLWVPROHFXODUZHLJKWLV u J&PROH
D 7KHPD[LPXPEXWDQHGLROSURGXFWLRQLVFRPSXWHGIURP
[6 t [ 3Y36 RU t Y36 RUY36 d 6RWKDWDWPRVW&PROHVRI%FDQEHSURGXFHGSHU&PROHRIOLJQLQ%XW&PROHVRI
OLJQLQ u JRI%SHU&PROHRIOLJQLQFRQVXPHG7KHUHIRUHJ%LV
SURGXFHGSHUJRIOLJQLQNJ%SHUNJRIOLJQLQLVWKHXSSHUOLPLWRI%SURGXFWLRQ
E 7KHRYHUDOOUHDFWLRQLV &+2 Y2 62 Y: 6+ 2 Y3 6&+ 2 Y& 6&2 Solutions to Chemical and Engineering Thermodynamics, 5h ed
Chapter 15
7KH&EDODQFHLV Y3 6 Y& 6 7KH+EDODQFHLV Y: 6 Y3 6 7KH2EDODQFHLV Y2 6 Y: 6 Y3 6 Y& 6 ZKLFKZKHQFRPELQHGZLWKWKHRWKHUHTXDWLRQVUHGXFHVWR
Y2 6 u Y3 6 $OWHUQDWLYHO\E\VWDUWLQJZLWK
u Y3 6
[ [ uY
Y2 6 6 3 3 6
u Y3 6 ZKLFKLVWKHVDPHUHVXOWDVREWDLQHGSUHYLRXVO\7KHFRPSOHWHUHVXOWVDUHVKRZQEHORZ
L \S ˜
L
\R L
\F \S
L
˜ \S ˜ \F \Z
L
L
L
L
L
\Z ˜ \S L
L
\T ˜ ˜ \S L
L
\F
\Z
\R
\S
\T
\S
Solutions to Chemical and Engineering Thermodynamics, 5h ed
Chapter 15
6WRLFKLRPHWU\ &+ 2 &2 o &+ 2 + 2 D 7HVWPDVVEDODQFHV& ZKLFKFKHFNVFDUERQDWRPVDUHFRQVHUYHG
ZKLFKFKHFNVK\GURJHQDWRPVDUHFRQVHUYHG
u u
ZKLFKFKHFNVR[\JHQDWRPVDUHFRQVHUYHG
2 u E *OXFRVH [ ; ' F * .DQG' F + .N-&PRO 6XFFLQLFDFLG [ .; ' F * .DQG' F + .N-&PRO $VVXPLQJQRELRPDVVSURGXFWLRQ ' F * 6 t <3 / 6 ' F * 3 VLQFH' F * + 2 ' F * &2
+ 6RWKDW . t <3 / 6 . Ÿ <3 / 6 d .. 7KHUHSRUWHGVWRLFKLRPHWU\LV<36 ZKLFKVDWLVILHVWKH
VHFRQGODZFRQVWUDLQW
F ' F +6 ' F + 3 4 Ÿ 4 N-&PROJOXFRVHFRQVXPHG u .
G )UDFWLRQRI*LQVXFFLQLFDFLG u .
D &EDODQFH . . ZKLFKFKHFNVFDUERQDWRPVDUHFRQVHUYHG
1EDODQFH . .u . . ZKLFKFKHFNVQLWURJHQDWRPVDUHFRQVHUYHG
2EDODQFH u . <: / 6 . u .u . Ÿ <: / 6 . &KHFNXVLQJWKH+EDODQFH
. u u <: / 6 .u . Ÿ <:6 . ZKLFKFKHFNVZLWKWKHDFFXUDF\RIWKHUHSRUWHGVWRLFKLRPHWU\VRK\GURJHQDWRPVDUHFRQVHUYHG:LOOXVH<:6 LQZKDWIROORZV
E 0HWKDQROLVQRWLQWKH7DEOHVRZLOOXVHGJHQHUDOL]HGGHJUHHRIUHGXFWLRQ
[ IRUELRPDVV[ . 6RWKHHQHUJ\EDODQFHLV
' F +6 .' F + 1 .' F + % 4 . . u . u . 4 4 N-PROH&+2+FRQVXPHG 1RWHWKDWE\GLUHFWFRPEXVWLRQ4 N-PROH&+2+FRQVXPHG .u .
u F )UDFWLRQRI*LQSURGXFW
Chapter 15
Solutions to Chemical and Engineering Thermodynamics, 5h ed
6WRLFKLRPHWU\LV
&+ <2 / 62 <1 / 6 1+ <: / 6 + 2 o <% / 6&+. 2. 1 . <& / 6&2 .EDU u .P
D )URPWKHLGHDOJDVODZPPHWKDQH 1
. u . u EDU ˜ P
.PRO &PROPHWKDQH PRO ˜ .
0:RIELRPDVV ¯¯ J&PRO7KHUHIRUHNJRIELRPDVV &PRO
DQG<%6 7KHUHIRUH
&EDODQFH<&6 <%6 1EDODQFH<16 ¯<%6 2EDODQFH¯<26<:6 ¯<%6¯<&6ĺ¯<26<:6 +EDODQFH¯¯<:6 ¯
ZKLFKOHDGVWR<:6 DQG<26 ' F +V .' F + 1 .' F + % 4
'F +V
' F + 0HWKDQH
.N-PRO' F + 1
' F + $PPRQLD
.N-PRO' F + %
. u .
E . . u . . u . 4; 4 N-PRO&+ FRQVXPHG
4 N-PRO&+ FRQVXPHG u
[
F 8VH
.PRO&+ P
. u IRUPHWKDQH
WKHQIUDFWLRQRI*LQELRPDVV u .
.
N
P &+ FRQVXPHG
.N-&PRO
Solutions to Chemical and Engineering Thermodynamics, 5h ed
D *O\FHURO &+ 2+&+2+&+ 2+ Ÿ &+ 2[
.' F *
.' F +
Chapter 15
. u
SURSDQHGLRO & + 2 Ÿ &+ 2 [
6WRLFKLRPHWU\ &+ 2 <2 / 6 2 <: / 6 + 2
. <3 / 6 &+ 2 <& / 6 &2 . . ZKLFKLVWKHPD[LPXPSURGXFWLRQRI
. SURSDQHGLROSRVVLEOH$WWKLVFRQYHUVLRQWKHVWRLFKLRPHWU\LV
&+ 2 <2 / 6 2 <: / 6 + 2 .&+ 2 <& / 6&2 6HFRQGODZFRQVWUDLQW!<36RU <3 / 6 d
&EDODQFH . <& / 6 Ÿ <&6 +EDODQFH <: / 6
. u Ÿ <: / 6 . 2EDODQFH u <2 / 6 . . u u . Ÿ <2 / 6 6RWKHVWRLFKLRPHWU\LV &+ 2 .&+ 2 .&2 + 2 &PROSURSDQHGLRO u
.J
&PROJO\FHURO .J
&PROJO\FHURO .JUHVXOWVLQ&PROSURSDQHGLRO JRISURSDQHGLRO
Ÿ NJJO\FHUROUHVXOWVLQNJSURSDQHGLRO
E *OXFRVHDVWKHVXEVWUDWH [ .VRWKHVHFRQGODZFRQVWUDLQWLV t <36 RU<36 d . $WWKLVFRQYHUVLRQWKHVWRLFKLRPHWU\LV
&+ 2 <2 / 6 2 <: / 6 + 2 .&+ 2 <& / 6&2 &EDODQFH . <& / 6 Ÿ <&6 +EDODQFH <: / 6
2EDODQFH u <2 / 6
Ÿ <: / 6 . u u . Ÿ <2 / 6
. u
6RWKHVWRLFKLRPHWU\LV &+ 2
.&+ 2 .&2 + 2 &PROSURSDQHGLRO u
&PROJOXFRVH J
.J
&PROJOXFRVH JUHVXOWVLQ&PROSURSDQHGLRO JRISURSDQHGLRO
Ÿ NJJO\FHUROUHVXOWVLQNJSURSDQHGLRO PD[LPXP
Solutions to Chemical and Engineering Thermodynamics, 5h ed
Chapter 15
6WRLFKLRPHWU\
&PROVXEVWUDWH<16 1+ <262 <:6 + 2 o <%6&+2 1 <36&+ 2 <26&2 & EDODQFH <%6 <36 <&6
1 EDODQFH<16 <%6
6HFRQGODZFRQVWUDLQW[6 <1 / 6[ 1 t [% <% / 6 [3 <3 / 6
,I QRELRPDVVSURGXFHG<%6 <16 DQGWKHUHPDLQLQJHTXDWLRQVUHGXFHWR
<36 <&6 DQG[6 t [3 <3 / 6 RU<3 / 6 d [6 / [3
[6 / .
$OVRWKHHQHUJ\EDODQFHLV<46 <36 [6
DQGWKHJHQHUDOL]HGGHJUHHRIUHGXFWLRQIRUPRIWKHR[\JHQEDODQFHLV
[6 [3 <36 [6 <36
6RWKDWHDFKRIWKHVHTXDQWLWLHVFDQEHSORWWHGDVDIXQFWLRQRI<36DQG [6 7KLVLVVKRZQLQWKHDWWDFKHGJUDSK
<26
Solutions to Chemical and Engineering Thermodynamics, 5h ed
Chapter 15
6LQFHWKHIHUPHQWDWLRQLVDQDHURELFY26 WKHPDVVEDODQFHV
&DUERQEDODQFH Y&6 Y36 Y%6
+\GURJHQEDODQFH Y16
Y36 Y:6 Y%6
2[\JHQEDODQFH Y&6 Y36 Y%6 Y:6
DUH
1LWURJHQEDODQFHY16
Y%6 DQGY%6
o Y16
6RWKDW
Y&6 Y36 RU Y&6 Y36
u Y36 Y:6 u RU Y36 Y:6 Y&6 Y36 u Y:6 RU
7KHVROXWLRQLV
Y&6 Y36 DQGY:6 Y&6 Y36 Y:6
*HQHUDOL]HGGHJUHHRIUHGXFWLRQ
[i
u Ci H i u Oi u N i u Si u Pi
Ci
RULIWKHPROHFXOHGRHVQRWFRQWDLQFDUERQ
[i H i u Oi u N i u Si u Pi
6LPSOHVWUHSUHVHQWDWLRQ
ϮнϮ,ϮKс,ϰнK
[& [ + 2 [&+ [&2
u u ƐĞĐŽŶĚůĂǁĂŶĂůLJƐŝƐƚĞůůƐƵƐƚŚĂƚǁŚŝůĞƚŚĞĐŽŶǀĞƌƐŝŽŶŽĨĐĂƌďŽŶƚŽŵĞƚŚĂŶĞǁŝƚŚŽƵƚĂŶLJĞdžƚĞƌŶĂů
ĞŶĞƌŐLJĨůŽǁŝƐŶŽƚŝŵƉŽƐƐŝďůĞ͕ŝƚŵĂLJŶŽƚŽĐĐƵƌƐƉŽŶƚĂŶĞŽƵƐůLJ;ďLJƚŚĞĂĐƚŝŽŶŽĨďĂĐƚĞƌŝĂͿƐŝŶĐĞƚŚĞƌĞŝƐ
ŶŽĚƌŝǀŝŶŐĨŽƌĐĞ͖ŝƚŝƐĨƌĞĞĞŶĞƌŐLJŶĞƵƚƌĂů͘
Solutions to Chemical and Engineering Thermodynamics, 5h ed
Chapter 15
7KHUHLVDSDSHUWKDWFODLPVWKDWLQWKHELRORJLFFRQYHUVLRQRIDQWKUDFLWHFRDOWRPHWKDQHWKH
PRODUUDWLRRIPHWKDQHWRFDUERQGLR[LGHSURGXFHGLVDSSUR[LPDWH8VLQJWKHDSSUR[LPDWH
FRPSRVLWLRQRIDQWKUDFLWHJLYHVWKHIROORZLQJVWRLFKLRPHWU\DQGVHFRQGODZDQDO\VLVXVLQJWKH
JHQHUDOL]HGGHJUHHRIUHGXFWLRQ
& + 2 16+ 2 t &+ &2 12 62 u u u u t u u [ u t 1RWDOORZHGE\WKHVHFRQGODZRIWKHUPRG\QDPLFV
& + 2 16+ 2 t &+ &2 1 2 u u u u t u u t $OVRQRWDOORZHGE\WKHJHQHUDOL]HGGHJUHHRIUHGXFWLRQDQDO\VLVRIWKHVHFRQGODZRIWKHUPRG\QDPLFV
EXWYHU\FORVH
+RZHYHUVLQFHXVHRIWKHJHQHUDOL]HGGHJUHHRIUHGXFWLRQLVDSSUR[LPDWHWKLVODVWUHDFWLRQPD\EHSRVVLEOH &+ 2 o &2 + 2
' I + R ' I * R LQN-PRO
&+ &2 + 2
2 ' rxn + R
R
R
' I + &2
' I + R+2 ' I + &+
u N-PRO&+ ' rxn * R
R
R
' I * &2
' I * R+2 ' I * &+
u N-PRO&+ :LWKVXFKDODUJH' rxn * R .HTLVYHU\ODUJHVRWKHUHFWLRQJRHVWRFRPSOHWLRQDQGGRQRWQHHGWRFRPSXWH
Solutions to Chemical and Engineering Thermodynamics, 5h ed
Chapter 15
JP
P ˜
PRO
7 5* ˜
˜.
PRO.
˜
Z ˜
˜
V
˜
[ ˜
Z ˜
V
˜
§ ·
¨
¸
¹
WHUP P˜ Z ˜
˜ §¨ ·¸
© ¹
U FP
˜
§ ·
¨
¸
©
¹
WHUP P˜ Z ˜
©
5*˜ 7
5*˜7
WHUP u P
L UL
L
˜ FP
˜
ºº
[L [H[S
˜ ªWHUP˜ ª U
U
L
¬
¬ ¼¼
ºº
[L [˜H[SªWHUP˜ª U
U
¬
¬ L ¼ ¼
[
[
U
ZL ˜
[L
7KHVHUHVXOWVDUHLQWHUPVRIPROHIUDFWLRQV
7RFRQYHUWEDFNWRJUDPVSHUJUDPRIZDWHU
ZL ˜
[L ˜
[L
[L ˜
Z
Z
LQXQLWVRIPJJRIZDWHU
Z DWUSP
U
Z DWUSP
JP
7 5* ˜
˜
˜. PD PRO
PRO.
˜
Z ˜
˜
˜ V
JP
PZ ˜
PRO
JP
PK ˜
PRO
U FP
˜
WHUPD PD˜ Z ˜
5*˜7
˜
WHUPK PK˜ Z
5*˜ 7
L UL
Chapter 15
Solutions to Chemical and Engineering Thermodynamics, 5h ed
§ ·
¨ ¸
©
PD
¹
[ § ·
¨ ¸ PD
PZ
© ¹
§ ·
¨ ¸
PK
© ¹
[ § · ¨ ¸
PZ
© PK¹
L
˜ FP
˜
ºº
[DL [˜H[SªWHUPD˜ª UL U
¬
¬
¼ ¼
ºº
[KL [˜H[SªWHUPK˜ ª UL U
¬
¬
¼¼
[D
[K
ULQFP
U
[DL
ZDL PD˜
[DL ˜PZ
7KHVHUHVXOWVDUHLQWHUPVRIPROHIUDFWLRQV
7RFRQYHUWEDFNWRJUDPVSHUJUDPRIZDWHU
ZKL PK˜
[KL
[KL ˜PZ
ZD
ZK
LQXQLWVRIPJJRIZDWHU
ZD U
ZK Solutions to Chemical and Engineering Thermodynamics, 5h ed
Chapter 15
*HQHUDOL]DWLRQRI(TQD WRPXOWLSOHLRQL]DWLRQVLWHV
D + D $
0 + 0 $
. D0 +$
. D
VRWKDW QHJOHFWLQJWKH0 WHUP 0 $
|
D +$
0 +$ 0 0 +
D + D $ 0 + 0 $ . D 0 $ . D. D 0 +$
VLPLODUO\. D
RU0 $ |
D $
0 $
0 +
0 +
DQGE\DQDORJ\
. D. D . D0 +$
0 $
0 $ 0 +
0 $ 6RWKDW 67
0 +
0 +$ ¦
L 0 +
DQG0 $
0 $L 6R
67 6R
6R
. D0 +$
0 +
. D
0 +
. D. D 0 +$
0 +
. D. D
0 +
6LQFH
S. ORJ . D S. 0 +
¦0
0 +$ VR67 6R
L $L
ª
º
« . D . D. D » 0
+$
»N
«0 0 + » 6R
¬« +
¼
ORJ . D DQGS.
ORJ 0 +
ª S.
S. S. º
« »
« S+
S+ »
«¬
»¼
7KHH[WHQVLRQWRVL[LRQL]DEOHVLWHVLVREYLRXV
67 6R
6R
. D. D . D. D . D . D 0 +$
,IWKHUHZHUHRQO\WZRLRQL]DEOHVLWHV
67 6R
. D. D . D. D . D 0 +$
. D. D . D. D 0 +$
Solutions to Chemical and Engineering Thermodynamics, 5h ed
([WUD3UREOHP
'HWHUPLQHWKHS+RIDFHWLFDFLGVROXWLRQVZLWKDGGHGVDOWV
)LOHLV3UREOHP
6RPHVDPSOHFDOFXODWLRQVILUVWZLWKRQO\DFHWLFDFLG
D & *LYHQ
D ˜&
D
D )LQG D
D
S+ ORJ D ˜ &
u S+
*LYHQ
D ˜ & ˜ H[Sª¬ ˜ ˜ D ˜ &
º
¼
D
D )LQG D
D
u S++ ORJª¬ D ˜ & ˜ H[Sª¬ ˜ D ˜ &
º º
¼¼
S++
*LYHQ
D ˜&
D
ª ª¬ ˜ ˜ D ˜ & º¼ º
»
«
»
D˜&
¬
¼
˜ H[S«
D )LQG D
D
u ª
ª
«
¬
«
¬
S++ ORJ« D ˜ & ˜ H[S« ˜
ºº
D˜&
»»
ª¬ D ˜ & º¼ »¼ »¼
S++
6HFRQGVDPSOHFDOFXODWLRQZLK1D&O
1RZDGG1D&O
0LVPRODOLW\RI1D&O
0 D *LYHQ
D ˜&
S+ ORJ D ˜ &
D
D )LQG D
D
u S+
*LYHQ
D
D ˜ & ˜ H[Sª¬ ˜ ˜ D ˜ & 0
º
¼
D
D )LQG D
S++ ORJª¬ D ˜ & ˜ H[Sª¬ ˜ D ˜ & 0
º º
¼¼
S++
*LYHQ
D ˜&
D
ª ª¬ ˜ ˜ D ˜ & 0 º¼ º
»
«
»
D˜& 0
¬
¼
D )LQG D
ºº
»»
ª¬ D ˜ & 0 º¼ »¼ »¼
S++
˜ H[S«
ª
ª
«
¬
«
¬
S++ ORJ« D ˜ & ˜ H[S« ˜
D˜& 0
D
u Chapter 15
Solutions to Chemical and Engineering Thermodynamics, 5h ed
,OOXVWUDWLRQ
SN 53 L
SN SN SK L
L
FKDUJHS L
L
FKDUJH FKDUJHS FKDUJHQ
L
53
L
53
L
L
LVRHOHFWULF FKDUJH
LVRHOHFWULF
L
SK L SN
L
51
L
SK ˜ L 51 FKDUJHQ L SK
SN SN
Chapter 15
Solutions to Chemical and Engineering Thermodynamics, th ed
Section 10.1
10.1
10.1-1
A simpler solution using Mathcad is available as a Mathcad worksheet.
(a) At the bubble point we have
xi Pi vap P where P
yi
5 bar
xET 0.05 ; xP 010
. , xNB 0.40 and xMP 0.45 . Procedure used was
i) Guess T,
ii) Compute each yi , and the sum ¦ yi . If ¦ yi 1 , guessed T is correct; if ¦ yi ! 1 ,
guessed T is too high; if ¦ yi 1 , guessed T is too low. If ¦ yi z 1 , we correct T and
K (bubble point) yET 0.4167 , yP 01730
,
recalculate.
Solution:
T 29366
.
.
and yMP 0.2502 .
yNB 01601
.
(b) The dew point calculation is similar. Here, yET 0.05 , yP 010
. , yNB 0.40 and yMP 0.45 .
P 5 bar, and T and the xi 's are the unknowns. Thus, here we guess the dew point
temperature, compute each of the xi 's from xi
Pyi Pi vap and evaluate ¦ xi . If ¦ xi
guessed temperature is the dew point temperature; if
1 , the
¦ xi ! 1 , guessed T is too low; if
¦ xi 1 , guessed T is too high. Solution (obtained using the computer) T
314.23 K (dew
and xMP 0.4409 .
point) xET 0.0039 , xP 0.0337 , xNB 05215
.
(c) The advantage of the Mathcad worksheet for the isothermal flash calculation is that one can use
the initial flash equations directly, rather than having to make the substitutions below.
For the isothermal flash vaporization calculation, we proceed as in Illustration 8.1-3. First, we
calculate the K factors, i.e.
. 4.402229
10 817.08 30315
,
KET
10185
.
5
and, similarly KP 2.238 , KNB 0546
and KMP 0.743 . Thus, the equations to be solved
.
are:
(1)
xET xP xNB xMP 1
(2)
yET yP yNB yMP 1 Ÿ 10185
. xET 2.238 xP 0546
. xNB 0.743xMP 1
Also,
xET L 1 KET KET 0.05 Ÿ xET 10185
. 9185
. L 0.05
(3)
and, similarly
(4)
xP 2.238 1238
.
L 010
.
(5)
xNB 0546
.
0.454 L 0.40
(6)
xMP 0.743 0.257 L 0.45
Solution procedure I used was to guess L, compute the xi 's from eqns. (3 to 6), and then
ascertain whether eqns. (1) and (2) were satisfied. After a number of iterations, I obtained the
following solution:
a
a
f
f
Solutions to Chemical and Engineering Thermodynamics, th ed
L
xET
xP
xNB
xMP
¦ xi
086667
.
0.0225
0.0858
0.4258
0.4659
.
1000
Section 10.1
V
013333
.
0.2289
01921
.
0.2326
0.3464
1000
.
yET
yP
yNB
yMP
¦ yi
(d) For an adiabatic flush vaporization, shown below, the energy balance must also be satisfied
liquid
liquid
X
vapor
pressure reducing valve or device
This is a (two-phase) Joule-Thomson expansion, so that the energy balance yields H in
H out ,
or
¦ xi H i T , P, x inlet conditions
L ¦ xi H i T , P, x
L
L
c
V ¦ yi H i T , P, y
V
h
outlet liquid conditions
outlet vapor conditions
This equation must be satisfied, together with the mass balances and phase equilibrium
equations of part c. Thus, we have one new unknown here, the outlet temperature, and an
additional equation from which to find that unknown.
10.1-2
(a) We start with eqn. (8.2-12b):
temperature SdT
¦ xi dGi SdT VdP
0.
And note that at constant
0 and
dGi
RTd ln f i
a
a ff
RTd ln xiJ i f i T1 P ,
so that
RT ¦ xi d ln xi RT ¦ xi d ln J i RT ¦ xi d ln fi VdP
0
However, for the pure fluid fugacity, we have, from eqn. (7.2-8a)
RTd ln f i
dGi
V i dP
Thus
b
g
RT ¦ xi d ln xi RT ¦ xi d ln J i ¦ xiV i V dP
Also
0
Solutions to Chemical and Engineering Thermodynamics, th ed
Section 10.1
¦ xiV i V ¦ xiV i ¦ xiV i ¦ xiV i
V ex Ÿ RT ¦ xi d lna xiJ i f V exdP
ex
Now assuming
i) Ideal gas-phase behavior: xiJ i Pi vap
yi P or xiJ i
0
yi P Pi vap and
ii) That PV ex / RT 1 we obtain
ex
a f ¦ x d lnc y P P h PVRT d ln P
¦ xi d ln xiJ i
vap
i
i
i
or
¦ xi d ln yi ¦ xi d ln P ¦ xi d ln Pi vap d PV ex RT id ln P
Now noting that ¦ xid ln Pi vap 0 , since Pi vap is a function of temperature only and
b g d ln P , yields ¦ x d ln y FGH PVRT 1IJK d ln P or
¦ xid ln P
a
y1 y2
x1 1 x1
y1 1 y1
i
ex
1
2
1
1
2
dy2
dy1 .
1
1
1
1
1
1,
f x a1 y f y a1 x f x y
y a1 y f
y a1 y f
a y x f dy d ln P
y a1 y f dx
dx
1
i
FG x x IJ dy FG PV 1IJ d ln P | d ln P
H y y K H RT K
x
x1
dy1 2 dy2
y2
y1
Since
ex
d ln P ¦ xi
1
1
1
1
1
1
y2
1
1
1
Also
1 y ,
and
x2
1 x1 ,
so
1
1
To obtain the x y diagram, I used the equation above in a finite difference form. Using
the argument i to denote the ith data point, the equation above becomes
y1 i x1 i
˜ y1 i y1 i 1
y1 i 1 y1 i
a
f
ln P i ln P i 1
y1 i is unknown, however, P1 i , P1 i 1 , x1 i are known. Also y1 1 is either 0 or 1
depending on which end of the data one starts with. In fact, I started at both ends, in two
separate calculations, to check the results. I solved this problem using the equation above
rewritten as
y1 i
where
B
B r B 2 4C
2
x1 i y1 i 1 ' ln P
and C
1 ' ln P
x1 i y1 i 1
1 ' ln P
and averaged the results from starting at the x1 0 and x1 1 ends.
Once x1 and y1 , were known, the activity coefficients were calculated from
J1
y1 P x1 P1vap and J 2
y2 P x2 P2vap . Results are given below.
Solutions to Chemical and Engineering Thermodynamics, th ed
b.
CCl4 + n-Heptane System
= exprmntl
x-y data
c.
Ethylene bromide + 1-nitropentane System
Azeotrope
= exprmntl
x-y data
Section 10.1
Solutions to Chemical and Engineering Thermodynamics, th ed
Section 10.1
(also available as a Mathcad worksheet)
10.1-3
T
69
p5
10.422
p6
26799
8.314 ( 273.15
p5
exp ( p5 )
p5
2.721
8.314 ( 273.15
T)
p6
exp ( p6 )
p6
1.024
35200
8.314 ( 273.15
T)
p7
exp ( p7 )
p7
0.389
x7
29676
10.456
p7
T)
11.431
x5
0.25
x6
0.45
P
x5 p5
x6 p6
x7 p7
( x5 p5 )
y5
0.3
P
1.258
( x6 p6 )
y6
P
y7
( x7 p7 )
P
Bubble point pressure
P
Bubble point compositions
P
1.258
y5
0.541
y6
0.366
x6
0.6
y7
0.093
Now on to dew point calculation
Initial guesses
z5
0.25
GIVEN
x5 x6
soln
x5
P
z6
1
0.45
x5
z7
x7
0.3
0.3
x5 p5 z5 P
x7 1
0.1
x6 p6 z6 P
x7 p7 z7 P
x7
soln 2
soln 3
P
0.768
FIND( x5 x6 x7 P )
soln 0
x6
soln 1
Dew point pressure
Dew point compositions
x5
0.071
P
x6
0.338
x7
0.592
Solutions to Chemical and Engineering Thermodynamics, th ed
Section 10.1
(also available as a Mathcad worksheet)
10.1-4
Solving for the bubble point pressure
T
P
69
p5( T )
26799
8.314 ( 273.15
exp 10.422
p6( T )
exp 10.456
p7( T )
y5
1.013
exp 11.431
0.33
y6
p5( T )
T)
35200
8.314 ( 273.15
T)
y7
p5( T )
P
T)
29676
8.314 ( 273.15
0.33
K5( T P )
2.721
p6( T )
p6( T )
P
p7( T )
K7( T P )
0.389
p7( T )
P
z5
0.33
K6( T P )
1.024
0.25
z6
0.45
z7
0.3
GIVEN
K6( T P ) z6
K5( T P ) z5
soln
y6 K6( T P ) z6
y5 K5( T P ) z5
K7( T P ) z7 1
y7 K7( T P ) z7
FIND( y5 y6 y7 P )
y5
soln 0
y5
0.541
y6
y7
soln 1
y6
soln 2
0.366
y7
P
soln 3
0.093
P
1.258
This is the bubble-point pressure solution. Now on to the dew-point pressure problem.
x5
0.33
x6
0.33
x7
Note that xi=yi/Ki
0.33
GIVEN
z5
z6
z7
K5( T P )
K6( T P )
K7( T P )
soln
1
x5
soln 2
P
z5
K5( T P )
x6
z6
K6( T P )
x7
z7
K7( T P )
FIND( x5 x6 x7 P )
x5
soln 0
x5
0.071
x6
soln 1
x6
x7
0.338
x7
0.592
soln 3
P
0.768
This is the dew-point pressure solution.
So for a mixture of the composition z5=0.25, z6=0.45 and z7=0.30, at a temperature of 69 C, the
mixture will be all liquid at pressures above 1.258 bar, and all vapor at pressures below 0.768 bar.
Vapor-liquid equilibrium will exist at this temperature only between 0.768 and 1.258 bar, so this is
the pressure range we will examine.
Solutions to Chemical and Engineering Thermodynamics, th ed
T
69
L
0.99
P
1.2
K5( T P ) x5
GIVEN
K6( T P ) x6
x5 ( L ( 1
K5( T P ) )
K5( T P ) ) z5
x6 ( L ( 1
K6( T P ) )
K6( T P ) ) z6
x7 ( L ( 1
K7( T P ) )
K7( T P ) ) z7
x6
x7
x5
soln 0
Section 10.1
soln 1
K7( T P ) x7
soln
soln 2
L
x7
0.32
0.223
x6
0.456
y5
K5( T P ) x5
y6
K6( T P ) x6 y7
K7( T P ) x7
y5
0.507
y6
0.389
0.104
P
1.1
L
K6( T P ) x6
x5 ( L ( 1
K5( T P ) )
K5( T P ) ) z5
x6 ( L ( 1
K6( T P ) )
K6( T P ) ) z6
x7 ( L ( 1
K7( T P ) )
K7( T P ) ) z7
x6
x7
x5
soln 0
soln 1
0.362
0.458
y5
K5( T P ) x5
y6
K6( T P ) x6 y7
y5
0.445
y6
0.427
y7
1
L
L
0.906
( x5
x6
x7 ) 0
FIND( x5 x6 x7 L)
soln 3
V
1
L
V 0.264
L
0.736
K7( T P ) x7
0.128
0.60
K5( T P ) x5
K6( T P ) x6
x5 ( L ( 1
K5( T P ) )
K5( T P ) ) z5
x6 ( L ( 1
K6( T P ) )
K6( T P ) ) z6
x7 ( L ( 1
K7( T P ) )
K7( T P ) ) z7
x6
x7
x5
soln
x7
x6
GIVEN
K7( T P ) x7
L
0.18
L
V
V 0.094
soln 2
x5
1.0
x7 ) 0
0.80
K5( T P ) x5
GIVEN
P
y7
x6
FIND( x5 x6 x7 L)
soln 3
x5
( x5
soln 0
soln 1
K7( T P ) x7
soln
soln 2
L
x7
0.414
soln 3
x5
0.141
x6
0.445
y5
K5( T P ) x5
y6
K6( T P ) x6 y7
K7( T P ) x7
y5
0.383
y6
0.456
0.161
y7
( x5
x6
x7 ) 0
FIND( x5 x6 x7 L)
V
1
V 0.451
L
L
0.549
Solutions to Chemical and Engineering Thermodynamics, th ed
P
0.9
L
K6( T P ) x6
x5 ( L ( 1
K5( T P ) )
K5( T P ) ) z5
x6 ( L ( 1
K6( T P ) )
K6( T P ) ) z6
x7 ( L ( 1
K7( T P ) )
K7( T P ) ) z7
x6
x7
x5
P
soln 0
soln
soln 2
L
x7
0.481
x6
0.412
y5
K5( T P ) x5
y6
K6( T P ) x6 y7
K7( T P ) x7
y5
0.323
y6
0.469
0.208
L
0.8
y7
K6( T P ) x6
x5 ( L ( 1
K5( T P ) )
K5( T P ) ) z5
x6 ( L ( 1
K6( T P ) )
K6( T P ) ) z6
x7 ( L ( 1
K7( T P ) )
K7( T P ) ) z7
x6
x7
soln 0
soln 1
K7( T P ) x7
soln
soln 2
L
x7
0.563
x6
0.359
y5
K5( T P ) x5
y6
K6( T P ) x6 y7
K7( T P ) x7
y5
0.267
y6
0.459
0.274
y7
K6( T P ) x6
x5 ( L ( 1
K5( T P ) )
K5( T P ) ) z5
x6 ( L ( 1
K6( T P ) )
K6( T P ) ) z6
x7 ( L ( 1
K7( T P ) )
K7( T P ) ) z7
x6
x7
soln 0
soln 1
K7( T P ) x7
soln
soln 2
L
x7
0.59
x6
0.339
y5
K5( T P ) x5
y6
K6( T P ) x6 y7
K7( T P ) x7
y5
0.251
y6
0.451
0.298
y7
K6( T P ) x6
x5 ( L ( 1
K5( T P ) )
K5( T P ) ) z5
x6 ( L ( 1
K6( T P ) )
K6( T P ) ) z6
x7 ( L ( 1
K7( T P ) )
K7( T P ) ) z7
x6
x7
x5
L
L
0.337
x6
x7 ) 0
FIND( x5 x6 x7 L)
V
1
L
L
0.09
( x5
x6
x7 ) 0
FIND( x5 x6 x7 L)
V
1
L
V 0.995
L
5.309 10
0.95
K5( T P ) x5
GIVEN
( x5
soln 3
0.071
L
1
V 0.91
x5
1.25
V
0.10
K5( T P ) x5
x5
FIND( x5 x6 x7 L)
soln 3
0.078
GIVEN
x7 ) 0
V 0.663
x5
L
x6
0.20
K5( T P ) x5
0.77
( x5
soln 3
0.107
x5
P
soln 1
K7( T P ) x7
x5
GIVEN
P
0.40
K5( T P ) x5
GIVEN
Section 10.1
soln 0
soln 1
K7( T P ) x7
soln
soln 2
L
x7
0.303
soln 3
x5
0.246
x6
0.451
y5
K5( T P ) x5
y6
K6( T P ) x6 y7
K7( T P ) x7
y5
0.536
y6
0.369
0.094
y7
( x5
x6
x7 ) 0
FIND( x5 x6 x7 L)
V
1
V 0.013
L
L
0.987
3
Solutions to Chemical and Engineering Thermodynamics, th ed
Section 10.1
0 1 9
i
0.768
0.071
0.338
0.592
0
.77
0.071
0.339
0.590
0.0053
.8
0.078
0.359
0.563
0.090
.9
0.107
0.412
0.481
0.337
1.0
PP
xx5
0.141
xx6
0.445
xx7
0.414
0.549
LL
1.1
0.180
0.458
0.362
0.736
1.2
0.223
0.456
0.320
0.906
1.25
0.246
0.451
0.303
0.987
1.258
0.250
0.450
0.300
1.0
(also available as a Mathcad worksheet)
10.1-5
Solving for the bubble point temperature
T
69
p5( T )
p6( T )
p7( T )
P
1.013
exp 10.422
exp 10.456
exp 11.431
26799
8.314 ( 273.15
p5( T )
K5( T P )
T)
29676
8.314 ( 273.15
2.721
P
p6( T )
1.024
K6( T P )
T)
T)
p6( T )
P
35200
8.314 ( 273.15
p5( T )
p7( T )
0.389
K7( T P )
p7( T )
P
Solutions to Chemical and Engineering Thermodynamics, th ed
y5
0.33
y6
0.33
y7
0.33
z5
Section 10.1
0.25
z6
0.45
z7
0.3
GIVEN
K5( T P ) z5
K6( T P ) z6
K7( T P ) z7 1
y5 K5( T P ) z5
y6 K6( T P ) z6
y7 K7( T P ) z7
FIND( y5 y6 y7 T )
soln
y5
soln 0
y5
0.548
y6
soln 1
y6
y7
soln 2
0.363
T
y7
soln 3
0.088
T
61.788
This is the bubble-point temperature solution. Now on to the dew-point temperature problem.
x5
0.33
x6
0.33
x7
Note that xi=yi/Ki
0.33
GIVEN
z5
z6
z7
K5( T P )
K6( T P )
K7( T P )
1
x5
soln 2
T
z5
x6
K5( T P )
z6
K6( T P )
x7
z7
K7( T P )
FIND( x5 x6 x7 T )
soln
x5
soln 0
x5
0.074
x6
soln 1
x6
x7
0.346
x7
soln 3
0.579
T
77.436
This is the dew-point pressure solution.
So for a mixture of the composition z5=0.25, z6=0.45 and z7=0.30, the mixture will be all liquid
at temperatures below 61.79 C, and all vapor at temperatures above 77.44 C. Vapor-liquid
equilibrium will exist only between 61.79 and 77.44 C, so this is the temperature range we
will examine.
T
62
L
0.99
P
1.013
K5( T P ) x5
GIVEN
K6( T P ) x6
x5 ( L ( 1
K5( T P ) )
K5( T P ) ) z5
x6 ( L ( 1
K6( T P ) )
K6( T P ) ) z6
x7 ( L ( 1
K7( T P ) )
K7( T P ) ) z7
x6
x7
x5
soln 0
soln 1
K7( T P ) x7
soln
soln 2
L
x7
0.303
soln 3
x5
0.246
x6
0.451
y5
K5( T P ) x5
y6
K6( T P ) x6 y7
K7( T P ) x7
y5
0.543
y6
0.367
0.09
T
65
L
0.80
y7
( x5
x6
x7 ) 0
FIND( x5 x6 x7 L)
V
1
V 0.012
L
L
0.988
Solutions to Chemical and Engineering Thermodynamics, th ed
x5 ( L ( 1
K5( T P ) )
K5( T P ) ) z5
x6 ( L ( 1
K6( T P ) )
K6( T P ) ) z6
x7 ( L ( 1
K7( T P ) )
K7( T P ) ) z7
x6
x7
x5
T
soln 0
x7
0.343
0.459
y5
K5( T P ) x5
y6
K6( T P ) x6 y7
y5
0.476
y6
0.41
L
y7
K6( T P ) x6
x5 ( L ( 1
K5( T P ) )
K5( T P ) ) z5
x6 ( L ( 1
K6( T P ) )
K6( T P ) ) z6
x7 ( L ( 1
K7( T P ) )
K7( T P ) ) z7
x6
x7
soln 0
soln 1
y5
K5( T P ) x5
y6
K6( T P ) x6 y7
y5
0.411
y6
0.444
y7
K6( T P ) x6
x5 ( L ( 1
K5( T P ) )
K5( T P ) ) z5
x6 ( L ( 1
K6( T P ) )
K6( T P ) ) z6
x7 ( L ( 1
K7( T P ) )
K7( T P ) ) z7
x6
x7
soln 0
( x5
L
L
0.813
x6
x7 ) 0
FIND( x5 x6 x7 L)
soln 3
V
1
L
V 0.365
L
0.635
K7( T P ) x7
0.144
soln 1
K7( T P ) x7
soln
soln 2
L
x7
0.443
0.124
x6
0.432
y5
K5( T P ) x5
y6
K6( T P ) x6 y7
K7( T P ) x7
y5
0.352
y6
0.464
0.183
y7
( x5
x6
x7 ) 0
FIND( x5 x6 x7 L)
soln 3
x5
V
1
L
V 0.551
L
0.20
K5( T P ) x5
K6( T P ) x6
x5 ( L ( 1
K5( T P ) )
K5( T P ) ) z5
x6 ( L ( 1
K6( T P ) )
K6( T P ) ) z6
x7 ( L ( 1
K7( T P ) )
K7( T P ) ) z7
GIVEN
1
0.40
K5( T P ) x5
L
V
V 0.187
soln
0.389
0.453
74
FIND( x5 x6 x7 L)
soln 3
K7( T P ) x7
x7
x6
x5
x7 ) 0
0.114
L
0.157
GIVEN
x6
K7( T P ) x7
soln 2
x5
L
( x5
0.60
K5( T P ) x5
71
soln
L
x6
68
K7 ( T P ) x7
soln 2
0.198
x5
T
soln 1
x5
GIVEN
T
K6 ( T P ) x6
K5 ( T P ) x5
GIVEN
Section 10.1
K7( T P ) x7
soln
( x5
x6
x7 ) 0
FIND( x5 x6 x7 L)
0.449
Solutions to Chemical and Engineering Thermodynamics, th ed
x5
T
soln
x7
soln 2
L
x7
0.504
soln 3
0.098
x6
0.398
y5
K5( T P ) x5
y6
K6( T P ) x6 y7
K7( T P ) x7
y5
0.301
y6
0.467
0.232
76
L
y7
K6( T P ) x6
x5 ( L ( 1
K5( T P ) )
K5( T P ) ) z5
x6 ( L ( 1
K6( T P ) )
K6( T P ) ) z6
x7 ( L ( 1
K7( T P ) )
K7( T P ) ) z7
x6
x7
soln 0
soln 1
K7( T P ) x7
soln
soln 2
L
x7
0.548
x6
0.369
y5
K5( T P ) x5
y6
K6( T P ) x6 y7
K7( T P ) x7
y5
0.27
y6
0.46
0.27
L
y7
L
L
0.251
x6
x7 ) 0
FIND( x5 x6 x7 L)
V
1
L
V 0.891
L
0.109
0.10
K5( T P ) x5
K6( T P ) x6
x5 ( L ( 1
K5( T P ) )
K5( T P ) ) z5
x6 ( L ( 1
K6( T P ) )
K6( T P ) ) z6
x7 ( L ( 1
K7( T P ) )
K7( T P ) ) z7
x6
x7
GIVEN
soln 0
soln 1
K7( T P ) x7
soln
soln 2
L
x7
0.57
0.077
x6
0.353
y5
K5( T P ) x5
y6
K6( T P ) x6 y7
K7( T P ) x7
y5
0.256
y6
0.453
0.291
L
y7
x6
x7 ) 0
FIND( x5 x6 x7 L)
V
1
L
V 0.966
L
0.05
K5( T P ) x5
K6( T P ) x6
x5 ( L ( 1
K5( T P ) )
K5( T P ) ) z5
x6 ( L ( 1
K6( T P ) )
K6( T P ) ) z6
GIVEN
( x5
soln 3
x5
78
( x5
soln 3
0.083
x5
1
V 0.749
x5
77
V
0.10
K5( T P ) x5
x5
T
soln 1
x5
GIVEN
T
x6
0
Section 10.1
K7( T P ) x7
( x5
x6
x7 ) 0
0.034
Solutions to Chemical and Engineering Thermodynamics, th ed
x7 ( L ( 1
x5
T
K7( T P ) ) z7
x6
x7
soln 0
soln 1
FIND( x5 x6 x7 L)
soln
soln 2
L
x7
0.592
soln 3
x5
0.071
x6
0.337
y5
K5( T P ) x5
y6
K6( T P ) x6 y7
K7( T P ) x7
y5
0.242
y6
0.445
0.313
61
L
y7
K6( T P ) x6
x5 ( L ( 1
K5( T P ) )
K5( T P ) ) z5
x6 ( L ( 1
K6( T P ) )
K6( T P ) ) z6
x7 ( L ( 1
K7( T P ) )
K7( T P ) ) z7
x6
x7
x5
V
1
L
V 1.045
L
0.045
0.9
K5( T P ) x5
GIVEN
i
K7( T P ) )
Section 10.1
soln 0
soln 1
K7( T P ) x7
soln
soln 2
L
x7
0.29
0.264
x6
0.446
y5
K5( T P ) x5
y6
K6( T P ) x6 y7
K7( T P ) x7
y5
0.566
y6
0.351
0.083
x6
x7 ) 0
FIND( x5 x6 x7 L)
soln 3
x5
y7
( x5
V
V
1
L
0.047
L
1.047
0 1 9
TT
61.788
0.25
0.45
0.3
1.0
62
0.246
0.451
0.303
0.988
65
0.198
0.459
0.343
0.813
68
0.157
0.453
0.389
0.635
71
xx5
0.124
xx6
0.432
xx7
0.443
LL
0.449
74
0.098
0.398
0.504
0.251
76
0.083
0.369
0.548
0.109
77
0.077
0.353
0.570
0.034
77.436
0.074
0.346
0.570
0.0
Solutions to Chemical and Engineering Thermodynamics, th ed
10.1-6
Section 10.1
Clearly, it is only water that condenses out at the dew point, since O2 and N 2 are far above their
critical temperatures. Thus, at the dew point
PHvap
2O
yH 2 O P
PH 2 O
partial pressure of H 2O in air.
[In writing this expression, all fugacity coefficients have been assumed equal to unity.]
From the data in Problem 5.12 we have, at the dew point,
5432.8
7.8086
26.3026 ln PHvap
2O
27315
. 256
.
and
PHvap
dew point
PH 2 O 24618
. Pa
2O
at the air conditions PHvap
T
2O
256
. qC
3354.9
PH 2 O
PHvap
T
2O
20.6q C
Ÿ relative humidity
PHvap
T
2O
256
. qC
u 100%
7338%
.
Solutions to Chemical and Engineering Thermodynamics, th ed
Section 10.1
10.1-7
Start by determining the pressure if only vapor was present using ideal gas law PV=NRT
5
1 u 8.315˜ 10
P
˜ 298.15
P
3
8.264
bar
3˜ 10
which is much higher than the vapor pressure of either pure component. Therefore, a vapor-liquid
mixture must be present. The molecular weight of butane is 58.124 and its density is 0.575 g/cc,
so that 0.1 moles of liquid butane would occupy 10.1 cc; the molecular weight of hexane is
86.178, and its density is 0.655 g/cc so that 0.9 moles of hexane would occupy 118.4 cc.
Therefore to proceed, I will assume that finally there will be about 2875 cc of vapor and 125 cc of
vapor.
LL 0.9
xb 0.075
P 0.5
yb xb ˜
V 1 LL
2.428
yh xh ˜
P
xh 1 xb
0.2
P
Given
xb ˜ 2.428
yb ˜ P
xh ˜ 0.2
P
equilibrium relation for butane
equilibrium relation for hexane
yh ˜ P
xb ˜ LL yb ˜ V
0.1
xh ˜ LL yh ˜ V
0.9
xb xh
1
yb yh
1
5
V˜ 8.314˜ 10
mass balance on butane
mass balance on hexane
sum of liquid mole fractions = 1
sum of vapor mole fractions = 1
Pressure from ideal gas equation of state
298.15
˜
3
2.875˜ 10
z find ( xb xh yb yh LL V P)
§ 0.081 ·
¨
¸
¨ 0.919 ¸
¨ 0.516 ¸
z ¨ 0.484 ¸
¨
¸
¨ 0.956 ¸
¨ 0.044 ¸
¨ 0.38 ¸
©
¹
xb z
xh z
0
V z
2
yh z
LL z
3
4
P z
5
6
yb
0.516
LL
0.956
Volume of liquid
yb z
1
yh
0.484
VL LL˜ §¨ xb ˜
©
58.124
0.575
xb
xh ˜
0.081
86.178 ·
¸
0.655 ¹
xh
0.919
VL
123.415
Close enough to the guess value of 125 cc that it is not worth iterating.
Solutions to Chemical and Engineering Thermodynamics, th ed
Section 10.1
10.1-8
p5 2.755
p6 1.021
p7 0.39
a) Bubble point pressure
P 0.55˜ p5 0.25˜ p6 0.2˜ p7
y5 0.55˜
y7 0.2˜
p5
P
1.8485
p6
y5
0.81972
y6 0.25˜
y7
0.0422
y5 y6 y7
P
p7
P
y6
P
0.13808
1
b) Dew point pressure
P 2
Initial guess
x5 0.55˜
P
x6 0.25˜
p5
P
x7 0.2˜
p6
P
p7
Given
0.55˜
x5
P
x6
p5
x5 x6 x7
0.25˜
x5 z
x5
0.20854
x6 z
x6
0.25578
x7
0.53569
1
x7 z
2
x5 x6 x7
p6
x7
P
0.2˜
p7
1
z find ( x5 x6 x7 P)
0
P
1
0.20854 ·
¨§
¸
0.25578 ¸
¨
z
¨ 0.53569 ¸
¨ 1.04459 ¸
©
¹
P z
3
P
1.04459
bar
Solutions to Chemical and Engineering Thermodynamics, th ed
Section 10.1
10.1-9
p5 2.755
a)
p6 1.021
p7 0.39
0.55
x5( L p ) x6( L p ) § 1 p5 · ˜ L p5
¨
¸
p
p ¹
©
0.25
p6
§1 ˜L ¨
¸
p
p ¹
©
p6 ·
0.20
x7( L p ) p7
§1 ¨
¸ ˜L p
p ¹
©
p7 ·
y6( L p ) x6( L p ) ˜
y5( L p ) x5( L p ) ˜
p6
y7( L p ) x7( L p ) ˜
p
p 0.9775
Given
L 0.1
x5( L p ) ˜ §¨ 1 0
©
p5
p
p7
p
initial guess for P, fix L
p6 ·
p7 ·
x6( L p ) ˜ §¨ 1 x7( L p ) ˜ §¨ 1 ¸
¸
¸
p ¹
p ¹
p ¹
©
©
p5 ·
p Find ( p )
p
1.129
x5( L p )
0.24
x6( L p )
y5( L p )
0.584
y6( L p )
0.274
0.247
x5( L p ) x6( L p ) x7( L p )
1
y5( L p ) y6( L p ) y7( L p )
1
x7( L p )
0.487
y7( L p )
0.168
b)
p 0.9775
Given
0
L 0.9
x5( L p ) ˜ §¨ 1 ©
initial guess for P, fix L
p6 ·
p7 ·
x7( L p ) ˜ §¨ 1 x6( L p ) ˜ §¨ 1 ¸
¸
¸
p ¹
p ¹
p ¹
©
©
p5 ·
p Find ( p )
L
0.9
p
1.789
x5( L p )
0.522
x6( L p )
y5( L p )
0.804
y6( L p )
0.261
0.149
x7( L p )
y7( L p )
0.217
0.047
Solutions to Chemical and Engineering Thermodynamics, th ed
Section 10.1
10.1-10
p5 2.755
a)
x5( L p ) p6 1.021
p7 0.39
0.55
x6( L p ) § 1 p5 · ˜ L p5
¨
¸
p
p ¹
©
0.25
p6
§1 ˜L ¨
¸
p
p ¹
©
p6 ·
0.20
x7( L p ) § 1 p7 · ˜ L p7
¨
¸
p
p ¹
©
y6( L p ) x6( L p ) ˜
y5( L p ) x5( L p ) ˜
p6
y7( L p ) x7( L p ) ˜
p
p 0.9775
Given
0
L 0.5
x5( L p ) ˜ §¨ 1 ©
p5
p
p7
p
initial guess for P, fix L
p6 ·
p7 ·
x6( L p ) ˜ §¨ 1 x7( L p ) ˜ §¨ 1 ¸
¸
¸
p ¹
p ¹
p ¹
©
©
p5 ·
p Find ( p )
p
b)
x5( L p )
0.386
x6( L p )
y5( L p )
0.714
y6( L p )
y5( L p ) y6( L p ) y7( L p )
1
0
x5( L p ) ˜ §¨ 1 ©
p
0.317
y7( L p )
0.203
1
L 0.27152
x7( L p )
0.297
x5( L p ) x6( L p ) x7( L p )
p 0.9775
Given
10.1-11 a)
1.491
0.083
Guess p, iterate on L
p6 ·
p7 ·
x6( L p ) ˜ §¨ 1 x7( L p ) ˜ §¨ 1 ¸
¸
¸
p ¹
p ¹
p ¹
©
©
p5 ·
1.285
x5( L p )
0.3
x6( L p )
y5( L p )
0.643
y6( L p )
0.294
0.234
x5( L p ) x6( L p ) x7( L p )
1
y5( L p ) y6( L p ) y7( L p )
1
x7( L p )
y7( L p )
0.406
0.123
Number of unknowns
x5, x6, x7, y5, y6, y7, TI, PI, TII, PII, V and L = 12
(4 constraints)
Number of constraints
TI = TII, PI = PII™[i ™\i = 1
VAP
x i Pi
yi P i 5, 6 and 7 (3 constraints)
Mass balances xi L + yi V = zi
(3 constraints) where z is the feed composition
p Find ( p )
Solutions to Chemical and Engineering Thermodynamics, th ed
Section 10.1
(Note summing the three mass balances gives L + V = 1, so that is not an independent
equation)
So the number of degrees of freedom is 12 – 4 – 3 – 3 = 2
Setting T = 69°C fixes one degree of freedom.
Therefore, can not also set one liquid phase composition and one vapor phase composition!
The system would be overspecified.
b) Joe has to release one degree of freedom. Let T vary, then can find a T and P to meet the
composition specifications.
Pivap
10.1-12
Ÿ
yi P 4.35 kPa and ln P (MPa) = 8.347 – 2644/T
T = 191.8 K
10.1-13
xBZ = 0.99
98% of benzene in the feed
xBZ = 0.4
xTMP = 0.6
105 mols/hr
xBZ = ?
Benzene mass balance
TMP mass balance
Product restriction
Also xC6 = 1 - xBZ
0.4 × 105 = 0.99 D + xBZB
0.6 × 105 = 0.01 D + xTMPB
0.98 × 0.4 × 105 = 0.99 D
D = 3.9596 × 104 moles/hr
B = 105–D = 6.0404 × 104 moles/hr
0.6 × 105 = 0.01 × 3.9596 × 104 + xTMP × 6.0404 × 104
6 0.039596 ×104
x TMP
0.98676
6.0404 u 104
x BZ
0.01324
10.1-14
Since the pressure in the smaller vessel is 0.6 bar, that is the partial pressure of the butane
in the vapor phase of the larger vessel. That is
0.6
yB P 0.6 xB PBvap xB u 2.583 so that xB
0.232 and xH 1 xB 0.768
2.583
Now P xB PBvap xH PHvap 0.232 u 2.583 0.768 u 0.218 0.767 bar
so that finally
0.6
0.6
yB
P 0.767
0.782 and yB 1 0.782 0.218
10.1-15 In the folder Aspen for Textbook>Chapter 10.1>Problems>Prob 10.15
Solutions to Chemical and Engineering Thermodynamics, th ed
Section 10.1
10.1-16
The Gibbs energy of a mixture consisting of N1 moles of species 1 and N 2 moles of species 2 is
G
N1 G1 RT ln x1 N 2 G 2 RT ln x2
o
o
§ o
§ o
N1 ·
N2 ·
N1 ¨ G1 RT ln
¸
¸ N 2 ¨ G 2 RT ln
N1 N 2 ¹
N1 N 2 ¹
©
©
So the change in Gibbs energy between a state I and a state II (without chemical reaction) is
'G
N1II G1 RT ln x1II N 2II G 2 RT ln x2II N1I G1 RT ln x1I N 2I G 2 RT ln x2I
o
o
o
o
Since there is no chemical reaction, N1II = N1I and N 2II = N 2I so this equation reduces to
'G
N1II RT ln x1II N 2II RT ln x2II N1I RT ln x1I N 2I RT ln x2I
Based on 1 mole of feed that is 0.5 moles of species 1 and 0.5 moles of species going to one stream
of 0.5 moles of a mixture 0.495 moles of species 1 and 0.005 of species 2, we have
'G
RT
0.495ln 0.99 0.005ln 0.01 0.5ln 0.5 0.5ln 0.5
= 0.495 u 0.01005 0.005 u 4.605 0.5 u 0.693 u 2
= 0.0050 0.0230 0.693 0.665
J
kJ
'G 0.665 RT 0.665 u 8.314
u 298.15 K=1.648
mol ˜ K
mol
This increase in free energy must occur as the result of work be supplied to the system.
Therefore, the work required for the separation is 1.648 kJ per mole of feed.
10.1-17 In the folder Aspen for Textbook>Chapter 10.1>Problems>Prob 10.17
10.1-18 In the folder Aspen for Textbook>Chapter 10.1>Problems>Prob 10.18
10.1-19 In the folder Aspen for Textbook>Chapter 10.1>Problems>Prob 10.1
10.1-20 In the folder Aspen for Textbook>Chapter 10.1>Problems>Prob 10.2
10.1-21 In the folder Aspen for Textbook>Chapter 10.1>Problems>Prob 10.3
10.1-22 In the folder Aspen for Textbook>Chapter 10.1>Problems>Prob 10.4
10.1-23 In the folder Aspen for Textbook>Chapter 10.1>Problems>Prob 10.5
10.1-24 In the folder Aspen for Textbook>Chapter 10.1>Problems>Prob 10.8
10.1-25 In the folder Aspen for Textbook>Chapter 10.1>Problems>Prob 10.9
Solutions to Chemical and Engineering Thermodynamics, th ed
10.1-26 In the folder Aspen for Textbook>Chapter 10.1>Problems>Prob 10.10
10.1-27 In the folder Aspen for Textbook>Chapter 10.1>Problems>Prob 10.13
Section 10.1
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
yi P VLQFHWKHSUHVVXUHLVORZHQRXJKWKDWIXJDFLW\FRHIILFLHQWFRUUHFWLRQV
f iV fi L Ÿ xi Pi YDSJ i
ZLOOEHVPDOO
D )RUWKHLGHDOVROXWLRQ J i IRUDOOVSHFLHV (% Q+ yi xi Pi YDS P 7KXV
y u DQG
y u ZKLFKLQGLFDWHVWKDWWKHLGHDOVROXWLRQDVVXPSWLRQLVLQYDOLG
¦ yi E 5HJXODUVROXWLRQEHKDYLRU V
a
I
I
OQ J y
f
xV xV xV G
FP
a
f FDO FF u f u u V I G G J 6LPLODUO\ J 7KHUHIRUH
RT
u y u DQG ¦ yi ZKLFK LV PXFK
FORVHUWRXQLW\
F 81,)$& XVLQJ WKH SURJUDP 81,)$& ZH KDYH J Ÿ y
u y
u DQG J ¦ yi
ZKLFKLVWRRKLJK
xi PYDS DQGWKHJLYHQYDSRUOLTXLGHTXLOLEULXPGDWDZLOOEHXVHG
G )LUVWWKHH[SUHVVLRQ J i yi P
WRFRPSXWHWKHVSHFLHVDFWLYLW\FRHIILFLHQWVLQWKHJLYHQVROXWLRQ
u VLPLODUO\ J u 8VLQJHTQV ZHREWDLQ E DQG D 7KXVXVLQJWKHYDQ/DDUHTQ
J
OQ J a
V
G FDO FF
FP
DW x
x x
J J DQG OQ J x x VR WKDW y
6LQFHQRQHRIWKHPRGHOV\LHOGV ¦ yi
y
DQG ¦ yi
QRQHRIWKHVROXWLRQPRGHOVLVFRPSOHWHO\FRUUHFW
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
6LQFHWKHUHJXODUVROXWLRQPRGHOFRPHVFORVHVWWRPHHWLQJWKLVFULWHULRQLWSUHVXPDEO\OHDGVWR
WKHEHVWSUHGLFWLRQV²KRZHYHUWKLVLVPHUHO\DK\SRWKHVLV
7KHYDQ/DDUHTXDWLRQZLOOEHXVHGWRILWWKHGDWDJLYHQ6WDUWLQJIURP
+ 2
J i yi P xi Pi YDS
)85)
DQG J 8VLQJHTQ ZHJHW
ZHREWDLQDWPROHZDWHU J D DQG E 7KXV
DQG OQ J OQ J x x
x x ZKLFKZHZLOODVVXPHLVYDOLGDWDOOWHPSHUDWXUHV
y u DQG
$WWKHQHZWHPSHUDWXUHZHKDYH xiJ i PYDS yi P RU xJ u xJ u y u ZKLFKPXVWEHVROYHGWRJHWKHUZLWKWKHDFWLYLW\FRHIILFLHQWH[SUHVVLRQV
DERYHDQGWKHFULWHULDWKDW x x DQG y y 6ROXWLRQSURFHGXUH,XVHGZDVWRJXHVVD
YDOXHRI x FRPSXWH x IURP x x FRPSXWH J DQG J IURPWKHH[SUHVVLRQDERYH yi a
IURP yi
xiJ i Pi YDS P IRU i
f
DQGDQGWKHQFKHFNWRVHHLI ¦ yi
3URFHHGLQJWKLVZD\
WKHIROORZLQJUHVXOWVZHUHREWDLQHG
FDOFXODWHG
PHDVXUHG
x
x
y
a
f
y
1RWH ¦ yi ZKLFKLVQRWTXLWHHTXDOWR7KHGLVFUHSDQF\EHWZHHQWKHFDOFXODWHGDQG
H[SHULPHQWDOUHVXOWVLQGLFDWHVWKHGDQJHUVRIXVLQJDSSUR[LPDWHVROXWLRQPRGHOV
7KHGHVLUHGUHVXOWPD\EHSURYHGDQXPEHURIGLIIHUHQWZD\V
wP
6LPSOHVWSURRI:HKDYHVKRZWKDWDWDQD]HRWURSLFSRLQW
XVLQJWKHWULSOHSURGXFWUXOH
wx T
FG IJ
H K
LQWKHIRUP
FG wP IJ F wT I F wx I
H wx K H wP K H wT K
T
P
\LHOGV
FG wP IJ
H wx K
T
x
6LQFHWKHUHLVQRUHDVRQWREHOLHYH
FG wT IJ F wP I
H wx K H wT K
P
F wP I
H wT K
x
RS
T
LQIDFW x
F wP I
H wT K
x
UV
W
P' H YDS
RT Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
FG wT IJ
H wx K
:HWKHQKDYH
$OWHUQDWHSURRI P
P
P
xJ PYDS xJ PYDS
RS
T
FG wT IJ
H wx K
LM
N
1RZ
w OQ PYDS
L
wx
FG
H
IJ
K
LM w OQ J w OQ P OP UV wx Q W
N wx
FG wT IJ
H wx K
P
FG IJ
H K
FG IJ FG IJ FG
H K H K H
L F w OQ J IJ OP Ÿ cJ P J P hM x G
N H wx K Q
wP
w OQ J w OQ J H ex wT
V ex
x
x
wx P
wx P
wx P
wx
T
o
o
RU J PYDS
YDS
IJ
K
P
P
J PYDS )URPKHUHRQLWLVWKHVDPHDUJXPHQWDVLQWKHWH[W
$OWHUQDWLYHWRSURRIDERYHVWDUWZLWK P
xJ PYDS xJ PYDS FG wP IJ { J P x J P RSFG w OQ J IJ FG w OQ P IJ UV J P
H wx K
TH wx K H wx K W
R
F
w OQ J I
w OQ P I U
F
x J P SG
TH wx JK GH wx JK VW
YDS
P
6LQFH WKH SXUH FRPSRQHQW YDSRU SUHVVXUH GRHV QRW GHSHQG RQ WKH PL[WXUH
P
YDS
LL *LEEV'XKHPHTQLV
YDS
FRPSRVLWLRQDWIL[HGPDQGTVLQFH
OP
Q
V
w OQ J w OQ PYDS
J PYDS
J PYDS xJ PYDS
R
wx
wx
xJ PYDS
RT V IRUDQLGHDOJDVSKDVH7KXV
P
YDS
YDS
P
YDS
P
YDS
YDS
P
P
1RZXVLQJDQDUJXPHQWVLPLODUWR L DERYHDQGDOVRXVLQJ LL JLYHV J PYDS J PYDS ,QJHQHUDOZHKDYH xiJ i Pi YDS yi P DQG ¦ xiJ i Pi YDS P $OVRIURPWKHH[SHULPHQWDOGDWDZH
YDS
NQRZ P(7
EDUDQG PBYDS
D ,GHDO VROXWLRQ
xi Pi YDS
EDU
yi P DQG
P x(7 u x( RU P
GLDJUDPVDUHJLYHQRQIROORZLQJSDJH
x(7 u DQG
P
x(7 &RQVHTXHQWO\ x y DQG P x ¦ xi Pi YDS
P 7KXV
y(7
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
E
5HJXODUVROXWLRQPRGHO
V i/
Gi
EHQ]HQH
(WKDQRO VROXELOLW\ SDUDPHWHUV DW q & ZDV FRPSXWHG XVLQJ ' H YDS
(7
'U YDS
(7
HWKDQRO
' H YDS
(7 RT DQG G (7
d'U
YDS
(7
V /(7
i FDOPRO
I %V (7 G (7 G % I (7 7KHVHDFWLYLW\FRHIILFLHQW
I % DQG OQ J % RT
H[SUHVVLRQV DUH XVHG ZLWK WKH JHQHUDO HTXDWLRQV DERYH WR REWDLQ WKH VROXWLRQ 7KH UHVXOWV DUH
JLYHQEHORZ
F 7KHSURJUDP81,)$&ZDVXVHGWRREWDLQWKHSUHGLFWLRQVVKRZQLQWKHILJXUHVEHORZ
G )LUVW ZH HYDOXDWH WKH DFWLYLW\ FRHIILFLHQWV DW WKH JLYHQ GDWD SRLQW XVLQJ J i yi P xi Pi YDS WR
7KXV OQ J (7
REWDLQ J (7 DQG J % 1H[W XVLQJ HTQV ZH REWDLQ D DQG
E ^7KLV LV WR EH FRPSDUHG ZLWK D DQG E LQ 7DEOH @ 7KXV ZH
REWDLQ
J (7 H[S
x(7 x(7 DQG
J % H[S
x(7 x(7 8VLQJWKHVHH[SUHVVLRQVLQWKHJHQHUDOHTXDWLRQVZHREWDLQWKHUHVXOWVSORWWHGEHORZ
RS
T
RS
T
UV
a f W
UV
a f W
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
D 8VLQJ WKH YDSRU SUHVVXUH GDWD 3ORWWLQJ OQ P YDS YV T , ILQG WKDW DW T
f DQG
a
YDS
P$&
| EDU DQG P7YDS | EDU 7KXV J $&
J7
DWWKHD]HRWURSLFFRPSRVLWLRQRI x7
DQGWUHDWLQJWROXHQHDVVSHFLHV,ILQG D
OQ J 7
x7 x0
DQG E
DQG OQ J $&
x$&
x
1H[WXVLQJHTQV 7KXV
7
7KHVHH[SUHVVLRQVKDYHEHHQXVHGWRREWDLQWKHUHVXOWVSORWWHGEHORZ
E f i L fiV Ÿ xiJ i Pi YDS yi P 7KXV J i xiJ i Pi YDS P DQG ¦ yi 3URFHGXUH,XVHGZDVIRU
HDFKYDOXHRI x7 WR
L *XHVVDQHTXLOLEULXPWHPSHUDWXUHT
LL &RPSXWH y7 DQG y$& DQGFKHFNWRVHHLI ¦ yi
LLL ,IQRWJXHVVDQHZYDOXHRITDQGUHSHDWWKHFDOFXODWLRQ
$VLPSOHUSURFHGXUHLVWRXVH0DWKFDGRUDQRWKHUFRPSXWHUDOJHEUDSURJUDP
5HVXOWV
([SHULPHQW
YDQ/DDU
x7 y7 T q& q & ,GHDO
,GHDOVROXWLRQUHVXOWVZHUHREWDLQHGLQDVLPLODUPDWWHUH[FHSWWKDWDOODFWLYLW\FRHIILFLHQWVZHUH
VHWHTXDOWRXQLW\
L
L
[L
:ULWHYDQ/DDUPRGHOWKLVZD\WRDYRLGGLYLVLRQE\]HUR
JDPW L
H[S
[L
[L
[L
JDPDF L
H[S
[L
[L
OQ JDPW L
OQ JDPDF L
[L
JDPW L
JDPDF L
[
L
[
L
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
[L JDPW L 3L
[L 3LGL
[L JDPDF L [L JDPW L \L
[L [L \LGL
3L
3LGL
\
3
L
\LG
L
3LG
[
L
dGi
[
L
RTd OQ f i
a
a ff
RTd OQ xiJ i f i T P RT ¦ xi d OQ xi RT ¦ xi d OQ J i RT ¦ xi d OQ fi VdP
dGi
V i dP 7KXV
b
g
RT ¦ xi d OQ xi RT ¦ xi d OQ J i ¦ xiV i V dP
+RZHYHUIRUWKHSXUHIOXLGIXJDFLW\ZHKDYHIURPHTQ D RTd OQ f i
$OVR
H[
¦ xiV i V ¦ xiV i ¦ xiV i ¦ xiV i
V H[ Ÿ RT ¦ xi d OQa xiJ i f V H[dP
1RZDVVXPLQJ
L ,GHDOYDSRUSKDVHEHKDYLRU xiJ i Pi YDS
LL 7KDW PV H[ RT ZHREWDLQ
RU
yi P RU xiJ i
yi P Pi YDS DQG
H[
a f ¦ x d OQc y P P h PVRT d OQ P ¦ xi d OQ xiJ i
i
i
i
YDS
¦ xi d OQ yi ¦ xi d OQ P ¦ xi d OQ Pi YDS
$QGQRWHWKDWDWFRQVWDQWWHPSHUDWXUH
VRWKDW
DQG
SdT
L
D :HVWDUWZLWKHTQ E ¦ xi dGi SdT VdP
L
d PV RT id OQ P H[
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
1RZ QRWLQJ WKDW ¦ xLd OQ Pi YDS
b g d OQ P \LHOGV ¦ x d OQ y FGH PVRT IJK d OQ P RU
¦ xLd OQ P
x
x
dy dy
y
y
6LQFH
a
H[
dy
dy a
y
y DQG
x
x OQ P i OQ P i y i LVXQNQRZQKRZHYHU P i P i x i DUHNQRZQ$OVR y LVHLWKHURU
GHSHQGLQJRQZKLFKHQGRIWKHGDWDRQHVWDUWVZLWK,QIDFW,VWDUWHGDWERWKHQGVLQWZR
VHSDUDWHFDOFXODWLRQVWRFKHFNWKHUHVXOWV,VROYHGWKLVSUREOHPXVLQJWKHHTXDWLRQDERYH
UHZULWWHQDV
ZKHUH
B
$OVR
f
y i
y i x i
˜ y i y i y i y i
f x a y f y a x f x y VR
y a y f
y a y f
a y x f dy d OQ P y a y f dx
dx
7RREWDLQWKH x y GLDJUDP,XVHGWKHHTXDWLRQDERYHLQDILQLWHGLIIHUHQFHIRUP8VLQJ
WKHDUJXPHQWiWRGHQRWHWKHiWKGDWDSRLQWWKHHTXDWLRQDERYHEHFRPHV
i
FG x x IJ dy FG PV IJ d OQ P | d OQ P H y y K H RT K
L
y y
x x
y y
H[
d OQ P ¦ xL
VLQFH Pi YDS LV D IXQFWLRQ RI WHPSHUDWXUH RQO\ DQG
B r B C
x i y i ' OQ P
DQG C
' OQ P
x i y i ' OQ P
DQGDYHUDJHGWKHUHVXOWVIURPVWDUWLQJDWWKH x DQG x HQGV
2QFH x DQG y ZHUH NQRZQ WKH DFWLYLW\ FRHIILFLHQWV ZHUH FDOFXODWHG IURP
J
E
y P x PYDS DQG J y P x PYDS 5HVXOWVDUHJLYHQEHORZ
CCl4 + n-Heptane System
= exprmntl
x-y data
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
F
Ethylene bromide + 1-nitropentane System
Azeotrope
= exprmntl
x-y data
F 0RGLILHG81,)$&SUHGLFWLRQ
-------------COMPONENT DATA
-------------Molecular Subgroups
---------------------------------------------------------------------Comp
Name
Number
Description
Frequency
---- -------------------------------- -----------------------1
carbon tetrachloride
53
CCl4
1
2
n-heptane
1
2
CH3
CH2
FDUERQWHWUDFKORULGHDQGQKHSWDQHDW.
[RIFDUERQWHWUDFKORULGH
FDUERQWHWUDFKORULGH
QKHSWDQH
2
5
Solutions to Chemical and Engineering Thermodynamics,5th ed
x(1)
GAM(1)
GAM(2)
GEX(J/MOL)
---------------------------------------------------0.0000
1.1693
1.0000
0.00
0.0250
1.1637
1.0001
9.54
0.0500
1.1582
1.0002
18.77
0.0750
1.1526
1.0006
27.70
0.1000
1.1471
1.0010
36.30
0.1250
1.1415
1.0016
44.56
0.1500
1.1360
1.0024
52.49
0.1750
1.1304
1.0034
60.05
0.2000
1.1249
1.0045
67.25
0.2250
1.1194
1.0058
74.07
0.2500
1.1140
1.0074
80.50
0.2750
1.1085
1.0091
86.53
0.3000
1.1031
1.0111
92.14
0.3250
1.0978
1.0134
97.32
0.3500
1.0925
1.0159
102.05
0.3750
1.0872
1.0187
106.31
0.4000
1.0820
1.0218
110.10
0.4250
1.0768
1.0252
113.40
0.4500
1.0717
1.0290
116.18
0.4750
1.0667
1.0332
118.43
0.5000
1.0618
1.0377
120.13
0.5250
1.0570
1.0427
121.27
0.5500
1.0522
1.0482
121.81
0.5750
1.0476
1.0541
121.74
0.6000
1.0431
1.0606
121.04
0.6250
1.0387
1.0676
119.68
0.6500
1.0345
1.0753
117.63
0.6750
1.0305
1.0837
114.88
0.7000
1.0266
1.0927
111.39
0.7250
1.0229
1.1026
107.13
0.7500
1.0194
1.1133
102.08
0.7750
1.0161
1.1249
96.20
0.8000
1.0130
1.1375
89.46
0.8250
1.0102
1.1512
81.83
0.8500
1.0077
1.1661
73.26
0.8750
1.0055
1.1823
63.73
0.9000
1.0036
1.1999
53.18
0.9250
1.0021
1.2191
41.58
0.9500
1.0010
1.2400
28.89
0.9750
1.0002
1.2627
15.04
1.0000
1.0000
1.2876
0.00
FDUERQWHWUDFKORULGHDQGQKHSWDQHDW.
[RIFDUERQWHWUDFKORULGH
FDUERQWHWUDFKORULGH
QKHSWDQH
Section 10.2
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
x(1)
0.0000
0.0250
0.0500
0.0750
0.1000
0.1250
0.1500
0.1750
0.2000
0.2250
0.2500
0.2750
0.3000
0.3250
0.3500
0.3750
0.4000
0.4250
0.4500
0.4750
0.5000
0.5250
0.5500
0.5750
0.6000
0.6250
0.6500
0.6750
0.7000
0.7250
0.7500
0.7750
0.8000
0.8250
0.8500
0.8750
0.9000
0.9250
0.9500
0.9750
1.0000
y(1)
0.0000
0.0697
0.1328
0.1901
0.2424
0.2903
0.3345
0.3752
0.4130
0.4481
0.4809
0.5115
0.5402
0.5672
0.5927
0.6167
0.6395
0.6611
0.6817
0.7013
0.7200
0.7379
0.7551
0.7716
0.7876
0.8029
0.8178
0.8323
0.8464
0.8601
0.8735
0.8866
0.8995
0.9122
0.9248
0.9373
0.9498
0.9622
0.9747
0.9873
1.0000
y(2)
1.0000
0.9303
0.8672
0.8099
0.7576
0.7097
0.6655
0.6248
0.5870
0.5519
0.5191
0.4885
0.4598
0.4328
0.4073
0.3833
0.3605
0.3389
0.3183
0.2987
0.2800
0.2621
0.2449
0.2284
0.2124
0.1971
0.1822
0.1677
0.1536
0.1399
0.1265
0.1134
0.1005
0.0878
0.0752
0.0627
0.0502
0.0378
0.0253
0.0127
0.0000
D [3YDS [ 3YDS 3 EXEEOHSRLQWSUHVVXUH (TXLPRODUPL[WXUHŸ[ [ 3 3YDS 3YDS 3YDS
E [3YDS \3 3YDS \
\
3YDS 3YDS
F * H[
$[ [ *H[
w § $1 1 ·
¨
¸
w1 © 1 1 ¹
*H[
57 OQ J Ÿ J
3YDS
3YDS 3YDS
$[ $[ [ § $[ ·
H[S ¨
¸ DQG J © 57 ¹
)RUDQHTXLPRODUOLTXLGPL[WXUH
§ $ ·
§ $ ·
J H[S ¨
J H[S ¨
¸
¸
© 57 ¹
© 57 ¹
P(bar)
0.060
0.063
0.066
0.069
0.072
0.074
0.077
0.080
0.083
0.085
0.088
0.090
0.093
0.095
0.098
0.100
0.103
0.105
0.107
0.109
0.112
0.114
0.116
0.118
0.120
0.122
0.125
0.127
0.129
0.131
0.133
0.135
0.136
0.138
0.140
0.142
0.144
0.146
0.148
0.150
0.151
$[ § $[ ·
H[S ¨ ¸ © 57 ¹
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
§ $ · YDS
§ $ · YDS
u H[S ¨
3 u H[S ¨
¸
¸ 3
© 57 ¹
© 57 ¹
§ $ · YDS
YDS
H[S ¨
3
3
¸ © 57 ¹
[ J3YDS [ J 3YDS
3
G [ J39$3
\ 3 [ J3YDS
3
\
§ $ · YDS
H[S ¨
¸ 3
© 57 ¹
§ $ ·
3YDS 3YDS H[S ¨
¸
© 57 ¹
3YDS
3YDS
DQG
\
YDS
YDS
3 3
3 3YDS
6RZKLOHWKHEXEEOHSRLQWSUHVVXUHFKDQJHVLQJRLQJIURPWKHLGHDOWRWKLVQRQLGHDOPL[WXUHWKH
EXEEOHSRLQWYDSRUFRPSRVLWLRQVDUHXQFKDQJHG
Ÿ \
3
YDS
D 6WDUWLQJ IURP xiJ i Pi YDS
yi P xi Pi YDS DQG XVLQJ WKH GDWD LQ WKH SUREOHP
yi P ZH REWDLQ J i
a
f
VWDWHPHQW ZH FDQ FRPSXWH HDFK J i DQG WKHQ OQ J 3 J $ DQG G H[ RT
7KHVHUHVXOWVWRJHWKHUZLWK G
a
H[
x3 OQ J 3 x$ OQ J $ RT x3 x$ DQGLWVLQYHUVHDUHWDEXODWHGRQWKHIROORZLQJSDJH
f
$OVR WKHUH LV D SORW RI OQ J 3 J $ YV x$ 7KLV SORW LQGLFDWHV WKDW WKH GDWD DSSHDUV WR EH
za
f
WKHUPRG\QDPLFDOO\ FRQVLVWHQW LH OQ J 3 J $ dx$ | WKRXJK WKH SRLQWV DW WKH FRPSRVLWLRQ
H[WUHPHV x$
DQG x$
ORRNVXVSHFW
E 6HH 3UREOHP WKH SORWV RI G H[ RT x3 x$ DQG x3 x$ G H[ RT DSSHDU RQ WKH IROORZLQJ
SDJH7KHIDFWWKDWQHLWKHULVOLQHDULQGLFDWHVWKDWQHLWKHUWKHWZRFRQVWDQW0DUJXOHVQRUWKHYDQ
/DDU HTXDWLRQ ZLOO DFFXUDWHO\ ILW WKH GDWD +HQFH RQH ZLOO KDYH WR XVH DW OHDVW D FRQVWDQW
5HGOLFK.LVWHUH[SDQVLRQIRUWKH*LEEVIUHHHQHUJ\WRREWDLQDJRRGILWRIWKHH[SHULPHQWDOGDWD
IRUWKLVV\VWHP
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
xA
xP
JP
JA
FG J IJ
HJ K
OQ
3
$
G H[
RT
G HV
RTx$ x3
x$ x3 RT
G H[
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
PPHDV
K3D 7KHQH[WVWHSLVWRILWSDUDPHWHUVLQWKH*LEEVIUHHHQHUJ\PRGHOVWRWKHH[SHULPHQWDOGDWD,
KDYHGRQHWKLVDVVXPLQJVPDOOHUURUVLQDOOWKHYDULDEOHV TPxDQGy DQGXVLQJWKHPD[LPXP
OLNHOLKRRGPHWKRG7KHUHVXOWVRIWKHGLIIHUHQWPRGHOVDUHJLYHQEHORZ
/ :LOVRQPRGHO / PFDOF
K3D TPHDV
R
&
TFDOF
R
&
xPHDV
xFDOF
yPHDV
yFDOF
ā
K3D EDU u 3D 7KHVXPRIVTXDUHVRIZHLJKWHGUHVLGXDOVIRUWKLVPRGHOLVWKHPHDQGHYLDWLRQLQ y LV
DQGLQPLV
157/PRGHO W W PFDOF
TPHDV
TFDOF
xPHDV
xFDOF
PPHDV
R
R
K3D K3D &
&
ā
yPHDV
yFDOF
ā
6XPRIVTXDUHVRIZHLJKWHGUHVLGXDOV 0HDQGHYLDWLRQLQ y LVLQ P 81,48$&0RGHO
7KHSDUDPHWHUVDUHW DQGW ±
0HDQGHYLDWLRQLQ y LVLQPLV
PPHDV
K3D PFDOF
K3D TPHDV
R
&
TFDOF
R
&
xPHDV
xFDOF
yPHDV
yFDOF
ā
ā
ā
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
PPHDV
K3D E YDQ/DDUPRGHO D VXPRIVTXDUHVRIZWUHVLGXDOV 0HDQ P y PFDOF
K3D TPHDV
R
&
TFDOF
R
&
xPHDV
xFDOF
yPHDV
yFDOF
B &RQVWDQW0DUJXOHV0RGHO A 0HDQGHYLDWLRQLQ P DQGLQ y DQGVXPRIVTXDUHVRIZHLJKWHGUHVLGXDOV PFDOF
TPHDV
TFDOF
xPHDV
xFDOF
yPHDV
yFDOF
PPHDV
R
R
K3D K3D &
&
6RRIWKHPRGHOVFRQVLGHUHGKHUHWKH:LOVRQPRGHOSURYLGHVWKHEHVWGHVFULSWLRQ RIWKHWZR
FRQVWDQWPRGHOV IRUWKLVGDWDVHW
8VLQJWKHSURJUDPIRUWKHRULJLQDO81,)$&PRGHOWDNLQJTDQG x3 DVNQRZQDQGFRPSXWLQJ
J 3 J $ DQGPDVZHOODV y3 ZHREWDLQ
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
x3
T q&
J FDOF
3
J FDOF
$
P FDOF
¦xJ P
L
EDU
A
L L
YDS
y3FDOF
y3H[S
PHDVXUHG
YDOXH EDU
:KLOHWKHSUHGLFWLRQLVQRWSHUIHFWLWLVUHODWLYHO\JRRG
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
)LUVWZHVKRXOGFKHFNWRVHHLIWKLVSUREOHPLVVROXEOH LHZHOOSRVHGLQWKHVHQVHRIWKH
*LEEVSKDVHUXOH 7KH*LEEVSKDVHUXOHHTQ LV F C M P 7KXVKHUHZHKDYH
F GHJUHHVRIIUHHGRP6LQFHWKHWHPSHUDWXUHDQGWZRLQGHSHQGHQWOLTXLG
SKDVHPROHIUDFWLRQVDUHVSHFLILHGWKHSUREOHPLVZHOOSRVHG
)RUWKHVROXWLRQRIWKLVSUREOHPWKHIROORZLQJVXEVFULSWVZLOOEHXVHG HWKDQRO EHQ]HQH DQG HWK\ODFHWDWH $VWKHILUVWVWHSFRPSXWHWKHSXUHFRPSRQHQWYDSRU
SUHVVXUHV7KLVLVGRQHE\XVLQJWKHYDSRUSUHVVXUHGDWDLQWKH³Chemical Engineers' Handbook´
PDNLQJSORWVRI OQ Pi YDS YV T DQGWKHQGHWHUPLQLQJ OQ Pi YDS DQGKHQFH Pi YDS DW T q & c T u .h ZHILQG
PYDS # EDU PYDS # EDUDQG PYDS # EDU
1H[WZHQHHGFRPSXWHWKHOLTXLGSKDVHDFWLYLW\FRHIILFLHQWV7KLVZLOOEHGRQHXVLQJWKHWHUQDU\
YDQ/DDUHTQ HTQ $ >6HHDOVR3UREOHPF@WRJHWKHUZLWKWKHHQWULHVLQ7DEOH
+HUHRQHKDVWREHFDUHIXODERXWWKHRUGHULQZKLFKWKHVSHFLHVDSSHDULQWKHWDEOH,REWDLQHGWKH
IROORZLQJ
D E Ÿ D D E Ÿ D D E Ÿ D 1RZIURPHTQ $ OQ J E D E D eD D D j}
{x D e j x D e j x x
x x e j x e j
x x xr
x m
E D ŸJ
E D E D x x x 7RREWDLQDQH[SUHVVLRQIRU J ZHLQWHUFKDQJHVXEVFULSWVDQGLQHTQ $ WRREWDLQ>VHH
VROXWLRQWR3UREOHPF@
D E E {x E e j x D e j x x e je jeE D D j}
x x e j x e j
x x x r
mx D D E E D D x x x OQ J E E D E E D E E D E D E x x x ŸJ
E D E E D D x x x $QH[SUHVVLRQIRU J LVREWDLQHGE\LQWHUFKDQJLQJLQGLFHVDQGLQHTQ $ WRREWDLQ>VHH
VROXWLRQWR3UREOHPF@
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
OQ J D E D E {x E e j x E e j x x e je jeE E E
x x e j x e j
x x x r
mx D Ÿ OQ J D E D J
DQG
E D j}
E x x x x x x :LWKWKHVHSUHOLPLQDULHVWDNHQFDUHRIZHFDQQRZSURFHHGRQWRWKHVROXWLRQ7KHHTXLOLEULXP
HTXDWLRQVDUH
xiJ i Pi YDS yi P DQG ¦ xiJ i Pi YDS P 7KHUHIRUH
u u xJ PYDS u u YDS
xJ P
u u EDU
P
EDU
y
DQG
y
y
1RWHWKDWWKHYDSRUFRPSRVLWLRQLVRQO\YHU\VOLJKWO\GLIIHUHQWWKDQWKHOLTXLGFRPSRVLWLRQ 7KLV
LVEHFDXVHWKHYDSRUSUHVVXUHVDQGDFWLYLW\FRHIILFLHQWVRIWKHVSHFLHVDUHDOOTXLWHVLPLODU $QDOWHUQDWLYHLVWRXVHWKHSURJUDP81,)$&WRHVWLPDWHWKHDFWLYLW\FRHIILFLHQWV8VLQJ
WKHSURJUDPZLWKHWKDQRO &+ &+ 2+ EHQ]HQH ±$&+ DQGHWK\ODFHWDWH
&+ &+ &+ &22 ZHREWDLQDWPROHHWKDQROPROHEHQ]HQHDQG
PROHHWK\ODFHWDWHDW q & WKDW
J J DQG J EDUDQG
7KHVROXWLRQLV P a
a
E xJ PYDS
D E y
f
y
f
DQG y
&OHDUO\WKLVUHVXOWLVGLIIHUHQWIURPWKHWHUQDU\YDQ/DDUSUHGLFWLRQ,QWKHDEVHQFHRIH[SHULPHQWDO
GDWDIRUWKLVWHUQDU\PL[WXUHLWLVGLIILFXOWWRVD\ZKLFKPRGHOLVEHWWHU
)RUWKHVLPSOHUPRGHOVLWLVSRVVLEOHWRVKRZE\VLPSOHPDWKHPDWLFVWKDWWKHPRGHOHLWKHUGRHVRU
GRHVQRWSHUPLWDGRXEOHD]HRWURSH)RUH[DPSOHWKHYDQ/DDUPRGHOLV
G H[ a xqxq a
DExx
DEx x
u
RT
xq xq
a Dx Ex Dx E x
1RZIRUWKHEHQ]HQHKH[DIOXRUREHQ]HQHV\VWHP G H[ KDVDQLQWHULRUPD[LPXPDQGDQLQWHULRU
a
a
f
f
PLQLPXP7KDWLV dG H[ dx LV]HURWZLFHLQWKHUHJLRQ d x d 7RVHHLIWKHYDQ/DDUPRGHO
SHUPLWVWKLVZHH[DPLQH
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
FG IJ DEa x f DEx DEx a x f D E
H K aDx Ea x ff Dx Ea x f Dx Ea x f
DE a x f
DEx a x f D E
Dx E a x f
dx E a x f
Ÿ a x x faDx Ex f x x D E d G H[
dx RT
RU
RU
Dx
Dxx Dx Exx Ex Dxx Exx
Ex Ÿ
D
E
FG x IJ RU x
Hx K x
r
D
E
1RZ DDQGE PXVWEHRIVDPHVLJQ RWKHUZLVHZHJHWWKHVTXDUHURRWRIDQHJDWLYHQXPEHU $OVRVLQFH d x d DQG d x d RQO\SRVLWLYHVLJQLVDOORZHG7KXV x x
dG
H[
dx
D E ZKHQ
$QGRQO\DQLQWHULRUPD[LPXP LI D ! DQG E ! RUDQLQWHULRUPLQLPXP LI
D DQG E FDQRFFXUEXWQRWERWK7KHUHIRUHYDQ/DDUPRGHOFDQQRWGHVFULEHWKH
REVHUYHGEHKDYLRU
6LPLODUO\REYLRXVO\WKHRQHFRQVWDQW0DUJXOHVPRGHO G H[ Axx FDQQRWJLYHERWKDQLQWHULRU
PLQLPXP PD[LPXPVRLWFDQQRWGHVFULEHREVHUYHGEHKDYLRU
,QVWHDGRIFRQWLQXLQJWKLVH[WUHPHDUJXPHQWZHZLOOORRNDWWKHUHVXOWVRIPHUHO\ILWWLQJWKH
H[SHULPHQWDOGDWD
7ZRFRQVWDQW0DUJXOHVPRGHO
PPHDV
K3D PFDOF
K3D xPHDV
xFDOF
yPHDV
yFDOF
RQO\D]HRWURSHDW x t Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
:LOVRQPRGHO
PPHDV
K3D PFDOF
K3D yPHDV
yFDOF
2QO\DVLQJOHD]HRWURSHSUHGLFWHGWRRFFXU
157/PRGHO
PFDOF
xPHDV
yPHDV
yFDOF
PPHDV
K3D K3D 1RD]HRWURSHUHVXOWVIRUPWKHOHDVWVTXDUHVILWWLQJRISDUDPHWHUV
81,48$&PRGHO
PPHDV
K3D PFDOF
K3D xPHDV
yPHDV
yFDOF
m
m
'RXEOHD]HRWURSHSUHGLFWHGDVLQGLFDWHG
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
7KHUHIRUHRIWKHPRGHOVFRQVLGHUHGRQO\WKH81,48$&PRGHOLVFDSDEOHRISURGXFLQJWKH
SHFXOLDUEHKDYLRURI G H[ IRUWKLVV\VWHP
)URP7DEOHZHKDYH DVVXPLQJDQLGHDOYDSRUSKDVH x
J
J
ORJ
FG J IJ
HJ K
6HHILJXUHEHORZ
/RJ
7KHWZRDUHDV,DQG,,DSSHDUWREHDSSUR[LPDWHO\HTXDOVRWKHGDWDZLOOEHSUHVXPHGWREH
WKHUPRG\QDPLFDOO\FRQVLVWHQW,QIDFWIURPSURSHUQXPHULFDODQDO\VLVZHILQGWKHGDWDWREHFRQVLVWHQW
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
P
xJ PYDS xJ PYDS FG wP IJ
H wx K
T
x o J o DQG
a
F I
GH JK
f
1RZ
DQG
FG wJ IJ P x J FG wP IJ FG wT IJ
H wT K H wx K H wx K
F wJ IJ P x J FG wP IJ FG wT IJ
J P x G
H wT K H wx K
H wx K
J PYDS x
P
YDS
DV x o J o DQG
xJ PYDS xJ PYDS P
wJ wx
PYDS J PYDS x o PYDS wP
wx
T x o FRQVWDQWWHPSHUDWXUHHEXOOLRPHWHU
YDS
P
J x o FG wP IJ
H wx K
DV
6R
T x o VRWKDW
wJ wx
FG wP IJ
H wx K
wJ wJ
wx
J PYDS x PYDS ZKHUHZHKDYHXVHGWKDW wx
wx
wx
J PYDS x PYDS
YDS
P
YDS
YDS
P
YDS
T
P
P
wJ o
wx
PYDS FG IJ J a x o f P H K
dPYDS wT
dT wx
YDS
P
RU
a
f
J x o PYDS FG dP IJ FG wT IJ
H dT K H wx K
YDS
PYDS
P x o FRQVWDQWSUHVVXUHHEXOOLRPHWHU
DOVRDYDLODEOHDVD0DWKFDGZRUNVKHHW &OHDUO\PDQ\GLIIHUHQWWKHUPRG\QDPLFPRGHOVFDQEHXVHG:HZLOOXVHWKHYDQ/DDUPRGHO
OQ J D
Dx Ex
DQG OQ J E
Ex
Dx
ZKLFKJLYHV OQ J f D DQG OQ J f E 8VLQJWKHGDWDLQWKHSUREOHPVWDWHPHQW
D OQ DQG E OQ 8VLQJWKHVHSDUDPHWHUYDOXHVLQWKHDFWLYLW\
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
FRHIILFLHQWHTXDWLRQVDERYHWRJHWKHUZLWK Pi
x x
y
y
xiJ i Pi YDS P
DQG P
P P DQG yi
Pi P JLYHVDW
EDU
$OVR
x
y
y
P
DOVRDYDLODEOHDVD0DWKFDGZRUNVKHHW 8VLQJ0DWKFDG,REWDLQHGWKHIROORZLQJUHVXOWV
T . K(2+ ($& T . K(2+ ($&
x(2+ x(2+ 7KXVWKHUHVXOWVH[KLELWVWURQJFRPSRVLWLRQandWHPSHUDWXUHGHSHQGHQFH)RUDQLGHDOVROXWLRQ
y
Pi YDS
yi xi
Pi YDS
xi Pi YDS yi P Ÿ i
Ÿ Kij
xi
P
y j x j PjYDS
7KXVIRUDQLGHDOVROXWLRQWKHUHODWLYHYRODWLOLW\ Kij KDVQRFRPSRVLWLRQGHSHQGHQFHEXWFDQEH
GHSHQGHQWRQWHPSHUDWXUH XQOHVVIRUWXLWRXVO\ Pi YDS DQG PjYDS KDYHWKHVDPHWHPSHUDWXUH
GHSHQGHQFHWKDWLV ' H iYDS
' H YDS
7KHFRPSRVLWLRQGHSHQGHQFHDULVHVIURPWKHQRQLGHDO
j
VROXWLRQEHKDYLRU6LQFHWKHDFWLYLW\FRHIILFLHQWVGHSHQGHQWRQWHPSHUDWXUHQRQLGHDOVROXWLRQ
EHKDYLRUDOVRFRQWULEXWHVWRWKHWHPSHUDWXUHGHSHQGHQFHRIWKHUHODWLYHYRODWLOLW\
D * H[ $[ [ ZKLFKLPPHGLDWHO\JLYHV 57OQJ $[ DQG 57OQJ $[ VR
7 [ 7 [
OQ J
DQG OQ J 57
57
E 6LQFH
w §*·
w § * H[ ·
w § 7 ·
Ÿ + H[ 7 + 7 ¨
¸ 7
¸ [ [ ¨
¸
w7 3 © 7 ¹
w7 3 © 7 ¹
w7 ©¨
7
¹
+ H[
[ [ DOVR * H[
6H[
+ H[ 76 H[
6H[
> 7 @ [ [
7
+ H[ * H[
7
[ [ 7 [ [ 7
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
PROHV u u u
PRO
½
­
'6 6H[ 5 ¦ 1 L OQ [ L u u ® OQ OQ ¾
¯
¿
'+
ª u º
H[S «
u u »
¼»
¬« u F J%
3%XEEOH
·
§
H[S ¨ u ¸
©
¹
xJ PYDS
F fI yP
H PK
D D y x
y x
a f J P a f Pf P e a f Pf
a f J P a f Pf P e a f Pf
P
i a f Pf P e d a f i a f Pf
e d
a f Pf P
a f Pf
P
P
f a f Pf
e a
a f Pf
P
;
J PYDS f P P
J PYDS f P P
A x x
YDS
YDS
YDS
YDS
RT
YDS Ax RT
YDS Ax RT
A x x RT
E ,GHDOPL[WXUHDWORZSUHVVXUH
f
A DERYHDQGDOVRDOO
P
F I H K
D PYDS T
PYDS T
TGHSHQGHQFH
D L
fi
6WDUWLQJIURP
L
f i ZHJHW IRUWKHLGHDOVROXWLRQ WKDWxi Pi YDS
yi P
1RZDGGLQJVXFKHTXDWLRQVIRUERWKFRPSRQHQWV ZHJHW
x PYDS x PYDS
PVRWKDW
YDS
yi
xi Pi
x PYDS x PYDS
a
,QDHTXLPRODUPL[WXUHyi x
E )RUWKHQRQLGHDOPL[WXUH
YDS
f P P P
YDS
A x RT
YDS
&RPSRVLWLRQ
TGHSHQGHQFHDQGTGHSHQGHQFH TDQGP
GHSHQGHQFH
YDS
YDS
H - PRO ˜ .
ª u º
§ ·
u u » H[S ¨
H[S «
¸ ¼
¬ u ©
¹
YDS
¦ [ L J L 3L u u u u EDU
J&
YDS
i
YDS
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
xiJ i Pi YDS yi P
1RZDGGLQJWKHVHHTXDWLRQVIRUERWKFRPSRQHQWV ZHJHW
xJ PYDS xJ PYDS
xiJ i Pi YDS
YDS
xJ P xJ PYDS
yi
PVRWKDW
)RUWKHRQH FRQVWDQW0DUJXOHVPRGHO
a
YDS
f
xi Pi H[S a xi
x PYDS H[S ax x PYDS H[S ax
,QDHTXLPRODUPL[WXUHZLWKWKH
yi
YDS
Pi YDS
YDS
YDS
PYDS PYDS
ZKLFKLVH[DFWO\WKHVDPHUHVXOWDVIRUWKHLGHDOVROXWLRQ+RZHYHUWKHVHWZR
GLIIHUHQWPRGHOVRQO\JLYHWKHVDPHYDSRUSKDVHFRPSRVLWLRQLQDQHTXLPRODU
PL[WXUH+RZHYHUHYHQLQWKLVFDVHWKHSUHVVXUHVIRUWKHLGHDODQGRQH
FRQVWDQW0DUJXOHVPL[WXUHVDUHGLIIHUHQW
a
yi x
f P H[Sa P H[S a P H[S a D 3
i
[ J3YDS [ J 3YDS G3
G7 [
[
GJ YDS
G3 YDS
GJ
G3 YDS
3 [ J [ 3YDS [ J G7
G7
G7
G7
G OQ J
G3YDS
G3YDS
YDS G OQ J [J3
[ J
[ J 3
[J
G7
G7
G7
G7
E\WKH*LEEV'XKHP(T
[GOQJ[GOQJ G OQ J
G 3 YDS
G 3YDS
G3
J3YDS J 3YDS [
[ J [ J G7 [
G7
G7
G7
' YDS +
' YDS + G OQ J
YDS
YDS
J3 J 3 [
[ J
[J
G7
57 57 §
·
G3 YDS
¸ E ,IFRPSRQHQWLVLQYRODWLOH ¨ 3YDS DQG G7
©
¹
' YDS +
G OQ J
G3
[ J3YDS
[ J
G7
G7
57 YDS
1RWHWKDWLIWKHFRPSRVLWLRQLVQRWKHOGFRQVWDQWWKLVSUREOHPLVPXFKPRUHFRPSOLFDWHG1HHGWR
NQRZKRZFRPSRVLWLRQFKDQJHVDORQJWKHHTXLOLEULXPOLQH
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
---------------------------------------------------x(1)
GAM(1)
GAM(2)
GEX(J/MOL)
---------------------------------------------------0.0000
1.4168
1.0000
0.00
0.0250
1.3889
1.0003
24.48
0.0500
1.3625
1.0010
47.51
0.0750
1.3376
1.0022
69.11
0.1000
1.3141
1.0039
89.29
0.1250
1.2919
1.0061
108.08
0.1500
1.2710
1.0087
125.49
0.1750
1.2512
1.0118
141.53
0.2000
1.2326
1.0153
156.23
0.2250
1.2150
1.0193
169.60
0.2500
1.1984
1.0236
181.66
0.2750
1.1827
1.0284
192.41
0.3000
1.1680
1.0337
201.89
0.3250
1.1541
1.0393
210.10
0.3500
1.1410
1.0454
217.05
0.3750
1.1287
1.0518
222.77
0.4000
1.1171
1.0587
227.26
0.4250
1.1062
1.0660
230.55
0.4500
1.0960
1.0737
232.63
0.4750
1.0865
1.0818
233.53
0.5000
1.0776
1.0903
233.26
0.5250
1.0692
1.0993
231.83
0.5500
1.0615
1.1086
229.26
0.5750
1.0543
1.1184
225.55
0.6000
1.0476
1.1285
220.72
0.6250
1.0414
1.1391
214.78
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
0.6500
0.6750
0.7000
0.7250
0.7500
0.7750
0.8000
0.8250
0.8500
0.8750
0.9000
0.9250
0.9500
0.9750
1.0000
1.0357
1.0305
1.0258
1.0215
1.0176
1.0141
1.0111
1.0084
1.0061
1.0042
1.0027
1.0015
1.0007
1.0002
1.0000
1.1501
1.1616
1.1734
1.1857
1.1985
1.2116
1.2253
1.2393
1.2538
1.2688
1.2843
1.3002
1.3166
1.3335
1.3509
207.74
199.62
190.42
180.16
168.84
156.48
143.09
128.68
113.25
96.83
79.41
61.01
41.63
21.29
0.00
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
x(1)
0.0000
0.0250
0.0500
0.0750
0.1000
0.1250
0.1500
0.1750
0.2000
0.2250
0.2500
0.2750
0.3000
0.3250
0.3500
0.3750
0.4000
0.4250
0.4500
0.4750
0.5000
0.5250
0.5500
0.5750
0.6000
0.6250
0.6500
0.6750
y(1)
0.0000
0.1186
0.2131
0.2903
0.3547
0.4094
0.4566
0.4978
0.5342
0.5667
0.5959
0.6224
0.6467
0.6689
0.6896
0.7087
0.7267
0.7435
0.7594
0.7745
0.7888
0.8025
0.8156
0.8282
0.8403
0.8520
0.8634
0.8744
y(2)
1.0000
0.8814
0.7869
0.7097
0.6453
0.5906
0.5434
0.5022
0.4658
0.4333
0.4041
0.3776
0.3533
0.3311
0.3104
0.2913
0.2733
0.2565
0.2406
0.2255
0.2112
0.1975
0.1844
0.1718
0.1597
0.1480
0.1366
0.1256
P(bar)
0.325
0.359
0.392
0.424
0.455
0.484
0.512
0.540
0.566
0.592
0.617
0.641
0.665
0.688
0.711
0.733
0.755
0.776
0.797
0.818
0.838
0.859
0.879
0.898
0.918
0.938
0.957
0.976
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
0.7000
0.8852
0.7250
0.8957
0.7500
0.9059
0.7750
0.9159
0.8000
0.9258
0.8250
0.9355
0.8500
0.9450
0.8750
0.9544
0.9000
0.9637
0.9250
0.9729
0.9500
0.9820
0.9750
0.9910
1.0000
1.0000
0.1148
0.996
0.1043
0.0941
0.0841
0.0742
0.0645
0.0550
0.0456
0.0363
0.0271
0.0180
0.0090
0.0000
1.015
1.034
1.053
1.072
1.091
1.111
1.130
1.149
1.169
1.188
1.208
1.227
5HJXODUVROXWLRQPRGHO
SKYDS YK SWYDS GK YW L
[K ˜ YK
L
SKK L
[K ˜ YK [K ˜ YW
L
SKW SKK L
L
L
GGHO GK GW ˜
OQJK ªYK ˜ SKK ˜ GGHOº
L
¬
¼
L
GW [K ˜ L
L JK H[S OQJK L
˜ OQJW L
YW ˜ SKW
L
L
˜ GGHO
˜ JW H[S OQJW L
L
3W [K ˜ JK ˜ SKYDS [K ˜ JW ˜ SWYDS L
L
L
L
L
3W
\K L
[K ˜ JK ˜ SKYDS
L
L
3W
L
\K
[K
[K
[K
3W 3W
[K \K
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
$VVKRZQLQWKH0$7+&$'ZRUNVKHHWEHORZWKHD]HRWURSLFFRPSRVLWLRQFKDQJHVDVWKH
WHPSHUDWXUHFKDQJHV7KHUHVXOWVDUH
[$=
3$= PP+J 7 R& 7 3PD 7 3PH 7 [PD 3PH 7
JE D
OQ JD
§ D ˜ [PD ·
¨
¸
E ˜ [PH ¹
©
] ILQG D E
L D ]
[ ˜ L
D
ª
ȼ
Ǥ
D˜[ · »
L ¸ »
« ¨ «¨
E ˜ [ ¸ »
L¹ ¼
©
JPP H[S«
E ]
[
E
§ E ˜ [PH ·
¨
¸
D ˜ [PD ¹
©
[ L
JE
E OQ JE
D
3PH D LQLWLDOJXHVVIRU9DQ/DDUSDUDPHWHUV
L
7 JD
3PD JLYHQ
3PD 7
[PH [PD
JD 7 E
[ [
L
L
E
ª
ȼ
Ǥ
E ˜ [ · »
L¸ »
« ¨ «¨
D˜[ ¸ »
L ¹ ¼
©
JPH H[S«
L
3 [ ˜ JPP ˜ 3PD [ ˜ JPH ˜ 3PH L
L
L
L
L
3
[
1RZWRILQGD]HRWURSHVDWGLIIHUHQWWHPSHUDWXUHV
)LUVWFKHFNZLWKJLYHQGDWD
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
ª
ª
º»
ǻ
D ˜ [[ º »
» »
« « E ˜ [[ ¼ ¼
¬¬
º»
ǻ
E ˜ [[ º »
« « D ˜ [[ » »
¬¬
¼ ¼
D
JPPP [[ H[S«
JPPH [[ H[S«
E
77 33 [[ [[JPPP
˜
[[ ˜ 3PD 77 [[ ˜ JPPH [[ ˜ 3PH 77
,QLWLDOJXHVVIRUD]HRWURSLFFRPSRVLWLRQ
[[ *LYHQ
[[ 0D[LPL]H 33 [[
[[ 33 [[
77 33 [[ [[JPPP
˜
[[ ˜ 3PD 77 [[ ˜ JPPH [[ ˜ 3PH 77
*LYHQ
[[ 0D[LPL]H 33 [[
[[ 33 [[
77 33 [[ [[JPPP
˜
[[ ˜ 3PD 77 [[ ˜ JPPH [[ ˜ 3PH 77
*LYHQ
[[ 0D[LPL]H 33 [[
[[ 33 [[
u 77 33 [[ [[JPPP
˜
[[ ˜ 3PD 77 [[ ˜ JPPH [[ ˜ 3PH 77
*LYHQ
[[ 0D[LPL]H 33 [[
[[ 33 [[
u 6XPPDU\RIUHVXOWV
7 q&
[$=
3$=
Solutions to Chemical and Engineering Thermodynamics,5th ed
D
L [L L
3L
7 3YDS H[S 7 3YDS H[S 7 7
7
[ L
JDP H[S
7
7
7
[ L
JDP H[S
7
3L [L JDP 3YDS [L JDP 3YDS
[
3
3LQEDU
3YDS H[S 3YDS H[S 7 7 7 7
7
[ L
JDPOL H[S
7
7
7
[ L
JDPL H[S
7
[ JDPO 3YDS
\L
L
L
3L
\L
\L
Section 10.2
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
3
L
[
L
E *HQHUDOL]HGIRUDQ\SUHVVXUH
L [L L
77L
7 '7 ZKLOH '7 ! 3YDS H[S 7 3YDS H[S 7 7
7
[ L
JDP
H[S
7
7
7
[ L
JDP H[S
7
3 [L JDP 3YDS
[L JDP 3YDS
'7 OQ
3 7 7 '7
77L 7
77L
+DYHWRUHFDOFXODWHYDSRUSUHVVXUHVDFWLYLW\FRHIILFLHQWVDQGYDSRUSKDVHPROHIUDFWLRQVVLQFH
WKHVHYDULDEOHVDUHRQO\GHILQHGZLWKLQWKHVXESURJUDP
3YDSL H[S 77L 3YDSL H[S 77L 77L
77L
[ L
JDP H[S
L
77L
77L
77L
[ L
JDP H[S
L
77L
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
[L JDPL 3YDSL
3
L
[L JDPL 3YDSL
[L JDPL 3YDSL
\L
3L
3L
[
JDP
[L JDPL 3YDSL
3
\L
\
77
JDP
3YDS
3YDS
77
L
\
[
L
L
[
L
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
DOVRDYDLODEOHDVD0DWKFDGZRUNVKHHW %HQ]HQHDFWLYLW\LQEHQ]HQHSRO\LVREXW\OHQH PL[WXUHV
1% 13,%
9% 9,% 5 F PE PSLE PE
:W%
:W% :W3,% :W%
PE PSLE
:W%
[E 9%
[E
[SLE [E
3KL%
3KL3 3KL%
:W% :W3,%
[E 9% [SLE 13,% 9,%
3KL% [SLE 3KL3 [E 13,% 9,%
P P
1% 9%
3KL%
OQ*%
OQ
3KL3 F 3KL3
J % H[S OQ*%
J % [E
P
3KL3
OQ*
OQ
P 3KL% F 3KL%
J 3 H[S OQ*
J3 [SLE
DE [E J %
DE OQ*
OQ*%
DFWLYLW\RIEHQ]HQH
3DUWLDOSUHVVXUHRIEHQ]HQH DE EDU
PE
PSLE PE
:W%
:W% :W3,% :W%
PE PSLE
:W%
[E 9%
[E
[SLE [E
3KL%
3KL3 3KL%
:W% :W3,%
[E 9% [SLE 13,% 9,%
[E 3KL% [SLE 3KL3 &DOFXODWLRQRISXUHFRPSRQHQWYDSRUSUHVVXUHRIEHQ]HQH
'DWDIURP3HUU\ V
3YDS PP+JDW7 &DQGPP+JDW&
% $ JLYHQ
%
%
OQ $
OQ $
FRQ ILQG $ %
$ FRQ $ % FRQ % H[S $
3YDS
%
3YDS
EDU
Solutions to Chemical and Engineering Thermodynamics,5th ed
Section 10.2
OQ*% OQ 3KL%
3KL3 F 3KL3
J % H[S OQ*%
J % [E
P
OQ* OQ 3KL3
P 3KL% F 3KL%
J 3 H[S OQ*
J3 [SLE
DE [E J %
DE OQ*
OQ*% DFWLYLW\RIEHQ]HQH
3DUWLDOSUHVVXUHRIEHQ]HQH DE EDU
PSLE PE
PE :W%
:W% :W3,% :W%
PE PSLE
:W%
[E 9%
[E
[SLE [E
3KL%
3KL3 3KL%
:W% :W3,%
[E 9% [SLE 13,% 9,%
3KL% [SLE 3KL3 [E 3KL%
3KL3 F 3KL3
J % H[S OQ*%
J % OQ*% OQ
[E
P
3KL3
P 3KL% F 3KL%
J 3 H[S OQ*
J3 OQ* OQ
[SLE
DE [E J %
DE OQ*% OQ*
DFWLYLW\RIEHQ]HQH
Solutions to Chemical and Engineering Thermodynamics,5th ed
Section 10.2
DOVRDYDLODEOHDVD0DWKFDGZRUNVKHHW 1& 13,%
9&
9,% 5 F PF PSLE PF
:W&
:W& :W3,% :W&
PF PSLE
:W&
[F 9&
[F [SLE [F
3KL&
3KL3 3KL&
:W& :W3,%
[F 9& [SLE 13,% 9,%
3KL&
[SLE 3KL3 [F 13,% 9,%
P
P 1& 9&
3KL&
OQ*& OQ
3KL3 F 3KL3
J & H[S OQ*&
J & [F
P
3KL3
OQ* OQ
P 3KL& F 3KL&
J 3 H[S OQ*
J3 [SLE
DF [F J &
DF OQ*
OQ*& DFWLYLW\RIF\FORKH[DQH
3DUWLDOSUHVVXUHRIF\FORKH[DQH DF EDU
PF PSLE PF
:W&
:W& :W3,% :W&
PF PSLE
:W&
[F 9&
[F [SLE [F
3KL&
3KL3 3KL&
:W& :W3,%
[F 9& [SLE 13,% 9,%
[F 3KL& [SLE 3KL3 3KL&
OQ*&
OQ
3KL3 F 3KL3
J & H[S OQ*&
J & [F
P
3KL3
OQ*
OQ
P 3KL& F 3KL&
J 3 H[S OQ*
J3 [SLE
DF [F J &
DF OQ*&
OQ*
DFWLYLW\RIF\FORKH[DQH
3DUWLDOSUHVVXUHRIF\FORKH[DQH DF EDU
Solutions to Chemical and Engineering Thermodynamics,5th ed
Section 10.2
PF
PSLE PF
:W&
:W& :W3,% :W&
PF PSLE
:W&
[F 9&
3KL&
[F
[SLE
[F
3KL3 3KL&
3KL& :W3,%
[F 9& [SLE 13,% 9,%
[F 3KL& [SLE 3KL3 OQ*& OQ 3KL&
3KL3 F 3KL3
J & H[S OQ*&
J & [F
P
OQ* OQ 3KL3
P 3KL& F 3KL&
J 3 H[S OQ*
J3 [SLE
DF [F J &
DF OQ*
OQ*& DFWLYLW\RIEHQ]HQH
3DUWLDOSUHVVXUHRIF\FORKH[DQH DF EDU
6RZKLOHQRWSHUIHFWWKHYDOXHRIWKH)ORU\SDUDPHWHUFKRVHQJLYHVDUHDVRQDEO\JRRG
GHVFULSWLRQRIWKHF\FORKH[DQHSRO\LVREXW\OHQHV\VWHP
&DOFXODWLRQRISXUHFRPSRQHQWYDSRUSUHVVXUHRIF\FORKH[DQH
'DWDIURP3HUU\ V
3YDS PP+JDW7 &DQGPP+JDW&
$ % JLYHQ
%
%
OQ $
OQ $
FRQ ILQG $ %
$ FRQ $ % FRQ % %
H[S $
3YDS
3YDS EDU
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
DOVRDYDLODEOHDVD0DWKFDGZRUNVKHHW 3HQWDQHDFWLYLW\LQSHQWDQHSRO\LVREXW\OHQH PL[WXUHV
13 13,%
93
9,% 5 F PS
PSLE PS
:W3
:W3 :W3,% :W3
PS PSLE
:W3
[S 93
[S
[SLE [S
3KL3
3KL3LE 3KL3
:W3 :W3,%
[S 93 [SLE 13,% 9,%
[SLE 3KL3LE [S 3KL3 P 13,% 9,%
P 13 93
3KL3
OQ*3
OQ
3KL3LE F 3KL3LE
J 3 H[S OQ*3
J 3 [S
P
3KL3LE
OQ* OQ
P 3KL3 F 3KL3
J 3LE H[S OQ*
J 3LE [SLE
D3 [S J 3
D3 OQ*3
OQ*
DFWLYLW\RIQSHQWDQH
3DUWLDOSUHVVXUHRIQSHQWDQH D3 EDU
PS
PSLE PS
:W3
:W3 :W3,% :W3
PS
PSLE
:W3
[S 93
[S
[SLE [S
3KL3
:W3 :W3,%
[S 93 [SLE 13,% 9,% 3KL3LE 3KL3
[S 3KL3 [SLE 3KL3 3KL3
3KL3LE F 3KL3LE
J 3 H[S OQ*3
OQ*3 OQ
[S
P
3KL3LE
OQ* OQ
P 3KL3 F 3KL3
J 3LE H[S OQ*
[SLE
DS [S J 3
OQ*
OQ*3 3DUWLDOSUHVVXUHRIQSHQWDQH DS EDU
DS
J3
J 3LE
DFWLYLW\RIQSHQWDQH
Solutions to Chemical and Engineering Thermodynamics,5th ed
Section 10.2
PSLE PS
:W3
:W3 :W3,% :W3
PS PSLE
:W3
[S 93
[S 3KL3
[SLE
[S
3KL3LE 3KL3
[S 93 [SLE 13,% 9,%
3KL3 :W3,%
[S 3KL3 [SLE 3KL3LE 3KL3
OQ*3 OQ
3KL3LE F 3KL3LE
J 3 H[S OQ*3
J 3 [S
P
3KL3LE
OQ* OQ
P 3KL3 F 3KL3
J 3LE H[S OQ*
J 3LE [SLE
DS [S J 3
DS OQ*3 OQ*
DFWLYLW\RIQSHQWDQH
3DUWLDOSUHVVXUHRISHQWDQH
DS EDU
6RZKLOHQRWSHUIHFWWKHYDOXHRIWKH)ORU\SDUDPHWHUFKRVHQJLYHVDUHDVRQDEO\JRRGGHVFULSWLRQ
RIWKHSHQWDQHSRO\LVREXW\OHQHV\VWHP
&DOFXODWLRQRISXUHFRPSRQHQWYDSRUSUHVVXUHRISHQWDQH
'DWDIURP3HUU\
V
3YDS
PP+JDW7
&DQGPP+JDW&
$ % JLYHQ
%
%
OQ $
OQ $
FRQ ILQG $ %
$ FRQ $ % FRQ % %
H[S $
3YDS
3YDS EDU
PS
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
:H ZLOO DVVXPH VLQFH +\GURFKORULF DFLG LV D VWURQJ DFLG WKDW WKH +&O PROHFXOH ZLOO EH
FRPSOHWHO\LRQL]HGDWDOOFRQFHQWUDWLRQV
/HW y ZW+&O y ZWRI+ 2 y
PRO+&O JUDPVVROXWLRQ
y
PRO+ 2 JUDPVVROXWLRQ
y
PRO+ &O LRQV JUDPVVROXWLRQ
y y
PROHIUDFWLRQRIZDWHU
y y y y
0ROHIUDFWLRQVIRUHDFKVROXWLRQDUHJLYHQLQWKHWDEOHEHORZ
1H[WZHXVHWKHSDUWLDOSUHVVXUH YDSRUOLTXLGHTXLOLEULXP GDWD)RUZDWHUZHKDYH
f :/ f :9 Ÿ x:J : P:YDS P: ZKHUHDJDLQZHKDYHQHJOHFWHGDOOIXJDFLW\FRHIILFLHQWFRUUHFWLRQV8VLQJIURP3UREOHPWKDW
IRU OQ P:YDS T LQ3D IRU EDU OQ P:YDS T DQG IURP WKH SUREOHP
VWDWHPHQWWKDW
B
OQ P: A T
ZHREWDLQ
OQ x:J : A B T
)URPZKLFKZHREWDLQWKHIROORZLQJUHVXOWV
y
x:J : J : x:J : x: x: OQ x:J : ±
±
±
±
1RWHWKDWWKHDFWLYLW\FRHIILFLHQWIRUZDWHULVVLJQLILFDQWO\OHVVWKDQXQLW\>1XPEHUVLQSDUHQWKHVHV
DUH J : FDOFXODWHGDVVXPLQJ+&OQRWLRQL]HG@
(TXLOLEULXP EHWZHHQ ZDWHU LQ DLU DQG ZDWHU LQ DTXHRXV VROXWLRQ UHTXLUHV WKDW f +/ 2 f +9 2 F
H
I
K
a
f
a
f
1HJOHFWLQJIXJDFLW\FRHIILFLHQWFRUUHFWLRQVZHKDYH
f +/ 2
x+ 2J + 2 P+YDS
2
f +9 2
f
y+ 2 P
P
x+ 2J + 2 EDU
AZW1D
HQWU\LQWDEOHIRU
&2
y+ 2 P
P+ 2
Ÿ J + 2
/HW W
P+ 2
x+ 2
ZW1D &2 0:1D &2 IRUP+ 2 LQEDU
J PRO Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
W
W DVVXPLQJ1D &2LV
FRPSOHWHO\LRQL]HG
PROHV1D &2 JVROXWLRQ
IJ
K
FG
H
PROHVLRQV JVROXWLRQ
W
W W W PROHV+ 2 JVROXWLRQ
PROHIUDFWLRQZDWHU
W
W
7KXV
P+ 2 W
J + 2
9DOXHVRI J + 2 FDOFXODWHGIURPDERYHHTXDWLRQDUHOLVWHGEHORZ
W
W
P+ 2 N3D J + 2 x+ 2 J + 2 DVVXPLQJ
1D &2 GLGQRWLRQL]H S H[
FG w G IJ
H wT K
H[
P x
w
x G H[ xGH[
w T P x m
w
RT x OQ J x OQ J w T P x
k
R ¦ xi OQ J i RT ¦ xi
H H[
p
FG w OQ J IJ
H wT K
i
P x
LM w dG T i OP RT w
¦ x OQ J
wT
MN w T PQ
F w OQ J IJ
RT ¦ x G
H wP K
FG w G IJ w ¦ x RT OQ J RT ¦ x FG w OQ J IJ
H wP K
H wP K wP
H[
T P x
P x
V H[
r
i
i
i
i
P x
H[
i
Tx
Tx
i
i
i
i
Tx
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
U H[
F w OQ J I PRT ¦ x FG w OQ J IJ
H wP K
L w OQ J I F w OQ J I OP
RT ¦ x MF
NH w OQ T K H w OQ P K Q
FG w H IJ RT ¦ x FG w OQ J IJ RT ¦ x FG w OQ J IJ
H wT K
H wT K
H wT K
H H[ PV H[
RT ¦ x GH w T JK
i
i
i
i
i
C3H[
i
P x
Tx
i
i
P x
Tx
H[
i
i
P x
i
i
i
P x
i
P x
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
§ ·
¨
¸
¨ ¸
¨ ¸
¨ ¸
¨
¸
¨ ¸
¨ ¸
¨
¸
¨
¸
GDWD ¨ ¸
¨ ¸
¨
¸
¨ ¸
¨ ¸
¨ ¸
¨
¸
¨ ¸
¨ ¸
¨
¸
© ¹
3YDS 3YDS L 3 GDWD
L
L [ [
L
L
\ ˜ 3
L L
JL [ GDWD
L
L
L
J
\ ˜ 3
L L
L
[
*H[ 5˜ 7˜ [ ˜ OQ JL [ ˜ OQ JL
L
L
L L
5 [ ˜ 3YDS
L
L
J
\ GDWD
\ \
JL [ ˜ 3YDS
L 7 Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
M )XQFWLRQWREHILWIJ
IJ [ ˜ [ ˜ 5˜ 7
M
M
9HFWRURIWHUPV
LQUHSFLURFDO9DQ
/DDUH[SUHVVLRQ
)[ 9HFWRURISDUDPHWHUV
6
M
*H[
M
§ [ ·
¨
¸
© [¹
'HWHUPLQDWLRQRISDUDPHWHUV
E\OLQHDUUHJUHVVLRQ
§ ·
¨
¸
© ¹
,QGHQWLILFDWLRQRIZRUNVKHHWDQGPRGHOSDUDPHWHUV
DOI EHW 6
6
6 OLQILW [ IJ )
&DOFXODWLRQRI*LEEVIUHHHQHUJ\JH[DQGWZRSDUWLDOPRODUH[FHVV*LEEVHQHUJLHVJH[ JH[
JH[ M
DOI˜ EHW ˜ [ ˜ [ ˜ 5˜ 7
M
M
JH[ M
[ ˜ DOI [ ˜ EHW
M
M
M
M
[ ˜ DOI [ ˜ EHW
M
M
JH[ M
§ JH[M ·
¸
© 5˜ 7 ¹
M
JDPLQI H[S EHW
JDPLQI
DOI ˜ EHW ˜ [ ˜ [ ˜ 5˜ 7
M
M
[ ˜ DOI [ ˜ EHW
M
JDP H[S¨
&DOFXODWLRQRIDFWLYLW\FRHIILFLHQWV
JDPLQI H[S DOI
DOI˜ EHW ˜ [ ˜ [ ˜ 5˜ 7
M
§ JH[M ·
¸
© 5˜ 7 ¹
JDP H[S¨
M
JDPLQI
&RPSDULVRQRIH[FHVV
*LEEVHQHUJ\IURP
H[SHULPHQWDOGDWDDQG
IURPYDQ/DDUILW
*H[
JH[
[
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
&RPSDULVRQRIDFWLYLW\
FRHIILFLHQWRIVSHFLHV
IURPH[SHULPHQWDOGDWD
DQGIURPYDQ/DDUILW
J
JDP
[
&RPSDULVRQRIDFWLYLW\
FRHIILFLHQWRIVSHFLHV
IURPH[SHULPHQWDOGDWD
DQGIURPYDQ/DDUILW
J
JDP
[
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
d
i G dT P x i i DQGDORQJWKHHTXLOLEULXPFXUYH
dG dT P x i dG dT P x i i $WHTXLOLEULXP Gi, T P x ,
i
,,
,,
,
,
i
,,
,,
7UHDWLQJTPDQGRQHPROHIUDFWLRQDVWKHLQGHSHQGHQWYDULDEOHVLQWKLVELQDU\V\VWHPZHREWDLQ
w Gi ,
dGi, Si, dT Vi ,dP dx
w x
i w Gi ,,
,,
,,
,,
dGi
dy
Si dT Vi dP w y
FG
H
(TXDWLQJ dGi, DQG dGi ,, Si,dT Vi , dP i
FG w G IJ
Hwx K
i
,
T P
IJ
K
FG
H
U|
|
IJ V|
K |W
Si,, dT Vi ,, dP dx
FG w G IJ
H wy K
i
,,
T P
dy i
1RZPXOWLSO\LQJE\ yi VXPPLQJDQGUHDUUDQJLQJJLYHV
c
h
Fw G I
¦ y GH w y JK
c
h
dy ¦ yi
FG w G IJ dx
Hwx K
¦ yi Si, Si,, dT ¦ yi Vi , Vi ,, dP
i i
i
,,
TP
E\WKH*LEEV
'XKHPHTXDWLRQ
c
h
Ÿ ¦ yi Vi , Vi ,, dP
6LQFH Gi,
TP
c
h
¦ yi Si, Si,, dT ¦ yi
FG w G IJ dx
Hwx K
i
,
TP
6LQFHx¶VDQGy¶VDUHPL[HG
WKH*LEEV'XKHPHTXDWLRQ
GRHVQRWDSSO\
Vi ,, !! Vi , DQG Vi ,, |
c H H h T ,
i
,,
i
RT
P
7KHUHIRUH
FG IJ
H K
RT
dP
¦ yi
P
i w Gi
dx
yi Hi, Hi,, dT ¦ yi
¦
T i w x T P i Ÿ RTd OQ P
FG w OQ P IJ
H wT K
)RU YDSRUOLTXLG HTXLOLEULXP ZLWK SKDVH , OLTXLG DQG SKDVH ,, YDSRU ZH KDYH DW ORZ DQG
PRGHUDWHSUHVVXUHVWKDW
RTd OQ P
,
Hi,, TSi,, RU Si, Si,,
Gi,, Hi, TSi,
i
x
c
h
' H YDS IRUyPROHV
FRPSRQHQWDQG
y yPROHVRI
FRPSRQHQWIURPVROXWLRQ
FG IJ
H K
w Gi
dx
' H YDSdT ¦ yi
T
w x T P i ' H YDS
RT Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
:HZLOOH[DPLQHWKLVSUREOHPWZRZD\V7KHILUVWLVWRXVHWKH9LVXDO%DVLF81,)$&SUHGLFWLYH
PRGHO7KHUHVXOWVDUHVKRZQEHORZ
-------------COMPONENT DATA
-------------Molecular Subgroups
---------------------------------------------------------------------Comp
Name
Number
Description
Frequency
---- -------------------------------- -----------------------1
isobutane
1
CH3
3
3
CH
1
2
furfural
62
furfural
--------------------------------------------------x(1)
GAM(1)
GAM(2)
GEX(J/MOL)
---------------------------------------------------0.0000
14.5135
1.0000
0.00
0.0250
12.2923
1.0021
167.41
0.0500
10.5312
1.0081
324.21
0.0750
9.1160
1.0179
470.84
0.1000
7.9644
1.0311
607.72
0.1250
7.0170
1.0478
735.21
0.1500
6.2294
1.0679
853.63
0.1750
5.5686
1.0913
963.27
0.2000
5.0095
1.1183
1064.38
0.2250
4.5327
1.1489
1157.17
0.2500
4.1231
1.1833
1241.85
0.2750
3.7690
1.2217
1318.59
0.3000
3.4609
1.2645
1387.52
0.3250
3.1913
1.3119
1448.78
0.3500
2.9542
1.3645
1502.46
0.3750
2.7447
1.4228
1548.64
0.4000
2.5586
1.4874
1587.40
0.4250
2.3928
1.5591
1618.77
0.4500
2.2443
1.6387
1642.77
0.4750
2.1110
1.7274
1659.42
0.5000
1.9909
1.8264
1668.69
0.5250
1.8823
1.9373
1670.55
0.5500
1.7840
2.0620
1664.94
0.5750
1.6946
2.2029
1651.79
0.6000
1.6132
2.3629
1630.99
0.6250
1.5390
2.5455
1602.42
0.6500
1.4713
2.7553
1565.90
0.6750
1.4093
2.9981
1521.25
0.7000
1.3526
3.2816
1468.23
0.7250
1.3008
3.6154
1406.59
0.7500
1.2534
4.0127
1335.98
0.7750
1.2102
4.4914
1256.04
0.8000
1.1709
5.0761
1166.31
0.8250
1.1354
5.8018
1066.25
0.8500
1.1035
6.7194
955.22
0.8750
1.0753
7.9057
832.46
0.9000
1.0509
9.4800
697.01
0.9250
1.0305
11.6368
547.71
0.9500
1.0146
14.7100
383.12
0.9750
1.0040
19.3101
201.36
1.0000
1.0000
26.6482
0.00
x(1)
0.0000
0.0250
0.0500
0.0750
0.1000
0.1250
0.1500
0.1750
0.2000
0.2250
y(1)
0.0000
0.9969
0.9982
0.9986
0.9988
0.9990
0.9990
0.9991
0.9991
0.9991
y(2)
1.0000
0.0031
0.0018
0.0014
0.0012
0.0010
0.0010
0.0009
0.0009
0.0009
P(bar)
0.005
1.536
2.628
3.411
3.972
4.374
4.659
4.859
4.996
5.085
1
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
0.2500
0.2750
0.3000
0.3250
0.3500
0.3750
0.4000
0.4250
0.4500
0.4750
0.5000
0.5250
0.5500
0.5750
0.6000
0.6250
0.6500
0.6750
0.7000
0.7250
0.7500
0.7750
0.8000
0.8250
0.8500
0.8750
0.9000
0.9250
0.9500
0.9750
1.0000
0.9991
0.9992
0.9992
0.9992
0.9992
0.9991
0.9991
0.9991
0.9991
0.9991
0.9991
0.9991
0.9991
0.9991
0.9990
0.9990
0.9990
0.9990
0.9990
0.9990
0.9989
0.9989
0.9989
0.9989
0.9989
0.9990
0.9990
0.9991
0.9992
0.9995
1.0000
0.0009
0.0008
0.0008
0.0008
0.0008
0.0009
0.0009
0.0009
0.0009
0.0009
0.0009
0.0009
0.0009
0.0009
0.0010
0.0010
0.0010
0.0010
0.0010
0.0010
0.0011
0.0011
0.0011
0.0011
0.0011
0.0010
0.0010
0.0009
0.0008
0.0005
0.0000
5.139
5.168
5.177
5.171
5.155
5.132
5.103
5.070
5.036
5.000
4.964
4.928
4.893
4.859
4.827
4.797
4.769
4.744
4.722
4.703
4.688
4.677
4.672
4.671
4.678
4.692
4.716
4.753
4.805
4.879
4.982
LVREXWDQHDQGIXUIXUDODW.
[\RILVREXWDQH
3[
3\
7KHSHFXOLDUEHKDYLRULVEHFDXVHOLTXLGOLTXLGHTXLOLEULXPRFFXUVDVGLVFXVVHGLQ6HFWLRQ
6HFRQGZHZLOOXVHWKH9DQ/DDUDFWLYLW\FRHIILFLHQWPRGHOZLWKα=2.62 DQGβ DQGXVLQJ
0$7+&$'
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
3YDS M 3YDS [ ˜ M
M
[
[ ·
§
¨ D ˜ M ¸
¨
E [M ¸
©
¹
3 [ ˜ J M˜ 3YDS [ ˜ J M˜ 3YDS
M
M
M
M
3
\
M
[
¨§ E ˜ M ·¸
¨
D [M ¸
©
¹
[ ˜ J M˜ 3YDS
\ [ [
J M H[S OQJ M
J M H[S OQJ M
\ \
M
M
[
\
ā
ā
ā
ā
ā
ā
ā
ā
ā
ā
ā
ā
ā
ā
ā
ā
M
M
3
3
[ \
3
M
M
E
OQJ M E D
OQJ M M
D [ M
M
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
6LQFHWKHPROHFXODUZHLJKWRISRO\HWK\OHQHLVDQGHWK\OHQHLVP §
§
·
I
I ·
[ OQ
* H[ 5 ¨ [ OQ [ OQ ¸ 5 ¨ [ OQ
¸
[
[ ¹
[ [
[ [ ¹
©
©
VLQFH I
[
DQGI
[ P[ P[ $OVR
[ P[ * H[
5 [ OQ [ [ [ OQ [ OQ [ [
5 ¬ª [ OQ OQ [ [ ¼º
OQ J
OQ J §I · §
[
·
DQG
OQ ¨ ¸ ¨ ¸ I OQ [ [ [ [
© [ ¹ © P ¹
§
·
[
OQ ¨¨
¸¸ [
[
[
[
¹
© 8VLQJ0$7+&$'
L [ ˜ L
ª
«
¬
L¼
L
J(L H[S« OQª [ ˜ [ º ¬L
[
º
»
[ ˜ [ »
L
L ¼
L
˜ [
º
ª
L
º
»
»
« [L ˜ [L
[ ˜ [ »
L ¼
¬ ¬
¼ L
J3L H[S« OQª«
J(
J3
[
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
7KHYDSRUSUHVVXUHRIZDWHULVEDUDWq&7RFRPSXWHWKHYDSRUSUHVVXUHRIDFHWRQHZH
ILW WKH GDWD LQ WKH SUREOHP VWDWHPHQW WR OQ P YDS A T B DQG ILQG A B YDS
q & EDU
DQG P$&
D 7RFRPSXWHDFWLYLW\FRHIILFLHQWVZHZLOOXVHWKHYDQ/DDU PRGHOZLWK D
JLYHQLQ7DEOH7KXV
D
OQ J $&
Dx$& E x$&
x$& x$& Ÿ J $& DW
x$& DQG
E
OQ J :
Ex: D x:
x: x: Ÿ J : DWx: $WYDSRUOLTXLGHTXLOLEULXP
YDS
x:J : P:YDS x$&J $& P$&
P
aa
ff
a a
aa
ff
a a
E
DV
ff
ff
6XEVWLWXWLQJWKHUHVXOWVIRU J i Pi YDS DERYH\LHOGV
P EDU
7KXVIRUDOOSUHVVXUHDERYHEDURQO\DOLTXLGRIFRPSRVLWLRQ x: x$& ZLOOEH
SUHVHQW
E 7KLVFDOFXODWLRQLVPRUHGLIILFXOWVLQFHZHFDQQRWFDOFXODWHWKHGHZSRLQW³SUHVVXUH´ DWIL[HG
WHPSHUDWXUH XQWLOWKHOLTXLGSKDVHFRPSRVLWLRQDQGDFWLYLW\FRHIILFLHQWVDUHNQRZQ7KXVWKH
SUREOHPLQYROYHVDWULDODQGHUURUVROXWLRQRIWKHHTXDWLRQV
x:J : P:YDS
YDS
y: P x$&J $& P$&
y$& P DQG x$& x:
ZKHUH x: x$& DQG PDUHWKHXQNQRZQV 7KH J i FDQEHFDOFXODWHGIURPWKH xi XVLQJWKHYDQ
/DDUHTXDWLRQV %\UHSHDWHGJXHVVHV,ILQG
DQG x$& 7KXVIRUDOOSUHVVXUHVEHORZEDURQO\WKHYDSRU RIFRPSRVLWLRQ y: y$& LVSUHVHQW
1RWH 2QH VKRXOG FKHFN WKH FRQGLWLRQV RI ERWK SDUWV D DQG E WR WKH DERYH SUREOHP IRU WKH
SRVVLEOHRFFXUUHQFHRIWZRFRH[LVWLQJOLTXLGSKDVHV
P
EDU x:
6LQFHWKHLQLWLDOFRQGLWLRQVDUHZWOLTXLGDQGZWYDSRUWKHRYHUFRPSRVLWLRQLV
(WKDQRO u u :DWHU u u DQGWKHWRWDOHQWKDOS\LV u u N-NJ 7KHSUHFLVHPHWKRGE\ZKLFKWKHSUHVVXUHLVUHGXFHGLVQRWVSHFLILHGVROHWXVDVVXPHWKHWZR
SKDVHPL[WXUHSDVVHVWKURXJKD-RXOH7KRPSVRQH[SDQVLRQYDOYHVRWKDWHQWKDOS\LVFRQVWDQW
,QWKLVFDVHWKHHTXDWLRQVWREHVROYHGDUH
Ö / u +
Ö
/u+
N-NJ
/
9
/ u [ ( / u \ ( DQG/ u [ ( / u \ ( 7KHVH HTXDWLRQV PXVW EH VROYHG E\ WULDODQGHUURU XVLQJ WKH WKHUPRG\QDPLF GLDJUDP LQ WKH
SUREOHPVWDWHPHQW7KHUHVXOWVREWDLQHGDW7 R&/ VRWKDW9 [( DQG\( Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
7 ·
§
¨ ¸
7 ¹
§ ·
¨
¸
7 ¹
3YDS ©
3YDS ©
3YDS
3YDS
1RWHFOHDUO\WKHV\VWHPKDVDPD[LPXPSUHVVXUHD]HRWURSHVLQFHWKH9/(SUHVVXUHLVKLJKHU
WKDWHLWKHURIWKHSXUHFRPSRQHQWYDSRUSUHVVXUHV
)LUVWILQGWKHDFWLYLW\FRHIILFLHQWVDW[ 3 EDU
˜
J J J
˜ 3YDS
> ˜ @
˜ 3YDS
J
D 1H[WILQGWKH9DQ/DDUSDUDPHWHUV
D E *LYHQ
J
º»
Ǥ
D · »
« ¨ E ˜ ¸ »
©
¹ ¼
ª
D
H[S«
J
ȼ
Ǥ
E · »
« ¨ D ˜ ¸ »
©
¹ ¼
ª
E
H[S«
] ILQG D E
]
§ ·
¨
¸
© ¹
E ]
D ]
EDQGF )LUVWGUDZ3[\DQG[\GLDJUDPVIRUWKHV\VWHP
M [ ˜ M
[ M
[
D
º
ª
«
»
[
·
§
Ǭ
M¸ »
D
« ¨ E ˜ [ ¸ »
M¹ ¼
©
M
M
3
\ \
M
M
[
\
ā
ā
M
M
[ ˜ J M˜ 3YDS
\ M
M
J M H[S
3 [ ˜ J M˜ 3YDS [ ˜ J M˜ 3YDS
M
[ [
E
º
ª
«
»
[
· »
« §¨
M¸
E
« ¨ D ˜ [ ¸ »
M¹ ¼
©
J M H[S
M
\
M
M
3
M
M
3UHVVXUHKDVDQ
LQWHULRUPD[LPXPVR
WKHUHIRUHWKHUHPXVWEH
DQD]HRWURSH
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
3
\
[
3
[
[ \
º»
Ǥ
D [ · »
« ¨ E ˜ [¸ »
©
¹ ¼
ª
D
JDP [ H[S«
ª
º
»
Ǥ
E [· »
« ¨ ˜ [ ¸ »
D
©
¹ ¼
E
JDP [ H[S«
33 [ [˜ JDP [ ˜ 3YDS [ ˜ JDP [ ˜ 3YDS
,QLWLDOJXHVVIRUD]HRWURSLFSRLQW
JLYHQ
33 [
G
33 [
G[
[ [ ILQG [
[
\ [ [˜ JDP [ ˜
3YDS
\ [
33 [
DVLWVKRXOGEH
\ [ \ [
G 'HZSRLQWRIDQHTXLPRODUPL[WXUH
*LYHQ
\ [
\ [
] ILQG [ 33 [
]
§ ·
¨
¸
© ¹
6RWKDWWKHGHZSRLQWOLTXLGKDVDPROHIUDFWLRQRIRIVSHFLHVDQG
WKHGHZSRLQWSUHVVXUHLVEDU
%XEEOHSRLQWRIDQHTXLPRODUPL[WXUH
33 \ EDU
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
D )ROORZLQJWKHVDPHSURFHGXUHDVLQWKHVROXWLRQWR3UREOHPWKHIROORZLQJLVREWDLQHG
Ethylene bromide + 1-nitropentane System
Azeotrope
= exprmntl
x-y data
Molecular Subgroups
---------------------------------------------------------------------Comp
Name
Number
Description
Frequency
---- -------------------------------- -----------------------1
ethyl bromide
1
CH3
1
2
CH2
1
65
Br
1
2
comp 2
1
2
56
CH3
CH2
CH2NO2
---------------------------------------------------x(1)
GAM(1)
GAM(2)
GEX(J/MOL)
---------------------------------------------------0.0000
1.6876
1.0000
0.00
0.0250
1.6593
1.0002
37.25
0.0500
1.6315
1.0009
73.25
0.0750
1.6042
1.0020
107.96
0.1000
1.5773
1.0036
141.35
0.1250
1.5509
1.0058
173.38
0.1500
1.5249
1.0085
204.01
0.1750
1.4995
1.0118
233.21
0.2000
1.4745
1.0157
260.93
0.2250
1.4500
1.0203
287.13
0.2500
1.4260
1.0257
311.77
0.2750
1.4024
1.0317
334.81
0.3000
1.3794
1.0387
356.18
0.3250
1.3569
1.0465
375.85
0.3500
1.3349
1.0552
393.76
0.3750
1.3135
1.0650
409.87
0.4000
1.2926
1.0759
424.10
0.4250
1.2722
1.0879
436.41
0.4500
1.2524
1.1013
446.72
0.4750
1.2331
1.1161
454.99
0.5000
1.2145
1.1324
461.14
0.5250
1.1964
1.1504
465.09
0.5500
1.1789
1.1702
466.79
0.5750
1.1621
1.1921
466.14
0.6000
1.1459
1.2161
463.08
0.6250
1.1304
1.2427
457.50
0.6500
1.1155
1.2719
449.34
0.6750
1.1014
1.3042
438.48
0.7000
1.0880
1.3398
424.85
1
1
1
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
0.7250
0.7500
0.7750
0.8000
0.8250
0.8500
0.8750
0.9000
0.9250
0.9500
0.9750
1.0000
1.0753
1.0635
1.0525
1.0424
1.0331
1.0249
1.0177
1.0116
1.0067
1.0031
1.0008
1.0000
1.3792
1.4228
1.4711
1.5248
1.5846
1.6514
1.7261
1.8099
1.9043
2.0110
2.1320
2.2698
408.32
388.79
366.16
340.29
311.06
278.34
241.98
201.83
157.73
109.52
57.00
0.00
HWK\OEURPLGHDQGFRPSDW.
[RIHWK\OEURPLGH
HWK\OEURPLGH
x(1)
0.0000
0.0250
0.0500
0.0750
0.1000
0.1250
0.1500
0.1750
0.2000
0.2250
0.2500
0.2750
0.3000
0.3250
0.3500
0.3750
0.4000
0.4250
0.4500
0.4750
0.5000
0.5250
0.5500
0.5750
0.6000
0.6250
0.6500
0.6750
0.7000
0.7250
0.7500
0.7750
0.8000
0.8250
0.8500
y(1)
0.0000
0.4618
0.6338
0.7237
0.7789
0.8163
0.8433
0.8638
0.8798
0.8927
0.9034
0.9123
0.9199
0.9264
0.9322
0.9372
0.9417
0.9458
0.9494
0.9528
0.9558
0.9587
0.9613
0.9638
0.9661
0.9683
0.9705
0.9725
0.9745
0.9765
0.9784
0.9803
0.9822
0.9841
0.9861
FRPS
y(2)
1.0000
0.5382
0.3662
0.2763
0.2211
0.1837
0.1567
0.1362
0.1202
0.1073
0.0966
0.0877
0.0801
0.0736
0.0678
0.0628
0.0583
0.0542
0.0506
0.0472
0.0442
0.0413
0.0387
0.0362
0.0339
0.0317
0.0295
0.0275
0.0255
0.0235
0.0216
0.0197
0.0178
0.0159
0.0139
P(bar)
0.153
0.278
0.398
0.514
0.626
0.734
0.839
0.940
1.037
1.130
1.220
1.307
1.391
1.472
1.550
1.625
1.698
1.768
1.836
1.901
1.965
2.026
2.086
2.144
2.201
2.256
2.311
2.364
2.417
2.469
2.521
2.574
2.626
2.679
2.732
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
0.8750
0.9000
0.9250
0.9500
0.9750
1.0000
0.9881
0.9902
0.9925
0.9948
0.9973
1.0000
0.0119
0.0098
0.0075
0.0052
0.0027
0.0000
2.787
2.844
2.902
2.963
3.026
3.093
HWK\OEURPLGHDQGFRPSDW.
[\RIHWK\OEURPLGH
3[
3\
6HHVROXWLRQWR3UREOHP
6HHVROXWLRQWR3UREOHP
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
§ ·
¨
¸
¨ ¸
¨ ¸
¨ ¸
¨
¸
¨ ¸
¨ ¸
¨
¸
¨ ¸
GDWD ¨ ¸
¨ ¸
¨
¸
¨ ¸
¨ ¸
¨ ¸
¨
¸
¨ ¸
¨ ¸
¨
¸
© ¹
JL \ ˜ 3
L L
JL [ ˜ L
5 L 3 GDWD
L
L
L \ GDWD
L
L
[ GDWD
L
L L [ [
L
\ \
L
L
\ ˜ 3
L L
[ ˜ L
*H[ 5˜ 7˜ [ ˜ OQ JL [ ˜ OQ JL
L
7 1 URZV GDWD
L
§ JL ·
¸
© JL ¹
OQ¨
6RWKHGDWDORRNVWKHUPRG\QDPLFDOO\
FRQVLVWHQWE\WKHLQWHJUDORUHTXDODUHD
WHVW
[L
Solutions to Chemical and Engineering Thermodynamics,5th ed
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
6HH3UREOHPIRULQLWLDOFDOFXODWLRQRI*H[
Section 10.2
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
)LWXVLQJDWKUHHWHUP5HGOLFK.LVWHUH[SDQVLRQ
ª º
«
»
« ˜ [ » ˜ [˜ [
« ˜ [ »
¬
¼
9HFWRURIWHUPV
LQ5.H[SDQVLRQ ) [ 'HWHUPLQDWLRQRISDUDPHWHUVE\OLQHDUUHJUHVVLRQ
9HFWRURI
SDUDPHWHUV
§ u ·
¨
¸
¨ ¸
¨
¸
© ¹
6
6 OLQILW [ *H[ )
M &DOFXODWLRQRI*LEEVIUHHHQHUJ\JH[DQGWZRSDUWLDOPRODUH[FHVV*LEEVIUHHHQHUJLHV
JH[ JH[
JH[ 6 ˜ [ ˜ [ 6 ˜ [ ˜ [ ˜ [ [ 6 ˜ [ ˜ [ ˜ [ [
M
JH[ M
JH[ M
[
ª
˜ 6 6 ˜ [ [ 6 ˜ [ [
M M
º
[
ª
º
M
M
¬ M
M
M
M
M
M
˜ 6 6 ˜ [ [ 6 ˜ [ [
¬ M
M
M
M
¼
¼
M
M
M
M
˜ [
M
¬ M
M
M
M¬ § JH[M ·
¸
© 5˜ 7 ¹
M
M¼
M
˜ [ ˜ ª 6 ˜ 6 ˜ [ [ º
JDP H[S¨
&DOFXODWLRQRIDFWLYLW\FRHIILFLHQWV
M
˜ [ ˜ [ ˜ ª 6 ˜ 6 ˜ [ [ º
M¼
M
§ JH[M ·
¸
© 5˜ 7 ¹
JDP H[S¨
M
&RPSDULVRQRILQSXW*H[DQGILWWHGJH[LQSXWDQGSUHGLFWHGSUHVVXUHDQG[\GLDJUDP
33 [ ˜ JDP ˜ [ ˜ JDP ˜ M
M
M
M
M
\\ [ ˜ JDP ˜ M
M
M
33
M
*H[M
3
JH[M
33
[M
J
JDP
[
J
JDP
[
[
\
\\
[
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
)LWXVLQJDWZRWHUP5HGOLFK.LVWHUH[SDQVLRQ
9HFWRURIWHUPV
LQ5.H[SDQVLRQ
)[ ª º
«
» ˜ [˜ [
¬ ˜ [ ¼
'HWHUPLQDWLRQRISDUDPHWHUVE\OLQHDUUHJUHVVLRQ
9HFWRURI
SDUDPHWHUV
§ u ·
¨
¸
© ¹
6
6 OLQILW [ *H[ )
M &DOFXODWLRQRI*LEEVIUHHHQHUJ\JH[DQGWZRSDUWLDOPRODUH[FHVV*LEEVIUHHHQHUJLHV
JH[ JH[
JH[ 6 ˜ [ ˜ [ 6 ˜ [ ˜ [ ˜ [ [
M
JH[ [
JH[ [
M
M
M
M
M
M
M
M
M
M
˜ ª 6 6 ˜ [ [ º ˜ [ ˜ [ ˜ 6
¬ M¼
M
M
M
˜ ª 6 6 ˜ [ [ º ˜ [ ˜ [ ˜ 6
¬ M¼
M
M
M
§ JH[M ·
¸
© 5˜ 7 ¹
JDP H[S¨
&DOFXODWLRQRIDFWLYLW\FRHIILFLHQWV
M
§ JH[M ·
¸
© 5˜ 7 ¹
JDP H[S¨
M
&RPSDULVRQRILQSXW*H[DQGILWWHGJH[LQSXWDQGSUHGLFWHGSUHVVXUHDQG[\GLDJUDP
33 [ ˜ JDP ˜ [ ˜ JDP ˜ M
M
M
M
M
\\ [ ˜ JDP ˜ M
M
M
33
M
*H[M
3
JH[M
33
[M
J
JDP
[
J
JDP
[
[
\
\\
[
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
6HH3UREOHPIRUFDOFXODWLRQRI*H[
,QLWLDOSDUDPHWHUJXHVVH V
§·
¨ ¸
©¹
YJ 9HFWRUFRQVLVWLQJRIIXQFWLRQWREHILW
GHULYDWLYHRIIXQFWLRQZUWVWSDUDPHWHU
GHULYDWLYHRIIXQFWLRQZUWQGSDUDPHWHU
X ˜ [ ˜ [˜ X
ª«
º»
«
»
X ˜[ X ˜ [
«
»
« X ˜ [ ˜[
»
X ˜X ˜[ ˜ [
«
»
) [ X « X ˜ [ X ˜ [
» ˜ 5˜ 7
ª¬ X˜ [ X ˜ [ º¼ »
«
«
»
X ˜ X ˜ [˜ [
« X ˜ [ ˜ [
»
«
»
« X ˜ [ X˜ [ ª X˜ [ X ˜ [ º »
¬
¬
¼ ¼
'HWHUPLQDWLRQRISDUDPHWHUVE\QRQOLQHDUUHJUHVVLRQ
3 JHQILW [ *H[ YJ )
9HFWRURISDUDPHWHUV
§ ·
¨
¸
© ¹
3
&DOFXODWLRQRI*LEEVIUHHHQHUJ\JH[DQGWZRSDUWLDOPRODUH[FHVV*LEEVHQHUJLHVJH[ JH[
3
3 ˜ 3 ˜ [ ˜ [ ˜ 5˜ 7
3
M
M
JH[ M
JH[ JH[ M
M
3 ˜ [ 3 ˜ [
3 ˜ [ ·
§
M
M
M
3
˜ [ ·
§
¨ ¸
¨ M ¸
¨
3 ˜ [ ¸
M
¨
3 ˜ [ ¸
©
¹
©
&DOFXODWLRQRIDFWLYLW\FRHIILFLHQWV
JDP H[S JH[
M
JDPLQI H[S 3
JDPLQI H[S 3
JDPLQI
M
JDP H[S JH[
M
M¹
JDP H[S JH[
M
JDPLQI
M
JDP H[S JH[
M
M
M
&RPSDULVRQRILQSXW*H[DQGILWWHGJH[LQSXWDQGSUHGLFWHGSUHVVXUHDQG[\GLDJUDP
33 [ ˜ JDP ˜ [ ˜ JDP ˜ M
M
M
M
M
\\ [ ˜ JDP ˜ M
M
M
33
M
*H[M
3
JH[M
33
[M
J
JDP
[
J
JDP
[
[
\
\\
[
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
)RUFDOFXODWLRQRI*H[VHH3UREOHP
,QLWLDOSDUDPHWHUJXHVVHVIRU:LOVRQHTXDWLRQ
9HFWRUFRQVLVWLQJRIIXQFWLRQWREHILW
GHULYDWLYHZUWVWSDUDPHWHU
GHULYDWLYHZUWQGSDUDPHWHU
YJ §·
¨ ¸
©¹
ª [˜ OQª¬ [ X ˜ [ º¼ [ ˜ OQª¬ [ X ˜ [º¼ º
«
»
˜ [ ˜ [
«
»
» ˜ 5˜ 7
[ X ˜ [
) [ X «
«
»
«
»
˜ [ ˜ [
«
»
[ X ˜[
¬
¼
'HWHUPLQDWLRQRISDUDPHWHUVE\QRQOLQHDUUHJUHVVLRQ
3 JHQILW [ *H[ YJ )
9HFWRURISDUDPHWHUV
3
§ ·
¨
¸
© ¹
&DOFXODWLRQRI*LEEVIUHHHQHUJ\JH[DQGWZRSDUWLDOPRODUH[FHVV*LEEVHQHUJLHVJH[ JH[
JH[ M
[ ˜ OQ [ 3 ˜ [ [ ˜ OQ [ 3 ˜ [ ˜ 5˜ 7
M
M
M
M
M
M
M
M
3
3
3
·¸
§
M ¨ [ 3 ˜ [
[ 3 ˜ [ ¸
M
M¹
M
© M
JH[ OQ [ 3 ˜ [ [ ˜ ¨
M
·¸
§
M ¨ [ 3 ˜ [
[ 3 ˜ [ ¸
M
M
M
M
©
¹
JH[ OQ [ 3 ˜ [ [ ˜ ¨
M
M
3
M
&DOFXODWLRQRIDFWLYLW\FRHIILFLHQWV
JDP H[S JH[
JDP H[S JH[
M
M
M
M
&RPSDULVRQRILQSXW*H[DQGILWWHGJH[LQSXWDQGSUHGLFWHGSUHVVXUHDQG[\GLDJUDP
33 [ ˜ JDP ˜ [ ˜ JDP ˜ M
M
M
M
M
[ ˜ JDP ˜ M
\\ M
M
33
M
*H[M
3
JH[M
33
[M
J
JDP
[
J
JDP
[
[
\
\\
[
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
6HWYDOXHRIDOSKDSDUDPHWHUDQGLQLWLDO
JXHVVHVIRUWDX XR DQGWDX X
$ § ·
¨ ¸
© ¹
YJ X
X
º
ª
«
»
« [˜ H[S $ ˜ X [ [ [ ˜ H[S $ ˜ X »
«
»
$ ˜ X ˜ [ ˜ H[S $ ˜ X [
«
»
» ˜ [˜ [ ˜ 5˜ 7
) [ X «
˜
[
˜
«
»
[
H[S
$
X
»
¼º
«
»
$
˜
X
˜
[
˜
H[S
$
˜
X
[
«
»
«
»
«
[˜ H[S $ ˜ X [ º
»
ª
¬
¬
¼
¼
9HFWRUFRQVLVWLQJRI
IXQFWLRQWREHILW
LWVGHULYDWLYHZUWWDX
LWVGHULYDWLYHZUWWDX
'HWHUPLQDWLRQRISDUDPHWHUVE\QRQOLQHDUUHJUHVVLRQ
3 JHQILW [ *H[ YJ )
9HFWRURISDUDPHWHUV
3
,GHQWLILFDWLRQRIZRUNVKHHWDQGPRGHOSDUDPHWHUV
§ ·
¨
¸
© ¹
W 3
W 3
J H[S $ ˜ W
J H[S $ ˜ W
&DOFXODWLRQRI*LEEVIUHHHQHUJ\JH[DQGWZRSDUWLDOPRODUH[FHVV*LEEVHQHUJLHVJH[DQGJH[
[ ˜ [ ˜ §¨
JH[ M
M
JH[ M
[
M
M
JH[ [
M
M
©
W˜ J
[ [ ˜ J
M
ª
«
˜ W˜ §¨
«
¬
©
ª
«
˜ W˜ §¨
«
¬
©
M
W˜ J
M
J
M
W˜ J
¹
[ [ ˜ J
M
J
M
º
» ˜ 5˜ 7
· M
W˜ J
¹
[ [ ˜ J
M
»
¼
º
» ˜ 5˜ 7
· [ [ ˜ J ¸
M
¹
M
[ [ ˜ J ¸
M
· ˜ 5˜ 7
[ [ ˜ J ¸
M
»
¼
&DOFXODWLRQRIDFWLYLW\FRHIILFLHQWV
§ JH[M ·
¸
© 5˜ 7 ¹
JDP H[S¨
M
§ JH[M ·
¸
© 5˜ 7 ¹
JDP H[S¨
M
&RPSDULVRQRILQSXW*H[DQGILWWHGJH[LQSXWDQGSUHGLFWHGSUHVVXUHDQG[\GLDJUDP
33 [ ˜ JDP ˜ [ ˜ JDP ˜ M
M
M
M
M
[ ˜ JDP ˜ M
\\ M
M
33
M
*H[M
3
JH[M
33
[M
J
JDP
[
J
JDP
[
[
\
\\
[
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
157/ZLWKWKUHHSDUDPHWHUV
,QLWLDOSDUDPHWHUJXHVVHV
§ ·
¨ ¸
¨ ¸
© ¹
YJ 9HFWRUFRQVLVWLQJRIIXQFWLRQWREHILWDQGLQRUGHULWVGHULYDWLYHV
ZLWKUHVSHFWWRDOSKD XR WDX X DQGWDX X X
X
ª
º
«
»
[˜ H[S X ˜ X [
[ [ ˜ H[S X ˜ X
«
»
«
»
«
»
X ˜ [˜ H[S X ˜ X
X ˜ [ ˜ H[S X ˜ X
«
˜ ˜ »
« ª[˜ H[S X ˜ X [ º
»
ª¬ [ ˜ H[S X˜ X [º¼
¬
¼
» ˜ [˜ [ ˜ 5˜ 7
) [ X «
«
»
X ˜ X ˜ [ ˜ H[S X ˜ X [
«
»
«
»
˜
[
[
H[S
X
˜
X
ª
º
¬
¼
«
»
«
»
X ˜ X ˜ [˜ H[S X ˜ X [
«
»
«
»
ª¬[˜ H[S X˜ X [ º¼
¬
¼
'HWHUPLQDWLRQRISDUDPHWHUVE\QRQOLQHDUUHJUHVVLRQ
3 JHQILW [ *H[ YJ )
9HFWRURISDUDPHWHUV
3
,GHQWLILFDWLRQRIZRUNVKHHW
DQGPRGHOSDUDPHWHUV
W 3
W 3
J H[S 3 ˜ W
§ ·
¨ ¸
¨
¸
© ¹
J H[S 3 ˜ W
&DOFXODWLRQRI*LEEVIUHHHQHUJ\JH[DQGWZRSDUWLDOPRODUH[FHVV*LEEVHQHUJLHVJH[ JH[
[ ˜ [ ˜ ¨§
JH[ M
M
M
©
W˜ J
[ [ ˜ J
M
M
W˜ J
· ˜ 5˜ 7
[ [ ˜ J ¸
M
M
¹
º
W˜ J
J
· » ˜ 5˜ 7
« © [M [M˜ J ¸¹
[ [ ˜ J »
M
M
¬
¼
º
J
W˜ J
ª
· » ˜ 5˜ 7
JH[ [ ˜ « W˜ §¨
¸
M
M
« © [M [M˜ J ¹
[ [ ˜ J »
M
M
¬
¼
JH[ [
M
M
ª
˜ « W˜ §¨
&DOFXODWLRQRIDFWLYLW\FRHIILFLHQWV
§ JH[M ·
§ JH[M ·
¸ JDPM H[S¨
¸
© 5˜ 7 ¹
© 5˜ 7 ¹
JDP H[S¨
M
&RPSDULVRQRILQSXW*H[DQGILWWHGJH[LQSXWDQGSUHGLFWHGSUHVVXUHDQG[\GLDJUDP
33 [ ˜ JDP ˜ [ ˜ JDP ˜ M
M
M
M
M
[ ˜ JDP ˜ M
\\ M
M
33
M
*H[M
3
JH[M
33
[M
J
JDP
[
J
JDP
[
[
\
\\
[
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
&RPSRQHQWLVZDWHU
U &RPSRQHQWLVGLR[DQH
/ I [ / [˜ T
T [ [˜ T [ ˜ T
[˜ U
I [ [˜ U [ ˜ U
66 U ˜ U T ˜ U T [ 66 T T ˜ U T ˜ U /
/
[ ˜ T
[˜ T [ ˜ T
[ ˜ U
[˜ U [ ˜ U
66
66
·¸ º» º»
ª § I [ · u T˜ OQ§ T [ · I [ ˜ § / U˜ / · º ˜ H[Sª« ª« T˜ OQ T [ T [ ˜ 66 T˜ T [ ˜ ¨§
¸
¨
¸
¨
¸
»
U ¹ ¼
««
¨ T [ T [ ˜ 66
T [ ˜ 66 T [ ¸ » »
©
¬ © [ ¹
© I [ ¹
¬¬
©
¹¼¼
JDP 66 [ H[S« OQ¨
66
66
Ἴ
ª § I [ · ˜ T˜ OQ§ T [ · I [ ˜ § / U˜ / · º ˜ H[Sª« «ª T˜ OQ T [ ˜ 66 T [ T u T [ ˜ ¨§
¸»»
¸
¨
¸
¨
¸»
U ¹ ¼
««
¨ T [ T [ ˜ 66 T [ ˜ 66 T [ ¸ » »
©
¬ © [¹
© I [ ¹
¬¬
©
¹¼¼
JDP 66 [ H[S« OQ¨
JH[ 66 [ 5˜ 7˜ > [˜ OQ JDP 66 [ [ ˜ OQ JDP 66 [ @
(UU 66 JH[ 66 [ *H[
(UU 66 JH[ 66 [
*H[
(UU 66 JH[ 66 [
*H[
(UU 66 JH[ 66 [ *H[
(UU 66 JH[ 66 [
*H[
(UU 66 JH[ 66 [
*H[
(UU 66 JH[ 66 [
*H[
(UU 66 JH[ 66 [
(UU 66 JH[ 66 [
*H[
(UU 66 JH[ 66 [
*H[
(UU 66 JH[ 66 [
*H[
(UU 66 JH[ 66 [
*H[
(UU 66 JH[ 66 [
*H[
(UU 66 JH[ 66 [
*H[
66( 66 (UU 66
66( 66 66( 66 (UU 66
(UU 66
(UU 66
(UU 66
(UU 66
(UU 66 JH[ 66 [
(UU 66
*H[
(UU 66
(UU 66
66( 66
(UU 66
66
(UU 66
(UU 66
(UU 66
(UU 66
66 (UU 66
66 3DUDPHWHUVIRUEHVWILW
*LYHQ
*H[
66 0LQHUU 66
§ ·
¨
¸
© ¹
M 33 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ M
\ M
M
M
[ ˜ JDP 66 [ ˜ M
M
33
M
M
JDP JDP 66 [
M
JDP JDP 66 [
M
M
JH[ 5˜ 7˜ ª [ ˜ OQ JDP [ ˜ OQ JDP º
M
M
M
¬
M
M
M
M¼
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
*H[M
3
JH[M
33
[
[M
J
JDP
J
JDP
[
[
\
\\
[
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
ZDWHUDQGGLR[DQHDW.
[RIZDWHU
ZDWHU
GLR[DQH
-------------COMPONENT DATA
-------------Molecular Subgroups
---------------------------------------------------------------------Comp
Name
Number
Description
Frequency
---- -------------------------------- -----------------------1
water
19
H2O
1
2
1,4-dioxane
81
c-CH2OCH2
---------------------------------------------------x(1)
GAM(1)
GAM(2)
GEX(J/MOL)
---------------------------------------------------0.0000
3.5435
1.0000
0.00
0.0250
3.4073
1.0005
83.65
0.0500
3.2761
1.0020
164.59
0.0750
3.1496
1.0047
242.75
0.1000
3.0277
1.0085
318.05
0.1250
2.9105
1.0135
390.41
0.1500
2.7976
1.0200
459.73
0.1750
2.6891
1.0278
525.92
0.2000
2.5849
1.0372
588.89
0.2250
2.4847
1.0484
648.55
0.2500
2.3885
1.0613
704.77
0.2750
2.2963
1.0763
757.47
0.3000
2.2078
1.0935
806.52
0.3250
2.1231
1.1132
851.81
0.3500
2.0419
1.1355
893.21
0.3750
1.9643
1.1608
930.59
0.4000
1.8901
1.1894
963.83
0.4250
1.8192
1.2218
992.76
0.4500
1.7517
1.2583
1017.25
0.4750
1.6873
1.2995
1037.15
0.5000
1.6260
1.3461
1052.27
0.5250
1.5677
1.3988
1062.46
0.5500
1.5124
1.4584
1067.53
0.5750
1.4601
1.5260
1067.28
0.6000
1.4105
1.6029
1061.52
0.6250
1.3638
1.6905
1050.04
0.6500
1.3199
1.7908
1032.60
0.6750
1.2787
1.9058
1008.97
0.7000
1.2402
2.0384
978.91
0.7250
1.2044
2.1920
942.13
0.7500
1.1713
2.3706
898.37
0.7750
1.1409
2.5797
847.32
0.8000
1.1132
2.8257
788.67
2
Solutions to Chemical and Engineering Thermodynamics,5th ed
0.8250
0.8500
0.8750
0.9000
0.9250
0.9500
0.9750
1.0000
1.0883
1.0662
1.0469
1.0308
1.0178
1.0081
1.0021
1.0000
3.1174
3.4656
3.8845
4.3930
5.0160
5.7872
6.7527
7.9759
722.07
647.17
563.58
470.90
368.69
256.47
133.76
0.00
ZDWHUDQGGLR[DQHDW.
[RIZDWHU
---------------------------------------------------x(1)
GAM(1)
GAM(2)
GEX(J/MOL)
---------------------------------------------------0.0000
3.5435
1.0000
0.00
0.0250
3.4073
1.0005
83.65
0.0500
3.2761
1.0020
164.59
0.0750
3.1496
1.0047
242.75
0.1000
3.0277
1.0085
318.05
0.1250
2.9105
1.0135
390.41
0.1500
2.7976
1.0200
459.73
0.1750
2.6891
1.0278
525.92
0.2000
2.5849
1.0372
588.89
0.2250
2.4847
1.0484
648.55
0.2500
2.3885
1.0613
704.77
0.2750
2.2963
1.0763
757.47
0.3000
2.2078
1.0935
806.52
0.3250
2.1231
1.1132
851.81
0.3500
2.0419
1.1355
893.21
0.3750
1.9643
1.1608
930.59
0.4000
1.8901
1.1894
963.83
0.4250
1.8192
1.2218
992.76
0.4500
1.7517
1.2583
1017.25
0.4750
1.6873
1.2995
1037.15
0.5000
1.6260
1.3461
1052.27
0.5250
1.5677
1.3988
1062.46
0.5500
1.5124
1.4584
1067.53
0.5750
1.4601
1.5260
1067.28
0.6000
1.4105
1.6029
1061.52
0.6250
1.3638
1.6905
1050.04
0.6500
1.3199
1.7908
1032.60
0.6750
1.2787
1.9058
1008.97
0.7000
1.2402
2.0384
978.91
0.7250
1.2044
2.1920
942.13
0.7500
1.1713
2.3706
898.37
0.7750
1.1409
2.5797
847.32
0.8000
1.1132
2.8257
788.67
0.8250
1.0883
3.1174
722.07
0.8500
1.0662
3.4656
647.17
0.8750
1.0469
3.8845
563.58
0.9000
1.0308
4.3930
470.90
0.9250
1.0178
5.0160
368.69
0.9500
1.0081
5.7872
256.47
0.9750
1.0021
6.7527
133.76
1.0000
1.0000
7.9759
0.00
Section 10.2
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
x(1)
0.0000
0.0250
0.0500
0.0750
0.1000
0.1250
0.1500
0.1750
0.2000
0.2250
0.2500
0.2750
0.3000
0.3250
0.3500
0.3750
0.4000
0.4250
0.4500
0.4750
0.5000
0.5250
0.5500
0.5750
0.6000
0.6250
0.6500
0.6750
0.7000
0.7250
0.7500
0.7750
0.8000
0.8250
0.8500
0.8750
0.9000
0.9250
0.9500
0.9750
1.0000
y(1)
0.0000
0.0645
0.1196
0.1671
0.2084
0.2446
0.2764
0.3046
0.3296
0.3520
0.3719
0.3898
0.4058
0.4202
0.4332
0.4449
0.4554
0.4649
0.4734
0.4811
0.4881
0.4944
0.5001
0.5054
0.5103
0.5149
0.5193
0.5238
0.5284
0.5335
0.5392
0.5460
0.5543
0.5650
0.5791
0.5983
0.6250
0.6639
0.7232
0.8204
1.0000
y(2)
1.0000
0.9355
0.8804
0.8329
0.7916
0.7554
0.7236
0.6954
0.6704
0.6480
0.6281
0.6102
0.5942
0.5798
0.5668
0.5551
0.5446
0.5351
0.5266
0.5189
0.5119
0.5056
0.4999
0.4946
0.4897
0.4851
0.4807
0.4762
0.4716
0.4665
0.4608
0.4540
0.4457
0.4350
0.4209
0.4017
0.3750
0.3361
0.2768
0.1796
0.0000
P(bar)
0.156
0.163
0.169
0.175
0.179
0.184
0.187
0.191
0.194
0.196
0.198
0.200
0.202
0.203
0.204
0.205
0.205
0.205
0.206
0.206
0.206
0.206
0.205
0.205
0.205
0.204
0.204
0.204
0.203
0.202
0.201
0.200
0.198
0.196
0.193
0.189
0.183
0.175
0.164
0.147
0.124
ZDWHUDQGGLR[DQHDW.
[\RIZDWHU
3[
3\
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
§ ·
¨ ¸
¨
¸
¨ ¸
¨ ¸
¨ ¸
¨
¸
GDWD ¨ ¸
¨ ¸
¨
¸
¨ ¸
¨ ¸
¨ ¸
¨
¸
© ¹
5 7 M 3 GDWD
M
\ [ GDWD
M M
M L '3 OQ 3 OQ 3
L
[ \ '3
% L L
[ ˜\
& '3
'3
[ \ '3
% '3
[ \ '3
% '3
'3
% ª%
¬
\ ˜ & º
¼
\
˜ & º
˜ & º
¼
\
¬
[ ˜\
& % ª%
\ '3
[ ˜\
& ¬
\ % ª%
¼
\
L [ \
L
% L
'3
L [ ˜\
L L & L
L
'3
'3
¬
L
\ L
L
% ª%
L
L
˜ &º
L¼
\
L [ \
L
% L
'3
L L
[ ˜\
L L & L
'3
'3
¬
L
\ L
L
% ª%
L
˜ &º
L
L¼
\
L
L [ \
L
% L
L '3
L
[ ˜\
L L
& L
'3
'3
¬
L
\ L
L
% ª%
L
L
˜ &º
L¼
\
L
L % [ \
L
L
'3
L L
& L
'3
L
[ ˜\
L L '3
L
\ % ª%
¬
L
L
L
˜ &º
L¼
\
L
L % [ \
L
L
'3
L L
& L
'3
L
[ ˜\
L L '3
L
\ % ª%
¬
L
L
L
˜ &º
L¼
\
L
L % L
[ \
L
L
'3
L
'3
& L
L
[ ˜\
L L '3
L
\ L
% ª%
L
¬
L
˜ &º
L¼
\
$V\RXZLOOVHHEHORZWKLVSRLQWLVQRWXVHG
L
L
\ 5H \
L % L
[ \
L
L
L
'3
L
]
'3
M \
M
& L
[ ˜\
L L '3
L
\ L
% ª%
L
¬
L
˜ &º
L¼
\
L
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
6WDUWIURPWKHRWKHUGLUHFWLRQEXWGRLWWKHOD]\ZD\E\UHYHUVLQJWKHFRPSRQHQWV
M \ [ [
M
33 3
M
M
[ \ '3
% % '3
'3
[ \ '3
% [ ˜\
& '3
\ '3
\ L % ª%
¬
'3
\ [ ˜\
& L
L
[ ˜\
& '3
[ \ '3
'3 OQ 33 OQ 33
M
˜ & º
¼
\
% ª%
¬
˜ & º
¼
\
% ª%
¬
˜ & º
¼
\
L [ \
L
% L
L '3
[ ˜\
L L & L
L
'3
'3
\ L
L
L
% ª%
¬
L
L
˜ &º
L¼
\
L % [ \
L
L
'3
L L
& L
'3
[ ˜\
L L '3
L
L
\ % ª%
¬
L
L
L
˜ &º
L¼
\
L
L % [ \
L
L
'3
L L
& L
'3
[ ˜\
L L '3
L
L
\ % ª%
¬
L
L
L
˜ &º
L¼
\
L
L % [ \
L
L
'3
L L
L
'3
[ \
L
L
'3
L L
& L
'3
[ ˜\
L L '3
L
L
L % & L L '3
\ ¬
L
L
[ ˜\
L
L
\ % ª%
L
˜ &º
L¼
\
% ª%
¬
L
L
L
˜ &º
L¼
L
% L
\
\ 5H \
L
L [ \
L '3
L
'3
[ ˜\
L L & L
'3
L
L
\ L
% ª%
L
¬
L
˜ &º
L
L
L
L¼
\
L
\ 5H \
L [ \
L
% L
L '3
L
'3
[ ˜\
L L & L
'3
L
L
]
M \
M
]
\ L
% ª%
L
¬
L
˜ &º
L¼
\
L
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
N \\ ]
N
N N \\ ]
N
M \\
N [ [
M
J M \\ M M
J M [ ˜ M
]
\\ ˜ 3
M
[
M
[ ˜ M
J
[ M
\\ ˜ 3
]
J
\\
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
·
¨§
¸
¨ ¸
¨ ¸
¨ ¸
¨
¸
GDWD ¨ ¸
¨ ¸
¨
¸
¨ ¸
¨ ¸
¨© ¸¹
5 7 M 3 GDWD
M
[[ [ GDWD
M M
[ [
M M
1 URZV GDWD
M
7KUHHSDUDPHWHU5HGOLFK.LVWHUILW
6 6 6 ª
JH[ 6 [[ [[ ˜ 6 6 ˜ ˜ [[ 6 ˜ ˜ [[ º
JH[ 6 [[ [[ ˜ ª 6 6 ˜ ˜ [[ 6 ˜ ˜ [[ ˜ [[ ˜ [[˜ ª 6 ˜ 6 ˜ ˜ [[ º
¬ ¬ ¼
º
¼
˜ [[˜ [[ ˜ ª 6 ˜ 6 ˜ ˜ [[ º
JDP 6 [[ H[S JH[ 6 [[
¬ ¬ ¼
¼
JDP 6 [[O H[S JH[ 6 [[O
33 6 [ ˜ JDP 6 [ ˜ [ ˜ JDP 6 [ ˜ 3
33 6 [ ˜ JDP 6 [ ˜ [ ˜ JDP 6 [ ˜ 3
33 6 [ ˜ JDP 6 [ ˜ [ ˜ JDP 6 [ ˜ 3
33 6 [ ˜ JDP 6 [ ˜ [ ˜ JDP 6 [ ˜ 3
33 6 [ ˜ JDP 6 [ ˜ [ ˜ JDP 6 [ ˜ 3
33 6 [ ˜ JDP 6 [ ˜ [ ˜ JDP 6 [ ˜ 3
33 6 [ ˜ JDP 6 [ ˜ [ ˜ JDP 6 [ ˜ 3
33 6 [ ˜ JDP 6 [ ˜ [ ˜ JDP 6 [ ˜ 3
33 6 [ ˜ JDP 6 [ ˜ [ ˜ JDP 6 [ ˜ 3
66( 6 33 6
33 6
33 6
33 6
33 6
33 6
33 6
33 6
33 6
3DUDPHWHUVIRUEHVWILW
*LYHQ
66( 6
6 0LQHUU 6
§ ·
6
¨ ¸
¨
¸
© ¹
33 [ ˜ JDP 6 [ ˜ [ ˜ JDP 6 [ ˜ M
M
M
M
M
[ ˜ JDP 6 [ ˜ M
\ M
M
33
M
33
3
\
[
[
[
JDP 6 [M
JDP 6 [M
7
W
5 GOL K .L W
ILW
[M
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
7ZRSDUDPHWHU5HGOLFK.LVWHUILW
66 66 JH[ 66 [[ [[ ˜ ª 66 66 ˜ ˜ [[ º ˜ [[˜ [[ ˜ 66
JH[ 66 [[ [[ ˜ ª 66 66 ˜ ˜ [[ º ˜ [[ ˜ [[˜ 66
¬
¬
¼
¼
JDP 66 [[ H[S JH[ 66 [[
JDP 66 [[O H[S JH[ 66 [[O
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
66( 66 33 66
33 66
33 66
66( 66 66( 66 33 66
33 66
33 66
33 66
33 66
33 66
3DUDPHWHUVIRUEHVWILW
*LYHQ
66( 66
66 0LQHUU 66
§ ·
¨
¸
© ¹
66
33 [ ˜ JDP 6 [ ˜ [ ˜ JDP 6 [ ˜ M
M
\ M
M
M
M
[ ˜ JDP 6 [ ˜ M
M
33
M
33
3
\
[
[
[
JDP 6 [M
JDP 6 [M
[M
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
9DQ/DDUILW
66 66 66
JH[ 66 [[ 66 ˜ [[ º
ª
« »
«
66 ˜ [[ »
¬
¼
66
JH[ 66 [[ 66 ˜ [[ º
ª
« »
«
66 ˜ [[ »
¬
¼
JDP 66 [[ H[S JH[ 66 [[
JDP 66 [[O H[S JH[ 66 [[O
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
66( 66 33 66
33 66
33 66
33 66
33 66
33 66
33 66
33 66
33 66
3DUDPHWHUVIRUEHVWILW
*LYHQ
66( 66
66 0LQHUU 66
§ ·
¨
¸
© ¹
66
33 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ M
M
\ M
M
M
M
[ ˜ JDP 66 [ ˜ M
M
33
M
33
3
\
[
[
[
JDP 66 [M
JDP 66 [M
[M
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
157/SDUDPHWHUILW
W 66
W 66
$ J 66 H[S $ ˜ 66
J 66 H[S $ ˜ 66
ª
66 ˜ J 66
º
J 66
º » ˜ 5˜ 7
»
»
¬ [[ [[ ˜ J 66 ¼
> [[ [[J
˜
66 @ ¼
JH[ 66 [[ [[ ˜ « W˜ ª«
JH[ 66 [[ [[ ˜ « W˜ ª«
ª
«
¬
JDP 66 [[ H[S§¨
©
«
¬
66 ˜ J 66
º
º » ˜ 5˜ 7
»
»
˜
66 ¼
¬ [[ [[J
> [[ [[ ˜ J 66 @ ¼
J 66
JH[ 66 [[ ·
JH[ 66 [[ ·
JDP 66 [[ H[S§¨
¸
¹
5˜ 7
©
¸
¹
5˜ 7
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
66( 66 33 66
33 66
33 66
66( 66 66( 66 33 66
33 66
33 66
33 66
33 66
33 66
3DUDPHWHUVIRUEHVWILW
*LYHQ
66( 66
66 0LQHUU 66
§ ·
¨
¸
© ¹
66
33 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ M
M
\ M
M
M
M
[ ˜ JDP 66 [ ˜ M
M
33
M
33
3
\
[
[
[
JDP 66 [M
JDP 66 [M
[M
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
:LOVRQPRGHOSDUDPHWHUILW
66
66
ª
ȼ
« [[ 66˜ [[
[[ 66 ˜ [[»
¬
¼
JH[ 66 [[ OQª [[ 66 ˜ [[ º [[ ˜ «
¬
¼
66
66
º»
ª
« [[ 66˜ [[ [[ 66˜ [[ »
¬
¼
JH[ 66 [[ OQª [[ 66 ˜ [[º [[˜ «
¬
¼
JDP 66 [[ H[S JH[ 66 [[
JDP 66 [[ H[S JH[ 66 [[
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
66( 66 33 66
33 66
33 66
33 66
66( 66 66( 66 33 66
33 66
33 66
33 66
33 66
3DUDPHWHUVIRUEHVWILW
*LYHQ
66( 66
66 0LQHUU 66
§
·
66 ¨
¸
© L u ¹
66 5H 66
7KHILWWLQJURXWLQHUHVXOWVLQDYHU\VPDOOLPDJLQDU\SDUWRIWKHVHFRQG:LOVRQSDUDPHWHU,ZLOOLJQRUH
WKHLPDJLQDU\SDUW
33 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ M
M
\ M
M
M
M
[ ˜ JDP 66 [ ˜ M
M
33
M
33
3
\
[
[
[
JDP 66 [M
JDP 66 [M
[M
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
&RPSRQHQWLVZDWHU
U &RPSRQHQWLVGLR[DQH
/ U ˜ U T ˜ U I [ / [˜ T
T [ T U T ˜ U [˜ U
/
[˜ T [ ˜ T
[ ˜ U
I [ [˜ U [ ˜ U
/
[ ˜ T
T [ [˜ T [ ˜ T
T ˜ [˜ U [ ˜ U
66
66
·¸ º» º»
ª § I [ · u T˜ OQ§ T [ · I [ ˜ § / U˜ / · º ˜ H[Sª« «ª T˜ OQ T [ T [ ˜ 66 T˜ T [ ˜ §¨
¸
¨
¸
¨
¸»
[
«
«
¨
U
I
[
T
[
˜
66
T [ ¸ » »
T
[
T
[
˜
66
¹
©
¹¼ ¬¬
¬ ©
©
¹
©
¹¼¼
JDP 66 [ H[S« OQ¨
66
66
Ἴ
ª § I [ · ˜ T˜ OQ§ T [ · I [ ˜ § / U˜ / · º ˜ H[S«ª «ª T˜ OQ T [ ˜ 66 T [ T u T [ ˜ §¨
¸»»
¸
¨
¸
¨
¸»
«
«
¨
U
[
T
[
˜
66
T [ ¸ » »
T
[
T
[
˜
66
I
[
¹
©
¹¼
¬ ©
©
¹
¬¬
©
¹¼¼
JDP 66 [ H[S« OQ¨
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
33 66 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ 3
66( 66 33 66
33 66
33 66
33 66
66( 66 66( 66 33 66
33 66
33 66
33 66
3DUDPHWHUVIRUEHVWILW
*LYHQ
66( 66
33 66
66 66 66 0LQHUU 66
§ ·
66 ¨
¸
© ¹
M 33 [ ˜ JDP 66 [ ˜ [ ˜ JDP 66 [ ˜ M
M
\ M
M
M
M
[ ˜ JDP 66 [ ˜ M
M
33
M
33
3
\
[
[
[
JDP 66 [M
JDP 66 [M
[M
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
D:HKDYHVKRZQWKDWIRUDQD]HRWURSHWRRFFXUWKHPYHUVXVxFXUYHPXVWKDYHDQLQWHULRU
PD[LPXPDQGPLQLPXPDVVKRZQEHORZ
§ wP ·
,IDQLQWHULRUPD[LPXP PD[LPXPERLOLQJD]HRWURSH WKHQ ¨
¸ VWDUWVRXWSRVLWLYHDWWKHLQWHULRU
© wx ¹T
§ wP ·
§ wP ·
D]HRWURSLFSRLQW ¨
¸ DQGWKHQ ¨
¸ LVQHJDWLYHSDVWWKHD]HRWURSLFSRLQW6LPLODUO\IRUDQLQWHULRU
x
w
© ¹T
© wx ¹T
§ wP ·
PLQLPXPLQWKH3YV[FXUYH PLQLPXPERLOLQJD]HRWURSH WKHQ ¨
¸ VWDUWVRXWQHJDWLYHDWWKHLQWHULRU
© wx ¹T
§ wP ·
§ wP ·
D]HRWURSLFSRLQW ¨
¸ DQGWKHQ ¨
¸ LVSRVLWLYHSDVWWKHD]HRWURSLFSRLQW6RIRUHLWKHUDPD[LPXP
© wx ¹T
© wx ¹T
§ wP ·
RUPLQLPXPERLOLQJD]HRWURSH ¨
¸ KDVRSSRVLWHVLJQVDWHDFKHQGRIWKHFRPSRVLWLRQUDQJH
© wx ¹T
E
w § G H[ ·
w
H H[
Rx x > $ x $ x @ D ¨¨
¸¸
wT © T ¹ P
wT
T
7KHUHIRUH H H[
E G H[
H H[ TS H[
F NG H[
RT
TS H[ Ÿ S H[
N N $ N $ N N N G H[
T
x x $ x $ x
N N $ N $ N
w
RT wN T P
N N GH[
RT OQ J RT ¬ª $ x x $ x x ¼º
S H[ Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
7KHUHIRUH OQ J $ x x $ x x
DQGE\V\PPHWU\
OQ J x $ x $ x x
G OQ J x o 6RWKDW J f
J f
H[S $ H[S $
x $ x $ x x $ DQG OQ J x o J x o f
J x o f
x $ x $ x
$ H[S DQG
H[S 'RHVWKLVV\VWHPKDYHDQD]HRWURSH"
(DVLHVWZD\WRGHWHUPLQHWKLVLVWRSORWPYHUVXVx(LWKHUDQLQWHULRUPD[LPXPRIPLQLPXPLQGLFDWHVDQ
D]HRWURSH$QLQWHULRUPD[LPXPDQGPLQLPXPLQGLFDWHV//(RUSHUKDSVEXWQRWOLNHO\DGRXEOH
D]HRWURSH1HLWKHULQGLFDWHVWKDWQRD]HRWURSHLVSUHVHQW6RRQHZD\LVWRFDOFXODWHDIHZSRLQWVDQG
GHWHUPLQHWKHVKDSHRIWKHFXUYH
)RUH[DPSOHDWx OQ J ˜ ˜ ˜ DQGJ DQG
OQ J ˜ ˜ ˜ DQGJ 6RWKDW P xJ PYDS xJ PYDS ˜ ˜ ˜ ˜ EDU DQGWKHUHIRUH
x ˜ J ˜ PYDS ˜ ˜ y
P x
7KLVLVVRFORVHWRxWKDWLWLVYHU\OLNHO\WKDWDQD]HRWURSHLVIRUPHG7KLVLVYHULILHGE\WKHFDOFXODWLRQRI
WKHFRPSOHWHP-x DQG x-y GLDJUDPVDUHVKRZQEHORZ
\
3
[
[
[
$QRWKHUZD\WRGHWHUPLQHZKHWKHURUQRWDQD]HRWURSHH[LVWVLVWRH[DPLQHWKHYDOXHVRI
§ wP ·
§ wP ·
DQG ¨
¨
¸
¸
© wx ¹T x © wx ¹T x ,IERWKRIWKHVHKDYHWKHVDPHVLJQWKHUHZLOOQRWEHDQD]HRWURSH7RGHWHUPLQHWKHFRPSRVLWLRQ
GHSHQGHQFHRIWKHSUHVVXUHZHXVH P xJ PYDS xJ PYDS VRWKDW
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
dP
dx
J PYDS x
d J YDS
dJ
P J PYDS x PYDS
dx
dx
d OQ J d OQ J J PYDS xJ PYDS
J PYDS xJ PYDS
dx
dx
+RZHYHUDVVKRZQEHORZ
d OQ J dx x &RQVHTXHQWO\ZHWKHQKDYH
d OQ J dx x DQG
DQG
d OQ J dx x d OQ J dx x 6LQFHWKHVLJQVRIWKHVHWZRWHUPVDUHRSSRVLWHDQD]HRWURSHIRUPV
1RZWRSURYHWKDWWKHGHULYDWLYHVRIWKHDFWLYLW\FRHIILFLHQWVDUHDVDVVXPHGDERYHZHFRPSXWHWKHPKHUH
d OQ J d ª
$ x x $ x x x ¼º $ x x x $ x x x dx
dx ¬
DQG
d OQ J d ª
$ x x $ x x x º¼ $ x x x $ x x x dx
dx ¬
7KHUHIRUH
d OQ J d OQ J $ $ DQG
dx x dx x DQG
d OQ J dx x DQG
d OQ J dx x $ $ 9DSRUSUHVVXUHRIZDWHU
OQ 3 YDS 7
[ L J L 3LYDS 7 EDU 9DSRUSUHVVXUHGDWD
3XUHZDWHU
01D&O
7.
3 EDU [+2J+2
7E
'7
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
D $WHTXLOLEULXPf i 9
fi / RUOQfi 9
OQ fi / %\WKHSUREOHPVWDWHPHQWWKHOLTXLGPL[WXUHLVLGHDO
VRfi /
xi Pi YDS DQGOQfi 9 LVJLYHQE\HTQ DV
xi Pi YDS
yi PIi9 DQGK i
yi
xi
Pi YDS
PIi9
E ,IWKHOLTXLGPL[WXUHLVQRQLGHDOWKHQfi /
xiJ i Pi YDS
yi PIi9 RUK i
yi
xi
fi 9
Ii9 7KHUHIRUH
yi P
xiJ i Pi YDS VR
J i Pi YDS
PIi9
Solutions to Chemical and Engineering Thermodynamics,5th ed
Section 10.2
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
$ % W 30( W % ·
§
¨ $
¸
W
&¹
©
3$=
JDP0( JDP0(
30( W
W 30( $ & 30( W
3$= % & W § $ % ·
¨
¸
W & ¹
©
30(. W 30(. W
JDP0(.
30(. JDP0(. 3$=
30(. W
8VLQJWKH9DQ/DDUHTXDWLRQ
[PH WHUP DOI [PHN [PH
[PHNOQ
˜ JDP0(.
[PH˜ OQ JDP0(
WHUP ˜ OQ JDP0(
DOI
EHW JDP( [ § · ˜ OQ JDP0(.
¨
¸
WHUP ¹
©
EHW
DOI
º
ª
««ª DOI˜ [ º» »
EHW ˜ [ ¼ ¼
¬¬
JDP. [ JDP( [ H[S JDP( [
EHW
º
ªª
«« > EHW ˜ [ @ º» »
DOI˜ [
¬¬
¼¼
JDP( JDP. [ H[S JDP. [
JDP. Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
[ 3 JLYHQ
3
[˜ JDP( [ ˜ 30( JDP. [ ˜ 30(. ˜ [
JDP( [ ˜ 30( 3
JDP. [ ˜ 30(. 3
] ILQG 3 [
§ ·
¨
¸
© ¹
]
L [[ ˜ L L
33 [[˜ JDP( [[ ˜ 30( JDP. [[ ˜ 30(. ˜ [[
L
L
L
L
L
33
\\ [[˜ JDP( [[ ˜
L
L
L
30( 33
L
]] [[
L
L
[[
\\
]]
[[
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
6XSSRVHZHXVHGDRQHFRQVWDQW0DUJXOHVLQVWUDGRI9DQ/DDU
[PH OQ JDP0(.
JDP( [ $ ˜ [
$
[PH
$
JDP( [ H[S JDP( [
JDP. [ $ ˜ [
[ JDP. [ H[S JDP. [
3 *LYHQ
3
[˜ JDP( [ ˜ 30( JDP. [ ˜ 30(. ˜ [
JDP( [ ˜ 30( 3
JDP. [ ˜ 30(. 3
] )LQG 3 [
[[ ˜ L L ]
L
§ ·
¨
¸
© ¹
33 [[˜ JDP( [[ ˜ 30( JDP. [[ ˜ 30(. ˜ [[
L
L
L
L
L
33
[[
\\ [[˜ JDP( [[ ˜
L
L
L
30( ]] [[
L
33
L
L
\\
]]
[[
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
DEDQGF
RT OQ J G H[
Ax RT OQ J f
A RT
RT OQ J f
RTx x
Ax x
OQ J J x
Ax
RTx
x J H[S x J H[S x
RT
RT
H[S DQGJ x H[S OLT
fi
YDS
$OVRx J P
YDS
yi
OLT
xiJ i Pi YDS f i
xiJ i Pi
P
YDS
yi P VRDWHTXLOLEULXP f i
xJ P
y
YDS
P
xiJ i Pi YDS
YDS
fi
yi P
u u u u EDU u u P
y
u u
P
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
)RUDVWDWLFFHOORSHUDWLQJDWFRQVWDQWWHPSHUDWXUHZHKDYH
J x o J f
§ wP ·
PYDS ¨
¸
© wx ¹T x o PYDS
+DYLQJQRRWKHULQIRUPDWLRQZHZLOODVVXPHWKDWWKLVV\VWHPFDQEHGHVFULEHGE\
WKHRQHFRQVWDQW0DUJXOHVHTXDWLRQ*
ex
Ax x VRWKDWRTOQJ Ax DQGRTOQJ Ax :LWKWKLVLQIRUPDWLRQDQGXVLQJWKH0$7+&$'ZRUNVKHHWEHORZZHFRQFOXGHWKDW
WKHEXEEOHSRLQWSUHVVXUHIRUDPL[WXUHRIx$ LVEDUDWR &DQGWKDWWKH
PL[WXUHKDVDQD]HRWURSHDWx$ DSUHVVXUHRIEDU
0$7+&$'ZRUNVKHHW
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
7KH*LEEV'XKHPHTXDWLRQDQGFRQVWDQWWHPSHUDWXUHDQGSUHVVXUHIRUDELQDU\PL[WXUH
UHGXFHVWR
§ w OQ J ·
§ w OQ J ·
x ¨
¸ x ¨
¸
© wx ¹T P
© wx ¹T P
RUHYHQPRUHVLPSO\
d OQ J x
x
d OQ J x
x P
E
x Pvap
vap
P
xJ P
vap
P
vap
P Pvap y vap
y P
P Pvap Ÿ y
x Pvap x Pvap
D
e
xJ P
A
RT
vap
RT
xe
P
x e
Ax
RT
Pvap
Pvap Pvap
A
Pvap
A
e RT Pvap e RT Pvap
e
A
RT
xe
vap
P
A x
RT
A
Pvap
y e
A
RT
e RT Pvap
vap
P
P
vap
A
y P
RUy
y e RT Pvap Pvap
Pvap
Pvap Pvap
RT OQ J i
H[
w Gi T
§ w OQ J i ·
R¨
¸
© wT ¹ P
H[
H[ ·
w § Hi
¨
Si ¸
¸
wT ¨ T
©
¹P
wT
H[
H[
H[
§ H i · § Si ·
Hi
¸ ¨
¸
¨
T
T ¨ wT ¸ ¨ wT ¸
©
¹P ©
¹P
P
H[
i
H
C P i C P i
T
T
T
VR
H
H[f
i
§ w OQ J if ·
RT ¨
¸
© wT ¹ P
Pvap Pvap
vap
G A x
vap
xJ P
H[
x § w OQ J ·
¨
¸
x © wx ¹T P
³
Gi
x
d OQ J 1RZLQWHJUDWLQJWKLVHTXDWLRQIURPx WRx JLYHVWKHGHVLUHGUHVXOW
x
OQ J x OQ J x
F § w OQ J ·
RU ¨
¸
© wx ¹T P
H[
Hi
T
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
SURSDQHLQQKHSWDQH
T
R
& J VRDWR & H
H[f
-PRO
H[f
VRDWR & H -PRO
6RRYHUDOODYHUDJHIRUSURSDQHLQQKHSWDQH -PRO
QKHSWDQHLQSURSDQH
T
R
& J
VRDWR & H
H[f
-PRO
VRDWR & H
H[f
-PRO
H[f
VRDWR & H -PRO
6RRYHUDOODYHUDJHIRUQKHSWDQHLQSURSDQH -PRO
1RWHGLIIHUHQFHLQVLJQV
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϭϬ͘ϮхWƌŽďůĞŵƐхWƌŽďϭϬ͘ϮͲϲϱ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϭϬ͘ϮхWƌŽďůĞŵƐхWƌŽďϭϬ͘ϮͲϲϲ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϭϬ͘ϮхWƌŽďůĞŵƐхWƌŽďϭϬ͘ϮͲϲϳ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϭϬ͘ϮхWƌŽďůĞŵƐхWƌŽďϭϬ͘ϮͲϲϴ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϭϬ͘ϮхWƌŽďůĞŵƐхWƌŽďϭϬ͘ϮͲϲϵ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϭϬ͘ϮхWƌŽďůĞŵƐхWƌŽďϭϬ͘ϮͲϳϬ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϭϬ͘ϮхWƌŽďůĞŵƐхWƌŽďϭϬ͘ϮͲϳϭ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϭϬ͘ϮхWƌŽďůĞŵƐхWƌŽďϭϬ͘ϮͲϳϮ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϭϬ͘ϮхWƌŽďůĞŵƐхWƌŽďϭϬ͘ϮͲϳϮ
dŚĞE/^dĚĂƚĂƐĞƚs>ϬϮϵǁĂƐƵƐĞĚĨŽƌƚŚĞĚĂƚĂĂƚϰϬŽ͕EŽƚĞƚŚĂƚŚĞƌĞĨŽƌĚĞŵŽŶƐƚƌĂƚŝŽŶ
ƉƵƌƉŽƐĞƐ͕ŽŶůLJĚĂƚĂĂƚƚŚĂƚƚĞŵƉĞƌĂƚƵƌĞ͕ĂŶĚŽŶůLJƚŚĞtŝůƐŽŶĂŶĚEZd>ŵŽĚĞůƐĂƌĞƵƐĞĚ͘/ƚŝƐ
ůĞĨƚƚŽƚŚĞƌĞĂĚĞƌƚŽĚŽƚŚĞŽƚŚĞƌĐĂƐĞƐ͘
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϭϬ͘ϮхWƌŽďůĞŵƐхWƌŽďϭϬ͘ϮͲϳϯ
'DWDVHW9/(
7KHUHVXOWVRIWKHFRUUHODWLRQIRUWKH:LOVRQPRGHODUH
Solutions to Chemical and Engineering Thermodynamics,5th ed
Section 10.2
7KHUHVXOWVRIWKHFRUUHODWLRQIRUWKH157/PRGHODUH
EZd>ŵŽĚĞů
Solutions to Chemical and Engineering Thermodynamics,5th ed
Section 10.2
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϭϬ͘ϮхWƌŽďůĞŵƐхWƌŽďϭϬ͘ϮͲϳϰ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϭϬ͘ϮхWƌŽďůĞŵƐхWƌŽďϭϬ͘ϮͲϳϱ
ƐĞdžĂŵƉůĞƐ͕ŽŶůLJƌĞƐƵůƚƐĨŽƌϰϬŽǁŝƚŚƚŚĞtŝůƐŽŶĂŶĚEZd>ŵŽĚĞůƐĂƌĞƐŚŽǁŶ͘ŽƚŚ
ƌĞƐƵůƚƐĂƌĞĨŽƌĚĂƚĂƐĞƚs>ϬϭϯŝŶƚŚĞE/^dĚĂƚĂďĂƐĞ͘
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϭϬ͘ϮхWƌŽďůĞŵƐхWƌŽďϭϬ͘ϮͲϳϲtŝůƐŽŶ
:LOVRQPRGHOUHVXOWV
7KHUHVXOVDUHQRWYHU\SUHWW\EHFDXVHWKHSORWJRHVWKURXJKWKHSRLQWV
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϭϬ͘ϮхWƌŽďůĞŵƐхWƌŽďϭϬ͘ϮͲϳϲEZd>
EZd>
Solutions to Chemical and Engineering Thermodynamics,5th ed
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϭϬ͘ϮхWƌŽďůĞŵƐхWƌŽďϭϬ͘ϮͲϳϳ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϭϬ͘ϮхWƌŽďůĞŵƐхWƌŽďϭϬ͘ϮͲϳϴ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϭϬ͘ϮхWƌŽďůĞŵƐхWƌŽďϭϬ͘ϮͲϳϵ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϭϬ͘ϮхWƌŽďůĞŵƐхWƌŽďϭϬ͘ϮͲϭ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϭϬ͘ϮхWƌŽďůĞŵƐхWƌŽďϭϬ͘ϮͲϰ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϭϬ͘ϮхWƌŽďůĞŵƐхWƌŽďϭϬ͘ϮͲϱ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϭϬ͘ϮхWƌŽďůĞŵƐхWƌŽďϭϬ͘ϮͲϲ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϭϬ͘ϮхWƌŽďůĞŵƐхWƌŽďϭϬ͘ϮͲϴ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϭϬ͘ϮхWƌŽďůĞŵƐхWƌŽďϭϬ͘ϮͲϵ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϭϬ͘ϮхWƌŽďůĞŵƐхWƌŽďϭϬ͘ϮͲϭϬ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϭϬ͘ϮхWƌŽďůĞŵƐхWƌŽďϭϬ͘ϮͲϭϭ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϭϬ͘ϮхWƌŽďůĞŵƐхWƌŽďϭϬ͘ϮͲϭϲ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϭϬ͘ϮхWƌŽďůĞŵƐхWƌŽďϭϬ͘ϮͲϮϬ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϭϬ͘ϮхWƌŽďůĞŵƐхWƌŽďϭϬ͘ϮͲϮϮ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϭϬ͘ϮхWƌŽďůĞŵƐхWƌŽďϭϬ͘ϮͲϮϵ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϭϬ͘ϮхWƌŽďůĞŵƐхWƌŽďϭϬ͘ϮͲϯϭ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϭϬ͘ϮхWƌŽďůĞŵƐхWƌŽďϭϬ͘ϮͲϯϳ
Section 10.2
Solutions to Chemical and Engineering Thermodynamics,5th ed
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϭϬ͘ϮхWƌŽďůĞŵƐхWƌŽďϭϬ͘ϮͲϯϴ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϭϬ͘ϮхWƌŽďůĞŵƐхWƌŽďϭϬ͘ϮͲϯϵ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϭϬ͘ϮхWƌŽďůĞŵƐхWƌŽďϭϬ͘ϮͲϰϱ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϭϬ͘ϮхWƌŽďůĞŵƐхWƌŽďϭϬ͘ϮͲϰϲ
/ŶƚŚĞĨŽůĚĞƌƐƉĞŶĨŽƌdĞdžƚŬхŚĂƉƚĞƌϭϬ͘ϮхWƌŽďůĞŵƐхWƌŽďϭϬ͘ϮͲϱϮ
Section 10.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
10.3-1
Section 10.3
Using the critical properties in the text, the program VLMU and the following interaction
parameters kC2 C3
0.001 ; kC2 C4
0.010 ; kC3 C4
0.003
I obtain the following
Bubble point, P bar
1
5
10
15
20
25
30
35
40
42
43
10.3-2
T K
.
23131
280.26
.
30817
.
32712
.
34199
354.47
.
36583
37527
.
384.56
388.31
390.31
yC2
0.3444
0.2222
.
01738
.
01460
.
01261
.
01102
0.0932
0.0835
0.0699
0.0632
0.0588
yC3
.
05907
0.6576
0.6693
0.6691
0.6642
0.6565
0.6434
0.6329
0.6133
0.6011
05919
.
yC4
0.0649
.
01202
.
01569
.
01849
0.2097
0.2333
0.2634
0.2836
0.3168
0.3357
0.3493
Using the same program and information as above, I obtain the following
Dew point, P bar
T K
xC2
xC3
xC4
1
5
10
15
20
25
30
32
35
38
40
42
43
25393
.
299.65
32527
.
342.32
35543
.
36615
.
37515
.
378.44
38300
.
387.09
389.57
39176
.
392.64
0.0038
0.0081
0.0117
0.0148
0.0180
0.0213
0.0257
0.0271
0.0296
0.0331
0.0361
0.0402
0.0434
0.2309
0.3128
0.3595
0.3924
0.4198
0.4446
0.4724
0.4806
0.4936
05103
.
05231
.
05389
.
05499
.
0.7653
0.6791
0.6289
05928
.
05623
.
05341
.
05019
.
0.4923
0.4768
0.4566
0.4408
0.4209
0.4068
See figure on following page.
10.3-3
Again we use the program VLMU and the data from Problem 10.3-1. Also, since, at 20 bar, the
bubble point of the mixture is 341.99 K and the dew point is 355.43 K, we only need to consider
temperatures between these two extremes. The results follow:
Solutions to Chemical and Engineering Thermodynamics, 5th ed
P
20 bar
T K
342.5
344.0
346.0
348.0
350.0
352.0
354.0
3550
.
x1
0.0478
0.0418
0.0352
0.0299
0.0258
0.0224
0.0197
0.0185
x2
.
05670
.
05564
.
05382
.
05170
0.4928
0.4668
0.4397
0.4259
x3
0.3852
0.4019
0.4263
0.4531
0.4814
0.5107
05407
.
05557
.
y1
0.0120
.
01069
0.0914
0.0789
0.0690
0.0609
0.0542
0.0512
Section 10.3
y2
0.6648
0.6641
0.6580
0.6464
0.6304
0.6106
05878
.
05756
.
y3
0.2143
0.2290
0.2506
0.2746
0.3007
0.3285
0.3580
0.3732
V L split
0.0303 0.9697
.
.
01265
08735
0.2635 0.7365
.
0.4095 05905
.
05612
0.4388
0.7175 0.2825
08796
.
01204
.
0.9628 0.0372
Note to instructor:
Re: Problems 10.3-1, 2 and 3
You should take time to discuss how the compositions are changing with pressure in each
of the cases above. For example, in Problem 10.3-1, at low bubble point pressures the vapor
composition is very different than the liquid, with the vapor greatly enriched in the light C2
component. However, as the pressure increases, and the critical point is approached, the vapor
composition becomes similar to that of the liquid. Analogous comments apply to the dew point
case of Problem 10.3-2.
a f
Vapor Liquid Equlibrium
40
30
P_bp
P in bar
i
P_dp
i
Bubble point
curve
20
Dew point
curve
10
0
220
240
260
280
300
320
340
360
380
400
T_bp T_dp
i
i
T in K
10.3-4
See solution to Problem 6.2. The derivation of Eqs. 10.3-8 is identical to the derivation of eqns.
6.4-29 & 30.
10.3-5 (a and b) These algorithms are incorporated into the program VLMU Examine that program to see
the algorithms used.
10.3-6 (a) The equations to be used to solve this problem are the mass balances, the equilibrium condition
(equality of fugacities) and the energy balance. Writing these equations for an open, steadyflow system, we have (for 1 mole of feed of compositions zi ).
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Section 10.3
Mass balances zi xi L yiV i 1, 2, ! , N
1 L V (summing equations above over all species)
phase equilibrium condition f i L f iV
Ÿ xi P
fi L
xi ˜ P
yi P
fi V
yi ˜ P
or
L
xi I i T , P, x
V
i
1, 2, ! , N
c
h
yi I i T , P, y
(1)
(2)
or
Ki
energy balance
L
Ii
yi
xi
Ki T , P, x
V
Ii
i
1, 2, ! , N
0
¦ b N H g ¦ b N H g ¦ b N H g
0
¦ b z H g V ¦ b x H g L¦ b y H g
or
V
i
i
in
i
i
i
out
i
i
in
i
i
Also, we have the summation conditions
¦ xi 1 and ¦ yi
i
L
out
V
i
(3)
L
(4)
i
(5)
1
(b) With the exception of Eq. (4), the other equations are the ones used in the isothermal flash
calculation (see Eqs. 10.1-14 & 15, and 10.3-4). Therefore the easiest algorithm to implement is
to use the one for the isothermal flash with an extra, outer loop which iterates on final
temperature. This is done by adding an enthalpy calculation (see eq. 10.3-8a) to the program,
and calculating the enthalpy of the feed stream, and the liquid and vapor streams. If eq. (4) is
not satisfied, the exit temperature is adjusted, and the calculation repeated.
(c) A Mathcad worksheet (10.3-6.MCD) is available for this calculation. This worksheet is also
available as the Adobe PDF file 10.3-6.pdf.
10.3-7
This problem is probably most easily solved using an equation of state, such as the Peng-Robinson
equation. Using the program VLMU, the critical properties in Table 6.6-1 and k12 0.01 (from
Table 9.4-1) we obtain the following
mole % ethane
Calculated
Measurement
in liquid
7.8
vapor mol %
6.62
P bar
39.68
vapor mol %
6.2
P bar
39.73
22.8
19.35
37.08
19.7
37.07
30.3
2581
.
3579
.
255
.
3560
.
59.0
5213
.
30.94
531
.
3213
.
89.0
8528
.
2589
.
854
.
2545
.
Thus we see, using the program VLMU, we obtain very accurate predictions in a simple manner
(though, of course, much work went into preparing the program). With the exception of the 59
mol % ethane liquid, the compositions are predicted to about 0.004 mole fraction and 0.4 bar
accuracy.
10.3-8
Again the program VLMU will be used with the critical properties in Table 6.6-1 and the value
k12 0.055 given in Table 9.4-1. The only question is how to use the program to get K values. I
have used the isothermal flash, since that is the only option that allows me to specify T and P.
One then has to choose a feed composition that assures that one is in the two-phase region (this
can be checked by a collection of bubble point or similar calculations, if desired). This
corresponds to methane compositions in the range
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Section 10.3
0.0711 d xC1 d 0.9916
In this range we find the following
x
y
K
methane 0.0711 0.9916
1395
.
benzene 0.9289 0.0084 0.009083
A
asked for in problem statement
10.3-9
We use the program VLMU, critical properties in Table 6.6-1, and a value of k12 0.003 given in
Table 9.4-1. The problem must be solved by trial-and-error. The easiest way is to start with the
bubble-point P program. This leads to the result that the bubble-point pressure is 8.47 bar.
Therefore, a lower pressure than 8.47 bar must be used to produce appreciable vapor. Likewise,
using the dew point pressure program gives a dew point pressure of 6.12 bar. Therefore, the
pressure must be between 6.12 bar and 8.47 bar to produce appreciable vapor. By trial-and-error
. 05
. 1 and
. K , P 7.234 bar V L 05
we find at T 31315
C3
nC4
Phase compressibility, Z
Feed
Liq
05
.
0.3699
05
.
0.6301
0.0268
Vapor
K
0.6301 1703
.
0.3699 05871
.
08594
.
10.3-10 Using the program VLMU, critical properties in Table 9.4-1 and the following interaction
parameters kC2 C3 0.001 ; kC2 nC4 0.010 ; kC2 iC 4 0.007 ; kC3 nC4 0.003 ;
kC3 iC4
0.007 and k nC4 iC4
0.0 we obtain the following
Component
ethane
propane
n - butane
i - butane
moles
compressibility
Feed
0.3100
0.3400
0.2100
01400
.
10
.
Liquid
.
01638
0.3372
0.3084
01907
.
0.4676
0.0548
Vapor
K
0.4384 2.677
0.3425 1016
.
01236
.
0.4008
0.0955 05008
.
05324
.
0.7952
Therefore, 53.24 mol % of tank contents will be vapor and the remaining 46.76 mol % will be
liquid. [If we believe the compressibilities, even though the liquid compressibility has some error,
then we have that 5.71% of the volume of the tank is filled with liquid, and the remaining 94.29%
is filled with vapor.]
10.3-11 The analysis is similar to that of Problem 10.3-6 except that the energy balance
0
¦ b z H g L¦ b x H g V ¦ b y H g
L
i
i
in
i
V
i
i
i
used there is now replaced with the entropy balance
0
¦ c z S h L¦ e x S j V ¦ e y S j
i
i
in
i
L
i
i
V
i
A Mathcad worksheet (10-3-11.MCD) is available for this calculation. This worksheet can be
viewed as the Adobe PDF file 10-3-11.pdf.
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Section 10.3
10.3-12 A reasonably good fit is obtained with kij = 0.11
Bubble point pressure at 273.13 K
35
30
25
20
15
10
5
0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
mole fraction of carbon dioxide
Vapor
Liquid
Bubble point pressure calculation using the Peng Robinson equation of state
Temperature = 273.13 K
T (K)
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
P (bar)
1.5559
2.0182
2.4785
2.9347
3.3918
3.8465
4.2990
4.7491
5.1970
5.6425
6.0857
6.5263
6.9645
7.4002
7.8332
8.2635
8.6911
9.1159
9.5379
9.9569
10.3729
10.7860
11.1959
11.6026
12.0062
12.4065
12.8034
13.1970
13.5872
13.9739
14.3570
14.7366
15.1126
15.4849
15.8534
16.2183
16.5794
16.9366
17.2900
17.6396
17.9852
18.3269
x1
0.0000
0.0100
0.0200
0.0300
0.0400
0.0500
0.0600
0.0700
0.0800
0.0900
0.1000
0.1100
0.1200
0.1300
0.1400
0.1500
0.1600
0.1700
0.1800
0.1900
0.2000
0.2100
0.2200
0.2300
0.2400
0.2500
0.2600
0.2700
0.2800
0.2900
0.3000
0.3100
0.3200
0.3300
0.3400
0.3500
0.3600
0.3700
0.3800
0.3900
0.4000
0.4100
y1
PV/RT(liq) PV/RT(vap)
0.0000 0.0064
0.9485
0.2237 0.0083
0.9497
0.3650 0.0102
0.9491
0.4623 0.0120
0.9478
0.5334 0.0138
0.9458
0.5876 0.0156
0.9435
0.6303 0.0173
0.9409
0.6648 0.0190
0.9382
0.6933 0.0207
0.9354
0.7172 0.0224
0.9325
0.7375 0.0240
0.9295
0.7551 0.0257
0.9264
0.7703 0.0273
0.9233
0.7837 0.0288
0.9202
0.7956 0.0304
0.9170
0.8062 0.0319
0.9139
0.8157 0.0334
0.9107
0.8242 0.0349
0.9075
0.8320 0.0363
0.9043
0.8391 0.0377
0.9011
0.8456 0.0391
0.8978
0.8515 0.0404
0.8946
0.8570 0.0418
0.8914
0.8621 0.0431
0.8882
0.8668 0.0444
0.8850
0.8712 0.0456
0.8818
0.8753 0.0469
0.8786
0.8791 0.0481
0.8754
0.8827 0.0492
0.8722
0.8861 0.0504
0.8690
0.8893 0.0515
0.8659
0.8923 0.0526
0.8627
0.8951 0.0537
0.8596
0.8978 0.0547
0.8565
0.9003 0.0557
0.8533
0.9028 0.0567
0.8503
0.9051 0.0577
0.8472
0.9073 0.0586
0.8441
0.9094 0.0596
0.8411
0.9114 0.0604
0.8381
0.9133 0.0613
0.8351
0.9151 0.0621
0.8321
Solutions to Chemical and Engineering Thermodynamics, 5th ed
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
273.13
18.6648
18.9986
19.3286
19.6545
19.9766
20.2947
20.6088
20.9191
21.2255
21.5280
21.8267
22.1216
22.4129
22.7004
22.9843
23.2648
23.5417
23.8153
24.0856
24.3528
24.6169
24.8780
25.1364
25.3921
25.6452
25.8960
26.1446
26.3912
26.6360
26.8791
27.1208
27.3612
27.6007
27.8394
28.0751
28.3156
28.5511
28.7918
29.0282
29.2704
29.5088
29.7537
29.9979
30.2444
30.4934
30.7453
31.0006
31.2596
31.5228
31.7907
32.0637
32.3424
32.6272
32.9189
33.2178
33.5246
33.8398
34.1638
34.4965
0.4200
0.4300
0.4400
0.4500
0.4600
0.4700
0.4800
0.4900
0.5000
0.5100
0.5200
0.5300
0.5400
0.5500
0.5600
0.5700
0.5800
0.5900
0.6000
0.6100
0.6200
0.6300
0.6400
0.6500
0.6600
0.6700
0.6800
0.6900
0.7000
0.7100
0.7200
0.7300
0.7400
0.7500
0.7600
0.7700
0.7800
0.7900
0.8000
0.8100
0.8200
0.8300
0.8400
0.8500
0.8600
0.8700
0.8800
0.8900
0.9000
0.9100
0.9200
0.9300
0.9400
0.9500
0.9600
0.9700
0.9800
0.9900
1.0000
0.9169
0.9186
0.9202
0.9218
0.9234
0.9248
0.9263
0.9276
0.9290
0.9303
0.9316
0.9328
0.9340
0.9352
0.9363
0.9375
0.9386
0.9396
0.9407
0.9418
0.9428
0.9438
0.9448
0.9458
0.9468
0.9478
0.9488
0.9498
0.9508
0.9518
0.9528
0.9538
0.9548
0.9558
0.9568
0.9578
0.9589
0.9600
0.9611
0.9622
0.9634
0.9646
0.9659
0.9672
0.9685
0.9699
0.9714
0.9730
0.9746
0.9764
0.9782
0.9802
0.9823
0.9846
0.9872
0.9899
0.9929
0.9963
1.0000
0.0630
0.0637
0.0645
0.0653
0.0660
0.0667
0.0673
0.0680
0.0686
0.0692
0.0698
0.0703
0.0708
0.0714
0.0718
0.0723
0.0727
0.0732
0.0736
0.0740
0.0743
0.0747
0.0750
0.0753
0.0756
0.0758
0.0761
0.0763
0.0765
0.0767
0.0769
0.0771
0.0772
0.0773
0.0774
0.0775
0.0776
0.0776
0.0777
0.0777
0.0777
0.0777
0.0776
0.0776
0.0775
0.0774
0.0772
0.0771
0.0769
0.0766
0.0764
0.0761
0.0758
0.0754
0.0750
0.0746
0.0742
0.0736
0.0731
Section 10.3
0.8291
0.8262
0.8233
0.8204
0.8175
0.8147
0.8119
0.8091
0.8063
0.8035
0.8008
0.7981
0.7955
0.7928
0.7902
0.7876
0.7850
0.7825
0.7800
0.7775
0.7750
0.7725
0.7701
0.7677
0.7653
0.7629
0.7606
0.7583
0.7559
0.7536
0.7514
0.7491
0.7468
0.7446
0.7424
0.7401
0.7379
0.7357
0.7335
0.7312
0.7291
0.7268
0.7246
0.7224
0.7202
0.7180
0.7158
0.7136
0.7114
0.7092
0.7070
0.7049
0.7027
0.7006
0.6986
0.6966
0.6947
0.6930
0.6915
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Section 10.3
Points are experimental data, and lines are correlation using the Peng-Robinson equation of state
with k12 = 0.11
40
25
0.9
35
0.8
30
20
0.7
0.6
15
20
15
k2
k1
P. bar
25
0.5
0.4
10
0.3
10
0.2
5
5
0.1
0
0
0
0.2
0.4
0.6
0.8
1
0
0
0.2
x 1, y 1
0.4
0.6
0.8
1
0
0.2
0.4
x1
0.6
0.8
1
x1
10.3-13 Clearly, because high pressures and a simple carbon dioxide + hydrocarbon system is involved,
and equation of state, such as the Peng-Robinson equation, should be used. Using the program
VLMU with kCO 2 nC6 011
. (Table 9.4-1) results in no solution at 140 bar and 75qC. However,
trying the bubble point and dew point pressure programs we obtain the following results (at
T 34815
. K ).
bubble point
P, bar
12.96
37.93
64.15
89.14
91.36
93.49
95.52
97.43
99.17
xCO 2
0.1
0.3
0.5
0.7
0.72
0.74
0.76
0.78
0.80
yCO 2
0.001
0.05
0.1
0.3
0.4
0.5
0.6
0.7
0.73
0.75
0.78
0.80
0.82
Program doesn’t
converge at higher
CO2 concentrations
dew point
P, bar
1.21
1.28
1.35
1.76
2.07
2.51
3.17
4.31
4.83
5.25
6.03
6.70
7.56
yCO 2
0.9
0.92
0.93
0.94
0.945
0.947
0.949
0.9495
dew point
P, bar
15.06
20.25
24.64
32.02
38.56
42.62
49.39
52.75
Program doesn’t
converge at higher
CO2 concentrations
120
100
80
bubble curve
60
dew curve
40
20
0
0
0.2
0.4
0.6
0.8
1
Since the program doesn’t converge at higher concentrations of CO2 along either the bubble point
or dew point curves, we have to make an estimate of the CO2 concentration based on the data
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Section 10.3
above. There are two possibilities: (1) The CO2 saturation of the liquid at 140 bar is in the
retrograde region at somewhere between xCO 2 of 0.8 and 0.95 [Note, simple equation of state
programs, such as VLMU typically do not converge in the retrograde region, and more sophisticated
algorithms and numerical methods must be used]; (2) at 140 bar only the vapor exists, that is, all the
hexane vaporizes.
Note the enormous solubility of carbon dioxide in hexane and, indeed, in reservoir crude! That is
why carbon dioxide has been used in enhanced oil recovery (crude oil swells so more is recovered,
and viscosity drops so the trapped oil in the earth matrix flows more easily.)
10.3-14 In the folder Aspen for Textbook>Chapter 10.3>Problems>Prob 10.3-14
10.3-15 In the folder Aspen for Textbook>Chapter 10.3>Problems>Prob 10.3-15
10.3-16 In the folder Aspen for Textbook>Chapter 10.3>Problems>Prob 10.3-16
10.3-17 In the folder Aspen for Textbook>Chapter 10.3>Problems>Prob 10.3-17
10.3-18 In the folder Aspen for Textbook>Chapter 10.3>Problems>Prob 10.3-18
10.3-19 In the folder Aspen for Textbook>Chapter 10.3>Problems>Prob 10.3-19
10.3-20 In the folder Aspen for Textbook>Chapter 10.3>Problems>Prob 10.3-20
10.3-21 In the folder Aspen for Textbook>Chapter 10.3>Problems>Prob 10.3-21
10.3-22 In the folder Aspen for Textbook>Chapter 10.3>Problems>Prob 10.3-22
10.3-23 In the folder Aspen for Textbook>Chapter 10.3>Problems>Prob 10.3-23
10.3-24 In the folder Aspen for Textbook>Chapter 10.3>Problems>Prob 10.3-24
10.3-25 In the folder Aspen for Textbook>Chapter 10.3>Problems>Prob 10.3-25
10.3-26 In the folder Aspen for Textbook>Chapter 10.3>Problems>Prob 10.3-26
10.3-27 In the folder Aspen for Textbook>Chapter 10.3>Problems>Prob 10.3-27
10.3-28 In the folder Aspen for Textbook>Chapter 10.3>Problems>Prob 10.3-28
10.3-29 In the folder Aspen for Textbook>Chapter 10.3>Problems>Prob 10.3-29
10.3-30 In the folder Aspen for Textbook>Chapter 10.3>Problems>Prob 10.3-30
10.3-31 In the folder Aspen for Textbook>Chapter 10.3>Problems>Prob 10.3-31
10.3-32 In the folder Aspen for Textbook>Chapter 10.3>Problems>Prob 10.3-32
10.3-33 In the folder Aspen for Textbook>Chapter 10.3>Problems>Prob 10.3-33
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
Section 11.1
11.1
11.1-1 (a) H V data are obtained from the Superheated Steam Tables. Also
H L T , P 01
. MPa | H L T , any pressure since, at low and moderate pressures, the liquid
enthalpy is independent of pressure to an excellent approximation. Therefore, H L T , P is
taken along the saturation line, and then extrapolated to high temperature, as shown below. 11.1-2
General: The starting point for solving this problem is
f NL2 f NV2 Ÿ xN 2 J N 2 f NL2 yN 2 P
PN 2
where PN 2 is the partial pressure of nitrogen in the gas phase. Also, f NL2
"liquid" nitrogen, is 1000 bar according to the problem statement.
(a) Ideal solution
J N 2 1 ; also PN 2 1 bar (problem statement)
thus
xN 2
(b) Non-ideal solution
PN 2
f NL2
1
1000
0.001
fugacity of pure
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
c
Here ln J N 2
0526
.
1 xN 2
Section 11.1
h . Since, from the ideal solution case above, x
2
N2
is quite small
(and will be even smaller here), it is reasonable to assume that 1 xN 2 | 1 , ln J N 2
J N2
, and
0526
.
. Thus
16922
.
PN 2
xN 2
J N 2 f NL2
1
.
u 1000
16922
. u 103
0591
The Henry's Law Constant is the constant in the expression
Hi xi
thus
1 bar
xN 2
HN 2
f i V , here, HN 2 xN 2
1 bar
1692 bar.
Also, since HN 2 { J N 2 f NL2 and V ex
0 , we have that the activity coefficient J N 2 is
independent of pressure. Thus, the only pressure variation of the Henry's law constant will be
through the Poynting pressure correction to the "liquid" phase of pure nitrogen.
11.1-3 (a) To compute the ideal solubility, we will use Shair's correlation, Fig. 11.1-1 first we need to
compute the reduced temperature
29815
.
Tr
1564
.
190.6
thus,
fL
# 365
. and f L
PC Fig. 8.3-1
365
. u 46.0 bar
167.9 bar
Therefore at 1 atm partial pressure of methane
PCH 4
ideal
xCH
4
(b) To
compute
J CH 4
L
f CH
4
the
L
1013
.
xCH 4 f CH
1013
.
bar
167.9 bar
activity
4
CH 4 - benzene: J CH 4
CH 4 - C6 H12: J CH 4
6.03 u 103 (all solvents)
coefficients,
we
use
L
xCH 4 J CH 4 f CH
4
1013
.
bar
or
6.03 u 103 xCH 4 The following results are obtained
2.91 CH 4 - CCl4: J CH 4
213
.
CH 4 - C6 H14: J CH 4
211
.
191
.
McDaniel
142
.
Guerry
(c) The regular solution model gives, for the CH 4 - C6 H14 mixture
J CH4
R|V cG G h I
expS
RT
|T
L
CH 4
2
CH 4
C6H14
U| expR52 u 568
. 7.3 U
. ,
V| ST 1987
V 126
u 298.2 W
.
W
2
2
C6H14
Since I C6 H14 | 1 . Note: CH 4 parameters from Table 11.1-1 C6 H14 parameters from Table 9.61.
This result is not in agreement with either set of data, but is distinctly closer to Guerry's
result!
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
f i G Ÿ xiJ i f i L
11.1-4 (a) Start from f i L
yi P
FfI
H PK
Section 11.1
i
Determination of nitrogen properties:
Tr N
From
Fig.
37315
. K
126.2 K
2
2.96 ; Pr N
F f I #1.
H PK
7.4-1
2
75 bar
33.94 bar
2.21
using
Shair's
Now
correlation
(Fig.
11.1-1)
N2
f NL2 P
1013
.
bar
61
. ; however, at 75 bar we have
PC
f NL2 P
f NL2 P
75 bar
PC
1013
.
PC
exp
R|V
S|
T
L
N2
.
P 1013
RT
6.6 Ÿ f NL2 P
61
. exp 0.0784
U|
V|
W
75 bar
224 bar
Determination of benzene properties:
Tr C H
6
6
From Fig. 9.8-1
. K
37315
. K
5621
0.664 and Pr C H
6
6
75 bar
48.96 bar
1532
.
f
# 0.23 and, for the liquid, we need the vapor pressure. From the
P vapor
bar at T 801
Chemical Engineers Handbook we have P vap 1013
.
. q C , and P vap 2.026 bar
A
at T 1038
.
. q C using ln P vap
bar.
B as the interpolating formula, we find P vap 1823
T
f
From Fig. 7.4-1, we have (with a little extrapolation) that
| 0.96 . [Along the saturation
P sat
line]. Thus
F I
H K
f CL6 H 6
PCvap
6H 6
.
UV 2159
R 89 u 73173
F f I expR|SV c P P h U|V 1823
.
bar
H P K |T RT |W . u 0.96 u expST8314
. u 37315
. W
L
C6 H 6
vap
sat
Thus, the equations to be solved are
xN 2 J N 2 224 bar
yN 2 75 bar
xC6 H 6 J C6 H 6 2.159 bar
xN 2 xC6 H 6
1
yN 2 yC6 H 6
1
together with
J N2
and
yC6 H 6 75 bar
R|V cG G h I
expS
RT
|T
L
N2
N2
C6 H 6
2 2
C6 H 6
U|
V|
W
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
Section 11.1
R|V cG G h I U|
expS
V|
RT
|T
W
L
C6 H 6
J C6 H 6
N2
C6 H 6
2 2
N2
where the N 2 parameters are gotten from Table 11.1-1 and the benzene parameters from Table
9.6-1. Because of the nonlinear nature of the equations (due to the composition dependence of
the activity coefficient), this problem is best solved by trial and error. I chose xN 2 0.0 as the
initial guess.
Solution obtained was:
xN 2
0.047 and yN 2
0883
.
xN 2
0.045 and yN 2
0.944
Measured values:
(b) At 100 bar the calculation is similar, however the numbers are a little different. I find
f NL2
f V
| 0175
.
;
| 6.76 and f CL6 H 6 | 2.290
P C6 H 6
PC
Solution obtained was:
xN 2
0.061 and
yN 2
0874
.
Measurement yields
xN2
0.0595 y N 2
0.968
In both cases the liquid compositions are in better agreement with experiment than the vapor
compositions!
Note: I have found that some students try to make a large extrapolation of the vapor pressure,
rather than using Shair's correlation ... it is a large extrapolation here, since the nitrogen critical
temperature is 126.2 K . If we extrapolate the low temperature vapor pressure, and make the
(small) fugacity coefficient correction, we obtain f NL2 1195 bar compared to 224 bar here at
75 bar. This leads to xN 2
0.0082 , compared to 0.047 calculated here and 0.045 observed
experimentally.
Moral: Use Shair's correlation instead of making large extrapolations of the vapor pressure.
11.1-5 (a) Suppose a small amount of liquid, 'N is, vaporized, then there are y'N moles of dissolved
gas in the vapor, and x 'x N 'N
moles of dissolved gas left in the liquid we are
interested in a different distillation, i.e., the case where 'x and 'N will be very small. Thus
x 'x N 'N
xN x'N N'x 'x 'N
| xN x'N N'x
Now writing a mass (mole) balance on the dissolved gas yields
xN
Thus
'x
'N
y'N x 'x N 'N | y'N xN x'N N'x
yx
dx
and taking the limit as 'N o 0 yields
N
dN
yx
N
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
(b) Using y
ax , N f
0
0
z
Hx
P x
dx
Hx
yields
dN
P
and x, N yields
x
x0
d ln x
N
F
H
x HP
N
P
Section 11.1
I or d ln x
K d ln N
z
HP N
d ln N or ln x ln x0
N0
P
HP
. Now integrating between
P
HP
ln N ln N 0
P
which can be rewritten as
x
x0
FG N IJ
HN K
( H P) P
or
0
N
N0
FG x IJ
Hx K
P HP
0
x
N
0.01 1 266 0.983 . Thus, only 1.7% of initial number of moles of
0.01 we have
x0
N0
liquid need be vaporized in a differential distillation (i.e., no violent boiling, otherwise
equilibrium will not be obtained) to remove 99% of the CO2 .
x
N
For
0.0001
0.0001 1 266 0.966
x0
N0
(c) For
Thus only 3.4% of initial number of moles of liquid need be vaporized to remove 99.99% of the
CO2 .
11.1-6
I used the bubble point option of program VLMU treating the liquid mole fraction of CO2 as an
adjustable parameter, until a CO2 partial pressure of 1.013 bar in the vapor phase was obtained.
The results appear below:
xCO 2
0.006
0.020
0.022
0.0221
0.0222
Ptot
0.31
0.96
105
.
105
.
106
.
yCO 2
PCO 2 yCO 2 ˜ Ptot
08680
.
0.269 bar
0.9562
0.918
0.960
1008
.
0.9601
10081
.
0.9603
1018
.
Therefore, the predicted solubility is xCO 2 | 0.02215 .
11.1-7 (a) This problem is treated in the same manner as the previous problem. The results are given
below:
k12 0
xCO 2
Ptot
yCO 2
0.015 141
.
0.6729
0.016 148
.
0.6869
0.0159 147
.
0.6856
PCO 2
yCO 2 ˜ Ptot
0.9488
101662
.
10078
.
Therefore, the predicted solubility is | 0.01595
experimental value of xCO 2
0.00328
xCO 2 . This is considerably higher than the
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
Section 11.1
(b) k12 0.2
xCO 2
Ptot
yCO 2
PCO 2 yCO 2 ˜ Ptot
0.015
4.94 0.9011
4.4514
0.003
136
0.6566
089298
.
.
0.0033 145
0.6775
0.9824
.
0.0034 148
0.6839
10121
.
.
In this case xCO 2 | 0.0034 which is quite close to the experimental value of 0.00328. This
illustrates the importance of the binary interaction parameter k12 .
11.1-8 (a) N
m
. K
u 27315
N G ˜ 8.314 u 105 bar
mol K
3
PV
RT
V
.
1013
bar
D u 1013
.
1 m3U W g m3
N
W
8.314 u 105 u 27315
.
18.015 g mol
NG
LM1 N OP
N NQ
1
1 NNWG
NG
NG NW
x
D
1
W
G
. KO
U O
L U u 8.314 u 10 u 27315
LM1 1244
.
u 10
x M1 P
.
D u 1013
D PQ
N 18.015
Q N
1
5
3
W
1
W
volume gas
; for simplicity, assume 1 m3 liquid
volume liquid
(b) L
N G ˜ 8.314 u 105 T
; NG
1013
.
V
1 m3
L
.
V ˜ 1013
8.314 u 105 T
LM1 N OP LM1 U u 8.314 u 10 T OP
.
Q
N N Q N 18.015 L u 1013
LM1 4.555 u 10 T U OP
L Q
N
1
5
W
x
1013
. L
8.314 u 105 T
1
W
G
6
(c) S
V
11.1-9
. Ku
N G ˜ 8.314 u 105 bar m3 mol K u 27315
106 cm3 m3
1013
.
bar
1013
.
1
; NW
8.314 u 27315
.
18.015
1.013
S u 8314
NG
. u 27315
.
NG
Su
x
NG NW
(d) xKH
1013
Ÿx
.
(e) N G
S0
; NW
MS
NG
NG NW
x
1
W
S u1.013
181.05
. u 27315
.
8314
S u 1013
.
S u 1013
1260599
.
.
1013
.
KH
100
18.015
S0
MS
18.015S0
18.015S0 100 MS
S0
18100
.015
MS
Condition for equilibrium as the solubility limit is f i L
'G eq
0
RT ln
fi L
fi V
S
S 1244
RT ln
a
f RT ln x RT ln
xisol f i L T , P, xi 1
f i V T , P, yi 1
a
f
f i V which implies
sol
i
a
f
f i L T , P, xi 1
f i T , P 1013
.
bar
V
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
Section 11.1
since the fugacity of the species in the vapor phase as this low pressure is given by the Lewis-Randall rule,
and just equal to the fugacity of the pure species at 1.013 bar. Now the last term on the right of this
equation is equal to the Gibbs free energy change of transferring the gas from the ideal gas state to a
solution of unit mole fraction. Therefore
f L T , P, xi 1
RT ln xisol 'G
'G eq 0 RT ln xisol RT ln V i
.
bar
f i T , P 1013
a
f
where 'G is the free energy changed asked for in the problem statement . Therefore 'G
A
ln xisol
w GT
wT
B
C ln T DT ET 2 , then 'G
T
G
H
2 wT
'
H
T
Ÿ
wT
T2
w
B
'H
R A C ln T DT ET 2
wT
T
'H 'G
H G
'S
G H TS Ÿ S
T
T
RS LM
T N
'S
CP
LM
N
RT A OP
Q
RT ln xisol , but
wG
B
C ln T DT ET 2 , also T1
wT
T
OPUVcT h R B CT DT 2 ET
QW
2
2
H or
3
R
B CT DT 2 2 ET 3 AT B CT ln T DT 2 ET 3
T
R
AT CT ln T CT 2 DT 2 3ET 3
T
R A C ln T C 2 DT 3ET 2
F wH I Ÿ 'C F w' H I R C 2 DT 6ET
H wT K
H wT K
2
P
P
11.1-10 a) We expect the solubility of bromine in water to be quite low, so that the molar concentration of
water will be essentially unchanged. Then the change in free energy for the bromine dissolution
process is
aq
'G
aq
f H O( M)
f Br2
M
554
.
RT ln
RT ln aq 2
f Br2 M 554
M 554
.
.
f
( M 0)
H 2O
M
147 M
5555
5555
.
. M
RT ln
RT ln
M 5555
M 5555
28.4
5555
.
.
.
where 55.55 is the molar concentration of water. Assuming M will be very small, the second term
can be neglected and we obtain
'G
M
147 M
RT ln
M 5555
.
28.4
The results are plotted below.
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
Section 11.1
1
0.307734
0
1
G
i
2
3
3.167164
4
0
0.01
b)
0.05
0.1
M
i
0.15
0.2
0.2
At saturation, the fugacity of liquid bromine equals that of bromine in solution. Therefore
aq
f Br2
f BrL2 ; or 147 M
28.4; so M
01932
.
Either by using this value in the equation above, or using the graph above, we see that the
Free energy change is zero if the liquid is saturated with bromine.
Z 0.225 TB 194.7 K
11.1-11 CO2
TC 304.2 K PC 7376
. bar
Toluene
591.7
41.13
0.257
383.8
From Prausnitz-Shair Correlation
fL
Tr 0.98 ;
# 0.6 . Assume very little toluene in vapor yCO 2 ~ 1
PC
xCO 2
a
L
T, P
J CO 2 f CO
2
At 10 bar Tr
0.98 , Pr
a f
1013
.
bar f expn V
10 1013
.
bar RTs
yCO 2 P f P
CO 2
10
7376
.
0136
.
f
Figure 7.4-1 | 0.96
P
L
f CO
0.6 u 73.76 44.26 bar
2
Now calculate the Poynting correction
55 cc mol u 8.987 bar
1 m3
u
exp
298.15 K u 8.314 u 105 bar m3 mol cc 106 cc
LM
N
LM 55. u 8.987 u 10 OP exp 0.02 102
.
N 2.9816 u 10 u 8.314 u 10 Q
1
exp
2
1
So ideal “Prausnitz-Shair” solubility
10 u 0.96
xCO2
0.213
44.26 u 102
.
OP
Q
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
Section 11.1
Now consider solution nonideality
Table 9.6-1
Table 11.1-1
G CO 2 6.0
G T 8.9
VT 107
VCO 2 55
ln J CO2
IT
a55 cc molfI 6.0 8.9 cal cc
2
2
T
0.781I 2T
8.314 J mol K u 29815
. K u 0.239 cal J
VT xT
VT xT VCO2 xCO2
c
107 1 xCO2
c
107 1 xCO2
c
h
h
107 1 xCO2 55xCO2
h
c
107 1 xCO2
107 107 55 xCO2
h
107 52 xCO2
So to find the nonideal solubility must solve
0.213
0.213
xCO2
exp 0.781I 2T
exp 0.781 107 1 xCO 107 52 xCO
c
{
which has the solution xCO2
2
hc
h}
2
2
(using MATHCAD)
0102
.
From Program VLMU with kCO 2 T
0.0
(using flash with equimolar feed)
Pressure
10.05 bar
x
y
0.2062
0.9952
0.7938
0.0048
yCO 2 u P 0.9952 u 10.05 ~ 10 bar
CO2
T
Note PCO 2
kCO2 T
010
.
CO2
T
PCO 2 ~ 10 bar
x
0.1174
0.8826
y
0.9949
0.0051
Which is reasonably close to the Prausnitz-Shair correlation result, especially given the difference in
the methods.
11.1-12 (a)
From the Steam Tables T 25q C , P 3169
.
kPa
(b) 1 atm = 1.-10 kPa
3169
.
yW
0.0313 bar ; remainder is oxygen and nitrogen.
1013
.
Initial partial pressure of N 2 0.79 u 1013
. 32
.
77.5 kPa
Initial partial pressure of O2 0.21 u 1013
. 32
.
20.6 kPa
Mole fraction N 2 in water xN 2 HN 2 PN 2
xN 2
77.5 kPa 100 kPa bar
8.48 u 104 bar mole fraction
0.775
8.48 u 104
0.0914 u 104 914
. u 106
Mole fraction O2 in water
20.6 100 bar
xO 2
4.35 u 104 bar mole fraction
20.6
100 u 4.35 u 104
4.74 u 106
11.1-13 a) Since the vapor pressures are low, we will assume the fugacity coefficients at saturation are
unity. Then the fugacity of methyl acetate (MA) and methanol (M) is
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
2 · vap
2
§ A
x MA exp ¨
1 x MA ¸ PMA
x MA exp 1.06 1 x MA 1.1260 bar
© RT
¹
2·
2
§ A
x M exp ¨
1 x M ¸ PMvap x M exp 1.06 1 x M 0.8465 bar
© RT
¹
vap
x MA J MA x MA PMA
fMA
fM
Section 11.1
x M J M x M PMvap
1.5
1
fugma
fugm
0.5
0
0
0.2
0.4
0.6
0.8
1
xx
methyl acetate mole fraction
b) The partial pressure of species i is gotten from
P
P
fi x i J i x i Pivap yi P Pi so that i J i x i Pivap and lim i
x o0 x i
xi
lim J i x i Pivap
x o0
§ A ·
Pivap exp ¨
¸
© RT ¹
§ A ·
Pivap exp ¨
¸ , so that the hypothetical standard state of a component in a mixture
© RT ¹
§ A ·
that obeys the one0constant Margules equation is Pivap exp ¨
¸ , while the actual pure component
© RT ¹
Therefore, Hi
fugacity is simply Pivap .
xair
11.1-14
xair
Pair
H
Henry’s
law
constant
Ÿ xH 2 O
mole fraction of air in water (liquid)
01333
.
kPa u 103 Pa kPa
# 0.3 u 107
4.3 u 104 bar u 105 Pa bar
1 (Do not have to consider air trapped in water)
At equilibrium
Ÿ xH 2 O PHvap
2O
f HL2 O
f HS2 O
f HV2 O ;
PHsub
2O
yH 2 O P
since xH 2 O
PHvap
2O
1
[Note: Because of the low pressures involved, we have neglected f P terms and Poynting
corrections.]
For comparison, in an air-free measurement, we have
PHvap
PHsub
PTP
2O
2O
where PTP is the true triple point temperature.
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
Since PHvap
2O
Section 11.1
has to be satisfied in both cases, we would obtain the same triple point
PHsub
2O
temperature in both the air-free experiment, and the measurement with air.
In the air-free experiment we measure P PH 2 O and get the triple point pressure. In the experiment
with air we measure P
PH 2 O Pair and, mistakenly, assume this is the triple point pressure;
actually PH 2 O is the triple point pressure. The error, 'P , is equal to the partial pressure of air; here
0.1333 kPa. Thus, we have
% error
'P
u 100
PHTP2 O
01333
.
u 100
0.6113
218%
.
[Note: From the Steam Tables, triple point pressure is 0.6113 kPa.]
11.1-15 The fugacity of a species 1 at infinite dilution in species 2 is, from eq. 9.4-10
b Pº
ª L
Z 1 2 2 »
ª
f1L
b1 L
a2
b1 º « 2
§ L b2P ·
RT
lim x1o0 ln
Z2 1 ln ¨ Z2 »
« 2 » ln «
¸
RT ¹ 2 2b 2 RT ¬
b 2 ¼ « ZL 1 2 b 2 P »
x1P1vap b 2
©
2
¬
RT ¼
so that
fL
Lim x1o0 1
H1
x1
we have that
ª
«
b
P1vap exp « 1
« b2
H1
b P ºº
ª L
Z 1 2 2 »»
ª
a2
b1 º « 2
RT
»»
« 2 » ln «
RT ¹ 2 2b 2 RT ¬
b 2 ¼ « ZL 1 2 b 2 P » »
2
¬
RT ¼ »¼
b P·
§
Z2L 1 ln ¨ Z2L 2 ¸ ©
¬«
11.1-16 We start with eqns. (9.9-11, 12 and 13) and have the following at infinite dilution of species 1
§ wND ·
a1
ln J f
*1
¨
¸
C
© wN1 ¹T,N ,x o0 b1RT
2
1
1 § wN 2Q ·
¨
¸
N ©¨ wN1 ¹¸T,N ,x o0
2 1
D x o0
1
a ·
§
2 ¨ b2 2 ¸
RT
©
¹
a2
b 2 RT
§ wNb ·
¨
¸
© N1 ¹T,N 2 ,x1o0
§
b 2 RT
a ·
b 22
a1
ln J f ·
§
2 ¨ b 2 2 ¸ RT
*1 ¸¸
¨¨1 b 2 RT a 2 ©
RT ¹
b 2 a 2 RT © b1RT
C ¹
2b 2 §
b 22
a RT ln J1f ·
¨¨ RT 1 ¸
b 2 RT a 2 ©
b1
C* ¸¹
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
1 § wN 2a ·
¨
¸
N ¨© wN1 ¸¹T,N ,x o0
2
1
Section 11.1
§
ª a RT ln J1f º
a 2 ª b 2 RT
b 22
a
ln J f · º
2 b 2 RT a 2 RT
«
¨¨1 1 *1 ¸¸ » b 2 « 1 »
b 2 «¬ b 2 RT a 2
b 2 RT a 2 © b1RT
C ¹ »¼
C* ¼
¬ b1
ª a1 RT ln J1f º
§
a2 ª
b 22
a
ln J f · º
« 2b 2 RT RT
¨¨1 1 *1 ¸¸ » b 2 « »
b 2 «¬
b 2 RT a 2 © b1RT
C* ¼
C ¹ »¼
¬ b1
These expressions are substituted into eq. (9.9-11) to obtain
ª 1 § wNb ·
º
b P·
§
ZL2 1 ln ¨ ZL2 2 ¸
« ¨
»
¸
RT ¹
©
« b1 © wN1 ¹T,N 2 ,x1o0
»
«
»
f1
b2P ·»
H1 Lim x1o0
P«
§ L
­ 1 1 § wN 2a ·
½ ¨ Z2 1 2
x1
a2
1 § wNb ·
°
°
«
RT ¸ »
ln ¨
¨
¸
¸»
®
¾
¨
¸
« 2 2b RT a N ¨ wN ¸
b
w
b
N
2P ¸
1 ¹T,N ,x o0
2©
1 ¹T,N 2 ,x1 o0 ° ¨ ZL 1 2
2
°¯ 2 ©
«
»
¿
2 1
2
©
RT ¹ ¼
¬
Solutions to Chemical and Engineering Thermodynamics, 5th ed.
Section 11.1
11.1-17
I will use the Prausnitz-Shair correlation
From Table 9.6-1
From Table 11.3-1
From Table 6.6-1
Tr ( 25 273.15)
Tr
TcO2
From Fig. 11.1-1
) ( x) GBr 11.5
VBr 51
GO2 4.0
VO2 33.0
TcO2 154.6
PcO2 50.46
1.929
fO2 5.6˜ PcO2
f/Pc=5.6 so that
x˜ VO2
)B( x) 1 ) ( x)
x˜ VO2 ( 1 x) ˜ VBr
ª ª¬ VO2˜ ) ( x) 2˜ 4.184˜ GO2 GBr 2º¼ º
»
8.314˜ 298.15
¬
¼
J ( x) exp «
Initial guess
given
x 0.002
x˜ J ( x) ˜ fO2
1
x find ( x)
x
3
3.539 u 10
fO2
282.576
Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
D )URPHTQ ZHKDYHDWHTXLOLEULXPWKDW xiI J iI x I
xiII J iII x II IRUDOOVSHFLHVi
d i
DQGIURPUHJXODUVROXWLRQWKHRU\ZHKDYH
DQG 7KHFULWLFDOVROXWLRQWHPSHUDWXUHLVIRXQGIURP
FG w G IJ
H wx K
FG w G IJ
H wx K
,0
f
IJ
K
TP
FG w G IJ
H wx K
a
,0
a
xV xV H[
a
f
xxV V G G ax V x V f
TP
I I a
f
f
V V G G xV xV a
E 7RILQGWKHXSSHUFRQVROXWHWHPSHUDWXUHWKDWLV TCPD[ ZHXVH
FG wT IJ
H wx K
RU
C
P
FG wT IJ V V aG G f RSx x x x aV V f UV H wx K a x V x V f T
ax V x V f W
FOHDUO\WKHWHUP k p PXVWEH]HURDWWKHXSSHUFRQVROXWHWHPSHUDWXUH7KXVZHREWDLQ
R
C
f
ZHREWDLQ
RTC
f
V V G G 7KXVVHWWLQJ
TP
FG
H
RT
w G H[
DQG
xx
wx T P
FG w G IJ
H wx K
T P
a
T P
%\WDNLQJGHULYDWLYHVZHILQGWKDW
,0
xG T P x G T P RT x OQ x x OQ x FG w G IJ
H wx K
ZKHUH
FG w G IJ
H wx K
H[
T P
G ,0
d i
P
Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
x
1RWH WKDW WKLV FRPSRVLWLRQ GHSHQGV RQO\ RQ WKH PRODU YROXPHV RI VSHFLHV DQG QRW WKHLU
DV VKRXOG EH H[SHFWHG
VROXELOLW\ SDUDPHWHUV 1RWH DOVR WKDW DV V o x o 6XEVWLWXWLQJWKLVUHVXOWLQWRHTQ
f
cV V V V h V aV V f
x
x
a
2QO\WKHQHJDWLYHVROXWLRQLVUHDOLVWLF7KXVWKHFRPSRVLWLRQDWWKHXSSHUFRQVROXWHSRLQWLV
V
V V V V r
V V V V RTC
DERYH\LHOGV
a f {a
fd
i V V } aV V f {dV V V V i aV V f}
V V G G V V V V V V F wd N G iI
GH wN JK
H[
Axx ZHKDYHIURP GiH[
RT OQ J i
DQG
, i
,, i
,, i
,,
i
DQGZHKDYHWZRHTXDWLRQVIRUWKHWZRXQNQRZQV xiI xiII E /LPLWRIVWDELOLW\FULWHULRQLV
G
FG w G IJ
H wx K
ZKHUH
TP
a
f
xG x G RT x OQ x x OQ x Axx G ,0
FG w G IJ
H wx K
VHWWLQJ
T P N j zi
,,
,,
i
,
i
,
i
Ax DQG
R| Ac x h U|
R x h U|
V| S| RT V| x H[S|S| AcRT
W
W
T
T
R U
R U
c x h H[S|S| AxRT |V| c x h H[S|S| AxRT |V| T W
T W
xi, H[S
7KDW RT OQ J Ax ,
D 7KH HTXLOLEULXP FXUYH LV RQH IRU ZKLFK Gi Gi RU xi,J i, xi,,J i,, IRU ERWK VSHFLHV LH
i 7KXVWKHHTXLOLEULXPSKDVHHQYHORSHLVWKHVROXWLRQWRWKHHTXDWLRQV
i
RT OQ J )RU G H[
TP
VROYLQJIRU x \LHOGV x
\LHOGV xx
G ex
RT
A
a
x x
RT
r
RU xi,
A
f x x RT
DQG xi,,
A
RT
A
Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
D 5HJXODU VROXWLRQ WKHRU\ VXSSRVH WKDW S H[ RU VLQFH G H[ H H[ T S H[ WKDW G H[ H H[ 7KLV LV WKH FDVH IRU WKH & + &&O V\VWHP EXW QRW IRU WKH & + &6 V\VWHP 7KHUHIRUH
UHJXODU VROXWLRQ WKHRU\ LV QRW DSSOLFDEOH WR WKH & + &6 V\VWHP 7R WHVW WKH +LOGHEUDQG
6FDWFKDUGPRGHOZHXVH
xV xV G H[
G G xV xV ZHREWDLQ
DW x u u u G H[
FDO PRO - PRO u FRPSDUHGZLWK | - PRO H[SHULPHQWDOO\7KXVZHFRQFOXGHGWKDWZKLOHWKH &&O & + a
f
V\VWHPKDV S H[ DQGWKXVPD\VDWLVI\WKHUHJXODUVROXWLRQmodelLWLVQRWZHOOUHSUHVHQWE\
6FDWFKDUG+LOGHEUDQGUHJXODUVROXWLRQtheory
E 6LQFH G ex LVDV\PPHWULFIXQFWLRQRIFRPSRVLWLRQIRUWKH &6 & + V\VWHPZHZLOOUHSUHVHQW
WKHFRPSRVLWLRQGHSHQGHQFHRI G H[ E\WKHRQHFRQVWDQW0DUJXOHVH[SUHVVLRQ G H[
a
H[
f
Axx ZLWK
A - PRO VRWKDW G
x - PRO DVLVREVHUYHGH[SHULPHQWDOO\)RUWKH
RQHFRQVWDQW0DUJXOHVHTQE\HTQ A
T8&
| . R u 7KXVLIDOLTXLGSKDVH V ZHUHWRH[LVWDWYHU\ORZWHPSHUDWXUHVLWZRXOGH[LVWDVWZRSKDVHV
EHORZ . DQG D VLQJOH VWDEOH SKDVH DERYH . +RZHYHU VLQFH . LV ZHOO EHORZ WKH
PHOWLQJSRLQWVRIHLWKHURIWKHSXUHFRPSRQHQWVDQGSUHVXPDEO\WKHHXWHFWLFSRLQWDVZHOO VHH
6HFWLRQ QROLTXLGOLTXLGSKDVHVHSDUDWLRQZLOOEHREVHUYHG>1RWHZHFDQLPSURYHRXU
HVWLPDWLRQ RI WKH XSSHU FRQVROXWH WHPSHUDWXUH E\ WDNLQJ LQWR DFFRXQW WKH WHPSHUDWXUH
w G
H
GHSHQGHQFHRIWKHH[FHVV*LEEVIUHHHQHUJ\,QSDUWLFXODUIURP T
ZHREWDLQ
wT
T
H[
H[
H[
T H
G
G
dT | H H[
T T T T
T T
T T
bg
H[
| - PRO 7KXV
a
G H[ x
DQGDW T
a
. G H[ x
IJ
K
ZKHUH ZH KDYH assumed IRU VLPSOLFLW\ WKDW H H[ LV WHPSHUDWXUH LQGHSHQGHQW $W x
H
FG
H
z
T
R UVOP S f T LMN WQ
T T f
. # - PRO DQG A T
. # - PRO LPSO\LQJDQXSSHUFRQVROXWHWHPSHUDWXUHRI
A
. T8&
R
*XHVVDJDLQ T8& T . a
G ex x
.
T
f .
8&
. ZKLFKLVFORVHWRJXHVV Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
6LQFHWKLVWHPSHUDWXUHLVVWLOOZHOOEHORZWKHPHOWLQJSRLQWVRIWKHVSHFLHVRXUFRQFOXVLRQGRHV
QRWFKDQJHWKHUHLVQRSKDVHVHSDUDWLRQ:HGRKRZHYHUVHHWKHLPSRUWDQFHRIDFFRXQWLQJIRU
WKHWHPSHUDWXUHGHSHQGHQFHRIWKHH[FHVV*LEEVIUHHHQHUJ\ DQGDFWLYLW\FRHIILFLHQWV @
F )RU YDSRUOLTXLG HTXLOLEULXP ZH KDYH f i L fiV Ÿ xiJ i Pi YDS yi P DW DQ D]HRWURSLF SRLQW
yi VRWKDW J i Pi YDS
xi
P IRURXUV\VWHP
u A
RT
DOVR
RU
RU
J & + P&YDS
+
x&6
OQ
YDS
J &6 P&6
P
YDS
OQ J &6 OQ P&6
OQ J & + OQ P&YDS
+ P&YDS
+ c
x&6
YDS
P&6
h
7KLVHTXDWLRQKDVDVLWVRQO\VROXWLRQ
x&6 # ZKLFKLVnotEHWZHHQDQG7KXVZHFRQFOXGHWKDWQRD]HRWURSHLVIRUPHG>1RWHZHFRXOG
JHW D EHWWHU HVWLPDWH RI G H[ DW WKH WHPSHUDWXUH RI LQWHUHVW E\ WDONLQJ LQWR DFFRXQW WKH
WHPSHUDWXUHGHSHQGHQFHRI G H[ DVZDVGRQHLQSDUWE7KHQZHILQGWKDW
G H[ T q & x - PRO 7KXVWKHVROXWLRQEHFRPHVHYHQOHVVnonLGHDODQGDQD]HRWURSHZLOOQRWEHIRUPHG@
D 6WDUWLQJIURP
a
xi,J i,
xi, H[S
f
LM Ac x h OP
MN RT PQ x J
, i
,, ,,
i i
xi,, H[S
LM Ac x h OP
MN RT PQ ,, i
ZKLFKZHFDQVROYHIRUA
e j c x h c x h
OQ
A
RT
, i
xi,,
xi,
,, i
c
8VLQJ WKH GDWD IRU EHQ]HQH x%,
DQGx%,,
h
\LHOGV
A
RT
ZKLOH XVLQJ WKH GDWD
A
6LQFHWKHWZRYDOXHVRIADUHGLIIHUHQWZHFRQFOXGH
RT
WKDWWKHRQHFRQVWDQW0DUJXOHVHTXDWLRQLVnotFRQVLVWHQWZLWKWKHH[SHULPHQWDOGDWD
E 5HJXODUVROXWLRQWKHRU\JLYHV
IRUSHUIOXRURQKHSWDQH\LHOGV
xi,J i,
xi, H[S
LMV cI h aG G f OP
RT
MN
PQ x J
i
, j
3
%
,, ,,
i i
xi,, H[S
LMV cI h aG G f OP
RT
MN
PQ i
,, j
3
%
Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
a
f
ZKLFKVROYLQJIRU G 3 G % \LHOGV
e j ZKHUH I
aG G f
V {cI h cI h }
3
%
xi,,
xi,
RT OQ
, j
i
,, j
)URP7DEOH V %
FF PRO V 3
I ,%
I ,3
I ,,%
x jV j
j
x3V 3 x%V %
P PRO IURPWKHSUREOHPVWDWHPHQW7KXV
DQG I ,,3
a
f
8VLQJ WKH EHQ]HQH GDWD ZH REWDLQ G 3 G % FDO PRO RU G 3 G % r RU 1RWH WKDW ZH FDQ QRW FKRRVH EHWZHHQ WKHVH WZR YDOXHV VROHO\ RQ WKH EDVLV RI WKH GDWD KHUH
$FWLYLW\FRHIILFLHQWGDWDRQSHUIOXRURQKHSWDQHLQRWKHUIOXLGVZRXOGEHQHHGHGWRIL[ G 3 7KH
YDOXHLVKRZHYHUTXLWHFORVHWRWKHYDOXHRIJLYHQLQ7DEOH
'RLQJ D VLPLODU FDOFXODWLRQ WR WKH RQH DERYH EXW QRW XVLQJ WKH SHUIOXRURQKHSWDQH GDWD
\LHOGV
a
f
G 3 G % RU G 3 G % r RU
7KXVUHJXODUVROXWLRQWKHRU\LVDOVRQRWFRPSOHWHO\FRQVLVWHQWZLWKWKHH[SHULPHQWDOGDWD
7KHFRQGLWLRQIRUPDWHULDOVWDELOLW\LVWKDW d G ! IRUDOOYDULDWLRQVDWFRQVWDQWTDQGP+HUH
WKLVLPSOLHV
w 'G PL[
w G
! RUHTXLYDOHQWO\
! wx T P
wx
TP
/RRNLQJDWWKHFXUYHLQWKHSUREOHPVWDWHPHQWZHVHHWKDWDWSRLQWVBDQGCWKLVGHULYDWLYHLV
]HURDQGEHWZHHQSRLQWVBDQGCLWLVQHJDWLYH7KLVLPSOLHVSKDVHLQVWDELOLW\RUSKDVHVHSDUDWLRQ
ZLWKSRLQWVBDQGCEHLQJWKHOLPLWVRIVWDELOLW\
7KHFRQGLWLRQRISKDVHHTXLOLEULXPLV
G,
DQG
'H PL[ x
TP
V V FG w' H IJ
H wx K
TP
FG w'G IJ
H wx K
TP
'G PL[ x
PL[
PL[
PL[
D H H G G 7KHUHIRUHWKHHTXLOLEULXPFRQGLWLRQVFDQEHZULWWHQDV
d
i
'G PL[ T P x , x,
FG w'V IJ
H wx K
VLPLODUO\ZHKDYH
G,, 'V PL[ x
G,, DQG G,
1RZIURP&KDSWHUZHKDYH
IJ
K
FG
H
FG IJ
H K
DQG
a
w 'G PL[
wx,
f 'G dT P x i x wa'G f PL[
,,
,,
PL[
wx,,
Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
d
i
'G PL[ T P x ,, x,
x,
DQGXVLQJ dx
a
f
a
w 'G PL[
w 'G PL[
x,
wx,
wx,
,,
PL[
wx,,
E f x wa'G f x wa'G f ,,
,,
PL[
wx,
PL[
wx,,
c x x h wa'wGx f c x x h wa'wGx f ,
RUVLQFH xi xi
,
,,
PL[
,
a
w 'G PL[
wx,
,,
i FGH w'wGx IJK
d
'G PL[ T P x , xi,
PL[
PL[
,,
f wa'G f PL[
wx,,
7KXVDWWKHHTXLOLEULXPVWDWHQRWRQO\LV Gi,
UHVXOWLQHTQ D JLYHVDWHTXLOLEULXP
,,
PL[
dx JLYHV
f 'G dT P x i x wa'G f 6XEWUDFWLQJWKHVHFRQGRIWKHVHHTXDWLRQVIURPWKHILUVW\LHOGV
a
w 'G PL[
wx,
TP
Gi,, EXWHTQ LVVDWLVILHGDOVR8VLQJWKLVODVW
i FGH w'wGx IJK
d
'G PL[ T P x ,, xi,,
PL[
TP
1RZHTQ LPSOLHVWKDWWKHVORSHRIWKH 'G PL[ FXUYHPXVWEHWKHVDPHDWWKHWZRSRLQWVDW
ZKLFKWKHSKDVHVDUHLQHTXLOLEULXP)XUWKHUIURPHTQV DQG ZHKDYHWKDWVLQFH Gi, Gi,, WKHWZROLQHVKDYHWKHVDPHLQWHUFHSW6LQFHWKHWZRWDQJHQWOLQHVKDYHWKHVDPHVORSHDQGWKH
VDPHLQWHUFHSWWKHOLQHVPXVWEHLGHQWLFDOLHWKHHTXLOLEULXPSRLQWVDUHRQDFRPPRQWDQJHQW
OLQH
7KH:LOVRQPRGHOLV
OP
Q
V
L cO O h OP H[S M
V
N RT Q
ZLWK
LM
N
RT ¦ xi OQ ¦ x j /ij G H[
L
j
L
i
/ij
j
ij
ji
ZKLFKIRUDELQDU\PL[WXUHUHGXFHVWR
f a
fp
G
x G x G RT a x OQ x x OQ x f x
x
F
I
G x G x G RT G x OQ
H x x / x OQ x / x JK G H[
DQG
VR
k a
RT x OQ x x / x OQ x/ x ,0
1RZZHORRNDWWKHGHULYDWLYHRI G Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
FG wG IJ
H wx K
x
x x /
G G RT OQ
TP
RTx
FG
H
x
x x / w
x
wx x x /
IJ
K
FG
H
x/ x w
x
x
RTx x
x/ x
wx x/ x
x
G G RT OQ
x x /
RT OQ
a
RT x x /
RT OQ
fRST x x / a xx a x // ff UVW
a
7KHQ
RT
x x /
RT
x/ x
a
a
FG w G IJ
H wx K
x
x x /
kf x x / x x / p RT OQ x / x x
f k x / x x / x p
FG
IJ
H
K a
f a / f
x/ x w F
RT
G x IJ RT/ a/ f
wx H x / x K a x / x f
x
RT
kax x / f x a / fp
x ax x / f
RT
kax / x f x a/ fp
x ax / x f
RT
TP
x
RT/
x x / x x /
x
RT/ RT OQ
x/ x x/ x
x x / w
x
RT/
wx x x /
x
x x / G G RT OQ
fRST x / x axx /a/ xff UVW
G G RT OQ
x
x/ x
RT x/ x
IJ
K
6R
FG w G IJ
H wx K
RT
TP
OP LM /
/
N x ax x / f x ax / x f Q
DWXSSHUFULWLFDOVROXWLRQWHPSHUDWXUH
1RZ d x d DQG d x d LQSDUWLFXODUQHLWKHU x QRU x LVQHJDWLYH$OVRFOHDUO\
Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
FG / IJ ! DQG FG / IJ ! Hx x / K
Hx/ x K
F w G IJ LVLI T . 7KXVWKHXSSHU
6RIRUWKH:LOVRQPRGHOWKHRQO\ZD\IRU G
H wx K
TP
FRQVROXWHWHPSHUDWXUHIRUWKH:LOVRQPRGHOLV T . DQGWKHUHLVQROLTXLGOLTXLGHTXLOLEULXP
D %XEEOHSRLQWSUHVVXUH$VVXPHYDSRUSKDVHLVLGHDO7KHQ
xiJ i Pi YDS yi P 6ROYHQW
I
x H[S OQ I xI PYDS y P Ÿ I H[S I xI PYDS y P x
m
m
3RO\PHU
I
x H[S OQ m I xI PYDS y P Ÿ I H[S I xI PYDS y P x
m
EXWIRUWKHSRO\PHU
PYDS a Ÿ y a y a 7KHUHIRUHIURPHTQ P PEXEEOH I H[S I xI PYDS m
SRLQW
LM
N
OP
Q
LMF
NH
I
K
OP
Q
OP
Q
LMF
NH
I
K
OP
Q
I
K
F
H
LM
N
LMF
NH
E
OP
Q
/LTXLGOLTXLGHTXLOLEULXP
I,
x, H[S OQ , I , x I , PYDS
m
x
LM
N
F
H
x,, H[S OQ
,,
,,
c h OPQ
I
K
LM I F II xcI h OP P
Q
N x H mK
Ÿ I H[SLMF I I xcI h OP I H[SLMF I I xcI h OP
NH mK
NH mK
Q
Q
,
,
6LPLODUO\
I , H[S m I , x I ,
I
K
c h
,,
,, , ,,
YDS
c h I ,, H[S m I ,, x I ,,
'DWDQHHGHG
D EXEEOHSRLQW YROXPHRIVROYHQW
YROXPHRISRO\PHU
VROYHQWSRO\PHU F SDUDPHWHU
YDSRUSUHVVXUHRIVROYHQW
E OLTXLGOLTXLGHTXLOLEULXP
YROXPHRIVROYHQW
YROXPHRISRO\PHU
VROYHQWSRO\PHU F SDUDPHWHU
,, ,,
Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
&RQVLGHUWKHFRQGLWLRQIRUOLTXLGOLTXLGSKDVHHTXLOLEULXPRIDVROXWHDVVXPLQJWKDWXQGLVVROYHG
VROXWHLVDOVRSUHVHQW
,
d
i f dT P x i f T P
d
i
f i T P x ,
RU
xi,J T P x , f i / T P
DQGWKHQ
d
xi,J T P x ,
,,
i
,,
i
/
d
i
xi,,J T P x ,, fi / T P
fi / T P i x J dT P x i ,,
i
,,
7KHUHIRUHLIWKHDFWLYLW\RIDVSHFLHVLQVROXWLRQLVHYHUJUHDWHUWKDQXQLW\DVHSDUDWHSKDVHSXUH
RUYHU\FRQFHQWUDWHG LQWKDWVSHFLHVZLOOIRUPDQGUHGXFHWKHDFWLYLW\RIWKHVSHFLHVLQWKHRWKHU
SKDVHVWRXQLW\
Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
DOVRDYDLODEOHDVD0$7+&$'ZRUNVKHHW 9 ˜ SKL [ 9 ˜ 9
9
FKL 7 9 9 P
[˜ 9
7
P
SKL [ SKL [
[˜ 9 [ ˜ 9
[˜ 9
SKL [
[
˜ 9 [ ˜ 9
[ 9
9
9
9[ SKL [
[SKL
7
SKL [
SKL [
9
[
·
SKL [ ·[
SKL [
§
SKL
¸ SKL [ SKL [ ˜ ¨ P ¸ FKL 7 ˜ SKL [ SKL [
© SKL [ ¹
©
[ 9 ¹ [ 9
SKL [
SKL [ [ [ 7 [ [
[ 9
[ [
*LYHQ
OQ§¨
SKL [ ·
¸ SKL [ SKL [ ˜ P FKL 7 ˜ SKL [ SKL [
© SKL [ ¹
OQ§¨
[
Y )LQG [ [
7 § [
·
¨
¸
© ¹
Y
[
[
·
SKL [ ·
§
¸ SKL [ SKL [ ˜ ¨ P ¸ FKL 7 ˜ SKL [ SKL [
© SKL [ ¹
©
¹
SKL [ ·
¸ SKL [ SKL [ ˜ P FKL 7 ˜ SKL [ SKL [
© SKL [ ¹
OQ§¨
Y )LQG [ [
7 § ·
¨
¸
© ¹
Y
*LYHQ
·
SKL [ ·
§
OQ§¨
¸ SKL [ SKL [ ˜ ¨ P ¸ FKL 7 ˜ SKL [ SKL [
© SKL [ ¹
©
¹
SKL [ ·
¸ SKL [ SKL [ ˜ P FKL 7 ˜ SKL [ SKL [
© SKL [ ¹
OQ§¨
Y )LQG [ [
7 OQ§¨
[
*LYHQ
OQ§¨
§ ·
¨
¸
© ¹
Y
*LYHQ
SKL [ ·
§
¸ SKL [ SKL [ ˜ ¨ © SKL [ ¹
©
·
¸ FKL 7 ˜ SKL [ SKL [
SKL [ ·
¸ SKL [ SKL [ ˜ P FKL 7 ˜ SKL [ SKL [
© SKL [ ¹
Y )LQG [ [
7 P¹
OQ§¨
§ ·
¨
¸
© ¹
Y
*LYHQ
SKL [ ·
·
§
¸ SKL [ SKL [ ˜ ¨ P ¸ FKL 7 ˜ SKL [ SKL [
© SKL [ ¹
©
¹
OQ§¨
SKL [ ·
¸ SKL [ SKL [ ˜ P FKL 7 ˜ SKL [ SKL [
© SKL [ ¹
OQ§¨
Y )LQG [ [
7 § ·
¨
¸
© ¹
Y
*LYHQ
·
SKL [ ·
§
OQ§¨
¸ SKL [ SKL [ ˜ ¨ ¸ FKL 7 ˜ SKL [ SKL [
P¹
© SKL [ ¹
©
SKL [ ·
¸ SKL [ SKL [ ˜ P FKL 7 ˜ SKL [ SKL [
© SKL [ ¹
OQ§¨
Y )LQG [ [
Y
§ ·
¨
¸
© ¹
7KHWHPSHUDWXUH.LVWKHKLJKHVWDWZKLFKDQRQWULYLDOVROXWLRQLVREWDLQHG$WKLJKHU
WHPSHUDWXUHRQWKHWULYLDOVROXWLRQRIERWKSKDVHVEHLQJHTXDOLVREWDLQHG7KXVWKH)+
PRGHOSUHGLFWVWKDW//(ZLOORFFXUXSWRWKLVWHPSHUDWXUH7KDWLVLVWKH8&67IRU
6$1DQG300$2IFRXUVHDWWKLVKLJKWHPSHUDWXUHWKHSRO\PHUVDUHOLNHO\WR
GHFRPSRVH
FKL 7
[
Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
D 7 7
*LYHQ
7
*LYHQ
H[
FOQ 7 *LYHQ
7* [[ D E7
*LYHQ
SKL
·[
* H[[
F
F 7 SKL
+ H[ SKL [ w §SKL
w § D[ SKL
SKL
[ SKL
·[[ § D FKL
· [
OQ OQ
SKL
[
SKL
FKL
777 SKL
[
SKL
SKL
[
OQ
SKL
[
SKL
[
FKL
SKL
[
SKL [
[
[
[
E
OQ
7
[
[
OQ
¨
¸
¨
¸
¨
¸
7
¸[
OQ wSKL
© 7
SKL
SKL [
P PP7FKL
7 ¨©SKL
7[
w7 © 7[ 7SKL ¹[
7
SKL
[ SKL
¹ [
¹
SKL [
P
+ H[ [[ > D F OQ 7 @
SKL
[[
SKL
SKL
[
OQ OQOQ
SKL
[[
[[
7 77SKL
[[
[[
SKL
SKL
SKL
H[ ·
SKL§ w[
SKL
[ SKL
SKL
[ P PPFKLFKL
FKL
SKL
[
SKL
[
SKL
[SKL
+[
OQ&SKL
SKL [ P FKL 7 SKL [
SKL [
H[ SKL
[
[ F[
SKL
[¸ SKL
¨
3
SKL¨ [
¸
7
© w7 ¹
Y YYILQGILQG
[[
[
[
[
H[ ILQG
H[[
Y 76ILQG
[
[
+
* H[ [[ > D YYF YY OQ 7 @ [[ D E7 F OQ 7 [[ E7 F OQ 7
E )RUDQXSSHUFRQVROXWHWHPSHUDWXUH
§ w * ·
§ DE7FOQ7 ·
t Ÿ *LYHQ
7 d ¨
¨¨ 7¸¸ ¸
5
©
¹
© w[ ¹73
7KHWHPSHUDWXUH.LVWKHKLJKHVWDWZKLFKDQRQWULYLDOVROXWLRQLVREWDLQHG$WKLJKHU
RU dSKL
D E5
[ 7FOQ7IRUDQXSSHUFRQVROXWHWHPSHUDWXUH
WHPSHUDWXUHRQWKHWULYLDOVROXWLRQRIERWKSKDVHVEHLQJHTXDOLVREWDLQHG7KXVWKH)+
OQ
SKL [
SKL [ FKL 7 SKL [
SKL [
PRGHOSUHGLFWVWKDW//(ZLOORFFXUXSWRWKLVWHPSHUDWXUH7KDWLVLVWKH8&67IRU
F )RUDQORZHUFRQVROXWHWHPSHUDWXUH
SKL [
P
6$1DQG300$2IFRXUVHDWWKLVKLJKWHPSHUDWXUHWKHSRO\PHUVDUHOLNHO\WR
§ w * ·
§ DE7FOQ7 ·
d Ÿ 7 t ¨
GHFRPSRVH
¸
¨¨ w[ ¸¸SKL
[
5 SKL
7 SKL [ SKL [ ©
¹
© OQ ¹73
SKL
[
[ P FKL
[
RU tSKL
D E5
7FOQ7IRUDQORZHUFRQVROXWHWHPSHUDWXUH
Y
ILQG [ [
7
OQ
OQ
Y
SKL [
SKL [
SKL [
SKL [
ILQG [ [
Y
*LYHQ
SKL [
SKL [ SKL [
SKL [ Y
P
P
FKL 7 SKL [
SKL [
FKL 7 SKL [
SKL [
Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
D ,PDJLQHWKHVHSDUDWLRQSURFHVVWRRFFXUFRQWLQXRXVO\DVEHORZ
x1 moles of pure species 1
1 mole of mixture
Separator
x2 moles of pure species 2
Q W
0DVVEDODQFH x x (QHUJ\EDODQFH H PL[ x H x H Q W Q
SJHQ T
6XEWUDFWLQJWKHSURGXFWRIWHPSHUDWXUHDQGWKHHQWURS\EDODQFHIURPWKHHQHUJ\EDODQFH\LHOGV
7KXV
DQG
(QWURS\EDODQFH aH
RU
W
S PL[ x S x S f x aH T S f x aH T S f W x G x G p 'G
RT ¦ x OQ x G PL[ T S PL[
k
G PL[
d
w G H[ T
RS
T
w Axx
T T
wT
8SSHUFRQVROXWHWHPSHUDWXUH T 8&
)RUDQHTXLPRODUPL[WXUH
Q
H[
i i
W H H[
RU
T FG IJ
H K
w G H[
wT T
UV Ax x
W
G H[ Ÿ S H[
RT ¦ xi OQ xi G H[ G H[ k
a
f Ax x RU
Ÿ RT x OQ x x OQ x
Axx
TW R x OQ x x OQ x
TW p
RT x OQ x x OQ x Q
E W
A
R
a fa f
a f a f
A R OQ u T 8&
OQ A
R OQ DQGIRUWKLVFDVH
Q
i
W H H[ i PL[
H H[
Ÿ H H[
T
P xi
Ÿ H H[
7KHUHIRUH
RT x OQ x x OQ x Axx i
wT
W H PL[ x H x H Q
+RZHYHU
W
R OQ ˜
T 8&
OQ F ,IALVDIXQFWLRQRIWHPSHUDWXUHWKHQ
RT 8&
A
Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
H H[
I
K
F w A IJ
G x x TG
HwTK
T F
H
w Ax x
wT
T
P x
H[
Q
DQGIRU W
a
f
a
f
RT x OQ x x OQ x xxT
ZHKDYHDVEHIRUH
P x
RT x OQ x x OQ x Ax x DVEHIRUHEXW
T
FG IJ
H K wA
Ax x
Txx
w T P x
T
W
T
EXW T 8& LVQRORQJHUHTXDOWR
A
R
A R
OQ FG w A IJ HwTK
P x
Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
7KHSURFHGXUHWKDWZLOOEHXVHGLVWRILWWKHSDUDPHWHUVLQDQDFWLYLW\FRHIILFLHQWPRGHOWRWKH//(GDWD
DWHDFKWHPSHUDWXUHWRFDOFXODWHWKHFRPSOHWH*H[FXUYH7KHQXVHDILQLWHGLIIHUHQFHDSSUR[LPDWLRQ
WRFDOFXODWH+H[DQGIURP*H[DQG+H[WRFRPSXWH6H[
1RWHWKDWWKHFRPSRVLWLRQVLQWKHWZRSKDVHVDUHTXLWHGLIIHUHQWWKHUHIRUHWKHRQHFRQVWDQW
0DUJXOHVHTXDWLRQLVQRWDSSOLFDEOH7KHUHIRUH,ZLOOXVHWKH9DQ/DDUHTXDWLRQ
DOI
º»
ǻ
DOI ˜ [ º »
«¬ «¬ EHW ˜ [ »¼ »¼
JDP [ DOI EHW H[Sª«
DOI LQLWLDOJXHVVHV
EHW 5 EHW
º»
ǻ
> EHW [ @ º »
« « DOI ˜ [ » »
¬¬
¼ ¼
JDP [ DOI EHW H[Sª«
7 [ [ [ [
[ [
*LYHQ
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
] )LQG DOI EHW
]
DOI ]
EHW ]
L § ·
¨
¸
© ¹
DOI
[ ˜ L
[ L
*H[ [ ˜ 5 ˜ 7˜ OQ JDP [ DOI EHW
L
L
L
EHW
[
[ ˜ 5 ˜ 7˜ OQ JDP [ DOI EHW
L
L
7 [ [ [ [
[ [
*LYHQ
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
] )LQG DOI EHW
DOI ]
]
EHW ]
§ ·
¨
¸
© ¹
DOI
EHW
Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
*H[ [ ˜ 5˜ 7˜ OQ JDP [ DOI EHW
L
L
L
[ ˜ 5˜ 7˜ OQ JDP [ DOI EHW
L
L
*H[
*H[
[
§ *H[L *H[L ·
¨
¸
7 ¹
© 7
'HULY L
)LQLWHGLIIHUHQFHDSSUR[LPDWLRQIRUG *H[7 G7ZKLFK
LVHTXDOWR+H[7A
+H[ ˜ 'HULY
L
L
+H[
[
*H[ ˜ *H[ *H[
L
L
6H[ L
L
$YHUDJHWRJHW*H[DWR&
+H[ *H[
L
L
6H[
[
Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
j
D
c
h CD’x
L w OQ J OP D’x x D M N x wx Q
CDR
’ P R RT OQ xJ RT
xDR ’ OQ x ’ OQ J E
R
LM w OQ J OP
N w OQ x Q
DR D
RT OQ J A x w OQ J w OQ x
w OQ J x
wx
w OQ J x
wx
FAxI
H RT K Ax x x
LM w OQ J OP D F A x x I N w OQ x Q H RT K
D
F ,QILQLWHGLOXWLRQ x o DQGD o DR A
$W//(FULWLFDOSRLQWT8&
DWx
R
u u R u T8&
D DR R u T8&
wx
w
DR R
LM
N
x
OP
Q
1HJDWLYHGHYLDWLRQVIURP5DRXOW¶VODZŸALVQHJDWLYH
A
D DR xx ZLWKAQHJDWLYH D ! DR RT
I
K
F
H
D
YDS
[ $ J $ 3$YDS
3 [ : J : 3:
u u u u J $
u u u u [ L J L 3LYDS \L 3 Ÿ \ :
J$
E //([ L, J L [ L,
DQG\ $
u u [ L,, J L [ L,, $VVXPHDFWLYLW\FRHIILFLHQWVDUHLQGHSHQGHQWRIWHPSHUDWXUHIURPWRR&
:DWHU u J : [ : Ÿ J : [ : $OFRKRO u J $ [ $ Ÿ J $ [ $ $FWLYLW\FRHIILFLHQWH[SUHVVLRQLVRIWKHIRUPRIWKHWZRFRQVWDQW0DUJXOHVH[SDQVLRQ HT VRWKDW
57 OQ J L
DL [ L
EL [ L DQG
57 OQ Jf
D E 5757 OQ J f
D E
57
ZLWKD $ % D $ % E %DQGE %
7KHVROXWLRQWRWKHVHHTXDWLRQVLV$ 57DQG% 57
1RZFKHFNIRU//(
Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
[, J [,
[,, J [,, DQG[ , J [ ,
[ ,, J [ ,, L [ ˜ L
L
OQJL ª ˜ [ ˜ [ º
L ¼
L
¬
OQJL ª [ ˜ [ º
L ¼
¬ L
I [ ˜ H[S OQJL
L
L
J
J
JL H[S OQJL
I JL H[S OQJL
[ ˜ H[S OQJL
L
L
I
I
[
JDP [[ H[Sª¬ ˜ [[ ˜ [[ ¼
6ROYHIRU//(
[ º
*LYHQ
[˜ JDP [
[˜ JDP [
[˜ JDP [
[ ˜ JDP [
] )LQG [ [ [ [
[
7KLVJUDSKLQGLFDWHV//(
VHH,OO
[ JDP [[ H[S [[ ˜ [[
[ [
[ [
[ [ [ [ § ·
¨
¸
¸
] ¨
¨ ¸
¨ ¸
©
¹
6RRQHOLTXLGSKDVHKDVPROHIUDFWLRQRIVSHFLHVDQGWKHVHFRQGOLTXLGSKDVH
KDVPROHIUDFWLRQ7KHUHIRUHDPL[WXUHZLWKDIHHGFRPSRVLWLRQRIRPROHIUDFWLRQ
ZLOOVSOLWLQWRWZROLTXLGSKDVHV
Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
JDPDLQI JDPELQI DOI OQ JDPDLQI
[ EHW OQ JDPELQI
[
YSE [ ˜ L
L L
DOI
º
«
»
DOI˜ [
º »
« «ª
L
»
««
EHW ˜ [ » »
L ¼ ¼
¬¬
JDP H[Sª
L
YSD EHW
º
«
»
ª¬EHW [L º¼ º» »
« «ª
««
DOI˜ [
» »
L
¬¬
¼ ¼
JDP H[Sª
L
JDP
JDP
3 [ ˜ JDP ˜ YSD [ ˜ JDP ˜ YSE
L
L
\ L
L
L
L
[ ˜ JDP ˜ YSD
L
L
3
L
[
\
3
[
[
6RV\VWHPGRHVQRWKDYHDQD]HRWURSHKRPHJHQHRXVRUKHWHURJHQHRXV
[
Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
3YDS( 3YDS$ L [ ˜ L
[ [
L
º
«
»
˜
[
« ª«
L º
» »
««
˜ [ » »
L ¼ ¼
¬¬
J$ L H[Sª
º
«
»
˜ [ º »
« «ª
L »
« « [
˜
» »
L ¼ ¼
¬¬
3 [ ˜ J$ L˜ 3YDS$ [ ˜ J(L˜ 3YDS(
L
L
J(L H[Sª
L
\ L
[ ˜ J$ L˜ 3YDS$
L
3
L
3
[
\
[
6RQHLWKHUDKRPRJHQHRXVRUKHWHURJHQHRXVD]HRWURSHLVIRUPHG
Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
DOI
JDP [ DOI EHW H[Sª«
ȼ
JDP [ DOI EHW H[Sª«
»
ǻ
DOI˜ [ º
«¬ «¬ EHW ˜ [ »¼ »¼
DOI EHW
ȼ
»
ǻ
> EHW [ @ º
«¬ «¬ DOI˜ [ »¼ »¼
EHW [ [ [ [
[ [
*LYHQ
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
] )LQG DOI EHW
]
[ § ·
¨
¸
© ¹
[ [ [
[ [
*LYHQ
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
] )LQG DOI EHW
[ ]
§ ·
¨
¸
© ¹
[ [ [
[ [
*LYHQ
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
] )LQG DOI EHW
[ ]
§ ·
¨
¸
© ¹
[ [ [
[ [
*LYHQ
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
] )LQG DOI EHW
]
§ ·
¨
¸
© ¹
Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
[ [ [ [
[ [
*LYHQ
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
] )LQG DOI EHW
[ § ·
¨
¸
© ¹
]
[ [ [
[ [
*LYHQ
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
] )LQG DOI EHW
[ ]
§ ·
¨
¸
© ¹
[ [ [
[ [
*LYHQ
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
] )LQG DOI EHW
[ ]
§ ·
¨
¸
© ¹
[ [ [
[ [
*LYHQ
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
] )LQG DOI EHW
[ ]
[ § ·
¨
¸
© ¹
[ [
[ [
*LYHQ
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
] )LQG DOI EHW
[ § ·
¨
¸
© ¹
]
[ [ [
*LYHQ
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
] )LQG DOI EHW
]
§ ·
¨
¸
© ¹
[ [
Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
] )LQG DOI EHW
§ ·
¨
¸
© ¹
]
L § ·
¨
¸
¨ ¸
¨ ¸
¨ ¸
¨
¸
¸
GDWD ¨
¨ ¸
¨ ¸
¨
¸
¨ ¸
¨ ¸
¨
¸
© ¹
D L GDWD
E L GDWD
7 GDWD
L
; 7
L
L L L L
1XPEHURIGDWDSRLQWV
Q URZV GDWD
Q
,QWHUFHSW
E LQWHUFHSW ; <
E
6ORSH
E VORSH ; <
E
3ORWV
I ; E E ˜;
I ;
D
;
,QWHUFHSW
E LQWHUFHSW ; =
E
6ORSH
E VORSH ; =
E
3ORWV
I ; E E ˜;
I ;
E
;
< DL
L
= EL
L
Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
ª
ª
W˜ *
º ª
º º» »º
»
¬ [ [ ˜ *¼ «¬ > [ [˜ *@ »¼ »¼ »¼
*
JDP [ * * W W H[S« [ « W˜ ª«
«
¬
«
¬
ª
ª
W˜ *
º ª
º º» º»
»
¬ [ [ ˜ *¼ «¬ > [ [ ˜ *@ »¼ »¼ »¼
*
JDP [ * * W W H[S« [ « W˜ ª«
«
¬
* «
¬
* [ W OQ *
W [ OQ *
[ [
[ [
*LYHQ
[˜ JDP [ * * W W
[˜ JDP [ * * W W
[˜ JDP [ * * W W
[˜ JDP [ * * W W
W
OQ *
OQ *
W
] )LQG * * W W
* ]
* ]
W [ OQ *
[ W §¨ ·¸
¸
] ¨
¨ ¸
¨ ¸
©
¹
OQ *
[ [
[ [
*LYHQ
[˜ JDP [ * * W W
[˜ JDP [ * * W W
[˜ JDP [ * * W W
[˜ JDP [ * * W W
W
OQ *
W
] )LQG * * W W
OQ *
·
¨§
¸
¸
¨
]
¨ ¸
¨ ¸
©
¹
Solutions to Chemical and Engineering Thermodynamics, 5th ed
[ [ [ [
[ [
*LYHQ
[˜ JDP [ * * W W
[˜ JDP [ * * W W
[˜ JDP [ * * W W
[˜ JDP [ * * W W
OQ *
W
OQ *
W
§ ·
¨
¸
¸
] ¨
¨ ¸
¨ ¸
©
¹
] )LQG * * W W
[ [ [ [
[ [
*LYHQ
[˜ JDP [ * * W W
[˜ JDP [ * * W W
[˜ JDP [ * * W W
OQ *
W
[˜ JDP [ * * W W
OQ *
W
] )LQG * * W W
[ § ·
¨
¸
¨ ¸
¨ ¸
¨
¸
© ¹
]
[ [ [
[ [
*LYHQ
[˜ JDP [ * * W W
[˜ JDP [ * * W W
[˜ JDP [ * * W W
W
OQ *
W
[˜ JDP [ * * W W
OQ *
]
] )LQG * * W W
[ § ·
¨
¸
¨ ¸
¨ ¸
¨
¸
© ¹
[ [ [
*LYHQ
[˜ JDP [ * * W W
[˜ JDP [ * * W W
[˜ JDP [ * * W W
[˜ JDP [ * * W W
W
OQ *
W
] )LQG * * W W
[ OQ *
]
§ ·
¨
¸
¨ ¸
¨ ¸
¨ ¸
©
¹
[ [ [
*LYHQ
[˜ JDP [ * * W W
[˜ JDP [ * * W W
[˜ JDP [ * * W W
[˜ JDP [ * * W W
W
OQ *
W
] )LQG * * W W
[ OQ *
[ [˜ JDP [ * * W W
W
OQ *
[ [
§ ·
¨
¸
¸
¨
]
¨ ¸
¨ ¸
©
¹
*LYHQ
[˜ JDP [ * * W W
W
[ [
[ [
[˜ JDP [ * * W W
[˜ JDP [ * * W W
OQ *
] )LQG * * W W
]
[ [
§ ·
¨
¸
¨ ¸
¨ ¸
¨ ¸
©
¹
Section 11.2
Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
[ [ [ [
[ [
*LYHQ
[˜ JDP [ * * W W
[˜ JDP [ * * W W
[˜ JDP [ * * W W
[˜ JDP [ * * W W
OQ *
W
OQ *
W
] )LQG * * W W
§ ·
¨
¸
¸
] ¨
¨ ¸
¨ ¸
©
¹
[ [ [ [
[ [
*LYHQ
[˜ JDP [ * * W W
[˜ JDP [ * * W W
[˜ JDP [ * * W W
[˜ JDP [ * * W W
OQ *
W
OQ *
W
] )LQG * * W W
§ ·
¨
¸
¸
] ¨
¨ ¸
¨ ¸
©
¹
L § ·
¨
¸
¨ ¸
¨ ¸
¨ ¸
¨
¸
¸
GDWD ¨
¨ ¸
¨ ¸
¨
¸
¨ ¸
¨ ¸
¨
¸
© ¹
7 GDWD
L
* GDWD
L L
; 7
L
1XPEHURIGDWDSRLQWV
Q
E LQWHUFHSW ; <
E
6ORSH
E VORSH ; <
E
3ORWV
I ; E E ˜;
* GDWD
L
L < *
L
L
= *
L
Q URZV GDWD
,QWHUFHSW
L L
L
u I ;
*
;
,QWHUFHSW
E LQWHUFHSW ; =
E
6ORSH
E VORSH ; =
E
3ORW
I ; E E ˜;
u I ;
*
;
Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
JDP [ $ % H[Sª¬ $ ˜ % ˜ [ ˜ %˜ [ º¼
JDP [ $ % H[Sª¬ $ ˜ % ˜ [ ˜ %˜ [ º¼
$ % [ [ [ [
[ [
[ [
[ [
*LYHQ
[˜ JDP [ $ %
[˜ JDP [ $ %
[˜ JDP [ $ %
[˜ JDP [ $ %
] )LQG $ %
]
[ § ·
¨
¸
© ¹
[ *LYHQ
[˜ JDP [ $ %
[˜ JDP [ $ %
[˜ JDP [ $ %
[˜ JDP [ $ %
] )LQG $ %
]
[ *LYHQ
§ ·
¨
¸
© ¹
[ [ [
[˜ JDP [ $ %
[˜ JDP [ $ %
[˜ JDP [ $ %
[˜ JDP [ $ %
] )LQG $ %
$ ]
]
[ [
§ ·
¨
¸
© ¹
% ]
[ [ [ [
[ [
*LYHQ
[˜ JDP [ $ %
[˜ JDP [ $ %
[˜ JDP [ $ %
[˜ JDP [ $ %
] )LQG $ %
[ ]
§
·
¨
¸
© u ¹
[ [ [
[ [
*LYHQ
[˜ JDP [ $ %
[˜ JDP [ $ %
[˜ JDP [ $ %
[˜ JDP [ $ %
] )LQG $ %
]
§ ·
¨
¸
© ¹
Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
[ [ [ [
[ [
[ [
[ [
[ [
[ [
[ [
[ [
[ [
[ [
*LYHQ
[˜ JDP [ $ %
[˜ JDP [ $ %
[˜ JDP [ $ %
[˜ JDP [ $ %
] )LQG $ %
[ § ·
¨
¸
© ¹
]
[ *LYHQ
[˜ JDP [ $ %
[˜ JDP [ $ %
[˜ JDP [ $ %
[˜ JDP [ $ %
] )LQG $ %
[ § ·
¨
¸
© ¹
]
[ *LYHQ
[˜ JDP [ $ %
[˜ JDP [ $ %
[˜ JDP [ $ %
[˜ JDP [ $ %
] )LQG $ %
[ § ·
¨
¸
© ¹
]
[ *LYHQ
[˜ JDP [ $ %
[˜ JDP [ $ %
[˜ JDP [ $ %
[˜ JDP [ $ %
] )LQG $ %
[ § ·
¨
¸
© ¹
]
[ *LYHQ
[˜ JDP [ $ %
[˜ JDP [ $ %
[˜ JDP [ $ %
[˜ JDP [ $ %
] )LQG $ %
]
§ ·
¨
¸
© ¹
Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
L § ·
¨
¸
¨ ¸
¨ ¸
¨ ¸
¨
¸
¸
GDWD ¨
¨ ¸
¨ ¸
¨
¸
¨ ¸
¨ ¸
¨
¸
© ¹
7 GDWD
L
< $
L
$ GDWD
L L
; 7
L
L
% GDWD
L L
L = %
L
L
L
1XPEHURIGDWDSRLQWV
Q URZV GDWD
Q
,QWHUFHSW
E LQWHUFHSW ; <
E
6ORSH
E VORSH ; <
E
3ORWV
I ; E E ˜;
I ;
$
;
,QWHUFHSW
E LQWHUFHSW ; =
E
6ORSH
E VORSH ; =
E
3ORW
II ; E E ˜ ;
II ; %
;
Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
U / T U ˜ U T ˜ U T [ I [ / [˜ T
[˜ U
I [ [˜ U [ ˜ U
U
/
T
/
[ ˜ T
T [ [˜ T [ ˜ T
T ˜ U T ˜ U [˜ T [ ˜ T
[ ˜ U
[˜ U [ ˜ U
W
W
ª § I [ · u T˜ OQ§ T [ · I [ ˜ § / U˜ / · º ˜ H[Sª ª T˜ OQ T [ T [ ˜ W T˜ T [ ˜ §
Ἴ
¸
¨
¸
¨
¸» ««
¨
¸»»
U ¹ ¼
©
¬ © [ ¹
© I [ ¹
¬¬
© T [ T [ ˜ W T [ ˜ W T [ ¹ ¼ ¼
JDP [ W W H[S« OQ¨
W
W
ª § I [ · ˜ T˜ OQ§ T [ · I [ ˜ § / U˜ / · º ˜ H[Sª ª T˜ OQ T [ ˜ W T [ T u T [ ˜ §
Ἴ
¸
¨
¸
¨
¸» ««
¨
¸»»
U ¹ ¼
©
¬ © [¹
© I [ ¹
¬¬
© T [ T [ ˜ W T [ ˜ W T [ ¹ ¼ ¼
JDP [ W W H[S« OQ¨
W W [ [ [ [
[ [
*LYHQ
[˜ JDP [ W W
[˜ JDP [ W W
[˜ JDP [ W W
[˜ JDP [ W W
] )LQG W W
§ ·
¨
¸
© ¹
]
[ [ [ [
[ [
*LYHQ
[˜ JDP [ W W
[˜ JDP [ W W
[˜ JDP [ W W
[˜ JDP [ W W
] )LQG W W
]
[ [ § ·
¨
¸
© ¹
[ [
[ [
*LYHQ
[˜ JDP [ W W
[˜ JDP [ W W
[˜ JDP [ W W
[˜ JDP [ W W
] )LQG W W
[ ]
[ § ·
¨
¸
© ¹
[ [
[ [
*LYHQ
[˜ JDP [ W W
[˜ JDP [ W W
[˜ JDP [ W W
[˜ JDP [ W W
] )LQG W W
]
[ [ § ·
¨
¸
© ¹
[ [
[ [
*LYHQ
[˜ JDP [ W W
[˜ JDP [ W W
[˜ JDP [ W W
[˜ JDP [ W W
] )LQG W W
]
[ [ [ [
§ ·
¨
¸
© ¹
[ [
*LYHQ
[˜ JDP [ W W
[˜ JDP [ W W
[˜ JDP [ W W
[˜ JDP [ W W
]
] )LQG W W
[ [ [ [
§ ·
¨
¸
© ¹
[ [
Solutions to Chemical and Engineering Thermodynamics, 5th ed
*LYHQ
[˜ JDP [ W W
[˜ JDP [ W W
[˜ JDP [ W W
[˜ JDP [ W W
] )LQG W W
[ [ W ]
§ ·
¨
¸
© ¹
]
[ [
[ [
W ]
*LYHQ
[˜ JDP [ W W
[˜ JDP [ W W
[˜ JDP [ W W
[˜ JDP [ W W
] )LQG W W
[ [ W ]
§ ·
¨
¸
© ¹
]
[ [
[ [
W ]
*LYHQ
[˜ JDP [ W W
[˜ JDP [ W W
[˜ JDP [ W W
[˜ JDP [ W W
] )LQG W W
§ ·
¨
¸
© ¹
]
[ [ [ [
[ [
W ]
W ]
*LYHQ
[˜ JDP [ W W
[˜ JDP [ W W
[˜ JDP [ W W
[˜ JDP [ W W
] )LQG W W
]
§ ·
¨
¸
© ¹
L § ·
¨
¸
¨ ¸
¨ ¸
¨ ¸
¨
¸
¸
GDWD ¨
¨ ¸
¨ ¸
¨
¸
¨ ¸
¨ ¸
¨
¸
© ¹
1XPEHURIGDWDSRLQWV
7 GDWD
L
W GDWD
L
; 7
L
Q URZV GDWD
E LQWHUFHSW ; <
E
6ORSH
E VORSH ; <
E
3ORW
I ; E E ˜;
L
L < W
L L
L
= W
L
Q
,QWHUFHSW
W GDWD
L L
L
u I ;
W
;
,QWHUFHSW
E LQWHUFHSW ; =
E
6ORSH
E VORSH ; =
E
3ORW
I ; E E ˜;
u I ;
W
;
Section 11.2
Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
DOI
JDP [ DOI EHW H[Sª«
ȼ
JDP [ DOI EHW H[Sª«
»
ǻ
DOI˜ [ º
«¬ «¬ EHW ˜ [ »¼ »¼
DOI EHW
ȼ
»
ǻ
> EHW [ @ º
«¬ «¬ DOI˜ [ »¼ »¼
EHW [ [ [ [
[ [
*LYHQ
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
] )LQG DOI EHW
]
[ § ·
¨
¸
© ¹
[ [ [
[ [
*LYHQ
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
] )LQG DOI EHW
[ ]
§ ·
¨
¸
© ¹
[ [ [
[ [
*LYHQ
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
] )LQG DOI EHW
[ ]
§ ·
¨
¸
© ¹
[ [ [
[ [
*LYHQ
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
] )LQG DOI EHW
]
§ ·
¨
¸
© ¹
Solutions to Chemical and Engineering Thermodynamics, 5th ed
[ [ [ [
[ [
*LYHQ
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
] )LQG DOI EHW
§ ·
¨
¸
© ¹
]
[ [ [ [
[ [
*LYHQ
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
] )LQG DOI EHW
§ ·
¨
¸
© ¹
]
[ [ [ [
[ [
*LYHQ
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
] )LQG DOI EHW
§ ·
¨
¸
© ¹
]
[ [ [ [
[ [
*LYHQ
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
] )LQG DOI EHW
§ ·
¨
¸
© ¹
]
L § ·
¨ ¸
¨
¸
¨ ¸
¨ ¸
GDWD ¨
¸
¨ ¸
¨ ¸
¨ ¸
¨
¸
© ¹
< DL
L
; 7
L
7 GDWD
L
L E L GDWD
L = EL
L
L
D
L D L GDWD
E
7
7
1XPEHURIGDWDSRLQWV
Q URZV GDWD
Q
,QWHUFHSW
E LQWHUFHSW ; <
E
6ORSH
E VORSH ; <
E
3ORWV
I ; E E ˜;
I ;
D
WUDFH
WUDFH
;
,QWHUFHSW
E LQWHUFHSW ; =
E
6ORSH
E VORSH ; =
E
3ORWV
I ; E E ˜;
I ;
E
WUDFH
WUDFH
;
Section 11.2
Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
ª
ª
W˜ *
º ª
º º» »º
»
¬ [ [ ˜ *¼ «¬ > [ [˜ *@ »¼ »¼ »¼
*
JDP [ * * W W H[S« [ « W˜ ª«
«
¬
«
¬
ª
ª
W˜ *
º ª
º º» º»
»
¬ [ [ ˜ *¼ «¬ > [ [ ˜ *@ »¼ »¼ »¼
*
JDP [ * * W W H[S« [ « W˜ ª«
«
¬
* «
¬
* [ W OQ *
W [ OQ *
[ [
[ [
*LYHQ
[˜ JDP [ * * W W
[˜ JDP [ * * W W
[˜ JDP [ * * W W
[˜ JDP [ * * W W
W
OQ *
OQ *
W
] )LQG * * W W
* ]
* ]
W [ OQ *
[ W § ·
¨
¸
¸
¨
]
¨ ¸
¨ ¸
©
¹
OQ *
[ [
[ [
*LYHQ
[˜ JDP [ * * W W
[˜ JDP [ * * W W
[˜ JDP [ * * W W
[˜ JDP [ * * W W
W
OQ *
W
] )LQG * * W W
OQ *
·
¨§
¸
¸
¨
]
¨ ¸
¨ ¸
©
¹
Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
[ [ [ [
[ [
*LYHQ
[˜ JDP [ * * W W
[˜ JDP [ * * W W
[˜ JDP [ * * W W
[˜ JDP [ * * W W
OQ *
W
OQ *
W
§¨ ·¸
¸
] ¨
¨ ¸
¨ ¸
©
¹
] )LQG * * W W
* ]
* ]
[ OQ *
W [ OQ *
W [ [
[ [
*LYHQ
[˜ JDP [ * * W W
[˜ JDP [ * * W W
[˜ JDP [ * * W W
[˜ JDP [ * * W W
OQ *
W
OQ *
W
§¨ ¸·
¸
] ¨
¨ ¸
¨ ¸
©
¹
] )LQG * * W W
[ [ [ [
[ [
*LYHQ
[˜ JDP [ * * W W
[˜ JDP [ * * W W
[˜ JDP [ * * W W
[˜ JDP [ * * W W
W
OQ *
W
OQ *
§ ·
¨
¸
¸
¨
]
¨ ¸
¨ ¸
©
¹
] )LQG * * W W
[ [ * ]
W OQ *
[ [
OQ *
W *LYHQ
[˜ JDP [ * * W W
[˜ JDP [ * * W W
[˜ JDP [ * * W W
[˜ JDP [ * * W W
OQ *
W
W
OQ *
§ ·
¨
¸
¸
¨
]
¨ ¸
¨ ¸
©
¹
] )LQG * * W W
[ [ * ]
* ]
W [ [
OQ *
[ [
W *LYHQ
[˜ JDP [ * * W W
[˜ JDP [ * * W W
[˜ JDP [ * * W W
[˜ JDP [ * * W W
W
OQ *
W
OQ *
OQ *
] )LQG * * W W
]
[ [
* ]
§ ·
¨
¸
¨ ¸
¨ ¸
¨ ¸
©
¹
Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
[ [ [ [
[ [
*LYHQ
[˜ JDP [ * * W W
[˜ JDP [ * * W W
[˜ JDP [ * * W W
[˜ JDP [ * * W W
OQ *
W
OQ *
W
§ ·
¨
¸
¸
] ¨
¨ ¸
¨ ¸
©
¹
] )LQG * * W W
L § ·
¨ ¸
¨
¸
¨ ¸
¨ ¸
GDWD ¨
¸
¨ ¸
¨ ¸
¨ ¸
¨
¸
© ¹
1XPEHURIGDWDSRLQWV
7 GDWD
L
* GDWD
L
; 7
L
Q
Q URZV GDWD
,QWHUFHSW
E LQWHUFHSW ; <
E
6ORSH
E VORSH ; <
E
3ORWV
I ; E E ˜;
* GDWD
L L
L < *
L L
L
= *
L
L
L
u I ;
*
;
,QWHUFHSW
E LQWHUFHSW ; =
E
6ORSH
E VORSH ; =
E
3ORW
I ; E E ˜;
I ;
*
u 1RWDYHU\DFFXUDWHILW
VKRXOGXVHDTXDGUDWLFIXQFWLRQRI7
;
Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
JDP [ $ % H[Sª¬ $ ˜ % ˜ [ ˜ %˜ [ º¼
JDP [ $ % H[Sª¬ $ ˜ % ˜ [ ˜ %˜ [ º¼
$ % [ [ [ [
[ [
[ [
[ [
*LYHQ
[˜ JDP [ $ %
[˜ JDP [ $ %
[˜ JDP [ $ %
[˜ JDP [ $ %
] )LQG $ %
]
[ § ·
¨
¸
© ¹
[ *LYHQ
[˜ JDP [ $ %
[˜ JDP [ $ %
[˜ JDP [ $ %
[˜ JDP [ $ %
] )LQG $ %
]
[ *LYHQ
§ ·
¨
¸
© ¹
[ [ [
[˜ JDP [ $ %
[˜ JDP [ $ %
[˜ JDP [ $ %
[˜ JDP [ $ %
] )LQG $ %
[ ]
[ [
§ ·
¨
¸
© ¹
[ [ [
[ [
*LYHQ
[˜ JDP [ $ %
[˜ JDP [ $ %
[˜ JDP [ $ %
[˜ JDP [ $ %
] )LQG $ %
]
§ ·
¨
¸
© ¹
Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
[ [ [ [
[ [
*LYHQ
[˜ JDP [ $ %
[˜ JDP [ $ %
[˜ JDP [ $ %
[˜ JDP [ $ %
] )LQG $ %
§ ·
¨
¸
© ¹
]
[ [ [ [
[ [
[ [
[ [
[ [
[ [
*LYHQ
[˜ JDP [ $ %
[˜ JDP [ $ %
[˜ JDP [ $ %
[˜ JDP [ $ %
] )LQG $ %
§ ·
¨
¸
© ¹
]
% ]
$ ]
[ [ *LYHQ
[˜ JDP [ $ %
[˜ JDP [ $ %
[˜ JDP [ $ %
[˜ JDP [ $ %
] )LQG $ %
§ ·
¨
¸
© ¹
]
[ [ *LYHQ
[˜ JDP [ $ %
[˜ JDP [ $ %
[˜ JDP [ $ %
[˜ JDP [ $ %
] )LQG $ %
§ ·
¨
¸
© ¹
]
L § ·
¨ ¸
¨
¸
¨ ¸
¨ ¸
GDWD ¨
¸
¨ ¸
¨ ¸
¨ ¸
¨
¸
© ¹
7 GDWD
L
L D L GDWD
L E L GDWD
L < DL
L
= EL
L
; 7
L
L
1XPEHURIGDWDSRLQWV
Q URZV GDWD
Q
Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
7 GDWD
L
D L GDWD
L E L GDWD
L < DL
L = EL
L
L
; 7
L
L
1XPEHURIGDWDSRLQWV
Q URZV GDWD
Q
,QWHUFHSW
E LQWHUFHSW ; <
E
6ORSH
E VORSH ; <
E
3ORWV
I ; E E ˜;
u I ;
D
WUDFH
WUDFH
;
,QWHUFHSW
E LQWHUFHSW ; =
E
6ORSH
E VORSH ; =
E
3ORWV
I ; E E ˜;
u 1RWDYHU\DFFXUDWHILWSHUKDSV
VKRXOGXVHDTXDGUDWULF
I ;
E
WUDFH
WUDFH
;
Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
(WK\OHVWHUSURSDQRLFDFLGKDVWKHVWUXFWXUH&+&+2&2&+&+
6RLWKDV&+JURXSV&+JURXSVDQG&22JURXS
U ˜ ˜ U
T ˜ ˜ T
/ / U T ˜ U T [ I [ [˜ T
U T ˜ U [˜ U
I [ [˜ U [ ˜ U
T /
/
[ ˜ T
T [ [˜ T [ ˜ T
U [˜ T [ ˜ T
[ ˜ U
[˜ U [ ˜ U
W
W
ª § I [ · u T˜ OQ§ T [ · I [ ˜ § / U˜ / · º ˜ H[Sª ª T˜ OQ T [ T [ ˜ W T˜ T [ ˜ §
Ἴ
¸
¨
¸
¨
¸» ««
¨
¸»»
U ¹ ¼
©
¬ © [ ¹
© I [ ¹
¬¬
© T [ T [ ˜ W T [ ˜ W T [ ¹ ¼ ¼
JDP [ W W H[S« OQ¨
W
W
ª § I [ · ˜ T˜ OQ§ T [ · I [ ˜ § / U˜ / · º ˜ H[Sª ª T˜ OQ T [ ˜ W T [ T u T [ ˜ §
Ἴ
¸
¨
¸
¨
¸» ««
¨
¸»»
U ¹ ¼
©
¬ © [¹
© I [ ¹
¬¬
© T [ T [ ˜ W T [ ˜ W T [ ¹ ¼ ¼
JDP [ W W H[S« OQ¨
W W [ [ [ [
[ [
*LYHQ
[˜ JDP [ W W
[˜ JDP [ W W
[˜ JDP [ W W
[˜ JDP [ W W
] )LQG W W
]
[ [ § ·
¨
¸
© ¹
[ [
[ [
*LYHQ
[˜ JDP [ W W
[˜ JDP [ W W
[˜ JDP [ W W
[˜ JDP [ W W
] )LQG W W
[ ]
[ § ·
¨
¸
© ¹
[ [
[ [
*LYHQ
[˜ JDP [ W W
[˜ JDP [ W W
[˜ JDP [ W W
[˜ JDP [ W W
] )LQG W W
[ ]
[ § ·
¨
¸
© ¹
[ [
[ [
*LYHQ
[˜ JDP [ W W
[˜ JDP [ W W
[˜ JDP [ W W
[˜ JDP [ W W
] )LQG W W
[ ]
[ § ·
¨
¸
© ¹
[ [
[ [
*LYHQ
[˜ JDP [ W W
[˜ JDP [ W W
[˜ JDP [ W W
[˜ JDP [ W W
] )LQG W W
]
§ ·
¨
¸
© ¹
Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
[ [ [ [
[ [
JLYHQ
[˜ JDP [ W W
[˜ JDP [ W W
[˜ JDP [ W W
[˜ JDP [ W W
] )LQG W W
§ ·
¨
¸
© ¹
]
[ [ [ [
[ [
*LYHQ
[˜ JDP [ W W
[˜ JDP [ W W
[˜ JDP [ W W
[˜ JDP [ W W
] )LQG W W
]
[ [ [ [
§ ·
¨
¸
© ¹
[ [
*LYHQ
[˜ JDP [ W W
[˜ JDP [ W W
[˜ JDP [ W W
[˜ JDP [ W W
] )LQG W W
]
RPLWWKLVSRLQWIURPWKHILWWLQJ
L § ·
¨
¸
¨ ¸
¨ ¸
GDWD ¨ ¸
¨
¸
¨ ¸
¨ ¸
¨ ¸
©
¹
7 GDWD
L
W GDWD
L
; 7
L
,QWHUFHSW
E LQWHUFHSW ; <
E
6ORSH
E VORSH ; <
E
3ORW
I ; E E ˜;
L § ·
¨
¸
© ¹
W GDWD
L
L < W
L L
L
= W
L
L
L
u I ;
W
;
,QWHUFHSW
E LQWHUFHSW ; =
E
6ORSH
E VORSH ; =
E
3ORW
I ; E E ˜;
u I ; W
;
Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
6HHVROXWLRQWR3UREOHPIRUIXUWKHUDQDO\VLVRIWKLVSUREOHP u . 6LQFH78& . $5IRUWKLVV\VWHP $ u DQG
PRO ˜ .
PRO
* H[ $[[ Ÿ 57OQJ $[ 57OQJ $[ 1RZFDOFXODWH//(E\VROYLQJ [, J, [,
IROORZV
7 . [,, J,, [,, DQG[ , J , [ ,
[ ,, J ,, [ ,, 7KHUHVXOWVDUHDV
[ 2, [ 2,, 7 . [ 2, [ 2,, Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
6WDUWE\GHYHORSLQJH[SUHVVLRQIRUWKH//(FULWLFDOSRLQWRIWKH)ORU\+XJJLQVHTXDWLRQ 1RWHWKDW
WKHDOJHEUDJHWVTXLWHFRPSOLFDWHGVRWKDWWKLVZDVGRQHZLWK0DWKHPDWLFD
1
P1 *
1 OQ I 1 OQ I FII 1 P1 ZLWKI
DQGI
57
1 P1 1 P1 1 P P1 1 P PF 1 P1 § P1 ·
OQ ¨
¸
© 1 P1 ¹
§ w* ·
¨
¸
57 © w1 ¹ 1
*
57
OQ I P I FI
VRWKDWOQJ OQ
§ w* ·
¨
¸
57 ©¨ w1 ¹¸
§ P 1
·
P ¸
1 P1 ¨
¨ 1 P1
¸
1
P1
P 1 1 P
© ¹
1 P1 P
1 P1 1
1 P1 I
P I FI
[
§
·
P 1
P
¸
1 P1 ¨ ¨ 1 P1 1 P1 ¸ P 1 1 F
P 1F
©
¹
P1 1 P1 1 P1 § w * ·
¨
¸
57 ¨© w1 ¸¹
1
P 1 1 1 P1 § P 1
·
P
¸
1 P1 ¨
¨ 1 P1
¸
1
P1
P ©
¹
P
1 P1 § P 1
·
P
P 1 P
¸
1 P1 ¨
¸
¨ 1 P1 1 P1 1 P1 ¹
1 P1 © P1 1
P 1 P
1 P1 P1 P 1 1 F
P 1F
1 P1 1 P1 § w* ·
1RZWRILQGWKHFULWLFDOSRLQWVHWWLQJ ¨ ¸
¨ w1 ¸
© ¹ 1
ELWRIDOJHEUD WKDWDWWKHFULWLFDOSRLQW
P
ZHREWDLQ DIWHUTXLWHD
§
·
¨ ¸ P
©
P
P¹
1RWHWKDWȤKDVWREHRUJUHDWHU GHSHQGLQJRQWKHYDOXHRIP IRUOLTXLGOLTXLGSKDVHVSOLWWLQJ
WRRFFXULQWKH)ORU\+XJJLQVPRGHO
I &
§ w * ·
DQG ¨ ¸
¨ w1 ¸
© ¹ 1
DQGF&
,Q3UREOHPZHVKRZHGWKDWWKHDFWLYLW\FRHIILFLHQWVIRUWKLVPRGHODUH
7KHUHIRUH OQ J x $ x $ x x DQGE\V\PPHWU\ OQ J x $ x $ x x
:HDOVR
VKRZHGWKDWWKHV\VWHPKDGDQD]HRWURSH6RKHUHDOOZHKDYHWRGRLVFKHFNIRU//(:LOOGRWKLVE\
FRPSXWLQJWKHXSSHUFRQVROXWHWHPSHUDWXUHIRUWKLVV\VWHPIROORZLQJWKHGHULYDWLRQLQ(TQVWR Solutions to Chemical and Engineering Thermodynamics, 5th ed
$ ˜ ˜ Section 11.2
$ ˜ ˜ *H[ [ [˜ [ ˜ > $˜ [ $˜ [ @
G
˜ [˜ [
˜ [
˜ ˜ [
*H[ [ o [ ˜ [
G[
G
G[
*H[ [ o ˜ [ 1RZQHHGWRILQG7XFIURPPD[LPXPYDOXHRI57 [ [ GA*H[G[A
I [ [˜ [ ˜ [
˜
G
˜ [˜ [
˜ [
˜ ˜ [
I [ o [ ˜ [
G[
[ JLYHQ
G
I [
G[
[ ILQG [
I [
78&
[
u 78& I .VRWKHUHLV
.7KLVLVEHORZWKHWHPSHUDWXUHRI&
QROLTXLGOLTXLGHTXLOLEULXP2QO\9/(ZLWKDQD]HRWURSHDVVKRZQLQ
SUREOHP
Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
7KHVWDUWLQJSRLQWLV
§ 1 ·
§ 1 ·
11 1* 1 * 157 OQ ¨
¸ 1 57 ¨
¸$
1
1
1
1
1
¹
¹
1
© © 1*
§ w1* ·
¨
¸
© w1 ¹731
§ 1 · 157
1 57
1 57
1
11 * 57 OQ ¨
$
$
¸
1
1
1
1
1
1
1
1
1
1 1 ¹
© § 1 ·
1 57
1
11 * 57 OQ ¨
$
$
¸ 57 57 1
1
1
1
1
1
1 1 ¹
© § w 1* ·
¨¨
¸
¸
© w1 ¹731 $1 $1 $1 1
57
57
1 1 1 1 1 1 1 1 1 57 1 1 1
$1 $1 1
1 1 1 1 1 1 1 571 $1 1 1 1 1 1 § w 1* ·
¨¨
¸
¸
© w1 ¹731
571 571 $1 1 1 1 1 1 1 1 1 57 ª¬ 1 1 1 11 º¼
1 1 1 1RZVHWWLQJ
§ w 1* ·
¨¨
¸
¸
© w1 ¹731 $1 1 1 57 ª¬ 11 1 º¼
1 1 1 $1 571 $1 Ÿ 57 1 1 1 1 1 1 1 1 1 $11 $[[ ZKLFKUHGXFHVWR57 $[[ RU7
5
DVZDVIRXQGLQ(TQE1H[WVHWWLQJ
57 ª¬ 11 1 º¼
§ w 1* ·
$1 $1 ¨¨
¸
¸
1 1 1 1 1 1 1 © w1 ¹731
Ÿ 57 ª¬ 11 1 º¼
1
1 1 Ÿ 57 1 1 $1 1 1 $1 1
1 1 RU1 1 $1 1
1 1 1 Ÿ 1
˜
57 1 1 $11 1 RU[
Ÿ 57 1 1 $11 8VLQJWKLVLQWKHHTXDWLRQ
157
[ Ÿ [
[
DVIRXQGHDUOLHU6RWKDWWKHFULWLFDORUXSSHUFRQVROXWHWHPSHUDWXUHLV 7
§ w 1* ·
%XWZHKDYHVKRZQWKDW ¨¨
¸
¸
© w1 ¹731
DERYHOHDGVWR
57 1 1 $
DVEHIRUH
5
Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
"
7KHSURFHGXUHWKDWZLOOEHXVHGLVWRILWWKHSDUDPHWHUVLQDQDFWLYLW\FRHIILFLHQWPRGHOWRWKH//(GDWD
DWHDFKWHPSHUDWXUHWRFDOFXODWHWKHFRPSOHWH*H[FXUYH7KHQXVHDILQLWHGLIIHUHQFHDSSUR[LPDWLRQ
WRFDOFXODWH+H[DQGIURP*H[DQG+H[WRFRPSXWH6H[
1RWHWKDWWKHFRPSRVLWLRQVLQWKHWZRSKDVHVDUHTXLWHGLIIHUHQWWKHUHIRUHWKHRQHFRQVWDQW
0DUJXOHVHTXDWLRQLVQRWDSSOLFDEOH7KHUHIRUH,ZLOOXVHWKH9DQ/DDUHTXDWLRQ
DOI
º»
ǻ
DOI˜ [ º »
«¬ «¬ EHW ˜ [ »¼ »¼
JDP [ DOI EHW H[Sª«
DOI EHW
ȼ
»
ǻ
> EHW [ @ º
«¬ «¬ DOI˜ [ »¼ »¼
LQLWLDOJXHVVHV
EHW 5 JDP [ DOI EHW H[Sª«
7 [ [ [ [
[ [
*LYHQ
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
] )LQG DOI EHW
]
DOI ]
EHW ]
L § ·
¨
¸
© ¹
DOI
[ ˜ L
[ L
*H[ [ ˜ 5˜ 7˜ OQ JDP [ DOI EHW
L
L
L
EHW
[
[ ˜ 5˜ 7˜ OQ JDP [ DOI EHW
L
L
7 [ [ [ [
[ [
*LYHQ
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
] )LQG DOI EHW
DOI ]
]
EHW ]
§ ·
¨
¸
© ¹
DOI
EHW
Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
*H[ [ ˜ 5˜ 7˜ OQ JDP [ DOI EHW
L
L
L
[ ˜ 5˜ 7˜ OQ JDP [ DOI EHW
L
L
*H[
*H[
[
§ *H[L *H[L ·
¨
¸
7
7 ¹
©
'HULY L
)LQLWHGLIIHUHQFHDSSUR[LPDWLRQIRUG *H[7 G7ZKLFK
LVHTXDOWR+H[7A
+H[ ˜ 'HULY
L
L
+H[
[
*H[ ˜ *H[ *H[
L
L
6H[ L
L
$YHUDJHWRJHW*H[DWR&
+H[ *H[
L
L
6H[
[
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Section 11.2
6ROXWLRQLQIROGHU$VSHQIRUWH[WERRN!&KDSWHU!6HF!3URE
6ROXWLRQZDVREWDLQHGXVLQJ0$7+&$'
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Section 11.2
G H[ T x
G x
b·
§
¨ a ¸ RTx x T¹
©
b·
§
xG xG x RT OQ x x RT OQ x ¨ a ¸ RTx x T¹
©
Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
)ROORZLQJWKHSURFHGXUHLQWKHWH[WERRN
wG x
b·
§
G G RT OQ x RT RT OQ x RT ¨ a ¸ RT x x
T¹
wx
©
w G x
wx
b·
§
G G RT OQ x RT OQ x ¨ a ¸ RT x x
T¹
©
RT RT §
b·
¨ a ¸ RT x
x ©
T¹
RT
b·
§
¨ a ¸ RT RU
x x
T¹
x x
©
b
a
T x x
b·
§
¨ a ¸ RU
x x
T¹
©
7KHPD[LPXPYDOXHRIx x RFFXUVDWx
b
T
a VR
b
a
T
x RU
b
KDVDPLQLPXPYDOXH
T
b
RUT
KDVDPD[LPXPYDOXH&RQVHTXHQWO\WKLVLVWKH8&67
a
7KHPLQLPXPYDOXHRIx x RFFXUVDWx x DQGLV]HUR7KHQ
b
f RUT T
7KXVWKLVPRGHOGRHVQRWKDYHD/&67
Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
G
H[
x x a bT G
x G x RT OQ x x G x RT OQ x x x a bT
x G x RT OQ x x G x RT OQ x x x a bT
x RT
§ wG ·
x RT
G RT OQ x x a bT
¨
¸ G RT OQ x x
x
© wx ¹
G RT OQ x G RT OQ x x a bT
§ w G · RT RT
§ w G ·
a
bT
IRUDQXSSHUFRQVROXWHWHPSHUDWXUH
¨ ¸
¨ ¸ x
© wx ¹ x
© wx ¹
RT
a bT 7KHODUJHVWYDOXHRIT RFFXUVZKHQx x VR
x x
RT
a bT RU Rb T a RUT
a Rb
1RWHWKDWXVLQJWKHSDUDPHWHUVLQWKHQH[WSDUWRIWKHSUREOHPT .
)URPWKLVUHVXOWZHFDQVHHWKDWWKHPRGHOKDVDQXSSHUFRQVROXWHWHPSHUDWXUHWRWKDWLW
DOORZVIRUOLTXLGOLTXLGHTXLOLEULXP
$OVRVLQFHWKHXSSHUFRQVROXWHWHPSHUDWXUHPXVWEHSRVLWLYHWKHFRQVWUDLQWVDUHWKDWa PXVWEH
SRVLWLYHDQGWKDWb PXVWEHOHVVWKDQR
7KHVSLQRGDOFRPSRVLWLRQVDUHIRXQGIURP
x RT
§ wG ·
x RT
G RT OQ x x a bT
¨
¸ G RT OQ x x
x
© wx ¹
G RT OQ x G RT OQ x x a bT G RT OQ x G RT OQ x x a bT
&DQQRWREWDLQQXPHULDOVROXWLRQVIRUSDUWVFDQGGXQWLOZHNQRZWKHYDOXHVRI G DQGG D &OHDUO\ZDWHU+0)DUHFRPSOHWHO\PLVFLEOH8VLQJmWRUHSUHVHQWPDVVIUDFWLRQ
ZDWHUDQGQ±KH[DQRODSSHDUWREHLPPLVFLEOHIRUVRPHFRPSRVLWLRQV
ZDWHUDWm DQGQ±KH[DQROm LVDOLPLWRIPXWXDOVROXELOLW\
ZDWHUDWm DQGQ±KH[DQROm LVWKHRWKHUOLPLWRIPXWXDOVROXELOLW\
E ,WLVQRWHDV\WRSLFNRXWWKHSODLWSRLQWVLQFHLWLVQRWLGHQWLILHGDQGLWLVWRUHDGWKHVPDOOGLDJUDP
DFFXUDWHO\7DNLQJWKHSODLWSRLQWWREHWKHKLJKHVWSRLQWRQWKH//(ERXQGDU\ ZKLFKPD\QRWEHWKHFDVH WKHSODLWSRLQWLVDSSUR[LPDWHO\
ZDWHUm QKH[DQROm DQG+0)m F 7KHLQLWLDOPL[WXUHKDVWKHIROORZLQJZHLJKWIUDFWLRQVZDWHU +0) DQGQKH[DQRO ,WLV
GLIILFXOWWRUHDGWKHWLHOLQHVRQWKLVOLWWOHGLDJUDPEXWDQHVWLPDWHLV
ZDWHUm QKH[DQROm DQG+0)m ZDWHUm QKH[DQROm DQG+0)m 7KHVHDUHWKHFRPSRVLWLRQVRIWKHFRH[LVWLQJSKDVHV1RZRQWRWKHPDVVEDODQFH/HW;EHWKHIUDFWLRQRI
NJLQSKDVH,DQG ; LVWKHIUDFWLRQRINJLQSKDVH0DVVEDODQFHRQWKHZDWHULV
; ; ; VRWKDW; 7KHUHIRUHNJDUHLQWKLVSKDVHWKDWKDVDPDVV
IUDFWLRQRIZDWHURI7KHVHFRQGSKDVHWKHQFRQWDLQV NJRIWKHSKDVHWKDWFRQWDLQVDPDVV
IUDFWLRQRIZDWHURI
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Section 11.2
%\ORRNLQJDWWKHGDWDLWLVFOHDUWKDWWKHGDWDLVQRLV\VRGRQRWH[SHFWDJRRG
FRUUHODWLRQ
7KHUHVXOWVXVLQJWKH157/PRGHODUHVKRZQEHORZ
Section 11.2
Solutions to Chemical and Engineering Thermodynamics, 5th ed
6\VWHPDFWLYLW\FRHIILFLHQWVDUHV\PHWULFVRG H[
RT OQ J 1RZXVLQJLQILQLWHGLOXWLRQGDWDA
Ax x RT OQ J Ax
u u OQ -PRO
.VR//(RQO\EHORZ.EXWWKDWLVEHORZWKH
u IUHH]LQJSRLQWRIERWKFRPSRQHQWVVRQR//(
D T8& HTQ
%XEEOHSRLQW %3 OQJ u u %3SUHVVXUH u u J EDU
xPJ P PPvap u u y% P
'HZSRLQWLVPXFKKDUGHUWRFDOFXODWHVLQFHOLTXLGFRPSRVLWLRQVDUHLQLWLDOO\XQNQRZQEXW
xiJ i Pi YDS
yi P y3
WKHYDSRUFRPSRVLWLRQLVNQRZQ3URFHGXUHLVWRJXHVVYDOXHRIx3 DQGx% VHHLIZHWKHQJHW
y3 y% § A · YDS
§ A · YDS
x3 H[S ¨
x% ¸ P3
y3 P DQGx% H[S ¨
x3 ¸ P%
© RT ¹
© RT ¹
§ A · YDS
§ A · YDS
x% ¸ P3 x%H[S ¨
x3 ¸ P%
DQGP x3 H[S ¨
© RT ¹
© RT ¹
6ROXWLRQLVx3 xE DQGP y% P ZLWK
A
RT
OQ F 6HYHUDOZD\VWRFKHFNIRUDQD]HRWURSH
,QHOHJDQWZD\LVWRFRPSXWH9/(DWDQXPEHURIFRPSRVLWLRQVDQGVHHLIDQD]HRWURSHRFFXUV x% 3
y% VRQRD]HRWURSHRFFXUV
1RWHWKDWx% LVEXEEOHSWRIHTXLPRODUPL[WXUHDQGy% LVGHZSRLQW
0RUHHOHJDQWZD\RIFKHFNLQJIRUD]HRWURSH
xJ PYDS
y P DQGxJ PYDS
,ID]HRWURSHRFFXUVx
P YDS
WKDWLPSOLHV YDS
P
J
J
y P
y DQGx
y
%XWWKHPD[LPXPYDOXHRIWKHUDWLRRIWKHJ RFFXUVZKHQx o DQGx o J x o J x o 7KHUHIRUHDQD]HRWURSHFDQQRWRFFXULQWKLVPL[WXUHVLQFHWKHVROXWLRQQRQLGHDOLW\
FDQQRWRYHUFRPHWKHYDSRUSUHVVXUHGLIIHUHQFH
D &OHDUO\ZDWHU+0)DUHFRPSOHWHO\PLVFLEOH8VLQJmWRUHSUHVHQWPDVVIUDFWLRQ
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Section 11.2
ZDWHUDWm DQGEXWDQROm LVDOLPLWRIPXWXDOVROXELOLW\
ZDWHUDWm DQGEXWDQROm LVWKHRWKHUOLPLWRIPXWXDOVROXELOLW\
ZDWHUDQGQEXWDQRODUHFRPSOHWHO\VROXEOHDQG+0)DQGQEXWDQRODUHFRPSOHWHO\VROXEOH
E ,WLVQRWHDV\WRSLFNRXWWKHSODLWSRLQWVLQFHLWLVQRWLGHQWLILHGDQGLWLVWRUHDGWKHVPDOOGLDJUDP
DFFXUDWHO\7DNLQJWKHSODLWSRLQWWREHWKHKLJKHVWSRLQWRQWKH//(ERXQGDU\ ZKLFKQRWEHWKHFDVH WKH
SODLWSRLQWLVDSSUR[LPDWHO\
ZDWHUm QEXWDQROm DQG+0)m F 7KHLQLWLDOPL[WXUHKDVWKHIROORZLQJZHLJKWIUDFWLRQVZDWHU +0) DQGQ
EXWDQRO ,WLVGLIILFXOWWRUHDGWKHWLHOLQHVRQWKLVOLWWOHGLDJUDPEXWDQHVWLPDWHLV
ZDWHUm QEXWDQROm DQG+0)m ZDWHUm QEXWDQROm DQG+0)m 7KHVHDUHWKHFRPSRVLWLRQVRIWKHFRH[LVWLQJSKDVHV1RZRQWRWKHPDVVEDODQFH/HW;EHWKHIUDFWLRQRI
NJLQSKDVH,DQG ; LVWKHIUDFWLRQRINJLQSKDVH0DVVEDODQFHRQWKHZDWHULV
; ; ; VRWKDW; 7KHUHIRUHNJDUHLQWKLVSKDVHWKDWKDVD
PDVVIUDFWLRQRIZDWHURI7KHVHFRQGSKDVHWKHQFRQWDLQV NJRIWKHSKDVHWKDWFRQWDLQVD
PDVVIUDFWLRQRIZDWHURI
6HHVROXWLRQWRWKLVSUREOHPLQWKHIROGHU$VSHQIRU7H[WERRN!&KDSWHU
6HHVROXWLRQWRWKLVSUREOHPLQWKHIROGHU$VSHQIRU7H[WERRN!&KDSWHU
6HHVROXWLRQWRWKLVSUREOHPLQWKHIROGHU$VSHQIRU7H[WERRN!&KDSWHU
6HHVROXWLRQWRWKLVSUREOHPLQWKHIROGHU$VSHQIRU7H[WERRN!&KDSWHU
6HHVROXWLRQWRWKLVSUREOHPLQWKHIROGHU$VSHQIRU7H[WERRN!&KDSWHU
Solutions to Chemical and Engineering Thermodynamics, 5th ed
$WWKHEXEEOHSRLQW DVVXPLQJDQLGHDOYDSRUSKDVH ¦ xiJ i Pi YDS ¦ yi P P 6R xBJ B PBYDS xWJ W PWYDS
P
D J B EDU Ÿ J B u u DQG y: u u
VR y%
x%,J ,% u E $W HTXLOLEULXP
RU
x%,J ,% x%,,J ,,% Ÿ J ,,%
,,
x%
J ,,:
, ,
x:
J:
,,
x:
u DOVR
xi,,J i,, DQG ¦ xi,,J i,, Pi YDS
F 6LQFH xi,J i,
J ,,%
¦ xi,J i, Pi YDS
P 7KH VHFRQG OLTXLG SKDVH ZLOO DOVR
KDYHDEXEEOHSRLQWSUHVVXUHRIEDU
7KRXJKWKHRYHUDOOFRPSRVLWLRQLVPROHLVREXWDQHLQIDFWWKHUHDUHUHDOO\WZRSKDVHVRQH
OLTXLGRIFRPSRVLWLRQPROHLVREXWDQHDQGWKHRWKHUOLTXLGRIFRPSRVLWLRQPROH
LVREXWDQH6LQFHDWOLTXLGOLTXLGHTXLOLEULXP xi,J i, xi,,J i,, DQG
ZHQHHGWRFDOFXODWHWKHEXEEOHSRLQWSUHVVXUHIRURQHSKDVHVLQFHWKHRWKHUSKDVHZLOOKDYHWKH
VDPHEXEEOHSRLQWSUHVVXUH$OVRVLQFHWKHSUHVVXUHVDUHQRWH[SHFWHGWREHYHU\KLJKZHZLOO
DVVXPHDQLGHDOYDSRUSKDVH6RWKHHTXDWLRQVZHZLOOXVHDUH
xiJ i Pi YDS
a
a
J LVRE xLVRE
J IXUI xLVRE
,
xIXUI
J ,IXUI
7KHUHIRUH
P
¦ yi P
P
IURPWKHSUREOHPVWDWHPHQW
f f "
+RZHYHUIURPWKHOLTXLGOLTXLGHTXLOLEULXPFRQGLWLRQZHKDYH
yi P DQG ¦ xiJ i Pi YDS
:HZLOOXVHWKHPROHOLTXLGIRUWKHFDOFXODWLRQV
¦ xi,J i, Pi YDS ¦ xi,,J i,, Pi YDS PEXEEOH
yLVRE
a
,,
xIXUI
J ,,IXUI Ÿ J ,IXUI xLVRE
¦ xiJ i P
,,
,,
IXUI IXUI
,
IXUI
f x xJ
u u u u u u u u
yIXUI
EDU Section 11.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
D $W//( xi,J i,
RS c h UV
RS c h UV
T
W
T
W
DQG
c x h H[SRST RTA c x h UVW c x h H[SRST RTA c x h UVW 7KHVHHTXDWLRQVDUHV\PPHWULFZLWKUHVSHFWWRWKHLQWHUFKDQJHRQWKHVXEVFULSWVDQG WKDWLV
UHSODFLQJ x E\ x x DQG x a x f E\ x \LHOGV H[DFWO\ WKHVDPH VHW RI HTXDWLRQV , ,
xi,,J i,, +HUHWKLVLPSOLHV
A
A
x, H[S
x,, H[S
x,
x,,
RT
RT
7KLV VXJJHVWV WKDW WKH HTXLOLEULXP LV V\PPHWULF 2I FRXUVH WKDW LV H[DFWO\ ZKDW ZH ZRXOG
H[SHFWZLWKWKHRQHFRQVWDQW0DUJXOHVH[SUHVVLRQ7KHUHIRUHZHKDYH
x+,
x+,,
,,
x(W2+
DQG
,
x(W2+
E +HUHDVLQWKHSUHYLRXVWZRSUREOHPVWKHEXEEOHSRLQWSUHVVXUHFDQEHFRPSXWHGIURPHLWKHU
OLTXLGSKDVHVLQFH xi,J i, xi,,J i,, $VVXPLQJDQLGHDOYDSRUSKDVHZHKDYH
xiJ i Pi YDS
,, ,,
yi P DQG P
¦ xiJ i Pi YDS ZKHUH WKH DFWLYLW\ FRHIILFLHQWV DQG YDSRU SUHVVXUH DUH JLYHQ LQ WKH SUREOHP VWDWHPHQW 7KH
VROXWLRQ SXWWLQJDOOWKHHTXDWLRQVLQWR0DWKFDG LV
P EDU y+ y(W2+ 7KHFRQGLWLRQIRUOLTXLGOLTXLGHTXLOLEULXPLV xi,J i, xi,,J i,, XVLQJWKHRQHFRQVWDQW0DUJXOHVHTQ
ZHKDYH RT OQ J i,
c
A xi,
a
J x
f OQ f J a x f Ÿ OQ JJ aaxx f
A
A
Ÿ
RT
RT
VRWKDW
J
h 1RZ
d a fi
H[Sda x f i
H[S x J
1RZWRFRPSXWHWKHSUHVVXUHLQWKHRQHSKDVHOLTXLGUHJLRQZHXVH
P
xJ PYDS xJ PYDS
d
a
fi
a
f c
h
u x H[S x x H[S x Section 11.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
P EDU
xDFHW
7ZR
SKDVH
UHJLRQ
6LQFHZHKDYHWKHFRQFHQWUDWLRQVRIWKHFRH[LVWLQJHTXLOLEULXPOLTXLGSKDVHZHFDQGHWHUPLQHWZR
ELQDU\SDUDPHWHUV$OVRVLQFHZHDUHLQWHUHVWHGLQWZRGLIIHUHQWWHPSHUDWXUHV //(DW q & DQG
q & ZHZDQWDQDFWLYLW\FRHIILFLHQWPRGHOZLWKVRPHEXLOWLQWHPSHUDWXUHGHSHQGHQFH
9/(DW RWKHUZLVHZHZLOOJHW//(ZLWKWKHVDPHFRPSRVLWLRQVDWDOOWHPSHUDWXUHV &RQVHTXHQWO\,ZLOO
XVHWKHWZRFRQVWDQW0DUJXOHVHTXDWLRQ
Ÿ RT OQ J x D Bx
RT OQ J x D B x
ZKHUH D i A i B Bi i B 7KHVHHTXDWLRQVDUHWREHXVHGZLWK x,J , x,,J ,, a
a
x, J ,
ILQG
x,,J ,, x,
x,
A
1RZXVLQJ T
x, xII
- PRO
B
f
f
x,,
x, 3XWWLQJDOOWKLVLQWR0DWKFDG,
- PRO q & WKHVDPHFRQVWDQWVDVDERYH,ILQG
Section 11.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
x0(.
P EDU
y0(.
$]HRWURSHLVSUHGLFWHGWRRFFXUDW x0(. | DQG P
EDU
1RWH//(GRHVQRWRFFXUDWWKLVKLJKHUWHPSHUDWXUH,ILWGLGWKHFDOFXODWHG P x GLDJUDPZRXOGKDYH
ERWKDQLQWHULRUPD[LPXPDQGPLQLPXPDVDIXQFWLRQRIWHPSHUDWXUHDQGWKHSUHGLFWHGxyGLDJUDPZRXOG
EHOLNHDVLGHZD\V6ZLWKWKHx-yOLQHFURVVLQJWKHx = yOLQHWZLFH
D &OHDUO\IURP
RT OQ J i x j ZHKDYH
G H[
xRT OQ J x RT OQ J xx x x
a
x
x x x N f x x PRO
7KHUHIRUHWKHXSSHUFRQVROXWHWHPSHUDWXUHIRUWKLVPRGHOLV
A
TXF
. R u Section 11.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
c
)LUVWGRWKH//(FDOFXODWLRQ$W//( xi,J i, xi,,J i,, +HUHWKLVLPSOLHV
A
A
x, H[S
x,, H[S
x,
x,,
RT
RT
DQG
A , A ,, x, H[S
x,, H[S
x
x
RT
RT
ZKHUH$ 7KHUHVXOWVXVLQJWKH0$7+&$'ZRUNVKHHWZLWKWKLVSUREOHPQXPEHUDUH
T . x+, F Ax I P
GH RT JK
1RZGR9/(FDOFXODWLRQ%DVLVLV xi H[S x+
RS c h UV
T
W
h RST c h UVW
RS c h UV
T
W
h RST c h UVW c
j
i
x+,, YDS
F Ax I P
GH RT JK
yi P DQGWKDW ¦ xi H[S i
j
i
YDS
7KHUHVXOWVXVLQJWKH0$7+&$'ZRUNVKHHWZLWKWKLVSUREOHPQXPEHUDUH
P EDU
P EDU P EDU
y+
T.
y+
T.
y+
P
T.
Section 11.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
//(
9DSRU
R
7HPSHUDWXUH &
3 EDU
/LTXLG
9DSRU
3 EDU
/LTXLG
9DSRU
/LTXLG
3 EDU
0ROHIUDFWLRQRInKH[DQH
9DSRU
R
7HPSHUDWXUH &
9/(
5HJLRQ
9/(5HJLRQ
9//(/LQH
/
/
//(5HJLRQ
0ROHIUDFWLRQRInKH[DQH
Section 11.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
7KHUHDUHPDQ\GLIIHUHQWDOJRULWKPVWKDWFRXOGEHXVHG2QHLVDVHTXHQWLDORQHRIILUVWWHVWLQJIRU
//(LI//(GRHVQRWRFFXUWKHQWHVWIRU9/(,I//(RFFXUVRQHPXVWDOVRWHVWIRU9//(HWF,Q
DOOWKHSRVVLELOLWLHVWKDWPXVWEHWHVWHGIRUDUHRQO\DOLTXLGSKDVHDVVWDEOHSKDVHRQO\DYDSRUDV
WKHVWDEOHSKDVH9/(ZLWKDOLTXLGULFKLQFRPSRQHQWDVWKHHTXLOLEULXPSKDVHV9/(ZLWKD
OLTXLGULFKLQFRPSRQHQWDVWKHHTXLOLEULXPSKDVHVRU9//(
Section 11.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
J3LQI J+LQI 3YDS3 3YDS+ 8VLQJWKHYDQ/DDUHTXDWLRQ
E OQ J+LQI
D OQ J3LQI
L [ ˜ L
L
D
º
ª
«
»
[
« ¨§
L ·
D
¸ »
« ¨ [ ¸ »
E
L¹ ¼
©
J3L H[S
E
º
ª
«
» 3L [L˜ 3YDS3 ˜ J3L [L ˜ 3YDS+ ˜ J+L
[
« ¨§
L·
E
¸ »
« ¨ »
D [L ¸
©
¹ ¼
J+L H[S
\3 3L
L
[ ˜ 3YDS3 ˜ J3L
L
3
L
[L
\3 L
[L
6RWKLVV\VWHPH[KLELWVHLWKHUD]HRWURS\RU//(7HVWIRU//(
[ [ *LYHQ
D
ª
ȼ
Ǥ
D [ · »
« ¨ E [¸ »
©
¹ ¼
[˜ H[S«
E
ª
ȼ
Ǥ
E [· »
« ¨ D [ ¸ »
©
¹ ¼
[ ˜ H[S«
D
º»
ª
Ǥ
D [ · »
« ¨ E [¸ »
©
¹ ¼
[˜ H[S«
E
ª
º»
Ǥ
E [ · »
« ¨ D [ ¸ »
©
¹ ¼
[ ˜ H[S«
\ )LQG [ [
[ \
[ \
[
[ [
[
[
[ [
[
[
6RWKHUHLV//(
Section 11.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Section 11.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
J 3LQI
J +LQI
3YDS3
3YDS+
8VLQJWKHYDQ/DDUHTXDWLRQ
D
OQ J 3LQI
L
J 3L
H[S
OQ J +LQI
E
L
[L
D
[L
D
E J +L
E
H[S
[L
E D
[L
3L
[L 3YDS3 J 3L
[L 3YDS+ J +L
[L
3
\3L
L
[
L
[L 3YDS3 J 3L
3L
Section 11.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
6RWKLVV\VWHPH[KLELWVHLWKHUD]HRWURS\RU//(7HVWIRU//(
[ [ *LYHQ
D
[ H[S
\3
[
DL
E E
[ H[S
E D
D
E [
[
[
\
[
D
[ H[S
[L [ H[S [
E
E D
[
[
ILQG [ [
[
\
[
[
[
[
[
\
[
[
[
[
[
6RWKHUHLV//(
Section 11.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
3YDS$ 3YDS% 3YDS$
GSG% JDP%LQI 3YDS$ GSG%
3YDS%
DOI OQ JDP$LQI
DOI
GSG$ JDP%LQI
JDP$LQI [ ˜ L
L L
[ 3YDS$
JDP$LQI
EHW
EHW
º
«
»
EHW ˜ [ º »
« ª«
L »
« « DOI˜ [ » »
L
¬¬
¼ ¼
L
\ L
[ ˜ J$ L˜ 3YDS$
L
3
L
\
3
[
J%L H[Sª
3 [ ˜ J$ L˜ 3YDS$ [ ˜ J%L˜ 3YDS%
L
3YDS% GSG$
[
DOI
º
«
»
˜
DOI
[
º
« «ª
L
» »
« « EHW ˜ [ » »
L ¼ ¼
¬¬
GSG$
EHW OQ JDP%LQI
J$ L H[Sª
L
3YDS%
GSG%
[
[
6\VWHPKDVDKRPRJHQHRXVD]HRWURSH
1RZFKHFNIRU//(
DOI
J$$ [[ H[Sª«
ȼ
»
ǻ
DOI˜ [[ º
«¬ «¬ EHW ˜ [[ »¼ »¼
[[ [[ [[
J%% [[ H[Sª«
EHW
ȼ
»
ǻ
EHW ˜ [[ º
« « DOI˜ [[ » »
¬¬
¼ ¼
[[ [[ [[
*LYHQ
[[J$$
˜
[[
[[J$$
˜
[[
[[J%%
˜
[[
[[J%%
˜
[[
] ILQG [[ [[
]
§ ·
¨ ¸
© ¹
6RRQO\VROXWLRQWRWKH//(FDOFXODWLRQ
LVQRWDUHDOVROXWLRQVRQR//(
7KHUHIRUHFRQFOXVLRQLVWKDWRQO\9/(
H[LVWVDQGV\VWHPKDVDQD]HRWURSH
Section 11.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
DOI
º»
ǻ
DOI˜ [ º »
«¬ «¬ EHW ˜ [ »¼ »¼
JDP [ DOI EHW H[Sª«
DOI JDP [ DOI EHW H[Sª«
EHW
º»
»
ǻ
> EHW [ @ º
«¬ «¬ DOI˜ [ »¼ »¼
EHW [ [ [ [
[ [
*LYHQ
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
] )LQG DOI EHW
DOI ]
]
§ ·
¨
¸
© ¹
EHW ]
3YDS$ 3YDS: L [ L
˜
[ [
L
DOI
º
«
»
DOI˜ [
º» »
« ª«
L
« « EHW ˜ [ » »
L ¼ ¼
¬¬
J$ L H[Sª
EHW
º
«
»
EHW
[
˜
º
ª
««
L » »
« « DOI˜ [ » »
L
¬¬
¼ ¼
J: L H[Sª
3 [ ˜ J$ L˜ 3YDS$ [ ˜ J: L˜ 3YDS:
L
L
L
\ L
[ ˜ J$ L˜ 3YDS$
L
3
L
//(UDQJHLVIURPDQLOLQHPROHIUDFWLRQRIWR M 3
\
3 3
M
\ \
M
7KLVLVSUHVVXUHLQ//(UHJLRQ
7KLVLVYDSRUPROHIUDFWLRQLQ//(UHJLRQ
3
3
[ \
Section 11.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
D E *
w §*·
¨ ¸
w7 © 7 ¹
+
VRWKDW+ H[
7
7 w § * H[ ·
w § 7 ·
¨
¸ 7 [$[%
¨
¸
w7 ©¨ 7 ¹¸
w7 ©
7
¹
[ $ [ % + H[ * H[
[ $ [ % 7
§ w [ $ [ % ·
¨
¸
w7
©
¹3[
+ 76VRWKDW6H[
§ w+ H[ ·
¨¨
¸¸
© w7 ¹ 3[
G 7KHHQHUJ\EDODQFHLV
F &H[
3
˜ + PL[ ˜ + ˜ + 4 ˜ + H[ ˜ + ˜ + ˜ + ˜ + 4 ˜ + H[ 4 ˜ ˜ ˜ 4
VRWKDW4 -KHDWUHPRYHGSHUPROHRI$DQGPROHV%
H 7KHHQWURS\EDODQFHLV
4
·
4
§
˜ 6PL[ ˜ 6 ˜ 6 6JHQ ˜ 6H[ ˜ 6 ˜ 6 5 ¨ ˜ OQ ˜ OQ ¸ ˜ 6 ˜ 6 6JHQ
7
¹
7
©
VR
· §
˜ ˜ ˜ 5 ¨ ˜ OQ ˜ OQ ¸ ˜ -.
¹ ©
DQG6JHQ -.SHUPROH$DQGPROHVRI%
I )ROORZLQJWKHDQDO\VLVRI(TQVWRZHREWDLQ
$[ $ [ %
7
EXWKHUH$ ˜ 7ZKLFKOHDGVWR
5
[ $ [ % 5 [ $ [ %
7
[ $ [ % 5 5 [ $ [ %
$VVKRZQRQWKH0$7+&$'ZRUNVKHHWEHORZ7KDVDPD[LPXPYDOXHDW[$ [% IRUZKLFK78& .
J 6LQFHWKHWHPSHUDWXUHRILQWHUHVWLVDERYHWKHXSSHUFRQVROXWHWHPSHUDWXUHZHQHHGFDOFXODWHRQO\9/(
QRW//(
6JHQ
Section 11.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
5 > ˜ [˜ [ @
78& [ 5 ˜ [˜ [
[ JLYHQ
G
78& [
G[
[ ILQG [
[
78& [
7 L [ ˜ L
ǻ
[
¬
5˜ 7
JDP H[S« $ ˜
L
$ ˜ 7
L
L
3YDS º
»
»
¼
L
º
ǻ
[
¬
5˜ 7 ¼
JDP H[S« $ ˜
L
»
»
3YDS 3 [ ˜ JDP 3YDS [ ˜ JDP ˜ 3YDS
L
L
\ L
L
L
L
L
[ ˜ JDP 3YDS
L
3
L
3
\
3
[
[ \
ª
33 ] ]˜ H[S« $ ˜
º
LQLWLDOJXHVVIRUD]HRWURSLFFRPSRVLWLRQ
§G
·
¨ 33 ] ¸
© G]
¹
33 ]
$VDFKHFN
[
º
ª
» ˜ 3YDS ] ˜ 3YDS˜ H[S« $ ˜ ] »
5˜ 7 ¼
¬ 5˜ 7 ¼
]
¬
*LYHQ
] ] ILQG ]
ª
]
¬
5˜ 7
]˜ H[S« $ ˜
\D] ]
º
» ˜ 3YDS
¼
\D]
33 ]
] )LQGHTXLOLEULXPOLTXLGFRPSRVLWLRQDW3 EDU
*LYHQ
33 ]
] ILQG ]
]
DQGYDSRUFRPSRVLWLRQ
ª
]
¬
5˜ 7
]˜ H[S« $ ˜
\\ ] º
» ˜ 3YDS
¼
33 ]
,QLWLDOJXHVVIRUIODVKYDSRUL]DWLRQ
JLYHQ
/
]˜ / \\ ] ˜ /
9 /
\\ ]
/ / ILQG /
9
Section 11.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
7KHSURFHGXUHWKDWZLOOEHXVHGLVWRILWWKHSDUDPHWHUVLQDQDFWLYLW\FRHIILFLHQWPRGHOWRWKH//(GDWD
DQGWKHQXVHWKLVPRGHOWRFDOFXODWHWKHYDSRUOLTXLGHTXLOLEULXP
1RWHWKDWWKHFRPSRVLWLRQVLQWKHWZRSKDVHVDUHTXLWHGLIIHUHQWWKHUHIRUHWKHRQHFRQVWDQW
0DUJXOHVHTXDWLRQLVQRWDSSOLFDEOH7KHUHIRUH,ZLOOXVHWKH9DQ/DDUHTXDWLRQ
DOI
º»
ǻ
DOI˜ [ º »
« « EHW ˜ [ » »
¬¬
¼ ¼
JDP [ DOI EHW H[Sª«
DOI EHW
º»
ǻ
> EHW [ @ º »
« « DOI˜ [ » »
¬¬
¼ ¼
JDP [ DOI EHW H[Sª«
LQLWLDOJXHVVHV
EHW 7 [ [ [ [
[ [
*LYHQ
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
] )LQG DOI EHW
]
DOI ]
§ ·
¨
¸
© ¹
EHW ]
DOI
L [ ˜ L
EHW
[ L
3YDS [
3YDS 3 [ ˜ JDP [ DOI EHW ˜ 3YDS [ ˜ JDP [ DOI EHW ˜ 3YDS
L
L
\ L
L
L
[ ˜ JDP [ DOI EHW ˜ 3YDS
L
L
M L
3
3 L
M
\ \
M
\
\
[
[
Section 11.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
3
3
[ \
33 [[ [[JDP
˜
[[ DOI EHW ˜ 3YDS [[ ˜ JDP [[ DOI EHW ˜ 3YDS
33 \\ [[ 33 [[JDP
˜
[[ DOI EHW ˜ 3YDS
33 [[
\\ \\ $PRUHFRPSOHWHSKDVHGLDJUDPLQFOXGLQJ//(DQG9//(UHJLRQV
Section 11.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
3YDS3 3YDS& 3YDS&
GSG3 JDP&LQI 3YDS3 GSG&
3YDS&
DOI OQ JDP3LQI
DOI
L GSG& JDP&LQI
JDP3LQI [ ˜ L
L
[ 3YDS3
JDP3LQI
EHW
EHW
º
«
»
EHW ˜ [ º »
« «ª
L »
« « DOI˜ [ » »
L
¬¬
¼ ¼
L
\ [ ˜ J3L˜ 3YDS3
L
L
3
L
\
3
[
J&L H[Sª
3 [ ˜ J3L˜ 3YDS3 [ ˜ J&L˜ 3YDS&
L
3YDS& GSG3
[
DOI
º
«
»
DOI˜ [
º» »
« «ª
L
««
EHW ˜ [ » »
L ¼ ¼
¬¬
GSG&
EHW OQ JDP&LQI
J3L H[Sª
L
3YDS3
GSG3
[
[
6\VWHPKDVDKRPRJHQHRXVD]HRWURSH
1RZFKHFNIRU//(
DOI
J$$ [[ H[Sª«
ȼ
ǻ
DOI˜ [[ º »
« « EHW ˜ [[ » »
¬¬
¼ ¼
[[ [[ [[
J%% [[ H[Sª«
EHW
ȼ
»
ǻ
EHW ˜ [[ º
«¬ «¬ DOI˜ [[ »¼ »¼
[[ [[ [[
*LYHQ
[[J$$
˜
[[
[[J$$
˜
[[
[[J%%
˜
[[
[[J%%
˜
[[
] ILQG [[ [[
]
7KHUHLVQRVROXWLRQWRWKH//(
FDOFXODWLRQDQGVRQR//(7KHUHIRUH
FRQFOXVLRQLVWKDWRQO\9/(H[LVWVDQG
V\VWHPKDVDQD]HRWURSH
Section 11.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
3YDS% 3YDS( JDP%LQI JDP$LQI 6LQFHWKHLQILQLWHGLOXWLRQDFWLYLW\FRHIILFHLQWVDUHLGHQWLFDODRQHFRQVWDQW0DUJXOHVH[SDQVLRQFDQ
EHXVHG
$ OQ L [ ˜ L
L
J%L H[Sª $ ˜ [
¬
L
º
J(L H[Sª $ ˜ [
¼
¬
3 [ ˜ J%L˜ 3YDS% [ ˜ J(L˜ 3YDS(
L
L
L
\ L
L
3
L
3
\
¼
[ ˜ J%L˜ 3YDS%
3
º
L
[
[ \
[
6\VWHPKDVDKRPRJHQHRXVD]HRWURSH
1RZFKHFNIRU//(
J$$ [[ H[Sª¬ $ ˜ [[ º¼
[[ [[ [[
J%% [[ H[S $ ˜ [[
[[ [[ [[
*LYHQ
[[J$$
˜
[[
[[J$$
˜
[[
[[J%%
˜
[[
[[J%%
˜
[[
] ILQG [[ [[
]
7KHUHLVQRVROXWLRQWRWKH//(
FDOFXODWLRQDQGVRQR//(7KHUHIRUH
FRQFOXVLRQLVWKDWRQO\9/(H[LVWVDQG
V\VWHPKDVDQD]HRWURSH
Section 11.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
D 6WDUWIURPWKH*LEEV'XKHPHTXDWLRQIRUHDFKSKDVH
S .dT V .dP ¦ xi.dGi .
ZKHUH.GHVLJQDWHVWKHSKDVHDQGLVHTXDOWR,,,RU,,,KHUH7KHFULWHULRQIRUHTXLOLEULXPLV
Gi,
Gi,,
Gi,,,
Gi QRQHHGWRGHVLJQDWHSKDVHRQ Gi $ORQJWKHHTXLOLEULXPFRH[LVWHQFHOLQH
dGi,
dGi,,
dGi,,,
dGi $OVR WKH SUHVVXUHV DUH HTXDO LQ HDFK SKDVH DV DUH WKH WHPSHUDWXUHV 7KXV ZH KDYH WKH WKUHH
HTXDWLRQV
dG
dG
dP
x, x,
dT
dT
dT
dP
dG
dG
V ,,
x,, x,,
dT
dT
dT
dP
dG
dG
x,,, x,,,
V ,,,
dT
dT
dT
V,
H . ¦ xi.Gi
dH G i .
i
FG
H
dGi dP
¦ xi.
Gi
dT T
dT i
IJ ¦ x H K K T
.
i
i
.
RULQPDWUL[IRUP
,
,,
,,,
¦ xi.Gi Ÿ T S .
LM V OP dP LM x OPF dG I LM x OPF dG G I LM ¦ x H OP
MMV PP dT MM x PPGH dT T G JK MM x PPGH dT T JK T MM ¦ x H PP N¦ x H Q
Nx Q
NV Q Nx Q
H . TS .
V.
S ,,,
8VLQJWKLVUHVXOWJLYHV
S ,, +RZHYHU
G
S,
,
,,
,,,
,
,,
,,,
,
i
,,
i
,,,
i
7KXVZHKDYHDOJHEUDLFHTXDWLRQVIRUWKHWKUHHXQNQRZQV
dP
dG G
dG G
DQG
dT
dT
T
dT
T
dP
8VLQJ&UDPHU¶VUXOHDQGVROYLQJIRU
JLYHV
dT
F I FG
H K H
IJ
K
FG
H
IJ
K
,
i
,,
i
,,,
i
Section 11.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
dP
dT
¦ xi, Hi, x, x,
T
¦ xi,, Hi,, x,, x,,
T
¦ xi,,, Hi,,, x,,, x,,,
T
x, x,
9,
9,, x,, x,,
9,,, x,,, x,,,
7KLVW\SHRIUHODWLRQVKLSZDVILUVWGHULYHGE\*LEEV
E 7KH*LEEV3KDVH5XOHLV F C P M L )RUOLTXLGOLTXLGPLVFLELOLW\ RQO\RQHOLTXLGSKDVH P YDSRUOLTXLG C DQG M F GHJUHHVRIIUHHGRP
7KXV LI DW IL[HG WHPSHUDWXUH WKH OLTXLG SKDVH PROH IUDFWLRQ LV YDULHG WKH WRWDO SUHVVXUH ZLOO
FKDQJH
LL OLTXLGOLTXLGLPPLVFLELOLW\ WZROLTXLGSKDVHV P YDSRUOLTXLGV 7KXV
F GHJUHHRIIUHHGRP
&RQVHTXHQWO\ DW IL[HG WHPSHUDWXUH WKH WZR SKDVH FRPSRVLWLRQV DQG WKH SUHVVXUH DUH IL[HG
9DU\LQJWKHDYHUDJHPROHIUDFWLRQZRXOGFKDQJHWKHPDVVGLVWULEXWLRQEHWZHHQWKHWZRSKDVHV
EXWZRXOG notFKDQJHWKHFRPSRVLWLRQRIHLWKHUSKDVHRUWKHWRWDOSUHVVXUH7KDWLVZKHQWZR
OLTXLGSKDVHVDQGDYDSRUSKDVHH[LVWLQDELQDU\PL[WXUHWKHHTXLOLEULXPSUHVVXUHGHSHQGVRQO\
RQWHPSHUDWXUHDQGQRWRQDYHUDJHFRPSRVLWLRQ
Section 11.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
3YDS( 3YDS($ $ 6LQFHWKHLQILQLWHGLOXWLRQDFWLYLW\FRHIILFHLQWVDUHLGHQWLFDODRQHFRQVWDQW0DUJXOHVH[SDQVLRQFDQ
EHXVHG
[ ˜ L
L L
J(L H[Sª $ ˜ [ º
L ¼
¬
J($L H[Sª $ ˜ [
¬
3 [ ˜ J(L˜ 3YDS( [ ˜ J($L˜ 3YDS($
L
L
L
¼
L
L
3
L
3
\
º
[ ˜ J(L˜ 3YDS(
\ 3
L
[
[ \
[
6\VWHPKDVDKRPRJHQHRXVD]HRWURSH
1RZFKHFNIRU//(
J$$ [[ H[Sª¬ $ ˜ [[ º¼
[[ J%% [[ H[S $ ˜ [[
[[ [[
[[ [[ [[
*LYHQ
[[J$$
˜
[[
[[J$$
˜
[[
[[J%%
˜
[[
[[J%%
˜
[[
7KHUHLVQRVROXWLRQWRWKH//(
FDOFXODWLRQDQGVRQR//(7KHUHIRUH
FRQFOXVLRQLVWKDWRQO\9/(H[LVWVDQG
V\VWHPKDVDQD]HRWURSH
] )LQG [[ [[
6RQRZZHPXVWILQGWKHHTXLOLEULXPFRPSRVLWLRQVRIWKHFRH[LVWLQJSKDVHVDWEDU
[[ / *LYHQ
[[/
˜ [[J$$
˜
[[ ˜ 3YDS(˜
33 /
33
[[ ˜ / [[ ˜ J%% [[ ˜ 3YDS($ ˜
/
]] ILQG [[ /
\\
]]
[[ ]]
\\ [[J$$
˜
[[ ˜ 3YDS(
33
§ ·
¨
¸
© ¹
Section 11.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
%DVHGRQ,OOXVWUDWLRQWKLVLVDSUREOHPLQYDSRUOLTXLGOLTXLG SKDVH HTXLOLEULXP$OVR
ZH KDYH WKDW WKH FRH[LVWHQFH SUHVVXUH LV FRQVWDQW RYHU WKH ZKROH UDQJH RI DYHUDJH RU WRWDO PROH
IUDFWLRQVIRUZKLFKWZROLTXLGSKDVHVH[LVW)URP,OOXVWUDWLRQRQHOLTXLGSKDVHLVSUHVHQWIRU
xLVREXWDQH x d DQG x t )RU overall PROH IUDFWLRQV LQ WKH UDQJH
d x d WZR OLTXLG SKDVHV H[LVW 7R FRPSXWH WKH 9// FRH[LVWHQFH SUHVVXUH LQ WKH
RQHOLTXLG SKDVH UHJLRQ ZH XVH
QHJOHFWLQJ IXJDFLW\ FRHIILFLHQW FRUUHFWLRQV YDS
YDS
YDS
xJ P xJ P
P ZKHUH P
N3D DQG PYDS N3D DQG J DQG J RU xJ DQG xJ DUHJLYHQLQ7DEOHLQ,OOXVWUDWLRQ$OVRWKHYDQ/DDUFRQVWDQWVDUHJLYHQWKHUHVR
J DQG J FDQEHFRPSXWHGDWRWKHUFRPSRVLWLRQV5HVXOWVDUHJLYHQEHORZ
P
x xJ PYDS xJ PYDS N3D
u u u u u u u u u u u p
p
p
p
u u u u Section 11.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Section 11.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
R&
Section 11.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Section 11.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
R&
Section 11.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Section 11.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
5 $ ˜ 5 ˜ ª $˜ [ º
»
JDP [ 7 H[S«
¬ 5˜ 7 ¼
§ $ ˜ [ ·
¸
© 5˜ 7 ¹
JDP [ 7 H[S¨
6LQFHWKH//(SKDVHERXQGDU\IRUWKHRQHFRQVWDQW0DUJXOHVHTXDWLRQZLOOEHV\PPHWULFZHZLOO
XVHWKHVLPSOHSURFHGXUHRIVHWWLQJWKHSKDVHFRPSRVLWLRQVDQGILQGLQJWKHFRH[LVWHQFHWHPSHUDWXUH
[ [ [˜ JDP [ 7
[˜ JDP [ 7
] ILQG 7
]
7 [ [ [
[˜ JDP [ 7
]
*LYHQ
[ ˜ JDP [ 7
[ [ [
[˜ JDP [ 7
] )LQG 7
]
*LYHQ
[ ˜ JDP [ 7
[ ˜ JDP [ 7
[ [ [
[˜ JDP [ 7
[˜ JDP [ 7
] )LQG 7
]
*LYHQ
[ [ [
[ ˜ JDP [ 7
[ ˜ JDP [ 7
*LYHQ
[˜ JDP [ 7
[˜ JDP [ 7
] )LQG 7
]
[ [ [
[ ˜ JDP [ 7
[ ˜ JDP [ 7
*LYHQ
[˜ JDP [ 7
[˜ JDP [ 7
] )LQG 7
]
[ [ [
[ ˜ JDP [ 7
[ ˜ JDP [ 7
*LYHQ
[˜ JDP [ 7
[˜ JDP [ 7
] )LQG 7
]
[ [ [
[ ˜ JDP [ 7
[ ˜ JDP [ 7
*LYHQ
[˜ JDP [ 7
[˜ JDP [ 7
] )LQG 7
]
[ ˜ JDP [ 7
[ ˜ JDP [ 7
[ [
[˜ JDP [ 7
[˜ JDP [ 7
] )LQG 7
]
[ [ ˜ JDP [ 7
[˜ JDP [ 7
[ [ ˜ JDP [ 7
[˜ JDP [ 7
] )LQG 7
*LYHQ
[ ˜ JDP [ 7
*LYHQ
[ ˜ JDP [ 7
[ ˜ JDP [ 7
[ [
*LYHQ
[˜ JDP [ 7
[˜ JDP [ 7
] )LQG 7
]
[ ˜ JDP [ 7
[ ˜ JDP [ 7
§ ·
¨ ¸
¨
¸
¨ ¸
¨ ¸
¨ ¸
¨
¸
GDWD ¨ ¸
¨ ¸
¨
¸
¨ ¸
¨ ¸
¨ ¸
¨
¸
© ¹
L 77 GDWD
L
L ; GDWD
L
L ; GDWD
L
L //(SKDVHERXQGDU\
77
77
; ;
Section 11.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
7KHSURFHGXUHWKDWZLOOEHXVHGLVWRILWWKHSDUDPHWHUVLQDQDFWLYLW\FRHIILFLHQWPRGHOWRWKH//(GDWD
DQGWKHQXVHWKLVPRGHOWRFDOFXODWHWKHYDSRUOLTXLGHTXLOLEULXP
1RWHWKDWWKHFRPSRVLWLRQVLQWKHWZRSKDVHVDUHTXLWHGLIIHUHQWWKHUHIRUHWKHRQHFRQVWDQW
0DUJXOHVHTXDWLRQLVQRWDSSOLFDEOH7KHUHIRUH,ZLOOXVHWKH9DQ/DDUHTXDWLRQ
DOI
JDP [ DOI EHW H[Sª«
º»
ǻ
DOI˜ [ º »
«¬ «¬ EHW ˜ [ »¼ »¼
DOI EHW
º»
ǻ
> EHW [ @ º »
« « DOI˜ [ » »
¬¬
¼ ¼
JDP [ DOI EHW H[Sª«
LQLWLDOJXHVVHV
EHW 7 [ [ [ [
[ [
*LYHQ
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
[˜ JDP [ DOI EHW
] )LQG DOI EHW
]
DOI ]
§ ·
¨
¸
© ¹
EHW ]
DOI
[ ˜ L
L EHW
[ L
3YDS [
3YDS 3 [ ˜ JDP [ DOI EHW ˜ 3YDS [ ˜ JDP [ DOI EHW ˜ 3YDS
L
L
\ L
L
L
[ ˜ JDP [ DOI EHW ˜ 3YDS
L
L
M L
3
3 L
M
\ \
M
\
\
[
[
Section 11.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
3
3
[ \
33 [[ [[JDP
˜
[[ DOI EHW ˜ 3YDS [[ ˜ JDP [[ DOI EHW ˜ 3YDS
33 \\ [[ 33 [[JDP
˜
[[ DOI EHW ˜ 3YDS
33 [[
\\ \\ $PRUHFRPSOHWHGLDJUDPVKRZLQJ//(DQG9//(UHJLRQV
6ROXWLRQLVLQWKHIROGHU$VSHQIRUWH[WERRN!&KDSWHU!6HF!3URE
6ROXWLRQLVLQWKHIROGHU$VSHQIRUWH[WERRN!&KDSWHU!6HF!3URE
Section 11.3
EDUDQG
7ZRSXUHFRPSRQHQWVDQGKDYHYDSRUSUHVVXUHV
&DQGPHOWLQJSRLQWV
&DQG
Section 11.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
EDUDW
&,QDFRQVWDQWWHPSHUDWXUH
EDUZDV
H[SHULPHQWDW&WKHVORSHRIWKHFKDQJHLQSUHVVXUH
PHDVXUHGDWLQILQLWHGLOXWLRQRIVSHFLHV
D $VVXPLQJWKLVREH\VWKHFRQVWDQW0DUJXOHVDFWLYLW\FRHIILFLHQWPRGHOFDOFXODWHWKH
LQWHUDFWLRQSDUDPHWHUA/RTDQGWKHLQILQLWHGLOXWLRQDFWLYLW\FRHIILFLHQWVRIERWK
FRPSRQHQWV
6ROXWLRQ
E 'RHVWKLVPL[WXUHIRUPDQD]HRWURSHDW&",IVRDWZKDWFRPSRVLWLRQ"
6ROXWLRQ
Solutions to Chemical and Engineering Thermodynamics, 5th ed
<HV
F $VVXPLQJWKHLQWHUDFWLRQSDUDPHWHU$LVLQGHSHQGHQWRIWHPSHUDWXUHILQGWKHXSSHU
FRQVROXWHWHPSHUDWXUH
6ROXWLRQ
)URPSDUWD
DW&
$ -PRO. . -PRO.
G :LOODOLTXLGOLTXLGSKDVHVHSDUDWLRQRFFXUIRUWKLVV\VWHP"
6ROXWLRQ
7KHXSSHUFRQVROXWHWHPSHUDWXUHLVIDUEHORZWKHPHOWLQJSRLQWVRIHLWKHURIWKHWZRSXUH
FRPSRQHQWV7KHUHIRUHWKHPL[WXUHZLOOIUHH]HORQJEHIRUHLWUHDFKHVWKHOLTXLGOLTXLG
SKDVHHQYHORSH
Section 11.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Section 11.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Section 11.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
^ŝŶĐĞƚŚĞ>>ŝƐƐLJŵŵĞƚƌŝĐ͕ĂŽŶĞͲĐŽŶƐƚĂŶƚDĂƌŐƵůĞƐĞƋƵĂƚŝŽŶŝƐƐƵĨĨŝĐŝĞŶƚƚŽĚĞƐĐƌŝďĞƚŚŝƐ
ƐLJƐƚĞŵ͕ƐŽƚŚĂƚ
ĂŶĚĂƚĞƋƵŝůŝďƌŝƵŵ
dŚĞƐŽůƵƚŝŽŶƚŽƚŚŝƐĞƋƵĂƚŝŽŶŝƐсϮ͘ϭϵϳ͘dŚĞƐŽůƵƚŝŽŶŽĨƚŚĞƌĞƐƚŽĨƚŚĞƉƌŽďůĞŵŝƐŐŝǀĞŶďĞůŽǁ͘
Section 11.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
3
\
[
ƵƚŶĞĞĚƚŽĐŽƌƌĞĐƚƚŚĞĐĂůĐƵůĂƚŝŽŶĨŽƌƚŚĞ>>ďĞƚǁĞĞŶŵŽůĞĨƌĂĐƚŝŽŶϬ͘ϮϱĂŶĚϬ͘ϳϱ͕ǁŚĞƌĞLJϭŝƐ
ĐŽŶƐƚĂŶƚĂƚϬ͘ϰĂŶĚWĂƚϭ͘ϳϮϭ
L 3 L
\ L
Section 11.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
3L
3L
[L \ L
\ L
[L
[L
Section 11.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
D 7KHIROORZLQJLQIRUPDWLRQLVDYDLODEOHDWR&
3KDVH
3KDVH
1LWURPHWKDQH 1RQDQH
:KHUHWKHQXPEHUVLQEROGDUHWKHUHSRUWHGH[SHULPHQWDOGDWD6LQFHWKHFRQFHQWUDWLRQRI
QLWURPHWKDQHLVTXLWHKLJKLQSKDVHDQGWKHUHLVRQO\OLPLWHGH[SHULPHQWDOGDWDIRUVLPSOLFLW\
ZHZLOODVVXPHLWDFWLYLW\LVXQLW\7KHQWKHDFWLYLW\FRHIILFLHQWRIQLWURPHWKDQHLQSKDVHLV
ϭͬϬ͘ϭϯϭ ϳ͘ϲϯϰ͘hƐŝŶŐƚŚĞƐĂŵĞůŽŐŝĐĨŽƌŶŽŶĂŶĞŝŶƉŚĂƐĞϮŐŝǀĞƐĂŶĂĐƚŝǀŝƚLJŽĨϭͬϬ͘ϬϮϰϳсϰϬ͘ϰϵ
&ŽƌƐŝŵƉůŝĐŝƚLJ͕ǁĞƚĂŬĞƚŚĞƐĞƚŽďĞƚŚĞŝŶĨŝŶŝƚĞĚŝůƵƚŝŽŶĂĐƚŝǀŝƚLJĐŽĞĨĨŝĐŝĞŶƚƐ͘
:HZLOOXVHWKHYDQ/DDUPRGHOZKLFKLV
OQJ f
a OQ DQGOQJ f
b
OQ 1RZWKH9DQ/DDUH[SUHVVLRQIRUG H[ LVG H[
ƚϵϬŽ
axbx
ax bx
7KHIROORZLQJLQIRUPDWLRQLVDYDLODEOHDWR&
3KDVH
3KDVH
1LWURPHWKDQH 1RQDQH
:KHUHWKHQXPEHUVLQEROGDUHWKHUHSRUWHGH[SHULPHQWDOGDWD6LQFHWKHFRQFHQWUDWLRQRI
Section 11.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
QLWURPHWKDQHLVTXLWHKLJKLQSKDVHDQGWKHUHLVRQO\OLPLWHGH[SHULPHQWDOGDWDIRUVLPSOLFLW\
ZHZLOODVVXPHLWDFWLYLW\LVXQLW\7KHQWKHDFWLYLW\FRHIILFLHQWRIQLWURPHWKDQHLQSKDVHLV
ϭͬϬ͘Ϯϭϰ ϳ͘ϲϯϰ͘hƐŝŶŐƚŚĞƐĂŵĞůŽŐŝĐĨŽƌŶŽŶĂŶĞŝŶƉŚĂƐĞϮŐŝǀĞƐĂŶĂĐƚŝǀŝƚLJŽĨϭͬϬ͘ϬϰϲϵсϰϬ͘ϰϵ
&ŽƌƐŝŵƉůŝĐŝƚLJ͕ǁĞƚĂŬĞƚŚĞƐĞƚŽďĞƚŚĞŝŶĨŝŶŝƚĞĚŝůƵƚŝŽŶĂĐƚŝǀŝƚLJĐŽĞĨĨŝĐŝĞŶƚƐ͘
:HZLOOXVHWKHYDQ/DDUPRGHOZKLFKLVDWR&
OQJ f
a OQ DQGOQJ f
b
OQ b
OQ $WR&
OQJ f
a OQ DQGOQJ f
1RZWKH9DQ/DDUH[SUHVVLRQIRUG H[ LVG H[
axbx
ax bx
Section 11.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
ĐͿdŚĞŶ͕^ŝŶĐĞ'с,Ͳd^͕^с;,Ͳ'Ϳͬd͕ĂŶĚƚŚĞƐĂŵĞĨŽƌƚŚĞĞdžĐĞƐƐƉƌŽƉĞƌƚŝĞƐ͘dŚĞƐĞǀĂůƵĞƐĂƌĞ
ƐŚŽǁŶŝŶƚŚĞƚĂďůĞďĞůŽǁ͘
ĚͿ&ƌŽŵƚŚĞĚĂƚĂŝŶƚŚĞƉƌŽďůĞŵƐƚĂƚĞŵĞŶƚ͕ƚŚĞs>>ƌĞŐŝŽŶƐĂƌĞŶŽŶĂŶĞŵŽůĞĨƌĂĐƚŝŽŶƐĨƌŽŵϬ
ƚŽϬ͘Ϯϭϰ͕ĂŶĚĨƌŽŵϬ͘ϵϱϯϭƚŽϭ͘ĞƚǁĞĞŶϬ͘ϮϭϰĂŶĚϬ͘ϵϱϯϭ>>ĞdžŝƐƚƐ͘dŚŝƐŝƐƐĞĞŶŝŶƚŚĞĨůĂƚ
ƌĞŐŝŽŶŝŶƚŚĞdžͲLJĚŝĂŐƌĂŵ͘
Section 11.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Section 11.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
6ROXWLRQWRWKHWKLVSUREOHPLVLQWKHIROGHU$VSHQIRUWH[WERRN!&KDS!6HFB
6ROXWLRQWRWKHWKLVSUREOHPLVLQWKHIROGHU$VSHQIRUWH[WERRN!&KDS!6HFB
6ROXWLRQWRWKHWKLVSUREOHPLVLQWKHIROGHU$VSHQIRUWH[WERRN!&KDS!6HFB
6ROXWLRQWRWKHWKLVSUREOHPLVLQWKHIROGHU$VSHQIRUWH[WERRN!&KDS!6HFB
6ROXWLRQWRWKHWKLVSUREOHPLVLQWKHIROGHU$VSHQIRUWH[WERRN!&KDS!6HFB
6ROXWLRQWRWKHWKLVSUREOHPLVLQWKHIROGHU$VSHQIRUWH[WERRN!&KDS!6HFB
6ROXWLRQWRWKHWKLVSUREOHPLVLQWKHIROGHU$VSHQIRUWH[WERRN!&KDS!6HFB
6ROXWLRQWRWKHWKLVSUREOHPLVLQWKHIROGHU$VSHQIRUWH[WERRN!&KDS!6HFB
6ROXWLRQWRWKHWKLVSUREOHPLVLQWKHIROGHU$VSHQIRUWH[WERRN!&KDS!6HFB
6ROXWLRQWRWKHWKLVSUREOHPLVLQWKHIROGHU$VSHQIRUWH[WERRN!&KDS!6HFB
6ROXWLRQWRWKHWKLVSUREOHPLVLQWKHIROGHU$VSHQIRUWH[WERRN!&KDS!6HFB
6ROXWLRQWRWKHWKLVSUREOHPLVLQWKHIROGHU$VSHQIRUWH[WERRN!&KDS!6HFB
Section 11.3
Section 11.4
Solutions to Chemical and Engineering Thermodynamics, 5th ed
1RWDWLRQ Ni
WRWDOQXPEHURIPROHVRIVSHFLHVi
NiJ
J
QXPEHURIPROHVRIVSHFLHViLQSKDVHJ
N
¦ NiJ
WRWDOQXPEHURIPROHVLQSKDVHJ
i
xiJ
NiJ
¦ N jJ
PROHIUDFWLRQRIVSHFLHViLQSKDVH J
j
D 7ZROLTXLGSKDVHV
(TXLOLEULXPFRQGLWLRQV f i , T P x ,
0DVVEDODQFHFRQVWUDLQWV Ni
7KXVZHKDYHVL[HTXDWLRQVIRUWKHVL[XQNQRZQV NiJ J
$OWHUQDWLYHO\ZHFDQWUHDW x, x, x,, x,, N , DQG N ,, DVWKHVL[XQNQRZQV
d
,,
,,
Ni, Ni,,
i
d
(TXLOLEULXPFRQGLWLRQ f, T P x ,
, ,, i
$OVR
i f dT P x i N, N,, ,,
0DVVEDODQFHFRQVWUDLQW N
N
N ,
NI
N, N ,
x,
N
N ,,
N ,,
N,, N ,,
x,,
,,
N,
N, N ,
N,,
N,, N ,,
%DVLFDOO\ZHKDYHWZRHTXDWLRQV> DQG @IRUWKHWZRXQNQRZQV N, DQG N,, 7KHRWKHU
HTXDWLRQVPHUHO\UHODWHWKHXQNQRZQ NI DQG NII WRWKHPROHIUDFWLRQVWKDWDSSHDULQHTQ F 7KUHHOLTXLGSKDVHV
i
E 7ZROLTXLGSKDVHVEXWVSHFLHVDQGFRPSOHWHO\LPPLVFLEOH
i f dT P x i i R| f dT P x i f dT P x i i HTXLOLEULXPFRQGLWLRQV S
|T f dT P x i f dT P x i i i
,
,
i
,
,
i
,,
,,
i
,,,
Ni, Ni,, Ni,,,
,,,
PDVVEDODQFHFRQVWUDLQWV Ni
7KXV ZH KDYH HTXDWLRQV IRU WKH XQNQRZQ NiJ i
i
x, x, x,, x,, x,,, x,,, N , N ,, N ,,, 3UHOLPLQDU\FDOFXODWLRQV
0:& + 2 J PRO C& + 2
PRO OLWHU SK\GUR[\EHQ]RLFDFLG 0:& + 2 J PRO C& + 2 PRO OLWHU 1RWDWLRQ
,, DTXHRXVSKDVH ,
GLHWK\OHWKHUSKDVH
J
RU
Section 11.4
Solutions to Chemical and Engineering Thermodynamics, 5th ed
$OVR
Ci,V , Ci,,V ,,
RUXVLQJHTQ Ci,V , Ni
DQG Ci,,
,
V KiV ,,
Ci,
C&,, + 2
C&,, + 2 c
C&, + 2 c
h
EDQGF 8VLQJEDWFKHVRIGLHWK\OHWKHUHDFKRIOLWHUV
VWEDWFK
C&, + 2
C&,, + 2
C&, + 2 C&,, + 2 C&, + 2
C&, + 2
RULQJHQHUDO
C&, + 2
$IWHUVHFRQGEDWFK
C&,, + 2
n
C&,, + 2
PRO OLWHU C&+ 2
K&+2
b g
C& + 2 K& + 2
,
K&n + 2 V ,, K n
,
n
,
n
K& + 2 V ,,
V
& + 2
C& + 2
,
K&n + 2 V ,, K V
K& +2
C& + 2
n
C&, + 2
C&, + 2 C& + 2
V
PRO OLWHU Ki Ni
V , KiV ,,
C&, + 2 PRO OLWHU KiCi,
h
C&, + 2
Ni 6LQFHGLHWK\OHWKHUDQGZDWHUDUHVRLQVROXEOHDQGVLQFHUHODWLYHO\VPDOODPRXQWVRIJDOOLFDFLG
DQGSK\GUR[\EHQ]RLFDFLGZLOOEHWUDQVIHUUHG,ZLOODVVXPHWKDW V , DQG V ,, DUHXQFKDQJHG
GXULQJWKHSDUWLWLRQLQJSURFHVV
)LQDOO\ Ni V ,, u Ci ZKHUH Ci FRQFHQWUDWLRQRIVSHFLHVLQWKHZDWHUSKDVHEHIRUHLW
LVFRQWDFWHGZLWKWKHGLHWK\OHWKHU
D +HUH
V , V ,, C&, + 2
PRO OLWHU Ci,V ,,
Ki
7KXV
LQLWLDOQXPEHURIPROHVRIVSHFLHVi
Ni
Ci,
Ki
Ci,
{ Ki RU Ci,,
Ci,,
& + 2
n
K& + 2 V ,,
V
ZLWK D VLPLODU H[SUHVVLRQ IRU SK\GUR[\EHQ]RLF DFLG 8VLQJ WKHVH HTXDWLRQV ZH REWDLQ WKH
IROORZLQJUHVXOWV
Section 11.4
Solutions to Chemical and Engineering Thermodynamics, 5th ed
LQLWLDO& + 2 LQLWLDO& +2
OLWHUEDWFK
n
V ,
LQDTSKDVH
OLWHUEDWFKHV
n V , V ,
5HJXODUVROXWLRQWKHRU\
+HUH
V /%U
aG G f OQ J
RT
/
V &&O
G %U
FF PRO
FF PRO G &&O OQ J %U
A
1RWHWKHKXJHSXULILFDWLRQLQILYHVWDJHV
OQ J V I OLWHUEDWFKHV
n
LQDTSKDVH
V I aG G f a
f
a
J &&O
%U
J +%U 2
&&O &&O J %U
x%U
+ 2
x%U
f
&&/ %U
&&/ %U
a f FF PRO G aFDO FFf
RT OQ J
V I cG G h RT
u u 5HJXODUVROXWLRQSDUDPHWHUV
%U V /
&6 V /
OP LM x
u MN u x x PQ a x f
I
F
G
H x a x fJK
%U
7KXV
x%U
FDO FF FDO FF %U
FF PRO G
%U
/
%U &6
FDO FF %U
&6
,Q &6 SKDVHOHW
x PRO &6 OLWHURIVROXWLRQ y PRO %U OLWHURIVROXWLRQ $VVXPLQJQRYROXPHFKDQJHRQPL[LQJZHWKHQKDYH
y
y x RU x
1RZDW
H[WUHPHVRI%U
FRQFLQ&6
F y Ÿ x x
GH y Ÿ x x
%U
%U
u u I &6 | I &6 | Section 11.4
Solutions to Chemical and Engineering Thermodynamics, 5th ed
7KXVZHFDQWDNH I &6 | RYHUWKHZKROHFRPSRVLWLRQUDQJHDQG
u RYHUZKROHFRPSRVLWLRQUDQJH1RZ
RU J &6
%U | OQ J &6
%U
ŸJ
K
,,
RUJDQLFSKDVH
,,
x%U LQRUJDQLFSKDVH
PROHV%U
PROHV%U PROHV+ 2
PROHV%U
PROHV%U PROHV&6
PROHV%U
PROHV+ 2
Ÿ K | PROHV%U PROHV&6 !! PROHV%U
PROHV&6
PROHV+ 2 !! PROHV%U
'DWDJLYHQLQSUREOHPVWDWHPHQWLVIRU
Kc
K
Kc
PROHV%U
OLWHUDTXHRXVVROXWLRQ
PROHV%U
OLWHU&6 VROXWLRQ
OLWHUV
PROZDWHU
OLWHUV
u PRO&6 $OVR
J +%U2
DTXHRXVSKDVH
x%U LQDTXHRXVSKDVH
7KXV
,
K
ZKHUH
1RZZHH[SHFW
J
,
ZKHUH
J ,,
J ,
x,
x,,
K
u u
K c J &6
%U
K c
)URPZKLFKZHREWDLQWKHIROORZLQJ
+ 2
y
x%U
J %U
m m
u u u u u u u m
+ 2
2
7RLQIHUDUHODWLRQVKLSEHWZHHQ J +
%U DQGWKHVDOWPRODULW\m,SORWWHG OQ J %U YHUVXVOQmDQG
YDULRXVSRZHUVRImXVLQJWKHGDWDDW y
2
,IRXQGWKDWWKHH[SUHVVLRQ OQ +
%U
Cm JDYHDYHU\JRRGILWRIWKHGDWD7KHSDUDPHWHUCLVDZHDNO\LQFUHDVLQJIXQFWLRQRIWKHEURPLQH
PROHIUDFWLRQ,GLGQRWDWWHPSWWRGHWHUPLQHWKLVGHSHQGHQFH
&RQVLVWHQF\UHODWLRQEHWZHHQ P YDS HDQGVROXELOLW\
xi Hi Pi xiJ i Pi YDS xiJ if Pi YDS Hi
J if Pi YDS Section 11.4
Solutions to Chemical and Engineering Thermodynamics, 5th ed
)RUDVXEVWDQFHDERYHLW¶VPHOWLQJSRLQW OLTXLGDVDSXUHFRPSRQHQW EXWRQO\RIOLPLWHGVROXELOLW\
Pi YDS
xiVDWJ if J if
Ÿ
H
i
xiVDW
xiVDW
1RZLQ7DEOHGDWDDUHIRUHLQ Ci Hic Pi C: xi Hic xi Hi VR
Pi YDS
Pi YDS
C: xiVDW
CiVDW
([DPSOHEXWDGLHQH 0:& + Hic
Hic
Hi
C:
N3D
PJ / u J PJ u / P u PRO J
a
h a
f c
u u N3DP PRO
u N3DP PRO
c
u EDUP PRO FRPSDUHGWR u ([DPSOH$QLOLQH
Pi YDS N3D u EDU u SSP
6ROXELOLW\ Hic
PJ / a
f b
g a
f
u u EDU PPRO
u EDU PPRO VRPHWKLQJZURQJLQWKLVFDVH FRPSDUHZLWK
u u EDU ([DPSOH'LFKORUREHQ]HQH P YDS N3D
6ROXELOLW\ PJ / 0: Hic
u EDU
PJ / u J PJ u / P u PRO a
c
f c
h a
u EDU P PRO FRPSDUHGWR u h
u EDU
u PJ / u J PJ u / P u PRO EDU P PRO FRPSDUHGWR
([DPSOH7ULFKORUREHQ]HQH 0:& + &O Hic
f
N3D P PRO
f
h
([DPSOH(WK\OEHQ]HQH P
N3D u EDU 0: u u u u EDU PPRO
Hic
YDS
cFRPSDUHGWR u h
([DPSOH1DSWKDOHQH P YDS u N3D u EDU 0: u u Hic
u EDU PPRO FRPSDUHGWR u EDU PPRO u EDU 0: ([DPSOH6W\UHQH P YDS N3D
u u Hic
u EDU PPRO FRPSDUHGWR u EDU PPRO c
h
b
g
Section 11.4
Solutions to Chemical and Engineering Thermodynamics, 5th ed
0ROHIUDFWLRQRIOHXFLQHLQZDWHULV
0: 0:
u FORVHWRLQILQLWHGLOXWLRQ
0ROHIUDFWLRQRIOHXFLQHLQHWKDQROLV
0( ˜ ˜ 0(
u $OVRFORVHWRLQILQLWHGLOXWLRQ
VRWKHUDWLRRIDFWLYLW\FRHIILFLHQWVRQOHXFLQHLQHWKDQROWRZDWHULVWKHHTXDO
WRWKHUDWLRRIOHXFLQHPROHIUDFWLRQVLQZDWHUWRHWKDQROZKLFKLV
$FWUDW 0:
0(
$FWUDW
DQGWKH*LEEVHQHUJ\RIWUDQVIHULV
'WUI*
u PRO
'WUI* ˜ ˜ OQ $FWUDW
Section 11.4
Solutions to Chemical and Engineering Thermodynamics, 5th ed
6WDUWE\JHWWLQJXQLWVLQWRDPRUHVWDQGDUGIRUPPROVGPA PROVOLWHU
7RSURFHHGZHZLOODVVXPHDOLWHURIVROXWLRQFRQWDLQVPROHVRIZDWHU
0ROHIUDFWLRQRIJO\FLQHLQZDWHULV
0: 0ROHIUDFWLRQRIJO\FLQHLQHWKDQROLV
0( 0:
˜ ˜ 0(
u 0:
7KHUDWLRRIDFWLYLW\FRHIILFLHQWVRQJO\FLQHLQHWKDQROWRZDWHULVWKHHTXDO
WRWKHUDWLRRIJO\FLQHPROHIUDFWLRQVLQZDWHUWRHWKDQROZKLFKLV
0:
$FWUDW $FWUDW
0(
u DQGWKH*LEEVHQHUJ\RIWUDQVIHULV
'WUI*
'WUI* ˜ ˜ OQ $FWUDW
-
u PRO
0ROHIUDFWLRQRIDODQLQHLQZDWHULV
0: 0ROHIUDFWLRQRIDODQLQHLQHWKDQROLV
0( ˜ ˜ 0(
u VRWKHUDWLRRIDFWLYLW\FRHIILFLHQWVRQDODQLQHLQHWKDQROWRZDWHULVWKHHTXDO
WRWKHUDWLRRIDODQLQHPROHIUDFWLRQVLQZDWHUWRHWKDQROZKLFKLV
0:
$FWUDW 0(
$FWUDW
DQGWKH*LEEVHQHUJ\RIWUDQVIHULV
'WUI*
'WUI* ˜ ˜ OQ $FWUDW
-
u PRO
0ROHIUDFWLRQRI$EDLQZDWHULV
0: 0ROHIUDFWLRQRI$EDLQHWKDQROLV
0( 0:
˜ ˜ 0(
u VRWKHUDWLRRIDFWLYLW\FRHIILFLHQWVRQ$EDLQHWKDQROWRZDWHULVWKHHTXDO
WRWKHUDWLRRI$EDPROHIUDFWLRQVLQZDWHUWRHWKDQROZKLFKLV
$FWUDW 0:
0(
$FWUDW
DQGWKH*LEEVHQHUJ\RIWUDQVIHULV
'WUI* u PRO
'WUI* ˜ ˜ OQ $FWUDW
0ROHIUDFWLRQRI$FDLQZDWHULV
0: 0ROHIUDFWLRQRIOHXFLQHLQHWKDQROLV
0( ˜ ˜ 0:
0(
u u VRWKHUDWLRRIDFWLYLW\FRHIILFLHQWVRQOHXFLQHLQHWKDQROWRZDWHULVWKHHTXDO
WRWKHUDWLRRIOHXFLQHPROHIUDFWLRQVLQZDWHUWRHWKDQROZKLFKLV
$FWUDW 0:
0(
$FWUDW
DQGWKH*LEEVHQHUJ\RIWUDQVIHULV
'WUI*
u PRO
'WUI* ˜ ˜ OQ $FWUDW
Solutions to Chemical and Engineering Thermodynamics, 5th ed
6ROXWLRQLVLQIROGHU$VSHQIRUWH[WERRN!&KDSWHU!6HF
6ROXWLRQLVLQIROGHU$VSHQIRUWH[WERRN!&KDSWHU!6HF
Section 11.4
Section 11.5
Solutions to Chemical and Engineering Thermodynamics, 5h ed
P
HQHUJ\EDODQFH
6\VWHP
P
6\VWHP
dV ,,
dV ,
dU
Q WS P ,
P ,,
dt
dt
dt
HQWURS\EDODQFH
dS Q SJHQ dt T
&RQVWUDLQWV WS V\VWHPLVLVRWKHUPDO P , DQG P ,, DUHFRQVWDQWWKXV d
dS
Q T
TSJHQ
TS TSJHQ dt
dt
DQG
dV ,,
dV ,
dU
d
P ,,
TS TSJHQ P ,
dt
dt
dt
dt
d
d
d
,
,
P ,,V ,,
PV TS TSJHQ dt
dt
dt
dG
d
Ÿ
TSJHQ
U TS P ,V , P ,,V ,,
dt
dt
ZKHUHZHKDYHGHILQHG G WREH G U TS P ,V , P ,,V ,, &RQVHTXHQWO\
dG
TSJHQ d Ÿ G PLQLPXPDWHTXLOLEULXP
dt
DQG
G U TS P ,V , P ,,V ,, U , U ,, T S , S ,, P ,V , P ,,V ,,
G , T P , N , G ,, T P ,, N ,,
c
c
c
h
h
c
h
h
c
c
h
h c
h
RQO\V\VWHPYDULDWLRQSRVVLEOHLVLQ Ni, DQG Ni,, IRUDOOVSHFLHViZKLFKFDQSDVVWKURXJKWKH
PHPEUDQH+RZHYHUVLQFH Ni, Ni,, FRQVWDQW dNi,, dNi, 7KXV
G
DQG
¦ N ,j G j, dT P, x , i ¦ N ,,j G j,, dT P,, x ,, i Section 11.5
Solutions to Chemical and Engineering Thermodynamics, 5h ed
FG wG IJ
H wN K
,
i T P N j zi
F wG dT P x iI
GH wN JK
L
F wG dT P x iI
MG dT P x i ¦ N G
M
H wN JK
AMN
d
,
j
i
Gi , T P , x , ¦ N ,j
i
,,
,
i
,,
,,
,
,
T P N j zi
,,
j
,,
j
,
,,
i
PLQXVVLJQDULVHVIURP dNi,,
T P N j zi
OP PPQ
dNi, 1RZERWKVXPPDWLRQWHUPVYDQLVKXSRQDSSOLFDWLRQRIWKH*LEEV'XKHPHTXDWLRQWRHDFKSKDVH
7KXVZHKDYHDWHTXLOLEULXPWKDW
FG wG IJ
H wN K
d
i
,
d
,
Ÿ Gi T P x
d
Gi , T P , x , Gi ,, T P ,, x ,,
,
i T P N j zi
i
i G dT P x i
,
i
,,
,,
,,
IRUHDFKSHUPHDEOHiVSHFLHVRUHTXLYDOHQWO\
fi , T P, x ,
f i ,, T P , x ,, d
/HW
i
d
i
C QXPEHURIFRPSRQHQWV
C p QXPEHURIFRPSRQHQWVWKDWFDQSDVVWKURXJKPHPEUDQH FOHDUO\ C t C p 7KHV\VWHPFRQVLVWVRIWZRKRPRJHQHRXVUHJLRQV WKDWFDQEHWKRXJKWRIDVWZRSKDVHV DQGWKHUH
DUHQRFKHPLFDOUHDFWLRQVRFFXUULQJ
6LQFH C YDULDEOHVDUHQHHGHGWRIL[WKHWKHUPRG\QDPLFVWDWHRIDVLQJOHSKDVHWKHUHDUH
C DSSDUHQWXQNQRZQVKHUH+RZHYHUZHKDYHWKHIROORZLQJHTXLOLEULXPFRQGLWLRQV
T,
Gi
T ,,
,
c
Gi T P I i s RT OQ xi
Ÿ I i s OQ xi
cC UHODWLRQVh p
h
)RUDELQDU\V\VWHPZLWKDPHPEUDQHSHUPHDEOHWRRQO\RQHRIWKHVSHFLHVZHKDYH
F ˜ GHJUHHVRIIUHHGRP
7KXVLIZHVSHFLI\WHPSHUDWXUHWKHWZRSUHVVXUHVDQGRQHSKDVHFRPSRVLWLRQZHFDQLQ
SULQFLSDOFRPSXWHWKHFRPSRVLWLRQRIWKHRWKHUSKDVH$OWHUQDWLYHO\DVLQWKHLOOXVWUDWLRQLQWKLV
VHFWLRQZHFDQVSHFLI\T x, x,, DQG P , DQGWKHQFRPSXWHWKHYDOXHRI P ,, QHHGHGWR
PDLQWDLQHTXLOLEULXP
D Gi T P x
UHODWLRQ
IRUi ! C p
:HDOVRNQRZWKHQXPEHURIPROHVRIWKHLPSHUPHDEOHVSHFLHVLQHDFKFHOO WKH\DUHWKHVDPHDV
LQWKHLQLWLDOVWDWH +RZHYHUVLQFHWKLVLVQRWLQIRUPDWLRQDERXWDQLQWHQVLYHSURSHUW\LWLVQRWRI
KHOSKHUH7KXVWKHDFWXDOQXPEHURIGHJUHHVRIIUHHGRPDUH
F C C p C C p Gi
,,
b
Gi RT OQ J i xi
OQ xi OQ J i
g
RU I i s OQ xi
OQ J i E PRODULWLHVRIWKHVDOWVLQVHDZDWHUDUHLQWKHIROORZLQJUDWLRV
m1D&O m.&O mPJ&O m0J62 Section 11.5
Solutions to Chemical and Engineering Thermodynamics, 5h ed
VLQFHWKHPDMRUVDOWFRPSRVLWLRQLV1D&OZHZLOOIRUVLPSOLFLW\DVVXPHRQO\1D&OLVSUHVHQW
:KHQFRQYHUWLQJWRPROHIUDFWLRQIURPLRQLFVWUHQJWKWKDWLVZHZLOODVVXPHDOO zi $OVR
ZH ZLOO DVVXPH WKDW WKH DGGLWLRQ RI VHD VDOW LQFUHDVHV WKH GHQVLW\ RI WKH ZDWHU VROXWLRQ EXW
PROHV OLWHU ,QWKLVFDVH
OHDYHVWKHZDWHUPRODUFRQFHQWUDWLRQXQFKDQJHGDW 1RZ
GVROYHQW
c
h
,,
VROYHQW
,
c
h
G VROYHQW T P , Ÿ I s RT OQ xs,,
c
c
h
c
hh V
VROYHQW
RU
I s OQ xs,, EDU
V VROYHQW
7KHRVPRWLFSUHVVXUH 'P LVJLYHQLQWKHWDEOHEHORZ
I
VROYHQW
s
G VROYHQW T P ,, G VROYHQW T P ,
s
,
,
s
ZKHUH ZH KDYH DVVXPHG WKH OLTXLG VROYHQW KHUH ZDWHU LV LQFRPSUHVVLEOH DQG ZH ZLOO WDNH
xs, 7KXVZHREWDLQ
G VROYHQW T P ,, I s RT OQ xs,,
,
VROYHQW
VROYHQW
s
,,
G VROYHQW
c
h G cT P x h aT P x f G T P I RT OQ x
cT P h G cT P h V c P P h GVROYHQW T P ,, xs,,
ZKHUH I LRQLFVWUHQJWK
8VLQJWKHGDWDLQWKHSUREOHPVWDWHPHQWDQGHTQV DQG ZHREWDLQWKHUHVXOWVLQWKH
WDEOHEHORZ
7RJHWWKHRVPRWLFSUHVVXUHZHXVHHTQ RUWKHHTXLYDOHQWUHODWLRQ
I
xs
'P
P ,, P ,
I s RT OQ
xs,,
OQ xs
OQ J s
,,
,
xs,,
Js
Ii
'P EDU
cP P h
7KHSURJUDP9/08LVXVHGIRUWKHFDOFXODWLRQVLQWKLVVROXWLRQ1RYDOXHRIWKHELQDU\
LQWHUDFWLRQSDUDPHWHUkijIRUK\GURJHQFRQWDLQLQJPL[WXUHVDUHJLYHQLQ7DEOH7KHUHIRUHD
QXPEHURIGLIIHUHQWYDOXHVZLOOEHWULHG7KHUHVXOWVDUHJLYHQEHORZ
+\GURJHQSURSDQHV\VWHPDW q & DQGEDU
x+ H[S
I
I FDOF
k + &
k + &
k + &
Section 11.5
Solutions to Chemical and Engineering Thermodynamics, 5h ed
+\GURJHQSURSDQHV\VWHPDW q & DQGEDU
x+ I H[S
I FDOF
k + &
k + &
k + &
k + &
I
H[S
I FDOF
k + &
k + &
k + &
+\GURJHQPHWKDQHV\VWHPDW q & DQGEDU
x+ k + &
7KHPRVWREYLRXVFRQFOXVLRQIURPWKHVHUHVXOWVLVWKDWWKH3HQJ5RELQVRQHTXDWLRQRIVWDWHGRHV
QRWILWWKHGDWDYHU\ZHOOXQOHVVYHU\ODUJHYDOXHVRIWKH kij SDUDPHWHURUELQDU\LQWHUDFWLRQ
SDUDPHWHUVDUHXVHG,QGHHGXQUHDOLVWLFDOO\ODUJHYDOXHVPXVWEHXVHG LHRIWKHRUGHURI
UDWKHUWKDQWKHRUOHVVIRUPDQ\RWKHURIWKHV\VWHPVZHKDYHFRQVLGHUHG 7KHPRVWSUREDEOH
UHDVRQIRUWKLVLVWKDWWKHWHPSHUDWXUHIXQFWLRQLQWKHa(T)WHUPRIWKH3HQJ5RELQVRQHTXDWLRQRI
VWDWHGRHVQRWUHSUHVHQWYHU\VXSHUFULWLFDOFRPSRQHQWVVXFKDVK\GURJHQDWWKHFRQGLWLRQVKHUH
YHU\ZHOO,QSDUWLFXODUWKHWHPSHUDWXUHIXQFWLRQ
aT
R Tc
D T D T
Pc
FG
H
N IJ
K
T
Tc
Z Z ZDVREWDLQHGE\ ILWWLQJGDWDDORQJWKHYDSRUOLTXLG
VDWXUDWLRQFXUYH LH Tr d ZKLOHIRUK\GURJHQZHKDYH Tr
7KLVLVDQ
HQRUPRXVH[WUDSRODWLRQDQGXQGRXEWHGO\JUHDWO\LQHUURU6HFRQGWKHDFHQWULFIDFWRUGHSHQGHQFH
RI NZDVIRXQGE\XVLQJYDSRUSUHVVXUHGDWDIRUDUJRQDQGPHWKDQH Z | DQGWKHQRUPDO
ZLWK N
SDUDIILQV Z ! 7KHUHIRUHWKHXVHRIWKLVIXQFWLRQZLWKK\GURJHQ Z DOVRUHSUHVHQWV
DQH[WUDSRODWLRQ6LQFHK\GURJHQLVSUREDEO\VRSRRUO\UHSUHVHQWHGE\WKH3HQJ5RELQVRQ
HTXDWLRQLWLVQRWVXUSULVLQJWKDWK\GURJHQFRQWDLQLQJPL[WXUHVDUHSRRUO\GHVFULEHG,IWKLV
VXJJHVWLRQLVWUXHWKHQWKHK\GURJHQPHWKDQHV\VWHPVKRXOGEHPRUHSRRUO\UHSUHVHQWHGWKDQWKH
K\GURJHQSURSDQHV\VWHPVLQFHLQWKHIRUPHUV\VWHPPHWKDQHLVDOVRVXSHUFULWLFDO
Tr
)LQDOO\ZHQRWHWKDWWKHVHQVLWLYLW\RIWKHIXJDFLW\FRHIILFLHQW RUWKHVSHFLHVIXJDFLW\ WR
WKHELQDU\LQWHUDFWLRQSDUDPHWHUVLVUHODWLYHO\ORZIRUWKHPL[WXUHVFRQVLGHUHGKHUHLQWKDWODUJH
YDULDWLRQVLQ k SURGXFHUDWKHUVPDOOYDULDWLRQVLQ I + F
H
I
K
Solutions to Chemical and Engineering Thermodynamics, 5h ed
Section 11.5
$PRODUVROXWLRQLVPROHVSHUOLWHURIVROXWLRQZKLFKDSSUR[LPDWHO\FRQWDLQVPROHV
RIZDWHUVRWKDWWKHPROHIUDFWLRQRIZDWHULV 6RWKDW
EDU ˜ P
u u .
PRO ˜ .
OQ EDU 3
u ZKLFKLVLQUHDVRQDEO\FORVHDJUHHPHQWZLWKWKHUXOHRIWKXPE
&RQVLGHUDV\VWHPZLWK&FRPSRQHQWVDQGWZRFRPSDUWPHQWV6LQFHLQDJHQHUDO&FRPSRQHQW
V\VWHPWKHUHDUH&GHJUHHVRIIUHHGRPDWILUVWJODQFHLWZRXOGDSSHDUWKDWWKHUHDUH & GHJUHHVRIIUHHGRP+RZHYHUWKHWHPSHUDWXUHPXVWEHWKHVDPHLQERWKFRPSDUWPHQWVZKLFK
HOLPLQDWHVGHJUHHRIIUHHGRP UHPHPEHUWKDWWKHSUHVVXUHVDUHQRWWKHVDPH $OVRWKHSDUWLDO
PRODU*LEEVHQHUJLHV RUFKHPLFDOSRWHQWLDOV RIWKH&VSHFLHVWKDWFDQSDVVWKURXJKWKH
PHPEUDQHPXVWEHHTXDOLQERWKFRPSDUWPHQWVZKLFKUHPRYHVDQRWKHU&GHJUHHVRIIUHHGRP
7KHSDUWLDOPRODU*LEEVHQHUJLHV RUFKHPLFDOSRWHQWLDOV RIWKH&VSHFLHVWKDWFDQQRWSDVV
WKURXJKWKHPHPEUDQHDUHXQFRQVWUDLQHG7KHUHIRUHWKHQXPEHURIGHJUHHVRIIUHHGRPDUH
) & & && &
D )LUVWZHQHHGWRGHWHUPLQHWKHSUHVVXUHGLIIHUHQFHGXHWRWKHKHLJKWRIPRIZDWHU
3 UJK)RUEDUK 3 UJ ANJPAVA ANJPA PVA P+2EDU
7KHUHIRUHIRUPRI+2ZHQHHG
'3 '3
EDURIRVPRWLFSUHVVXUH
$VVXPLQJDQLGHDOVROXWLRQIURPHTQ ZHKDYH
[SRO\ ' 3 9VROYHQW 5 7 9VROYHQW 9VROYHQW
u [SRO\ '3˜ 9VROYHQW
˜ PDVVSRO\ [SRO\
P
PRO
u ˜ ˜ [SRO\
˜ [SRO\ [SRO\ ˜ PDVVSRO\
RUJUDPVSRO\PHUSHUJUDPRIZDWHU
RUJUDPVRISRO\PHUSHUOLWHURIZDWHU
E 1HWUHVXOWRIWKHSURFHVVLVWRVWDUWZLWKSXUHZDWHUDQGPRYHLWXSPZLWKRXWGRLQJDQ\ZRUN7KLV
ZRXOGEHDSHUSHWXDOPRWLRQPDFKLQHDQGFHUWDLQO\YLRODWHVWKHVHFRQGODZRIWKHUPRG\QDPLFVDQGHYHQ
WKHILUVWODZ2QHSUREOHPLVWKDWLQWKHZHOOWKHZDWHUSUHVVXUHLVEDUVRWKHUHLVQRGULYLQJIRUFHIRU
LWWRSDVVWKURXJKWKHVHPLSHUPHDEOHPHPEUDQH6RWKDWWKHZHOOZDWHUZRXOGKDYHWREHSUHVVXUL]HGIRU
WKHV\VWHPWRIXQFWLRQEXWWKLVZRXOGUHTXLUHVXSSO\LQJZRUN LQZKLFKFDVHLWZRXOGEHHDVLHUWRSXPSWKH
ZDWHUWRWKHVHFRQGIORRU,IWKHULJKWPHPEUDQHVFRXOGEHIRXQGLQRUGHUIRUWKHV\VWHPWRZRUNWKH
SRO\PHUFRQFHQWUDWLRQZRXOGKDYHWREHXQLIRUPWKURXJKRXWWKHV\VWHP7KLVZLOOQRWKDSSHQGRWR
FRQFHQWUDWLRQSRODUL]DWLRQDVDUHVXOWRIWKHIROORZLQJZDWHU
Section 11.5
Solutions to Chemical and Engineering Thermodynamics, 5h ed
§ ·
¨ ¸
¨
¸
¸
¨
GDWD ¨ ¸
¨ ¸
¨
¸
© ¹
; GDWD
1XPEHURIGDWDSRLQWV
Q
§ ·
¨ ¸
¨
¸
¸
¨
<
¨ ¸
¨ ¸
¨
¸
© ¹
<
L
L
< GDWD
Q URZV GDWD
L < ¢²
;
L
¢²
6FDWWHUVXJJHVWVWKDWWKHRVPRWLFVHFRQG
YLULDOFRHIILFLHQWLVQRWJRLQJWREHYHU\DFFXUDWH
DQGZLOOEHVPDOOLQQXPHULFDOYDOXH
5HJUHVVLRQ6WDWLVWLFV
,QWHUFHSW
E LQWHUFHSW ; <
E
6ORSH
E VORSH ; <
E
I ; E E ˜;
P
%
˜ ˜ E
P
u %
%HTXDOVE P P57
E ˜P
˜ 3D /J A
I ;
<
;
Section 11.5
Solutions to Chemical and Engineering Thermodynamics, 5h ed
Solutions to Chemical and Engineering Thermodynamics, 5h ed
Section 11.5
Section 11.5
Solutions to Chemical and Engineering Thermodynamics, 5h ed
PO JZDWHU PROVZDWHU
JIUXFWRVH PROVIUXFWRVH
7RWDOPROHV xZDWHU xIUXFWRVH 6LQFHZDWHULVDWYHU\KLJKPROHIUDFWLRQJ ZDWHU EDUîP
î
î.
RT
PROî.
D 3 OQ u EDU
FF
P
Vwater
î
PRO FF
î .
RTm
PROî.
u .
E 'T Tm T f OQ xZDWHU
' fus H
PRO
R
6RWKHIUHH]LQJSRLQWZLOOEH &
vap
T
F $WWKHERLOLQJSRLQWxZDWHU PZDWHU
OQ T
vap
T
EDUVRPZDWHU
EDU
T
T
.VRWKDWWKHERLOLQJSRLQWHOHYDWLRQLV.
)LUVWFDOFXODWHWKHK\GUDXOLFSUHVVXUHDWNP P
NJ
P
1
N3D
EDU
P U gh u EDU u P u u u
P
V
NJîPV
1P
N3D
'U\ZHLJKWRIVKULPSLVVRNJVKULPS JIOXLGVDQGJLVVNHOHWDO DQG
VDOWVWKDW
,ZLOOQHJOHFW 6RWKHDFWXDOFRQFHQWUDWLRQRIRVPRO\WHVLV PPRONJDWVHD
OHYHO
DQG PPRONJDWNPGHHS$OVRIOXLGVRIDOODQLPDOVLVZWVDOWVR
WKDW
JURIVHDZDWHUKDVJUDPVVDOWDQGJRIZDWHU 0VDOW PROHVRI
LRQVDQG PROHVRIZDWHU
6R QHJOHFWLQJWKHHIIHFWRIWKHRVPRO\WHRQGHQVLW\ NJRIIOXLGLQVKULPSDWVHDOHYHO
0RVPRO\WH0LRQVPROHVZDWHU WRWDOPROHVDQG
PROH
IUDFWLRQRIVROXWHLV DQGPROHIUDFWLRQRIZDWHULV
5HSHDWLQJWKHFDOFXODWLRQDWNP
7RWDOPROHVLV 0RVPRO\WH0LRQVPROHVZDWHU WRWDOPROHVDQG
PROHIUDFWLRQ
Solutions to Chemical and Engineering Thermodynamics, 5h ed
Section 11.5
RIVROXWHLV DQGPROHIUDFWLRQRIZDWHULV
7KHRVPRWLFSUHVVXUHDWVHDOHYHOLV
u u OQ EDU QHJOHFWLQJ
3
u WKHDFWLYLW\FRHIILFLHQWRIZDWHUZKLFKZRXOGEHYHU\QHDUXQLW\DWWKHKLJKPROHIUDFWLRQ
RI
ZDWHU 5HSHDWLQJWKHFDOFXODWLRQDWNP
u u OQ EDU 3
u
7KHGLIIHUHQFHLVRQO\EDUZKLFKGRHVQRWFRPSHQVDWHIRUWKHEDUSUHVVXUH
GLIIHUHQFH
7KHIUHH]LQJSRLQWGHSUHVVLRQLV
RTP
'T
xZDWHU
xZDWHU xZDWHU ' IXV H TP
$WVHDOHYHO
'T R &DQGDWNP'T R & 6LQFHWKHVROXWLRQZLWKSHVWLFLGHLVLVPROZDWHUZHZLOODVVXPHLWVDFWLYLW\FRHIILFLHQW
LVXQLW\
7KHVWDUWLQJSRLQWLVHTQ RT
,,
,,
J VROYHQW
3 P ,, P , / OQ xVROYHQW
VVROYHQW
EDUP
u .
u u PRO.
OQ
OQ
EDU
FP
P
u u
PRO
FP
u OQ EDU u EDU
6RWKHRVPRWLFSUHVVXUHGLIIHUHQFHEHWZHHQWKHWZRVLGHVRIWKHPHPEUDQHLVJUHDWHUWKDQ
EDUQRWWKHEDU
\RXUHPSOR\HUUHTXHVWHG
u 7KHRVPRWLFSUHVVXUHIRUDVPDOOPROHFXOHLQDTXHRXVVROXWLRQKDVEHHQIRXQGWREH
JLYHQE\WKH
IROORZLQJH[SUHVVLRQ
x RT
3 P ,, P ,
B x B x V
ZKHUHLVWKHVROYHQWDQGLVWKHVPDOOPROHFXOHVROXWH2EWDLQH[SUHVVLRQVIRUDFWLYLW\
FRHIILFLHQWVRIWKHVROYHQWDQGVROXWHDVDIXQFWLRQRIPROHIUDFWLRQ
Section 11.5
Solutions to Chemical and Engineering Thermodynamics, 5h ed
6ROXWLRQ
P ,, P ,
3
x RT
B x B x
V
OQ J x B x B x OQ x
OQ J x B x B x OQ x
w OQ J wx
B x B x x
w OQ J wx
RT
OQ xJ V
x
x B x B x x
w OQ J wx
x
B x B x x
x
w OQ J wx
B x B x x
x
w OQ J E\WKH*LEEV'XKHPHTXDWLRQ
wx
x
B x B x x x
x
x x
B x B x x
x
7KLVODVWH[SUHVVLRQPXVWEHLQWHJUDWHGWRREWDLQWKHH[SUHVVLRQIRU J ·
·
§
§
OQ x B ¨ x x ¸ B ¨ x x ¸ C
¹
¹
©
©
ZKHUHC LVWKHFRQVWDQWRILQWHJUDWLRQ
OQ J § ·
§ ·
B ¨ ¸ B ¨ ¸ C B B © ¹
© ¹
VRWKDWC B B DQG
·
·
§
§
OQ J OQ x B ¨ x x ¸ B ¨ x x ¸
¹
¹
©
©
OLP OQ J x o
Section 12.1
Solutions to Chemical and Engineering Thermodynamics, 5th ed
)ROORZLQJ,OOXVWUDWLRQZHKDYH OQ x
ZKHUH I x
%HQ]HQH V a
f
G I u V
xV xV FDOFF E\LWHUDWLRQZHILQGWKDW
FF PRO G x H[S W 7ROXHQH V G E\LWHUDWLRQZHILQGWKDW x
&DUERQWHWUDFKORULGH V G H[S W E\LWHUDWLRQZHILQGWKDW x
H[S W &KORUREHQ]HQHLVDVRPHZKDWPRUHFRPSOLFDWHGFDOFXODWLRQLQWKDWWKHUHJXODUVROXWLRQSDUDPHWHUV
DUHQRWDYDLODEOH)URP3HUU\ V&KHPLFDO(QJLQHHU V+DQGERRNWKHIROORZLQJGDWDLVREWDLQHG
PROHFXODUZHLJKW VSHFLILFJUDYLW\ P YDS 3D T q&
8VLQJWKLVGDWDZHILQG V G
FG 'U IJ
H V K
YDS FF PRO a
'H YDS
RT . 'U YDS
- PRO DQG
f
FDO FF ZLWKWKHVHSDUDPHWHUVE\LWHUDWLRQZHILQG x
H[S W 7KXVWKHRUGHURIRXUSUHGLFWLRQVLVFRUUHFWEXWRXUSUHGLFWLRQVDUHEHWZHHQ
DQGWRRKLJK
:HZLOODJDLQXVHUHJXODUVROXWLRQWKHRU\)ROORZLQJLOOXVWUDWLRQZHKDYH
IRUDVLQJOHVROYHQW
OQ x G I DQG
a
OQ x
f
c
G
IRUPL[HGVROYHQWV
h ZKHUH
G
¦ I G DQG I ¦ x V x jV j
j
j
j
j
l
l
l
DQGWKHVXPVH[WHQGRYHUDOOVROYHQWDQGVROXWHVSHFLHV7KHSDUDPHWHUVXVHGLQWKHFDOFXODWLRQ
DUHOLVWHGEHORZ
Section 12.1
Solutions to Chemical and Engineering Thermodynamics, 5th ed
a
V FF PRO
n KH[DQH
&&O
6LQFHWKHPROHIUDFWLRQRIQDSKWKDOHQHDSSHDUVRQWKHULJKWKDQGVLGHRIHTQV DQG WKURXJK
WKHYROXPHIUDFWLRQ DQGVROXELOLW\SDUDPHWHUWHUPVWKHVHHTXDWLRQVPXVWEHVROYHGE\WULDODQG
HUURU7KHUHVXOWVDUHJLYHQEHORZ
&RPSRVLWLRQRILQLWLDO (TXLOLEULXPVROXELOLW\ (TXLOLEULXPFRPSRVLWLRQ
PL[WXUH
xH x&&/
f G aFDO FFf
RIQDSKWKDOHQH
x1$
RIPL[WXUH
x+
x&&/
,QWKHERRN7KH3URSHUWLHVRI*DVHVDQG/LTXLGVWKHGLWLRQE\5HLG3UDXVQLW] 3ROLQJ
0F*UDZ+LOO ZHILQGWKHIROORZLQJSURSHUWLHVIRUELSKHQ\O TC . PC EDU Z TB . 7KHYDSRUSUHVVXUHLVJLYHQE\
P YDS
OQ
x x x x x PC
T
ZKHUH x OLTXLGUDQJH TC
$OVRWKHOLTXLGGHQVLW\LV J FF DQG 0: 7KH+DQGERRNRI&KHPLVWU\DQG
3K\VLFVJLYHVWKHIROORZLQJIRUWKHVXEOLPDWLRQSUHVVXUHRIELSKHQ\O
ORJ P PP+J IRU q & d T d q & T
1HLWKHURIWKHVHH[SUHVVLRQLVJRRGIRUWKHWHPSHUDWXUHUDQJHRILQWHUHVWWRXVKHUHEXWZHZLOOXVH
WKHH[SUHVVLRQDERYH,QWKLVZD\WKHIROORZLQJUHVXOWVDUHREWDLQHG
T
q &
PBYDS
P EDU
P3R\QWLQJ
f % EDU
y% H[S
y% calc
u u u u u Section 12.1
Solutions to Chemical and Engineering Thermodynamics, 5th ed
q&
T
u u u u u u u u T
q &
u u +HUHWKH3R\QWLQJIDFWRUZDVFDOFXODWHGIURP H[S
LM c P P h ˜ OP 7RSURFHHGZHQRZPXVW
˜T
PQ
MN V
B
FKRRVHD k&2 % YDOXH,KDYHQRWGRQHDQH[WHQVLYHVWXG\EXWEDVHGRQWKH T
k&2 %
q & LVRWKHUP
VHHPVOLNHDUHDVRQDEOHFRPSURPLVHYDOXH7KLVYDOXHZDVXVHGWRREWDLQWKH
SUHGLFWHGYDSRUPROHIUDFWLRQVE\XVLQJ9/08 VSHFLHVIXJDFLW\RSWLRQ DQGE\WULDODQGHUURU
FKRLFHVRIWKHFRPSRVLWLRQVPDWFKLQJWKHELSKHQ\OIXJDFLW\>1RWHLI\RXRU\RXUVWXGHQWVREWDLQ
DEHWWHUFRUUHODWLRQSOHDVHOHWPHNQRZ@&OHDUO\WKHRYHUDOODJUHHPHQWLVQRWYHU\JRRG,WLV
TXDOLWDWLYHO\EXWQRWTXDQWLWDWLYHO\FRUUHFW
D )URPWKHGDWDLQWKHSUREOHPVWDWHPHQWWKHIXJDFLW\RIVROLGQDSKWKDOHQHDW q & DQGEDULV
u EDU u
f 1 u h u EDU ˜ P NPRO
c |UV |RS
. u EDU ˜ FP NPRO u P FP u PRO NPRO |W
|T H[S
u EDU
7KHIXJDFLW\FRHIILFLHQWIRUWKHYLULDOHTXDWLRQRIVWDWHLV
P
OQ I i ¦ y j Bij B
ZLWK B ¦ ¦ yi y j Bij RT
1RZ ZH ZLOO DVVXPH WKDW y N LV VPDOO VR WKDW y N | ,Q WKLV FDVH B # B&2 DQG
c
h
F ¦ y B BI c u B B
h VR
GH
JK
R
P
EDU
OQ I S u u
NPRO
. u u u T
1 &2 j ij
j
&2 &2 Ÿ I1
PRO
NPRO
EDUP
PRO.
VR
f 1sat H[S
y1
LM d
N
V P P sat
RT
I1
iO
u QP u U|
V| W
Section 12.1
Solutions to Chemical and Engineering Thermodynamics, 5th ed
E 1RZXVLQJWKH3HQJ5RELQVRQ(TXDWLRQRIVWDWH:HXVHWKHSURJUDP9/08 VSHFLHVIXJDFLW\
RSWLRQ DWWKHVSHFLILHGTDQGPZLWK k&2 1 DVLQ,OOXVWUDWLRQ DQGDGMXVWWKH
u EDU LV
FRPSRVLWLRQ RI QDSKWKDOHQH XQWLO D YDSRU SKDVH IXJDFLW\ RI QDSKWKDOHQH RI REWDLQHG%\WULDODQGHUURU,ILQG y1 ZKLFKLVPXFKORZHUWKDQWKHYLULDOHTXDWLRQ
UHVXOW6LQFHDWVRPHZKDWGLIIHUHQWWHPSHUDWXUHVDQGSUHVVXUHVWKH3HQJ5RELQVRQPRGHOZDV
LQUHDVRQDEO\JRRGDJUHHPHQWZLWKH[SHULPHQWDOGDWD,KDYHPRUHFRQILGHQFHLQWKLVUHVXOW
D 6WDUWLQJIURP(T
OQ [ 1
' +§
7·
OQ J 1 IXV ¨ ¸
57 © 7I ¹
OQ J 1 OQ J 1 PRO
·
§
¨ ¸
¹ ©
˜ .
PRO ˜ .
H[S DQGJ 1
[1
J1
J1
8VLQJ WKLV HTXDWLRQ DQG WKH VROXELOLW\ GDWD ZH REWDLQ WKH UHVXOWV EHORZ $OVR JLYHQ DUH WKH
81,)$& SUHGLFWLRQV WKRXJK VRPH FOHYHUQHVV LV QHHGHG LQ FKRRVLQJ IXQFWLRQDO JURXSV LQ WKLV
PRGHODVGLIIHUHQWFKRLFHVUHVXOWLQGLIIHUHQWSUHGLFWLRQV
6WUXFWXUHRIQDSKWKDOHQHDQGDFHWRQH
6RWKDW [ 1
$&+
$&
$FHWRQH &+ & &+ 81,)$& J 6ROYHQW PROHIUDFWLRQ
FDOFXODWHG J 1LWUREHQ]HQH
[1 $FHWRQH
0HWKDQRO
:DWHU u 6R81,)$&JLYHVWKHFRUUHFWWUHQGVDQGRUGHUVRIPDJQLWXGHEXWLVDELWWRRORZIRUWKHILUVWWKUHH
VROYHQWVDQGDIDFWRURIWRODUJHIRUZDWHU
7 'IXV+ ˜ §¨
6 7˜ 7 ·
§ 6 ·
¸ ˜ OQ¨ ¸
© 6 ¹
© 7 7 ¹
7 6 7˜ 7 ·
§ 6 ·
¸ ˜ OQ¨ ¸
© 7 7 ¹ © 6 ¹
'IXV+ ˜ §¨
2
7 'IXV+
6 u 7 'IXV+
PRO
6 u PRO
Solutions to Chemical and Engineering Thermodynamics, 5th ed
Section 12.1
)URPWKHGDWDJLYHQLQWKHSUHYLRXVWZRSUREOHPVZHVHHWKDWWKHVROXELOLW\RILVROHXFLQHDWR&
LVJNJRIZDWHU7KHUHIRUHWKHOHXFLQHLVROHXFLQHVROXWLRQFDQEHFRROHGWRMXVWXQGHUR&
ZLWKRXWFU\VWDOOL]DWLRQRIWKHLVROHXFLQH$WWKDWWHPSHUDWXUHWKHVROXELOLW\RIOHXFLQHLVDERXW
JNJ ZDWHU 7KHUHIRUH JNJ ZDWHU RI SXUH OHXFLQH FU\VWDOV FDQ EH UHFRYHUHG
&RQVHTXHQWO\RI WKH OHXFLQH LQ WKHRULJLQDO VROXWLRQ FDQ EH UHFRYHUHG DV SXUHFU\VWDOV LQ
WKLV SDUWLDO FU\VWDOOL]DWLRQ SURFHVV ,I WKH WHPSHUDWXUH LV ORZHUHG WR UHFRYHUHG DGGLWLRQDO OHXFLQH
FU\VWDOVRILVROHXFLQHZLOODOVRIRUPVRWKDWWKHSXULW\RIWKHOHXFLQHZLOOEHFRPSURPLVHG
$VVXPHDFWLYLW\FRHIILFLHQWLVLQGHSHQGHQWRIWHPSHUDWXUHRYHUWKLVWHPSHUDWXUHUDQJHWKHQIURP
(TE
77
[ 7
77
[ 7
5 OQ
' IXV + 5 OQ
7 7 [ 7
7 7 [ 7
)URPGDWDDWDQGq&
u ' IXV + u
OQ
PRO
)URPGDWDDWDQGq&
u OQ
' IXV + u
PRO
)URPGDWDDWDQGq&
u u ' IXV +
OQ
PRO
$YHUDJHRYHUDOO -PRO
7KLVVXJJHVWVWKDWUHSRUWHGYDOXHDWq&LVWRRKLJK²EXWDYHUDJHLVSUREDEO\2.
Section 12.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
/HW TNB
QRUPDOERLOLQJWHPSHUDWXUHRIWKHVROYHQW TB
DFWXDO HOHYDWHG ERLOLQJWHPSHUDWXUH
RIVROYHQW$WWKHDFWXDOERLOLQJWHPSHUDWXUH a
fL TB P x
a
V
B
f x J x f aT Pf f aT Pf
xJ x
f aT Pf
fL TB P x
6RWKDW
V
L
DQG
OQ xJ x
OQ
L
V
6LQFHWKHYDSRUDWWKHYDSRUOLTXLGLQWHUIDFHRIHLWKHUDSXUHERLOLQJVSHFLHVRUDVKHUHDOLTXLG
ZLWKDQRQYRODWLOHVROXWLRQLVWKHSXUHVROYHQWVSHFLHV1RZ
f f cT P yh f aT Pf a
a
fV TB P
fL TB P
B
B
B
f G aT P f G aT P f RT
f
V
L
B
B
B
1RZIROORZLQJWKHFRQVWDQWSUHVVXUHSDWKEHORZ
T T% /LTXLG
9DSRU
/LTXLG
9DSRU
T T1%
ZHREWDLQ
TB
' YDS H TB P
' YDS H TNB ³ 'C dT
3
TNB
TB
' YDS S TB P
' YDS S TNB ³
TNB
'C3
dT
T
ZKHUH
'C3
6LQFH ' YDS G TNB P
C3V C3L %\HTXLOLEULXPUHTXLUHPHQWIRUSXUH
' YDS H TNB TNB ' YDS S TNB
IOXLGDWWKHQRUPDOERLOLQJSRLQW
' YDS S TNB
DQG
' YDS H TNB
TNB
Section 12.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
OQ xJ x
' YDS H TNB ª
TB º
« »
RTB
T
RT
NB ¼
B
¬
³
TNB
'C3 dT TB 'C3
dT R TNB T
³
)XUWKHUPRUHIRUVPDOOVROXWHFRQFHQWUDWLRQV
J x | DQG OQ x OQ x | x VRWKDW
' YDS H TNB ª
TB º 'C3 ª TNB
T º
x
OQ B » « »
« RTB
R ¬
TB
TNB ¼
¬ TNB ¼
a f
a
f
DQGZKHQWKH 'C3 WHUPVPD\EHQHJOHFWHG
x
' YDS H TNB § TNB TB ·
¨
¸
RTNB
© TB
¹
RU
TB TNB 'TB
#
n
ERLOLQJSRLQWHOHYDWLRQ
RTBTNB x
RTNB
x
|
' YDS H TNB
' YDS H TNB
FRPSDUHZLWKIUHH]LQJSRLQWGHSUHVVLRQHTXDWLRQ
Tm T f
TB
' YDS H TNB ª
TB º 'C3 ª TNB
T º
OQ B » « »
« RTB
R ¬
TB
TNB ¼
¬ TNB ¼
OQ xJ x
³
FRPSDUHZLWKHTQ ,I 'C3 LVLQGHSHQGHQWRIWHPSHUDWXUHZHREWDLQ
³
VRWKDWZHREWDLQ
TB
TB 'C
ª
T º
3
dT ' YDS H TNB « B » 'C3 dT TB
T
T
T
T
NB
NB
NB ¼
¬
' YDS G TB P
'T
n
IUHH]LQJSRLQWGHSUHVVLRQ
#
RTm x
' IXV H Tm
6LQFH ' YDS H ! ' IXV H LWIROORZVWKDWWKHDGGLWLRQRIDVROXWHGHSUHVVHVWKHIUHH]LQJSRLQWWRD
JUHDWHUH[WHQWWKDQLWUDLVHVWKHERLOLQJSRLQW
6LQFHPHWKDQROHWKDQROJO\FHURODQGZDWHUDOOKDYH2+JURXSVWKH\DUHLQVRPHVHQVHVLPLODU
$OVRVLQFHQRDFWLYLW\FRHIILFLHQWGDWDDUHJLYHQZHZLOODVVXPHWKDWWKHPL[WXUHVLQYROYHGDUH
LGHDO>2QHUHDOO\VKRXOGJRWRWKHOLEUDU\DQGWU\WRILQGDFWLYLW\FRHIILFLHQWGDWDIRUWKHVH
V\VWHPVRUGDWDIURPZKLFKDFWLYLW\FRHIILFLHQWVFDQEHFRPSXWHGRUXVHWKHYDQ/DDUFRQVWDQWVLQ
7DEOH@$VVXPLQJLGHDOVROXWLRQEHKDYLRUZHREWDLQIURPHTQ ' H Tm ª Tm T f º 'C3 ª Tm
T º
OQ m » OQ x IXV
«
»
« R
R «¬ T f
T f »¼
«¬ TmT f »¼
1RZ ' IXV H - PRO DQGIURP3UREOHP
'C3 C3L C3S
- Jq & u J PRO - PROq & Section 12.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
7KXVIRU Tm
OQ x
7KXV x+2
LM
N
LM
N
OP
Q
OQ
u OP Q WRFDXVHDIUHH]LQJSRLQWGHSUHVVLRQWR q & WKDWLVZHPXVWDGG
x
y
m
y
m
RU y
PROHFXODUZHLJKWRI
m 1RZIRU PHWKDQRO m y g
HWKDQRO m y g
JO\FHURO m y g DUHQHHGHGSHUJUDPV + 2 WRORZHUIUHH]LQJSRLQWWR q & 1RWHVXSSRVHZHWRRNLQWRDFFRXQWVROXWLRQQRQLGHDOLWLHVVLQFHWKHYDQ/DDU
FRHIILFLHQWVIRUWKHVHV\VWHPV VHH7DEOH DUHSRVLWLYHWKHDFWLYLW\FRHIILFLHQWVZLWKEH
JUHDWHUWKDQXQLW\DQG OQ J + 2 ! 7KXVLQVWHDGRI OQ x+2 ZHZLOOKDYH
OQ x+2J +2
q &
PROHIUDFWLRQRIVROXWH
/HW y JUDPVRIVROXWHWKDWQHHGEHDGGHGWRJUDPV + 2 m
VROYHQW
7KXVZHQHHG
. DQG Tf
q &
RU OQ x+2
OQ J +2 VRWKDW x+ 2 ZLOOEHOHVVWKDQKHUHDQGPRUH
VROXWHPXVWEHDGGHG
)RUWKHVHVROXWLRQVWKHHIIHFWVRIQRQLGHDOLWLHVDUHQRWWRRODUJH8VLQJYDQ/DDU
SDUDPHWHUVLQ7DEOHIRUPHWKDQRO + 2 DW q & ZHILQG x | LQVWHDGRI
KHUH VRWKDW yPHWK # JUDPV J + 2 LQVWHDGRIJUDPVKHUH %HORZDUHIUHH]LQJ
SRLQWGHSUHVVLRQGDWDIRUYDULRXVDTXHRXVVROXWLRQV PHWKDQRODQGHWKDQROGDWDIURPWKH
+DQGERRNRI&KHPLVWU\DQG3K\VLFVJO\FHUROGDWDIURP*O\FHURO HGLWHGE\&60LQHUDQG
11'DOWRQ5HLQKROG3XEOLVKLQJ1HZ<RUNS )URPWKHVHGDWDZHILQGWKDWWR
ORZHUVROXWLRQIUHH]LQJWHPSHUDWXUHWR q & ZHQHHGZWPHWKDQRO YVZWLQRXU
FDOFXODWLRQV ZWHWKDQRO YVZWFDOFXODWHG DQGZWJO\FHURO ZW
FDOFXODWHG Section 12.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
$VVXPHLGHDOVROXWLRQVWREHJLQDOVRQHJOHFW 'C3 WHUP
' H Tm § Tm T f ·
§ T f ·
OQ x+ 2 IXV
¨
¸ ¨
¸
¨ TmT f ¸
R
¨© u T f ¸¹
©
¹
(*)UHH]LQJ
J&DOFDW±R&
+ 2 IUHH]LQJRXW
Tf Tf J + 2 J (* x
x+5($/
x+,' 2 2 OQ x(*
F
GH
Tf
u Tf
I
JK
+ 2
Section 12.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
)RUQRQLGHDOVROXWLRQV x
x LGHDO
J
7 I.
LGHDODQGUHDO
UHDO
LGHDO
(*
IUHH]HV
RXW
ZDWHU
IUHH]HV
RXW
[+2
Section 12.3
Solutions to Chemical and Engineering Thermodynamics, 5th ed
' IXV H
-PRO
x
Tf FRPSRQHQW
Tf FRPSRQHQW
Download