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Heat and Thermodynarnics
1,12
5 l1 t{lo.tic Theory of Gasec
The continuous collision of the molecules of thc gas with the
rvalls of the containing vessel and their reflection from thc u,alls
results in the change of momentum of the molecules. According to
Newton's second law of motion, the rate of change of momentum
per unit area of the $'all surface corresponds to the force exerted by
the gas per unit arca. The force per unit area measures t-he pressure
15t
Nature o! Eeat{
The component .of the ^velqcity with which the molecule P
BCFG is ur^and the momentum of the
*itt stri[i it" lppotitt"fag'c
'
is reflected back with the samc
molecule
Tr.,
*.i.."ti i" -rr.'
momentum nltll \r ah opposite directign and after traversing a
;i;;;; I rvill itrike thi"oirpoiitC face ADEE'
of the gas.
"/
of the trinetic theory of gaseq{
,r4P r'loetulatcs
gas iis composed of small indivisible particles called
The g.,
rr,.
.z(d1
"(1
'mol ecules. LThggripqrtier, gljhejrrdlvi d ual molecul es are t he sam e
--G|Fil
distance between the molecules is large as compared
to thai of a solid or liquid and hence[_ghc forces of inter-molicu]arl
attragtio.n are negl
arc continuously in. .motion _with varying
46;ryfnr-hnd molecules
the molecules move in straight lines betweer, 'ur,!
velocities
two consecutive collisionsf The collisions do not alter the molecular
density of the gas, i.e.,lo.n the average the number of molecules pre.
sent in a unit volume reinains the same.{ AIso, the molEdulEs?oio-t
t
of the gas.
i-ccur;iilaG;f@me
W^rl6rl The size of the molecules is infinitesimally small an
compared to the average. distance.trav.ersed by. a molecule betwees
any 1wo consecutive collisions3 The distance between any two 6q1:n
secutive collisions is called free path and the average distance i:-
called the mean free path. The mean free path is depindent on tt is
pressure of the gas. If the pressure is high the mean free path
- is le:le
'ss
ind if the pressure is low the mcan free path is more.
ern
mu1-(-mut\ :2mur
As the velocity of the molecule iszr, the time interval between
cwo successive impacts on the wall BCIG is
--
Ig9"!o t!1e_-Unle
IE_ *gtSe gtrurs-p-er&slll nju{.gq$j9_!phsr9!
whole ol therr energy ls Elnellc.
..!q
Fig. 5'7
The change in momentum produced due to the impact is
4n
d he
t
2l seconds
ur
.'. No. of imPacts Per second
I
t. srir s'p".;io;;-t .hli."",nt. of a Gae
2t
ct of the molecules
A.
Y on the walls of the containing vessel accounc ior the pressurc of th,s.
,r$
u "
gar..
na:fr
",
:2rur
ne
(Co4;idcr a cub-rqallLesg!-,4 B-}\\!OE of side I cm conrainin.
thc g)s
il.'ig. 5.7). The volume of tEe veffiT)C
-is 13
cc. Let n and ,zt represent- th_e very large number he
fas
ioleculcs present in the vessel and the mass ofeach moleculf
Is
resPcctively.
P
movi.ng-.in
a
random
a
direction witl a
Consider molecule
velocity Cr. Thggggllylan be reiiilvEd
axes respecti
comPon.llt-s 1Ll, ?'t
Therefore,
C12
:
Change in momentum produced in one second due to the im-
pact of thiJmolecule is
z*u'x#
ry
The force Fx due to the impact of all the n molecules in one
second
--lq---
u1,1-vr.2*wrz
=
: +luf*uzz*"'+"J I
lU
Heat and Thermoilynamict
t.',rrce l,er unit area on the walil BCIO or ADEE is equal to the
1rr
errure Py
Px :
e:
13tr
- Vp
or
squ-aree
rh"!;;;'id;;i"if"iifr-.*ind
"f
moloculea.]
TABLE
Molecular Velociticr et OoC
given by
,, : # (ur21arr;......+r"2)
and the prcssure P7 on the walls ABCD and EFGE is given by
Molccvlar Wciiht
fr {rrr+rrz+......+wnzl
18.4 x 10r
Ilelium
1
13.l x l0'
Nitrogen
?a
4.95 x lO'
20
a.ut * ro.
ArSOn
.10
.1.14 x l0r
1- (aaz + o s2 + wzz) + ......
Carbon dioxide
41
3'95 x lo'
* (u,2+r,"r*.,r)
Chlorine
7L
3.11 x l0r
3
tt, f
(urz { x rz 4 ut') + (ur' +u"'. + r'r"l
5F L
]
: #[ cf +c22+cJ+......c",
]
...(d),
But volume, Y _: lr. Let C be the root.mean-square velocity ol
the molecules (R.M.S. velocity).
C, :
C r'
Orygon
<: X.l*S. vclocitY of hYdrogen
-'-4r/ The densiw of hydrosen at N.T.P. is 0'000089 glcc' There'
&/ forc C for hydrolen can bJcalculated as follor'r's :
c- rT
*Cr' *Cr'*.-....CJ
7L
or
nCz
3tzox lr6te8l
: V/ ----T-OoO0-89-_
- Cr2+C22+Ct21.......Cn2
Substituting this value in equation (i), v".e get
D _ m.n0z
'-
3v-
...(ii)
Brrt .4f : rntl rvhere .0f is the mass of the gas of volume V, m is
the mass o[ each molecule and z is the number of molecules in a
volume [/.
: l'B4x 105 cm/s
(b) For oxygen the density it N'T'P'
: l6x0'000089
: f sx%;T56tggr
C for oxYgen
L;From
in cm/s
2.016
p _ Px*Pv*Pz
Then
Root mcan cqtare wlocitY
Eydrogen
a" $qpresrurc of a gas is the same in all directions, the mean
pressure P is given by
:
...lirr'
tNotc. R.M.S. volocity C ia tho Equaro root of tbo moan of tbe of tJre
it,ienot'oqual to tbe meenvolooily
.|
#r(urr zrrfurr+...... +z,r)
Similarly the pressurc Py on the watls CDEF and lBO.& is
Pz -
155
Noture ol deot
fJ
: 4'6x-e;-O
l0{ cm/s
oOOOeO-
(c) For air the densitl' at N'T'P' is 0'001293 g/cc
I _____o:ooizoz__
3 x 76;l3G;(e8t
c for air
: {
/
-- 4'850 x 10' cm/s
of moleculee in one c.c. of
a#xarople 5.4 calculote the number
ilata
orgjri "'t W?f .p- using the lollowing
:-
tctt -
i
t5(;
iq L,l<,*l /-lt.r)t /<i ,t,.,,
(
H at an it o*- or rlr**,
e
^"
\ r! rt , <rJ Eeat
157
Let the mzuls of each molbcule be rz and Avogadro's number be
.rY.
M:mXN
Dcnsity of merarry : 1J.6 glcmt
R.M.S. uelocity of ozygen molcculn ot 0.C
: 4.62 y 10t cmft
Mass o! one tuolecule o! orygen
- SZ.8:r.l0-u f
J-og,
P:
-f, *No
tu,
| '"o
2N
* o,
3
Let the toass of each molecule be m and the number of
molecules in one cc
- z
i-J rnnC2
P:
mc2
- x 13.6xgB0 dynes/cm2
76
m : 52'B X l0-2{ g
C : 4.62 x lff cm/s
'\
1l$tr
, oe
r(insfi6
Kn
Energy per Unit Volume of a Gas
.y',rr
u:4,-oJrxluu
n : t'S$J
(q\ L;( Kv*?: J
I wt
: T'*
,o%o*
.
Also from equation (ii1, G a, T
It means that the root mean square vclocity of the molecules is
also directly proportional to the square root of the absolute
PCz
3
temPerature
r"
fi' -culeshtoelbeaome root meon s$urrctoelocitg ae tlut olhgibqet
: i-E
moleculee at
2
w"hgrc E : l.pC! and is equal to the Linetic energy perunit volume
ot the gas. p is.the mz*s per unit volume. Hencil'the pressure of a
nymerically
.cgual to tu'o-thirds of the ..uriEir,"iic energy of
-C_T_i:
" jiru':.$.'ilH"l[:::|;;ffi:'#J.' c q'<'"' t'^ r
\\''
Dhe
*r".rg*
r\c"o
For hydrogen molecules
f, "'rcf : ]*r,
.".(i)
For oxygen molecules
Pcz
J, *Ps :
P- +!f
{vr,
...(i0
Dividing (d) by (di)
PV:*'"
Consider I gram molecule of the gas at a temperature ? K
PY:RT
J" uo: RT
iro * *o,
?
i ,,a: +o
pressure of a gas, according to the kinetic theory, is
P:i- I
-10trC
The enerry of a gas molecule ir,
b
*S
yO bO ,{r".{ C
zero temperature, the kinetic energy should be zero. It means at
absolute zero tempcrature, the molecules are in a perfect statL of rest
and have no kinetic energ'y. But beforc the absolute zero tempcrature is reached, all gases change their state to liquids and solids.
3x76x13.6x980
h:
.r"ryp
Thus, from equation (;i), thc mean kinetic cnerg'y of a molecule
is directly proportional to the absolute tcmperature of a gas. When
the tenpe;ture of the gas ir increased, the mean kinetic energy ofl
the moleculcs increases. When heat is withdrawn from a gas, the
mean kinctic energy of the moleculer decreases- Thus, at absolute
3P
?L:
Herc
Here & is c^llcd Boltzmann,s
I
P:
...(dr)
.
.. (t)
Here
mr0f
fr
Cy :
Ca
@7:T
*
:;,r1,""r,,
...(dir)
H eat ond T her modynamic,,
Nature oJ Heat
: 173 K
T. :?
:16
lf,t
t'-
I
Py:;MC2
,)1.2
From equation (iii1
mrOrtT,
Consider one gram molecule of-a gas at an absolute tempera'
rure ?. The meanlnersy of the molecules
- mtCt'
T2 : 16xl73
: 2768 K
?2
-lMc..
2
: )- N'nc'
rv : \' Nmc'
T2 :2768-273
: 2{950c
: I tt.-f,'.*c'
E*.r-plq.s,&d Culculate rle RMS velocity o! the oxygen rnole-
culcs al 27"C.
First calculate the n.dfS velocity of oxygcn at N.T.p.
n
"-
Here
: LmCz
2
p
,:Jmr
: 16x0'000089
C : 4'6x 10. cm/s
Let the 8.tr48 velocity of the molecules at 27oC be C1
+:'[T
c1
PY:-|rv.*r,
But
Ct
.
: 4'6t,* Js-
C, : {.8*Xl0t cm/t
5,17 Derivatioo of Gas Equation
' F.o* kinetic theory,
P: +
pc2
PV:frT
...(d,
In the gas cquation
PV:RT
P is in dynes/sq cm
C : 4.6X l0r cm/s
:300K
PY - NkT
/Vx& :.8
Note on the Gas Equation.
:'cy J T
T :273 K
?r :27oC
: 27 *273
/
:3kT
2
P : 76X 13.6 X980 dynesTcmr
P
Here
Mean kinetic energy of a molecule
l-gP
- V
t )I nt
3v
I : B'31X I0? ergs/g mol'K
?isinK
/ is the volume in cc per gram molecule.
Eraople 5'7 Calculale the volume occupied by 3'2 gtams of
oaygetu at ?6 cm o! Hg and 27"C,
Here
P : 76x 13'6 x9B0 dYnes/sq cm
r :27,+273
:300K
B : 8'31 x l0? ergs/g mol-K
PV:Bf
8'1lxl0?x300 cc
g-mor
l : %
xT36x98o Per
I/ - 2{610 cc per g mol
Heat and Thennoily namiu
Volume for 3'2 *
From equation (dd)
: 2461 cc
PY - NKT
7
is
When
constant,
Pc., lI
of Gas Laws
fi'l{tDerivation
(i) ggP\Iw(t*)
v
According to the kinetic theory,
It means that for a given mass of gas, the presurc is dircc4y
proportional to its absolute temPerature provided the volume remainr
':ig
constant.
P* *#*.
ooluma of an iileal gaa ie gfuen by
At a constant tempcratureT, O is constant, rn.r"ro..
constant tempcrature
PN
O: -ffi
"r(']
t
J MC': constant
Hence
Li;l -gMk*I&;D-
G9
whera N io Aaogadro's number.
For an ideal gas, for one gram molecule of a gas,
-at constant tetnperature.
But
Accoidirrg io the kinetic theory
":
P:nkT
P- *#*
Consider oue gram molccule of a gas at
P
o: -Tr
akolute temperature
But
M:mN
pv: Iy,lr.r.
J
-
...(d)
PN
: nT
Erample 5'9. Calatbte the nunfrer of molcculca in one ctlia
melre of an iileol gaa ot N.T.P.
3 --
Z *C': ;AI
Let the number of molecules per cc be to.
mCn :3bT
PAI
O: 'ffi
Substituting this value in equation (i),
PY : NHI
...(rr)
where lV is the Avogadro's uumber and & is the Boltzmann's
Number of molecules in one cubic mcue volumc
,:
constant.
If
-n
*:-fr
fl,
The mean Linetic energy of a molecule
I
?V:RT
R: NK
PY : NKT.
Let z be the number of moleculcs per cc. fn that case,
*.,
PY:**o
T.
/
r'.'ample 1.8 Shoo tha,t n, the nuwber of nolcoula pr u;nit
Pr- **o
PV - const.
IOZ
(iii1 Begnault'a Lau
"t "at.rX;rr* r.,
32
,ao
Noture of fleof
P is constant,
Yc.T
Ir means that for a given mass of gas, the volume is directly
pr<>g;rt ional to the absolutc temperature provided the pressure
rcrnains c()nstant. This i: Charles' Law. 9*
JI.)
la .L e,v*A
ilere
nXlS
PIV x l6c
u: -m-
P :76x l3'6x980 dynes/cmr
.l[ * 6.023x lOa
I : B'31 x l0? crgig rnol'K
Naturc of Ecot
Eeat anil Thermdynaniu
762
Hcrc
T :273K
76x 13'6x980i6'023x lOax 106
4t:
n1
: 2'7 X l0re
Pt : 76x l3'6 x 980 dYnes/cur
P1
: lQ-e mm of Hg
l0-? cm of Hg
:dYnes/cmt
- : r.Utt*t,,u
&
Erample 5'10 Calculate the number of moleaules in one like
oJ an iileal gae al 136'd Q temperoture onil 3 armoepheres prellure.
fi1 :
Let the number of molecules Per cc :
Plv
(ddt Let thc R.M.S. velocity at OoC be C1 and at 39oC bc Ot'
Numb'cr of molecules Per cc at OaC and l0-0 mm of Hg pressure
e tl,tr
and at 39oC and l0-c mm of Hg pressure
Number of molecules in one litre,
c:7,X103
P][x lOE
:
t: --w-
Here presure is the same in both the cases
E - 8'31xl07 ergr/g mol-K
I' : (273+ 136'5)
: 409.5 K
ImnP;:Imn,cg,
n1?1t :
3 x 76 x l3'6x980 x6'023 x lOax l0r
7lg :
B'31 x l0?x409'5
/,
--t','U*'0"
number of moleculea per cc ol o gae ie
The
5'11
,Zfzlcrote
'/., *r*, at-N.t.P. Calcul,ate the number of rnoleculee per ce of lhe
But
gu.
' () of 0C anil 10'6 mm pressu?e of mercury and
(id) al 39oC ond f0-6 mm pressure 'nercuty.
For a unit volume of a gas
'
P:LmnCN
5
From (d)
or
"td
Pr :
,rt :
n,
n,
nrxPl
-FI
C,'
W
C e' t/-T
C,, T,
tt': rl2xTr
-Tlr\: 3'553 x l01o
f1 :0o C
:273+39
... (i)
At OoC and l0-'mm Pressure
mnror
tl2
:273K
At N.T.P.
I
Hcre
tus0gs
fs:39oC
(i) At 0"C, R.M.S. velocity is equal to C
P, : -g- mn1 Cr
fL,
P:!mnC'
J
P:3x76x13'6x980 dYnes/cmr
-l[:6'023x104
Pt : i
(dd)p,
2'7x l01e x l0-?x 13'6x980
------7il13i-;580--
z1 :3'553 ll0ro
7t
n-w
Hcre
l0-z x l3'6x980
.. . (ii)
:312K
3'553x1010x273
ms: _*____m_
\:
3'109 x lolo
ro
I 5't9 ,,Avoga&o's HYPothesis
tJL, " konrider two gases A and B at a pressure P and each having a
1 vo)ume I/'
: m1
r\VC Z
Mass of each molecule of the first gas
Number of molecules of the first gas : rLL
Mean square velocitl of the molecules of
the fint gas _ qt
For thc 6nt gar
p: I e,e.
:+
Similarly for the
.
'#
P:
Nalarc of Eeal
g?ret.
Here, r, and r, represent the rates of
,*-,;ff;?l;,u;;;::1"#;ti::'tototranitomhineticenersvo!onc
_+ryL
Total random kinetic energ.y per gram-molecule
$ r,4,
I
e
= ; kT.N
: +. #,,
From (d) and (dr),
l
or
**C!__
I .mrnr1sz
c
- J-R?
2
3
n*t7f _ ,irnrCrr'
the twogasesn g at- the same
,.,_^^,lf
temperaturc ?, the;:::
f,rnettc energy of the morecures
oi [.iiiirr" gases is the same
+m,c,,-|wc,"
From (r:di) and (iu),-
"
\ 6i\-8 i
h
...(du)
::r.;-;
.--- r-:'""' "qpd ":lfn :,: :.,#=
conditions or
;:ffifi :lX"U:J,;ff f$;91.";;;l,u*u..;;.i;il.il'tni,
5.20 Grahan's La,w of Difueion of
According to the trinetic tfr"orf
or
^c
Geeeg
: 3'735 X I0ro ergs
- S7Sijoules _/
ffi ill"tlgdH'"".H#:o,:r1ffi ff :.?"i:'#:t-'H:ffi ;1'"'""",*",#I
cvle
,ffff':;:',!a!,;];*:;'!;ov.'*'kineticenersvo!omote-
.---=---r
Average K.E. of a molecule
@
=j-m@:
Here & is Boltzuann,s constant.
Noto. The aven
f *l.3Bx l0_rcx300
: 6'2r x ro-r' ergs
m:*ild&,'H?fl ,YirlT*""i,""9,"k.:;';ffi '"#"TI.iT"#;
- It means *, *:311,,:J[r"re vcrocity of the morecures
3:1,ffr .,"#,x,mr:::lt{;t,:1T..d,",?::Lb,}"Tu
P,
+-/f
*o* -
I.38x l0-rc erg/molecule-deg
Averagc K.E. of a molecule :
/F
'Cn l
C,_
. : f *e sx r0?x300
&:
P:*r*
T
c'-Vp,
of oxygen
' , m@xN
...(,,)
where zn. ,spres@l the ,nass
of each molecule,. z, the
molecules atia crr tue meu,,
:rumber of
rq""i"".r.1iry ot.the morecules.
T ---7-:
diffusion of the two
...(,)
second gas
.
165
EaTplc 5.14. Colculnte the mean bandationol bhetic cnery!
i;;r:; n":';.JZ joulcs/motc_K
gw mol,eculc of a gaa at |Zt"e,
Aoogailro'e
rrerc
,
-/
nurfier,
N - 6.06x10u
(Dcki' 1974)
i::?*:ii-il"1*"
iI:6.06xIOD
8.32 \,jouler/molecule'K
L_!:l
T - (606;10-"/
Mean tranrlational tinctic cnergy per molecul
" - |Ul
Eeat anil llhermollynamice
166
3x8'32x1000
: T;TT6;TT
: 2.059 a l0-ro joule
Erampte 5'15- Colculate the lotal ranilom kinetic energy oJ one
gram of nitrogen at 300 K.
!ii:-l
Nalure oJ HuAV
Let il bc the Avogadro's number.
Then j IenNCs : ;- kNf
I MCz
--- : 3--
v
;Rr
c:r{ry
Total random energy for one gram molecule of nitrogen
q
Here the molecular rveight of mercu\ M -
: ;RT
C:.T@
v
Total random energy for I gram ofnitrogen
3Rr
: -2Af
whcre the molecular weight of nitrogen y
3RT
n:-
: 28 g
2M
-3x8'3x107x300
- --ETE-
: 133.4 x 107 crgs
: 133.4 Joulc
: 1.93 a 10r crn/e
//
Eramply'5 18. Wilh what speed woulil one grom molecul,e ot
orygen at 300 K be moaing in oril,er that the translational, kinetia
enegry of its cenlre of mass is equol to the totol ranilom binetic enetgg
oJ all, ila molecul,es 2. Molecular wei,ght of ozygen (M) : 32.
Total random kinetic energy tf I gru--*olecule of olygen
-ezi-
- f,i'nr : +xB'3xlo?x3oo
- 3.735 x l01o ergs
Kinetic energy of .ilf grams of oxygen moving with a velocity o
Notc. Tbc total resdom kinotio energy for I gram of a gar h difierun!
for diferent gaaes at the gamo temporature.
-Eranple
- iMas
1.16. Colculote tlu I'otal random hinctic cnugg o! 2 g
of helium at 200 K.
g'Jggy lQro
|*r,:
Energy for I g of helium
: -zfr3RT
j
I
xSZxu2 : 3'735 x l01o
.
Energy for 2 g of helium
2x3xRT
: ----mi3RT
:__fr*
3x8.3x107x200
:____l-
. Exarople 5'19. Calculale the temperature at whiah the r.m.e.
tselocity of o hyilrogen mol,ecule will be equal to the opeeil oJ the eorth'e
first aatellite (i.e. o : 8 km/s).
Energy for I gram molecule of hydrogen
:!uor:1ar
- 2"-'2
m
,:-57
Moz
:1245 joulee
2x(8x105)r
: T;3:3;-i3r*
_ 1'.n- ple 5.17. Calailate the root mcan aquo?e oelocjty o! a
molecule of mercury aqpoltr at 300 K.
tc
:+mO-;W
3'735 x l01o
__--]6-
r : 4.8 X 10r crn/s
: 1245x 107 ergs
Mean Li.ctic encrgy of one molecule of.mercury
221W
./.
: 5'14x1$ K
Exaople.520. At what temperalure, ptessure remaining e;a.tu,
tant, wil,l the r.m.s, oelocitg oJ a gas be halJ its talue oA OoC I
lDel,hi lEone.) 1975J
Heal anil Thermoilynamia
c, lT
C, E.V T
.WE
/ l'?2 Atom.icity of Gasec
Mono'-otomic- ga.E. A mono-atohic gar molecule
fe\(l
has one
Atom'-tach molecure has thrcc d.gr;;r;i?..doa* io-i."rif,tory
r
T:_1,r-r;
E5'
,:o,
+
Encrgy associated with each degree of freedom _
273
:;
-2(/r'75'c '/
*
.
5.21
Degreee
of
Freedom and r![arwett's
R,
--- - Law
--:"6iof Fquipartition of Eaergy
-'
AJ
,"r:eryyg:
-rYx |r*
)*o': * o,
C, -
JI7x,t
[But,
nil *
:*;
$:: *l:g.H, il;:*f i1$*
I:##*ii::"fl''
c,-#:3ra
*qoe{td
lm(ur):Imgf):lm(wr)
I mC' : 3 [l m(ur)i: 3 B rn(u!)]
: 3 [] m(w')7
q
:; KT
lmut:lbT
...(i0
I moa : lkT
...(ddi)
*^rd: *lil
...(iu)
Thgreforc,
the
encrgy associated with each
.degree of freedomaverage-linctic
J yf
.BJ
"
..-(')
all equivalent, mean square velocities arong
:
a:$ar
ti
ur-oE-rd
or
Consider one gram molecule of a gas.
-a.ffi;*s*m
**[,].T:l*
ir;l";;r
td*
I. *
of freedom of rotationl It has in au nve e;;;;;;';'i;;;d:B
-Z;
"na;;fl;H
$ ;Y^*:*:x r,:ffi:;":H??jffi:];he mean,,^,oi d(n
nas ?ffi-"i;e
mol ec uI e" h as thr ee"re
d-egr ees
*",#"?*Xli#f
bT
:{ g*4r
ffiffi#Hfl,
.m#;d =;iffi
But
| W
Energy associated with three degrees offrcedom
"r:+:68'25K
?z:
o'
I
-t"i.ryf
This represents the theorem of equipartition of energy
?t: ?
Tr:273 K
,\
rcg
Tn,rs(he energy associated with- Ew
each dqgree
u?trEE .,r
of rrcec
frcedom
(whether trinsratory 3i
i, iA:)
ncr
"r: T
Here
Nature of Eeat
(#
- the increase in internar energy per unit dqpec
,"-R".",r""
But .
)
Cp-Cv:B
Cp : Cv*n
: t**n: f,n
For a mono-atomic gas
Co
a9v
q
\
TR : 167 )
: -a--
]a
-/
rise of
,,
\^e
170
Eeat anil Thermoilynamia
The value of T is found to be uuc experimentally for monoatomic gases like argon and helium.
\fi 6 Oiot*r;" g*. A diatomic gas molecule has two atoms.
SuchY molecule has three degrees of freedom of translation anC two
degrees of freedom of rotation.
Energy associated with each degree of freedom
: 2l*r
Energy associated with 5 degrees of freedom :
(b) A triatomic gas having 7 degrees of freedom has an energy
asociated with I gram molecule : N x -2 kT :
But
C?-Cv :
vr_
: i.o** - tp n
R
Cv
5.2t Marwell's Law of Dirtribution of Velocity
u-C'
r- Cv
1
+BA 1.40
: +-:
\
rp
)
-/
\o\
v$
The value of 1 : l'40 has been found to be true experi'
mentally for diatomic gases like hydrogen, oxygen, nitrogen etc.
1ii$ Triatomia gae. (a) A triatornic gas h-aving 6 degrees of
freedom has an cnergy associated with I gram molecule
- .Irx $*r : *r
At a particular temperature, a gas molecule has a fixed mean
kinetic eneiry. It does not mean that the molecule is moving with
,n. r.*" spJid th.oughout its movement. After each encounter thc
."."d of ihc molecule changes and due to a large number ol
i=oitirio*, the speed is different at different instants. But the root
*""r, rouur" vilocity (r.m.s.) C remains the same at a fixed temperaare no-t.moving wit! the same
i"r.. ei iny instant, all the molecules
vclocity
highcr than C and the
a
with
moving
.r"io"iw. So*".t"
;tL*'*ittr " velocity lowei than C- But the mean kinetic energy of
all thc molecules remains constant at a given temperaturc'
I)erivation of Maxwell'e law of Distribution of Molecular
velocitiee
The mean slluare velocity of molecules is defined by the
equation
c":-!fi:sn
3E*8 - 4.B
'R+B:
cP
;
Thus the value of 1, Ce and Cv can be calculated depending
upon the degrees offreedom of a gas molecule'
R
Cy: CvlR
Cp-Cy: R
Cp - Cy+R
:
7(l
: ffi:1.28
q
a :3RT
R
Cp: CylR
RT
wv:
^dU _EF
2'-
B
7
:
Cp-Cy :
;
u:-;-Rr
1- n
a.,: !!
dT2:
a:;Rr
:;R
, But
Cv
: #:r.33
.(
But
cP
vr-
$ Vt
Consider one gram molecule of gas.
Energy associated with I gram molecule of a diatomic gas
: il*i (5kT :;
lft
Nalurc of Ecal
c,: |J] * ar
Here dIV is the number of molecules having velocities between
c and oadc. If the total number of rnolecules is iv, then a fracticn
$wiff
have the comPonents of velocities in r direction in the range
r76
'lr><-i't+<(n-\tr1
"')(,\de())l)(
+r)(crrt J rtc . 1.o'(1.
r':frv)-\c,r-', i', eru(.i'c(
When cesium atoms strike the wire they get ionized and reevaporate. They.are collected by a negatively-charged detecting
cyllnder. su-rroundrng- the tungsten wire. The magnitude oI the
Naturc
*
EeaC and Thermodynomico
end of the plate. After-a sbort time, sufficicnt quantity
of
deporited on the plate
Gid;;p.IirJ-pr,",.'-.t"i,'tr,". silvcr ir
p.
r"r"iir.
ft,h
current indicates the intensity of the atorns at variouipositions. Thc
detector can be ooved to differeot positions of the wiri. The atoms
reaching.at Dl have higher velocitf than rhose reaching at Dr. Thc
verticai height of the derecLor iepresenu the magiitude- of the
velocity and the ionization current indicates the- nu-mber of atomr
striking. the wire at a particular point. A graph is drawn betweea
the ronlzation current along the y-axis and the vertical height (spceC
of the atolps) of the detector along the c-axis. The velocity iistri.
bution is found to be in ag.eemerit with the Maxwellian disiribution
I
I
t-
lalof velocity.
_!t_
-!ts_
*i*
S25 Mean Free Peth
I
A cnr.r
n.',cLi.:,-.f-o
5v
a-_u
Iq de-riving the expression for the pressure of a gas on thc basir
^. .
ot'kinetic
theory, it was assumed that the molecules aie of negligible
U
Fig.6.0
intensity
silver- on the- prate p is studied
and this represents the
--
-of
velocity di_strilutioa
of tht molecules. Th; graph repie...rii's tlre
number of nolecules and verocity agre* *itn'Ml**Lil iiriau"rri""
of velocity.
In 1947, Estermann, Simpson and Stern desisned a more
precise apparatus to study the veiocity aistribution.
Cesium atoms from the oven emerge from the opening z{
(Fig. 5'10).. B is-a slit and D is a hot tuigstcn wire. - rie *-riore
_appbratus is enclosed in an evacuated chamb?.i;r;;;
fb:i** of
HJ). The opcning 4-and the slit s.* t*i^"ifu.--i" ti..
uilr."""
ol a gravitational 6eld, cesium atoms will strike the wire at D.
But due to the graviational field, the path ir;-p;;rili".'.il.
"io*
sizc. They were asumed to- be geometrical points. A geornetri.rl
point has no cimensions and hence inter-molecular colli.eiins will not
be possible. But, a molecule has a frnite size (though small) and
mover in the space of the -vlisel containing it. tt colliies *itfi iG*
molecules and the walls of the containing vcssel. (Thc path iourr"a
by a moleculo.betwten-any tw.? ..consecutive colliiionsis
,r."ight
"
line, and is called the frcc p.tL, The direction of the moleculS
L
changed afrer every collision. After a number of collisions, the totrl
pa-tF app_?B to bc zig-za1 and the free
.path is not constant 1Fig.
5.lly. -Therefore, a.tero(ucrn.frcc prth is ured to indicateihc
3
\ \\
i:-\\ \
\\
Fig. 6.10
qa-ttr p_ao not rcach the wire. Thr atoms
soins
f,.:g:lTg-11"
along
thc pathr I and 2 reach at_ D, and D3.respectivity.
city of the atoms in path I is highei tr,"nifii pitfr i. - , ft "i,ito-.
mean distance traveiled bi a llolecule between two coltisions. It'ihc
-fi."
total distance travelled aftcr JV collisions is 8, then ttr" *."n
pathtrirgivenby
B )
r_
^-T
f
"'t;l
Lct thc molccules he assumed-ry'be spheres of diaarcter d. A
collision betweeo two molecules will'take place if the djstanr:e between the centres of the- two molecules is d.-, coilirion wiii also ociur
if the colliding molecule has a diameter 2dland the other moleculc
is simpl'7 a geornetricat point. 'I'hrir, assuritine all ot!:er moleculcs
to be gromerrlc.rl p,'ints and.the c()lli1ri1q n:oiec,;lr c:'C .rr::crrr ! j
thls molccr.r je wili '."'ir.'cr a volume rd?t tn c'r:.* rr,:?: I j
r"rrr.]:ipx .,,!r tr., ih,c vr.l 'rirl,i :.l'a cylinder of diamrrr:: it i *:
yE
Eul otil TlurmodYnnmiat
llolurc o! Ecal
179
Detcrmiaation of Ereea free 1nth. As dlacussed in article
5.27, the relation berween coefficient of tiscosity and the mean free
pattr of a molecule is given bY
(l.t ,t bc the numbcr of molecules Per cc.
ihco, the number of molecules Present is a volume ndlu
.l
r: i
: ril,tvXn
This value also represents the number of collisions made by the molc.
cule in one second.
The distance moved iir one second :
cbllisionr in one second : rd\t v.n.
For unit volume,
u and the number of
mt:
,-\ .'.1 u. rfree path ^ : #r*
r...(i)
This equation was deduced by Clausius'
,*#
...(i0
Thc mean &cc path is inversely proportional to the sqrrare of
thc diameter of the molecules.
Let m bc the mass of each molecule'
Then'
'llllll : P
l--
m
(d).
5'26 TranePort Phenomear
According to Maxwell's law of distribution of velocity
ilN
But
Also
^:h have the
that all
molecules
He essumed
Maxwell derived the exPression,
samc average rpecd.
)
..{o)
He calculated the value of I on the basis
of thc law of diltri-
^-T.r*
POSSeSses
TAsLE
I
n",
l-
Hrurogon
...(i)
...(;i)
ma$ mouon.
u-uo, u-us and w-ll)s
corresponding to thermal motion without mass motion, similar quantitiis with lnass motion arc
and
A - u-uo
| - a-as
W - $-wo
From equation (ddi;, Maxwell's law of distribution of velocitl
can be wiitten as
2.18 x l0-8 om
l.t!x lO-. om
0.S{lxlH oro
2.85xlflom
3.ag X10-{ sDo
0.009 x l0-! ors
?'l1x lf cna
&6Ox l0-t om
,26,
uo and rao be the components of the mass vclocity'
Let uo,
-the
actual vilocity of a molicule consists of two- Parts :
Therefore,
(i) the mass velocity comPonentS oe, us and rao
ldi) the random thermal velocity conPonente
bution of velocities
Meaa Srcc Petb (tr)
+*NAa
"-4ct
4* crilc : du ihs dw
du ilv ilw
dlv : .u/s
"-bo'
.y'+-ts2+.toz
*
6t
dN : NAs e-J (rrr+rs+otl du & dw ...(ddd)
Equation (irid) has to be modified in case the gas as a wholc
the
-- gas.
...(iu)
..'0L//
pC
the mein free pith ol a molecule can be calculated lrom equation
The mean frce path is inversely proportional to the denrity of
'Th" expression for the mean path according to Boltzmann ir
3't
The root mean square velocity C of a molecule can becalculat'
.ed knowing pre$ure, d=ensity and iemperature. The coefficient of
,ir".ti,, .itire gas is determined expeiimentally' Hcncc the value of
...(did)
7.o"P
P
r: Iecl
.
:#)
mnC)t
d,N : IiAsc-6(u8+vt+'')d(l dY dw...qixy
good only if ao, zo, tto, T and 'Y are consholds
Equation 1io)
taDt throughout the gas.
If the gas is not in an equilibrium state, there are three possi'
bitiiies occurring singly or jointly.
/,\J
Eeal anil Thumodyramict
Nature of Eeot
: /VxgkT:gR?
U :3RI
d:
J
,du -m
aH:
: 3.8 : 5'96 cals/g.mole K
Thus, the atomic heat of solids (.4s) is 38 and this value agrcct
with the Dulong and Petit's Lav.,.
E=arople 5'21. In an experiment, lhc aiacceilg o! the gc,a wa,
Jou,,d to l,e 1'04x70-t d1'nssi(:r-rt per unit oelocity gradieat. The
.B.ff.S. rclocity of the moleculea ie 4'5 xJ0trcmis. The denaity ol
d : 3;10-8 ca
Eraaple 5.22. Im an experiment the oiacooitg of tie got woa
tound to be 2.25 x 10 -t CGS uzrts. The BMB aelocity of the moleculea
i.s 4'5x l0r ernlt. The density of the guis .l gram per litre, Colculcte the mean Jrce path oJ the moleculee .
Here
I :2'25 x l0-' CGS units
C : 4.5 x lS cm/r
p:
lhe gaa ie 1'25 grams ,?e/ litre.
. 3zt
nec
3_x!
4>< lr'
", _- l0-rx4'5x
l0'
l-15 1[0-ccm
Calcalate 1i1 the nean lree path oJ thc molecu,lcs o! the gac, (ifi
t,equeneg of collisiott anil (iii) molecular diameter of the gd* malccules.
Ifere
(r)
! : l'56x l0-'units
C : 4'5 x 10. cm/s
p : 1.25x l0.r g/cc
. grl
n@
. : 3xl'66x10-.
< *4"-etc
,/,
'>
5'2t. \alaulale the meon lree ytath of a gaa moleouk,'
giuen lllo;t ibe molcaubr diameter ie 2 x l0-s cm ond the tumbcr of
moleautre pet cc ie 3x10t.
A:-mI
.:@
rU5;TOTIZE TO.
^
i : 9110-c c-n
(ii) Frequen",
"t.",I'iAi."IlS.;f
=
X:9
Ir : gllg'
Ix
,t
lO-c
l{ - 5xlff
x loB
Noto. Thc moco froc peth ir lor than iho wlvclcngth of lighf in tbc
viaiblc rpootrum
Eramptc 5'24. Calculote the meon lree path of gaa molecvlee
in o c,tnnrbei of 10'o mm o.f rbercury 1)reEsure, uauming thc molcculor
ilianckr to be 2L. Ane ryam molecule. of the gos occupica 22'4litres ol
N.T;P. Toke thc lemperolure of thc chamber to bc 27 3 K. (Agro 19? 5l
At 760 mm Hg pre$ure and 273 K temperalure, the number
of molecules in 22'4 litres of a gas
- 6.023 x lOs
Therefore, the number of molecules per cmt in the chamber at
r0-! mm pressure and 273
"J:rTiXi;:ll,o_,
Nurnber of moieculeg p;;"r;r;"
22400 x 760
22+00
According to Maxwell's relatioa
-l
^*Jm
,l--l
-
I
l:3yl(}rcrn
coilisions per second
Ti6-freem
liid; Avogadro's numlru.o23
I g/titre
r'|4iffix'fiI
Mean frce Path,
r - 3'538 x l0ro molecules/cml
d - 2L: 2x l0-8 cm
-l
^-r,dro
I
3'l,t x (2 x l0-r1t* 3'538 x I01s ==
2'25 X l0 cm
Eeat ord Thanoilynamia
194
Below the Boyle temperature, the gases are highly compressible
and this suggests the eristence of inter-rnolecular attrattions. Beyond
c-i>fttoz t'ltia
Natlrre o!
Ecat'-'\'i '1 ' '-'
(
(
(J- ,'.,u ,
o,uh
-t-
196
llere c is a constant and I/ is the vol '*c of the gar
Hence correct pressure
: (P+il: (r*# )
I
BoYLE
rl--
/ TEMPERATURS
wherc P I ihe observed pressure.
Q)f Corr""tion for Volume. The fact that the molcc.rles
have 6hite size shows that the acrual space for the movenrent of thc
molecules is less than the volume of tb6 vessel. The morecules have
the_1p!9le_-ol-rl_9q9l"q aro-und the'.n- and duC-o t6,is-Ectoi, rtx
the
i6rrection
sorrcL'rton -foi-vilirEre
ror volume ls
6 lr'nere
o
rr&ere 6li-a
O rS approamateiy
imateiy
four
lOUr
times
ttmes
the
-tS
actual volume or the molecuies.
Thercfoie the corrected volumc of
the gas :
(f-b). .11 a-
-Dtk, the radius6tdne'motecule be r.
Thc volume of the rnolecule
L
T "''
The centre of auy two moleculo
P-------+
The volume of the sphere of influence of each molecule,
Fig. 6.18.
A
B-in(2r)!:Br
the Bgyle temperature, Boyle's law is obeyed and intermolecular
(". a,.t7&,tions are less significant.
5t36 Van der \flaals Equation of State
Al\ 5/36
y
(lVntt. deriving the prefect gas equation PV : RT on the basis
^- oi kjnetic theory, it was assumed that (r) the size olthe molecule of
(S'\
(ii) the forces ol inter-molecular attraction
qas
nesliEible and (ii1
/AY-/
cas is neeligible
/)\'-/
t the
are absent. But in actual practice, at high pre$ture, the size of the
molecules of the gas becomes significant and cannot be neglected in
comparison with the volume of the gas. Also, at high pressure, the
rnolecules come closer and the forces of intermolecular attracti'lc are
appreciable. Therefore, correct;";* hculd be applied to the ga!
e(luatlon;l
1i) Correction for Presgure. A molecule in the intcrior of a
qas expericnces forces ofattraction in all directions and the resultant
Iohesive force is zero, A molecule near the walls of the container
ue to
experi enc es a resu I ran t fo@helaattF=D
.thi!'...'9nttr.@gasisIessthan-theactual
.
p..r*r.dThe :orrecliotr foi prestute p depJnds upon (d) the number
bf molectrTes striking unit aiea of the walls of the container Per
second and (id) the n"umber of molecules Present in a giveri volume.
Both these far:tors depend on the density of tha gas.
. .'. Correction fcrr pressure p * pt -+
p:-7,o
cach othcr only
".o.pp$,.ch
by a 'rinim"m distance of 2r ri.c., the
dia-meter of cach moleculi.
coarider a container of volume 7. If thc molecules are a[ouied
to enter one by one,
The volume available for first molecule
:Y
Volume available for second noolecule
: Z-B
Volume available for third molecule
: F_25
Volume a,railable
.rjt}:#:il
i
Average spacc available for each molecule
:
n
: v- $l+z*3......(n-rI
: ti,'-g
-i'
OT - NB
(n-'l)n
---T-
2fn
B
196
Eeal anil Thermoilyrnmict
Nalurc of Eeat
Ar thc number of molcculca ir vcry tu.g"
$
ca. be neglected-
Dl
decrease in oressure. It is not possible in actual practicc. The rtater
18 and I'D, though ,r*t^ti.,'".r, be ,"at;'.d i_n practice by carcful
.'. Average rpace available for cach molecule
: f-g
(ButS-Bc)
'2r--
expcrimentalion.
tempcraturg, the theoretiial and
- At higher
cxperimenLal isothermals
ire similai.
__ Y_n(ul
until now qany as 56 differcnt equations of state bave beeo
"s single equation satisfie;aII
suggested. But no
the observed fact!.
'2
: Y-4(ns)
: y-b
b : 4{tt*l - tH
Dicterice (1901) has suggested an equation
Jl?iL,*.
t- 'l- ,O* the Vau dcr Waals equation of state for a gas is
)- t-'
(r**)(r-6) : Rr
...(i) ) ,
- Rr "-#
(r+t:*)v-b) : Rr
5.37 GriticelConstante
(rn.
temperature and the corresponding ,ulo". . Jf
-f6\"
";ti.ul
''/Jprerrure.and
volume at the critical point are catied tajcritical.cons.
rants. At the crrtrcal point, the rate of chnngc of pressurc with
From the Van der Waals equation of statq
(;;+F;;:;)?;"^ $rt
-/-
-b)
Berthelot has suggeEted an equation
.wherc c and b are Van der Waals constants.
, : ,+u_;,
P(y
actual volume ol
I
...(rt)
volume (#)is
zero. This point is calledthc point of inflexion.
r-/Accorriing to Van der Waals equation
( ,*#) rr-,r : Rr
,:l#)-+
U
g
:l
{l
Ul
Ir.r t
E
:r.q
...(0
...(j0
Differentiating P with respect to 7
iIP
_RT
7V : (7:[]r * -v-
ol
At the critical ooint *
dy
2a
...(di0
: n
v-To
V0LUME
Y -Y"
------+
Fig. 5.19
Graphs betwcen Pressure and volume at various temperaturcs are
<irawn using equation (d). The graphs are as shown in Fig. 5'19.
In the graph, rhe horizontal portion is absent. But in its place, the
curve ABCDll is obtained. This docs not agree u'ith the experimental
isothermals {or C0, as obtained by Andreu's. Houever, the portion
1J3 has been explairrcd a.s due to supercooling of the vapouts anci the
prrrtion .ED due to super-heating of thc liquid. IJut the portion BCD
c rnnot be explained because it shows decrease in volume with
.
or
2o
-RT, - Vr' -- v
(7r-D)8T
I
-
2o
tt
W : O;D;-.),
R?o
...(iu)
Differentiating equarion lrrir;
dzP
@ : ip=rf _ 'vi
2Rr
tu
Eeat onil Thermollynamice
,os
Noture of Hedl
At thc critical noint -jS- : o,
l
TASLE
Critical Temperature and Pressure of Comraon Gasee
ry_rv
V:Y"
Stfrata,rnoe
zRro
6a
(7"-D)t
7] :
6o
IIelium
2RT"
...(u)
[[ydrogon
Nitrogen
Dividing (tu) by (u)
Ait
Y, V "-b
32
zVo - 3l'r-3b
Oxygen
.. .(r'i)
Substituting the value of
2a
fis:
: 3b in equation (ic)
@z
--24t0"c
t2.80
-146.C
_140.c
33.60
...(uir)
ExBa
D
'o:
TEW - 56r- \
^
P": h'.
/V
f
48.00
73.00
Ammonie
130'c
15.00
Chlorino
I4b-U
76.00
Carbon dioxide
218.00
r,r* \(\^, (,"/ t',:3b
t
^-J,
P":;o
'qg
t$
T,,
P,
Y,
W: N;
+: #
lgl
lal
,.,1,1i1)
61 az
(27 )' il':bz
n
27t'2
@
6*a
:TIR'
o
IDCatrg
Yr
...(rd)
Fr'.,m s*r.,ions (rioi; and (ii)
')"'o""
critical volume and critical temperature have the same vilue. Ii
Pz
...(i)
: D-Ei'"msa
a
'lt
Po,
60'00
-118.C
3I'C
Two gases are said-to be in corresponding rtates if the rarios ol
their actual pressurc, vqlprg and temperatura and cridcal pressure,
Pr
::Ed
39.00
5.94 C<refficient of Van der Waals Constants
Substituting these values of 7" and ?" in equation (ii)
5.38 Corresponding stater
2.26
-2Bgoc
Wator
RT"
mBa
t":i_Rb
cnd
(atn?p{phcrro)
Argon
Yo:3b
Vo
Critical Preacr.rr.c
t-t
w: v;6)e
id..
Criticol
lfcnpnraturc t0
.:
o1
1)24
2
t
1t
Ll
"6+
-F:
C
...(iu)
Dividing (t'di) by (dr')
f, :
P;
8o
27ba
TEi X-a-
: lEB6
--- Ri"
gP"
r,
...(u)
Eeal and Thcrmodynomica
?10
Ako
Nahrc ol Ecat
RT,
-m
- Tfre@
R?, -t8
-P};:
The quantity
"^-- -8lE
-,
O:-
...(ui)
)
*T * calted the critical coe$.cient of a gas.
i-i-rt
Its calculated value :
RT,
(;i)
B (tu) .27U
33 and it is
i the same for all gases.
L.,) \
l.,+-:r.,,aC!
O(.
or
5.+0 Reduced Equation of State
the pressure, volume and tcmpcrature of a gas in tcrmr
-Taking
.of- reduced
prcssure, volume and temperature,
PVT A, *O- :
: T
-F:-:
F, 7,t c
f 2
14
gases is given below.
P:a.P",T-PTo,T:\To
TABI,E
F.rperimeatal Values of Gttical GoefEcient -$#
T"inK
in atm.
Spccit oolulma
dn cmlg
7Vo
Ilelium
6l
2.26
I6.4
3'lt
Eydrogen
33.1
r2.8
2.)-o
8.28
Nitrogoa
t26.9
33'6
3.21
8.12
Orygen
164.2
19.7
2.52
g.a
Carbon dioxido
so4
7Ztg
2.17
3.{r
Water
u7
2r8
3.181
4.30
Srbel;l;ncc
Notc, Eoro V
"-
Pa
RTo
moloouler w0 x apecif,c volumo.
The experimental value of the critical coefficient of all gases is
greater than thc theoretical value of2'67.
- -E,-gple 5'2{. Calculate the Yan iler Waols constanta lor dry
?6 : 732 K, P" - 37'2 atmosplwreo,
B per mole : 82'07 cml atmos K-1.
(0
Pr:37'2 atmospheres
llo :
R:
132 K
82'07 cm! atmos K-r
":#ry
o:- /11\(t3'ozl'032)'
\3rl-r7Ea:
lt.3l x l(F etuog clqr
According to Van der Waals eguation
(r*#) rr-rr : Rr
...(d)
( -r"*&)rF r,"-u) - Ry r"
But
P, :;i,
Y":3b
and
T": h
. ly.:,+)re.3a_6t:y#
b'rPz
127
t
[.* p', ] [3B-r] : sy
...(i;)
This is the reduced equation of state ior a gas. If two gases
have the same val rres o[ a, B and 1 they are said to ie in corresfrnding states.
oir, gioen that
Here
8 x37'2
D:36'41 crnr
The experimental values of the critical coefficient for different
0,
82'07 x I32
@elhi 1975)
5 4l Properties of Matter Near Critical poiat
Based on the experiments of Andreu,s, Amagat and others, the
state of matrer near the critical pcint can bi srrmmarised zrs
follows :--
(l) The-dersities of the vapour and, the liquid gradualy
approach each other and their ciensities
"r" .qrril at t$'c critical
point.
(21 At the. critical point or just aear the critical point, the
line of demarcation between the liquid and the vapcur d-lsappears.
Consequentiy, there must exist riutual <iiEusion'bet*r.cn thl r*o
phases and the surlhce tension mrrgt tre zero. This alro meaus rhat
the forces of inter-molecular atrraction fur the liquid and vapour
stater must be equal at the critical point-
.
Eeat anil Thermoilgnamict
2tl
its centre of mass is equal to the total- random kinetic energy of all
: 2l- ^^
its nrolecules ? (N'lolicular weight of hydrogen
-[Ao".
l'93 x 106 cm/s]
61. Calculate the temPerature at which the r.m.s' velocitv of
a helium molecule will be equal to the speed of the earth's 6rst
satellite i.e., 7, : B km/s.
[Ans. 10'28 x lOp K]
62. Calculate the mean kinetic energy of a molecuie of a gas
at 1,000'C. Given,
E : B'31 x l0? ergs/gram mol-K
N : 6.02x lOa
(Delhi 1969) lltta. 2'07x l0-rr ergpl
63. If the density of nitrogen is l'25 g/litre at N'T'P', calculate the R.M.S. velocity of its molecules.
cm/sl
lDelhi 1972 ;'Delhi (Eons.\ 19731 lLne. 4'95xlS
mole'
of
oxygen
speed
is
the
R.M.S.
6{. At what temperature
at27"C?
cules twice their R.M.S. speed
'
(Delhi 1973) [Ane. 927'C]
of the molecules of hydro'
velocity
R.M.S.
the
65. Calculate
gen at 0oC. It4olecular weight of hydrogen : 2'016 and
-
R:
8'31 x I0? ergs/gram mole
oC
(Dith; 197 1\ lLtr.
I 8'4 x I S cm/rl
66. Calculate the R.M.S. velocity of the hydrogen molecules
at room t€mperature, given that one^ Iitre of the gas at room temPc'
rature and normal pre-ssure weiglrs-0'086 g-'
(Oetni /9761 1Aoe. l'BB x lS cm/s)1
67.
Write short notes on :
Mean free path
(ii) joule-Thomson Effect
(i)
lAgro 1962 ; Delhi (826.) 1966)
(Agro 1962 ; Delhi 1974, 7 5\,
Co;rtinuity of state
(iu) Rowland's experimcnt.for finding J
(r) Van der W"als equation of state
(oi) Pressure exerted bY an ideal gas
(oii1 Critical constants
luiii\ Degrees of freedom
(ir) Atomicity of gases
(r) Maxwell's iaw of distribution of velocity. (Delhi 1975'S
(ril Andrelr's' experiments
(dii)
(rii1 Amagat's expcriments
(riii\
Halborn's experiments
{ria) Behaviour ofgases at high Prcssure
(lcu) Critical Point
\rl,il ftrresponding states
(roiil Intermolecular attraction
(x',tiii) Tcopcraturc of invcrsion
lair\
@r\
Reduced equation of state for a gas lDelii lfiorta.) 19761
Porous plug cxPerimcnt"
Thermodynamics
6'l Therraodynam,ic Systern
A thermodynamic sysi em is one which can be described in
terms of the thermodynam,c co-ordinates. The co.ordinates of a
thermodynamic system'can be specified by any pair of quantities ufz.,
pressure (P), volume (Z), temperature (f) and entropy (B). The
thermodynamic systems in engineering are gas, vapour, steam, mixture of gasoline vapour and air, ammonia vapo'-:rs and its liquid. In
Physics, thermodynamics includes besides the abovc, systems likc
stretched rvires, thermocouples, magnetic materials, e]ectrical condenser, electrical cells, solids and surface films.
Examples : 1. Stretched wire. In a stretched rvire, to find
the Young's modulus of a wire by stretching, the complete thermodynamic co-ordinates are
(o) thc stretching force /
(D) the length of the stretching wire and
(o) lhe temperature of the wire.
The pressure and volume are considered to be constant.
2. Surface Fihas. For liquid films, in the study of surface
tension, the thermodynamic co-ordinates are
(@) the surface tension
(b) the area of the film and
(c) the temperature.
3. Revcreible Cells. The thermodynamic coordinates to
completely describe a reversible cell are
(a) thc E.M.F. of the cell
(b) the charge that flows and
(c) thc temPeraturc.
215
rntrrne-tc4 ecl
216
lleat and Thermoirlnamica
62 Therrnal equitibriurn and Conceot of Temperature
.\ thernrodl'narnic s)'stem is said to lle in rherrhal equilibrirtn
if arry trvo c,l its independent tlrermrclvnamic co.ordinatei ,t and I:
remain c()nstant as long as the e\lernal conditions remain unaltered.
Consider a 3as enclosed in a cylinder fittcd rtirh a piston. If the
pressure and volume of the enclosed mass of gas are P and 11 at the
tempcrature o[ the surroundings, rhese values o[ P and I/ u,ill
remain constant as long as the external conditions t'iz. temperature
and pressure remain unaltered. The gas is said to be in thermal
equilibrium with tlie surroundings.
The zeroth law of thermodynamics u,as formulated alter the
first and the second laws of thermodynamics have been enunciated.
This lau, helps to define the term temperuture of a system.
This law states that if , of three syslems, A, B ar;d, C, A and B
ore separately in thernol, equilibrium utith C, then A anil B are olso
in thermal equilibrium with one a.nother:
Conversely the law can be stated as follows :
,hrt.y' or more systems are in thermal contact, each to each,
by roeans of diathermal walls and are all in thermal "equilibrium
together, then any two systems taken separately are in thermal
equilibrium with one another.
Consider three fluids .4, B and C. Let Pa, Pa represent the
pressure and volume of -4, Ps, /s, the l)ressure and volume of B,
ind Pc, Ysare the pressure and volume of C.
If .4 and B are in thermal equilibrium, then
4f
Sr(P*Vn) : 6r(Ps,Ysl
or
Ir[Pl, Y7., Ps,7a] : 0
...(d)
Expression (d) can be solved, and
Ps:/r[Pa, V*,Ys)
...(ir)
If B and C are in thermal equilibrium
or
flz(Ps, Ps) : #slPc,Ycl
frtPr, Ya, Pc,Zcl : 0
...
From equations (zi; and (diz) for .r{ and C to be
equilibrium separately,
.fr(Pe, Ye,Vs) : lzlra, Pc,Yc)
(iir)
in thermal
...(iu)
If d and C are in thermal equilibrium with I separately. then
according to the zeroth law, A and C are also in thermil equiiibrium
with one another.
.'. .F'r[Pl, Ve,, Pc,7c] : 0.
Equation (it') conta;ns a variable 7s,
does not contain the variable 7s. It means
$r(P* V s) : ps(Pc, I'c)
Thirmoilynam;cs
217
In general,
d,(Ie, I'a) - cr(,I'o, I'n) : dr(Pc, I'c)
....r,ii)
These three functions have the same numerical valrre thougi,
the parameters (P, [/1 of each are diffcrent. This nurnerical valire
is termed as tentperatur': (T) af the body.
...(riii;
This is called the equation o[ stare o[ the nuid.
-,Q'
Therefore, the temperature of a system can be defined as
the property that de termines ,rvhether or not the body is in thermal
equilibrium with the neighborrring sy!rems. If a nuntl,rer. oIs,vstems
are in therm"] tquiiibrium, this cor,:rnon property of the system can
be represented by a single numerical value called the temperature,
It means that if two systems are rrot in thermal equilibrium, t{rey
are at different temperatur r.
_ Example. fn a nr"rcury in glass thermometer, the pressure
above the mercury column is zero and volume of mercury measures
the temperature. If a thermometer shorvs a constant reading.in two
systems. A and_B separately,. it will show the same reading even
when .r{ and B are brought in contact.
6'3 Concept of Heat
Heat is defined as energy in transit. As it is not possible to
speak of work in a body, it is also not possible to speak oi heat in a
body. Work is either done on a body or by a body. Similarly, heat
can flow from a body or to a body. If a body is at a constant t"mperature, it has both mechanical and therrnal energies dqe to the
molecular agitations and it is not possible to separaie them. So, in
this case, we cannot talk of heat energy. It mtans, if the flovr of
heat stops, the word heat cannot be used.' It is only used when
there is transfer of energy between two or more systems.
Consider two systems.r{ and B in thermal contact $,ith one
another and surrounded by adialatic walls.
For the sJstem /,
Ps: f2l?s, Pc, I'c]
AIso
-_ it.z$2ruzr,-
...(t )
.
whereas eguation (r)
...(ui)
fl : Ar-arlW
...(r)
energy
transferred,
U1
is the initial internal
where II is the heat
€nergy, U2 is the frnal internal energy and W is the work done.
Similarly for the sPtem B, '
E', - U2',-Ur',*W',
...{r4
(ii)
Adding (i) and
E + E' : (Ur-Ur)+ril + (U 2'
-At') +W'
fl +E' - llU t*a r')-(Ur*Ur'))+ (f +W')
... (idr)
Thc total change in the internal energy of the composite system
[(u.*ur,)_ (4*u i))
2t8
H eat anil ThcrmoilYnamics
The work Lrne by the cor^rposite system : W +W'
It means that the heat transferred by the composite s1'stem
: E +H'. But the composite system is surrounded by adiabatic
walls and the net heat ransferred is zero.
u*'; jo-u,
...(iu)
Thus, for two systems A arrd B in thermal contact with each
other, and the composite system sur-rounded b-y ad.iaba.tic walls, the
heat gained by one-system' is equal to the heat lost by the other
system,
6'4 Quasietatic Proceee
A system in thermodynamical equilibrium must satisfy the
following requirements strictly :(i) Mechanical Equilibriuro. For a system to be in mecha-
nical equilibrium, there should be no unbalanced forces acting on
p.it of the system or the system as a whole'
".ry (di)
Thermal Equilibrium. For a system to be in thermal
equiliLrium, rhere should be no-temperature difference between the
pirts of the system or between the system and the surroundings.
(iid; Cheaical Equilibrium. {or- a system to be in chemical
equiliLrium, there should be no chemical reaction r.r'ithin the systero
.rid ulro no movemerlt of any chemical constituent from one part of
the system to the other
When a system is in thermodynamic equiiibrium and the
surroundings are kept unchanged, there will be no motion and also
oo *c.k wi'il be dorie. On theother hand, if the sum of the exter'
nal forces is altered, resulting in a finite unbalanced force actir:3 on
the system, the condition for mechanical eguilibrium wiil not be
satisfied any longer. This results in thc following :-
(i) Due to unbalanced forces within the -system, turbulence,
waves'etc. may be set uP. Thesystem as a whole may possess an
accelerated motion(dt) Due to turbulence, acceleration etc' the temperature disuibution within the system may not be uniform. There may also
exist a finite temperature difference between the systelo and the
surrouncl'ngs.
(iii.y Due to the presence of unbalanced fprces and difference
in temperature, chemical reaction may take place or there may be
Thermodynamicc
21'1
A quasistatic process is definei as the Pr99c: in which -the
deviation from theimodynamic equilibrium is infinitesimal and aU
the states through which the system passes during a quasistatic
process can be considered as equilibrium states.
In actual practice' many proc$s.e-s :losely approach a quasista'
tic orocess and mav be trcated as such with no signihcant eror.
Consider the exparsion ofa gas in-a closed-cylinder fitted witha
piston. Initially'weights are on the piston and the pressure of the gas
inside the cylinder ii trigtrer than the atmospheric pressure. If the
wershts are'small and aie taken off slowly one by one, the process
considered quasistatic. If, however, all the -weights are re'
"^rii" at once, expinsion takes place-suddenly and it-will
be a non'
'moved
equilibrium p.oceis. The system will not be in equilibrium at any
tiiee during this Process.
A quasistatic process is an idcal concept that is applicable to
all thermodynamic iystems includin_g electric and magnetic systems.
It should bi noted that conditions for such a process can never bc
satisfied rigorouslY in Practice.
6'5 IIeat-A Path Function
Heat is a path function. When a syst€m chalSes from a state
I to state 2, thiquantity of heat transferred will depend upon thc
intermediate stagls through which the- systern passes i;e., its path.
Hence heat is an ineract differential and is written as 8II.
On integrating, we get
[*m:E'o la
lla
Here, 1EI, represents the heat transferred -during thc given
procss bctwien the states I and 2 along a pardcular Path A.
6'6 \f,orlr-A Path Function
Suppose that a systen is taken from an- initial equilibrium
state I to'a final equilibrium state 2 by- two different paths A and B
tfig. O't). The prbcesses are quasistatic.
t
I
P
movement of a chemical constituent.
From this discussion, it is clear that a finite unbalanced force
may cause the system to pass through non'eguilibrdurn states. If
during a thermodynamic process, it is desired-to describe every state
of a system by thermodynamic co_ordinates_referred to-the system ru
a whole, the process should not be brought about by a finite unbalanced force.
V _---+
a[. 0.1
Heat and Thermodynomica
250
ll'A:l
r 3'\
I pdt'
J r.{
rlB
r28
J
For the path B
;r-e
r2A
6lf:
l.{
I
- re 8rr/ : JIB
PdV
...(i)
..
It is customary to represent, \.vork done by the st'stem as {"r'.,
heat flowing into tl,e s,'stem ras
work done on the syste:-o as
the system as
and heat flowing out of -ve,
^1-ve,
First Law of Thermodynarnics
vJ
x' frr)rt
.l
lo represettt
: []l:, BtF : ty, _ wl
... i;ii)
JW'
It may be pointed out rhat it is meaningless to say "rvork in a
system or rvork of a svstem". Work cann^t be interpreted similar to
temperature or pressure of a system.
In terms of calculus 817 is an inexact differential. It means
that F is not a property of the systen, una j 817 cannot be express.
ed as the difference between two quantities that depend entirely
on the initial arrd the 6rral states.
_ Hence, heat and work are path functioas and they depend
only on the process They are not point functions such ai preisure
or temperature. Work done in taking ttre slstem from state I to
state 2 will be different for differenr paths.
6'7 Gomparisoo of Heat and l{ort
There are many similarities between heat and work. These
\
I. Heat and work are both transient phenomena. Systems do
4. Heat and work represent the energy crossing the boundary
of the system.
5. Heat and work are path functions and hence they are inexact
differentials. They are wriuin as 8,8 and 8I7.
6. (a) Eeut.is defined as the form of energy that is traruf€rred
across a boundary by virtue of difference of iimpcrature or temperature gradient.
(b) Work is said to be done by a system if the sole effect on
things external to the system could be the raising of a weight.
:
7m')
@
condition. Therefore for a cyclic process
and
.are :-
not possess heat or work.
2. When a system tindereoes a change, heat transfer or work
done may occur.
3. Heat and work are boundary phenomena. They are observed at the boundary of the sysiem.
-ve.
heat produced. It is true rvhen the whole ol the rvork done is used
in producing heat or t'ice oersa. Here,W : JE rvhere J is theJoule's
mechanical equivalent o[ heat. But in practice, rvfen a certain
quantity of heat is supplied to a svstem the whole of ttr-e heat energy
may not be converted into 'a'ork. Part of the heat may bc used in
doing exrernal rvork and the rest ()[ the heat :nighr be used in
increasing the internal enerqv of the molecule(Qflet the quantity
of heat supplied to a system be 8I/, the amorrntYF-external rvork
donp be 87 and the increase in internaI energy of the molecules be
rlu)/Ine te rm U represenrs the internal energy"of a gas due to molecylar agitation as well as due to the forces of inter-molecular attraction. lflathematically
w aa
...(d)
.(i,)
The values of II'e and Il'6 are not equal. Therefore work
cannct be expressed as a difference between the values of some
property of the system in the two states. ThereJore, it is rtot cotect
Ir,
22t
Thermodynarnics
l'lrc;.r'els rrndcr these curves are different and hence the
lrrrntitics of rvor k clorte .rle :rlso different.
For the p;ulr .\,
$ r': fa'
$aU
: O
...(14
lBc,th arc erpressed in l'reat urrits].
'I-lris erluation represents
Joule's law.
For a svsrem carried through a cyclic process, its initial and
linal internal' energies are cqual.' From the first law of ti,.rmodv"amics, for a system undergoing any number of complete cvcles
ur-ut : sl
" :
Y:r
D,H
16 rrr
H : llt
[Both are in heat units]
6.9 First Law of Thermodynarnice for a Change ia State of
a Closed System
,
Fo.r a clo.sed. system during
thermodynamrcs rs lvrrttell as
" aa : rli
Yr
a complete cycle, the first lau. of
st;,
Heat and Thermodyrumiu
229
In practice, however, we are alco conccrned with a PJocclB
Let thi system undergo a cycle, changing its
rathcr than a cYcle.
Tlxrmodgnamict
223
(8.e-8If) depends only on the initial and thc final states of the
svsrem and is independent of the path followed between the two
states,
d.E :
Let
(8H-Etr)
From the above logic, it can be seen that
2
I an : constant and is indepcndent of the path.
J
I
H;-
state from I to 2 along the path :{ and from ? to I aiong-the -path B'
This cyclic Process is iepresented in the P-Y diagcam (Fig' 6'2)'
According to the first law of thermodyaamicl
2A
Similarly, ,}[. cannot be written as (W1-W1), because it also
dcpends upon the path.
Here
lEIs represents the heat transferred,
18
I,rt'* J ta - {,t'* I,:'
...(0
lo
2a
lo
l,** [:': [,'** I;*
...(dr)
Subtracting (id) from (;)
IB
LO
18
ro
[,,
I*'"- [:'- L''- )ro
or
l8
LO
l,Ittr-tr):l(DE-Dr)
zo
1fl1 represents the work done,
E, represents the total encrgy of the system in
Now, consider the second cycle-in lvhich the-rystemchanges
from state'l to statc 2 along the paih /, and returns from state 2 to
rtate I along the Path C. For this cyclic process
2A.
tflt - (E,_E)+LW,
[Notc. 1EI1 carmot be writton ae (EI1-II1), bocsueo it dependa upou
ftr-{tw
r8
state 2
the pathl.
For the comPletc cYclic Process
2A
This naturally suggests that E is a point function and dE rr an
.cxact differential.
The point function .O is a iroperty of the system.
Here dD is the derivative of E and. it is an eract differential.
8H-8W : dE
...(du)
8H : d0qtW
...(r,)
Integtating equation (o), from the initial state I to the final
... (iii )
Ilere I and C represent arbitrary processe between the states
I and 2. ?herefore, it-can be concluded that thc quantity (88*8W)
ig the same for all processes between the stdtes I and 2. The quantity
I
state 2,
E, represints the total energy of the system in
state l.
At this point, it is worthwhile discussing what this E c^n
possibly mean. With reference to the system, the^ energies_crossing
i:he boundaries are all taken care of in the form of E and W. Foi
.dimensional stability of Eq. (u), this f mustbe energy and this must
belong to the system. Therefore,
E2 represents the energy of the system in state 2
E, represents the energy of the system in state I
This energy E acquires a value at arry given equilibrium condition by virtue of its thermodyna-mic state. The working substance,
for example a gas, has molecules moving in all randbm fashion.
The moleiules have energy associated by virtue of mutual attraction
and this part is similar to the potential energy ofa body in macro.
scopic terms. _ They also have ve locities and hence kinetic energy.
'fhis energy E therefore can be visualised as comprisine ctf moleculir
potential and kinetic energies in addition- to maLioscopic potential
ind. kinetic energies. The first part, which owes its rxrstence ro the
224
II eal and lhei.modynamics
Tkrmodytwrtiu
Erpr : AB-U^+W
E - 50*10 : 60 jouler
lhermodynel,,tc r'l:l,rrre is often cailcd tl,c internal enercy which is
comFletely deper,derrt on thc ihermodynamic stare. and' the other
trvo depend o,t mecllarric;rl or phvsicaI surte oI thc system
E:
nature etc.
(D) For the curvcd path frcm B to A,
U yKE + PE +Othcls whiclr depenrl upon chemical
W:
For a closed s),stem inon-chemical) rhe clranges in all othcrs
except U are insignificant and
8H
: dU +EW
-2}joules
: -50-20 : -I0.Jouler
(-ve sign shows that heat is liberated by thc syrteur)
Dr : 0,
Up - 40 joulca
0s-U1 : 56
dE:dU
From equation (r)
Ot! t
a&d
(c)
.
..("i)
Flere all thq quantities are in consisrenr unirs
Era:nple 6 l.
ll,hen o .ey.\tem is taken, lrom the state A to tLe
stalc B, alonT the Vath ACB,80"ioules oJ heat f.ows ilto thr systrm,
anil the system dois S0joules o/Lortc 1i,;g. A.S\.
(a) y7w much heat fl.owa into the syslem along llrc ltullt ADB,
iJ the work ilone is J0 jouies.
Or :
from.r{, to D is J-10 joules";d};;; J';;
For A-D,
ia returned, lrom the state B lo th,c stute A along
- (6) The -system
lhe curaeil poth.
The work d,one-ott. lhe syslem is 20 joules. Docs the
systenx absorb or liberale heat anil how mich ?
(c) l! Ua
--C, UD: 40 joules, finit the ltcat absurbed in the
?roeess AD anil DB.
50 jouler
In the Drocess /?A, l0 joules of wort is done. WorL
is zero.
.EIrp: (Up_Ag{W
- 40+I0 - 50 joules
For DB
trIpa
- Ac-Ao*W
: 50-40+0 : l0 joutee
Applications of Firat Law of Thermodynqnic!
q
Speei6c lleat of a Gae (T aod, v Independent)
Thc internal energy of a system is a single valued
function of
the state variables or:2., pressure, volume.
6.10
case of a gas, any two
t
*ifffr,,,*.,,,
i
*,o,li'Si*:# i.h-*T#.1'JHl *;
a:fiY,r)
P
...(d)
Differentiating equation (r)
oo: (#), rr*
Fig. 0.3
AJong the parh ACB,
.s.
A:+80joules
P : $30 joules
+80 : as-Ua*3O
U s-t t:
80-30 : 50 jouler
(a) Along the Path -r{DB,
W : *l0joules
(#),*
...(,')
ff anamountof heat gEI is supplied to. a thcrmodyaamical
lyrtem' say an idcar gas and if the'iorume lncrer*e!
by ily at a
pressure P, theu according
ffXo",
Ilacs : Uy*U1!W
Ilere
dane
Here
to the 6rst law;i-6il;;.-
8E: da+EW
8W : P.d,V
8E : dU+P.i|Y
Substituting the value of dU from equation (d0
,u - (#),ur*{#\,rr*r*
.. . (r:r'd)
H cot ond Thennodgnumict
?hermodynaniu
Herc Cp, Cy and I are expressed in the same units.
Dividing both sidcs bY dT
#: (3.r)..(#),#*#
(# ) : G+)" "[ '*(sur),){,
From cquation ldir;
'r: ( #)"rr*lr* (#) ,)uo
...idr)
For a process at constant temPeraturc
iI the gas is heated at constant volume,
, EH\ : t'
ilT:g
[77 .)u
(a^a)r
dv :o
^ -l
dT
(# ),: (-37;":,.
...(u)
For a reversible adiabatic process
8E :9,
From equation (iu)'
Therefore, from equation (fr),
o: cv ur+lr. (#) ,T,
c" : c"*f
cydr
..'("t)
,,(#) :-[r*(*#),]
g. Even thougt thc
So 8.8 : 0 : dU +0. Thercfore, dA :
there is no
coostant,
is
tcmperature
the
o,.'-rlrrml- cnu'gea while
energ'Y'
internal
in
cirange
": +(#),
\ aF-1, -
",: (#),
lirora ttre ideal gas eguation
PY ':' R?
/:lL
F'r,t il(t'^,,
'.
ce-.cv
...(utO
-
: r, l# ),*(p), (# ),
l,y *'-
T
,
,* "H
D
o * F--F
(*f
),:
'*
(#), * ("#')-,
"..(ri;i)
*(e,#'):-[on(iF;,1
...qrio|
-:t
1
=,.( ;i)u:o
Cp-Cv
-' (-# ),
aV
\
L'P* {)!
...(rid)
The isobaric volume coefficient of expansion
(ao_\:0"
cp-.c1
:- [r* (#),y,
Dividing throughout by dV,
From Joule's experiment, for an ideal gas on opcning thc stop'
*n.k, i-ro *o"tk u'as done and no heat transfer took placc'
i:r P (*F;,:o
...(t)
This equation rePresents the amount of heat energy -supplied
to a system ii an isothlrmal reversible process and is equal - to thc
s,'.m of the work done by the systeE aird the increase in its internal
(#),:",
,,: (#)"*[ r*(#),](#),
()r
- P(irltr+ (#),ruo,
energy.
\\Ihen the gas is heated at constant Pressure,
,*(#),1(#),
c,-cv: [ ,*( #),](*f ),
...(d")
.. . t.ilii i
or
Heal ond ThcntoilY*amio
228
:-(#)
"(#)
From equations (rid) and (rio)
or
/ a" \
Cv-Qt
,..(rt)
\aTJ:-W
good for an adiabatic reversible
Thir expression holds
Proccss.
6'11 lrotLcrmal Procege
to the surroundings and
ftf "rrt.- is pcrfectly conducting
the process, it is called
throughout
"
cotirtarrt
tl" t.g-p.rli"r. *-"i*
229
Tlwmodynomir"r
substance, there is rise in temPerature because the extcrnal worlt
done on the working substance increases its internal erre-rgy. \{}rr:n
work is done by thJworking substance, it is done at the cr:st of its
internal energy. As the system is perfectly insulated frorn the
surroundings, there is fall in temPerature.
[Ttrrr, during an adiabatic process, the working substance r:
rrerlettly insulated from the surroundings. AII along the process,
ihere is change in temperatureJ A curve between pressure and
volume during the adiabatic pro"cess is called an adiabatic curve or
an adiabatic.
Examples. l. The compression of the mi,xture of oil vapour
eld air during compression stroke of ao internal combustioa is an
adiabatic proCess and there is rige in temPeralure.
2. The expansion of the combustion product! during the
working stroLe of an engine is an adiabatic process and there is fail
in temperafitre.
3. The sudden bursting of a cycle tube is an adiabatic
Processr.
Apply the first law of thcrnodyuamics to an adiabatic Process,
$.EI
V--+
8.8: dU+Etr
0
frg. 0'a
certain
anisotherma!process)_9'yidf 1,3i"i*,c--':b::1ll'"::"; by the
u"a nt"i"g a vol-ume represented
;fi;""*ir;-l.o4t"t.
6'a).
(Fig.
a
inint
w.orking sub'
Pressure is decreased and work is done_ by the
farl
be
uE 14'
shor{d
Juuutu
ulerc
and
an<t
tt'"tgy
energy
i"t"t"tl
rnt€rnal
its
oIcost
the
at
.*r,".^"[f.-";;;i-i"
stance
*eie
in
F;+F{{ts.*::l::r*?:f*.*,,r:d::'f
'-':,**::x1:
f, ;:-;;;;'#J[3f i;;Ptheiurroundilq-3:d-::'::::::::::
Thus from A to B the temperature remalns cons'
ilit t.-p"totr...
--;i^iaothermai'
a Il io
c-ur've or uothermal,
--ieotbermal curve
-ollarl t}'e
a.a
it Lrr.a
the iaotLcrmal
^.,^,^
[",.*Ttl,:ai,-. t
point B and let the
Consi<ler the working substance -at-the
on
the workiog sub'
is
done
work
oror,rif-. i""r*"a. Ex"ternal
5;:;;Jtu.t. tuo"rd te rise in temperatlT",',
!i13",-1v.t::Ti
ji"1';r,'ffi
}? : "'
{:*:E
*iis::,
temPerature
itl :H:::"ff:J?*k,
and x
hrrroundings
of the
/fnrrt. durinc the isothermd process, tle temperature ol
*orfio g t.rLsta nc i rem ai ns ronltant. . It.-?lisothermal procers is
for an ?-1Y:jS:iSi:
ru. eguation
il;;;.d;-r;;J[8r.
:
conltant
PY : BT
[For oae gram molecule of a 88s']
For r gram moleculer of a gar PY : *nf )
6.12 Adiabrrtc Procc!'
- 0,
-du+8IF
...(d)
The procesd that takc place edlilenly or quickly are adiabatic
Proccsses.
6.13 Iroc.horicProcesr
fif ,U" working substance is takea in a non'expanding cham.ber,
the h-eat supplied riill increase thc presstrre and teryperature. The
volume of thc substance will rcmain coultant. Such a process is
crllsd at iaochoria groceee.l The work done ir zero because there ir
nochange in voluml. Th"c whole of the. heat .supplied ineeases
thc inteil,al energy. Therefore, during thc isochoric Process 8F :0.
8E:dA
...(0
The heat transferred in guch a Process
Efl : A#r
crilT - da
"..(id)
Hence C, is the specific heat ficr one grarn'molecule of a gas ar
constant volume.
6.f.1 Ieobtrlc Procer
fif O"workingsubstanceis tahen ia aq crpanding chamber
t.ptLJ. constant !r.rrrrr", the pocesr is called an is.rir-^" prt''c:s)
Hire, the temperaturc and voluml change. If an amount of heaf
Ef,I is-civen to-the working substance' it is partly used in increasilg
^'
the teilperature of the wolking substaace by ilI and 'r-'t
gra*
arrrt'ul[
one
Considering
work.
doin-g
external
in
substance,
Eeol ond Thermodgnamice
2E0
8EI
But
...(ii)
8E - OydT
P.dY : r'dT
Dividing by C,PY,
... (iii)
gmm of a gas
Here C, and O, rePrescnt the specific heats for I
constanL
gas
ordinarY
and r is the
If C, and Crtte the gram-molecular specifrc heatu of gas' then
+
...(io)
the universal gas corutant'
-ttS/ Gas Equation Duriog rn Adlebedc Process
v
I gr-am cf the work-ing substance (ideal gas) perfectly
Corrrid.r
o-'?
'-v
gas at thc cost of its inter'
d;;;i' i"u it temPeratdrc bY??'
r
.
For an idcal gas
^
P.dYqv.dP : r'itT -'.1? -B#
Differcntiating,
(di)'
Substituting thc valuc of dT it equation
a1%*ff=o
colP.&Y+Y;d4+r.'i' : o
But,
| : c'-c,
rYl-r : const.
Y
Also
..(dd)
,t rTPJ1'r :
"'(;;;1
flr'f
:
PFT
Pf .L :
T
r?.
-_{
_l
lx?,xilT
- Pv:q
fTYr-t : Colutt.
in volume'
...(4
As the external work is done by the
c,dr+!+:9'
,W+\.fu)"*r
log PYt : coost.
PY't - const. VT--'--"
This is the equation ccnnecting pressure and volume tl 'rr:rr:
adiabatic process.
PY:rT
Taking
/rT\
(-r-,) ' i'Y: const'
: da+!-*!
dA -
{)
But r is const.
dP is the change
where P is the preslure of thc gas and
";;#,
Y
Integrating, log P1Y log Y - const.
8.E[ : da +8w
,"1
II
P
Applying the 6rst law of thermodynamics
8E: g
8W : P'dY
co : ,
C,
dPJY --::
dV
gas be 87.
o
dP
But
done by the
ioruf"iJ-iro* tfrE r"r."""ai"gt. Lei the external work
But
and
ilv
i'-r-+-P:u
o'-cn: +
Hffiis
Ce.P.dV+Co.V.dP -'0
c)
- C..d7+ +
o,-c, :
C,.P.dV +C,.Y 'dPlCr.PdV -CoPdl: : o
.. . (i)
- I xlrdz+P$
Caill
?hermodynarnic*"
CCflSt.
COIrSt,
COISI.
Thu)during an adiabatic Process
'$ ,, PPI : const'
$rY TY-'."- const. and
t .ft't
const.
lit\/b:
o preslure of 2 ar,r;
i,rzrrot" 6'2. A rtotor oar tYre Ifhaa
tgre audilenig bu'the
27oC'
of
ternperature
tln-room
"niult
tertPerolure.
reaulting
the
furd
Eeal and ?he.rmodyaamict
Tltcrrnodgnamiu
Pr: 2 atmospberes
,v_
rl
-
D
It:
qt_
-t
adiabatically
to ha$ ite
. Erlmple 6 {. 4i, it.in compreased
t'olume. Calculate the clmnge
ita lemperature.
@eth; lbAgl
Let the initial temperature be ?, K anC the 6nal tenperature
27? +27
300 K
I atmosphere
T,K,
Initial volume
Prr-,
-TJ-
-a-art
: Vt
: V2
:vl
(+)'-'
( ?,\'
During an adiabatic process
,: -
t
Final volumc
1.4
Prr -r
2
\T" J
TrYrt-r : Trl/rr-t
l', -17-r
m
r2: ^f
r,Ly,
/ 7', 1t'r
( f,)": \300/
T, :
0'4log (0.5) : l'4 fiog
300]
"1-log
:
I'4
los
fi-3.4080
-0'1204
But
log ?. -
3.3476
3.3+76
-lT-
Change in temperature
T,: 2.16.1 K
: -26.9.C
(ii) temperatire.
I atarosphere ; ps : ?, 7 :
1,4
Yr- +
Tr121o.to
l'319 ?t
perfect gas ot 2)"C ie adiabatically compieaaeil in a ,"r"rr;bi D"rx,cos
trom an initiol pressure o! 1 atmoaltierc to a final grcssr;c of d0
otrnos7lherea,Calculatethe.resultingdifferenceintimp'eiotutc.
ll
tl
l,'
Duriag sudden compression, the progess is adiabatic
P,Y,, : P,Y,T
Pr: Pr[
t)'
: I[2]1..
In a reversible adiabatic process
or
Here,
2.636 atmoqpheres
yr:
+
rr- 300 K i?t: I
t: 1.4
Trl7r;t-r
iii)P1 -P;
Ir lYryt-t
,r=- ?r121t.t-t
i
J
LDcthi (Eo*.11978)
Ptz-r P't-r
-Tr't : V;-
(+)'-':(+)'
Pr: 50,
&- l,
Tt: 273*27
:300K
Tt:?
Y'3
5
(50;an_(#-)'"
2
T- Iog (50) : -i- ft.S fl-log 3001
?r - 1,{3{ K
: 11161r
q
30q2y.t
39s.9 K
l2i.9.C
l'40
Tr121r'ro-t
n- Exampl"- 6.9. I g_ra1n molecule o! a monoatomic 1y : 5l3l
.,2-m:aple 6.3. A ou.antitg_o!.air at-or;a1i:not
Z\"C anil atmoapheric pret.,
ccmpre.iaeil rc-naiy ;ta
ool.ume. Iind thc
.euilenly
..i preasvre anil
:
J
Tr121t-t
* Tr-Tt
: l.3lg rr_T,
: 0.319 Tr K
2.39t I
Yt:Yi
_7 for air :
?, :
T1 :
?t :
l'4 log T, : 3.4680-0.1204
td) Pr :
giE
234
Heat and ThermodYnamirs
rXlanople 6'6. A quantity o! dry air at 27'C- is compresacd
(il elowiuanld G;l auddenli to 113 r,7 its rolume. Finil the clnnge in
iifrTature in each c,aae, a.*su.n1.ing 7 to be i'4 lor dry air. -- .
lAgra 1969 ; Delhi 71,7 i)
When the process is slovu'. the tenrperature of thejlslgm
-
(l)
thCre is ng:1,"$-.:-.,
@
temoerature.
Y1
:V,
nr:I
?1 :300K,
't : l'4 ^'
V
ufirlopee of Adiabatics and Isothermrls
(v
^ ,Y
In an isothermal Process
PV:const.
Differentiating,
" vJt =-?A{
PdV+VdP : O =)
#: -+
..(t
In an adiaba""
!17it .or,r,.
?
''Xi"*T|1)u, : o .=7v
?r-?
Tr lrrlt'r - Trlrrlv-r
-
12:,1;l;1,,
?2
;fifnermodynamia // z--
or
(2) When the compressiou is sudden, tire process is aCiabatic.
.Here
L@
: --3oo I gll' '
LV J
dP
-dT : -,n''1P
w =Px v{*t A{
.. .(tr)
T'herefore, the slope of an adiabatic is Y times the slope of the
isothermal.
: 300 [3]t'r-t
^ i8i:i"t
The temperature of air
= increases by
192.5-27 : 165'5'C or f65'5 K
/Eranple 6'7. A aertain mass oJ gas cf NTP is
-ea9tanded,
lo
Calculote the
-threc times-ita t)olwme uniler adiabatio conditions.
femperoture onil preature. '( for the gas ia t. 40.
"e;ffiiig
lDelhi (Eons.) i5!
:
3l'
(l) Here, Y1o 7,
-V,
Tr.: 273 K
?rrrt-r : IlYrt'r
12: 11
T2 *
Fig. 6'6
?
[+]' '
rz: 27s[+]"-'
Ts*176K:-97'C
l't : 3V
(2) Here, Y1 : Y,
Pr: latmosphere, Pr:'!
PrVrt - P1V1l
P2: P1[+]'
,Pz: | (+I'
V*
P, : 0.21{8 atmosphere J
Hence, tlre adiabatic curve is steeper- than the isothermal curve
(Fig. 6$, utL poi"t where the two curves intersect each other' \ *
.L7 Work Done During an Isotherrral P:eggss
When a gas is allowed to expand isothermally, work is done
bv ir.
k I t.o, the initial and final volumes be 71 and 73 respectively. In
of the shade{.s-1rip represents the work done for
ri;.Y'6, ih. .t". volume
d7. When the volume changes from 71
in
"'i"^fi'"to"g"
to Yr,
...(r)
[:t P . dY J arca aBba
Work dcne
-^ )vr
Fis. 6'6 represents the indicator diagram' Considering one
gram m6lecule of the
orv
rii
: A,
oRT
Thcdytonkt
Ecat ond llffiYrrr;mia
During an adiabatic Proc€:ts'
W- u\i"+
'o::
r': f
,__VT
or
w
l- -l
l-J. !
: r-716-F:rJ
P
Since
Fig. 6'6
V.t
- RI log" V,
l)1v1 = Pllt'
\'1
lr2
P1Y1t: PrYra:K
and
pt
d
...(ii)
Taking'f1 and ?ras the temPeratures u! q.-poinc AandB
repectively;nd considering one Sram molecule of the gas
P1Y1 : RI1
I'l
r\= h
I r PrVrY &717'1
r=7-LVp=r--7rFi- J
: * [''n-'r']
. (i,,)
T
w = RTx2.3026* togro
the temperature remains constant). So the heat transferred is equal to the work
done.
d.uwork Done During an Adiabatic Process
shown by the indicator diagram (Fig. 6.7) the work done for an increase tn
PlYs: BT7
Substituting these values in equation (di)
\__--...(i\,) t
w
. Here, the change in the internal energy of the system is zero (because
nl)
/ and B Iie on ttre same adiabatic
n:
.. 1ii )
W = RTx 2.3026 log,,,
or
...(d)
w:1+l#^-#,=]
V ----+
Also
: Kli:#I
.
: #[nn-ar, ]
...(id0
Here, heat transferred is zero because the systen 5 thernally
insulated fiom the surroundings. The decrease ir the internal
*.rgy of the system (due to,fall in temperature) ir equal to the
a$e !Qr8o.
work done by thc system
^Dd
>
@ Irnrg. Relrtroa Bstween Adiabetlc end rrothcrrorl Etgtlctttcr
l. IeotherrnrlElesticitY
During an irotherual Procers
PP : const
Diffeentiating,
PitY+vitP
o Q v ol'g = ?
y.ilP -
-=aJF- : 'D
,--t
6.7.
Fig.
From the definition of elasticity of a gar
c(
.olume dV = P.dV Work done when the gas expands fioln V1 to l/: rs uiven
by,
tY=l
l,
a':
" I',
PdV=AreaAlllta
?dp
Ei,- _
:#
=trfV
Av
-..(i)
Eeol and Thcrmoilynamiae
238
From (d) and (di)
l
E6 n Pr/
...(di0
2. Adlebedc Eleetlctty
During an adiabatic proces!
P77 : const
Differentiating, P'(Yt'rdV +Yt dP :
O
YdP
...(iu)
4V-1P
oermodynamica
Eeitt: #_,
...(o)
rlre atmospheric pressure be Po. Tbe;ressure of air inside the versel
is Pr.
The stopcock B is suddenly opened and closed just at the
moment when the levels of the liquid on the two gid$ of the mano'
tDeter are the sarne. Some quantiiy .of air escapes to the atm.osphere.
The air insidc the vessel exfands adiabatically. The tenPerature of
air inside the vessel falls due to adiabatic expansion. The air inride
the vessel is alloli,ed to gain beat from the surroundingp and it finally
attains the temperaturelf the surr oundings. Let the pressure at the
cnd be Pr and ihe diffe.ence in levels on the two sides of the mano'
Theory. Consider a fixed mass of air left in the vessel in tlc
cnd. l'his mass of air has expanded from volume 71 (less than the
volume of the vessel) at preisure P, to volumeT3 at pressure P3.
The process is adiabitic aishown by the curve /B (Fig. 6'9).
.
From (du) and (o),
fr.4:7P
239
meter be [.
From the definition of elasticity of a gas
:#
T
...(ui)
Comparing (iii) and (oi)
Er41 :7E1,s
Thus, the adiabatic elasticity of a gas is T times the isothermal--elasticity.
\.\_7-
6'20 Clement aud Desorpes Method-Determinatioa of 1
, Clement and Desormes in 1Bl9 designed an experiment to find
PrYrr - PoYrt
*i: (*)'
...(0
A and C are at thc
Finally thc poipt C is reached. The points
'considercd as an isotherroom tempelaturi. Thcrefore AC can be
rnal.
P1Y1 :
P2V2
Y2
Pl
v;: -4
...(r0
'f, the ratio between the two specifc heats of a gas.
c(4,v2)
'---:1
:--1
B (8,v2)
rj
r-:
--j.l
:-l
E
:::I
---:I
r-l:
L:
Fig.6.0
in equation (d),
Substituting the value d +
Y1
rig.0.8
The vessel .d has a capacity of 20 to 30litres and is fitteC in
a iro:r containing cotton and wool. At the top end, three tubes
are fitted as shown in Fig. 6'8. Through 8r, dry air is forced into
the vrssel d. Ttre stop cock B1 is closed when the pressure inside
.4 i: ;li6htiy greater thau the atmospheric pressure. Let the
,.{ifierence i* }evr;l arr the two sides of, the ruanometer be .& and
t:(*)'
Taking Iogarithms,
log P1-log .l3r : Y[ioq Pr.- log Pg]
..| * iog /', *iot i-'o
losT;1G"""
E e,tt and ?hermodynont:t
But
P, :
.
v_
_tggt&+_{l-_l9gj._' - Iog(Po1I/1-lcg (Po*t)
Po1 H and Ps -
/Po*E\
\-7; /
*,_,o,
-'tg
the Wheatstone's bridgc arrangement
The vessel is surrounded by a constant temperature bath. Let
the -initial pressure
-and temperature be P1 and'?1 (room tcmpera-
(t#)
r.s (r.
+)
-;4;H-)
t
Approximately, ,:&-#u
_P;
y:=.8,
- E-h
241
It is controll..d. !y rpfuS.arrange'n-ent (Fig. 6.10). Dry air (or gas)
at a pressurc higher" than the arn cspheric plessure is loiced into'ihe
vessel ,{ and the srop-cock I is closed. - The oil manometer .ll{ is
used to measure the pressure of air inside the vessel ;t. '.lhe bcloT.Le_l .B (a platinum u,ire) and a sensitive galvanomerer are used in
Po*lr
,
Hence
Ihcrmoilynamice
...(iu)
Similarly, 1 for any gas can be determined by this method.
Ilrawbacks. When the stop-cock is opened, a series of oscillations are set up. This is shown by the up and down movement of
the liquid in the manometer. Therefore, the exact moment
when the stopcock should be closed is nor known. The pressure may
nbt be equal to the atmospheric pressure when the stop-cock is
closed. It may be higher or iess than the atmospheric pressure.
Thus the result obtained rvill not be accurate.
6'2f Pertington'cMethod
Lummer, Pringsheim and Partington designed an apparatus
ture). The bridge is kept slightly disturbed from the baranced
position. ,The valve .L is suddenly opened and closed. The wheatstones bridge is at once adjusted for balanced position. Tire remrlerature of air inside ,{ has decreased due to adiibatic expansion oiair.
Let the remperarure inside be ?o- and the atmospheric p.ess,.re Fo. rr
the ap-pararus is allowed to rcmain as such for some time. it will iai.n
heat from the surroundings and the balance point gets aLtr.u"ll In
order that the balance point 'emains undistuibed, iome piec.s oiice
are added into the watcr surrounding the vessel ,tl. whtn the icmperaturc of water-bath is the same as that of air just
"rt.r rJi"b"tic
exparuion, thc bridge will rcmain balanccd.
?j of the bath represents the temperature of
. ^The-tempgTtu_re
air aftcr
the adiabatic expansion
Por-,
&, .
T:-T;r
p,
( T, \,
1
\r-r_
:
\Tl \7;/
(r:lXlog P1-logPq) : y [og f1-log ?o]
log P1-log Po
vt- _ _
T, arc known, y can lre calculated. The
. As Pp P9, l7"C
value of l for air at
", :$-is found to Uc i.iO:+
(l) Due to thc large volume of the vessel, the
to determine the value of 1. In this method, the pressure and
4dqot$:s. expansion is adiabatic.
erParuron.
tl) tne
.TA.adlaDatrc
,,gTpT"tur€s
- r arter
ano
expansion.
temperature are measured accurately beforc and after the adiabatic
are measured accurately just before
Drewbechs. This method cannot be used to find the varue
of 1 at .$sh.. temperatur* because it is not possiblc t" a.tr-'-"
AIR
the cooling correction accurately.
6'22 Ruclherdt's Experiodent
r_ In 1929, Ruchhardt designgd an apparatus
find the value of
y. It is based on tbe principlJof mechan-ics. Air toloiglj
i, .""r"*a
in a. big -jar - (Fig. 6'i I ). h tube of uniform
of ils. .""tio, is
fitted and a ball of mass 6ts. in thS tube just".."
like a pirt"n.-in tf,.
equilibrium position, the rybaU h at the poin:t j. th.';;;;;.'p
"f
air inside the vessel, is given by
Fis.6.10
Tne apparatus consisl of a vessel / hav{tg a capqcity between
130 and 150 litrcs. The valve .t can be opened and closed suCdenly.
P:
Poq
!f-
244 /
q3/ krenersible Process
E eat tn iI Thernoilgn amict
The rhermodynamical state of a system can be defined rvirh
the help of the thermodynamical coordinates of rhe s)'srem. The
slate of a system can be changed by altering the thermodynamical
coordinates. Changing from one state to the other by changing the
thermodynamical coordinates is called a ptocess.
Consider two states of a system ie., state Aand state B.
Change of state fiorn z{ to B or r:icc ocrsa is a process and the direction of the process u'ill depend upon a new thermodynamical coordinatc called entropy. All processes arc not possible in the universe.
Consider the following processes :
_(l) _Le1 two blocks .z{ and I at different tcmperatures ?, and
Tr(Tr;Tr\ be kept in contact but the system as a whole is insulated from the surroundings. Conduction of heat takes place berween
the blocks, the temperaiure of I falls and rhe temperatur'e of B
rises and thermodynamical equilibrium will be reached.
(2) Consider a flywheel rotati.ng with an angular velocitv -.
Its initial kinetic energy is |1<.,r. After some timJthe wheel comes
to rest and kinetic energy is utilised in overcoming friction at the
bearings. The temperature of the wheel and the blarings rises and
the increasc in their internal energy is equal to the origiial kinetic
energy of the fly wheel.
(3) Consider two flasks r{ and 3 connecrcd by a glass tube
p_rovided with a stop cock. Let / contain air at high pres-sure and
B is evacuated. The system is isolated from the surroundinss. Il
the stop cock-is-opencd, air rushes from .d to 8, the prorrr.i i., .4
decpeases and the volume of air incrcases.
All rhe above three examples though different, are thermodynamical processes involving change in thermodynamical coordinares_
,Also, in accordance wirh the fint law of thermodynamics. the princi_
pl^e-of c-onservation of ene-rgy is no-t violated-becaurg the total inergy
of the system is conservcd. rtis also clcar that, with the inirial co*ditions described above, the three processes will take place.
Let us consider thc possibility of the above rhree processes
taking place in the reverse direction. rn the first case, if the reverse
process is possible, the block I should transfcr heat to ,{ and initial
conditions should bc restored. rn the second case, if rhe revcrse
process is possible, the
-heat energy must again change to kinetic
energ'y and the fly wheel should stait rotating with the initial angular velocity_ar. In,the third case, if the revcrse process is possible
the air in B must flow back to / and the initial corrdition shluld be
obtained.
- But, it is a matter of common experiencc, that none of the
above conditions for the reverse processei are riached. rt
-."n,
that the direction of the pr-ojess carxngrbe determi"ia Uy L.*i"g
the thermodynamical coordinates in the two end states. 'r" a.t.rmine the direction of the proccss a sew thcrmodynamical coordinate
hrs been devised by Cliusius and this ir c"tfe-a'Ge .-..Ji.T,n.
sysrem. similar to internal cDGrg[r entropy is also a functi6n of rhe
2t5
Thermoilynamice
rrate of a syst:.rn. . For any possible process, the entropy of an isolatrd lystem should increase or remain constant. The piocess in r.r'hich
there is a possibility ofdecrease in entropy cannot take place.
- If th-e enrropy of an isolated system is_ maximum, any changc
of statc will mean decrcase in entropy and hence that change6f
rtate will not take place.
To conclude, procases in wi;ch the enlrol;y oJ an iaolateil
sudem ilecreaaes ilo nol lttke place or tor oll processea toking place
in an isolateil syatem the entropu oJ the system shoulil increase or
remrin constont. It means a _process is irreversible if thc entropy
decreases when the direction of rhe process is reversed. A processi!
qid to be irreversiblc if it cannot be retraced bggL3fi.e"flI1x_iE-.o'ffigin!
an lrrt-vCisibic pro.efi hJat-iiiffy is
al ways used to overcomE-fiiEtion. -Encrgyft also 6i$ipa iEfi n:flre
form-of-gonrdu-c1ion and'i-aaliatioil. This loss of ;iieiEy alw-efi'=tekes
p l'ate w.liiihir-ttre-cnginE'woiTilin ,one *di.recrion or ihe revirse direction, Such energy cannot be regained. In actual practiie all
th? engines are irreversible. If electric current is passdd through a
rvire, heat is produced. If the direction of the current - is reversed,
heat is again produced. This is also an example of an irreversible
'process. All chemical reacrions are irreversible. In general, all
natural processes are irreversible.
.
,
6'21 Revcrsibte Proceeg
or.
r1$t rgl,. .-Ig+is -pro€esr, the inirial conditions of.the working
subSTanEE c5_ be obtainedl
Consider a cylinder, containing a gas at a certain pressure and
temperature. The cylinder is fitted with a frictlonless piston. If
the pressure is decreased, the gas expands slowly and maintains'a
constant temperature (isothermal process). The energy required for
th-is,expansion is continuously drawn from the sourcc (surroundings).
If the presrure on the piston is increased-, the -gas contracts sloi[y
and maintains constant temperature (isothermal process). The energy
liberatecl during compression is given to the sink (surroundin$).
This is also true for an adiabatic process provided the process takes
place infinitely slorvly.
The process rvill not be reversible if there is any loss of heat
due to friction, radiation or conduction. If the changes take place
rapidly, the procgss wp-not be revcrsible. The energy used in over-
6fbe retraced.
The
can Dg-stat
ldi.tlons of reversibility for any heat engine or process
as fsllows :-
he pressure and tcmperature of thc working substanec
rLe-1c,11-\/ I o n- - z)(' (4 6 r4.,1
-<taJt<-Wr
Ecot ond ?hermoilYnamico
246
must not dih-er appreciably from those of the surrotmdings at any
stage of the_cycle of operation.
Jq)rAii p/ pro"or.s taking place in the cycle of o.peration
must bi inEditely slor.'.
*orking parts of the engine must be completely free
from
d.re to conduction
operation
of
the
cycle
radiaiion
during
or
It should be remembered that the complete reversible Process
.
or cycle ofoperation is only an id-eal -case. In an. actual Process,
therc is alwiys loss of heat-due to friction, conduction o,r radiationThe temperaiure and pressure of the working substance differ apprcciably from those of the surroundings.
,61-laS
Second Lew of Theraodynaaice
tDr "
A heat engine is chiefly concerned with the conversion ofheat'
cnersv into mec-hanical work. A refrigerator is a device to cool a
certa"i; space below the temperature ot'its surroundings' - The first
is a qualitative statement which does not
law of thermodynamics
of the-existence of either a heat engine or a
oreclude the poisibility
'rcfriserator. 'Thc firsi law does not contradict the Cxistence of_a
i60"7;ffi;i.nt hlar-ngine or a self'acting refrigerator'
In practice, these two arc not attainable. These phenomena
are recognized and this lcd to the formulation -of a law governing
these tw6 dcvices. It is called second law of thermodynamics.
A new tcrm reservoir is used to explain the second law. A
reservoir is a device having infinite thermal capacity and which
can absorb, retain or reject inlimited qiiantity of heat without any:
pf
. ' f:SlylS-Plalgk statemcnt the second law is as follows ;
is impossible to get a continuous-supply-of-work from a
"It
(or
which can transfer heat with a single heat reservoir.T'
bodv
ensine)
'Th( ii a nelgative statement. According to this statcment, a single
i single temperature cannot continuously transfer heat
into work. It me-ans thal there should be two reservoirs for any heat
' engine. Onc-rcser+reir{callcd -thc s9.ur-cc! it t"F:l at a lrigher tem'
peiutu.e and the oiher reservoir (called the sink) is taken at a Iower
, reservoir at
temp6
--l**
this statemen t, zero degree atsolute t"-p.."tur"
not attainablE because no heat is rejected to the sink at zero deg'
is
--tccordlfrg-ia,
ree Kelvin. If an engine works between any temPeraturc higher
than zero degree Kelvin and zero degrec Eelvin, it means it uses a
sinsle reservoir which contradicts Belvin'Plancl('s statement of the
sec"ond l,aw. Similarly, no engine can bc 100% efficient.
. In a heat engine, the engirle draws heat from the source attd
after doing some external work, it rejegts thc remaining-hcat to the
sinE Thi source and sink arc of infinite thermal capacity and they
rraintain constant tcmP€rature.
Thcrmoilynamiu
24?
Firet Part. According to Kelvin, the second law can also be
stated as follows :
'-L'#'
'/ "lt is imppSible to-_ge!_p_ c_qe!n!_oU!_ llpply of work from a
pody, !r eeoling_ft to a temperature_lower than that of its surround-
-l!gy--
In a heat engine the working substance does some work and
rejects the remaining heat to the sink. The temperature of the
source must be higher than the surroundings and the engine will not
work when the temperatures of the source ind the sink ire the same.
Take the case of a steam engine. The steam (working substance) at
tigh pressure is introduced into the cylinder' of theingine. Steam
expands, and it,does external work. The contents remaining behind
after doing work are rejected to the surroundings. The teriperature
of the woiking substanie rejected to the surrouidingr is higier than
the temperature of the surroundings.
If this working substance rejectedby the first engine is used in
another enCil-e,_ it can do work and the temperature oi the working
substance will fall further.
I It means that the working substance can do work only
--- I if- its
temperarure is higher than thai of the surroundings.
Second Part. Accordil1to
_C_!""riu. :
is
"It !-mpq,q!ible-to_m_a_Le b_eat_{q!y from a body at a lower
te-rnp:I?lirle ro a. body at a higher temperature withbdt doing extirnal work on the working substance." _
V
'
This part is applicable in the case-of ice plants and refriserators. Heat itself cannot flow from a body at a lower temperatuie to
a body ata higher temperature. But, it ii possible, if some external
work is done on the working substance. Take the case of ammonia
ice plaut. Ammonia is the working substance. Liquid ammonia at
low pressure takes heat from the br-ine solution in the brine tank and
is converted to low pressure vapour. External work is done to compress the-ammonia vapous to_ high pressure. This ammonia at high
pressure is pased through coils over which water at room temperature is poured. Ammooia vapour gives heat to water at room tempeI3tulg arrd gets itself converted inio liquid again. This high pressure
Iiquid ammonia is throttled to low prdsure liquid ammon'ia. In the
whole process ammonia (the working substairce) takes heat from
brine solution (at a lower temperature) and gives heit to water at
room temperature (at a-higher temperature). This is possible only
due to the external work done on ammonia by the piston in compressing it.. The only work of electricity in thi ammonia ice plant'is
to.move the piston to do external work on ammonia. If the external work is not done, no ice plant or refrigerator will r.r'ork. Hence,
it is possible to make heat flow from a body at a lower temperature
to a. body at a higher temperature by doing exterilal work on the
worklng substance.
_Thus, the second law of thermodynamics plays an imporrant
part for prlctical devices e,g.,heat engines and-refrigerators'. The
frnt law of thermodynamics only gives the rclation- between the
Heat atd Thermodunamica
l\'.-)rk done 76d the heat produced. But the second larv of thermo(.1\ namics
ives the conditions undcr v''hich heat can be converted
Thetmoilynamics
absorbe.d by rhe rvorking substance be I/1 at the
The point J? is obtained.
tcmperaturc ?,',.
I
Considcr one gram molecule of the working sub!tance.
Work done from A to B (isothermal process)
/cgr-g!-B-er.er.ibl"&et.
.--.-' Heat cngines are used to convert heat into mechanical u'ork.
Sadi Carnot (Frcnch) conceived a theoretical engine which is free
from all the dclects of practical engines. Its e{Eciency is maximum
and rt is an ideal heat engine.
For any e'rgine, there are three essential requisites :
(l) Source. The source should bc at a fixed high temperature
?1 from u'Ii-rch tlie heat engine can draw heat. It has in6nite thermal
capacity and any amount of heat can be drawn from it at constant
temperature ?r.
(2) Sink. The sink should be at a fixed lower temperature
Z, to which any amount of heat can be rejected. It also has infinite
tirermal capacity and its temPerature remains constant at fr.
219
aY,: \i',' dv : Rr, ',c,+
:
arca ABGD
(2) Place the engine on the stand having
""' irrr,ur.a',o[l
Decrease the pressure on the working substance.
The volume
(3) I{orking Substancc. A cylinder with non-conducting
sides and conducting bottom contains the perfect gas as the wotking
sttbstanae.
I
I
P
(B , V3)
CYLINDER
WORKING
SUBSTANCE
m
COI.IDUCTING;
T-----'a
AT Tt
ffi
ATT2
Fig. 8.12.
A pcrfect non-conducting and frictionless piston is fitted into
the cylinder. The- working substance undergoes a complete cyclic
operation (Fig. 6'12).
A perfectly non.conducting stand is also provided so that
the working substance can undergo adiabatic operation.
-2
ffi C.a.leo.'T(rycle
,Y\g-b
tr{ Plaee the engine containing the workin} substance over
D' fth" source
at temperature Tr. The working sub.starice is
also at a
.
temperature, 1r. _]J. pressure is Pt and _volume is 71 as shown by
the point d in Fig. 6'13. Decrease the pressure. The volume
cf the working substance increases. lVork is done by -the working
substance. As the bottom is perfectly conducting to the source at
ternperature 7r, it absorbs heat. The process is completely isotherm;^1. The temperature remains constant. Let the amount of heat
t,r
EFG
.V.-..----.-.-}
Fig.6.l3
increases. The process is completely adiabatic. Work is done by
the working substance at the cost of its ,:aternal energy. The tem-
perat Te fa1ls. . Tle.working substance^undergoes- adiabatic change
from I ro C. At C the temperature is
(Fig. 6.13).
-
"1
Work done from B to C (adiabatic process)
P . dv
1 But PV't : constant : 'K
\Yi':
Pzv'-Rra
\i:
tvtd;
i"
Prv, - RT,
I
l.u_r_ Y'
9"-'-RV't-r:
-Ptr"r:PzYzr-R
I
E I+I{a
| -"t,
RVr-Tr}
: __T=r__ _: *_RLTr-rzl
:ll
1
"/_l
ff1 : Area B0flG
,
...(di)
250
Ecd and Thcrmodynamia
261
Thermodynamia
The points I and C arc on the same adiabatic
(3) Place the engine on the sink at temperature ?;. Increare
the pressure. The s'ork is done on the working substance. Ar the
base is cgnducting to the sink, the proces is isothermal. A quantity
of heat IIs is rejecled to the sink at temperature ?r. Finilly tht
TrIrl-r - rzyrY'r
rr / 7, 1r-r
.
T: \7;/
point D is reached.
From (ui) and 1uid1
Work dogc from C to D (isothermal process)
\il: ff,' ,u,
(+ )": (#:)"
Yr l',
v;:
V,
T:N;
_ RTr,,s :+
vt
Y"
- -RTrbS+
r:
Ys
...(rdd)
area CEID
From equation (u)
-ve sign indicates that work is done on thc working
substance.)
w: *l**I,,-,, l
(4) Place the engine on the insulating stand. fncreace the
pressure. The volume decreases. The process is completely
adiabatic. The temperature rises and finally the point d is reached.
Work done from D to u4 (adiabatic process).
W- Hr-H,
Efficiency
'OU
Useful outDut
,', : ---l;pui----------------
n
.ufrr: ATIaDIEA
...(?).^ b:i"
and
and-cancel
Waare
equal
opposite
each
othe..f
[]71and
9H
The net work done by the working substance in one
,
E1: RTIdC +
rl E
"o*-pl.t.
: Area ABGEIAreaBCEO-A., O*!-OOrea
DIEA
: Area ABCD
The net amount of heat absorbed by the working substance
- fl1-flg
Net work : WL+Wy+W"*W,
-
RrL
br+*W-Rr,bs +,-^T={,,
w : nr, rcsft-ar,tor h
...(o)
The points A atd D are on the same adiabatic
TrYr:r'r : TrYr'r-r
?,
W
T;
Heat is supplicd from the source from L to B only.
._ _ R(Tt-T]l
Y-l
/'-
cycle
yr
W: nr,bslL-n?,log T
(The
WI: ,f,,
...(uit)
/ 7t 1r't
E: \-%-/
...(t i)
12
W H,_8.
E;:-tr;RLr,-r,).-(+
) : *-T
:
"rlGG)- -2 \-Z
,H2
: l- -E;
rl: !-T T"
-1,
-
T
u'-
tI
...(adrT)
The Carnot's engine is perfectly reversiblc. It can be operated
in the revcrse dircction also. Then it works as a refrigcrator.
The heat IIs is taken from the sink and external work is donJon the
working subitance and heat II1 is given to the sourcc at a higher
temlrcrature.
The isothermal process will take place only when the piston
mcves very slowly to iive enough time for the heat transfer ti akc
placc. The aaiabatiC Prcccrs will take placc whcn the piston tnovet
Thcrmoilynamics
o<0
Z5A
E2
H eot anil Thermodynamice
H
extremely fast to avoid heat transfer. Any practical engine cannot
satis[y t]rese conditions.
. AII practical engines have an efficicncy less than the Carnot's
300 - 200
does an amciunt of work lY and rejects an amount of heat .&, to the
sink a_t temperature ?r. Wtren it rvorks as a refrigerator, it absorbs
heat E2 from the sink at temperature Tr. W amount of work is done
on it by some external means and rejects'heat Hr to the source at a
temperature 7r (FiS.6'14). Ir, the iecond case heat flows from a
body at a lower temperature to a body at a higher temperature,
with the help of external work done on the vvorking substance and it
works as a refrigerator. This will not be possible if the cycle is not
completely reversible.
than l000/6 bu-t in'the-case of a refrigeiator, the
formance can be much higher than 1OOo1.
Here f/, is the desired refrigerating effect.
of per-
/'
.r-ig+ D ample 6'8. Iinil the efi.ciency of the Carnot't etgine aorbT ing detween the ateam point anil the ice point.
Tr :273*100 : 373 K
I\
'::',':t:273K
Tr
i 27g
' :t-zlT:
t00
Wg
o/eefficiency:
Coefficient of Perfotmance. The amount of heat absorbed
at the lower temperature is 112. The amount of work done by the
external process (input energy) : W and the amount of heat rejected
: Hr
:,
Therefore the coefHcient of performance of a refrigerato r : 2.
In the case of a heat engine, the efficiency'coefficient
cannot be more.
engrne.
6'27 Caroot's Engine and Refrigerator
'Carnot's cycle is perfectly reversible, It can r,r'ork as a heat
engine and also as a refrigerator. When it works as a heat engine,
it absorbs a quantit,v of heat .t/, from the source at a temperaturi 7r,
r-H,
,*
$xfOO
: 26.91%
Eraaple 6.9. .?inil the effwieacy oJ o Carnot'c cnginc worlcing
between 127"C onil 27"C.
It:273*127 : 46911
Tt :273+27 : 300 K
I:L+
: l-jP400 - 0.25
//
/o cfficieucy :
?5.o/o
6'f0. A Cornot'e ctgirtc whoae tcmperalurc of tlrc
_ lxlmple
aarice is 400 K takee 200 caloriea oJ hcal at lhil tcmpqraturc and
rejecls 150 coloriea ol hcot to thc sjn(. What ia lhe lemperottre of
lhe sinb I Also ulculole tl* eficieocy oJ the enginc. " i
(I) HEAT
(iil nernce narrcn
ENGINE
Fig. 6.14
Coefficient of performance
.ErH,
: =W' : Er-r,
Suppose 200joules of energy is absorbed at the lower tempcrature and 100 joules of work is done with external help. Then
200+1001: 300 joules are rejected at the higher temperatuie.
The coefficient of preformance
E2
: -ly:
JIr:200cal;
Tr -*00 K ;
tt
E,
trI:150cal
Ir:
-7i: T,
rt
: _f,i "*,
?r: ffi*O* :
3{X) K
I
Ecal and Thcrmoilgnanice
26t
{lrlrnodyramia
,-r{+
&: Er. +
: 1-# :0.25
0/6 efficiency
z, 500x300 :576.92ca1
ra,:
'
260
:
lf Er-Et: 76'92 cal
joula
76'92x4.2
: 323.08joulu
:25%
z' Exraple 6'11. A Carnot'a engine- !9 oyg'qted belween two
necntoirr oilemperattrea of 150 K anil J50 K. IJ the cngine reccilrtce
1000 caloriee of heat troh the source in e-och cycle, c,alculate tha
omou,nt of fuat- rejecteil to the aink in eoch- cgcle. Calculate lhe
eficiency-ot the engine ond the work ilone bg thc engine in eoch cycle,
(I calorie 4'2 joules).
-
fr:350K
rr *450K;
Er : 1000 cal'i - E1 :
1
E,
Tr
E;
Era_mple-6.!3. A Carnol'a_retrigerator lahea heot lrom ualq al
0'C anil iliacarda it to a room ot 22"C. 1 kg of woter at-A"C h to bc
clwngeil into ice ot.0'q. Eow mony caloriTe of heat ore diacorilcil to
thc room I what is the uork-ilone by the retrigirator in thie proceaa I
'
What is the coeficienl oJ perlormonie ol thi machine ?
[Dclhi 19741
Et:?
IIr : l000x80 : 80,000 car
ll
Ez* ,r* #,
?r:300K
-tg#
!:
:272.77 Lrs
(l)
.71
'rr
_
3s0
l-m-86
T:41
- E#'
Jl
80,000 x 300
- ---2
Er _ S7,9oo g$
0/6 efrciency :22'22o/o
WorL done in each cYcle
(2) Work done by the refrigerator
: Er-flt
: t000-777.77
- 222'23 cal
:W:J1E\_ES
W : 4'2 (87,900-80,000)
W - 4.2x79A0
F : 3.I8ilxl0.Joutc.
:222'23x4'2 joules
:9l?'rgjoutce
Errnplc 6.12. A Carnot'a enCill working aa- a refrigerator
bclwacn "ZN K and 800 K receioee ,500 calorics of luot lrotn the
ruscruoir dl thc bwer lemperatute. Calculate thc amount of heat rclicctcd
tln rceatoir at the higher tempetalure, Oolculate aho thc amoual
of worlc donc hr eaa,h cycle to operate thc rc7'rigeralq. -Ao
-
T- n;;
?t :273K
flr
TL
E,
100
:0'2222
E1: I
fr - 300 E
lrr
Er
246
197aJ
lDelhi lflona.l
E|: 5(X) cal
Ir: 260 K
(3) CoeGcient of perforu.ance,
Er
: E;g90,000
-- 97,900-80,000
80,000
- 7900
: 10.13
,tS
Eeat onit ?hermoitynamica
,/-
'".)Unt"T_pIe 6I{. A carnot engine uhose lou temperalure rescr-
coir ia ot 7"c t,,s on efi,ciency ,I 50%. It is desired ii
;iiir.rf, tn,
0o/o; ns-how i""y.a"[]iu-_rmrW tii ir*i*i",i'
:{-r!r:?v.to
the hqh temperatutc reaqroir be increased t
"t
(Oetht lgliy
Thennodynamiu
7
Ii(Eciency of the engine d
In the first case
.- 50o/o : g'5, Tr : 273*2 : 2g0 K.
It-?
:?
"
Er-8,
--E;- :4w
-1:
: t-TJ
tl
or
0'5.: I--
or
7r : 560 'K
3
In the se@nd case
1'- 70!s: U7,
Tt:280 K'
rr':!
,' :
Fig.6.t6.
l-\ m,
Efficicncy of the engine B
: .r- '?gT,,'
T'' : 840 E
0.?
Increale in temperaturca Stl0-560
\\
-
280 K
- 71':
E"-E{
W
- El
flr,
Since
I>rl';E{)Er
Also,
'W
: Er-Ht: E{-E{
Thr.rs, for the two cngines A arrd
l
i;r r.i*fr#'i*t rfl },,i:oJ. :
tem, (4
tr t L..q,.
t
"'ti i, tt.
tcmperature ?2 and (Er,-Hrl
o"""i;
r vs
Consider two reversible engines d aud B, working between the
temperature limie ?, and ?r (Fig.6.15)- .d and A"are coupledSuppose.r{-is more etcient than 8. The engine d workr ., . t..t
elglne 11rd^8 as a refrigerator. Thc engine? absorb,s an amouat
oI hcat Irr ,rom the source at a teEperature [.
It does exterua]
work P and transfers it to 8. The heat r-jected to the sink is E. at
a temperature &. The engine B absorbs Eeat Ea, from the sink' at
temperature fi and. I[ amount of work is done on the workinf substance. The heat given to the source at tempcraturc ?1 is .E1,."
Supposc tbe engine / is more efficient than B.
+,i
rytltl**i,"-.r,*.1$;ffi?i;[:":J".1_"#;i#:
ffirll*
lilthp', i::: i H:#,.x,# r*rH.Ifl{
done on the svstem. This is
secgnd law
of thermodynamics. Thus, a
"orrto.V'toihe
L;;;,'..*,iL
n) The two ensines
(reversibre) working between
""r;"1 *,J*-.'t*o-tempe.ature
timits iave
the same efficiencyf fl\{;r;;;;;;t"
1';.';"r;
rh
i, ; ;' i.,,' d' r, 1", a -,l i.i"ii;"
;""!
";
--
"
".
;:,* glr$ :,.e-.l'ff
ETi: " efit?
the efficiency
depends :$
."1'y ;p";ih;
i:# #:LT;
r-*I i._p"rarre ri,nits.
In a practical
engine there is alwavs
.
tricticil;ft
,";ffi :i"La;",i"".r".:;;;.-*"':l.Jd:,."Hr_o"l;r;:
I.wer thar. that of a Carnot's ."*,"r=f '";'
-n
'o
i"I,tf TJ''*:lSi.;
.*',u'r'
11G_) \
Gt1''r\
Thctmoilytlzmico
272
H eat aail Thermodynumicd
For the same compression ratio, the efficiency of an Otto
engine_ is mor_e than a 'diesel engine. fn practice, th.
ratio for an otto engine is from i to g.rrh fo.. ai.r.r."sl"."it
"ornpr.rrio.i.
from l5 to 20. Due to-theJ-righer
ratro, an actual dieser
en,gine has higher efficiency lhan"o*p..*ror
the' Otto (petrcrfl .nein.. -fhe
cyrrncter must bc strong enough to withstand very high p.6ss,r...
6'37 Multicylin6.l Enginea
With an engine,havin.g one cylinder, the engine works only
d.uring the u'orking stroke. The piston movcs a,r.ing'ihe ,"rioi,t,.
tlrree strokes due ro the momentum of the shaft. i"""
--tir.
-"iti""t-ina..
engine (say 4-c_ylinder.engine) the four cyri*ders
".. "o"11.J1
working of each cylinder is given below ifdret
8@nd
Eirst quartor
Workilg
Erhaust
Compreasion
Charging
Ssooad quBrtor
ErhausL
Charging
T9orking
Coopreesioo
Third quarter
Charging :
Comprossion
ExheuaC
Working
Sourtb quartor
Compreesion
YPorking
Charging
Erheust
?h;rd
Fourth
In this wa-y, -the poyqr of the engine increases and the shaft
gets Eomentum during each qgarter cycl-e.
- Thc cycle AND reprcscnts a comptetc cycle and Carnot's
theorem can be ^ applied. Suppose the volume it the point z{ is p1
and temperarure h f 1d?. '[he pressure is just below iu raturation
prcssure and- the liquid begins ro evaporate and at the point B the
volume_ is 72. .The substance is in the uapour state. Suppose the
mass of the liquid at I is one gram. The amount of heat a-bsorbed
is IIr, Here E1:L1d!, rvhere L+dL is the latent heat of thc liquid
at temperaturc (T lit?1,
At the point B, the prcssure is decreased by dP. The vapour
will-expand and its tempcrature falls. The tcmperature at C it f .
Ac this pressure and temperature f, the sas bceins to condense and
is convcrted into the liquid stare. At thc-point D, the substance is in
the liquid srate. From-c to D, rhe amount of hlat reiected (civen
out) is II1. Here II1 : .t where Z is the latcnt heat .i t".ociit,r."
'/'. By in-crealtlS_t1r9 pressure a little, the original point z{. is iestored.
-Applying
The cycle ABCDA is co-mpletely
thc principle
-cycle reversible.
of the Carnot's reversible
flt
H,
Here,
[,9#Clap_evroor,eteq-1H,9a!-F-.{gi-tio.n
tT)-*""ider
thei*ott..*"ii-iaen ., 'i.ilr.o,ure
and
auafl at temperature ?. }Ierc EA and.ED show the f+d?
liquid state
2f3
.
-T:
rl
Tl)
Er
Tr
:Er: r;
Er_E, Tr_T,
-T_
-Z;:
flr - L+ilL, E1 : L,
T1 : T{iII, T1
lI
Er-flr: LailL-L -: ilL
Tr-?r-T+/IT_T:ilT
dll itf
_T-T
The area of thc figure
ABOD-fl1_82-ill
.
i,P
L
@ :--T (VL-V)
P
Yz
v'------;
Fig.6.23
ol the substance- At I and D the substance is purely in the Iiqtrid
state (Fig. 6'23). From I to I or:D to c trre subitance is in tra'nsi-
fr.T the liq.uid to-the gaseous state and o;cic ieiii.--ai'Cl"a
IITthe substance is purely in the gaseous
u
state. From B to .F or C
to u the substance is in the gaseo,s state. Join I to D and B to c
by dotted
lines.
_ dP(Yr_Vrl
dP (vt-vi dr
...(,
This is called thc Clapeyron's latent heat equation.
. Applicetloar. $ eficct of clwnge o! gtreaeure on thc mclting
point..
When a solid is convertcd into a liquid, therc is change in
volume.
(i) If % is greatcr than 71
dP
fiV ls a positive quantity. It means that thc rate of change of
274
E eal and Thermodynaniet
?lvr'modynamice
pressure with respect lo temperatrrre is positive." In such cases, rhe
melting point of the substance rvill increase rvjth increase in pressure
and lice lersc.
(ii)' If l'o is iess than Ir,.
dP,
6,Vis
a negative quantity. It me ans ihat the rare of chenge
of prcssure u'ith respect to temperature is neqative. In such cases,
the me lting point of tlre substance u'ill decrease rvith increase in
presture and ricc t,er,sn. ln the case of meltinq ice, the volrrme of
rvater formed is less than the volume of ice taken. Ilence 1tr q l"r.
Therefr:re, the melting pcint ol ice decreases rvilh increase in
pressure. I{ence ice u'ill rnelt at a lemperature lorver than zertr
degree centigrade at a pressure higlrer than the normal pressure.
76 x 13.6 x 980 x 273(t
:0.0074K-0.007{"c
* - -!,rerrplc 6 18. Find lhc incrcase in the boiling ltoid"o! uoter
1!0:C when the ple"lure ia inueated by one atmiipherc." Lalent
ltheat
o! tuporisation
lf.1team ia t40 callgrim and I gram o! eteam
c,ccupieE a talrrme of 16?7 cms.
dP :
:373K
, : 5{0x4'2x l0? ergs
l'. : l'000 cms
I/r : 1677 cmo
dPL:
-a7 TW_I1
ur_ dPxT(Y'-Vi
..-_-_-Z-
the liquid r.e. I', > l'r.
{ve quentity'.
lVith increase in pressure, the boiling point of a substance in.
creases and tice uer.-ra, Tire liquid n'ill Lroil at a lorver temperatLrre
under reduced pressure. In the case of water, the boiling poirrt irrcreases w'ith inclease in pressure and z.lce uersa. Water boils at 100'C
only at 76 cm of IJq pressure. In the laboratories, rvhile preparing
steam, the boiling point is less than 100'C because the atmospheric
pressure is less than 76 cm ol I{g. In pressure cookers, the liquid
br.rils at a higher te mperature. irecause the pressure inside is more
than the atmospheric pressure.
76x I3.6x980x373x 1676
540x4.2x10?
:27'91"C
Tlrerefore, theincreuae in the boiling point of water u,ith an
increase in pressure of one atmosphere
: 27.92.C
: 27.92 K
-,
Example 6'17. Aolculatc the depreeeion. in the melting point oJ
ice Ttroiluced by one otmosTshere increase of prensure. Gitten latenl
he.ct t{ ice : 80 cal ner gram rt.nd the specific uolurtues o! 1 gram of
ice anil wq,ter at 0'C are l'09 I cnts and l'000 cmxrespecliuelg.
\Panjab 19{)S)
Here
L: B0 cal : B0x4 2 x l0? ergs
1' :273K
dP : I atmosPhere
: 76 x I3'6 x 980 dynesfcmt
I'r : l'091 cms
l', : l'00C cmg
dPL:
dr
?:(|:Y,
I'1)
)o .= dP.f.(l'r__
*r
T_
76 x t3.6 x 980 dynes/cmr
T : 100*273
Ice melts at 0"C only at a pressure of 76 cm of Hg.
, 12) Eflect o! clange tf yre.ssure on the boiling point.
' When a liquid is cortverted into a gaseous state, the tr.,lrr:re I's
of tlie gas is aln'ays qrearer rhan the corresponding volume I'1 of
dP
'Iherefrrre, j7
is a
- I.09t )
B0x4'2xl0z
: _0.007{ K
Therefore, the d.ecrease in the ntclrinq point of ice wirh an
increase in pressure of one atmosphere
Erample 6.19. C.alculate lhe-ehange.-itt temperalure o! boiting
water whe n tlee pressure i,s inqcrceil W 27.12 mm o/ Hg. Thi normil
boiling point o! waler at utmocpheric pressure ;s l00.Cl
Latent heat o! ateam
anrl specifc volume o! ateatn
: 537 callg
- IA74 cmt
(Delhi 197 4)
dP : Z.7l2x 13.6x980 dynes/cmr
? : 100+273 :373K
L : 537x4'2x l0? ergs
Irr : l'000 cm8
['r :
,
dPL
1674 cm8
d:f : 16;-11
n8
Ecal aad Thermod;yta,amict
Thermodynamicc
d7: -l K'
T :273 K
I'.-I', : -0'091 cmi
L : 79.6 cal/g
- 79.6 x 4.l B x l0? ergs/g
)p _ L. dT
"'
:2.792K
:2792"C
.
Therefore, the increase in thc boiling pint of r+,ater rvith an
increase'of 0. I atmosphere pressurc
:2.792,(
:2.792C
- ru;w
dP :'gq#Haf
6.23. Calculol;e the c,h.,a,ruge in lhe mclring point ol ice
.when
-"- P:!-ple
il h
pressurcof
wbjectcd lo a
100 otmosphere,e
: 0.917 gfcms and
Lalent heat of ice : 336 Jlg
Densitg of ice
d.p:.
(Delhi 1972)
m : rv;n
dPL
: 135.2+ I
: f36.2 atrnoepberes
6'25. lf ater boils a! a tem1teroture o! 101"C ur a
pressure ol 787 mm o[ Hg. i gram o! water occupties 7,60 I cm3 on
euaporution, Calculate the lotenl huct d steam. J : 4.2X 107 ergs/cal.
, Ll
Wl,L*ple
T _273K
dPL:
m
(vt-vi: ,- #i7
336 x l0?
-0.7326"C
The decrease in the melting point olice rvith a pressure of 100
atmospheres
..t
: 0'7326"C
'- Erary.ple-6 2{- Calculate the pTessure requireil to l<twc
melling point o!_icc W l"C.
79.6 c?llg, apecific aolume o! wuter al 0"C 1.000 cm.
- pressare
:
' lOrin;-iiiA1
: 1'013 xl0c dyn.r7.*rr. r 09r cm3 and I armosTthere
dPL
ii: ro;q
fv;Tt
:27mmofHg
dT- 273 x99x76x 13.6 x 9B0x (-0.091)
=
lDelhi (Hons.) 197 tl
dP : 787-760
0'083
: -- 0.917
* -0.091 cm3
_ r dPvr_yrl
d?':--J-
.\L
atmospheres
Pressure required
: 99 atmospheres
iIP : 99 x 76 x 13.6 x 980 dynes/cm2
L : 336Jlg
: 336 x l0z ergs/g
specqrc_aolq?ne o[ ice ar 0"c
79.6x4.18x 107
_______
273 x0 091 x I 013 x l0 o
dP : 135.2 atrnospheres
dP : ld0- I
dr* -0.7326 K
dynes/cmr
'
- 2'7 cm of Hg
: 2'7 x l3'6 x 980 dynes/cmr
dT :1'g: lK
T :373K
Yr-Vr: 1,601- I : 1,600 cms
L:1
T dP 17r-Vry
u, : ----mL-
373 x2'7 X l3'6 x 980 x 1,600
I
ergs/g
r._ 373x2.7 x I3'6x9B0x 1,600 callg
4'2 x l0?
tr : 511'3 cal/g
Exarnple 6'26. lYhen. leail is melteil at atmoepheric preasurc,
Ithe melting gtoittl is A00 K\ lhe density d,ecreases lrom 1.1'01lo 10'65
glcms and the latent heot of lusion is 2* 5 J,rg. tllwr is the melling
Troint nt a pressure oJ 100 atmospheres
?
iDelhi (Hons.l 1972)
.Eed aill lf&rlrrpitrrtrj.t
Il1 - -{-l(XX) jouler
fr:500K
f3:300K
H rofl) -800
T
5m
-.1-o
: -;
0tg
Now consider the reversible cycle from state I to state 2 aloog
thc path d and from state 2 to state I along the path ()'
For this reversible cyclic process
E: : -8(X)joules (since hcet is rejected!
--
Thrrmoilyr.omict
:o
\1:4.1',;$
and (ii)
...(d0
From equations (i)
300
fla sE : rlc 8E
joulc/degrec
lzn-T
Jzo-T-
... (d;i)
(3) Conridcr a C.arnots reversible engine working betwcen the
tenpcratura 500 K and 300 IL Suppose 1000joules ofheat energy
is drewn from the high temperature rescrvoir.
flr
E,
T,: T
Herc
l0o0 _ E!
500 3m
^E : 600joulcs
trfl _ flr. H,
T, ' T,
4?
-il0C0joules
:
Er
\
?
__
_-_-4
Er :-600jouler
?l:sCgi;
?r:300K
fi
l&lJ, . ,-6001
T
500 ' '3m
This shows tn"t
tt iniieper.jent of
I Sthc path and is a function of the end states only, thereforc it is a
paths from state 2 to state 1. Thc quarttity
E
T T:O
proPcrty.
This example showr
casc and in no casc
t-'6.*2. Eotropy
f#
This property is callcd entrpP!' Entropy is a thermodynamicd
: o, only in the limiting
pnoperty ana ii dc6ned by the relation
is grcater thaa zero.
>#
or
ud tLe Sccoad llw of Thcrmody-a-icr
Considcra cloacd systcm undcrgoing a rcversible proccss from
statc I to rtatc 2 along thc path .{ and from state 2 to state I along
.thc path I (Flg. 6'25). As thir is a reverrible cyclic process
{4:o
2t
EE
t#
,.lr.:';:
,* the sane value for atl rhe reversible
frD &a
,r7+ lrr'7:
...(0
81-81
: f +
...(is)
...(o)
The quantity Br-Br rePrescnts the change in entropy of the
system whc'n it is 6hangcd from state I to statc 2'
,.
&f:, Eatropy changee of r Clorcd Syrtco Duriog aB
Irrcvcrdblc Procccc
to Z
o
,r: +
Consider a rgversible cycle wherc the stat! is clr'ngeC from
aand 2 to I r.long thc path A tFiS' 6'26)'
"f""ltfrip.tn
For a reversible cYclic Process
3[ar:o
I
80
Eedt and llhermodgna*iu
lza EE rrB -?-:
8E
Ju TT)za
o
...(r)
Thermoilynamiu
291
Equation (ia) shows that the eflect of irreversibility is always
to increase the entropy of a system.
fu**
Entropy
W-.".'
Consider adiabatics L and,.df on the P-7 indicator diagram
iFig. 6'27). All along the adiabatic f,, with change in pressure
H.\
-.-^
P
"-.-._.Fig. 6.20
Now considcr an irreversibre path o frorn state 2
to state r.
Applying Clausius inequality for the cycle of processe!
rg
Jz<o
A aad O
V____+
F.ig.6.27
. t'o 8^E , fro
Irt f - )zo 7"-< o
8^A
...(j0
Frorn equations (d) and (ir:)
rta 8E' tro 8I1
Je" --:F- -lzo -7- 7 o
,$ince path B is reversiblc and entropy
['B _qL:
J*r f
r
Br-o ,
+
i:
D.R
U-rr: J; #
ad for an irreversible process
:- f jg-
there is change in volume and temperature. This shorvs that all
along the adiabatics L or M, there is change of temperature" Consider the isothermals at tcmperatures f1, T, and, Ts. ABCD represents
the Carnot's reversible cycle. From ,4 to 8, heat energy .&1 is
absorbed at ternperature f1. From C to D, heat energy Il, is
rejected at temperature ?s.
!!^:
-fl'-aI
:
t7
Sirnilarly considering the cycle DCEI
...(iio
To conclude,
For a revereible procers
8,,-..e.
is a property
Jj| as - ll] as
ur>
12
...(ttu)
Er:Hx
Tz
TN
*:"*r:
!l': *constant
From one adiabatic to the other adiabatic, heat enerey is
either absorbrd or rejecteri. The qr"rantity of hc'at absorbed or
rejected is not constant but it depends upon the temperature. Fligher
the temperature, more is ihe heat energv absorbed or rejected and
oiccoerso. The quantity H,'T betrteen nvo adiabatics is conslanr
and this is callcd the change in cntropy. Let the entrrlpy lirr thc
ediahatics L and. M be .s1 and S1 rcspectivcly.
Here ,S, and 8, are arbiirary quantities.
,Ss*.S,: i.{'-.,,nrron,.
"
ooo
If the adiabatics
,..1." t.a
;,la ;; Hili.,Y,?rit",
' Change
and the heat absorbed or
: J:: d's: &-Br :
tB -f8E
: rsr represents
I: Lg
tt.
,i,:ol'ifixl:,1lf
All alons tt
Er E,
"il.Jffi :i.*"r.f*:o"i'.11,'iir",.nt
rWfriWmyeeytt*rxx;t*
in a
Entropv
!r":r".i'ia
o
Er
Rcversibre proceee
Ez
r;: 4
Hence the total change in entropy of the working substance in
a complete reversible Procesg
...12',
is carred
rt
But for a complete reversible procds
...( I )
the thermodynamic co.ordinate
lo
J"rUU
of a
Eystcm' This integrar refers
to the value of the function at
ut
Liii""r ',1'",". This runcrion the final
:f,[#:::tf I3]:
'ti.i'ii,iJiif
by the working substance in thc cycle
lt
In general, the change in entropy
6'*.
the total gain in entroPy
ABCDA
in entropy
dB: I{1l
Yrf
293
Thermodynamice
Eeat anil Thermoilgnamict
: f'r - F: -ff :
r.-.
w6,*O
o'
Ghange in Eatropy in an frreversible Procesg
In an irreversible process like conduction or radiation, heat is
'lost by . Uoay at a hightr temperaturc f1 and is gained by the body
at a lower teriperatrr.:e ?s. Hire Zlr is greater than ?2'
Let the quantity of heat given out by a body at a tern?eratrtre
Tl be E and the heat gained by the body at a temPerature 'I'g be l' '
Consider the hot and the cold bodics as one system'
Loqs in
(qryl
SntroPY
of thc hot bodY -
Gain in entroPy of the qgld bodY -
ffi Tff r,,l.8 i.":.I:?::#tr:i: IHHI r,t jil?f
E
71
t
Therefore, the total increase in entropy of the system
: Hq
r;-N
It is a positive quantity because- 71 is less than fr' Thus the
entropy of tlie system increases in all irreversfble processes'
( 6fi
Thtrd Lew of TteroodYnaraics
V \ U all heat engines, there is always loss of heat in the form o,f
radiatiin and friction. Therefore, in actual heat engines
"o"i""iion,
El
.^ E,
-V/
i
l!i:,:'.rt:T T:rr3 Ei fll : ffi d,"il,,t-.r,,o* or, e work ins
h
ill:{i:lh##ri:{r:#t::rl#i,.il#
;i:"..
reiected
ins substance ar a t.*i*jr,,ll,
Tir"rBL!, rs
f
by the worl-
ii,u'i,u*#i+,i"fi
Zr).
D ," a ,r,.rllr.nl,.rrurrg.
f{f li'jJtt$:,i}*ii;ii
ro
temperature
From
enrropy. Thus
t
-. ----^, to
1
7; is not equar T '" )
/.- but it is a positive quar^titv' when
*-*ir,,o,
of the system increases. and
cvcle after cycle is repeatcd, the entropy
-the
system has attained the
tir^*i-"ti val'te. When
;:ili;.
and no work can
reached
is
stagnancy
of
a
3tage
maxi*rr- value,
'In this universe
the entroPy
t. aorr. by the engine-at thisltage.
a
maximum
will
reach
also
rlniverse
thc
and
u-ltimately
irl"..."ri"g
when no ivork will be possible. With the increase
""i".
"f "niropy
thi disorder of the molccules of a substance increases'
i" ."t-rr.
T-h;;;#dy is also a tucarure of the disorder of the syttem' With
294
Eeal anil Tl*rmodrynamia
decrease in entropy, tbc disorder d,ecreasant/At absorure zero
Here Br:ur:+-+-ffi
,',4SfU - E,-n,
.-. Area ABID :
tem.
r)etot*te, the entropg lettih lo zero and the moleculee oy a subsianie or
?^'-y-t::!..:::ye!ect order (welt arrensedl. Thia is'the tiird ii,t
et*f moa!tuatntcE.rt
Erample. The molecules are more free to move in the
gaseous.state than in the liquid state. The enropy ;, mo..-'in
Therefore, the area ABCD represents the encrgy convcrted to
ttre
gaseous state thaa in the Iiquid state. The molecutt u.l---lr.'rr..
to movc in thc liq-uid stare lhan in the solid state. rir.
more in tjre liquid stare than in the sorid. Thus when ,"u.iJ;i"
""i.oo, i,i,
converted from a solid to a hquid and.then from the i;q"ia']o",t,.
^
solid state, the entropy increajes and oice ueraa. Wh;"-i;;l;
ted into water and the.n into steam, the entropy and disordcr"i"nr.rmolecules increase. when steam is converted'i"to *"t..
"i:ir,.
l"a"it".,
into ice,.the entropy and disorder of the molecut"r a.".IrrJ.""E"n",
entropy ia o mcasure of the ilisorder of the moleculio ijlir-iisii.'
is impossiblc to bring any systern to
_L , By any ideal procedure,-it
aDsotute
atur.e performing a finite numbi,
opl"utio"r.
1:rg
!:*p.:
r nrs rs.calre.l thc
_prrnciple ol unattainability of absolute"fzeio. Thus
according to Fowrer an.r,Guggenheim, the
i".uiiiry'pii"iipr"
is called the third law of theiirodynamics. "".tt
6'48 Temperature-Entropy piegram
,.ti,li",l[*r;'"i:.1rrfi
---
29t
Thermodgtomia
worL
: +? -t-'*:r-+
Efficiency
Here Hs is the unavailable energY.
H,: !^' x1t:s1 x?r
|"
change in entropy at
The unavailable energy depends on the
r
temperature ?1 and the temperature 7r'
6;{9 EntroPY of Perfect Gas
""a
Consider one gram of a perfect gas-at a prcssure P, volumeP
t.-p.."turc ?."Lct the quantity of heat given to the gas bc t;1.
: da+8w
8E: lxcuxdr+P#
8H
ifi ,?.1illfu ?,j,1
*="fi;4'rd:x',lJ:-.'iff
From .d to Il, heat energy .H1 js absorbed at templrature?r.
increase in entropv Bs
i.f,.
ti[* place from z to d-ffi-o'igi;ili'r*-
gH : TdS
rdB :cvar+
,\ = ,lo
P: n-
Also
or
c
Fig.6.20
B to C, there is no change in^ entr-opy. Thc tcmpetature decreases
at conltant entropy. From c to D, -:here is decreise in ent.oo, -7,3,J.
fi. Irom D to'd, thcre is
;H;g; i;
i:.":::T:.temp^e_iature
entropy but the terDperaturc increases.
";
.o""Jlixffi jjg?#":H'try:t?jiH:;s:Hffif au,.ir;f **
The area ABCD : Sr (\-!I.)
But
ff
...(i0
: C"df +ry
ds:cy#*3+
* :," !iis*+li"+
rntesratins,
Ix:
7:dl
=
H
P
...(d)
-
'8, (Tr_Ttl
Br:+andB1 :t
B,-s, - cy log,
-**i
be
+
...(idJ)
Br-8, : cnx2'3026 loglo ft+|"r'3026'logr ft ...(ig)
The changc in entropy can be calculaied in terms of presurc
alro'
Difierentiating
PY - rT
PiIYlYilP - ril?
PdY - rilT-YdP
296
Ecat anil Thermodyzamhe
: CyxdT4J!! - ry
ras: (e"4. *) n'- '*
f ,iS
into steam at l(X)oC
f:t,
Cv*
dT vdP
n -TdB: ",
----iT
Also
8^E
._Foo:
Y
f
_T:
T
:?24A c.t/E
: +2700000
373
as:cr$*! #
\..o"Ersmptc 6 31. Calcdate tlrr- increuc in entropg-wt*n f Eam
of iJc d -ly1 ia conoedeil i*to dteom ar 100'C. Spca',{'a l*at of icc
:-.510
2-0:\ latenl heal oJ ice - 80 callg,
- latcnt -heat.ot etun
Integrating
Iar
B3-81 :
lBonboY 1971; Delhi 1973t
<xrlle.
(l) Iucrease in entropy when the tcnPeraturc of I gram of
"ff:+-+l:;g
cptoso
#-+r"&+.
Br-8, : cpx2.3o26xrogroft -
: 5000x540
: 2700000 cal
8II : 2700000 cal
The gain irr entroPY
PY - rT
["'ds :
29?
E:emple 6'30. Calculatc lhe chong-c itt cntropry vfur 5 kg ol
ualer ot 100'C b anuertcd inlo deom al the aaru lcmlreroh.rc.
Heat absorbed by 5 kg of water at l00oC whcn it is convcfted
Substituting the value of PdY ia equation (ii1
But
lknwitynomico
icc iniriass from -10"C to OoC
...(u)
u:l;"+
l*z.zo2orog,.f ...1ur)
t* il?
: *lrrT
Notc. r ir tho onflinary gae co-net1a,nt and
to be takon ia unite of
wort, o" rclnrento tbo apooiflo heat for I gram of-hae
e gas rt oon"teoi-piJerr"e.
T,
: tB tOSo -flf
If Cp reprcsenls-gpm 'nolecular specific heat of a Aas at
: ,rtE x2.3026 k"ft
Br-Br - ctx2-3026 logp
: I x0'5x2'3026 t"et#
coastant pressure and B the universal gas constant, then
*-+x2.3026rosro+ ...(udd)
of ice at fC h ootoqleil itu ualer at the eame teipiroture. 6'(punjah 1963, Delhi lg75l
Heat abaorbed by l0 g^of icc at 0"c when it is converted into
wttcr at OoC : 10X80 :
800 cal
8trI : ggtg
?: OoC ""1
:273K
The gain in entropy
0'01865 callK
(2) Increaec in entropy when I gram of ice at fC is converted
into waier at 0"C.
fU: +
: ,tr : o.re3
cal,rK
(3) Incrcase in enboPy vrhen the teraperature of I g of water
,r: #
=#
:
is rsis;d from OoC to 100'C.
:2.e3 crryB
ou:1;:+
298
Eeal anil Thermodgnamiw
(d)rrtr:50g;?r:273K
mr:50gi?t:353K
f,
: msx2.3026 log,o7;
.
299
Thermodyrnmice
: Ixtx2.3026loCr.#
Let the final temperature of the mixture be ? K
mr sX (T-?rl
- m, s(Tr-T).
50x I x (T -273) : 50x I x(353-7;
?:3lgK
^ il
: 0.312 cal/K
(4) Increase in entropy when I g watcr at
100"C is. converted
into steam at 100"C
,":+
(ii) Change in entropy by 50 g of water when its temperaturc.
rises from 273K to 313 K.
8r/
-T-
540
: ffi
:1.4*T callK
iT dT
*t .l*rT
Total increase in enfopy
: 0.01865+0.293+0.3 l2 + t.+47
: 2'07065 cal/K
/
6'32. one grammorecule oJ a gas expan*a ieothermar-ty '/to ETanple
tourtimea it.g aoturie. catc"iiii;i;;h";;;t;-liri"#ffi, ;,
termc oj the
:50xlXtof" #
..
- 50x2'302Oxf"Srr{l}
gaa conslant.
lYcrk done
: [o' Pav
Herc, thc fve sign indicates gain in entropy.
(ddd) Changc in entropy by 50 g of water when its temperatur(
)Vr
But
PV:R?
DRr
w - Rr
: f6.829 callK
falls from 353 K to 313 K
8.8
tr dT
: T_:
-r l rrT
:50xtxlos"{}f,
Ii:$
- R? r"r"#
Here
- 50x2.3026xlogro ffi
fi:r
:
: _6.023 cal/K
Here, the -ve sign indicates loss in entropy.
Therefore, the total gain in entropy of the system
W
RT x2.3026log1s (4)
H.erc, W anct I are in the units of work
Gain in entropy :
: 6.829-6.023
: 0'806 cal/K
t#
:#:ys#tu3
: I.387 I
v Erampte 6.33.
J
J Example 6'34. Calculate the change in entropg wh,en 50 graynr
of uater al 15e, ia mixeil with 60 grams o/ water at 40"C. Bpecifia
(Ilajosthan 19611
heat ol woler may be aasumeil to be equol to 1.
(t)
caUK
50 grams o! water at 0"C ic- ,nixeil
uilh an
-;n
qual maas oJ water at 83'C. Calculate the resultant ;"criiie
crotroptg.
(punjab 19681
mr:50s
?r : 15+273 : 288 K
mr : l0 grams
. Tt: 40+273: 313 K
300
Eeat ond Themwilynamice
301
Thermoilynamiet
Let the fiual temperature bc f K.
rrlrxdx (T-"r) - mrxE x(?r-f)
50x I x ("-288) : 80x I x(313-?)
: l0o x 2'3026 X log,o (#+)
: a 19.32 cal/K
? : 303.4 K
(iii) Change in entropy uhen l0 gram! of steam at 373K ir
(ii) Change in entropy when the tcmperature of 50 g of water
rises from 288 K to 303.4 K
8E :
: -T
condensed to water at 373 K
t? itT
^t !*, ,-
(-ve sign indicates decrease in entropy)'
(iu) Change in entropy when l0 grams of water at 373 K is
cooled to water at 331'2 K
8E :
tr dr
: +2.602 cal/K.
(ri{ phan-ge- in_ entropy w-hen the remperature of 80 g of water
decreases from 313 K to 303:4 K
-
_r!
: l0x2'3026 log,^
\
r^u f9-^!!
\ 373 )
J
303'4
: Boxlx2'302b y logroTT3: _2.497 caUK
: -l'188 cals/K
Net change in entroPY
Therefore, the net change in the entropy of the system
: t9.32-14.47-l.l88
: a3'662 callK
: +2.602-2.4at
: +0.llc cal/K
Ifence thc net increase in the cntroPy of the system
Hcnce the net increasc in the entropy ol'the system
3'E62 cal/K
:-:
at 20'c is conaerteil into ice at
1d/ Erample 6.36. 1 g oI water c.rytocity
Zonslant. preasure. Heat
ot
for I g o! water ia 4'2
-i|'C anil that of ice ia Z'l.Ile'R. Heat of
tusion of ice at
lls-R
'Colaulate
"'03C
the total changa in the entropy of the
335 J[g.
-
f'.-e'nplg 6.35. 10 g o! eteam at 700"Cis bloun hdo g0 srdmr
-of" watu ! (PC, cotzhineil in a enlorimeter of uater equivatint
I0
grarns. llhe uhok o! thc ateam ia conilenceil. ialcu,late tie increoae
in.the cr*ropry of thc eyatem.
tDelhi (Eo*t) filq
(i)
mr -
fr :
*t )rr-:F
-T
8E : _, tr iy|
: _T
r,-T_
: 0.ff5 caliK
sgstem.
l0g
(l) Changc in entropy when the temPerature of I g of water at
293 K falls to 273 K.
l00oC : 373 K
,rh : 90+10 : 100 g
?t:273K
ds: + :*tl;:+
Let the final temperatue be ? K
l0x5110*lOi.?n-fi : rciif -273)
? : 331'2 K
dT
_ I x u., t273 .r_
]rr,
: 4'2x2'3026 logro (#)
(di) Change in entropy rvhen the temperature of water and
calorimeter rigcs from 273 K to 331.2 K
8fl :
T
: -0'2969 J/K
t? dT
(ir) Change in entropy whcn I g of warer at 273 K is convert'
^, I4.T
ed into ice at 273 K
rcolx''q
'''-&
l0x5,t0
273
-\ r )
caUK
- -14't7
:50xlx2.3026xtoSroffi
,
/ 8E \
_t-.t:_
d^s
d
gH
??r
: -1.227 JiK
: --ffi
-t
: -?
;z
809
Ecat otd llhcrmoilynamia
2?s K(tl,:,:"lgf
in entropy when the tcmperature of I g of ice at
dB: 8.4:
T
^, I,,T
: _0.07834J/K
.6'5f Negative Teaperatures
The specific heat of a substa'ce decreases with increase in
Total change in entropy of the system
: _0.2969 _ 1.227 _0.07834
_ _r.6022{ J/E
Negative sign shows that there is decrease in entropy of the
Ue 6.37.
wa.tet ot Z7g K ia brought
,h, ,f)"yr:.at
system.
in contact
"i"ig"-ii i,irr-rii ,r
ia the chonge in entropy o! (i) the resen)oi,r ond (ii)
(l) Increase in urtro_py wh_en the temperature of 1000 g of
'water is raised from 273 K io 373
K
,u : [?r 8.8
Jr, T
,,sx2.gl26rorro
#l
:1000x1x2.3026 lo1ro!ffi
the tempgrature of {o and
thermodyna-;"r, tt.
-In
meter l/f is rnore significant -e.
than ?.
fr. negative temperatures are hotter than the positive temoe.
ratures and milu_s
(.-0) is . the hottest t.*p.rlt,.ri ;"4-'H;
?:1".
zero ({0) is the coldest
temperature.
The
negative
temperature
is not possible with. the system. as a
. .
whole and is only an exception to the iule that only posiiive t.*o.ratures cxist. The ncgative te-mperarures-are possiutd
lo. iroii.
ble.sub-systems. For all normal purposes the temperat*ir'ar.
""ry ut*n"!s
q. U9.52 Marwell's Thermodynamical Relarlonr
> "L/
From the two laws of thermodynamics, u"**''"tt was abre
(2) (0 Change in entropy of the reservoir,
)
-8.8
'" - -T373
temperature, However, the specific heat does not tend to zero as the
temperature rends to infiniry. This shows that the t.*p"",ii"* ir"i
a + ve sign only.
But recentexperimenrs by Ramsey (1956) have shown that a
part of a system i.e., the nucleus of a so[id,-can-have a
perature. This sub-system is considered isolated from the'
".gatiu;-t..1
marn s/stem (f.e-.,.solid lattice), The specific heat of the sub.system tends to
zero at high temperatur-e. A imall amount of hcat
tcnds to
raise the tcmperature of the system to infinity. It is"rtrgy
posiibte to .aJ
still pore.energy to the.sub-sysrem at infinity and it f6rces the subl
system rnto the negarive temperature region. rt has been shown bv
microscopic statistical anarysii that there-is no distin"tio"-uit*.Ii
-pu-rl'.
positive.
- 312 calK
_ -- l000xtxt00 :
6'50 Zero Point Encrjy
motion and hence possess energy. The eneigy df tli;;;i;J.;';;
absolute zero .emplrature is caiied zero pofit i.;;:-
- I x2.l x2.3026 .r,r(#i )
..,
-268.1 caliK
Negative sign shows decreasc in entropy
to
derive six fundamental_thermodynari.rical reiations. The gtate of a
'qxsrem c1p.be specified by.gny piir of quantities udz. pressur.
ipi
volume (lz.)' temperaturc (?) and entropy (s). In solvihg u,y tt.ii
modynamical problem, the most suitabte' pair is chosin uird the
quantities constituting the pair are takt,n as'inriependent variables.
From the first law of thermodynamics
(2) (d, change t':"#1#;universe
* ,{3.9 cal/K
Therefc-ire, ihe net increase in entropy of the universe
: 43.S caliK
E0t
According to Kinetic theory, the energy of a system at absolute
zero should be zero. It means the moleculei of the' system do not
motion. But according ro rhe modern corrc.pt, cven at
n::::..?"y
absolute
zero, rhe morecures are not compretery depriJed of their
t4 itr
- |!^e_Z!
with a+"-.
heat reacnoir at.iTiK
(l1what
ti
water uhen ite temperalure reaiiee Af g _i_s
Kg
Thermoilynamict
or
8E: dU1gtr
8U : }E*PLV
tE : dLt+pdy
From the second law of thermoCynamics,
,t: t#*
8Ii = '?'ri"9
Heal ond ?hcrmodYtomi*
301
lfhermoilynomirl
ft meaas ilA is z perfect diffcrential
?'a : ?tD
Subnrituting this value of 8II in the 6rst equation
...(i)
iru : Tdg-PdY
Considering 8, O and 7 to bc functions of two independcnt
y" tt lt" x and y can be any two variables out of P, 7'
ardy
(# .(:#), ..
"ari"Ufal-""i
7 and Sl,
.
t
)
\
oOtP
)" -' oyo"
iu - (g:), r"*(H), r,
#;,)rfUr { ., aray
: (#),a,n(Zl).uu
-ff),(#),
uo
atB
(#), au+(fi)y: ,[(#),,,*(#) .uo7
-'l(#),u"*(#). * l
Here r and y cao be any two variables out of P, Y, T and B.
I)crlvrtion of Rclrtloar
(l) TaLing f and 7 as independent variabler and
: [' (H) ,-'(#))*
s: lI
g-Y
+.[, (#).-r(#)J*
#:r,#:r,
'#-0, -'o
{
Comparing the cocficicuu of ilr arrd dy, wG get
(*),-n(#),-'(#), "(i+
('#). :'(#). -r
ardy
(#t(H),-(*).(Y"),
: (f.),(#).-(H ),(%). ...(do)
Substitutirrg thegc valud in cquation (i)
t#)"u"*(#).uu
o l'V
Simplifying,
* : t*),u,*(1#),uo
(#). "'('ii)
Subatituting thcic naluer in equation (io)
(#),:(#),
Diffcrcntiatiag equation (ii) with respect to y and equatioo
(iid) *'ith resPcct to ,
But
#* (#).(#),*,#.
-\:#)"(#),-,#
1
#:(#),(#).*,#
-(t),('#),-,#
about by, changing 7
The chanse iD iDtcrnal energy
-dVbrought
first and f W df later or t'ica
and f whether? is changcd W
rerao is thc same.
azN "od
...(o)
aS:S
...(o0
(2) Tating ? and P as independent variablec and
z-T
l*P
end
dt
-ar
oT'
as
:_1.
AP
oc -"
VP
: o- at
E eal and T hermoilynamice
79. Derive the lollowing reli.tions :
c,_a.-: _.? (
(al
(r)
rds
(c)
GPlaTls
: car*r($),ar
[Delhi 1Hons.), 19781
r--t
:
/rb) Carnot's engine
j$ Carnot's theorem
lDelhi, 19751
(ui) Second Larv of thermodynamics
(oii) Clement and l)esormes' rnethod
(uiri) Ruchhardt's experiment for y
is operated between two reservoirs at
temperatures of 500 _elgine
K and 40b K. If the ."gin.
..*i".", iooo
calories of heat from the source i., ea"h cycle,
calcul:_te (a) the
amount of heat reiected to tlre sink in each ;i;i;; (/,)'";,1'".rf,11.,,.y
of the engine and"(c) the .rork ;;;;';y-r ;;-"- engine in
each cycte in
(i) joules (ii) kilo-Watt hours.
[Ans. (a) I600 catories, (b)2}o/.o, (c) (r) t780joutes,
(ir 4.941x lO n-tWfri
87. A Carnot,s. engine^lvorkilg a-s a refrigerator between
Absolute gas scale
250 K and 300 K receives"r000 caroriei
Jn.ut -"*"""i'"ii.".,
from the reservoir
at the lower temnerature.. (4- cal;;late
*dperature.
the
,.jected to the reservoir-at the 'highe.
(if) carcurate
-i" -.u"^r,
also the amount of
(z) Rankine cycle
1ri; Diesel engine
(a.ii) Steam engine
(riiil Otto cycle
work a"ie
refrigerator.
n-.\riul Entropy is a measure of disorder
(ru) Entropy tends to a maximum
,---'lroi1 Third Law of thermodynamics
(rrii) Absolute zero temPeraturc
y,\*aiiil Entropy of a perfect gas
(rir) Tempcrature'Entropy diagram
fDelhi (flona.l 19771
"y"i.',J'.ip#i" tr,.
[Ans. (4 1200 cal, (irl) 840 joules]
_
(/5w.' ,\ calculate
the depression in the melting poi't -of i".
plod[ced by 2 atmospt e.er ii,-".e]*-of
fr"rr,r... Given latent heat
of ice : B0 cal/g
the specific u'oiul., ol. I gram of
-andcms and
water at OoC are I.091
1.000 cm3 iespectire&.
ice and
@
ri'a the increase ," ,lr",o"i,,ti];r::'Jt-:."J::i::,?
-.-!
whenJhe pressure i, il;;*;.;'-6;;;il:rpheres.
Latent heat of
i.; fr,t""* -.ll'iii"l"ibzz
Hl,tgg. of steam is s+o car/i J"i [Ane.
ss'B4K;;'dir+:cf
/6f,-:
.
(zr) Thermodynamic systcm
UlAf
?
[Ans. (r) 320 K, (i;) 20o/L)
86. A Carnot,s
(iii') Isochoric Process
p$.l
83. Find the eflicienq, ol rhe Carnot's ensline . rvorking
[Ans. 23.640";)
84. Find the efficiency of a Carnot,s engine u.orking betwecn
227'C and 27"C.
tAns. 402,1
85. A Carnot's
rvhose
temperature
of the source is 400
Sngine
K takes 500 calories of,hrat
ar rhit terrf'e.uture and ,"j..t, +ti,l'.urories of heat to the sink. What is the' temperaturq
of the sink
- --
Calculate the efficiency of the engine.
Write short llotes on
r^
n\"-4 ;-{-i) Isothermal Process
/
t--@l Adial,ratic Process
'
311
betrveen 150'C and 50"C.
f+ )"(+# \,
Y
Thermailynamica
Thcrmal Equilibrium
(rzii) Conccpt of Tcmperature
\/.(*oiii) Concept of Heat
1-,@rio) Zeroth Law in Thermodynamics.
(rcu) Phase changes of the second order. [Delhi (Hon's.) 751
' 80. A motor car tyre has a pressure of 3 atmospheres at the
room teinperature of 27"C. If the tyre suddenly,bursts what is
the resulting temPerature ?
[Aas. 218'6 K :'-54'4'C]
81. A quantity of air (Y : l'4) at 27oC is compressed sud'
denly to I of its original volume. Find the final temperature. -
[Aos. 522'3K-249'3"C]
82. A quantity of air at 27'C and atmosPheric pressurc is
suddenly compressed to I of its original volume' Find (i) the final
pressure
and (ii) the final temperature.
'
[Ans. (i) 8'29 atmospheres (rr) 571'1 K : 298'l'C]
"__
^.
,l:"l11,' r. ,,]. *:i;; *ffi;il:l:
@ calcurare
j:ti"]1"T**giffi
r,
lttii'fff
&4rs,L
3#i'i"1,:.,'?"0,,:T'[.i;e,::]
near oI lusron
-i- is 4563 cal/mol and increase"i".*iu*"
on fusion is I8.7 cmr/moi.-
iii :"+:i'{ib, ."g..
[Ans. _0 0697Y6 K or _0.06976.C]
9r. calculate the temperature at which ice wiil freeze
if the
pressure is increased by 135.2
i,oi"-. ;r,." ; il;.
";ilh..*:
i;;u;;. il" ff :,"148i.",;,:o.B::
atmospheric p*rrurr : "l_*1,.;'
06-dynes7"*;.""i.,"nt
of fusion of ice
@..*;t,"0., : r*rn1';;V;;i. ua' heat [Ans.
_ l.o.c1
:
^l
Given that
in specific vorume *(."
-the "fr"ng"
o".
'[-."i';F'G;il?*l".r'?[l',lt'-1
;';;_::',1lr,li."llX?rl
is 1676
i,s
1676-cms.
cms. rut""i
Latent t."t oF-"uo;#;^.i, nr a+o^* e,^ '"'.",?I
lxf _ tt
J,I : 4.2x
4.2 x r0?
l0? ergs/car
ergs/cal .;J;,;;##;i".;;Hii#.3iq;ffi;,:[9.
.;J;r;;##;i".;;;;.rrr.e _
:
;.il;l'il;,
r os rr,rna",,^_2
JAne. I50"Cl