Physics 41 HW Set 11 Chapters 20 and 21
Chapter 20
1. An ideal gas initially at Pi, Vi, and Ti is taken through a cycle as
shown.
(a) Find the net work done on the gas per cycle.
(b) What is the net energy added by heat to the system per cycle?
(c) Obtain a numerical value for the net work done per cycle for 1.00
mol of gas initially at 0°C.
.
.
(a)
The work done during each step of the cycle equals the
negative of the area under that segment of the PV curve.
W  WDA  W AB  WBC  WCD
W  Pi Vi  3Vi   0  3Pi  3Vi  Vi   0  4PV
i i
(b)
The initial and final values of T for the system are equal.
Therefore, Eint  0 and Q W  4PiVi .
(c)
W  4PV
i i  4nRTi  4  1.00  8.314  273   9.08 kJ
2. Find the work done on a fluid that expands as
indicated in the figure.
(a)
W   PdV
W    6.00  106 Pa   2.00  1.00  m 3 
  4.00  106 Pa   3.00  2.00  m 3 
  2.00  10 6 Pa   4.00  3.00  m 3
Wi  f  12.0 MJ
(b)
W f i  12.0 MJ
3. A sample of ideal gas is expanded to twice its original
volume of 1.00 m3 in a quasi-static process for which P = V 2,
with  = 5.00 atm/m6, as shown. How much work is done on the
expanding gas?
f
Wif    PdV
i
The work done on the gas is the negative of the area under the curve
2
P  V between Vi and V f .
f

1
Wif     V 2 dV    V f3  Vi3
3
i

V f  2Vi  2  1.00 m 3   2.00 m 3
3
3
1
Wif    5.00 atm m 6 1.013  10 5 Pa atm   2.00 m 3   1.00 m 3    1.18 MJ


3
4. An ideal gas is carried through a thermodynamic cycle consisting of two
isobaric and two isothermal processes as shown. Show that the net work
done on the gas in the entire cycle is given by
W net  P1 V2  V1 ln
P2
P1
W  W AB  W BC  W CD  W D A
B
C
D
A
C
D
W    PdV   PdV   PdV   PdV
A
B
B
W   nRT1 
A
C
D
A
dV
dV
 P2  dV  nRT2 
 P1  dV
V
V
B
C
D
V 
V 
W   nRT1 ln  B   P2 VC  V B   nRT2 ln  2   P1 V A  V D 
 V1 
 VC 
Now P1V A  P2VB and P2VC  P1VD , so only the logarithmic terms do
not cancel out.
Also,
VB P1
V
P

and 2  2
V1 P2
VC P1
W
P 
P 
P 
P 
P 
 nRT1 ln  1   nRT2 ln  2    nRT1 ln  2   nRT2 ln  2   nR T2  T1  ln  2 
 P2 
 P1 
 P1 
 P1 
 P1 
Moreover P1V2  nRT2 and P1V1  nRT1
W
P 
  P1 V 2  V1  ln  2 
 P1 
4. The inside of a hollow cylinder is maintained at a temperature Ta while the outside is at a lower
temperature, Tb . The wall of the cylinder has a thermal conductivity k. Ignoring end
effects, show that the rate of energy conduction from the inner to the outer surface in
the radial direction is
 T  Tb 
dQ
 2Lk  a

dt
ln b / a
(Suggestions: The temperature gradient is dT/dr. Note that a radial energy current
passes through a concentric cylinder of area 2rL.)
For a cylindrical shell of radius r, height L, and thickness dr, the equation for thermal
conduction,
dQ
dT
becomes
 kA
dt
dx
dQ
dT
 k  2 rL 
dt
dr
Under equilibrium conditions,
dQ
is
dt
constant; therefore,
dT  

Tb
Ta
dQ  1   dr 

 
dt  2 kL   r 
dT  
dQ  1  b dr


dt  2 kL  a r
Tb  Ta  
But
and
dQ  1   b 

 ln  
dt  2 kL   a 
Ta Tb ,
so
dQ 2 kL Ta  Tb 

dt
ln  b a
5. In an insulated vessel, 250 g of ice at 0°C is added to 600 g of water at 18.0°C. (a) What is
the final temperature of the system? (b) How much ice remains when the system reaches
equilibrium? The latent heat of fusion is 3.33x105 J/kg and the specific heat of water is 4186
J/kg C°.
(a)
Since the heat required to melt 250 g of ice at 0°C exceeds the heat required to cool 600 g of
water from 18°C to 0°C, the final temperature of the system (water + ice) must
be 0C .
(b)
Let m represent the mass of ice that melts before the system reaches equilibrium at
0°C.
Qcold  Qhot
mL f  m w cw  0C  Ti 
m  3.33  10 5 J kg     0.600 kg  4 186 J kg C   0C  18.0C 
m  136 g, so the ice rem aining  250 g  136 g  114 g
6. An ideal gas initially at 300 K undergoes an isobaric expansion at 2.50 kPa. If the
volume increases from 1.00 m3 to 3.00 m3 and 12.5 kJ is transferred to the gas by heat,
what are (a) the change in its internal energy and (b) its final temperature?
(a)
Eint  Q  PV  12.5 kJ  2.50 kPa  3.00  1.00  m 3  7.50 kJ
(b)
V1 V 2

T1 T2
V
3.00
T2  2 T1 
 300 K   900 K
V1
1.00
Chapter 21
7. A cylinder contains a mixture of helium and argon gas in equilibrium at 150°C. (a) What is the
average kinetic energy for each type of gas molecule? (b) What is the root-mean-square speed of each
type of molecule?
(a)
K
(b)


3
3
kBT  1.38  1023 J K  423 K   8.76  1021 J
2
2
K
1 2
m vrm s  8.76  1021 J
2
vrm s 
so
1.75  1020 J
m
For helium,
m 
4.00 g m ol
6.02  1023 m olecules m ol
 6.64  1024 g m olecule
m  6.64  1027 kg m olecule
Similarly for argon,
m 
39.9 g m ol
6.02  1023 m olecules m ol
 6.63  1023 g m olecule
m  6.63  1026 kg m olecule
Substituting in (1) above,
we find for helium,
vrm s  1.62 km s
and for argon,
vrm s  514 m s
(1)
8. A 1.00-mol sample of hydrogen gas is heated at constant pressure from 300 K to 420 K. Calculate (a)
the energy transferred to the gas by heat, (b) the increase in its internal energy, and (c) the work done
on the gas.
We us the tabulated values for C P and C V
(a)
Q  nC P T  1.00 m ol 28.8 J m ol K   420  300 K  3.46 kJ
(b)
Eint  nCV T  1.00 m ol 20.4 J m ol K  120 K   2.45 kJ
(c)
W  Q  Eint  3.46 kJ 2.45 kJ 1.01 kJ
9. A 2.00-mol sample of a diatomic ideal gas expands slowly and adiabatically from a pressure of 5.00
atm and a volume of 12.0 L to a final volume of 30.0 L. (a) What is the final pressure of the gas? (b)
What are the initial and final temperatures? (c) Find Q, W, and Eint.
(a)


PV
i i  PfV f

1.40
V 
 12.0
Pf  Pi i   5.00 atm 
 1.39 atm

 30.0
 Vf
(b)
Ti 
Tf 
(c)


5.00 1.013  105 Pa 12.0  103 m
PV
i i

nR
2.00 m ol 8.314 J m ol K 
PfV f
nR



3
1.39 1.013  105 Pa 30.0  103 m
2.00 m ol 8.314 J m ol K 

3

365 K
253 K
The process is adiabatic: Q  0
  1.40 
C P R  CV
5

, CV  R
CV
CV
2
5

Eint  nCV T  2.00 m ol  8.314 J m ol K   253 K  365 K   4.66 kJ
2

W  Eint  Q  4.66 kJ 0  4.66 kJ
10. Fifteen identical particles have various speeds: one has a speed of 2.00 m/s; two have speeds of
3.00 m/s; three have speeds of 5.00 m/s; four have speeds of
7.00 m/s; three have speeds of 9.00 m/s; and two have speeds of 12.0 m/s. Find (a) the average speed,
(b) the rms speed, and (c) the most probable speed of these particles.
(a)
vav 
(b)
 nivi 
N
v 
2
av
1
1 2  2 3  3 5  4 7  3 9  212   6.80 m s
15 

 nivi2  54.9 m 2
N
so vrm s 
(c)
v 
2
s2
 54.9  7.41 m s
av
vm p  7.00 m s
11. In an ultrahigh vacuum system, the pressure is measured to be 1.00  10–10 torr (where 1 torr = 133
Pa). Assuming the molecular diameter is 3.00  10–10 m, the average molecular speed is 500 m/s, and
the temperature is 300 K, find (a) the number of molecules in a volume of 1.00 m 3, (b) the mean free
path of the molecules, and (c) the collision frequency.
(a)
 N 
PV N A
so that
PV  
RT and N 
RT
 N A 
1.00  10  1331.00 6.02 10  
10
N 
(b)

23
 8.314 300
1
nV  d2 21 2

V
N  d2 21 2
 779 km
(c)
f
v
 6.42  104 s1

 3.21 10
12
3.21 1012 m olecules
1.00 m
 
3
m olecules  3.00  1010 m

2
 21 2