605
PROB LE M S
ice ϩ original water. (a) What is the equilibrium temperature of the
system? What are the entropy changes of the water that was originally the ice cube (b) as it melts and (c) as it warms to the equilibrium temperature? (d) What is the entropy change of the original
water as it cools to the equilibrium temperature? (e) What is the
net entropy change of the ice ϩ original water system as it reaches
the equilibrium temperature?
Module 20-2 Entropy in the Real World: Engines
•23 A Carnot engine whose low-temperature reservoir is at 17ЊC
has an efficiency of 40%. By how much should the temperature of
the high-temperature reservoir be increased to increase the efficiency to 50%?
•24 A Carnot engine absorbs 52 kJ as heat and exhausts 36 kJ as
heat in each cycle. Calculate (a) the engine’s efficiency and (b) the
work done per cycle in kilojoules.
•25 A Carnot engine has an efficiency of 22.0%. It operates between constant-temperature reservoirs differing in temperature by
75.0 C . What is the temperature of the (a) lower-temperature and
(b) higher-temperature reservoir?
•26 In a hypothetical nuclear fusion reactor, the fuel is deuterium
gas at a temperature of 7 ϫ 10
8
K. If this gas could be used to operate a Carnot engine with T L ϭ 100 C, what would be the engine’s
efficiency? Take both temperatures to be exact and report your
answer to seven significant figures.
•27
A Carnot engine operates between 235 C and
115 C, absorbing 6.30 10
4
J per cycle at the higher temperature.
(a) What is the efficiency of the engine? (b) How much work per
cycle is this engine capable of performing?
••28 In the first stage of a two-stage Carnot engine, energy is absorbed as heat Q 1 at temperature T 1 , work W 1 is done, and energy
is expelled as heat Q 2 at a lower temperature T 2 . The second stage
absorbs that energy as heat Q 2 , does work W 2 , and expels energy as
heat Q 3 at a still lower temperature T 3 . Prove that the efficiency of
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••16
An 8.0 g ice cube at Ϫ10ЊC is put into a Thermos flask containing 100 cm
3
of water at 20 C. By how much has the entropy of the
cube – water system changed when equilibrium is reached? The specific heat of ice is 2220 J/kg иK.
••17 In Fig. 20-25, where V 23 ϭ
3.00V 1 , n moles of a diatomic ideal
gas are taken through the cycle with
the molecules rotating but not oscillating. What are (a) p 2 /p 1 , (b) p 3 /p 1 ,
and (c) T 3 /T 1 ? For path 1 : 2, what
are (d) W/nRT 1 , (e) Q/nRT 1 ,
(f) ⌬E int /nRT 1 , and (g) ⌬S/nR? For
path 2 : 3, what are (h) W/nRT 1 ,
(i) Q/nRT 1 , (j) ⌬E int /nRT 1 , (k)
⌬S/nR? For path 3 : 1, what are
(l) W/nRT 1 , (m) Q/nRT 1 , (n)
⌬E int /nRT 1 , and (o) ⌬S/nR?
••18
A 2.0 mol sample of an
ideal monatomic gas undergoes
the reversible process shown in
Fig. 20-26. The scale of the vertical
axis is set by T s ϭ 400.0 K and the
scale of the horizontal axis is set by
S s ϭ 20.0 J/K. (a) How much energy
is absorbed as heat by the gas?
(b) What is the change in the internal energy of the gas? (c) How much
work is done by the gas?
•••19 Suppose 1.00 mol of a monatomic ideal gas is taken from initial pressure p 1 and volume V 1 through two steps: (1) an isothermal
expansion to volume 2.00V 1 and (2) a pressure increase to 2.00p 1 at
constant volume. What is Q/p 1 V 1 for (a) step 1 and (b) step 2? What
is W/p 1 V 1 for (c) step 1 and (d) step 2? For the full process, what are
(e) ⌬E int /p 1 V 1 and (f) ⌬S? The gas is returned to its initial state and
again taken to the same final state but now through these two steps:
(1) an isothermal compression to pressure 2.00p 1 and (2) a volume
increase to 2.00V 1 at constant pressure. What is Q/p 1 V 1 for (g) step 1
and (h) step 2? What is W/p 1 V 1 for (i) step 1 and (j) step 2? For the
full process, what are (k) ⌬E int /p 1 V 1 and (l) ⌬S?
•••20 Expand 1.00 mol of an monatomic gas initially at 5.00 kPa
and 600 K from initial volume V i ϭ 1.00 m
3
to final volume V f ϭ
2.00 m
3
. At any instant during the expansion, the pressure p and volume V of the gas are related by p ϭ 5.00 exp[(V i Ϫ V )/a], with p in
kilopascals, V i and V in cubic meters, and a ϭ 1.00 m
3
. What are the
final (a) pressure and (b) temperature of the gas? (c) How much
work is done by the gas during the expansion? (d) What is ⌬S for the
expansion? (Hint: Use two simple reversible processes to find ⌬S.)
•••21
Energy can be removed from water as heat at
and even below the normal freezing point (0.0 C at atmospheric
pressure) without causing the water to freeze; the water is then
said to be supercooled. Suppose a 1.00 g water drop is supercooled until its temperature is that of the surrounding air, which
is at Ϫ5.00ЊC. The drop then suddenly and irreversibly freezes,
transferring energy to the air as heat. What is the entropy change
for the drop? (Hint: Use a three-step reversible process as if the
water were taken through the normal freezing point.) The specific heat of ice is 2220 J/kg иK.
•••22
An insulated Thermos contains 130 g of water at 80.0ЊC.
You put in a 12.0 g ice cube at 0 C to form a system of
Њ
Њ
Њ
2
Isothermal
V 23
V 1 Volume
Pressure
Adiabatic
1
3
Figure 20-25 Problem 17.
Entropy (J/K)
Temperature (K)
0
T s
S s
Figure 20-26 Problem 18.
V 0 , p 0
Volume
Pressure
b
a
c
d
V, p
Figure 20-27 Problem 29.
the engine is (T 1 Ϫ T 3 )/T 1 .
••29
Figure 20-27 shows a reversible cycle through which 1.00 mol
of a monatomic ideal gas is taken.
Assume that p ϭ 2p 0 , V ϭ 2V 0 , p 0 ϭ
1.01 ϫ 10
5
Pa, and V 0 ϭ 0.0225 m
3
.
Calculate (a) the work done during
the cycle, (b) the energy added as
heat during stroke abc, and (c) the
efficiency of the cycle. (d) What is
the efficiency of a Carnot engine operating between the highest and lowest temperatures that occur in
the cycle? (e) Is this greater than or less than the efficiency calculated in (c)?
••30 A 500 W Carnot engine operates between constanttemperature reservoirs at 100 C and 60.0 C.What is the rate at which
energy is (a) taken in by the engine as heat and (b) exhausted by the
engine as heat?
••31 The efficiency of a particular car engine is 25% when the engine does 8.2 kJ of work per cycle. Assume the process is reversible.
What are (a) the energy the engine gains per cycle as heat Q gain
from the fuel combustion and (b) the energy the engine loses per
cycle as heat Q lost ? If a tune-up increases the efficiency to 31%,
what are (c) Q gain and (d) Q lost at the same work value?
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PROB LE M S
ice ϩ original water. (a) What is the equilibrium temperature of the
system? What are the entropy changes of the water that was originally the ice cube (b) as it melts and (c) as it warms to the equilibrium temperature? (d) What is the entropy change of the original
water as it cools to the equilibrium temperature? (e) What is the
net entropy change of the ice ϩ original water system as it reaches
the equilibrium temperature?
Module 20-2 Entropy in the Real World: Engines
•23 A Carnot engine whose low-temperature reservoir is at 17ЊC
has an efficiency of 40%. By how much should the temperature of
the high-temperature reservoir be increased to increase the efficiency to 50%?
•24 A Carnot engine absorbs 52 kJ as heat and exhausts 36 kJ as
heat in each cycle. Calculate (a) the engine’s efficiency and (b) the
work done per cycle in kilojoules.
•25 A Carnot engine has an efficiency of 22.0%. It operates between constant-temperature reservoirs differing in temperature by
75.0 C . What is the temperature of the (a) lower-temperature and
(b) higher-temperature reservoir?
•26 In a hypothetical nuclear fusion reactor, the fuel is deuterium
gas at a temperature of 7 ϫ 10
8
K. If this gas could be used to operate a Carnot engine with T L ϭ 100 C, what would be the engine’s
efficiency? Take both temperatures to be exact and report your
answer to seven significant figures.
•27
A Carnot engine operates between 235 C and
115 C, absorbing 6.30 10
4
J per cycle at the higher temperature.
(a) What is the efficiency of the engine? (b) How much work per
cycle is this engine capable of performing?
••28 In the first stage of a two-stage Carnot engine, energy is absorbed as heat Q 1 at temperature T 1 , work W 1 is done, and energy
is expelled as heat Q 2 at a lower temperature T 2 . The second stage
absorbs that energy as heat Q 2 , does work W 2 , and expels energy as
heat Q 3 at a still lower temperature T 3 . Prove that the efficiency of
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••16
An 8.0 g ice cube at Ϫ10ЊC is put into a Thermos flask containing 100 cm
3
of water at 20 C. By how much has the entropy of the
cube – water system changed when equilibrium is reached? The specific heat of ice is 2220 J/kg иK.
••17 In Fig. 20-25, where V 23 ϭ
3.00V 1 , n moles of a diatomic ideal
gas are taken through the cycle with
the molecules rotating but not oscillating. What are (a) p 2 /p 1 , (b) p 3 /p 1 ,
and (c) T 3 /T 1 ? For path 1 : 2, what
are (d) W/nRT 1 , (e) Q/nRT 1 ,
(f) ⌬E int /nRT 1 , and (g) ⌬S/nR? For
path 2 : 3, what are (h) W/nRT 1 ,
(i) Q/nRT 1 , (j) ⌬E int /nRT 1 , (k)
⌬S/nR? For path 3 : 1, what are
(l) W/nRT 1 , (m) Q/nRT 1 , (n)
⌬E int /nRT 1 , and (o) ⌬S/nR?
••18
A 2.0 mol sample of an
ideal monatomic gas undergoes
the reversible process shown in
Fig. 20-26. The scale of the vertical
axis is set by T s ϭ 400.0 K and the
scale of the horizontal axis is set by
S s ϭ 20.0 J/K. (a) How much energy
is absorbed as heat by the gas?
(b) What is the change in the internal energy of the gas? (c) How much
work is done by the gas?
•••19 Suppose 1.00 mol of a monatomic ideal gas is taken from initial pressure p 1 and volume V 1 through two steps: (1) an isothermal
expansion to volume 2.00V 1 and (2) a pressure increase to 2.00p 1 at
constant volume. What is Q/p 1 V 1 for (a) step 1 and (b) step 2? What
is W/p 1 V 1 for (c) step 1 and (d) step 2? For the full process, what are
(e) ⌬E int /p 1 V 1 and (f) ⌬S? The gas is returned to its initial state and
again taken to the same final state but now through these two steps:
(1) an isothermal compression to pressure 2.00p 1 and (2) a volume
increase to 2.00V 1 at constant pressure. What is Q/p 1 V 1 for (g) step 1
and (h) step 2? What is W/p 1 V 1 for (i) step 1 and (j) step 2? For the
full process, what are (k) ⌬E int /p 1 V 1 and (l) ⌬S?
•••20 Expand 1.00 mol of an monatomic gas initially at 5.00 kPa
and 600 K from initial volume V i ϭ 1.00 m
3
to final volume V f ϭ
2.00 m
3
. At any instant during the expansion, the pressure p and volume V of the gas are related by p ϭ 5.00 exp[(V i Ϫ V )/a], with p in
kilopascals, V i and V in cubic meters, and a ϭ 1.00 m
3
. What are the
final (a) pressure and (b) temperature of the gas? (c) How much
work is done by the gas during the expansion? (d) What is ⌬S for the
expansion? (Hint: Use two simple reversible processes to find ⌬S.)
•••21
Energy can be removed from water as heat at
and even below the normal freezing point (0.0 C at atmospheric
pressure) without causing the water to freeze; the water is then
said to be supercooled. Suppose a 1.00 g water drop is supercooled until its temperature is that of the surrounding air, which
is at Ϫ5.00ЊC. The drop then suddenly and irreversibly freezes,
transferring energy to the air as heat. What is the entropy change
for the drop? (Hint: Use a three-step reversible process as if the
water were taken through the normal freezing point.) The specific heat of ice is 2220 J/kg иK.
•••22
An insulated Thermos contains 130 g of water at 80.0ЊC.
You put in a 12.0 g ice cube at 0 C to form a system of
Њ
Њ
Њ
2
Isothermal
V 23
V 1 Volume
Pressure
Adiabatic
1
3
Figure 20-25 Problem 17.
Entropy (J/K)
Temperature (K)
0
T s
S s
Figure 20-26 Problem 18.
V 0 , p 0
Volume
Pressure
b
a
c
d
V, p
Figure 20-27 Problem 29.
the engine is (T 1 Ϫ T 3 )/T 1 .
••29
Figure 20-27 shows a reversible cycle through which 1.00 mol
of a monatomic ideal gas is taken.
Assume that p ϭ 2p 0 , V ϭ 2V 0 , p 0 ϭ
1.01 ϫ 10
5
Pa, and V 0 ϭ 0.0225 m
3
.
Calculate (a) the work done during
the cycle, (b) the energy added as
heat during stroke abc, and (c) the
efficiency of the cycle. (d) What is
the efficiency of a Carnot engine operating between the highest and lowest temperatures that occur in
the cycle? (e) Is this greater than or less than the efficiency calculated in (c)?
••30 A 500 W Carnot engine operates between constanttemperature reservoirs at 100 C and 60.0 C.What is the rate at which
energy is (a) taken in by the engine as heat and (b) exhausted by the
engine as heat?
••31 The efficiency of a particular car engine is 25% when the engine does 8.2 kJ of work per cycle. Assume the process is reversible.
What are (a) the energy the engine gains per cycle as heat Q gain
from the fuel combustion and (b) the energy the engine loses per
cycle as heat Q lost ? If a tune-up increases the efficiency to 31%,
what are (c) Q gain and (d) Q lost at the same work value?
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