Problems
701
Carnot engine is
In 1816 Robert Stirling, a Scottish clergyman,
patented the Stirling engine, which has found a wide variety
of applications ever since. Fuel is burned externally to
warm one of the engine’s two cylinders. A fixed quantity of
inert gas moves cyclically between the cylinders, expanding
in the hot one and contracting in the cold one. Figure
P22.57 represents a model for its thermodynamic cycle.
Consider n mol of an ideal monatomic gas being taken
once through the cycle, consisting of two isothermal
processes at temperatures 3T
i
and T
i
and two constant-
volume processes. Determine, in terms of n, R, and T
i
,
(a) the net energy transferred by heat to the gas and
(b) the efficiency of the engine. A Stirling engine is easier
to manufacture than an internal combustion engine or a
turbine. It can run on burning garbage. It can run on the
energy of sunlight and produce no material exhaust.
57.
W
eng
!
mc(T
h
1/2
"
T
c
1/2
)
2
58.
An electric power plant has an overall efficiency of 15.0%.
The plant is to deliver 150 MW of power to a city, and its
turbines use coal as the fuel. The burning coal produces
steam that drives the turbines. This steam is then con-
densed to water at 25.0°C by passing it through cooling
coils in contact with river water. (a) How many metric tons
of coal does the plant consume each day (1 metric
ton ! 10
3
kg)? (b) What is the total cost of the fuel per
year if the delivered price is $8.00/metric ton? (c) If the
river water is delivered at 20.0°C, at what minimum rate
must it flow over the cooling coils in order that its temper-
ature not exceed 25.0°C? (Note: The heat of combustion of
coal is 33.0 kJ/g.)
59.
A power plant, having a Carnot efficiency, produces
1 000 MW of electrical power from turbines that take in
steam at 500 K and reject water at 300 K into a flowing river.
The water downstream is 6.00 K warmer due to the output of
the power plant. Determine the flow rate of the river.
60.
A power plant, having a Carnot efficiency, produces elec-
tric power ! from turbines that take in energy from steam
at temperature T
h
and discharge energy at temperature T
c
through a heat exchanger into a flowing river. The water
downstream is warmer by %T due to the output of the
power plant. Determine the flow rate of the river.
61.
An athlete whose mass is 70.0 kg drinks 16 oz (453.6 g) of
refrigerated water. The water is at a temperature of 35.0°F.
(a) Ignoring the temperature change of the body that re-
sults from the water intake (so that the body is regarded as
a reservoir always at 98.6°F), find the entropy increase of
the entire system. (b) What If ? Assume that the entire
body is cooled by the drink and that the average specific
heat of a person is equal to the specific heat of liquid
water. Ignoring any other energy transfers by heat and any
metabolic energy release, find the athlete’s temperature
after she drinks the cold water, given an initial body
temperature of 98.6°F. Under these assumptions, what is
the entropy increase of the entire system? Compare this
result with the one you obtained in part (a).
62.
A 1.00-mol sample of an ideal monatomic gas is taken
through the cycle shown in Figure P22.62. The process
A : B is a reversible isothermal expansion. Calculate
(a) the net work done by the gas, (b) the energy added to
the gas by heat, (c) the energy exhausted from the gas by
heat, and (d) the efficiency of the cycle.
63.
A biology laboratory is maintained at a constant tempera-
ture of 7.00°C by an air conditioner, which is vented to the
air outside. On a typical hot summer day the outside
temperature is 27.0°C and the air conditioning unit emits
energy to the outside at a rate of 10.0 kW. Model the unit
as having a coefficient of performance equal to 40.0% of
the coefficient of performance of an ideal Carnot device.
(a) At what rate does the air conditioner remove energy
from the laboratory? (b) Calculate the power required for
the work input. (c) Find the change in entropy produced
by the air conditioner in 1.00 h. (d) What If ? The
outside temperature increases to 32.0)C. Find the
fractional change in the coefficient of performance of the
air conditioner.
64.
A 1.00-mol sample of an ideal gas expands isothermally,
doubling in volume. (a) Show that the work it does in ex-
Isothermal
processes
P
V
V
i
2V
i
T
i
3T
i
Figure P22.57
5
Isothermal
process
1
10
50
V(liters)
B
C
A
P(atm)
Figure P22.62