124
2 Macroscopic Thermodynamics
Rubin equation of state, and compare the result obtained with the experimental
value P expt = 506.6 bar, and with the pressures predicted by the ideal gas, Van
der Waals, and Redlich–Kwong equations of state (as obtained in Problem 16).
18. Show, using Eqs. (2.8.11), that B 2 (T ) = RT B
2 (T ) quite generally, and obtain
an expression for B 3 (T ) in terms of B
2 (T ) and B
3 (T ).
19. Carry out the calculations for the Carnot cycle using a Van der Waals fluid for
the working fluid and show that the final expression for the efficiency η for the
Carnot cycle is the same as that for an ideal gas. (Hint: follow the heat trail.)
20. Use the ideal gas equation of state and the adiabaticity condition that P V γ is
constant, with γ ≡ C P /C V the heat capacity ratio, to verify the values given
for P 2 , P 3 , P 4 , T min , and T low for the CCW Otto cycle in Example 2.5.
21. Obtain expressions for W i→f and Q i→f for the four steps making up a
counterclockwise Carnot cycle using an ideal gas for the working substance.
How are Q CCW
import and Q CCW
export related to Q CW
import and Q CW
export ? Show explicitly
that η CCW defined as η CCW ≡ W CCW (cycle)/Q CCW
import is meaningless and that
ε Carnot
COP must be larger than ε COP for any other refrigeration cycle.
22. We have seen that for a refrigeration cycle in which the goal is to remove heat
(by importation) from the low-temperature heat reservoir and to export heat to
the high-temperature heat reservoir, the appropriate definition of the coefficient
of performance, ε COP , for a refrigeration cycle is ε COP ≡ Q import /W , with W
the work done on the working substance. If, however, the goal of a refrigeration
cycle is the importation of heat into the high-temperature heat reservoir (i.e.,
of Q export from the working fluid), then the appropriate definition of ε COP for
a heat pump must be ε COP ≡ Q export /W . Obtain an expression for ε COP for a
Carnot heat pump in terms of the temperatures T high and T low of the two heat
reservoirs.
23. Obtain the equations for x opt and y opt that arise from the optimization conditions
(2.5.29) and use them to obtain the quadratic equation (2.5.31) that determines
x opt /T high .
24. By examining the smaller root of Eq. (2.5.31), show that x opt /T high is given
by Eq. (2.5.32), and employ your expression for x opt /T high together with
Eq. (2.5.30) to obtain the expression given in Eq. (2.5.32) for y opt /T low . Employ
these results to obtain expressions for T 1,opt and T 2,opt , and show that the
efficiency of the Curzon–Ahlborn endoreversible heat engine is indeed given
by Eq. (2.5.33b).
25. Utilize the relations obtained in Example 2.4 for the ideal gas volumes and
pressures, P i , V i (i = 2, 3, 4), and temperatures T min , T low , T max in terms
of V 1 , P 1 , and T high to sketch plausible P V - and T S-diagrams pertaining
to the clockwise (CW) idealized Otto (engine) cycle that takes an ideal gas
through the closed thermodynamic cycle represented by (V 1 , P 1 , T high )
isochor
−→
(V 1 , P 2 , T max )
adiabat
−→ (5V 1 , P 3 , T low )
isochor
−→ (5V 1 , P 4 , T min )
adiabat
−→ (V 1 , P 1 , T high ).
Employ your T S-diagram to argue that the engine efficiency η Otto is less than
the efficiency η Carnot of a Carnot engine that operates between heat reservoirs
2 Macroscopic Thermodynamics
Rubin equation of state, and compare the result obtained with the experimental
value P expt = 506.6 bar, and with the pressures predicted by the ideal gas, Van
der Waals, and Redlich–Kwong equations of state (as obtained in Problem 16).
18. Show, using Eqs. (2.8.11), that B 2 (T ) = RT B
2 (T ) quite generally, and obtain
an expression for B 3 (T ) in terms of B
2 (T ) and B
3 (T ).
19. Carry out the calculations for the Carnot cycle using a Van der Waals fluid for
the working fluid and show that the final expression for the efficiency η for the
Carnot cycle is the same as that for an ideal gas. (Hint: follow the heat trail.)
20. Use the ideal gas equation of state and the adiabaticity condition that P V γ is
constant, with γ ≡ C P /C V the heat capacity ratio, to verify the values given
for P 2 , P 3 , P 4 , T min , and T low for the CCW Otto cycle in Example 2.5.
21. Obtain expressions for W i→f and Q i→f for the four steps making up a
counterclockwise Carnot cycle using an ideal gas for the working substance.
How are Q CCW
import and Q CCW
export related to Q CW
import and Q CW
export ? Show explicitly
that η CCW defined as η CCW ≡ W CCW (cycle)/Q CCW
import is meaningless and that
ε Carnot
COP must be larger than ε COP for any other refrigeration cycle.
22. We have seen that for a refrigeration cycle in which the goal is to remove heat
(by importation) from the low-temperature heat reservoir and to export heat to
the high-temperature heat reservoir, the appropriate definition of the coefficient
of performance, ε COP , for a refrigeration cycle is ε COP ≡ Q import /W , with W
the work done on the working substance. If, however, the goal of a refrigeration
cycle is the importation of heat into the high-temperature heat reservoir (i.e.,
of Q export from the working fluid), then the appropriate definition of ε COP for
a heat pump must be ε COP ≡ Q export /W . Obtain an expression for ε COP for a
Carnot heat pump in terms of the temperatures T high and T low of the two heat
reservoirs.
23. Obtain the equations for x opt and y opt that arise from the optimization conditions
(2.5.29) and use them to obtain the quadratic equation (2.5.31) that determines
x opt /T high .
24. By examining the smaller root of Eq. (2.5.31), show that x opt /T high is given
by Eq. (2.5.32), and employ your expression for x opt /T high together with
Eq. (2.5.30) to obtain the expression given in Eq. (2.5.32) for y opt /T low . Employ
these results to obtain expressions for T 1,opt and T 2,opt , and show that the
efficiency of the Curzon–Ahlborn endoreversible heat engine is indeed given
by Eq. (2.5.33b).
25. Utilize the relations obtained in Example 2.4 for the ideal gas volumes and
pressures, P i , V i (i = 2, 3, 4), and temperatures T min , T low , T max in terms
of V 1 , P 1 , and T high to sketch plausible P V - and T S-diagrams pertaining
to the clockwise (CW) idealized Otto (engine) cycle that takes an ideal gas
through the closed thermodynamic cycle represented by (V 1 , P 1 , T high )
isochor
−→
(V 1 , P 2 , T max )
adiabat
−→ (5V 1 , P 3 , T low )
isochor
−→ (5V 1 , P 4 , T min )
adiabat
−→ (V 1 , P 1 , T high ).
Employ your T S-diagram to argue that the engine efficiency η Otto is less than
the efficiency η Carnot of a Carnot engine that operates between heat reservoirs