2.9 Problems for This Chapter
125
at temperatures T max and T min . How is your result supported by the outcomes
from this particular idealized Otto engine cycle?
26. By carrying out expansions of the type that we used for the Van der Waals
equation in Sect. 2.8.2, show that the second and third virial coefficients for a
Redlich–Kwong gas are given by
B 2 (T ) = B −
A
RT
3
2
; B 3 (T ) = B
2
+
AB
RT
3
2
,
in terms of the A and B Redlich–Kwong constants.
27. Calculate the P and T values for which H 2 (g) is in a corresponding state with
Xe(g) at a temperature of 450.0 K and a pressure of 85.0 bar. You will need the
critical temperatures and pressures for these gases: T c (Xe) = 289.74 K, P c (Xe)
= 58.40 bar; T c (H 2 ) = 32.94 K, P c (H 2 ) = 12.93 bar.
28. The compressibility Z for a Van der Waals gas has a value Z = 1.00084 at
temperature T = 298.15 K and pressure P = 1 bar. The Boyle temperature for
the gas is T B = 125 K. Use Z = 1 + B VdW
2
(T )/V and approximate V using
the ideal gas law (this is reasonable, as P = 1 bar is a pressure at which many
gases behave essentially as ideal gases). Estimate values for the Van der Waals
parameters a and b.
29. Using Eq. (2.8.17b) the fugacity coefficient can be expressed in terms of the
molar volume V as
RT ln γ =
P
P ideal
V −
RT
P
dP .
Show that ln γ may also be given as
RT ln γ = P V − RT −
V
V ideal
P dV − RT ln
P
P ideal
.
Show further that for a Van der Waals fluid this expression can be evaluated to
give
ln γ =
b
V − b
−
2a
RT V
− ln
1 −
a(V − b)
RT V
2
,
as both a and b are small in comparison with V ideal for P 1 bar.
30. Employ the expressions for the Van der Waals parameters a and b in terms of
the critical constants to obtain the reduced form of the equation for ln γ .
31. By noting that for a pure substance the molar Gibbs energy is the chemical
potential, i.e., G(T , P ) ≡ μ(T , P ), show that the fugacity for a Van der Waals
fluid may be obtained from
125
at temperatures T max and T min . How is your result supported by the outcomes
from this particular idealized Otto engine cycle?
26. By carrying out expansions of the type that we used for the Van der Waals
equation in Sect. 2.8.2, show that the second and third virial coefficients for a
Redlich–Kwong gas are given by
B 2 (T ) = B −
A
RT
3
2
; B 3 (T ) = B
2
+
AB
RT
3
2
,
in terms of the A and B Redlich–Kwong constants.
27. Calculate the P and T values for which H 2 (g) is in a corresponding state with
Xe(g) at a temperature of 450.0 K and a pressure of 85.0 bar. You will need the
critical temperatures and pressures for these gases: T c (Xe) = 289.74 K, P c (Xe)
= 58.40 bar; T c (H 2 ) = 32.94 K, P c (H 2 ) = 12.93 bar.
28. The compressibility Z for a Van der Waals gas has a value Z = 1.00084 at
temperature T = 298.15 K and pressure P = 1 bar. The Boyle temperature for
the gas is T B = 125 K. Use Z = 1 + B VdW
2
(T )/V and approximate V using
the ideal gas law (this is reasonable, as P = 1 bar is a pressure at which many
gases behave essentially as ideal gases). Estimate values for the Van der Waals
parameters a and b.
29. Using Eq. (2.8.17b) the fugacity coefficient can be expressed in terms of the
molar volume V as
RT ln γ =
P
P ideal
V −
RT
P
dP .
Show that ln γ may also be given as
RT ln γ = P V − RT −
V
V ideal
P dV − RT ln
P
P ideal
.
Show further that for a Van der Waals fluid this expression can be evaluated to
give
ln γ =
b
V − b
−
2a
RT V
− ln
1 −
a(V − b)
RT V
2
,
as both a and b are small in comparison with V ideal for P 1 bar.
30. Employ the expressions for the Van der Waals parameters a and b in terms of
the critical constants to obtain the reduced form of the equation for ln γ .
31. By noting that for a pure substance the molar Gibbs energy is the chemical
potential, i.e., G(T , P ) ≡ μ(T , P ), show that the fugacity for a Van der Waals
fluid may be obtained from