126
2 Macroscopic Thermodynamics
ln |f | = ln
RT
V − b
+
b
V − b
−
2a
RT V
.
32. For CO 2 considered as a Van der Waals fluid, plot the fugacity as a function of
pressure for pressures between 0 and 90 bar at temperature T = 300 K. The
Van der Waals a and b parameters for CO 2 are a = 3.6551 dm
6 bar mol
−2 and
b = 0.042816 dm
3 mol
−1 .
33. Obtain expressions for the thermal expansion coefficient α, the isothermal
compressibility coefficient κ T , and dU for a Van der Waals fluid. Show that
dU for the Van der Waals fluid reduces to the expression for the ideal gas in the
ideal gas limit (a = b = 0).
34. The equation
(b
2 P i + a +
3
2 bRT i )
2
= 8a bRT i
is the molar version of Eq. (2.8.35), and is the form normally encountered in
thermodynamics texts. Show that the first derivative of the inversion pressure
P i with respect to the inversion temperature T i for a Van der Waals fluid is given
by
dP i
dT i
=
1
2b
2
⎡
⎣
8a bR
T i
− 3bR
⎤
⎦ .
Verify that for P i,max = a/(3b
2 ), the inversion curve equation has only one
solution, namely, T i = 8a/(9Rb), and that the approximate result T i
2a/(Rb) is obtained formally as the upper inversion temperature at P i = 0.
35. Employ the inversion equation (2.8.35) to compute the upper and lower
inversion temperatures for N 2 treated as a Van der Waals fluid at a pressure of
100 bar, given that a = 1.3661 dm
6 bar mol
−2 and b = 0.038577 dm
3 mol
−1 .
36. Employ the result from Problem 29, together with the Van der Waals equation
of state in reduced form to plot γ against P R for T R = 1.20, T R = 2.00, and
T R = 3.00.
37. Obtain an expression for G(ξ ) and the affinity A(ξ ) for the generic chemical
reaction ν A A + ν B B → ν C C + ν D D.
38. By considering explicitly the reaction for the formation of water from molecular
hydrogen and oxygen, namely,
2H 2 (g) + O 2 (g) 2H 2 O(g) ,
obtain expressions for the Gibbs energy, G(ξ ), and the affinity, A(ξ ), as functions of the extent of reaction, ξ . Show further, that the equilibrium constant,
2 Macroscopic Thermodynamics
ln |f | = ln
RT
V − b
+
b
V − b
−
2a
RT V
.
32. For CO 2 considered as a Van der Waals fluid, plot the fugacity as a function of
pressure for pressures between 0 and 90 bar at temperature T = 300 K. The
Van der Waals a and b parameters for CO 2 are a = 3.6551 dm
6 bar mol
−2 and
b = 0.042816 dm
3 mol
−1 .
33. Obtain expressions for the thermal expansion coefficient α, the isothermal
compressibility coefficient κ T , and dU for a Van der Waals fluid. Show that
dU for the Van der Waals fluid reduces to the expression for the ideal gas in the
ideal gas limit (a = b = 0).
34. The equation
(b
2 P i + a +
3
2 bRT i )
2
= 8a bRT i
is the molar version of Eq. (2.8.35), and is the form normally encountered in
thermodynamics texts. Show that the first derivative of the inversion pressure
P i with respect to the inversion temperature T i for a Van der Waals fluid is given
by
dP i
dT i
=
1
2b
2
⎡
⎣
8a bR
T i
− 3bR
⎤
⎦ .
Verify that for P i,max = a/(3b
2 ), the inversion curve equation has only one
solution, namely, T i = 8a/(9Rb), and that the approximate result T i
2a/(Rb) is obtained formally as the upper inversion temperature at P i = 0.
35. Employ the inversion equation (2.8.35) to compute the upper and lower
inversion temperatures for N 2 treated as a Van der Waals fluid at a pressure of
100 bar, given that a = 1.3661 dm
6 bar mol
−2 and b = 0.038577 dm
3 mol
−1 .
36. Employ the result from Problem 29, together with the Van der Waals equation
of state in reduced form to plot γ against P R for T R = 1.20, T R = 2.00, and
T R = 3.00.
37. Obtain an expression for G(ξ ) and the affinity A(ξ ) for the generic chemical
reaction ν A A + ν B B → ν C C + ν D D.
38. By considering explicitly the reaction for the formation of water from molecular
hydrogen and oxygen, namely,
2H 2 (g) + O 2 (g) 2H 2 O(g) ,
obtain expressions for the Gibbs energy, G(ξ ), and the affinity, A(ξ ), as functions of the extent of reaction, ξ . Show further, that the equilibrium constant,