1.3 Real Gases
25
intermolecular forces and the parameter a describes the effect of attractive intermolecular forces. For higher temperatures the second term is relatively unimportant, and the
compression factor will exceed unity for all values of y. For temperatures below the
Boyle temperature the second term becomes relatively more important, and a value of
Z less than unity will occur if y is not too large.
E X A M P L E 1.10
a. Find an expression for the Boyle temperature of a van der Waals gas.
b. Find the value of the Boyle temperature of nitrogen gas as predicted by the van der Waals
equation.
Solution
a. Since y is proportional to P for small values of P, we seek the temperature at which
∂Z
∂y
y0
0
b
(1 − by) 2 −
a
RT
y0
b −
a
RT
where the subscript y 0 indicates the value of y at which the derivative is evaluated.
The Boyle temperature is
T Boyle a/Rb
b. For nitrogen,
T Boyle
0.1408 Pa m 2 mol −1
8.134 J K −1 mol −1
3.913 × 10 −5 m 3 mol −1
433 K
Exercise 1.9
a. Find an expression for the Boyle temperature of a gas obeying the Dieterici equation of state.
b. Find the value of the Boyle temperature of nitrogen according to the Dieterici equation of
state.
c. Find the expression for the molar volume at which Z 1 for the van der Waals gas for a
given temperature below the Boyle temperature. Hint: Find the nonzero value of y in Eq.
(1.3-6) that makes Z 1.
d. Find the value of the molar volume and the pressure at which Z 1 for nitrogen at 273.15 K,
according to the van der Waals equation.
P R O B L E M S
Section 1.3: Real Gases
1.26 For the van der Waals equation of state, obtain formulas
for the partial derivatives (∂P/∂T ) V ,n , (∂P/∂V ) T ,n , and
(∂P/∂n) T ,V .
1.27 For the virial equation of state,
a. Find the expressions for (∂P/∂V ) T ,n and (∂P/∂T ) V ,n .
b. Show that (∂ 2 P/∂V ∂T ) n (∂ 2 P/∂T ∂V ) n .