Physical Chemistry Third Edition

1.3 Real Gases 25

intermolecular forces and the parameteradescribes the effect of attractive intermolec-

ular forces. For higher temperatures the second term is relatively unimportant, and the

compression factor will exceed unity for all values ofy. For temperatures below the

Boyle temperature the second term becomes relatively more important, and a value of

Zless than unity will occur ifyis not too large.

EXAMPLE1.10

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.Sinceyis proportional toPfor small values ofP, we seek the temperature at which

(

∂Z

∂y

)∣∣

∣

∣y 0 ^0 

(

b

( 1 −by)^2

−

a

RT

)∣∣

∣

∣

y 0

b−

a

RT

where the subscripty0 indicates the value ofyat which the derivative is evaluated.

The Boyle temperature is

TBoylea/Rb

b.For nitrogen,

TBoyle

0 .1408 Pa m^2 mol−^1

(

8 .134JK−^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 whichZ1 for the van der Waals gas for a

given temperature below the Boyle temperature. Hint: Find the nonzero value ofyin Eq.

(1.3-6) that makesZ1.

d.Find the value of the molar volume and the pressure at whichZ1 for nitrogen at 273.15 K,

according to the van der Waals equation.

PROBLEMS

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,nand (∂P/∂T)V,n.

b. Show that (∂^2 P/∂V ∂T)n(∂^2 P/∂T ∂V)n.