Physical Chemistry , 1st ed.

At very low pressures (which is one of the conditions under which real gases
might behave somewhat like ideal gases), the volume of the gas system will be
large (from Boyle’s law). That means that the fraction b/Vwill be very small,
and so using the Taylor-series approximation 1/(1 x) (1 x)^1  1 
xx^2   for x , we can substitute for 1/(1 b/V) in the last expres-
sion to get

Z 1 

V

b



V

b



2



RT

a

V

  

where successive terms are neglected. The two terms with Vto the first power
in their denominator can be combined to get

Z 1 b

R

a

T



V

1



V

b



2

  

for the compressibility in terms of the van der Waals equation of state. Compare
this to the virial equation of state in equation 1.17:

Z

p

R

V

T

 1 

V

B





V

C

 2   

By performing a power series term-by-term comparison, we can show a cor-
respondence between the coefficients on the 1/Vterm:

Bb

R

a

T

 (1.22)

We have therefore established a simple relationship between the van der Waals
constants aand band the second virial coefficient B. Further, since at the Boyle
temperature TBthe second virial coefficient Bis zero:

0 b

R

a

TB



we can rearrange to find that

TB

b

a

R

 (1.23)

This expression shows that all gases whose behavior can be described using
the van der Waals equation of state (and most gases can, at least in certain
regions of pressure and temperature) have a finite TBand should behave
like an ideal gas at that temperature, if higher virial equation terms are neg-
ligible.

Example 1.6

Estimate the Boyle temperature of the following. Use the values ofaand b

from Table 1.6.

a.He

b.Methane, CH 4

Solution

a.For He,a0.03508 atmL^2 /mol^2 and b0.0237 L/mol. The proper nu-

merical value for Rwill be necessary to cancel out the right units; in this case,

we will use R0.08205 Latm/molK. We can therefore set up

1.6 Nonideal Gases 15