Physical Chemistry , 1st ed.



T

p



V, n



V

nR

nb



We can also determine the volume derivative of pressure at constant tempera-
ture and amount



V

p



T, n



(V

n



RT

nb)^2



2

V

an

3

2



Both terms on the right side survive this differentiation. Compare this to the
equivalent expression from the ideal gas law. Although it is a little more com-
plicated, it agrees better with experimental results for most gases. The deriva-
tions of equations of state are usually a balance between simplicity and ap-
plicability. Very simple equations of state are often inaccurate for many real
situations, but to accurately describe the behavior of a real gas often requires
complicated expressions with many parameters. An extreme example is cited
in the classic text by Lewis and Randall (Thermodynamics, 2nd ed., revised by
K. S. Pitzer and L. Brewer, McGraw-Hill, New York, 1961) as

p

RTdB 0 RTA 0 

C

T

^0

d

(^2) (bRTa)d (^3) a d (^6) c
T
d
2
2
(1 
d^2 )e
d
2
where dis the density and A 0 ,B 0 ,C 0 ,a,b,c, , and are experimentally de-
termined parameters. (This equation of state is applicable to gases cooled or
pressurized to near the liquid state.) “The equation ...yields reasonable agree-
ment, but it is so complex as to discourage its general use.” Maybe not in this
age of computers, but this equation of state is daunting, nonetheless.
The state variables of a gas can be represented diagrammatically. Figure 1.10
shows an example of this sort of representation, determined from the equation
of state.
1.6 Nonideal Gases 17
(V/p)T 0
p 1 p
p 0
T 0 T 2
p 2
(V/T)p 0
(V/p)T 2
V
T
(V/T)p 2
Figure 1.10 The surface that is plotted represents the combination ofp,V, and Tvalues that
are allowed for an ideal gas according to the ideal gas law. The slope in each dimension repre-
sents a different partial derivative. (Adapted with permission from G. K. Vemulapalli,Physical
Chemistry,Prentice-Hall, Upper Saddle River, N.J., 1993.)