Modern Thermodynamics

- Chapter 1

27

value of Tr and pr, the value of Z must be the same for all gases. The plot shown in Fig. 1.6 indicates that

the validity of the law of corresponding states is fairly general.

Van der Waals equation and the law of corresponding states, however, have their limitations that

van der Waals himself noted in his 1910 Nobel Lecture:

"But closer examination showed me that matters were not so simple. To my surprise I realized that the amount by

which the volume must be reduced is variable, that in extremely dilute state this amount, which I notated b, is

fourfold the molecular volume* – but that this amount decreases with decreasing external volume and gradually falls

to about half. But the law governing this decrease has still not been found."

* Molecular volume is the actual volume of the molecules

(NA4πr3/3

for a mole of spherical molecules of radius r)

van der Waals also noted that the experimental value of Zc=(pcVmc/RTc) for most gases was not 3/8 =

0.375, as predicted by his equation, but was around 0.25 (0.23 for water and 0.29 for Ar). Furthermore, it

became evident that the van der Waals constant a depended on the temperature -- Rudolf Clausius even

suggested that a was inversely proportional to T. Thus, the parameters a and b might themselves be

functions of gas density and temperature. As a result, a number of alternative equations have been

proposed for the description of real gases. For example, engineers and geologists often use the following

equation, known as the Redlich-Kwong equation:

p =

NRT

V − b

−

a

T

N2

V(V − Nb)

=

RT

Vm − b

−

a

T

1

Vm (Vm − b)

(1.5.7)

The constants, a and b, in this equation differ from those in van der Waals equation; they can be related to

the critical constants and they are tabulated just as van der Waals a and b are (exc 1.12). We will discuss

other similar equations used to describe real gases in chapter 6.

The limitation of van der Waals type equations and the principle of corresponding states lies in

the fact that molecular forces and volume are quantified with just two parameters, a and b. As explained

below, two parameters can characterize the forces between small molecules fairly well but larger

molecules require more parameters.