Modern Thermodynamics

- Chapter 1

26

The Law of Corresponding States

Every gas has a characteristic temperature Tc, pressure pc, and volume Vmc which depend on the

molecular size and intermolecular forces. In view of this, one can introduce dimensionless reduced

variables defined by:

Tr = T/Tc Vmr = Vm/Vmc pr = p/pc (1.5.5)

van der Waals showed that if his equation is rewritten in terms of these reduced variables, one obtains the

following "universal equation" (exc 1.18) which is independent of the constants a and b:

pr =

8Tr

3V

r

−1

−

3

Vr2

(1.5.6)

This is a remarkable equation because it implies that gases have corresponding states: at a given value of

reduced volume and reduced temperature, all gases have the same reduced pressure. This statement is

called the law of corresponding states or principle of corresponding states which van der Waals

enunciated in a 1880 publication. Noting that the reduced variables are defined wholly in terms of the

experimentally measured critical constants, pc, Vmc and Tc, he conjectured that the principle has a general

validity, independent of his equation of state. According to the principle of corresponding states then, at a

given Tr and Vmr the reduced pressures, pr, of all gases should be the same (which is not necessarily the

value given by (1.5.6)).

The deviation from ideal gas behavior is usually expressed by defining a compressibility factor

Z =

Vm

Vm,ideal

=

pVm

RT

, which is the ratio between the actual volume of a gas and that of the ideal gas at a

given T and p. Ideal gas behavior corresponds to Z = 1. For real gases, at low pressures and temperatures

it is found that Z 1; but for higher pressures and temperatures, Z 1. It is also found that there is a

particular temperature, called Boyle Temperature, at which Z is nearly 1 and the relationship between p

and V is close to that of an ideal gas (exc 1.11). One way to experimentally verify the law of

corresponding states is to plot Z as a function of reduced pressure pr at a given reduced temperature Tr.

The compressibility factor Z can be written in terms of the reduced variables: Z = (pcVmc/RTc)(prVmr/Tr);

if the value of (pcVmc/RTc) = Zc is the same for all gases (for the van der Waal gas Zc= (pcVmc/RTc) = 3/8

(exc 1.17)) then Z is a function of the reduced variables. Experimental values of Z for different gases

could be plotted as a functions of pr for a fixed Tr. If the law of corresponding states is valid, at a given