Review of the Van der Waals theory

Recall that the Van der Waals equation of state was derived earlier by perturbation theory. The unperturbed Hamiltonian describes a system of hard spheres and is given by

and a perturbation of the form

where is the hard-sphere potential given by

and is an arbitrary attractive potential. In the low density limit, we had

and the free energy was determined to be

with

The equation of state takes the form

The critical point is defined by the conditions:

which leads to the following values for the critical pressure, temperature and density:

The critical exponents predicted by the theory are as follows:

1.

The internal energy is given by

from which it can be seen that the heat capacity is

.

2.

The isothermal compressibility can be expressed as

and

so that

3.

By Taylor expanding the equation of state about the critical pressure and density, it is easy to show that

4.

The exponent can be computed using the Maxwell construction (see problem set 11), which attempts to fix the fact that the Van der Waals equation has an unphysical region where . The Maxwell construction is illustrated below:

  
Figure 3:

When the Maxwell construction is carried out, it can be shown that

.

The following table compares the Van der Waals exponents to the experimental critical exponents:

  
Figure 4:

Thus, one sees that the Van der Waals theory is only qualitatively correct, but not quantitatively. It is an example of a mean field theory.