# Van der Waals equation

The Van der Waals equation can be obtained considering the atoms of a gas as hard spheres. In this case, in fact, the mean field has the form:

Therefore, we will have:

## Critical point of Van der Waals equation[edit | edit source]

The behaviour of the Van der Waals isotherms is the following:

As we can see this changes with the temperature and resembles that of real isotherms (see Phase transitions and phase diagrams); however, Van der Waals isotherms are always analytic and have a non physical behaviour in certain regions of plane, called *spinodal curves*, if : for some values of we have , which is physically impossible. This is a consequence of the roughness of the approximation we have made, since it can be shown that it doesn't ensure that the equilibrium state of the system globally minimizes the Gibbs free energy. As we will shortly see, however, this problem can be solved "by hand" with *Maxwell's equal area rule,* or *Maxwell's Construction*

Let us now see how to determine the critical point of a system obeying Van der Waals equation.
First of all, from the representation of the isotherms we can see that the critical point is a flex for the critical isotherm (i.e. the one with ); in other words, we can determine the critical point from the equations:

Equivalently, we can note that the equation is cubic in . In fact, we can rewrite the Van der Waals equation as:

This model has also an interesting property, since it predicts that:

*universal number*, independent of and and so of the particular fluid considered. Experimentally this ratio is approximately 0.29 for Argon, 0.23 for water and 0.31 for . Therefore, even if it is very rough, this model leads to reasonable conclusions.

## Law of corresponding states[edit | edit source]

We can also rewrite Van der Waals equation in a dimensionless form, rescaling the thermodynamic quantities of the system. In particular, defining:

*when rescaled by their critical thermodynamic properties, all fluids obey the same state equation*. This is the

*law of corresponding states*that we have already encountered in Critical exponents and universality. This is a form of universality, but substantially different from the one we have seen until now, which applies only in the neighbourhood of a critical point: in fact, the law of corresponding states applies

*everywhere*on the phase diagram. It can even be shown that this law is a consequence of dimensional analysis, and is more general than what might seem: experimentally the law of corresponding states is well satisfied also by fluids which do not obey Van der Waals equation.

## Maxwell's equal area rule[edit | edit source]

As we have previously anticipated, *Maxwell's equal area rule* is a method to "manually" remove the unphysical regions of Van der Waals isotherms.

From Phase coexistence and general properties of phase transitions we know that at the coexistence of two phases the chemical potentials and the pressures of the two phases must be equal; furthermore, from Thermodynamic potentials we also know that the chemical potential is the Gibbs free energy per particle, namely , and in general we have also:

## Critical behaviour[edit | edit source]

Let us now study the behaviour of systems obeying Van der Waals equations near the critical point, computing one of the critical exponents.

### Exponent beta[edit | edit source]

This exponent^{[1]} can be computed from the shape of the coexistence curve for ; this can be done using the law of corresponding states. In fact, defining:

Therefore, from Maxwell's equal area rule we have:

^{[2]}:

In fact, if we compute all the other critical exponent, we get exactly:

## A more precise approximation[edit | edit source]

We have seen that the problem of Van der Waals equation comes from the rough approximation that we have made in . A better formulation of Van der Waals mean field theory can be done using the potential:

- ↑ Remember that by definition describes the behaviour of the order parameter in the neighbourhood of the critical temperature, so we will have .
- ↑ If we didn't neglected the term of the expansion of , we would have found:
Again, the terms linear in can be neglected since is small (and , are just numbers).