van der Waals equation of state derivation

This attempts to account for some of the assumptions of the perfect gas equation of state. It is a good example of scientific thinking about a mathematical model, "model building" in other words. a and b are empirical parameters, but they can also be estimated.

First we take the ideal gas equation and try to account for volume taken up by the molecules themselves, by subtracting nb from the volume, where b is a constant.

This approximates molecules as hard spheres. So each molecule has a sphere of exclusion around it of 2r, anything less and the spheres would penetrated eachother.

This gives a total excluded volume of (4/3)π(2r)³ per molecule, which is 8V_molecule.

This number is then halved "to prevent overcounting" to give 4V_molecule. I have never been able to understand why this step is done.

But the end result is that b is roughly 4VN_A. Since many molecules are quite soft, this number is typically the upper limit of empirical measurements.

The term on the right is because attractive forces will reduce both the rate of collisions of the molecules with the side of the container, and reduce the speed of these collisions. These forces are found to act with a strength proportional to the square of the molar concentration (n / V) of the molecules.

Remember that this is overwhelming an empirical law. And the justification given for it is only vague. There are other more satisfactory ones which you may come across later.