The van der Waal's equation is not the most accurate description of a non-ideal gas. In chemical engineering applications there are many more accurate descriptions. However, the van der Waal's equation has a simple physical interpretation in terms of two parameters, a and b.
a is the intermoleculer attraction due to London dispersion forces
b is the molar volume of the gas when compressed to a solid
Fortunately, we can calculate all of the needed derivatives of the van der Waal's gas to apply the machinery of Maxwell's relations, the thermodynamic equation of state and the two physical properties alpha and kappa.
alpha is the expansion coefficient, which is a measure of how much a material expands as it is heated.
kappa is the isothermal compressibility, which is a measure of how much the molar volume is decreased as pressure is a applied.
We show the derivation of all of these quantities in the following segment.