Van der Waals gas

In the ideal gas law we assumed that gas is made up of non-interacting particles. The Van der Waal gas is a slightly better model of gas that takes into account short and long range particle interactions.

At very small length scales, molecules repel each other. We can approximate this to say that each molecule has some volume $b$ it takes up in space, and that molecules don’t intersect. Effectively this decreases the volume available to a gas particle by $Nb$ , since each of the $N$ other particles takes up some volume. We thus add the correction $V \to V - Nb$ .

At longer length scales, atoms attract each other. We can think of this attraction as being a potential proportional to the number of particle-particle interactions, and inversely proportional to distance.

The potential felt by a single atom is $\sim N/V$ , so we write the total potential $\mathcal U = \frac{aN^2}{V}.$ We thus say

\begin{align*} \Delta P &= - \frac{\partial\mathcal U}{\partial V} = -\frac{\partial}{\partial V}\left(\frac{aN^2}{V}\right) \\ &= -\frac{aN^2}{V^2} \\ P &\to P -\frac{aN^2}{V^2} \\ P &= \frac{Nk_BT}{V-Nb} - \frac{aN^2}{V^2} \end{align*}

This gives us the Van der Waals equation of state.

Nk_bT = \left(P + \frac{aN^2}{V^2}\right)\big(V - Nb \big).