Van der Waals Force

Abstract

We derive the equation of state for an ideal gas in 3D, pV= NkBT, and in 2D, πA= NkBT. We then show how these gas laws are modified for a real gas and derive the result found by van der Waals for 3D, (p+ a/ V2) (V- b) = NkBT. We also derive the 2D version, (π+ a/ A2) (A- b) = NkBT. The correction constant b is due to the fact that the gas atoms take up a finite fraction of the volume, thus reducing the free volume. The factor a, which is of interest here, is due to the attractive force between the atoms, reducing the pressure exerted on the walls of the container. We derive the result for an attractive potential between the atoms of a general power law form. We end by deriving the van der Waals interaction-potential between two atoms, both at zero temperature and at finite temperature. We give numerical results for alkali-metal dimers.