# Van der waals forces interaction

My slides say:

A gas with low density in which you can consider the attractive energy the sum of the isolate atoms couple $$u=\int^S_\sigma{4\pi\frac{N}{V}}\left(\frac{\beta}{r^6}\right)r^2\mathrm{d}r=-\frac{4\pi N\beta}{V}\int^S_\sigma{r^{-4}} \mathrm{d}r=-\frac{4\pi N\beta}{3V}\left(\frac{1}{s^3}-\frac{1}{\sigma^3}\right)$$

and then continue:

$$=-\frac{4\pi N\beta}{3V{\sigma^3}}$$

where

• $$\sigma$$ is the diameter of the atom;

• $$\frac{N}{V}$$ is density.

Question: How was $$\large{\frac{1}{s^3}}$$ canceled to get from $$\large{\frac{4\pi N\beta}{3V}\left(\frac{1}{s^3}-\frac{1}{\sigma^3}\right)}$$ to $${\large{-\frac{4\pi N\beta}{3V{\sigma^3}}}}\,?$$

• Seems like a low-density gas would tend to have particles far apart, right? If we take that to the limit, then $\lim\limits_{s \to \infty}{\left(\frac{1}{s^3}\right)} = \, ?$ – Nat Feb 9 at 22:50
• Classic case of a bad slice. They should have said, that this is an approximation – user8408080 Feb 10 at 2:07