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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 ${\large{-\frac{4\pi N\beta}{3V{\sigma^3}}}}\,?$

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

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