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In a statistical physics lecture, I found an equation describing the entropy $S$ of an ideal gas and it is said that equation can be obtained thanks to elementary thermodynamics formulae. The equation is the following: $$ S = N\left(c_\mathrm{v}\ln\left(\frac{U}{N}\right) + k_\mathrm{b}\ln\left(\frac{V}{N}\right)+ constant\right)$$ With

  • $N$ the particle's number,
  • $c_\mathrm{v}$ the heat capacity at constant volume ($dU = Nc_\mathrm{v}dT$) and $c_\mathrm{v} = \frac{3}{2}k_\mathrm{b}$ for monoatomic perfect gas, supposed T-independent
  • $k_\mathrm{b}$ the Boltzmann constant
  • and $V$ the volume of the gas

The problem is I cannot obtain that equation. I succeed in getting $\Delta S$ as a function of $(P,T)$, $(T,V)$, or $(P,V)$ but not as a function of $U$ and $N$. Could you help me?

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  • $\begingroup$ It has to be U, V, and N, not just U and N. If U an d N are constant, S can still increase if V is increased. $\endgroup$ Sep 14 at 12:09
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I think this is just the Sackur-Tetrode equation in disguise.

Taking the form from wikipedia we can quickly rearrange to your form:

$$ \frac{S}{K_B N} = \ln \left[\frac{V}{N}\left(\frac{4\pi m}{3 h^2}\frac{U}{N} \right)^{3/2} \right] + \frac{5}{2} $$

or,

$$ S = N \left(K_B \ln \frac{V}{N} + \frac{3 K_B}{2} \ln \frac{U}{N} + \mathrm{constants} \right) $$

which is the form you give.

The full derivation is the resolution of the Gibbs Paradox, and the linked wikipedia page contains the derivation.

However, if you've already got a form containing T rather than U, you're probably almost there. For a monoatomic ideal gas you have the relation:

$$ U = - \frac{\partial}{\partial \beta} \ln Z = \frac{3}{2}K_B T $$

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