# Entropy of perfect gas function of internal energy and number of particles [closed]

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?

• 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. Sep 14 at 12:09

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.

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