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I know that the entropy of an ideal gas is given by the Sackur-Tetrode equation, but is there also a way to take into account that even the ideal gas will acquire some volume $v_0$? Or is it then just the correct way to look at $S(E,V-v_0,N)$ in order to describe this?

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If you want the gas particles to have extended size then the first approximation would be to use the hard sphere gas model. Assume you have $N$ non-interacting hard spheres each of volume $v_0$ in a box of volume $V$. Then the entropy can be calculated as $$S = Nk_B \ln \left[\frac{e}{N}\left(V - \frac{Nv_0}{2}\right)\left(\frac{4\pi mEe}{3Nh^2}\right)^{3/2}\right].$$

C.f. Kardar "Statistical Physics of Particles" problem 4.4.

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If you consider a small volume $v_0$ then the gas is no longer ideal by definition. If you modify the Sackur-Tetride to take this into account (with a hard sphere model for example as suggested by FenderLesPaul) you obtain a Van der Waals equation of state.

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