# Partition function, Ideal gas, Entropy

## Homework Statement

For a diatomic gas near room temperature, the internal partion function is simply the rotational partition function multiplied by degeneracy $Z_e$ of the electronic ground state.
Show that the entropy in this case is
$S = Nk\left[ \ln \left( \frac{VZ_eZ_\text{rot}}{Nv_Q}\right) + \frac{7}{2}\right].$

## Homework Equations

The entropy of an ideal gas is given by
$S = -\left( \frac{\partial F}{\partial T}\right)_{V,N} = Nk\left[ \ln \left( \frac{V}{Nv_q}\right) + \frac{5}{2} \right]-\frac{\partial F_\text{int}}{\partial T}.$

The rotational partion function should be
$Z_\text{rot} = \frac{kT}{2B}$
where $B$ is the rotational constant.

Helmholtz free energy
$F = -kT \ln Z$.

## The Attempt at a Solution

The internal free energy is
$F_\text{int} = -kT\left[ \ln Z_\text{rot} + \ln Z_e\right].$
Which gives
$-\frac{\partial F_\text{int}}{\partial T} = k\left[\ln Z_\text{rot}Z_e+1\right].$
Apparantly this is for a single molecule and scaling this with $N$ give the correct answer.

However what I don't understand is if I want to derive it from the full partition function I should have $Z_\text{int} = (Z_eZ_\text{rot})^N$. But this is the partition function for distinguishable particles. Shouldn't I instead have $Z_\text{int} \approx \frac{1}{N!} (Z_eZ_\text{rot})^N$ for undistinguishable particles?

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It's been quite a number of years since I worked through the details of some of these, but your last line looks correct for Maxwell-Boltzmann type statistics. You use Stirling's formula to evaluate $ln (N!) =N ln(N)-N$.
It's been quite a number of years since I worked through the details of some of these, but your last line looks correct. You use Stirling's formula to evaluate $ln (N!) =N ln(N)-N$.
When thinking about it now, I think I see the reason for that. The indistinguishable particle part is already accounted for in the original equation (the $N$ there is indeed from stirlings approximation). And then it doesn't matter that the internal states are the same since we already accounted for that.