Consider an ideal gas of classical particles of mass $m$ in uniform potential $\xi$ in 3d. The gas $N$ molecules, volume $V$ and is at temperature $T$. I believe that the Hamiltonian of this system is $H=\sum_{i=1}^N \frac{p_i^2}{2m}+\xi$.

The partition function is $$\begin{align} Z &= \sum_{\Gamma}e^{-\beta \left(\sum_{k=1}^N \frac{p_k^2}{2m}+ \xi\right )} \\ &\rightarrow \frac{1}{N!}\int \prod_{i=1}^N \frac{d^3p_i d^3q_i}{h_0^3} e^{-\beta \left(\sum_{k=1}^N \frac{p_k^2}{2m}+ \xi\right )} \\ &=\frac{1}{N!}\int \prod_{i=1}^N \frac{d^3p_i d^3q_i}{h_0^3} e^{-\beta \left(\sum_{k=1}^N \frac{p_k^2}{2m}\right)}e^{-\beta N \xi} \\ &=\frac{V^N e^{-\beta N \xi}}{N!h_0^{3N}} \prod_{i=1}^N \int d^3p_i e^{-\beta \left(\frac{p_i^2}{2m}\right)} \\ &=\frac{V^N e^{-\beta N \xi}}{N!h_0^{3N}} \left(\sqrt{\frac{\pi}{\frac{\beta}{2m}}}\right)^{3/2} \\ &=\frac{V^N e^{-\beta N \xi}}{N!h_0^{3N}} \left( \frac{2m\pi}{\beta}\right)^{3/2} \\ \end{align}$$

So the Helmholtz free energy is $$\begin{align} F=-k_BT \ln \left[ \frac{V^N e^{-\beta N \xi}}{N!h_0^{3N}} \left( \frac{2m\pi}{\beta}\right)^{3/2} \right] \end{align}$$

But apparently $\mu=\xi+k_BT\ln \left[\frac{N\lambda^3}{V}\right]$

So I take $$\begin{align}\mu &= \left(\frac{\partial F}{\partial N}\right)_{T,V} \\ &= \frac{\partial}{\partial N}\mid_{T,V} -k_BT \ln \left[ \frac{V^N e^{-\beta N \xi}}{N!h_0^{3N}} \left( \frac{2m\pi}{\beta}\right)^{3/2} \right] \\ &= -k_BT \frac{\partial}{\partial N}\mid_{T,V} \left( \ln \left[ \left( 2m\pi k_BT\right)^{3/2} \right]+\ln \left[ V^N e^{-\beta N \xi} \right] - \ln \left[ N!h_0^{3N} \right] \right) \\ &= -k_BT \frac{\partial}{\partial N}\mid_{T,V} \left( \ln \left[ V^N e^{-\beta N \xi} \right] - \ln \left[ N!h_0^{3N} \right] \right) \\ &= -k_BT \frac{\partial}{\partial N}\mid_{T,V} \left( N\ln V +-\beta N \xi- \ln \left[ N!h_0^{3N} \right] \right) \\ &= -k_BT \left( \ln V +-\beta \xi- \frac{\partial}{\partial N}\mid_{T,V} \ln \left[ N!h_0^{3N} \right] \right) \\ \end{align}$$

but the $N!$ is not very tractable so I dont see how the indistinguishability can be incorporated. Or should it be there at all? If not why not?


1 Answer 1


You will probably find the Stirling approximation useful here. For your purposes it will be sufficient to use the form $\ln N! \approx N\ln N - N$, which is valid for $N\gg 1$.


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