The pressure of an ideal gas can be obtained from the translational partition function.

\begin{equation} P=NKT\left(\frac{\partial lnq_{tr}}{\partial V}\right)_T \end{equation}

We write the translational partition function:

\begin{equation} q_{tr }=\left(\frac{2\pi mkT}{h^2}\right)^{3/2}V \end{equation}

Taking natural logarithms

\begin{equation} lnq_{tr}=ln\left( \frac{2\pi mkT}{h^2}\right)^{3/2}+lnV \end{equation}


\begin{equation} \frac{dlnq_{tr}}{dV}=\frac{ 1}{V} \end{equation}

Substituting into the pressure formula and taking into account that $Nk=nR$

\begin{equation} P=\frac{NkT}{V}=\frac{nRT}{V } \end{equation}

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