[NOTES/SM-04019] Properties of an Ideal Gas

Partition function
We recall the result that the canonical partition function for an ideal gas is

\begin{eqnarray}
Z = z^N = V^N \left(\frac{2\pi m}{\beta h^2}\right)^{3N/2},
\end{eqnarray}
and therefore
\begin{equation}
\log Z = N \log V - \frac{3N}{2} \log \beta + \frac{3N}{2} \sigma
\end{equation}
where \( \sigma\) is a constant independent of \(T,V, N\) given by \[\sigma= \log(2\pi m/h^2)\]

Internal Energy
The internal energy of an ideal gas is given by
\begin{equation}
U = -\frac{\partial\log Z}{\partial\beta} = \frac{3}{2}NkT.
\end{equation}
It is to noted that the internal energy of an ideal gas depends only on temperature.

Helmholtz free energy
The Helmholtz free energy is related to the partition function by

\begin{equation}
F = - kT \log Z, \quad Z= e^{-\beta F},
\end{equation}

Equation of State
The pressure is obtained from the partition function by

\begin{equation}
p=kT \big(\pp[\log Z]{V}\Big) = kT(N/V).
\end{equation}
This gives the equation of state as
\begin{equation}
pV - NkT.
\end{equation}
If \(N_0\) denotes the Avogadro number and if \(n\) is the number of moles, we write \(N=nN_0, R=N_0k\) to get we the ideal gas equation in the form

\begin{equation}
pV = nRT.
\end{equation}

Entropy
The entropy can be computed using
\begin{eqnarray}
S &=& k\big(\log Z + \beta U\big) +\frac{3Nk}{2} \text{const} \\
&=& k N \Big(\log V +\frac{3}{2} \log T\Big)+\frac{3Nk}{2}\text{const}
\end{eqnarray}
where const is independent of \(T,V,N\).