[NOTES/SM-04014] Partition Function of an Ideal Gas

Partition function of a perfect gas

For perfect gases at low density, the potential energy can be neglected. For a mono atomic gas, the total energy is just the sum of kinetic energy of all molecules. Hence $$ E=\sum_{i=1}^N\frac{1}{2} m\vec{v}_i^2 $$

\begin{align*}
f(E) = & Z^{-1}e^{-\beta E}\\
=&Z^{-1}\exp\Big(-\sum\beta \epsilon_i\Big)\qquad\qquad
\epsilon_i=\frac{1}{2} mv^2_i\\
=&Z^{-1}e^{-\beta\epsilon_i}e^{-\beta\epsilon_2}\cdots e^{-\beta
\epsilon_N}
\end{align*}

We can now compute the partition function as follows.

\begin{align*}
Z = & \int_V\cdots\int d^3v_1\cdots
d^3v_N~e^{-\beta\epsilon_1}\cdots
=\prod_{k=1}^N\left(\int d^3r_k\int d^3v_k e^{-\beta\epsilon_k}\right)\\
=&V^N\left(\int d^3v_1e^{-\beta\frac{1}{2}m\vec{v}_1^2}\right)^N
=z^N
\end{align*}
where \(z\), called {\tt single particle partition function}, is given by

\begin{align*}
z=&V\int dv_xdv_ydv_y e^{-\beta\frac{1}{2}m(v_x^2+v_y^2+v_z^2)}\\
=&V\left(\frac{2\pi}{\beta m}\right)^{3/2}
\end{align*}