This lecture introduces the equipartition theorem.
Write an explicit expression for the total classical Newtonian energy of the material in terms of the independent free variables.
For the ideal monatomic gas, we have \[ E = \frac12mv_1^2 + \frac12mv_2^2 + \frac12mv_3^2 + \cdots \text{(total of $N$ atoms)} \] where if we write things in terms of Cartesian coordinates gives us \[ \frac12mv_1^2 = \frac12mv_{1x}^2 + \frac12mv_{1y}^2 + \frac12mv_{1z}^2 \] so you end up with \(3N\) terms that depend on independent free variables.
Count the number of indepdendent free variables that are squared in the expression for the energy (“quadratic terms”). We call this the number of degrees of freedom \(f\).
For the ideal monatomic gas \(f=3N\).
The equipartition theorem predicts that \begin{align} U_{\text{classical}}(T) &= \frac{f}{2}k_BT \end{align}
For the ideal monatomic gas \(U_{\text{classical}}(T) = \frac{3}{2}Nk_BT\).
After this we do something else.