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## Homework Statement

(a) The rotational energy of a diatomic molecule such as [tex]H_{2}[/tex] is given by

[tex]\frac{J^2_{\|}}{2I_{\|}}+\frac{J^2_{\bot}}{2I_{\bot}}[/tex].

[tex]J^2_{\|}[/tex] stands for the angular momentum with respect to the axis of symmetry of the molecule and [tex]J^2_{\bot}[/tex] the angular momentum perpendicular to that axis. If you have a gas with N molecules held at temperature T, what is the mean energy per molecule?

(b) A long thin needle floats within a gas at constant temperature. What will be the mean orientation of the needle's angular momentum vector? Parallel or perpendicular to the axis of symmetry of the needle?

## Homework Equations

Equipartition theorem

[tex]\left\langle x_i \frac{\partial \mathcal{H}}{\partial x_j}

= kT\delta_{ij} \right\rangle [/tex]

## The Attempt at a Solution

(a) Because of the equipartition theorem, each degree of freedom gives [tex]\frac{1}{2}kT[/tex] to the energy, so I think the mean energy per molecule should be [tex]\frac{3}{2}kT[/tex] (the free particle contribution) plus [tex]\frac{2}{2}kT[/tex] (the rotational energy).

(b) I think the angular momentum vector will be oriented perpendicular to the axis of symmetry of the needle, but I don't have an argument for it.

Thanks for your attention!