- #1
luisgml_2000
- 49
- 0
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!