Distance between hydrogen molecules in H2 using specific heat capacity

[enjoi668]

Homework Statement

Below about 80 K the heat capacity at constant volume for hydrogen gas (H2) is 3/2k per molecule, but at higher temperatures the heat capacity increases to 5/2k per molecule due to contributions from rotational energy states. Use these observations to estimate the distance between the hydrogen nuclei in an H2 molecule.

Homework Equations

boltzmann distribution (possibly)
C = deltaEatom / deltaT C=specific heat capacity

The Attempt at a Solution

I honestly do not even have an idea on where to start this problem, any help at all, or hints on where to start, or some guidance on the equations I should use will be more than appreciated. Thanks

 

[ideasrule]

Here's a hint: the rotational energy states "freeze out" at low temperatures because individual H atoms in H2 lose their identity and become, very loosely speaking, a single particle. Classically, that's never going to happen: two balls connected with a stick isn't going to look any less like two balls and a stick at low temperatures. So think quantum mechanically. At what distance, roughly, would the hydrogen atoms lose their identities?

 

[ehild]

When the temperature is high enough, the rotation gives 2 more degrees of freedom, so the average energy of the molecules increases by kT. This term arises from the rotational energy, which is E=0.5 Iw^2 (I is the moment of inertia and w is the angular speed of rotation). The angular momentum of rotation is L=Iw, and it can be an integer multiple of h/(2pi) according to Bohr's quantum condition. The molecules can be in their lowest excited rotational quantum state, so L=h/(2pi). The average energy for the rotation is 80k. The moment of inertia of the hydrogen molecule is 2mr^2. From all of this, you can estimate the distance between the hydrogen atoms.

ehild

 

[enjoi668]

thank you very much