Inter-atomic force in a Hydrogen molecule - finding spring stiffness

[Username1234]

Inter-atomic force in a Hydrogen molecule -- finding "spring" stiffness

Homework Statement

At about 2000 K the heat capacity (at constant volume =) increases to (7/2)k per molecule due to contributions from vibrational energy states. Use these observations to estimate the stiffness of the "spring" that approximately represents the inter-atomic force between the two H atoms in a Hydrogen gas molecule (H2).

Homework Equations

K=.5Iω^2
Ι=2mr^2
E=mCΔT
Y=(ks,i)/d

These are possibilities...not sure whether they are all the right equations to use...

The Attempt at a Solution

Stress of interatomic bond = (ks,i)(s)/(d^2) where s is the stretch of the interatomic spring and d is the "original distance", which in this case would be the length of the interatomic bond between the two Hydrogen atoms in the H2 molecule (or the distance between the hydrogen nuclei in H2). I calculated (hopefully correctly) d already, and would only need stress and stretch to solve this equation. However, I have no idea if it even makes sense to be using this relationship (with stress and stretch) in this problem, one, because it doesn't make a lot of sense, and two, because it is from a much earlier chapter that we are not going through right now in class.

If you could help, I would greatly appreciate it!

Thanks.

 

[ideasrule]

A good approximation to the hydrogen molecule is a quantum harmonic oscillator. The ground state potential energy of the bond would then be (1/2)*h-bar*omega, where omega is sqrt(k/m). Since this vibrational energy state only starts contributing to the total energy at 2000K, you can figure out the numerical value of the energy state using equipartition and set it equal to (1/2)*h-bar*omega.

 

[Username1234]

Oh, thank you! That helps a lot.