Equipartition Theorem and Microscopic Motion

[ghoultree]

Homework Statement

What is the typical rotational frequency f_rot for a molecule like N₂ at room temperature (25°C)? Assume that d for this molecule is 10^-10m. Take the atomic mass of N₂ to be m_N2=4.65x10^-26kg. You will need to account for rotations around two axes (not just one) to find the correct frequency.

Homework Equations

rms angular speed: ω=sqrt{(2kT)/(md²)}
f_rot=ω/(2\pi)
k=1.381x10^-23 J/(molecule*K)

The Attempt at a Solution

For my first attempt, I plugged the given numbers into sqrt{(3kT)/(md²)}. I used 3 instead of 2 because I was accounting for the two different axes. I'm not sure if the was the correct way to go about it. Secondly, I divided the given mass by 2. Then, I divided my answer by 2pi.

 

[voko]

The formula you have is derived from equating \frac 1 2 kT to rotational energy, which is \frac {J \omega^2} {2}, where J = \frac {md^2} {2} is the moment of inertia of two masses m on a weightless rod d. But you have two degrees of freedom, so you must equate kT. That means instead 2kT you have in the formula you must take 4kT, not 3kT.

As follows from the above, you should not divide m by two, because you are already given the atomic mass.