What is the typical rotational frequency \"f_not\" for a molecule like \"N_2\" a

Question

What is the typical rotational frequency "f_not" for a molecule like "N_2" at room temperature (25 C)? --- Assume that "d" for this molecule is " 2 A = 2 *10^-10 ". Take the atomic mass of "N_2" to be " m = 4.65*10^-26 ". You will need to account for rotations around two axes (not just one) to find the correct frequency. The angular velocities add like vectors, so " w^2= (w_x)^2 + (w_y)^2 " They have units of radians/sec. Express "f_not" numerically in Hz (revolutions/second) to three significant figures.

Explanation / Answer

A bimolecular gas particle like the nitrogen molecule has two degrees of freedom for rotational motion.
E = (2/2)kT = kT

The rotational kinetic energy of object of moment of inertia J rotating at angular velocity is:
E = (1/2)J²

Consider the nitrogen molecule as two point masses connected by thin massless rod. Take the middle of the rod as axis of rotation. Moment of inertia equals mass times squared distance to the axis of rotation
J = 2 (m/2)(d/2)² = (1/4)md²

Angular velocity and frequency are related as:
= 2f

Therefore
E = (1/2) (1/4)md² (2f )² = (1/2)²md²f²
and
kT = (1/2)²md²f²
=>
f = [ 2kT/(²md²) ]
= (2 1.38×10²³JK¹ 298.15K / (² 4.65×10²kg (10¹m)²) ]
= 1.34×10¹² Hz

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