Define the Diatomic Molecules in Two Dimensions?

You have been transported to a two-dimensional world by an evil wizard who refuses to let you revisit to your beloved Columbia unless you can determine the thermodynamic properties for a swivel heteronuclear diatomic molecule constrained to move only in a plane (two dimensions). You may suppose in what follows that the diatomic molecule doesn't undergo translational motion.  Indeed, it merely has rotational kinetic energyabout its center of mass. The quantized energy intensity of a diatomic in two dimensions are with degeneracies gj=2, for J not equal to zero, and gj=1 when J = 0.

ε_j=hcBJ²        J=0, 1, 2, 3,.......

As usual, B=h/π²Ic, where I is the moment of inertia

Hint: For reaching out of the wizard's evil clutches, treat all stage as having the same degeneracy and after that.... Oh, no! He's got me, too!

a) Assuming t>> hcB derive the partition function Z_rot for a solitary diatomic molecule in two dimensions.

b) Determine the thermodynamic energy E and heat capacity C_v in the limit, where t>> hcB, for a set of independent, indistinguishable, heteronuclear diatomic molecules constrained to rotate in a plane. Contrast these results to those for an ordinary diatomic rotor in three dimensions. Comment on the differences and examine briefly in terms of the number of degrees of freedom required to describe the motion of a diatomic rotor confined to a plane.