Free Rotation kT

Free Rotation (kT Vo) The energy levels, ef, for free internal rotation can be derived from quantum mechanics. They are 

With equations (10.138) and (10.139) the partition function for free rotation can be written. However, when the internal rotation can be described by a 

Because the quantity (/ 2/87r/r/c) is small at most T, the summation can be replaced by an integral over K in a procedure similar to that used to evaluate the rotational and translational partition functions earlier. The result is 

The thermodynamic properties can be calculated from Zm, f using the equations derived earlier. For example, the contribution to the heat capacity can be shown to be 5 R. 

As an example of the application of this procedure, Pitzer3 has calculated the thermodynamic properties for dimethylcadmium, including free rotation. At 298.15 K, he obtained the following result for the entropy  

---

Internal rotational energy levels are present in some (nonlinear) molecules, in which rotation about a bond in the molecule replaces a vibrational motion. The contribution of the internal rotation to the thermodynamic functions is determined by the magnitude of kT, the energy available to thermally excite the molecule, relative to Vo, the height of the potential barrier. For free rotation (kT> Vo) the energy levels are given by... 

---