Entropy for rotational degree of freedom

Here no closed-form expression is available except at the limits of very high and very low temperatures. Another comphcation is that water has three principal moments of inertia, which are 1.09,1.91, and 3.0 10 g cm. The rotational partition function of water is a product of three partition functions with the respective moment of inertia. The partition function of each rotational mode can be obtained by summing the Boltzmann term over the rotational quantum number J, as given below 

Here the factor (2J+1) is the degeneracy factor associated with the rotational quantum number, J. If we approximate the water molecule as a rigid rotator, then the energy level of each rotational mode is determined hy its moment of inertia and is given hy 2 

However, the calculation needs to be carried out numerically, as mentioned earlier. That is, we need to go through the process of calculating the rotational partition function Q(fi) and then use Eq. (19.A.1) to obtain the entropy. 

The above description of rotational entropy is somewhat approximate because the rotational energies depend on the vibrational state of the molecule through vibration-rotation coupling, as the molecule is not fully rigid. At ambient eondi-tions, this correction is non-negligible. 

In the classic limit and under rigid rotator approximation, one can easily derive the following expression for the partition function for one rotator with moment of inertia given by /, 

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