B Entropy for vibrational degree of freedom

The partition function of a single harmonic oscillator is given by 

The total number of modes of vibration in a water molecule is three. We have already discussed this before. Each mode will contribute its own share of entropy. The total partition function is the product of the entropies of each mode and the entropy is the sum of the entropy contribution from each mode. 

The entropy of the molecular vibrational degree of freedom is usually small because the vibrational frequencies are pretty large. For a water molecule, these vibrational frequencies are more than ISk T, so the exponential terms in Eq. (19.B.2) above are all very small. 

In liquid water there are a number of eoUeetive inter-molecular vibrational modes, such as the HB excitation around 200 cm, which is like a breathing mode involving displacement of many molecules. Another example is the libration mode at around 600 cm, which is a restricted rotation, due to hydrogen-bonding. These low-frequency modes contribute to the entropy and specific heat of liquid water. But these modes are highly anharmonic, with a short lifetime, so the above description based on a harmonic oscillator cannot be used to describe them. 

This difficulty does not arise in solid, where we have an energy spectrum corresponding to its 3N - 6 modes of vibration. If this spectrum is discrete, then the addition of the weight functions for each of these 3N - 6 modes of vibration gives the entropy of a solid (Einstein s theory of heat capacity). 

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