High-Temperature Limit for Vibrational Heat Capacity

HIGH-TEMPERATURE LIMIT FOR VIBRATIONAL HEAT CAPACITY 

We can see the vibrational heat capacity approaches R asymptotically but the example for HCl shows that at room temperature, the actoal heat capacity can be quite low. What does the formula 

Thus we see that mathematically the vibrational heat capacity does tend toward R at high temperatures but if the bond is very stiff the value can be considerably smaller than R at room temperature. 

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