Partition function for ideal gas for "medium" temperature

[rockyleg]

Looking for the heat capacity of ideal gas due to rotational degrees of freedom.
If the temperature of the gas is much higher than the temperature corresponding to the energy differential between states,the partition function can be written as the integral over the density of states.
If the temperature is much smaller,then the higher terms of the partition function can be ignored.
Is there a usual method for when the two temperatures are close to each other?

 

[DrClaude]

If the temperature is much smaller,then the higher terms of the partition function can be ignored.
Is there a usual method for when the two temperatures are close to each other?

Actually, when the temperature is much smaller, then only the ground state is significantly populated. It is when the temperature is close to the rotational temperature (rotational constant divided by k_B) that you can sum the first few terms.

You should try it out: consider that the temperature is approximately equal to the rotational temperature and see how many terms you need to consider in the partition function. You will see that the values decrease very fast.