In the low-temperature limit (kT « E), each term in the rotational partition function (equation 6.30) is much smaller than the one before. Since the first term is independent of T, cut off the sum after the second term and compute the average energy and the heat capacity in this approximation. Keep only the largest T-dependent term at each stage of the calculation. Is your result consistent with the third law of thermodynamics? Sketch the behavior of the heat capacity at all temperatures, interpolating between the high-temperature and low-temperature expressions.
Zrot = j=0∑∞ (2j+1)e-E(j)/kT = j=0∑∞ (2j+1)e-j(j+1)e/kT