The heat capacity of an ideal gas has been shown to be calculable directly by statistical mechanics if the energies of the quantum states are known. However, unless one makes careful calculations, it is not easy for a student to understand the qualitative results. Why there are maxima (and occasionally minima) in heat capacity-temperature curves and where they occur are questions that are sometimes hard to answer. Fortunately, the statistical mechanical equation can be transformed to an exact variant that expresses the heat capacity as the sum of jumps from one quantum state to a higher one, and, with this, explanations become more transparent. In particular, figures show how the low-temperature rotational heat capacity arises from the sum of a very few jumps. Moreover, it is shown that the heat capacity reaches a virtually classical value long before the energy spacings become small in comparison with k[subscript B]T. (Contains 14 figures.)