# Homework Help: Trying to understand heat capacity

1. Apr 23, 2010

### benbenny

For a system of distinguishable particles Im reading that at the limit of high temperatures, and at the limit of low temperatures the heat capacity with respect to constant volume, goes to zero. Ive read that the reason for this is that at high temperatures, and low temperatures, the spacing of the energy levels for the system are $$\Delta\epsilon >> kT$$, and $$\Delta \epsilon << kT$$ respectively, and therefore, the system cannot adsorb energy at these limits. Im trying to reconcile this information with my knowledge of the heat capacity to be the change in temperature of a system per added unit of energy.

I would imagine that the large spacing between the energy levels means that inputting enough energy into the system will cause a relatively large change in the energy of the system compared with the temperature of the system, and thus that the heat capacity would be large at low temperatures, and vice versa in the case of high temperatures. This is also how understand the definition of heat capacity $$C_v = \frac{\partial U}{\partial T}$$ which to me implies that big differences of energy compared with small differences in temperature imply large heat capacity. Can someone help me understand why I am wrong?