Why Does Heat Capacity Approach Zero at Temperature Extremes?

[benbenny]

For a system of distinguishable particles I am 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. I've 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. I am 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?

Thanks in advance.

 

[NateD]

The issue is that at high and low temperatures, the spacing of energy levels is so large that adding additional energy does not significantly affect the temperature of the system. At high temperatures, the energy levels are so far apart that even a large amount of energy does not cause much of a change in the energy of the system, and thus the temperature of the system. At low temperatures, the energy levels are so close together that adding additional energy does not cause a significant increase in the energy of the system, and thus the temperature of the system. This means that at high and low temperatures, the heat capacity will be low because the change in temperature per unit of energy will be small.