# zero specific heat for infinite temperature

Why in a two possible energies system heat capacity goes to zero as the temperature goes to infinity? I thought as T increases heat capacity approached to a constant positive number, however I'm getting something like the figure

In the two-level system, at very high temperature one reaches the situation where the populations of the states no longer change when the temperature is raised, so no more uptake of energy is possible. The states become equally populated (all the Boltzmann factors $$\propto\exp(-E_i/k_BT)$$ are the same as $$T\rightarrow\infty$$). The energy vs temperature curve saturates at a constant value, and it follows that the heat capacity tends to zero at high temperature. Many other model systems which have an upper bound to the energies of the states, and either a finite number of discrete states or a continuum of states with a reasonably behaved density of states, should behave in the same way.