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This is a two-part question...

Firstly, models of the specific heat capacity $C$ (i.e. Debye, Einstein) in relation to the temperature $T$ give $C$ as steadily increasing with $T$. I assume that the change in $C$ is due to the heat the system gains being stored in its degrees of freedom- so why doesn't it increase in steps?

Secondly, what value does the temperature refer to? If the heat supplied raises the temperature from $T_1$ to $T_2$, does $C$ refer to $T_1$, $T_2$ or somewhere in between? If, in experiment, a large temperature difference was used, can it be assumed that $C$ changes over the range and so the answer is an average/useless?

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2 Answers

Even if you assume a classical situation, you still have the Boltzmann distribution factor which is exponential, and a smooth function of T. So C won't change in discrete steps.

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When you talk about the heat capacity increasing in steps, I assume you're referring to the equipartition principle, which says that the heat capacity is ${1\over 2}k_B$ times the number of degrees of freedom. As more degrees of freedom become available, the heat capacity goes up in steps of ${1\over 2}k_B$.

The thing is that equipartition only applies in some circumstances, roughly speaking when the degrees of freedom "look classical," in the sense that the separation between energy levels is small compared to $k_BT$ (among other conditions). Those assumptions don't hold for the Debye and Einstien models. In particular, as the temperature is raised, more quantum levels become available, and those newly-available ones, more or less by definition, aren't in that classical regime.

For your second question, the heat capacity only applies to infinitesimal increments. That is, instead of writing $$ \Delta Q=C\,\Delta T, $$ you have to write $$ dQ=C\,dT. $$ if $C$ varies significantly over the temperature interval in question, you have to integrate $$ \int_{T_1}^{T_2}C(T)\,dT $$ to get the required heat input.

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