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Well I am not an expert on this topic, but I did find something close to the answer to this. Here goes.

The heat capacity of metals is a combination of the electronic contribution and the phonon contribution. The vibrations in the lattice give 3R to the heat capacity at high temperatures and something close to planck statistics as they approach the 'Debye Temperature' of about 100Kelvin.

For Metals they have this T^3 term that is due to phonon heat capacity (which

asymptotically approaches the value of 3R at temperatures above TDebye). But they also have a T^1 term that is due to the relationship between electrical conductivity and thermal conductivity. So the real question for you is, for the metal you are looking at, what is the temperature, and what are those coefficients of Cv=AT^1 + BT^3. If you find those, you can answer your question.

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