How specific heat of a solid depends on temperature?
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In solids, it’s because more phonon (quantized vibrational) modes become accessible at higher temperature. The Boltzmann distribution tells you how probably a given mode is to be excited at different temperatures; if it’s very unlikely (low T or high vibrational energy) then it doesn’t contribute much to the specific heat at that T. At higher T more of the higher energy modes are populated and you can store more heat.
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