Heat Capacity within the Canonical Ensemble

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The heat capacity is a coefficient that gives the amount of energy to raise the temperature of a substance by one degree Celsius. The heat capacity can also be described as the temperature derivative of the average energy. The constant volume heat capacity is defined by

using the notation that á Eñ = U - U(0) where U(0) is the energy at zero Kelvin. The molar internal energy of a monatomic ideal gas is á Eñ = 3/2RT. The heat capacity of a monatomic ideal gas is therefore C_v = 3/2R.

For a monatomic gas there are three degrees of freedom per atom (these are the translations along the x, y, and z direction). Each of these translations corresponds to ½RT of energy. For an ideal diatomic gas some of the energy used to heat the gas may also go into rotational and vibrational degrees of freedom. For solids there is no translation or rotation and therefore the entire contribution to the heat capacity comes from vibrations. Given their extended nature the vibrations in solids are much lower in frequency than those of gases. Therefore, while vibrations in typical diatomic gases typically contribute little to the heat capacity, the vibrational contribution to the heat capacity of solids is the largest contribution. As the temperature is increased, there are more levels of the solid accessible by thermal energy and therefore Q increases. This also means that U increases and finally that C_v increases. In the high temperature limit in an ideal solid there are 3N vibrational modes that are accessible giving rise to a contribution to the molar heat capacity of 3R.