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My chemistry book says that heat capacity varies with temperature. There is no further discussion. Does the heat capacity of a given amount of substance really vary with temperature? If yes, why is it so?

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The heat capacity is the slope of the plot of internal energy U with temperature T. The internal energy is energy that due to the rotation and vibrational energy a molecule possesses and as the temperature increases more rotational and vibrational energy levels becomes excited and so the internal energy increases.

Initially the internal energy is small and changes slowly as energy is added, mainly because there is not enough energy to excite many rotations or vibrations. Thus the slope of U vs. T is small and so is the heat capacity. Then as temperature increases more of the energy levels become excited & the internal energy rises rapidly and so does the slope of U vs T and so heat capacity increases.

At very high temperatures many, many energy levels are populated and so the rate of increase becomes constant as the temperature increases and so the heat capacity becomes constant, i.e. it levels out to a maximum value. The plot shows the change of heat capacity with energy just for vibrational levels, but it has the same general shape when rotational and translational energy are added.

U-vs-T

internal energy vs temperature

cv-vs-T

vibrational heat capacity vs temperature

enter image description here

low temperature region of heat capacity vs temperature.The 'bumpy' plot is due to the discrete values of vibrational energy levels

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