Temperature Variation of Heat Capacity for an Ideal Diatomic Gas

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This Demonstration considers the variation of heat capacity of an ideal diatomic gas, specifically hydrogen, with temperature. Both classical and quantum points of view are considered. The quantum-mechanical theory shows a monotonic increase in molar heat capacity with temperature. This approaches a constant value at high temperature, namely J/mol K. This limit corresponds, in fact, to the prediction of classical statistical mechanics, which takes into account translational, rotational and vibrational contributions to heat capacity, equal to , and , respectively, independent of temperature. In quantum statistical mechanics, on the other hand, the rotational and vibrational contributions become active only above certain characteristic temperatures, which leads to the observed, somewhat stepwise increase of heat capacity with increasing temperature.

Contributed by: Hayley Petit and Siddharth Madapoosi (June 2013)
Additional contributions by: Eitan Geva (University of Michigan)
Open content licensed under CC BY-NC-SA


Snapshot 1: classical and quantum heat capacities at a low temperature, where only the translational contribution to heat capacity is active

Snapshot 2: classical and quantum heat capacities at a moderate temperature, where both the translational and rotational contributions to the heat capacity contribute

Snapshot 3: classical and quantum heat capacities at a high temperature, where translational, rotational and vibrational components all contribute


[1] D. A. McQuarrie, Statistical Mechanics, Sausalito: University Science Books, 2000 chapters 6–7.

[2] D. A. McQuarrie and J. D. Simon, Physical Chemistry: A Molecular Approach, Sausalito: University Science Books, 1997 chapter 18.

Submission from the Compute-to-Learn course at the University of Michigan.


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