Low-Temperature Heat Capacity of Hydrogen Molecules

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Hydrogen is the lowest boiling molecular species, remaining a gas down to 20K. At and above room temperature, , the rotational degree of freedom is fully excited; thus the rotational contribution to heat capacity approaches its equipartition value, per mole. Owing to the exceptionally small moment of inertia of , rotation becomes inactive at temperatures below about 50K. However, the heat capacity behaves anomalously as the temperature is lowered. This anomaly was first explained by Dennison in 1927. Since is a homonuclear molecule, only half of its rotational states are accessible. In the singlet nuclear-spin state, known as parahydrogen (p-) only even- rotational states are accessible; in the triplet nuclear-spin state, known as orthohydrogen (o-) only odd- rotational states are accessible. The molecular partition functions representing the rotational and nuclear spin degrees of freedom are given by




The rotational energies are given by with -fold degeneracies. It is convenient to define the rotational characteristic function , equal to 87.57 for and 65.70 for HD. The factors 1 and 3 represent the degeneracies of the para and ortho nuclear spin states, respectively.

The rotational contribution to heat capacity per mole can be calculated using . This can be plotted for o-, p- and a 3:1 mixture which exists in hydrogen gas at room temperature. The two forms do not interconvert unless a catalyst, such as activated charcoal or platinum is present, so the 3:1 ratio will persist as the temperature is lowered. In the presence of a catalyst, the partition function can be represented by its equilibrium value , with the sum running over both even and odd . This will be reflected in a heat capacity that reaches a maximum in excess of around . Para in the state, with a purity around 99.7%, can be obtained by cooling the equilibrium mixture down to 20K. (There also exist elaborate procedures for obtaining pure o-.)

The isotopomer HD is a heteronuclear diatomic molecule, with the nuclear spin-molecular rotational partition function given by

, K.

The nuclear-spin degeneracy equals , where the spins of the proton and deuteron are and 1, respectively.

You can select any combination of five heat-capacity curves over a temperature range. These can be identified using the tooltip.


Contributed by: S. M. Blinder (March 2011)
Open content licensed under CC BY-NC-SA



Snapshot 1: ortho and para rotational heat capacities as functions of temperature

Snapshot 2: 3:1 mixture and equilibrium mixture with catalyst

Snapshot 2: rotational heat capacity of heteronuclear HD molecule

Reference: S. M. Blinder, Advanced Physical Chemistry; A Survey of Modern Theoretical Principles, New York: Macmillan, 1969 pp. 475–478.

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