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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.
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