Wolfram Demonstrations Project

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-