Applying the
rigid rotor approximation, the energies and degeneracies of the rotational states are given by: : E_J = \frac{J(J + 1)\hbar^2}{2I};\quad g_J = 2J + 1. The rotational
partition function is conventionally written as: : Z_\text{rot} = \sum\limits_{J=0}^\infty{g_J e^{-E_J/k_\text{B} T\;}}. However, as long as the two spin isomers are not in equilibrium, it is more useful to write separate partition functions for each: : \begin{align} Z_{\text{para}} &= \sum\limits_{\text{even }J}{(2J + 1)e^{{-J(J + 1)\hbar^2}/{2Ik_\text{B} T}\;}}\\ Z_{\text{ortho}} &= 3\sum\limits_{\text{odd }J}{(2J + 1)e^{{-J(J + 1)\hbar^2}/{2Ik_\text{B} T}\;}} \end{align} The factor of 3 in the partition function for orthohydrogen accounts for the spin degeneracy associated with the +1 spin state; when equilibrium between the spin isomers is possible, then a general partition function incorporating this degeneracy difference can be written as: : Z_\text{equil} = \sum\limits_{J=0}^\infty{\left(2 - (-1)^{J}\right)(2J + 1)e^{{-J(J + 1)\hbar^2}/{2Ik_\text{B} T}\;}} The molar rotational energies and heat capacities are derived for any of these cases from: : \begin{align} U_\text{rot} &= RT^2 \left( \frac{\partial \ln Z_\text{rot}}{\partial T} \right) \\ C_{v,\text{ rot}} &= \frac{\partial U_\text{rot}}{\partial T} \end{align} Plots shown here are molar rotational energies and heat capacities for ortho- and parahydrogen, and the "normal" ortho:para ratio (3:1) and equilibrium mixtures: Because of the antisymmetry-imposed restriction on possible rotational states, orthohydrogen has residual rotational energy at low temperature wherein nearly all the molecules are in the
J = 1 state (molecules in the symmetric spin-triplet state cannot fall into the lowest, symmetric rotational state) and possesses nuclear-spin
entropy due to the triplet state's threefold degeneracy. The residual energy is significant because the rotational energy levels are relatively widely spaced in ; the gap between the first two levels when expressed in temperature units is twice the characteristic
rotational temperature for : : \frac{E_{J=1} - E_{J=0}}{k_\text{B}} = 2\theta_\text{rot} = \frac{\hbar^2}{k_\text{B}I} \approx 174.98\text{ K}. This is the
T = 0 intercept seen in the molar energy of orthohydrogen. Since "normal" room-temperature hydrogen is a 3:1 ortho:para mixture, its molar residual rotational energy at low temperature is (3/4) × 2
Rθrot ≈ 1091 J/mol, which is somewhat larger than the
enthalpy of vaporization of normal hydrogen, 904 J/mol at the boiling point,
Tb ≈ 20.369 K. Notably, the boiling points of parahydrogen and normal (3:1) hydrogen are nearly equal; for parahydrogen ∆Hvap ≈ 898 J/mol at
Tb ≈ 20.277 K, and it follows that nearly all the residual rotational energy of orthohydrogen is retained in the liquid state. However, orthohydrogen is thermodynamically unstable at low temperatures and spontaneously converts into parahydrogen. This process lacks any natural de-excitation radiation mode, so it is slow in the absence of a catalyst which can facilitate interconversion of the singlet and triplet spin states. At room temperature, hydrogen contains 75% orthohydrogen, a proportion which the liquefaction process preserves if carried out in the absence of a
catalyst like
ferric oxide,
activated carbon, platinized asbestos, rare earth metals, uranium compounds,
chromic oxide, or some nickel compounds to accelerate the conversion of the
liquid hydrogen into parahydrogen. Alternatively, additional refrigeration equipment can be used to slowly absorb the heat that the orthohydrogen fraction will (more slowly) release as it spontaneously converts into parahydrogen. If orthohydrogen is not removed from rapidly liquified hydrogen, without a catalyst, the heat released during its decay can boil off as much as 50% of the original liquid. == History ==