Molecular hydrogen occurs in two nuclear isomeric forms, orthohydrogen with the nuclear spins of its two protons aligned parallel to each other, and parahydrogen with its two proton spins aligned antiparallel. These two forms can be called spin isomers or more specifically nuclear spin isomers. Parahydrogen is in a lower energy state than orthohydrogen. At room temperature and thermal equilibrium, thermal excitation causes hydrogen to consist of approximately 75% orthohydrogen and 25% parahydrogen. When hydrogen is liquified at low temperature, there is a slow spontaneous transition to a predominantly para ratio, with the released energy having implications for storage. Essentially pure parahydrogen form can be obtained at very low temperatures, but it is not possible to obtain a sample containing more than 75% orthohydrogen by heating. A 50:50 mixture of ortho- and parahydrogen can be made in the laboratory by passing it over an iron(III) oxide catalyst at liquid nitrogen temperature (77 K) or by storing hydrogen at 77 K for 2–3 hours in the presence of activated charcoal. In the absence of a catalyst, gas phase parahydrogen takes days to relax to normal hydrogen at room temperature while it takes hours to do so in organic solvents.
Nuclear spin states of H2 Each hydrogen molecule (H2) consists of two hydrogen atoms linked by a covalent bond. If we neglect the small proportion of deuterium and tritium which may be present, each hydrogen atom consists of one proton and one electron. Each proton has an associated magnetic moment, which is associated with the proton's spin of 1⁄2. In the H2 molecule, the spins of the two hydrogen nuclei (protons) couple to form a triplet state known as orthohydrogen, and a singlet state known as parahydrogen. The triplet orthohydrogen state has total nuclear spin I = 1 so that the component along a defined axis can have the three values MI = 1, 0, or −1. The corresponding nuclear spin wavefunctions are | ↑↑ ⟩ {\displaystyle \left|\uparrow \uparrow \right\rangle } , 1 2 ( | ↑↓ ⟩ + | ↓↑ ⟩ ) {\displaystyle \textstyle {\frac {1}{\sqrt {2}}}(\left|\uparrow \downarrow \right\rangle +\left|\downarrow \uparrow \right\rangle )} and | ↓↓ ⟩ {\displaystyle \left|\downarrow \downarrow \right\rangle } . This formalism uses standard bra–ket notation; the symbol ↑ represents the spin-up wavefunction and the symbol ↓ the spin-down wavefunction for a nucleus, so ↑↓ means that the first nucleus is up and the second down. Each orthohydrogen energy level then has a (nuclear) spin degeneracy of three, meaning that it corresponds to three states of the same energy (in the absence of a magnetic field). The singlet parahydrogen state has nuclear spin quantum numbers I = 0 and MI = 0, with wavefunction 1 2 ( | ↑↓ ⟩ − | ↓↑ ⟩ ) {\displaystyle \textstyle {\frac {1}{\sqrt {2}}}(\left|\uparrow \downarrow \right\rangle -\left|\downarrow \uparrow \right\rangle )} . Since there is only one possibility, each parahydrogen level has a spin degeneracy of one and is said to be non-degenerate.
Allowed rotational energy levels Since protons have spin 1⁄2, they are fermions and the permutational antisymmetry of the total H2 wavefunction imposes restrictions on the possible rotational states of the two forms of H2. Orthohydrogen, with symmetric nuclear spin functions, can only have rotational wavefunctions that are antisymmetric with respect to permutation of the two protons, corresponding to odd values of the rotational quantum number J; conversely, parahydrogen with an antisymmetric nuclear spin function, can only have rotational wavefunctions that are symmetric with respect to permutation of the two protons, corresponding to even J. The para form whose lowest level is J = 0 is more stable by 1.455 kJ/mol than the ortho form whose lowest level is J = 1. The ratio between numbers of ortho and para molecules is about 3:1 at standard temperature where many rotational energy levels are populated, favoring the ortho form as a result of thermal energy. However, at low temperatures only the J = 0 level is appreciably populated, so that the para form dominates at low temperatures (approximately 99.8% at 20 K). The heat of vaporization is only 0.904 kJ/mol. As a result, ortho liquid hydrogen equilibrating to the para form releases enough energy to cause significant loss by boiling.
Thermal properties
Applying the rigid rotor approximation, the energies and degeneracies of the rotational states are given by:
E J = J ( J + 1 ) ℏ 2 2 I ; g J = 2 J + 1 {\displaystyle E_{J}={\frac {J(J+1)\hbar ^{2}}{2I}};\quad g_{J}=2J+1} . The rotational partition function is conventionally written as:
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