Spin isomers of hydrogen

Parahydrogen is in a lower energy state than is 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 mixture or 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)^[4] or by storing hydrogen at 77 K for 2–3 hours in the presence of activated charcoal.^[5] 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.^[5]

Nuclear spin states of H₂

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Each hydrogen molecule (H
₂) 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 H
₂ 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 M_I = 1, 0, or −1. The corresponding nuclear spin wavefunctions are ${\displaystyle \left|\uparrow \uparrow \right\rangle }$, ${\displaystyle \textstyle {\frac {1}{\sqrt {2}}}(\left|\uparrow \downarrow \right\rangle +\left|\downarrow \uparrow \right\rangle )}$ and ${\displaystyle \left|\downarrow \downarrow \right\rangle }$. This formalism uses standard

The singlet parahydrogen state has nuclear spin quantum numbers I = 0 and M_I = 0, with wavefunction ${\displaystyle \textstyle {\frac {1}{\sqrt {2}}}(\left|\uparrow \downarrow \right\rangle -\left|\downarrow \uparrow \right\rangle )}$. 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 H
₂ wavefunction imposes restrictions on the possible rotational states of the two forms of H
₂.^[1] 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.^[1]

The para form whose lowest level is J = 0 is more stable by 1.455 kJ/mol

Thermal properties

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Applying the rigid rotor approximation, the energies and degeneracies of the rotational states are given by:^{[9][page needed]}

${\displaystyle E_{J}={\frac {J(J+1)\hbar ^{2}}{2I}};\quad g_{J}=2J+1}$.

The rotational partition function is conventionally written as:^{[citation needed]}

${\displaystyle Z_{\text{rot}}=\sum \limits _{J=0}^{\infty }{g_{J}e^{-E_{J}/k_{\text{B}}T\;}}}$.

However, as long as the two spin isomers are not in equilibrium, it is more useful to write separate partition functions for each:^{[citation needed]}

${\displaystyle {\begin{aligned}Z_{\text{para}}&=\sum \limits _{{\text{even }}J}{(2J+1)e^{{-J(J+1)\hbar ^{2}}/{2Ik_{\text{B}}T}\;}}\\Z_{\text{ortho}}&=3\sum \limits _{{\text{odd }}J}{(2J+1)e^{{-J(J+1)\hbar ^{2}}/{2Ik_{\text{B}}T}\;}}\end{aligned}}}$

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:^{[citation needed]}

${\displaystyle Z_{\text{equil}}=\sum \limits _{J=0}^{\infty }{\left(2-(-1)^{J}\right)(2J+1)e^{{-J(J+1)\hbar ^{2}}/{2Ik_{\text{B}}T}\;}}}$

The molar rotational energies and heat capacities are derived for any of these cases from:^{[citation needed]}

${\displaystyle {\begin{aligned}U_{\text{rot}}&=RT^{2}\left({\frac {\partial \ln Z_{\text{rot}}}{\partial T}}\right)\\C_{v,{\text{ rot}}}&={\frac {\partial U_{\text{rot}}}{\partial T}}\end{aligned}}}$

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:^{[citation needed]}

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.^{[citation needed]} The residual energy is significant because the rotational energy levels are relatively widely spaced in H
₂; the gap between the first two levels when expressed in temperature units is twice the characteristic rotational temperature for H
₂:^{[citation needed]}

${\displaystyle {\frac {E_{J=1}-E_{J=0}}{k_{\text{B}}}}=2\theta _{\text{rot}}={\frac {\hbar ^{2}}{k_{\text{B}}I}}\approx 174.98{\text{ K}}}$.

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) × 2Rθ_rot ≈ 1091 J/mol,^{[citation needed]} which is somewhat larger than the enthalpy of vaporization of normal hydrogen, 904 J/mol at the boiling point, T_b ≈ 20.369 K.^[10] Notably, the boiling points of parahydrogen and normal (3:1) hydrogen are nearly equal; for parahydrogen ∆H_vap ≈ 898 J/mol at T_b ≈ 20.277 K, and it follows that nearly all the residual rotational energy of orthohydrogen is retained in the liquid state.^{[citation needed]}

However, orthohydrogen is thermodynamically unstable at low temperatures and spontaneously converts into parahydrogen.

Modern isolation of pure parahydrogen has since been achieved using rapid in-vacuum deposition of millimeters thick solid parahydrogen (p–H
₂) samples which are notable for their excellent optical qualities.[17]

Use in NMR and MRI

When an excess of parahydrogen is used during

Signal amplification by reversible exchange (SABRE) is a technique to hyperpolarize samples without chemically modifying them. Compared to orthohydrogen or organic molecules, a much greater fraction of the hydrogen nuclei in parahydrogen align with an applied magnetic field. In SABRE, a metal center reversibly binds to both the test molecule and a parahydrogen molecule facilitating the target molecule to pick up the polarization of the parahydrogen.^[21][22][23] This technique can be improved and utilized for a wide range of organic molecules by using an intermediate "relay" molecule like ammonia. The ammonia efficiently binds to the metal center and picks up the polarization from the parahydrogen. The ammonia then transfers the polarization to other molecules that don't bind as well to the metal catalyst.^[24] This enhanced NMR signal allows the rapid analysis of very small amounts of material and has great potential for applications in magnetic resonance imaging.

Deuterium

Diatomic deuterium (D
₂) has nuclear spin isomers like diatomic hydrogen, but with different populations of the two forms because the deuterium nucleus (deuteron) is a boson with nuclear spin equal to one.^[25] There are six possible nuclear spin wave functions which are ortho or symmetric to exchange of the two nuclei, and three which are para or antisymmetric.^[25] Ortho states correspond to even rotational levels with symmetric rotational functions so that the total wavefunction is symmetric as required for the exchange of two bosons, and para states correspond to odd rotational levels.^[25] The ground state (J = 0) populated at low temperature is ortho, and at standard temperature the ortho:para ratio is 2:1.^[25]

Other substances with spin isomers

Other molecules and functional groups containing two hydrogen atoms, such as

Molecular oxygen (O
₂) also exists in three lower-energy triplet states and one singlet state, as ground-state paramagnetic triplet oxygen and energized highly reactive diamagnetic singlet oxygen. These states arise from the spins of their unpaired electrons, not their protons or nuclei.

References