Specific relative angular momentum

File:Spinisomersofmolecularhydrogen.gif

Spin isomers of molecular hydrogen

Molecular hydrogen occurs in two isomeric forms, one with its two proton spins aligned parallel (orthohydrogen), the other with its two proton spins aligned antiparallel (parahydrogen). At room temperature and thermal equilibrium, hydrogen consists of approximately 75% orthohydrogen and 25% parahydrogen.

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 M_I = 1, 0, or −1. The corresponding nuclear spin wavefunctions are $|\uparrow \uparrow \rangle ,1/{\sqrt {2}}(|\uparrow \downarrow \rangle +|\downarrow \uparrow \rangle )$ and $|\downarrow \downarrow \rangle$ (in standard bra-ket notation). Each orthohydrogen energy level then has a (nuclear) spin degeneracy of three, meaning that it corresponds to three states of the same energy, although this degeneracy can be broken by a magnetic field.

The singlet parahydrogen state has nuclear spin quantum numbers I = 0 and M_I = 0, with wavefunction $1/{\sqrt {2}}(|\uparrow \downarrow \rangle -|\downarrow \uparrow \rangle )$. Since there is only one possibility, each parahydrogen level has a spin degeneracy of one and is said to be nondegenerate.

The ratio between the ortho and para forms is about 3:1 at standard temperature and pressure – a reflection of the ratio of spin degeneracies. However if thermal equilibrium between the two forms is established, the para form dominates at low temperatures (approx. 99.8% at 20 K). Other molecules and functional groups containing two hydrogen atoms, such as water and methylene, also have ortho and para forms (e.g. orthowater and parawater), although their ratios differ from that of the dihydrogen molecule.

Thermal properties

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 the two forms of H₂ can adopt. Orthohydrogen, with symmetric nuclear spin functions, can only have rotational wavefunctions that are antisymmetric with respect to permutation of the two protons. Conversely, parahydrogen with an antisymmetric nuclear spin function, can only have rotational wavefunctions that are symmetric with respect to permutation of the two protons. Applying the rigid rotor approximation, the energies and degeneracies of the rotational states are given by

${\begin{aligned}&E_{J}={\frac {J(J+1)\hbar ^{2}}{2I}};{\text{ }}g_{J}=2J+1\\\end{aligned}}$.

The rotational partition function is conventionally written as

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

However, as long as these two spin isomers are not in equilibrium, it is more useful to write separate partition functions for each,

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

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

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

The molar rotational energies and heat capacities are derived for any of these cases from

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

Plots shown here are molar rotational energies and heat capacities for ortho- and parahydrogen, and the "normal" ortho/para (3:1) and equilibrium mixtures:

• 
  Molar Rotational Energies.
• 
  Molar Heat Capacities.

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 H₂; the gap between the first two levels when expressed in temperature units is twice the characteristic rotational temperature for H₂,

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

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) x 2Rθ_rot = 1091 J/mol, which is somewhat larger than the enthalpy of vaporization of normal hydrogen, 904 J/mol at the boiling point, T_b = 20.369 K.[1] 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. It follows that nearly all the residual rotational energy of orthohydrogen is retained in the liquid state. Orthohydrogen is consequently unstable at low temperatures and spontaneously converts into parahydrogen, but the process is slow in the absence of a magnetic catalyst to 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, or supply additional refrigeration equipment to absorb the heat that the orthohydrogen fraction will release as it spontaneously converts into parahydrogen. If orthohydrogen is not removed from liquid hydrogen, the heat released during its decay can boil off as much as 50% of the original liquid.

The first synthesis of pure parahydrogen was achieved by Paul Harteck and Karl Friedrich Bonhoeffer in 1929.

Modern isolation of pure parahydrogen has been achieved utilizing rapid in-vacuum deposition of millimeters thick solid parahydrogen (pH2) samples which are notable for their excellent optical qualities.

Further research regarding parahydrogen thinfilm quantum state polarization matrices for computation seems a likely future prospect for these material sets.

Use in NMR

When an excess of parahydrogen is used during hydrogenation reactions (instead of the normal mixture of orthohydrogen to parahydrogen of 3:1), the resultant product exhibits hyperpolarized signals in proton NMR spectra. This effect is called PHIP ("Parahydrogen Induced Polarisation") or PASADENA ("Parahydrogen and Synthesis allow dramatically enhanced nuclear alignment" – named this way as the first recognition of the effect was done by Bowers and Weitekamp of Caltech in Pasadena) and has been utilized to study the mechanism of hydrogenation reactions. See references .

References

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