Summary
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).P. Atkins and J. de Paula, Atkins' Physical Chemistry, 8th edition (W.H.Freeman 2006), p.452 At
room temperature and
thermal equilibrium, hydrogen consists of 25% parahydrogen and 75% orthohydrogen.
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. The proton has an associated
magnetic moment, which is associated with the proton's spin. In the H2 molecule, the spins of the two hydrogen nuclei (protons) couple to form a
triplet state (I = 1, α1α2, (α1β2 + β1α2)/21/2, or β1β2 for which MI = 1, 0, −1 respectively — this is
orthohydrogen) or to form a
singlet state (I = 0, (α1β2 – β1α2)/21/2 MI = 0 — this is
parahydrogen). The ratio between the ortho and para forms is about 3:1 at
standard temperature and pressure - a reflection of the spin degeneracy ratio, but if
thermal equilibrium between the two forms is established, the para form dominates at low temperatures (approx. 99.8% at 20 KRock, Peter A. "Chemical Thermodynamics", MacMillan 1969, p.478). Other molecules and functional groups containing two hydrogen atoms, such as
water and
methylene, also have ortho and para forms, although their ratios differ from that of the dihydrogen molecule.
Thermal properties
The permutational antisymmetry of the H2 wavefunction (protons are
fermions) 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 byF. T. Wall (1974). Chemical Thermodynamics, 3rd Edition. W. H. Freeman and Company.
\begin{align}
& E_{J}=\frac{J(J+1)\hbar ^{2}}{2I};\text{ }g_{J}=2J+1 \\
\end{align}.
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^/{2Ik_{B}T}\;}}\text{ ; }Z_{\text{ortho}}=3\sum\limits_{\text{odd }J}{(2J+1)e^/{2Ik_{B}T}\;}}.
The factor of 3 in the partition function for orthohydrogen accounts for the spin degeneracy associated with the states with odd values of J. When equilibrium between the spin isomers is possible, then a general partition function incorporating this degeneracy difference can be written as
\begin{align}
& Z_{\text{equil}}=\sum\limits_{J=0}^{\infty }{(2-(-1)^{J})(2J+1)e^/{2Ik_{B}T}\;}} \\
\end{align}
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)\text{ ; }C_{v,\text{ rot}}=\left( \frac{\partial U_{\text{rot}}}{\partial T} \right) \\
\end{align}
Plots shown here are molar rotational energies and heat capacities for ortho- and parahydrogen, and the "normal" ortho/para (3:1) and equilibrium mixtures.
Image:ortho-para_H2_energies.jpg|Molar Rotational Energies.
Image:ortho-para_H2_Cvs.jpg|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 can not 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
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. This residual energy, 1091 J/mol, is somewhat larger than the
enthalpy of vaporization of normal hydrogen, 904 J/mol at the boiling point, Tb = 20.369 K (this refers to the "normal", room-temperature, 3:1 ortho:para mixture).
link 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. 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
Ortho-Para conversion. Pag. 13 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.
The first synthesis of pure parahydrogen was achieved by
Paul Harteck and
Karl Friedrich Bonhoeffer in 1929.
When used during hydrogenations, parahydrogen gives rise to abnormally high signals in the NMR spectrum. This effect is called PHIP or PASADENA effect and was simultaneously discovered at two laboratories in Los Angeles (USA) and Bonn (Germany) in 1995. It was subsequently utilized to study the mechanism of hydrogenation reactions.
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.
Rapid Vapor Deposition of Millimeters Thick Optically Transparent Solid Parahydrogen Samples for Matrix Isolation Spectroscopy
Further research regarding parahydrogen thinfilm quantum state polarization matrices for computation seems a likely future prospect for these material sets.
References
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}}
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Bonhoeffer KF,
Harteck P
| title = Para- and ortho hydrogen
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| publisher = AIR FORCE RESEARCH LAB EDWARDS AFB CA PROPULSION DIRECTORATE WEST
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