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What is the mechanism by which the ortho- and para- spin isomers of hydrogen interconvert?

If such a mechanism exists, does this mean that ortho-hydrogen increases in concentration on increasing temperature?

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The first question is:

What is the mechanism by which spin isomers of hydrogen switch between the ortho and para forms?

There is some explanation in ChemPhysChem 2006, 7 (3), 551–554 (non-paywall version here):

One can define three situations. In the first, a magnetic conversion occurs without bond elongation or breaking. For example, in solid dihydrogen dipolar magnetic nuclear interactions are responsible for an extremely slow spin conversion.[4] In the presence of unpaired electrons the conversion is accelerated by the magnetic hyperfine interaction.[5] The second situation also involves a magnetic conversion mechanism, but is assisted by an intermediate H–H bond elongation, for example, by binding to a transition metal center. Finally, in the third situation, a H–H bond splitting and re-formation with other hydrogen atoms occurs, which corresponds to a chemical spin conversion.

The second question is:

If such a mechanism exists, why does ortho-hydrogen not increase in concentration on increasing temperature?

The equilibrium fraction of ortho-hydrogen does increase from 0% at absolute zero to 75% at high temperature.

para-Hydrogen is the lowest energy, most stable form, since it is capable of accessing the $J=0$ rotational level (whereas ortho-hydrogen cannot). There are three microstates corresponding to ortho-hydrogen:

$$|\!\uparrow\uparrow\rangle, |\!\downarrow\downarrow\rangle, |\!\uparrow\downarrow\rangle + |\!\downarrow\uparrow\rangle,$$

but only one microstate for para-hydrogen:

$$|\!\uparrow\downarrow\rangle - |\!\downarrow\uparrow\rangle.$$

So, according to the Boltzmann distribution, the equilibrium fraction of ortho-hydrogen goes from 0% at 0 K to 75% at infinite temperature (in practice, a temperature at which the effects of rotational quantisation are not seen).

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