Problems and Solutions on Thermodynamics and Statistical Mechanics

Statistical Phyaicn 209

(e) When T = 00/2, the partition functions for ortho- and parahydro-
gen are

where X = h2/4.1r2md2kT. It does not appear possible to solve these and
calculate C, accurately, but we can estimate them using the approximate
results of (d).

2043

Molecular hydrogen is usually found in two forms, orthohydrogen

(“parallel” nuclear spins) and parahydrogen (“anti-parallel” nuclear spins).

(a) After coming to equilibrium at “high” temperatures, what fraction

of H2 gas is parahydrogen (assuming that each variety of hydrogen is mostly

in its lowest energy state)?

(b) At low temperatures orthohydrogen converts mostly to parahydro-

gen. Explain why the energy released by each converting molecule is much

larger than the energy change due to the nuclear spin flip.

( was co ns in)

Solution:

(a) For the two kinds of diatomic molecules of identical nuclei, the

vibrational motion and the degeneracy of the lowest state of electron are

the same for both while their rotational motions are different. The identical

nuclei being fermions, antisymmetric nuclear spin states are associated with

rotational states of even quantum number 1, and symmetry nuclear spin

states are associated with rotational states of odd quantum number 1 (the

reverse of bosons). Thus

where s is the half-integer spin of a nucleon (for the hydrogen nucleus, s =

1/2), s(23+l) is the number of antisymmetric spin states and (s+1)(2s+l)