A ^7 N14 nucleus has nuclear spin I=1. Assume that the...

A ${ }^7 \mathrm{~N}_{14}$ nucleus has nuclear spin $I=1$. Assume that the diatomic molecule $\mathrm{N}_2$ can rotate but does not vibrate at ordinary temperatures and ignore electronic motion. Find the relative abundances of the ortho- and para-molecules in a sample of nitrogen gas. (Ortho $=$ symmetric spin state; Para $=$ antisymmetric spin state). What happens to the relative abundance as the temperature is lowered towards absolute zero? (Justify your answers!)

Key Concepts

Spin Isomers in Homonuclear Diatomic Molecules

Spin isomers refer to different molecular forms arising from the symmetry of the nuclear spin wavefunctions. In the case of nitrogen, the ortho-molecules possess symmetric nuclear spin states and therefore must occupy rotational eigenstates of even J, while the para-molecules possess antisymmetric spin states and occupy odd J states. The relative abundances of these isomers are affected by both the intrinsic statistical weights of the spin combinations and the thermal energy distribution across the corresponding rotational energy levels, resulting in distinct behavior at different temperatures.

Boltzmann Distribution and Partition Function

This concept explains how at a given temperature, different energy levels are populated according to their energy and degeneracy, following the Boltzmann distribution. The rotational energy levels, which depend on J, contribute to the overall partition function for the molecular rotations. At high temperatures, many rotational levels are thermally accessible and the relative populations are determined by both the degeneracy (including nuclear spin factors) and the Boltzmann factors. In contrast, as the temperature is lowered towards absolute zero, only the lowest energy rotational state (typically J = 0, the symmetric state) remains significantly populated, leading to an imbalance in the ortho/para ratio.

Nuclear Spin Statistics

This concept deals with the way individual nuclear spins combine and how their combined symmetry under particle exchange affects the overall quantum state. For diatomic molecules composed of identical nuclei, the total nuclear spin states can be classified as symmetric or antisymmetric. These classifications determine the allowed combinations with rotational states so that the overall wavefunction has the correct symmetry (symmetric for bosons, antisymmetric for fermions). In the problem, the nitrogen nuclei (spin I = 1) combine to give different total spins, and the symmetric (ortho) states and antisymmetric (para) states influence the statistical weights of the spin isomers.

Rotational Energy Levels and Wavefunction Symmetry

In diatomic molecules, the rotational wavefunctions are characterized by the quantum number J, where even and odd values have definite symmetry properties under exchange of the nuclei. The overall molecular wavefunction must be symmetric for bosonic nuclei. Therefore, when the nuclear spin state is symmetric (ortho), the rotational state must also be symmetric, corresponding to even J values; similarly, antisymmetric (para) nuclear spin states pair with odd J rotational levels. This coupling between rotational quantum numbers and spin symmetries is crucial in determining the relative populations of spin isomers.

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