Chapter 12, Problem 12A.3E

Interpretation Introduction

Interpretation:

The ratio of the populations of a linear rotor in the levels with $\text{J}\text{=}0$ and $\text{J}\text{=}\text{5}$ has to be calculated.

Concept Introduction:

The ratio of the population can be found using the Boltzmann distribution formula:

$\begin{array}{l}\frac{{\text{N}}_{\text{2}}}{{\text{N}}_{\text{1}}}\text{=}\frac{\raisebox{1ex}{${\text{Ng}}_{\text{2}}{\text{e}}^{\text{-}\raisebox{1ex}{${\in }_{\text{2}}$}\!\left/ \!\raisebox{-1ex}{$\text{kT}$}\right.}$}\!\left/ \!\raisebox{-1ex}{$\text{q}$}\right.}{\raisebox{1ex}{${\text{Ng}}_{\text{1}}{\text{e}}^{\text{-}\raisebox{1ex}{${\in }_{\text{1}}$}\!\left/ \!\raisebox{-1ex}{$\text{kT}$}\right.}$}\!\left/ \!\raisebox{-1ex}{$\text{q}$}\right.}\\ \text{Cancelling}\text{N}\text{and}\text{q}\\ \frac{{\text{N}}_{\text{2}}}{{\text{N}}_{\text{1}}}\text{=}\frac{{\text{g}}_{\text{2}}}{{\text{g}}_{\text{1}}}{\text{e}}^{\raisebox{1ex}{$-\left({\in }_{\text{2}}\text{-}{\in }_1\right)$}\!\left/ \!\raisebox{-1ex}{$\text{kT}$}\right.}\end{array}$

$\begin{array}{l}{\text{N}}_{\text{2}}\text{is}\text{the}\text{final}\text{population}\\ {\text{N}}_{\text{1}}\text{is}\text{the}\text{initial}\text{population}\\ {\text{g}}_{\text{2}}\text{degeneracy}\text{for}\text{the}\text{quantum}\text{level}\text{J}\text{=}\text{2}\\ {\text{g}}_{\text{1}}\text{degeneracy}\text{for}\text{the}\text{quantum}\text{level}\text{J}\text{=}\text{1}\end{array}$

$\begin{array}{l}{\in }_{\text{1}}=\text{Energy}\text{of}\text{the}\text{quantum}\text{level}\text{J}\text{=1}\\ {\in }_2=\text{Energy}\text{of}\text{the}\text{quantum}\text{level}\text{J}\text{=2}\\ \text{k=}\text{Boltzmann}\text{constant}\text{=}\text{1}\text{.381}{\text{×10}}^{\text{-23}}{\text{JK}}^{\text{-1}}\\ \text{T}\text{=}\text{Temperature}\end{array}$