Question

The vibrational temperature of a molecule prepared in a supersonic jet can be estimated from the observed populations of its vibrational levels, assuming a Boltzmann distribution. The vibrational frequency of HgBr is 5.58 × 10¹² s^-1 and the ratio of the number of molecules in the n=1 stateto the number in the n=0 state is 0.127. Estimate the vibrational temperature under these conditions.

Short answer

The vibrational temperature of the molecule$\text{HgBr}$ is$12.98K$

Step-by-step solution

Vibrational Temperature

Vibrational temperature can be simply defined as temperature of a vibrating molecule, often used in thermodynamics and is given by the equation-

${\mathbf{\text{θ}}}_{\mathbf{\text{vib}}}\mathbf{=}\frac{\mathbf{\text{hν}}}{{\mathbf{\text{k}}}_{\mathbf{\text{B}}}}$

Where, ${\mathbf{\text{k}}}_{\mathbf{\text{B}}}$is Boltzmann’s constant, $\mathbf{\text{ν}}$is characteristic frequency of the oscillator and $\mathbf{\text{h}}$is Planck’s constant.

Given information and formula used

Vibrational frequency of $\mathit{H}\mathit{g}\mathit{B}\mathit{r}$is $\mathbf{5}\mathbf{.}\mathbf{58}\mathbf{\times }{\mathbf{10}}^{12}{\mathbf{s}}^{\mathbf{-}\mathbf{1}}$

Ratio of the number of molecules in the$\mathit{n}\mathbf{=}\mathbf{1}$ state to the number in the$\mathit{n}\mathbf{=}\mathbf{0}$ state is$\mathbf{0}\mathbf{.}\mathbf{127}$

The ratio of population of two levels is given by-

$\frac{{\mathbf{\text{N}}}_{\mathbf{\text{2}}}}{{\mathbf{\text{N}}}_{\mathbf{\text{1}}}}\mathbf{=}\mathit{e}\mathit{x}\mathit{p}\left(-\frac{\text{hν}}{{\text{k}}_{\text{B}}\text{T}}\right)$

Ratio of population

Rearrange the above equation.

$\begin{array}{c}-\frac{\text{hν}}{{\text{k}}_{\text{B}}\text{T}}=\ln \left(\frac{{\text{N}}_{\text{2}}}{{\text{N}}_{\text{1}}}\right)\\ \text{T\hspace{0.17em}}=-\frac{\text{hν}}{{\text{k}}_{\text{B}}\ln \left(\frac{{\text{N}}_{\text{2}}}{{\text{N}}_{\text{1}}}\right)}\end{array}$

Where, $\frac{{\text{N}}_{\text{2}}}{{\text{N}}_{\text{1}}}$is ratio of molecules in n =1 and =0'

From given information: $\frac{{\text{N}}_{\text{2}}}{{\text{N}}_{\text{1}}}=0.127$.

Calculation of vibrational temperature 

Substitute the values in the equation.

$\begin{array}{c}\text{T}=-\frac{\text{hν}}{{\text{k}}_{\text{B}}\ln \left(\frac{{\text{N}}_2}{{\text{N}}_{\text{1}}}\right)}\\ =-\frac{6.626\times {10}^{-34}\times 5.58\times {10}^{12}}{1.38\times {10}^{-23}\times \ln \left(0.127\right)}\\ =12.98\text{\hspace{0.17em}K}\end{array}$

Thus, the vibrational temperature of molecule is $12.98K$