Question

In Example 20.4, we determined that the moment of inertia of the molecule is $\mathbf{5}\mathbf{.}\mathbf{70}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{47}}\mathbf{kg}{\mathbf{m}}^{\mathbf{-}\mathbf{2}}$.

(a) Calculate the relative population of the J = 5 level and the ground state at${\mathbf{25}}^{\mathbf{o}}\mathbf{C}$.

(b) Calculate the relative population of the J = 15 level and the ground state at ${\mathbf{25}}^{\mathbf{o}}\mathbf{C}$.

(c) Calculate the relative population of the J = 25 level and the ground state at ${\mathbf{25}}^{\mathbf{o}}\mathbf{C}$.

Short answer

(a) The relative population of the J =5 level and the ground state is 0.107.

(b) The relative population of the J = 15 level and the ground state is$1.95\times {10}^{-8}$.

(c) The relative population of the J = 25 level and the ground state is$9.64\times {10}^{-22}$.

Step-by-step solution

Relative population

The energy difference from the ground state and the system's temperature determine the relative population. A larger energy differential results in a smaller population in the higher energy state.

Elevated temperatures, on the other hand, result in a higher population in a higher energy state.

Given information and formula used

The moment of inertia of NaH is $5.70\times {10}^{47}\mathrm{kg}{\mathrm{m}}^{-2}$.

Temperature is ${25}^{\mathrm{o}}\mathrm{C}$= 298 K.

The ratio of the population of two levels is given by:

localid="1663598966142" $\frac{{\mathbf{N}}_{\mathbf{2}}}{{\mathbf{N}}_{\mathbf{1}}}\mathbf{=}\mathbf{exp}\left(\mathbf{-}\frac{\mathbf{\epsilon }}{{\mathbf{k}}_{\mathbf{B}}\mathbf{T}}\right)$ ……. (i)

localid="1663599130638" $\mathbf{\epsilon }$is Boltzmann’s constant

The energy difference between states is given by:
localid="1663599006691" $\mathit{\Delta }\mathit{E}\mathbf{=}\frac{{\mathbf{h}}^{\mathbf{2}}}{\mathbf{8}\mathbf{\pi I}}\mathbf{\left[}{\mathbf{J}}_{\mathbf{f}}\mathbf{(}{\mathbf{J}}_{\mathbf{f}}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{-}{\mathbf{J}}_{\mathbf{i}}\mathbf{(}{\mathbf{J}}_{\mathbf{i}}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{\right]}$ …….. (ii)

where, J is the energy level of the electron and h is Planck’s constant.

Calculation of energy difference

The energy between J = 5 and ground state.

Substitute the values in equation (ii).

$\begin{array}{rcl}\mathrm{\Delta E}& =& \frac{{(6.63\times {10}^{-34})}^2}{8\mathrm{\pi }(5.70\times {10}^{-47})}\left[5(5+1)-0(0+1)\right]\\ \mathrm{\Delta E}& =& \frac{43.95\times {10}^{-68}}{8\mathrm{\pi }(5.70\times {10}^{-47})}\left[5\left(6\right)-0\left(1\right)\right]\\ \mathrm{\Delta E}& =& \frac{43.95\times {10}^{-68}}{8\mathrm{\pi }(5.70\times {10}^{-47})}\times 30\\ \mathrm{\Delta E}& =& 9.2\times {10}^{-21}\mathrm{J}\end{array}$

Similarly, substitute the values J = 15 and J = 25 in the above equation.

∴$△\mathrm{E}$for from ground state will be localid="1663592254800" $7.36\mathrm{x}{10}^{-20}\mathrm{J}$.

∴ $△\mathrm{E}$for from ground state will be $1.99\mathrm{x}{10}^{-19}\mathrm{J}$.

Calculation of relative population for  J = 5

a) Substitute the value $\mathrm{\Delta E}=9.2\times {10}^{-21}\mathrm{J}$in the equation (i).

localid="1663593594932" $$\begin{gathered} \frac{N_2}{N_1}=\exp \left(-\frac{9.2\times {10}^{-21}}{1.38\times {10}^{-23}\times 298}\right) \\ \frac{N_2}{N_1}=\exp \left(-2.23\right) \\ \frac{N_2}{N_1}=0.107 \end{gathered}$$

Calculation of relative population for J = 15

b) Substitute the value $\mathrm{\Delta E}=7.36\times {10}^{-20}\mathrm{J}$in the equation (i).

$$\begin{gathered} \frac{N_2}{N_1}=\exp \left(-\frac{7.3\times {10}^{-20}}{1.38\times {10}^{-23}\times 298}\right) \\ \frac{N_2}{N_1}=\exp \left(-17.75\right) \\ \frac{N_2}{N_1}=1.95\times {10}^{-8} \end{gathered}$$

Calculation of relative population for J = 25

c) Substitute the value $\mathrm{\Delta E}=1.99\times {10}^{-19}\mathrm{J}$in the equation (i).

$$\begin{gathered} \frac{N_2}{N_1}=\exp \left(-\frac{1.99\times {10}^{-19}}{1.38\times {10}^{-23}\times 298}\right) \\ \frac{N_2}{N_1}=\exp \left(-48.39\right) \\ \frac{N_2}{N_1}=9.64\times {10}^{-22} \end{gathered}$$