Quantum Chemistry/Example 20

An electronic transition of H atom from an energy level to the ground state is observed with a corresponding wavelength of 102.5 nm. Determine the initial state of the electron from which transition has occurred.

The Rydberg's phenomenological equation will be used to solve the energy transition problem

Rydberg's phenomenological equation for hydrogen $\lambda ={ℝ}_H\left(\frac{1}{n_1^2}\right)-\left(\frac{1}{n_2^2}\right)$

where the $R_H$ is Rydberg constant for hydrogen and is equal to 109737 $c{m}^{-1}$. $n_1$ is the final energy level, $n_2$ is the initial energy level for hydrogen transition. Both $n_1$ and $n_2$ are integers and $n_2>n_1$.

The wavelength is given as 103nm

Concert to SI units:

$\lambda =103nm*1^{-7}\frac{cm}{nm}=1.03^{-5}cm$

The H atom's transition is from unknown energy level to the ground state.

So we can know the final state is the ground state which means $n_1=1$

The $n_2$ now can be calculated:

$\frac{1}{n_2^2}=\frac{1}{n_1^2}-\frac{1}{\lambda *R_H}$

$\frac{1}{n_2^2}=\frac{1}{1}-\frac{1}{0.00103cm*109737c{m}^{-1}}$

$n_2^2=8.67\approx 9$

Because the energy level always $>0$

$n_2=3$

Therefore, the initial state of the electron from which transition has occurred is energy level 3.