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 $$

where the $$R_H$$ is Rydberg constant for hydrogen and is equal to 109737 $$cm^{-1}$$. $$n_1$$ 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:

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

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

$$n_2^2=8.67\approx{9}$$

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.