extension questions\n24. the hydrogen spectral lines in model 2 are only the wavelengths of light that are…

extension questions\n24. the hydrogen spectral lines in model 2 are only the wavelengths of light that are in the visible range and therefore \seen\ by the naked eye. however, many other wavelengths can be detected with special equipment.\na. propose a hydrogen - electron transition that involves light with a wavelength in the ultraviolet (uv) range (10 - 400 nm).\nb. propose a hydrogen - electron transition that involves light with a wavelength in the infrared (ir) range (1000 - 106 nm).

extension questions\n24. the hydrogen spectral lines in model 2 are only the wavelengths of light that are in the visible range and therefore \seen\ by the naked eye. however, many other wavelengths can be detected with special equipment.\na. propose a hydrogen - electron transition that involves light with a wavelength in the ultraviolet (uv) range (10 - 400 nm).\nb. propose a hydrogen - electron transition that involves light with a wavelength in the infrared (ir) range (1000 - 106 nm).

Answer

Explanation:

Step1: Recall Rydberg formula for hydrogen - atom

The Rydberg formula for the wavelength of light emitted or absorbed during a hydrogen - electron transition is $\frac{1}{\lambda}=R_H\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)$, where $R_H = 1.097\times10^{7}\ m^{- 1}$, $n_1$ and $n_2$ are principal quantum numbers with $n_2>n_1$.

Step2: Determine UV - range transition

For the ultraviolet range ($\lambda$ between 10 - 400 nm). When $n_1 = 1$ and $n_2>1$, the Lyman series is in the UV range. For example, when $n_1 = 1$ and $n_2 = 2$, $\frac{1}{\lambda}=1.097\times10^{7}\left(\frac{1}{1^2}-\frac{1}{2^2}\right)=1.097\times10^{7}\left(1 - \frac{1}{4}\right)=1.097\times10^{7}\times\frac{3}{4}=8.2275\times10^{6}\ m^{-1}$. Then $\lambda=\frac{1}{8.2275\times10^{6}\ m^{-1}}\approx121.5\ nm$. So a transition from $n = 2$ to $n = 1$ is in the UV range.

Step3: Determine IR - range transition

For the infrared range ($\lambda$ between 1000 - 106 nm). When $n_1 = 3$ and $n_2>3$, the Paschen series is in the IR range. For example, when $n_1 = 3$ and $n_2 = 4$, $\frac{1}{\lambda}=1.097\times10^{7}\left(\frac{1}{3^2}-\frac{1}{4^2}\right)=1.097\times10^{7}\left(\frac{1}{9}-\frac{1}{16}\right)=1.097\times10^{7}\times\frac{7}{144}\approx5.33\times10^{5}\ m^{-1}$. Then $\lambda=\frac{1}{5.33\times10^{5}\ m^{-1}}\approx1875\ nm$. So a transition from $n = 4$ to $n = 3$ is in the IR range.

Answer:

a. A transition from $n = 2$ to $n = 1$ b. A transition from $n = 4$ to $n = 3$