a hydrogen atom emits a photon as its electron changes from n = 5 to n = 1. what is the wavelength of the…

Explanation:

Step1: Use Rydberg formula

The Rydberg formula for hydrogen - like atoms is $\frac{1}{\lambda}=R_H\left(\frac{1}{n_f^2}-\frac{1}{n_i^2}\right)$, where $R_H = 1.097\times 10^{7}\ m^{-1}$, $n_f = 1$ (final energy level), and $n_i = 5$ (initial energy level).

Step2: Calculate the reciprocal of wavelength

Substitute the values into the formula: $\frac{1}{\lambda}=1.097\times 10^{7}\left(\frac{1}{1^2}-\frac{1}{5^2}\right)$. First, calculate $\frac{1}{1^2}-\frac{1}{5^2}=1 - \frac{1}{25}=\frac{24}{25}$. Then $\frac{1}{\lambda}=1.097\times 10^{7}\times\frac{24}{25}=1.097\times 10^{7}\times0.96 = 1.05312\times 10^{7}\ m^{-1}$.

Step3: Find the wavelength

$\lambda=\frac{1}{1.05312\times 10^{7}\ m^{-1}}\approx94.96\times 10^{-9}\ m = 94.96\ nm$.

Step4: Determine the region of the spectrum

Photons with wavelengths in the range of approximately 10 - 400 nm are in the ultraviolet (UV) region of the electromagnetic spectrum.

Answer:

$\lambda = 94.96\ nm$ These photons fall in the ultraviolet region.