The hydrogen atom can absorb light of wavelength \(1094 \mathrm{nm}\). (a) In what region of the electromagnetic spectrum is this absorption found? (b) Determine the initial and final values of \(n\) associated with this absorption.

To determine the region of the electromagnetic spectrum where light with a wavelength of \(1094 \mathrm{nm}\) belongs, we can refer to the classification of the electromagnetic spectrum: - Radio waves: \(10^3\) to \(10^5 \mathrm{nm}\) - Microwaves: \(10^2\) to \(10^3 \mathrm{nm}\) - Infrared (IR): \(7 \times 10^2\) to \(10^5 \mathrm{nm}\) - Visible: \(400\) to \(700 \mathrm{nm}\) - Ultraviolet (UV): \(10\) to \(400 \mathrm{n}\) - X-rays: \(0.01\) to \(10 \mathrm{nm}\) - Gamma rays: < \(0.01 \mathrm{nm}\) Since \(1094 \mathrm{nm}\) falls in the range of the Infrared region (\(700\) to \(10^5 \mathrm{nm}\)), the absorption occurs in the Infrared region of the electromagnetic spectrum.

To find the values of the initial and final principle quantum numbers (n) associated with this absorption, we can use the Balmer-Rydberg equation: \(\frac{1}{\lambda} = R_H \left( \frac{1}{n_{1}^2} - \frac{1}{n_{2}^2} \right)\) Where \(\lambda\) is the wavelength of absorbed light, \(R_H\) is the Rydberg constant for hydrogen (\(1.097 \times 10^7 \mathrm{m^{-1}}\)), and \(n_1\) and \(n_2\) are the initial and final principal quantum numbers, respectively. Given the wavelength \(\lambda = 1094 \mathrm{nm} = 1.094 \times 10^{-6} \mathrm{m}\), we can plug it into the Balmer-Rydberg equation and find the difference between the inverses of the squared initial and final quantum numbers: \[\frac{1}{1.094 \times 10^{-6}} = 1.097 \times 10^7 \left( \frac{1}{n_{1}^2} - \frac{1}{n_{2}^2} \right)\] Now, we can simplify this equation and solve for the difference: \(\frac{1}{n_{1}^2} - \frac{1}{n_{2}^2} = \frac{9.138 \times 10^{-7}}{1.097 \times 10^7}\) \(\frac{1}{n_{1}^2} - \frac{1}{n_{2}^2} \approx 8.33 \times 10^{-14}\) Since this absorption process involves a hydrogen atom, we know that the initial and final quantum numbers are integers. Now, we need to find the closest integer values of \(n_1\) and \(n_2\) that satisfy the above condition. Through trial and error, we can find that when \(n_1 = 1\) and \(n_2 = 5\), the difference \(\frac{1}{1^2} - \frac{1}{5^2}\) equals approximately \(8.33 \times 10^{-14}\). So, the initial value of n associated with this absorption is \(n_1 = 1\), and the final value of n is \(n_2 = 5\).