The hydrogen atom can absorb light of wavelength \(2626 \mathrm{nm}\). (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, we first need to know the wavelength in meters. Given the wavelength of 2626 nm, we can convert it to meters using the following conversion factor: 1 nm = 1 x 10^(-9) m So, 2626 nm = 2626 x 10^(-9) m = 2.626 x 10^(-6) m Now, we can check the electromagnetic spectrum to find the region corresponding to this wavelength. The different regions and their approximate wavelength ranges are: - Radio waves: > 1 x 10^(-1) m - Microwaves: 1 x 10^(-3) - 1 x 10^(-1) m - Infrared: 7 x 10^(-7) - 1 x 10^(-3) m - Visible light: 4 x 10^(-7) - 7 x 10^(-7) m - Ultraviolet: 1 x 10^(-8) - 4 x 10^(-7) m - X-rays: 1 x 10^(-11) - 1 x 10^(-8) m - Gamma-rays: < 1 x 10^(-11) m Looking at these ranges, it can be concluded that the wavelength (2.626 x 10^(-6) m) falls in the infrared region of the electromagnetic spectrum.

To find the initial and final energy levels, we will use the Rydberg formula for the hydrogen atom: \(1/λ = R_H(1/n^2_i - 1/n^2_f)\) Where: - λ = wavelength (in meters) = 2.626 x 10^(-6) m - R_H = Rydberg constant for hydrogen ≈ 1.097 x 10^7 m^(-1) - n_i = initial energy level - n_f = final energy level We are given the wavelength and asked to find n_i and n_f values. Since the hydrogen atom is absorbing light energy, the electron must be transitioning from a lower energy level (n_i) to a higher energy level (n_f). First, let's isolate \(n^2_i\) term in the Rydberg formula: \(n^2_i = 1/(λR_H + 1/n^2_f)\) Now we'll use the trial and error method to find the closest integer values of n_i and n_f that satisfy the equation: Let's start with n_f = 2. \(n^2_i = 1/(2.626\cdot10^{-6}\cdot1.097\cdot10^7 + 1/4)\) \(n^2_i ≈ 1/(4.86 + 0.25)\) \(n^2_i ≈ 1/5.11 = 0.195\) Now since we know that ni must be an integer value, we can deduce that ni = 1, as it is the closest integer value to 0.195. So, the initial energy level (n_i) is 1. To find the final energy level (n_f), we can use the rearranged Rydberg formula: \(n^2_f = 1/((1/λ) - (1/R_H)(1/n^2_i))\) \(n^2_f = 1/((1/(2.626\cdot10^{-6})) - (1/(1.097\cdot10^7))(1/1^2))\) \(n^2_f ≈ 1/(3.81\cdot10^5 - 1.097\cdot10^7)\) \(n^2_f ≈ 4.02\) Now, since n_f must also be an integer value, we can deduce that n_f = 2, as it is the closest integer value to 4.02. Therefore, the initial energy level n_i is 1 and the final energy level n_f is 2.