The Lyman series of emission lines of the hydrogen atom are those for which \(n_{f}=1 .\) (a) Determine the region of the electromagnetic spectrum in which the lines of the Lyman series are observed. (b) Calculate the wavelengths of the first three lines in the Lyman series- those for which \(n_{i}=2,3,\) and 4 .

The Lyman series of emission lines of the hydrogen atom are those for which nf = 1. It means that electrons are transitioning from higher energy levels to the lowest energy level (ground state). The Lyman series emissions are found in the UV region of the electromagnetic spectrum since the transitions correspond to large energy changes, which result in shorter wavelengths.

The wavelengths of the Lyman series can be determined using the Rydberg formula: \[ \frac{1}{\lambda} = R_H\Big(\frac{1}{n_{f}^2} - \frac{1}{n_{i}^2}\Big) \] Where: - λ is the wavelength of the emission line - \(R_H\) is the Rydberg constant, approximately equal to \(1.097 \times 10^7 m^{-1}\) - nf is the final energy level, which should be 1 in the case of the Lyman series - ni is the initial energy level. We can now calculate the wavelengths of the first three lines in the series (those for which ni = 2, 3, and 4).

Step 1: For \(n_i = 2\): \[ \frac{1}{\lambda_{1}} = R_H\Big(\frac{1}{1^2} - \frac{1}{2^2}\Big) \] \[ \lambda_{1} = \frac{1}{R_H\left(\frac{3}{4}\right)} \] \[ \lambda_{1} = \frac{1}{(1.097 \times 10^7)(\frac{3}{4})} \] \[ \lambda_{1} = 1.215 \times 10^{-7}m \] Step 2: For \(n_i = 3\): \[ \frac{1}{\lambda_{2}} = R_H\Big(\frac{1}{1^2} - \frac{1}{3^2}\Big) \] \[ \lambda_{2} = \frac{1}{R_H\left(\frac{8}{9}\right)} \] \[ \lambda_{2} = \frac{1}{(1.097 \times 10^7)(\frac{8}{9})} \] \[ \lambda_{2} = 9.724 \times 10^{-8}m \] Step 3: For \(n_i = 4\): \[ \frac{1}{\lambda_{3}} = R_H \Big(\frac{1}{1^2} - \frac{1}{4^2} \Big) \] \[ \lambda_{3} = \frac{1}{R_H\left(\frac{15}{16}\right)} \] \[ \lambda_{3} = \frac{1}{(1.097 \times 10^7)(\frac{15}{16})} \] \[ \lambda_{3} = 9.574 \times 10^{-8}m \] In conclusion, the first three lines in the Lyman series (ni=2, 3, and 4) have wavelengths of approximately \(1.215 \times 10^{-7}m\), \(9.724 \times 10^{-8}m\), and \(9.574 \times 10^{-8}m\), respectively. These lines are observed in the UV region of the electromagnetic spectrum.