The Lyman series of emission lines of the hydrogen atom are those for which \(n_{\mathrm{f}}=1\). (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 \(n_{1}=2,3,\) and \(4 .\)

The Rydberg formula for the hydrogen atom emission lines is given by: \[\frac{1}{\lambda}=R_H\left(\frac{1}{n_{f}^2}-\frac{1}{n_{i}^2}\right)\] where: - \(\lambda\) is the wavelength of the emitted light; - \(R_H\) is the Rydberg constant for hydrogen, approximately equal to \(1.097 \times 10^7\,m^{-1}\); - \(n_f\) is the final energy level of the electron; - \(n_i\) is the initial energy level of the electron. For the Lyman series, \(n_f = 1\).

Now, we will calculate the wavelengths of the first three lines in the Lyman series, i.e., for \(n_i = 2, 3\), and \(4\). Using the Rydberg formula for each value, we get: For \(n_i = 2\): \[\frac{1}{\lambda_1}=R_H\left(\frac{1}{1^2}-\frac{1}{2^2}\right)\] \[\lambda_1=\frac{1}{R_H\left(\frac{3}{4}\right)}\] For \(n_i = 3\): \[\frac{1}{\lambda_2}=R_H\left(\frac{1}{1^2}-\frac{1}{3^2}\right)\] \[\lambda_2=\frac{1}{R_H\left(\frac{8}{9}\right)}\] For \(n_i = 4\): \[\frac{1}{\lambda_3}=R_H\left(\frac{1}{1^2}-\frac{1}{4^2}\right)\] \[\lambda_3=\frac{1}{R_H\left(\frac{15}{16}\right)}\] We can now calculate the wavelengths numerically for each case: \[\lambda_1 \approx \frac{1}{1.097\times10^{7}\left(\frac{3}{4}\right)} = 1.215\times10^{-7}\,m\] \[\lambda_2 \approx \frac{1}{1.097\times10^{7}\left(\frac{8}{9}\right)} = 1.025\times10^{-7}\,m\] \[\lambda_3 \approx \frac{1}{1.097\times10^{7}\left(\frac{15}{16}\right)} = 9.733\times10^{-8}\,m\]

Now that we have the wavelengths of the first three lines of the Lyman series, we can determine the region of the electromagnetic spectrum in which these lines are observed. The ranges for different types of electromagnetic radiation are as follows: - Radio waves: \(\lambda > 10^{-1} \,m\) - Microwaves: \(10^{-1}\,m > \lambda > 10^{-3}\,m\) - Infrared: \(10^{-3}\,m > \lambda > 7 \times 10^{-7}\,m\) - Visible light: \(7 \times 10^{-7}\,m > \lambda > 4 \times 10^{-7}\,m\) - Ultraviolet: \(4 \times 10^{-7}\,m > \lambda > 10^{-8}\,m\) - X-rays: \(10^{-8}\,m > \lambda > 10^{-11}\,m\) - Gamma rays: \(\lambda < 10^{-11}\,m\) As we can see from the calculated values, the first three lines of the Lyman series fall within the ultraviolet range (specifically, all three wavelengths are between \(4 \times 10^{-7}\,m\) and \(10^{-8}\,m\). So the final answers are: (a) The Lyman series is observed in the ultraviolet region of the electromagnetic spectrum. (b) The first three wavelengths in the Lyman series are approximately \(1.215 \times 10^{-7}\,m\), \(1.025 \times10^{-7}\,m\), and \(9.733 \times 10^{-8}\,m\).