The series of emission lines of the hydrogen atom for which \(n_{1}=3\) is called the Paschen series. (a) Determine the region of the electromagnetic spectrum in which the lines of the Paschen series are observed. (b) Calculate the wavelengths of the first three lines in the Paschen series - those for which \(n_{1}=4,5,\) and \(6 .\)

The Rydberg formula for hydrogen is given by: \[ \frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \] where \(\lambda\) is the wavelength of the emission line, \(R_H\) is the Rydberg constant for hydrogen (\(R_H \approx 1.097 \times 10^7 \, m^{-1}\)), \(n_1\) is the principal quantum number of the lower energy level, and \(n_2\) is the principal quantum number of the higher energy level. For the Paschen series, \(n_1 = 3\).

To find the wavelengths of the first three lines in the Paschen series, we need to use the Rydberg formula and set \(n_1 = 3\) and \(n_2 = 4, 5, 6\). 1. For the first line, \(n_2 = 4\): \[ \frac{1}{\lambda_1} = R_H \left( \frac{1}{3^2} - \frac{1}{4^2} \right) \] 2. For the second line, \(n_2 = 5\): \[ \frac{1}{\lambda_2} = R_H \left( \frac{1}{3^2} - \frac{1}{5^2} \right) \] 3. For the third line, \(n_2 = 6\): \[ \frac{1}{\lambda_3} = R_H \left( \frac{1}{3^2} - \frac{1}{6^2} \right) \]

Now we can plug in the values for \(R_H\) and calculate the wavelengths for the first three lines. 1. For the first line, \(\lambda_1\): \[ \frac{1}{\lambda_1} = 1.097 \times 10^7 \left( \frac{1}{9} - \frac{1}{16} \right) \] \[ \lambda_1 = \frac{1}{(1.097 \times 10^7)(\frac{7}{144})} \] \[ \lambda_1 \approx 1.875 \times 10^{-6}\, m \] 2. For the second line, \(\lambda_2\): \[ \frac{1}{\lambda_2} = 1.097 \times 10^7 \left( \frac{1}{9} - \frac{1}{25} \right) \] \[ \lambda_2 = \frac{1}{(1.097 \times 10^7)(\frac{16}{225})} \] \[ \lambda_2 \approx 1.282 \times 10^{-6}\, m \] 3. For the third line, \(\lambda_3\): \[ \frac{1}{\lambda_3} = 1.097 \times 10^7 \left( \frac{1}{9} - \frac{1}{36} \right) \] \[ \lambda_3 = \frac{1}{(1.097 \times 10^7)(\frac{27}{324})} \] \[ \lambda_3 \approx 1.093 \times 10^{-6}\, m \]

The electromagnetic spectrum is divided into several regions based on the wavelength. Since all three of our wavelengths are in the range of \(10^{-6}\,m\), the region of the electromagnetic spectrum for the Paschen series is in the infrared region. So the answer is: (a) The Paschen series is observed in the infrared region of the electromagnetic spectrum. (b) The wavelengths of the first three lines in the Paschen series are approximately \(1.875 \times 10^{-6}\, m\), \(1.282 \times 10^{-6}\, m\), and \(1.093 \times 10^{-6}\, m\).