The series of emission lines of the hydrogen atom for which \(n_{f}=4\) is called the Brackett series. (a) Determine the region of the electromagnetic spectrum in which the lines of the Brackett series are observed. (b) Calculate the wavelengths of the first three lines in the Brackett series - those for which \(n_{i}=5,6,\) and 7.

The Rydberg formula for hydrogen 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, \(R_H\) is the Rydberg constant for hydrogen \((R_H = 1.097 \times 10^7 \, m^{-1})\), \(n_f\) is the final principal quantum number, and \(n_i\) is the initial principal quantum number.

To determine the region of the electromagnetic spectrum for the Brackett series, we will first need to find the minimum and maximum wavelengths of the series. The minimum wavelength corresponds to the limit as \(n_i\) approaches infinity, and the maximum wavelength corresponds to the case where \(n_i = n_f + 1\). Minimum wavelength (\(n_i \rightarrow \infty\)): \[ \frac{1}{\lambda_{min}} = R_H \left( \frac{1}{n_f^2} \right) \] \[ \lambda_{min} = \frac{1}{R_H \cdot \frac{1}{n_f^2}} \] Maximum wavelength (\(n_i = n_f + 1\)): \[ \frac{1}{\lambda_{max}} = R_H \left( \frac{1}{n_f^2} - \frac{1}{(n_f+1)^2} \right) \] \[ \lambda_{max} = \frac{1}{R_H \cdot \left( \frac{1}{n_f^2} - \frac{1}{(n_f+1)^2} \right)} \] For the Brackett series, \(n_f = 4\). Substitute this value into the equations for \(\lambda_{min}\) and \(\lambda_{max}\), and compute their numerical values to determine the region of the electromagnetic spectrum.

Now we will use the Rydberg formula and the given values of \(n_i\) (5, 6, and 7) to calculate the wavelengths of the first three lines in the Brackett series. For \(n_i = 5\): \[ \frac{1}{\lambda_1} = R_H \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) \] \[ \lambda_1 = \frac{1}{R_H \cdot \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)} \] For \(n_i = 6\): \[ \frac{1}{\lambda_2} = R_H \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) \] \[ \lambda_2 = \frac{1}{R_H \cdot \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)} \] For \(n_i = 7\): \[ \frac{1}{\lambda_3} = R_H \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) \] \[ \lambda_3 = \frac{1}{R_H \cdot \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)} \] Substitute the given values of \(n_i\) and \(n_f\) into the equations for \(\lambda_1, \lambda_2\), and \(\lambda_3\), and compute their numerical values.