Question

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.

Short answer

(a) The Brackett series is observed in the infrared region of the electromagnetic spectrum with a minimum wavelength of \(\lambda_{min} \approx 1.46 \, \mu m\) and a maximum wavelength of \(\lambda_{max} \approx 4.05 \, \mu m\). (b) The wavelengths of the first three lines in the Brackett series for \(n_i = 5, 6\), and 7 are approximately: \(\lambda_1 \approx 1.94 \, \mu m\) \(\lambda_2 \approx 2.17 \, \mu m\) \(\lambda_3 \approx 2.29 \, \mu m\)

Step-by-step solution

Recall the Rydberg formula for hydrogen

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.

Determine the region of the electromagnetic spectrum for the Brackett series

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.

Calculate the wavelengths of the first three lines in the Brackett series

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.

Key concepts

Rydberg formula

The Rydberg formula is an important equation in physics, particularly in the study of atomic spectra. It is used to predict the wavelengths of light in the electromagnetic spectrum, which are emitted or absorbed by electrons in a hydrogen atom. The formula is expressed as:\[\frac{1}{\lambda} = R_H \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)\]Here, \(\lambda\) stands for the wavelength of light, \(R_H\) is the Rydberg constant for hydrogen, approximately equal to \(1.097 \times 10^7 \, m^{-1}\), \(n_f\) is the final principal quantum number, and \(n_i\) is the initial principal quantum number. These quantum numbers are integers representing the energy levels that the electron transitions between. By using the Rydberg formula, you can calculate the wavelengths of emissions of light that occur when an electron falls from a higher energy level to a lower one within a hydrogen atom. This calculation is crucial for understanding the discrete lines observed in atomic spectra and plays a significant role in our grasp of quantum mechanics.

Electromagnetic spectrum

The electromagnetic spectrum encompasses all types of electromagnetic radiation, ranging from low-energy radio waves to high-energy gamma rays. Each type of radiation is characterized by its wavelength or frequency. The spectrum is typically divided into several regions such as:

• Radio waves
• Microwaves
• Infrared radiation
• Visible light
• Ultraviolet light
• X-rays
• Gamma rays

The Brackett series, which is a set of emission lines produced when electrons transition to the fourth energy level of a hydrogen atom (\(n_f = 4 \)), falls within the infrared region of the electromagnetic spectrum. The infrared region is characterized by wavelengths longer than that of visible light but shorter than microwaves. Thus, the Brackett series lines are not visible to the human eye but are incredibly useful in astrophysics and other scientific fields for studying the properties of distant stars and galaxies.

Hydrogen atom emission lines

Emission lines are a critical aspect of atomic spectra, particularly in the context of hydrogen, the simplest and one of the most abundant elements in the universe. When an electron in a hydrogen atom transitions from a higher energy level to a lower one, it emits a photon of light. The wavelength of this light can be calculated using the Rydberg formula, as previously discussed.For hydrogen, these emission lines are categorized into several series based on the final energy level reached by the electron. Some well-known series include:

• Lyman series (\(n_f = 1\)) - ultraviolet region
• Balmer series (\(n_f = 2\)) - visible region
• Paschen series (\(n_f = 3\)) - infrared region
• Brackett series (\(n_f = 4\)) - infrared region

These lines are significant because they allow scientists to identify the presence of hydrogen in celestial objects like stars and gas clouds. By observing these lines, astronomers can develop insights into the composition, temperature, density, and dynamics of cosmic environments, enhancing our understanding of the universe.