Problem 91 - Chemistry The Central Science In Si Units

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.

Step-by-Step Solution

Answer

The Brackett series is observed in the infrared region. The wavelengths for transitions from \( n_i = 5, 6, \) and 7 to \( n_f = 4 \) are 4050 nm, 2620 nm, and 2170 nm, respectively.

Step 1: Understanding the Brackett Series

The Brackett series of the hydrogen atom refers to the emission lines that occur when an electron falls to the principal quantum number level \( n_f = 4 \). This series is part of the infrared region of the electromagnetic spectrum.

Step 2: Using the Rydberg Formula

The Rydberg formula is used to calculate the wavelengths of the spectral lines: \[\frac{1}{\lambda} = R_H \left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)\]where \( R_H \) is the Rydberg constant \((1.097 \times 10^7 \text{ m}^{-1})\), \( n_f = 4 \), and \( n_i \) is the initial quantum number.

Step 3: Calculate Wavelength for \( n_i = 5 \)

Substitute \( n_i = 5 \) into the Rydberg formula: \[\frac{1}{\lambda} = 1.097 \times 10^7 \left(\frac{1}{16} - \frac{1}{25}\right)\]Calculate the difference: \[\frac{1}{16} - \frac{1}{25} = \frac{9}{400}\]Now calculate \( \lambda \):\[\frac{1}{\lambda} = 1.097 \times 10^7 \times \frac{9}{400} = 2.469 \times 10^5 \]Thus, \( \lambda \approx 4.05 \times 10^{-6} \text{ m} \) or 4050 nm.

Step 4: Calculate Wavelength for \( n_i = 6 \)

Substitute \( n_i = 6 \) into the Rydberg formula: \[\frac{1}{\lambda} = 1.097 \times 10^7 \left(\frac{1}{16} - \frac{1}{36}\right)\]Calculate the difference: \[\frac{1}{16} - \frac{1}{36} = \frac{5}{144}\]Now calculate \( \lambda \):\[\frac{1}{\lambda} = 1.097 \times 10^7 \times \frac{5}{144} = 3.816 \times 10^5 \]Thus, \( \lambda \approx 2.62 \times 10^{-6} \text{ m} \) or 2620 nm.

Step 5: Calculate Wavelength for \( n_i = 7 \)

Substitute \( n_i = 7 \) into the Rydberg formula: \[\frac{1}{\lambda} = 1.097 \times 10^7 \left(\frac{1}{16} - \frac{1}{49}\right)\]Calculate the difference: \[\frac{1}{16} - \frac{1}{49} = \frac{33}{784}\]Now calculate \( \lambda \):\[\frac{1}{\lambda} = 1.097 \times 10^7 \times \frac{33}{784} = 4.617 \times 10^5 \]Thus, \( \lambda \approx 2.17 \times 10^{-6} \text{ m} \) or 2170 nm.

Key Concepts

Hydrogen Atom Emission Lines

The hydrogen atom is a quite fascinating model in atomic physics. Emission lines are a key concept here, as they represent the specific wavelengths of light emitted when an electron falls to a lower energy level in the atom. Each transition has a characteristic line in the spectrum, creating unique series of lines, named after their discoverers.

When an electron in the hydrogen atom moves from a higher energy level (higher quantum number) to a lower one, it emits energy in the form of light. This energy translates into distinct emission lines, useful in identifying the presence of hydrogen in various cosmic and laboratory settings.

• The Brackett series involves transitions that terminate at the quantum number level \( n_f = 4 \).
• Each series of lines falls within specific regions of the electromagnetic spectrum. For instance, the Lyman series lies in the ultraviolet region, while the Brackett series can be observed in the infrared spectrum.

This emission of light at specific wavelengths provides a fingerprint that helps astronomers and physicists understand more about the fundamental nature of hydrogen and its behavior under different conditions.

Rydberg Formula

The Rydberg formula is a crucial mathematical tool used to calculate the wavelengths of the emitted or absorbed light for electron transitions in hydrogen. It's a wonderful example of how mathematics can help us understand physical phenomena.

The formula 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 light emitted or absorbed.
• \( R_H \) is the Rydberg constant, approximately \( 1.097 \times 10^7 \text{ m}^{-1} \).
• \( n_f \) and \( n_i \) are the final and initial quantum numbers respective to an electron transition.

With this formula, one can calculate the exact wavelengths for transitions like those in the Brackett series and identify their placement within the electromagnetic spectrum. It's particularly valuable not only in studying hydrogen but also in understanding general atomic behavior.

Infrared Spectrum

The infrared spectrum is a segment of the electromagnetic spectrum with wavelengths longer than visible light. While invisible to the human eye, infrared is crucial in scientific research thanks to its ability to reveal information invisible in other spectral regions.

The Brackett series of hydrogen falls within this infrared region. This is because the energy changes involved in the electron transitions to the \( n_f = 4 \) level correspond to light with wavelengths longer than those in the visible range.

• Infrared is often used in astrophysics to observe celestial objects obscured by interstellar dust that blocks visible light.
• It's highly useful in remote sensing, astronomy, and even certain medical imaging technologies.

Observing the Brackett series in the infrared spectrum allows scientists to analyze hydrogen present in astronomical objects and broaden our understanding of the universe.

Quantum Numbers

Quantum numbers are fundamental to understanding the structure of an atom. They describe the distribution and behavior of electrons in an atom, providing insights into their energies and orbits.

For hydrogen, and other atoms, there are four quantum numbers essential for this explanation:

• Principal Quantum Number \( n \): Determines the energy level of an electron and its average distance from the nucleus. Larger values mean higher energy levels.
• Azimuthal Quantum Number \( l \): Describes the shape of the orbital, ranging from 0 to \( n-1 \).
• Magnetic Quantum Number \( m_l \): Describes the orientation of the orbital in space.
• Spin Quantum Number \( m_s \): Describes the intrinsic angular momentum (spin) of the electron.

Understanding these numbers allows scientists to predict the likelihood of an electron being in a particular position, a core concept in the quantum mechanical model of the atom. For emission lines like those in the Brackett series, the quantum numbers help determine which transitions will lead to the emission of specific wavelengths of light.