Question

Between which two orbits of the Bohr hydrogen atom must an electron fall to produce light of wavelength \(1876 \mathrm{nm} ?\)

Short answer

The electron must fall from the \(n_i = 3\) orbit to the \(n_f = 2\) orbit to produce light of the given wavelength.

Step-by-step solution

Convert wavelength to meters

Since \(\lambda\) is given in \(\mathrm{nm}\) (nanometers), it needs to be converted to meters for use in the formula. \(1 \mathrm{nm} = 1 \times 10^{-9} m\). Thus, \(\lambda = 1876 \times 10^{-9} m \approx 1.876 \times 10^{-6} m\).

Solve the Rydberg formula for \(n_i\)

Rearrange the Rydberg formula to solve for \(n_i\): \(\frac{1}{n_i^2} = \frac{1}{\lambda R} + \frac{1}{n_f^2}\).

Substitute the known values into the formula and solve for \(n_i\) for each \(n_f\)

As we are only dealing with hydrogen, \(n_f\) will start from 1 (lowest possible orbit) and increment until we find \(n_i\). Plugging in the value of \(\lambda = 1.876 \times 10^{-6} m\) and \(R = 1.097 \times 10^7 m^{-1}\), we calculate the value of \(n_i\) for each \(n_f\). Since \(n_i\) and \(n_f\) are quantum numbers, they can only be positive integers, so we round \(n_i\) to the nearest integer. The correct \(n_f\) is the one that gives us a value of \(n_i\) greater than \(n_f\). Repeat this until we identify the correct values.

Key concepts

Rydberg Formula

The Rydberg formula is a critical tool used to calculate the wavelengths of spectral lines for hydrogen and hydrogen-like elements. It is essential for understanding electron transitions between different energy levels in an atom. The general form of the formula is:\[ \frac{1}{\lambda} = R \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) \]where:

• \( \lambda \) is the wavelength of the emitted or absorbed light,
• \( R \) is the Rydberg constant, approximately \(1.097 \times 10^7 m^{-1}\) for hydrogen,
• \( n_i \) is the initial energy level,
• \( n_f \) is the final energy level.

In practice, this formula helps students and scientists determine which electron transitions produce specific wavelengths of light, such as when an electron falls from a higher energy orbit to a lower one, emitting energy as light in the process.
The calculation requires knowledge of both \( n_i \) and \( n_f \) to uncover the wavelength or vice versa.

Electron Transition

Electron transitions involve changes in the energy levels of electrons within an atom. When an electron transitions from a higher energy level to a lower one, it emits energy in the form of light. This light is characterized by a specific wavelength, which can be calculated using the Rydberg formula. Several key points about electron transitions:

• Higher to lower energy transitions result in photon emission.
• Lower to higher energy transitions require photon absorption, typically needing external energy sources such as heat or light.
• These transitions are quantized, meaning electrons can only move between distinct energy levels.
• In hydrogen, these transitions result in spectral lines that can be observed and measured.

Understanding electron transitions is crucial because they reveal important information about atomic structure and behavior, providing insights into chemical reactions and processes. Combining observations with theoretical models helps predict and explain how electrons behave in various elements' atoms.

Wavelength

Wavelength is the distance between two consecutive peaks (or troughs) of a wave and is usually denoted by the symbol \( \lambda \). In the context of the Bohr model and electron transitions in atoms, it represents the light emitted or absorbed during these transitions.Some key points about wavelength:

• Measured in units of length, commonly in meters or nanometers (nm).
• The wavelength of emitted light depends on the energy difference between the initial and final electron energy levels.
• Shorter wavelengths correspond to higher energy transitions, while longer wavelengths correspond to lower energy transitions.
• Wavelength is inversely related to frequency and directly influences the color of the light; for example, shorter wavelengths appear blue/violet while longer wavelengths appear red.

In calculations involving electron transitions, knowing the wavelength allows us to determine which energy levels the electron moved between. Converting wavelengths to the appropriate units is necessary to use them in formulas like the Rydberg equation.