Question

Arrange the following \(\mathrm{H}\) atom electron transitions in order of decreasing wavelength of the photon absorbed or emitted: (a) \(n=2\) to \(n=\infty\) (b) \(n=4\) to \(n=20\) (c) \(n=3\) to \(n=10\) (d) \(n=2\) to \(n=1\)

Short answer

b, c, a, d

Step-by-step solution

Understand the relationship between energy levels and wavelength

The wavelength of a photon absorbed or emitted during an electron transition is inversely proportional to the energy difference between the two levels. This is described by the equation \[ \Delta E = \frac{hc}{\lambda} \] where \( \Delta E \) is the energy difference, \( h \) is Planck's constant, \( c \) is the speed of light, and \( \lambda \) is the wavelength.

Calculate energy differences for each transition

For the hydrogen atom, the energy difference between two levels is given by: \[ \Delta E = R_H \left( \frac{1}{n_i^2} - \frac{1}{n_f^2} \right) \] where \( R_H \) is the Rydberg constant, \( n_i \) is the initial energy level, and \( n_f \) is the final energy level.

Determine \( \Delta E \) for each transition

(a) \( n=2 \) to \( n=\infty \): \[ \Delta E_a = R_H \left( \frac{1}{2^2} - \frac{1}{\infty^2} \right) = \frac{R_H}{4} \](b) \( n=4 \) to \( n=20 \): \[ \Delta E_b = R_H \left( \frac{1}{4^2} - \frac{1}{20^2} \right) = R_H \left( \frac{1}{16} - \frac{1}{400} \right) = R_H \left( \frac{25}{400} - \frac{1}{400} \right) = \frac{24R_H}{400} = \frac{3R_H}{50} \](c) \( n=3 \) to \( n=10 \): \[ \Delta E_c = R_H \left( \frac{1}{3^2} - \frac{1}{10^2} \right) = R_H \left( \frac{1}{9} - \frac{1}{100} \right) = R_H \left( \frac{91}{900} \right) = \frac{91R_H}{900} \](d) \( n=2 \) to \( n=1 \): \[ \Delta E_d = R_H \left( \frac{1}{1^2} - \frac{1}{2^2} \right) = R_H \left( 1 - \frac{1}{4} \right) = \frac{3R_H}{4} \]

Rank the transitions by energy difference

The energy differences from largest to smallest are: (d) \[ \frac{3R_H}{4} \], (a) \[ \frac{R_H}{4} \], (c) \[ \frac{91R_H}{900} \], (b) \[ \frac{3R_H}{50} \].

Determine decreasing wavelength

Since photon wavelength is inversely proportional to energy difference, the order of decreasing wavelength is: (b), (c), (a), (d).

Key concepts

Photon wavelength

In the context of the hydrogen atom, the wavelength of a photon is critical in determining electron transitions. Wavelength refers to the distance between two consecutive peaks of a wave. When an electron moves between energy levels in a hydrogen atom, it either absorbs or emits a photon with a specific wavelength.
The wavelength \( \lambda \) of the photon is inversely proportional to the energy difference \( \Delta E \) between the two levels: \[ \Delta E = \frac{hc}{\lambda} \] Here, \( h \) is Planck's constant and \( c \) is the speed of light. This means that larger energy differences correspond to shorter wavelengths and vice versa.
Understanding this relationship helps us predict the order of electron transitions by observing the wavelengths of the emitted or absorbed photons.

Energy levels

Energy levels in an atom are essentially the specific energies that electrons can have. In a hydrogen atom, these levels are designated by principal quantum numbers denoted as \( n \). When we talk about an electron transition, we mean the movement of an electron from one energy level to another.
Each level has a specific amount of energy, and this energy decreases as you move further from the nucleus of the atom. The equation for the energy difference \( \Delta E \text{for a hydrogen atom} \) between two levels is given by: \[ \Delta E = R_H \left( \frac{1}{n_i^2} - \frac{1}{n_f^2} \right) \] where \( n_i \) and \( n_f \) are the initial and final energy levels, respectively.
This equation helps us calculate the energy difference for transitions, important for finding the corresponding photon wavelength.

Rydberg constant

The Rydberg constant \( R_H \) is a fundamental constant used in atomic physics. It is crucial for calculating the wavelengths of spectral lines of hydrogen and other elements. Its value is approximately \( 1.097 \times 10^7 \ \text{m}^{-1} \).
In the context of electron transitions, the Rydberg constant (\