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Chapter 7: Problem 30

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

Expert verified
b, c, a, d

Step by step solution

01

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.
02

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.
03

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 \( n=1 \): \[ \Delta E_d = R_H \left( \frac{1}{1^2} - \frac{1}{2^2} \right) = R_H \left( 1 - \frac{1}{4} \right) = \frac{3R_H}{4} \]
04

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) \[ \frac{3R_H}{50} \].
05

Determine decreasing wavelength

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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 (\

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Most popular questions from this chapter

The flame tests for sodium and potassium are based on the emissions at \(589 \mathrm{nm}\) and \(404 \mathrm{nm},\) respectively. When both elements are present, the \(\mathrm{Na}^{+}\) emission is so strong that the \(\mathrm{K}^{+}\) emission can be seen only by looking through a cobalt-glass filter. (a) What are the colors of the \(\mathrm{Na}^{+}\) and \(\mathrm{K}^{+}\) emissions? (b) What does the cobalt-glass filter do? (c) Why is \(\mathrm{KClO}_{4}\) used as an oxidizing agent in fireworks rather than \(\mathrm{NaClO}_{4} ?\)

Rank these photons in terms of increasing energy: blue \((\lambda=\) \(453 \mathrm{nm}) ;\) red \((\lambda=660 \mathrm{nm}) ;\) yellow \((\lambda=595 \mathrm{nm})\)

For each of the following, give the sublevel designation, the allowable \(m_{l}\) values, and the number of orbitals: (a) \(n=4, l=2\) (b) \(n=5, l=1\) (c) \(n=6, l=3\)

Fish-liver oil is a good source of vitamin \(\mathrm{A},\) whose concentration is measured spectrometrically at a wavelength of \(329 \mathrm{nm}\) (a) Suggest a reason for using this wavelength. (b) In what region of the spectrum does this wavelength lie? (c) When \(0.1232 \mathrm{~g}\) of fish-liver oil is dissolved in \(500 . \mathrm{mL}\) of solvent, the absorbance is 0.724 units. When \(1.67 \times 10^{-3} \mathrm{~g}\) of vitamin \(\mathrm{A}\) is dissolved in \(250 . \mathrm{mL}\) of solvent, the absorbance is 1.018 units. Calculate the vitamin A concentration in the fish-liver oil.

Use the Rydberg equation to find the wavelength (in \(\mathrm{nm}\) ) of the photon emitted when an electron in an \(\mathrm{H}\) atom undergoes a transition from \(n=5\) to \(n=2\)
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