Textbook Question
The familiar phenomenon of a rainbow results from the diffraction of sunlight through raindrops. (a) Does the wavelength of light increase or decrease as we proceed outward from the innermost band of the rainbow?
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Ch.6 - Electronic Structure of Atoms
Chapter 6, Problem 7b3
Consider the three electronic transitions in a hydrogen atom shown here, labeled A, B, and C. (b) Calculate the energy of the photon emitted for each transition.
Calculate the energy of the photon emitted for transition C.
Verified step by step guidance1
Identify the initial and final energy levels for transition C. From the diagram, transition C is from n=5 to n=4.
Use the formula for the energy levels of a hydrogen atom: E_n = -13.6 eV / n^2, where n is the principal quantum number.
Calculate the energy for the initial level (n=5) using the formula: E_5 = -13.6 eV / 5^2.
Calculate the energy for the final level (n=4) using the formula: E_4 = -13.6 eV / 4^2.
Determine the energy of the photon emitted by finding the difference between the initial and final energy levels: ΔE = E_5 - E_4.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Energy Levels in Hydrogen Atom
In a hydrogen atom, electrons occupy discrete energy levels, denoted by quantum numbers (n). Each level corresponds to a specific energy, with lower levels being more negative and closer to the nucleus. When an electron transitions between these levels, energy is either absorbed or emitted in the form of photons, with the energy difference between the levels determining the photon's energy.
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Hydrogenation Reactions
Photon Energy Calculation
The energy of a photon emitted during an electronic transition can be calculated using the formula E = hf, where E is the energy, h is Planck's constant (6.626 x 10^-34 J·s), and f is the frequency of the emitted photon. Alternatively, the energy can also be calculated using the difference in energy between the two levels involved in the transition, E = E_final - E_initial.
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Transition Notation
Transitions in a hydrogen atom are often labeled (e.g., A, B, C) to indicate the specific energy level changes. For example, transition C represents an electron moving from a higher energy level to a lower one, resulting in the emission of a photon. Understanding the notation and the corresponding energy levels is crucial for calculating the energy of the emitted photon for each transition.
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Related Practice
Textbook Question
A certain quantum-mechanical system has the energy levels shown in the accompanying diagram. The energy levels are indexed by a single quantum number n that is an integer. (b) Which quantum numbers are involved in the transition that requires the least energy?
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Textbook Question
Consider the three electronic transitions in a hydrogen atom shown here, labeled A, B, and C. (a) Three electromagnetic waves, all drawn on the same scale, are also shown. Each corresponds to one of the transitions. Which electromagnetic wave (i), (ii), or (iii), is associated with electronic transition C?
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Textbook Question
Consider the three electronic transitions in a hydrogen atom shown here, labeled A, B, and C. (c) Calculate the wavelength of the photon emitted for each transition. Do any of these transitions lead to the emission of visible light? If so which one(s)?
Calculate the wavelength of the photon emitted for transition B.
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Textbook Question
Consider a fictitious one-dimensional system with one electron. The wave function for the electron, drawn below, is c1x2 = sin x from x = 0 to x = 2p. (b) At what value or values of x will there be the greatest probability of finding the electron?
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Textbook Question
The contour representation of one of the orbitals for the n = 3 shell of a hydrogen atom is shown here. (a) What is the quantum number l for this orbital?
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