Textbook Question
Is energy emitted or absorbed when the following electronic transitions occur in hydrogen? a. from n = 4 to n = 2
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Ch.6 - Electronic Structure of Atoms
Brown15th EditionChemistry: The Central ScienceISBN: 9780137542970
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Table of contents
- Ch.1 - Introduction: Matter, Energy, and Measurement111
- Ch.2 - Atoms, Molecules, and Ions166
- Ch.3 - Chemical Reactions and Reaction Stoichiometry138
- Ch.4 - Reactions in Aqueous Solution127
- Ch.5 - Thermochemistry104
- Ch.6 - Electronic Structure of Atoms109
- Ch.7 - Periodic Properties of the Elements107
- Ch.8 - Basic Concepts of Chemical Bonding102
- Ch.9 - Molecular Geometry and Bonding Theories120
- Ch.10 - Gases122
- Ch.11 - Liquids and Intermolecular Forces79
- Ch.12 - Solids and Modern Materials102
- Ch.13 - Properties of Solutions99
- Ch.14 - Chemical Kinetics153
- Ch.15 - Chemical Equilibrium83
- Ch.16 - Acid-Base Equilibria109
- Ch.17 - Additional Aspects of Aqueous Equilibria117
- Ch.18 - Chemistry of the Environment60
- Ch.19 - Chemical Thermodynamics109
- Ch.20 - Electrochemistry113
- Ch.21 - Nuclear Chemistry79
- Ch.22 - Chemistry of the Nonmetals52
- Ch.23 - Transition Metals and Coordination Chemistry57
- Ch.24 - The Chemistry of Life: Organic and Biological Chemistry7
Brown15th EditionChemistry: The Central ScienceISBN: 9780137542970Not the one you use?Change textbook
Chapter 6, Problem 39a
a. Using Equation 6.5, calculate the energy of an electron in the hydrogen atom when n = 2 and when n = 6. Calculate the wavelength of the radiation released when an electron moves from n = 6 to n = 2.
Verified step by step guidance1
Identify the formula for the energy of an electron in a hydrogen atom: \(E_n = -\frac{R_H}{n^2}\), where \(R_H\) is the Rydberg constant (2.18 \(\times\) 10^{-18} \(\text{ J}\)) and \(n\) is the principal quantum number.
Calculate the energy of the electron when \(n = 2\) using the formula: \(E_2 = -\frac{R_H}{2^2}\).
Calculate the energy of the electron when \(n = 6\) using the formula: \(E_6 = -\frac{R_H}{6^2}\).
Determine the energy difference between the two states: \(\Delta E = E_2 - E_6\). This energy difference corresponds to the energy of the photon released.
Use the relationship between energy and wavelength: \(\Delta E = \frac{hc}{\lambda}\), where \(h\) is Planck's constant (6.626 \(\times\) 10^{-34} \(\text{ J s}\)) and \(c\) is the speed of light (3.00 \(\times\) 10^8 \(\text{ m/s}\)), to solve for the wavelength \(\lambda\).
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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 the principal quantum number 'n'. The energy of an electron in these levels can be calculated using the formula E_n = -13.6 eV/n², where E_n is the energy at level n. This means that as n increases, the energy becomes less negative, indicating that the electron is less tightly bound to the nucleus.
Photon Emission and Wavelength
When an electron transitions between energy levels, it can emit or absorb a photon, which corresponds to the energy difference between the two levels. The energy of the emitted photon can be calculated using the equation ΔE = E_initial - E_final. The wavelength of the emitted radiation can then be determined using the equation λ = hc/ΔE, where h is Planck's constant and c is the speed of light.
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Planck's Constant and the Speed of Light
Planck's constant (h) is a fundamental constant that relates the energy of a photon to its frequency, given by E = hf, where f is the frequency. The speed of light (c) is the speed at which light travels in a vacuum, approximately 3.00 x 10^8 m/s. These constants are essential for converting energy differences into wavelengths, allowing for the calculation of the wavelength of emitted radiation during electron transitions.
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Related Practice
Textbook Question
The visible emission lines observed by Balmer all involved nf = 2. (b) Calculate the wavelengths of the first three lines in the Balmer series—those for which ni = 3, 4, and 5—and identify these lines in the emission spectrum shown in Figure 6.11.
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Textbook Question
Is energy emitted or absorbed when the following electronic transitions occur in hydrogen? b. from an orbit of radius 2.12 Å to one of radius 8.46 Å
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Textbook Question
The visible emission lines observed by Balmer all involved nf = 2. (a) Which of the following is the best explanation of why the lines with nf = 3 are not observed in the visible portion of the spectrum: (i) Transitions to nf = 3 are not allowed to happen, (ii) transitions to nf = 3 emit photons in the infrared portion of the spectrum, (iii) transitions to nf = 3 emit photons in the ultraviolet portion of the spectrum, or (iv) transitions to nf = 3 emit photons that are at exactly the same wavelengths as those to nf = 2.
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Textbook Question
Indicate whether energy is emitted or absorbed when the following electronic transitions occur in hydrogen: a. from n = 3 to n = 6
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Textbook Question
Consider a transition of the electron in the hydrogen atom from n = 4 to n = 9. b. Will the light be absorbed or emitted?
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