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Ch 41: Quantum Mechanics II: Atomic Structure
Young & Freedman Calc - University Physics 14th Edition
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook
All textbooksYoung & Freedman Calc 14th EditionCh 41: Quantum Mechanics II: Atomic StructureProblem 35
Chapter 41, Problem 35

Estimate the energy of the highest-ll state for (a) the LL shell of Be+ and (b) the NN shell of Ca+.

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Understand the problem: The goal is to estimate the energy of the highest-l state for the given shells of the ions Be+ and Ca+. The energy levels in an atom or ion are determined by the principal quantum number (n), the azimuthal quantum number (l), and the effective nuclear charge (Z_eff). The highest-l state corresponds to l = n - 1 for a given shell.
Recall the formula for the energy of an electron in a hydrogen-like ion: \( E_n = - \frac{Z_{\text{eff}}^2 \cdot 13.6 \text{ eV}}{n^2} \), where \( Z_{\text{eff}} \) is the effective nuclear charge and \( n \) is the principal quantum number. For multi-electron atoms, \( Z_{\text{eff}} \) accounts for the shielding effect of inner electrons.
For part (a), the L shell corresponds to \( n = 2 \). The highest-l state is \( l = n - 1 = 1 \). Estimate \( Z_{\text{eff}} \) for Be+ (atomic number 4) by considering the shielding effect of the inner electrons. Use \( Z_{\text{eff}} \approx Z - \text{shielding} \). Substitute \( n = 2 \) and the estimated \( Z_{\text{eff}} \) into the energy formula.
For part (b), the N shell corresponds to \( n = 4 \). The highest-l state is \( l = n - 1 = 3 \). Estimate \( Z_{\text{eff}} \) for Ca+ (atomic number 20) by considering the shielding effect of the inner electrons. Use \( Z_{\text{eff}} \approx Z - \text{shielding} \). Substitute \( n = 4 \) and the estimated \( Z_{\text{eff}} \) into the energy formula.
Compare the results for the two ions. Note that the energy of the highest-l state depends on both \( Z_{\text{eff}} \) and \( n \). The higher the \( Z_{\text{eff}} \), the more tightly bound the electron is, resulting in a lower (more negative) energy.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Energy Levels in Atoms

Energy levels in atoms refer to the discrete energy states that electrons can occupy. Each level corresponds to a specific shell, denoted by quantum numbers, and is associated with a certain amount of energy. The energy of an electron in a given shell is influenced by the nuclear charge and the electron's distance from the nucleus.
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Quantum Numbers

Quantum numbers are sets of numerical values that describe the unique quantum state of an electron in an atom. The principal quantum number (n) indicates the energy level, while the azimuthal quantum number (l) describes the shape of the orbital. For example, in the L shell, l can be 0 (s orbital) or 1 (p orbital), affecting the energy and spatial distribution of electrons.
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Ionization Energy

Ionization energy is the energy required to remove an electron from an atom or ion in its gaseous state. It varies depending on the electron's energy level and the effective nuclear charge experienced by the electron. For ions like Be+ and Ca+, the ionization energy can be used to estimate the energy of electrons in specific shells, such as the L and N shells.
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Related Practice
Textbook Question

The energy of the 2s2s state of lithium is −5.391-5.391 eV. Calculate the value of ZeffZ_{eff} for this state.

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Textbook Question

The doubly charged ion N2+ is formed by removing two electrons from a nitrogen atom. What is the ground-state electron configuration for the N2+ ion?

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Textbook Question

The energies for an electron in the KK, LL, and MM shells of the tungsten atom are −69,500-69,500 eV, −12,000-12,000 eV, and −2,200-2,200 eV, respectively. Calculate the wavelengths of the KαK_{\(\alpha\)} and KβK_{\(\beta\)} x rays of tungsten.

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Textbook Question

Estimate the energy of the least strongly bound level in the LL shell of N2+.

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