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Ch 41: Quantum Mechanics II: Atomic Structure
Young & Freedman Calc - University Physics 14th Edition
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoYoung & Freedman Calc 14th EditionCh 41: Quantum Mechanics II: Atomic StructureProblema 35
Capitolo 41, Problema 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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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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Intro to Energy & Types of Energy
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