Ch 41: Quantum Mechanics II: Atomic Structure

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 41: Quantum Mechanics II: Atomic Structure

Problema 33b

Capitolo 41, Problema 33b

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

Guida verificata passo dopo passo

Step 1: Understand the problem. The question asks for the energy of the least strongly bound level in the L shell of N2+. This involves calculating the binding energy of an electron in the L shell of a nitrogen ion with a charge of +2.

Step 2: Recall the formula for the energy levels of a hydrogen-like ion: $ E_n = - \frac{Z^2 \cdot R_H}{n^2} $, where $ Z $ is the atomic number, $ R_H $ is the Rydberg constant, and $ n $ is the principal quantum number. For the L shell, $ n = 2 $.

Step 3: Determine the effective nuclear charge $ Z_{eff} $. For N2+, the atomic number $ Z $ of nitrogen is 7, but since two electrons have been removed, the effective nuclear charge experienced by the remaining electrons will be higher due to reduced electron shielding.

Step 4: Substitute $ Z_{eff} $, $ R_H $, and $ n = 2 $ into the formula $ E_n = - \frac{Z_{eff}^2 \cdot R_H}{n^2} $. This will give the energy of the least strongly bound level in the L shell.

Step 5: Interpret the result. The negative sign in the energy indicates that the electron is bound to the nucleus. The magnitude of the energy represents how strongly the electron is bound, with smaller absolute values corresponding to less strongly bound levels.

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Binding Energy

Binding energy is the energy required to remove an electron from an atom or ion. In the context of atomic physics, it indicates how strongly an electron is held by the nucleus. The least strongly bound level corresponds to the electron that is easiest to remove, which is crucial for understanding ionization processes and energy levels in multi-electron systems.

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Intro to Energy & Types of Energy

Quantum Levels

Quantum levels, or energy levels, refer to the discrete energy states that electrons can occupy in an atom. Each level is associated with a specific quantum number and can hold a limited number of electrons. For nitrogen ions like N2+, the L shell corresponds to the second energy level, which is important for determining the energy of electrons in that shell.

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Sound Intensity Level and the Decibel Scale

Ionization Energy

Ionization energy is the amount of energy needed to remove an electron from an atom or ion in its gaseous state. It is a critical concept in understanding the stability of electrons in various shells and how they interact with external energy sources. For N2+, estimating the energy of the least strongly bound level involves calculating the ionization energy for the L shell electrons.

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Intro to Energy & Types of Energy

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Concetti chiave

Binding Energy

Quantum Levels

Ionization Energy

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