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

Ch 41: Quantum Mechanics II: Atomic Structure

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Young & Freedman Calc 14th Edition

Ch 41: Quantum Mechanics II: Atomic Structure

문제 34a

41장, 문제 34a

The energy of the $2 s$ state of lithium is $negative 5.391$ eV. Calculate the value of $Z_{e f f}$ for this state.

검증된 단계별 안내

Understand the concept of effective nuclear charge (Zeff): Zeff is the net positive charge experienced by an electron in a multi-electron atom. It accounts for the shielding effect caused by other electrons.

Use the formula for the energy of an electron in a hydrogen-like atom: $ E_n = -13.6 \frac{Z_{\text{eff}}^2}{n^2} $ (in eV), where $ E_n $ is the energy of the electron, $ Z_{\text{eff}} $ is the effective nuclear charge, and $ n $ is the principal quantum number.

Rearrange the formula to solve for $ Z_{\text{eff}} $: $ Z_{\text{eff}} = \sqrt{\frac{-E_n \cdot n^2}{13.6}} $.

Substitute the given values into the formula: $ E_n = -5.391 \ \text{eV} $ and $ n = 2 $ (since the electron is in the 2s state).

Simplify the expression to calculate $ Z_{\text{eff}} $: Perform the square root and division operations to find the effective nuclear charge for the 2s state of lithium.

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주요 개념

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Effective Nuclear Charge (Zeff)

Effective Nuclear Charge (Zeff) is the net positive charge experienced by an electron in a multi-electron atom. It accounts for the shielding effect caused by other electrons, which reduces the full nuclear charge. Zeff can be calculated using the formula Zeff = Z - S, where Z is the atomic number and S is the shielding constant. Understanding Zeff is crucial for predicting electron behavior and energy levels in atoms.

Energy Levels in Atoms

Energy levels in atoms refer to the quantized states that electrons can occupy. Each energy level corresponds to a specific energy value, which is influenced by the nuclear charge and electron-electron interactions. The energy of an electron in a given state can be calculated using the formula E = -Zeff^2 * 13.6 eV/n^2, where n is the principal quantum number. This concept is essential for understanding electron configurations and transitions.

Quantum Mechanics and Electron States

Quantum mechanics describes the behavior of particles at atomic and subatomic levels, including electrons in atoms. It introduces the idea that electrons exist in discrete states or orbitals, each with specific energy levels. The principles of quantum mechanics, such as wave-particle duality and uncertainty, are fundamental for analyzing electron configurations and predicting the properties of elements, including their ionization energies.