QuestionMay 6, 2025

Calculate the effective nuclear charge on a valence electron in an aluminum atom.

Calculate the effective nuclear charge on a valence electron in an aluminum atom.
Calculate the effective nuclear charge on a valence electron in an aluminum atom.

Solution
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Answer

Z_{\text{eff}} = 3.5 Explanation 1. Identify the atomic number and electron configuration Aluminum has an atomic number of 13. Its electron configuration is 1s^2 2s^2 2p^6 3s^2 3p^1. The valence electron is in the 3p orbital. 2. Apply Slater's rules to calculate shielding For the 3p electron: - Electrons in the same group (3s, 3p): Each contributes 0.35. - Electrons in the n=2 shell (2s, 2p): Each contributes 0.85. - Electrons in the n=1 shell (1s): Each contributes 1.00. Shielding constant S = (2 \times 0.35) + (8 \times 0.85) + (2 \times 1.00) = 0.7 + 6.8 + 2 = 9.5. 3. Calculate effective nuclear charge **Effective nuclear charge** Z_{\text{eff}} = Z - S, where Z is the atomic number. Z_{\text{eff}} = 13 - 9.5 = 3.5.

Explanation

1. Identify the atomic number and electron configuration<br /> Aluminum has an atomic number of 13. Its electron configuration is $1s^2 2s^2 2p^6 3s^2 3p^1$. The valence electron is in the $3p$ orbital.<br /><br />2. Apply Slater's rules to calculate shielding<br /> For the $3p$ electron:<br />- Electrons in the same group ($3s, 3p$): Each contributes $0.35$.<br />- Electrons in the $n=2$ shell ($2s, 2p$): Each contributes $0.85$.<br />- Electrons in the $n=1$ shell ($1s$): Each contributes $1.00$.<br /><br />Shielding constant $S = (2 \times 0.35) + (8 \times 0.85) + (2 \times 1.00) = 0.7 + 6.8 + 2 = 9.5$.<br /><br />3. Calculate effective nuclear charge<br /> **Effective nuclear charge** $Z_{\text{eff}} = Z - S$, where $Z$ is the atomic number.<br />$Z_{\text{eff}} = 13 - 9.5 = 3.5$.
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