Questions: The shielding of electrons gives rise to an effective nuclear charge, Zeff, which explains why boron is larger than oxygen. Estimate the approximate Zeff felt by a valence electron of boron and oxygen, respectively?

The shielding of electrons gives rise to an effective nuclear charge, Zeff, which explains why boron is larger than oxygen. Estimate the approximate Zeff felt by a valence electron of boron and oxygen, respectively?
Transcript text: The shielding of electrons gives rise to an effective nuclear charge, $Z_{\text {eff }}$, which explains why boron is larger than oxygen. Estimate the approximate $Z_{\text {eff }}$ felt by a valence electron of boron and oxygen, respectively? View Available Hint(s) +5 and +8 +3 and +6 +5 and +6 +3 and +8 +1 and +4
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Solution

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Solution Steps

Step 1: Understanding Effective Nuclear Charge

The effective nuclear charge, \( Z_{\text{eff}} \), is the net positive charge experienced by an electron in a multi-electron atom. It accounts for the actual nuclear charge \( Z \) minus the shielding effect caused by other electrons.

Step 2: Calculating \( Z_{\text{eff}} \) for Boron

Boron has an atomic number \( Z = 5 \). The electron configuration is \( 1s^2 2s^2 2p^1 \). The valence electron in the 2p orbital is shielded by the 2 core electrons in the 1s orbital and the 2 electrons in the 2s orbital.

Using Slater's rules:

  • Electrons in the same group (2p) contribute 0.35 each.
  • Electrons in the n-1 shell (1s) contribute 0.85 each.

Thus, the shielding constant \( S \) is: \[ S = 2 \times 0.85 + 2 \times 0.35 = 1.7 + 0.7 = 2.4 \]

Therefore, the effective nuclear charge for boron is: \[ Z_{\text{eff}} = Z - S = 5 - 2.4 = 2.6 \]

Step 3: Calculating \( Z_{\text{eff}} \) for Oxygen

Oxygen has an atomic number \( Z = 8 \). The electron configuration is \( 1s^2 2s^2 2p^4 \). The valence electrons in the 2p orbital are shielded by the 2 core electrons in the 1s orbital and the 2 electrons in the 2s orbital.

Using Slater's rules:

  • Electrons in the same group (2p) contribute 0.35 each.
  • Electrons in the n-1 shell (1s) contribute 0.85 each.

Thus, the shielding constant \( S \) is: \[ S = 2 \times 0.85 + 2 \times 0.35 + 3 \times 0.35 = 1.7 + 0.7 + 1.05 = 3.45 \]

Therefore, the effective nuclear charge for oxygen is: \[ Z_{\text{eff}} = Z - S = 8 - 3.45 = 4.55 \]

Final Answer

The approximate \( Z_{\text{eff}} \) felt by a valence electron of boron and oxygen are:

\[ \boxed{Z_{\text{eff}} \text{ for Boron} = 2.6} \] \[ \boxed{Z_{\text{eff}} \text{ for Oxygen} = 4.55} \]

Given the multiple-choice options, the closest match is: \[ \boxed{+3 \text{ and } +6} \]

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