Question

The effective nuclear charge of an element with three valence electrons is 2.60. What is the atomic number of the element?

• A. 1
• B. 2
• C. 3
• D. 4
• E. 5

Hint:

For \(n=1\) electrons, the shielding constant is 0.30. For \(n \geq 2\), the electrons in the same shell shield by 0.35.

Answer 1

Answer: (E)

Concept: Effective nuclear charge (\(Z_{eff}\)) is calculated using Slater's Rules: \(Z_{eff} = Z - \sigma\), where \(Z\) is the atomic number and \(\sigma\) is the shielding (screening) constant.

Step 1: Test the options.
An element with 3 valence electrons likely belongs to Group 13. Let's check Boron (\(Z=5\)). Electronic configuration of Boron: \(1s^2, 2s^2, 2p^1\).

Step 2: Calculate \(\sigma\) using Slater's Rules for a \(2p\) electron.

• Electrons in the same group (\(2s, 2p\)): There are 2 other electrons (\(2s^2\)). Each contributes 0.35. \[ 2 \times 0.35 = 0.70 \]
• Electrons in the \((n-1)\) shell (\(1s\)): There are 2 electrons. Each contributes 0.85. \[ 2 \times 0.85 = 1.70 \] Total \(\sigma = 0.70 + 1.70 = 2.40\).

Step 3: Calculate \(Z_{eff}\).
\[ Z_{eff} = Z - \sigma = 5 - 2.40 = 2.60 \] This matches the value given in the question. Thus, the atomic number is 5.