Ge (Z = 32) in its ground state electronic configuration has \( x \) completely filled orbitals with magnetic quantum number \( m_l = 0 \). The value of \( x \) is _________.
Detailed Step-by-Step Solution
To solve this problem, we need to write the full electronic configuration of Germanium and then count the specific orbitals satisfying two conditions: they must have \( m_l = 0 \) and they must be completely filled (contain 2 electrons).
Step 1: Electronic Configuration of Ge (Z = 32)
Using the Aufbau principle, the ground state electronic configuration is:
Step 2: Analyze Orbitals with \( m_l = 0 \)
Let's remember how the magnetic quantum number (\( m_l \)) works for different subshells:
- s-subshell (\(l=0\)): Has 1 orbital, and its \( m_l = 0 \).
- p-subshell (\(l=1\)): Has 3 orbitals (\( m_l = -1, 0, +1 \)). Exactly one has \( m_l = 0 \).
- d-subshell (\(l=2\)): Has 5 orbitals (\( m_l = -2, -1, 0, +1, +2 \)). Exactly one has \( m_l = 0 \).
Step 3: Count the Completely Filled Orbitals
Now we evaluate each subshell in Germanium to see if its \( m_l = 0 \) orbital is completely filled (has 2 electrons).
| Subshell | Total Electrons | Is the \(m_l=0\) orbital completely filled? | Count |
|---|---|---|---|
| \(1s\) | 2 | Yes (2 electrons) | 1 |
| \(2s\) | 2 | Yes (2 electrons) | 1 |
| \(2p\) | 6 | Yes (All 3 orbitals are full) | 1 |
| \(3s\) | 2 | Yes (2 electrons) | 1 |
| \(3p\) | 6 | Yes (All 3 orbitals are full) | 1 |
| \(4s\) | 2 | Yes (2 electrons) | 1 |
| \(3d\) | 10 | Yes (All 5 orbitals are full) | 1 |
| \(4p\) | 2 | No! By Hund's Rule, the 2 electrons occupy separate orbitals singly (e.g., \(p_x^1, p_y^1\)). No orbital in \(4p\) has 2 electrons. | 0 |
Conclusion: Adding the counts together (\(1+1+1+1+1+1+1\)), we find there are 7 completely filled orbitals with \( m_l = 0 \). Therefore, the value of \( x \) is 7.
Master Hund's Rule and Quantum Numbers
Questions like this are designed to test your attention to detail. Many students correctly identify that every subshell has an \( m_l = 0 \) orbital, but they forget to apply Hund's Rule of Maximum Multiplicity to the outermost \( 4p^2 \) electrons, mistakenly counting it as a filled orbital.
To master orbital filling rules (Aufbau Principle, Pauli Exclusion Principle, and Hund's Rule), be sure to review our comprehensive guide on the Structure of Atom Class 11 Chemistry.
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