Not considering the electronic spin, the degeneracy of the second excited state (\( n = 3 \)) of H atom is 9, while the degeneracy of the second excited state of H\(^-\) is _________.
Detailed Conceptual Analysis
This is a brilliant conceptual trap! The question contrasts a single-electron system (H atom) with a multi-electron system (H\(^-\) ion).
Step 1: Understand the H atom (Single-Electron System)
For a single-electron system like the Hydrogen atom, the energy of an orbital depends only on the principal quantum number (\(n\)).
Therefore, for the second excited state (\(n = 3\)), the \(3s\), \(3p\), and \(3d\) subshells all have the exact same energy (they are degenerate).
Number of orbitals = \(1\) (from \(3s\)) + \(3\) (from \(3p\)) + \(5\) (from \(3d\)) = 9.
Step 2: Understand the H\(^-\) Ion (Multi-Electron System)
The Hydride ion (H\(^-\)) has two electrons. In multi-electron systems, electrons repel each other (shielding effect), which causes the energy levels to split. The energy now depends on both \(n\) and \(l\) according to the Aufbau \((n+l)\) rule.
The increasing order of energy levels for H\(^-\) is:
Step 3: Determine the States of H\(^-\)
Let's map out the ground and excited states for the two electrons in H\(^-\):
- Ground State: Both electrons are in the lowest energy level: \( 1s^2 \).
- First Excited State: One electron gets promoted to the next available energy level: \( 1s^1 \ 2s^1 \).
- Second Excited State: The electron is promoted to the next higher energy level after \(2s\): \( \mathbf{1s^1 \ 2p^1} \).
Step 4: Calculate the Degeneracy of the Second Excited State
The second excited state configuration is \( 1s^1 \ 2p^1 \).
The question explicitly asks to "Not consider the electronic spin." Therefore, we only look at spatial degeneracy (the number of available orbitals for that specific configuration).
- → The \(1s\) electron is in the only available \(s\) orbital (1 option).
- → The \(2p\) electron can be in any of the three \(p\) orbitals (\(p_x, p_y, \text{ or } p_z\)) which all have the exact same energy (3 options).
Conclusion: The degeneracy of the second excited state of H\(^-\) (ignoring spin) is exactly 3.
Mastering Degeneracy and Energy Levels
This question highlights why JEE Advanced is considered one of the toughest exams. It tests if you blindly apply formulas or if you truly understand the physics. The moment a system has more than one electron (like H\(^-\), He, Li\(^+\)), the subshells (\(s, p, d\)) within a principal shell are no longer degenerate due to inter-electronic repulsions and varying penetration effects.
To solidify your understanding of the \((n+l)\) rule, orbital penetration, shielding effects, and quantum mechanics, we highly recommend reading our detailed conceptual guide on the Structure of Atom Class 11 Chemistry.
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