Which of the following configuration is violating Pauli's exclusion principle? — Atomic Structure Chemistry Question

💡 Solution & Explanation

### Step 1 - Pauli's Exclusion Principle and Quantum Numbers According to Pauli's exclusion principle, no two electrons in a single atom can have the same set of all four quantum numbers: 1. Principal quantum number ($n$) 2. Azimuthal quantum number ($l$) 3. Magnetic quantum number ($m_l$) 4. Spin quantum number ($s$) If two electrons reside in the same spatial orbital, they share the identical set of orbital quantum numbers ($n$, $l$, and $m_l$). Therefore, they must possess different spin quantum numbers ($s$). Each orbital can accommodate a maximum of two electrons with opposite (antiparallel) spins. --- ### Step 2 - Analysis of Configuration (a) In configuration (a), the $\ce{2s}$ orbital contains two electrons with parallel spins, both pointing upwards ($\uparrow\uparrow$). Quantum numbers for each electron in the $\ce{2s}$ orbital: * First electron: $(n, l, m_l, s) = (2, 0, 0, +1/2)$ * Second electron: $(n, l, m_l, s) = (2, 0, 0, +1/2)$ Because both electrons have the exact same set of four quantum numbers, configuration (a) directly **violates Pauli's exclusion principle**. --- ### Step 3 - Analysis of Configuration (b) In configuration (b), the $\ce{2s}$ orbital contains two electrons with opposite spins ($\uparrow\downarrow$): * $(2, 0, 0, +1/2)$ and $(2, 0, 0, -1/2)$ Every electron has a unique set of four quantum numbers. Configuration (b) is valid — does **not** violate Pauli's exclusion principle. --- ### Step 4 - Analysis of Configuration (c) In configuration (c), the $\ce{2s}$ orbital has one electron and the $\ce{2p}$ subshell has one paired orbital (opposite spins) and two singly occupied orbitals. Every electron has a unique set of quantum numbers. Configuration (c) violates Hund's rule, but does **not** violate Pauli's exclusion principle. --- ### Step 5 - Systematic Analysis of the Options * **Option (A):** Correct — Configuration (a) violates Pauli's exclusion principle (two parallel spins in same $\ce{2s}$ orbital). * **Option (B):** Incorrect — Configuration (b) is a perfectly valid configuration obeying all quantum mechanical rules. * **Option (C):** Incorrect — Configuration (c) violates Hund's rule only, not Pauli's exclusion principle. * **Option (D):** Incorrect — Since neither (b) nor (c) violates Pauli's exclusion principle, this combination is incorrect. $$\boxed{\text{A}}$$