The total number of orbital for (n + l) = 4 is — Atomic Structure Chemistry Question

💡 Solution & Explanation

**Step 1 - Identify the Subshell from Azimuthal Quantum Number** The azimuthal quantum number ($l$) defines the subshell type and shape within a principal energy level. The standard designations are as follows: * For $l = 0$: $s$-subshell * For $l = 1$: $p$-subshell * For $l = 2$: $d$-subshell * For $l = 3$: $f$-subshell Given that $l = 3$, we are dealing with the $f$-subshell. **Step 2 - Calculate the Number of Orbitals in the Subshell** The number of degenerate orbitals ($N_{\text{orbitals}}$) in any given subshell is determined by the azimuthal quantum number ($l$) according to the formula: $$N_{\text{orbitals}} = 2l + 1$$ Substituting the value $l = 3$: $$N_{\text{orbitals}} = 2(3) + 1$$ $$N_{\text{orbitals}} = 6 + 1 = 7 \text{ orbitals}$$ Thus, the $f$-subshell consists of $7$ distinct orbitals. **Step 3 - Calculate the Maximum Number of Electrons** According to Pauli's Exclusion Principle, any single orbital can accommodate a maximum of $2$ electrons, which must have opposite spins ($m_s = +1/2$ and $m_s = -1/2$). Therefore, the maximum number of electrons ($N_{\text{electrons}}$) that a subshell can hold is twice the number of orbitals in that subshell: $$N_{\text{electrons}} = 2 \times (2l + 1)$$ Substituting the values: $$N_{\text{electrons}} = 2 \times 7$$ $$N_{\text{electrons}} = \boxed{14}$$ --- **Step 4 - Evaluation of the Options** * **Option (A) 7**: This is incorrect because $7$ represents the number of *orbitals* in the $f$-subshell ($l=3$), not the maximum number of electrons. * **Option (B) 10**: This is incorrect because $10$ is the maximum number of electrons accommodated in a $d$-subshell, where $l = 2$ and $2(2(2)+1) = 10$. * **Option (C) is incorrect**: $12$ electrons does not correspond to the capacity of any standard atomic subshell. * **Option (D) 14**: This is correct because it matches the calculated capacity of $14$ electrons for the $f$-subshell ($l = 3$).