If the numbers of orbitals of a particular type were (3l + 1), but spin quantum numbers were only +1 — Atomic Structure Chemistry Question

💡 Solution & Explanation

### Step 1 - Understand the Hypothetical Rule for Orbital Count In standard quantum mechanics, the number of spatial orbitals in a subshell characterized by the azimuthal quantum number \(l\) is given by the formula \(2l + 1\). In this hypothetical problem, we are given a new rule where the number of orbitals (\(N_{\text{orbitals}}\)) for a subshell of type \(l\) is determined by the formula: $$N_{\text{orbitals}} = 3l + 1$$ --- ### Step 2 - Identify the Azimuthal Quantum Number for \(d\)-type Orbitals The azimuthal quantum number (\(l\)) corresponds to the shape and type of the subshell. The standard values of \(l\) for different subshells are: * \(s\)-subshell: \(l = 0\) * \(p\)-subshell: \(l = 1\) * \(d\)-subshell: \(l = 2\) * \(f\)-subshell: \(l = 3\) For the \(d\)-type orbitals, we identify: $$l = 2$$ --- ### Step 3 - Calculate the Number of Orbitals in the \(d\)-subshell Using the hypothetical formula from Step 1, we substitute the value of \(l = 2\) to find the total number of orbitals in the \(d\)-subshell: $$N_{\text{orbitals}} = 3l + 1$$ $$N_{\text{orbitals}} = 3(2) + 1$$ $$N_{\text{orbitals}} = 6 + 1 = 7\text{ orbitals}$$ Under this modified rule, the \(d\)-subshell consists of exactly \(7\) spatial orbitals instead of the usual \(5\) orbitals. --- ### Step 4 - Calculate the Maximum Electron Capacity of the \(d\)-subshell According to Pauli's Exclusion Principle, each spatial orbital can accommodate a maximum of \(2\) electrons, provided they have opposite spins. The problem states that the spin quantum numbers (\(m_s\)) are still restricted to only the two standard values: $$m_s \in \left\{+\frac{1}{2}, -\frac{1}{2}\right\}$$ Since there are only \(2\) possible spin states, each orbital continues to hold a maximum of \(2\) electrons. Therefore, the maximum electron capacity (\(N_{\text{electrons}}\)) of the \(d\)-type subshell is: $$N_{\text{electrons}} = N_{\text{orbitals}} \times 2 = 7 \times 2 = \boxed{14}$$ --- ### Step 5 - Systematic Analysis of the Options * **Option (A) 10:** Incorrect. This is the standard real-world maximum capacity of a \(d\)-subshell (\(5 \times 2 = 10\)). * **Option (B) 14:** Correct. Under the hypothetical rule \(3l + 1\), the \(d\)-subshell has \(7\) orbitals → maximum \(14\) electrons. * **Option (C) 7:** Incorrect. This is the number of orbitals, not electrons. * **Option (D) 5:** Incorrect. This is the normal number of \(d\)-orbitals. $$\text{Correct Option: } \boxed{\text{B}}$$