When the value of the azimuthal quantum number is 3 the maximum and the minimum values of the spin multiplicites are

Options

A. 4,3
B. 8,1
C. 1,3
D. 8,2

Answer

To solve the problem of determining the maximum and minimum values of the spin multiplicities when the azimuthal quantum number (L) is 3, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Orbital Type**:
- The azimuthal quantum number \( L = 3 \) corresponds to the F orbital.

2. **Determine the Number of Orbitals**:
- The F subshell has a total of 7 orbitals. These are labeled as \( f_{-3}, f_{-2}, f_{-1}, f_{0}, f_{1}, f_{2}, f_{3} \).

3. **Calculate Maximum Spin Multiplicity**:
- To achieve maximum spin multiplicity, we need to maximize the number of unpaired electrons. This is done by singly occupying each of the 7 orbitals.
- Each electron has a spin quantum number of \( +\frac{1}{2} \).
- Therefore, if all 7 orbitals are singly occupied, the total spin \( S \) is:
\[ S = \frac{7}{2} \]
- The formula for spin multiplicity is given by:
\[ \text{Spin Multiplicity} = 2S + 1 \]
- Substituting the value of \( S \):
\[ \text{Spin Multiplicity} = 2 \left(\frac{7}{2}\right) + 1 = 7 + 1 = 8 \]

4. **Calculate Minimum Spin Multiplicity**:
- To achieve minimum spin multiplicity, all electrons must be paired. In the case of the F orbital, with 7 electrons, we can pair 6 electrons (3 pairs) and leave one electron unpaired.
- When electrons are paired, their spins cancel each other out. Thus, the total spin \( S \) becomes:
\[ S = 0 \]
- Using the spin multiplicity formula again:
\[ \text{Spin Multiplicity} = 2(0) + 1 = 1 \]

5. **Conclusion**:
- The maximum spin multiplicity when \( L = 3 \) is **8**.
- The minimum spin multiplicity when \( L = 3 \) is **1**.

### Final Answer:
- Maximum Spin Multiplicity: **8**
- Minimum Spin Multiplicity: **1**