in `Sc^(3+),Ti^(2+),Ti^(3+),V^(2+)` increasing order of spin only magnetic moment is:

Options

A. `Sc^(3+)ltTi^(2+)ltTi^(3+)ltV^(2+)`
B. `Sc^(3+)ltTi^(3+)ltTi^(2+)ltV^(2+)`
C. `Ti^(2+)ltSc^(3+)ltTi^(3+)ltV^(2+)`
D. `Sc^(3+)ltTi^(2+)ltV^(2+)ltTi^(3+)`

Answer

To determine the increasing order of spin-only magnetic moments for the ions Sc^(3+), Ti^(2+), Ti^(3+), and V^(2+), we will follow these steps:

### Step 1: Identify the electronic configurations of the ions
1. **Scandium (Sc)**: Atomic number = 21
- Electronic configuration: [Ar] 3d¹ 4s²
- For Sc^(3+), we remove 3 electrons (2 from 4s and 1 from 3d):
- Configuration: 3d⁰ (0 unpaired electrons)

2. **Titanium (Ti)**: Atomic number = 22
- Electronic configuration: [Ar] 3d² 4s²
- For Ti^(2+), we remove 2 electrons (both from 4s):
- Configuration: 3d² (2 unpaired electrons)
- For Ti^(3+), we remove 3 electrons (2 from 4s and 1 from 3d):
- Configuration: 3d¹ (1 unpaired electron)

3. **Vanadium (V)**: Atomic number = 23
- Electronic configuration: [Ar] 3d³ 4s²
- For V^(2+), we remove 2 electrons (both from 4s):
- Configuration: 3d³ (3 unpaired electrons)

### Step 2: Count the number of unpaired electrons
- **Sc^(3+)**: 0 unpaired electrons
- **Ti^(2+)**: 2 unpaired electrons
- **Ti^(3+)**: 1 unpaired electron
- **V^(2+)**: 3 unpaired electrons

### Step 3: Apply the formula for spin-only magnetic moment
The formula for the spin-only magnetic moment (\( \mu \)) is given by:
\[ \mu = \sqrt{n(n + 2)} \text{ Bohr Magneton} \]
where \( n \) is the number of unpaired electrons.

1. **Sc^(3+)**:
- \( n = 0 \)
- \( \mu = \sqrt{0(0 + 2)} = 0 \text{ Bohr Magneton} \)

2. **Ti^(2+)**:
- \( n = 2 \)
- \( \mu = \sqrt{2(2 + 2)} = \sqrt{8} = 2.83 \text{ Bohr Magneton} \)

3. **Ti^(3+)**:
- \( n = 1 \)
- \( \mu = \sqrt{1(1 + 2)} = \sqrt{3} = 1.73 \text{ Bohr Magneton} \)

4. **V^(2+)**:
- \( n = 3 \)
- \( \mu = \sqrt{3(3 + 2)} = \sqrt{15} = 3.87 \text{ Bohr Magneton} \)

### Step 4: Arrange the ions in increasing order of magnetic moment
- Sc^(3+): 0 Bohr Magneton
- Ti^(3+): 1.73 Bohr Magneton
- Ti^(2+): 2.83 Bohr Magneton
- V^(2+): 3.87 Bohr Magneton

### Final Order
The increasing order of spin-only magnetic moments is:
\[ \text{Sc}^{3+} < \text{Ti}^{3+} < \text{Ti}^{2+} < \text{V}^{2+} \]