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

Options

A. 1. `V^(+) lt Ti^(2+) lt Ti^(3+) lt Sc^(3+)`
B. 2. `Sc^(3+) lt Ti^(3+) lt V^(2+) lt Ti^(2+)`
C. 3. `Sc^(3+) lt Ti^(3+) lt Ti^(2+) lt V^(2+)`
D. 4. `Ti^(3+) lt Ti^(2+) lt Sc^(3+) lt V^(2+)`

Explanation

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 number of electrons in each ion
- **Sc^(3+)**: Scandium has an atomic number of 21. Therefore, Sc^(3+) has 21 - 3 = 18 electrons.
- **Ti^(2+)**: Titanium has an atomic number of 22. Therefore, Ti^(2+) has 22 - 2 = 20 electrons.
- **Ti^(3+)**: Ti^(3+) has 22 - 3 = 19 electrons.
- **V^(2+)**: Vanadium has an atomic number of 23. Therefore, V^(2+) has 23 - 2 = 21 electrons.

### Step 2: Determine the electron configuration for each ion
- **Sc^(3+)**: The electron configuration is 1s² 2s² 2p⁶ 3s² 3p⁶ (completely filled) → 0 unpaired electrons.
- **Ti^(2+)**: The electron configuration is 1s² 2s² 2p⁶ 3s² 3p⁶ 3d² → 2 unpaired electrons.
- **Ti^(3+)**: The electron configuration is 1s² 2s² 2p⁶ 3s² 3p⁶ 3d¹ → 1 unpaired electron.
- **V^(2+)**: The electron configuration is 1s² 2s² 2p⁶ 3s² 3p⁶ 3d³ → 3 unpaired electrons.

### Step 3: Calculate the spin-only magnetic moment using the formula
The formula for the spin-only magnetic moment (μ) is:
\[ \mu = \sqrt{n(n + 2)} \]
where n is the number of unpaired electrons.

- **Sc^(3+)**: n = 0
\[ \mu = \sqrt{0(0 + 2)} = 0 \, \text{Bohr magneton} \]

- **Ti^(2+)**: n = 2
\[ \mu = \sqrt{2(2 + 2)} = \sqrt{8} = 2\sqrt{2} \, \text{Bohr magneton} \]

- **Ti^(3+)**: n = 1
\[ \mu = \sqrt{1(1 + 2)} = \sqrt{3} \, \text{Bohr magneton} \]

- **V^(2+)**: n = 3
\[ \mu = \sqrt{3(3 + 2)} = \sqrt{15} \, \text{Bohr magneton} \]

### Step 4: Arrange the magnetic moments in increasing order
Now we can summarize the magnetic moments:
- Sc^(3+): 0 Bohr magneton
- Ti^(3+): √3 Bohr magneton
- Ti^(2+): 2√2 Bohr magneton
- V^(2+): √15 Bohr magneton

To compare:
- 0 < √3 < 2√2 < √15

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