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 moment for the ions Sc^(3+), Ti^(2+), Ti^(3+), and V^(2+), we will follow these steps:

### Step 1: Determine the electronic configurations of the ions
1. **Scandium (Sc)** has the atomic number 21. Its electronic configuration is:
- Sc: [Ar] 4s² 3d¹
- For Sc^(3+), we remove 3 electrons (2 from 4s and 1 from 3d):
- Sc^(3+): [Ar] 4s⁰ 3d⁰ (0 unpaired electrons)

2. **Titanium (Ti)** has the atomic number 22. Its electronic configuration is:
- Ti: [Ar] 4s² 3d²
- For Ti^(2+), we remove 2 electrons (both from 4s):
- Ti^(2+): [Ar] 4s⁰ 3d² (2 unpaired electrons)
- For Ti^(3+), we remove 3 electrons (2 from 4s and 1 from 3d):
- Ti^(3+): [Ar] 4s⁰ 3d¹ (1 unpaired electron)

3. **Vanadium (V)** has the atomic number 23. Its electronic configuration is:
- V: [Ar] 4s² 3d³
- For V^(2+), we remove 2 electrons (both from 4s):
- V^(2+): [Ar] 4s⁰ 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: 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.

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

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

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

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

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

Thus, the increasing order of spin-only magnetic moment is:
\[ \text{Sc}^{3+} < \text{Ti}^{3+} < \text{Ti}^{2+} < \text{V}^{2+} \]

### Final Answer
The increasing order of spin-only magnetic moment is:
**Sc^(3+) < Ti^(3+) < Ti^(2+) < V^(2+)**