Transition elements show magnetic moment due to spin and orbital motion of electrons. Which of the following metallic ions have almost same spin only magnetic moment?

Options

A. `Co^(2+)`
B. `Cr^(2+)`
C. `Mn^(2+)`
D. `Cr^(3+)`

Explanation

To solve the question regarding which metallic ions have almost the same spin-only magnetic moment, we will follow these steps:

### Step 1: Understand the Formula for Spin-Only Magnetic Moment
The spin-only magnetic moment (μ) can be calculated using the formula:
\[ \mu_s = \sqrt{n(n + 2)} \]
where \( n \) is the number of unpaired electrons.

### Step 2: Analyze Each Ion
We will analyze the given ions one by one to determine the number of unpaired electrons.

#### 1. Cobalt (Co²⁺)
- Cobalt has the electron configuration: \[ [Ar] 3d^7 4s^2 \]
- For Co²⁺, it loses 2 electrons (from 4s), leading to: \[ 3d^7 \]
- The distribution of electrons in 3d subshell for Co²⁺ is:
- ↑↓ ↑↓ ↑ ↑ ↑
- Number of unpaired electrons (n) = 3

#### 2. Chromium (Cr²⁺)
- Chromium has the electron configuration: \[ [Ar] 3d^5 4s^1 \]
- For Cr²⁺, it loses 1 electron from 4s and 1 from 3d, leading to: \[ 3d^4 \]
- The distribution of electrons in 3d subshell for Cr²⁺ is:
- ↑ ↑ ↑ ↑
- Number of unpaired electrons (n) = 4

#### 3. Manganese (Mn²⁺)
- Manganese has the electron configuration: \[ [Ar] 3d^5 4s^2 \]
- For Mn²⁺, it loses 2 electrons (from 4s), leading to: \[ 3d^5 \]
- The distribution of electrons in 3d subshell for Mn²⁺ is:
- ↑ ↑ ↑ ↑ ↑
- Number of unpaired electrons (n) = 5

#### 4. Chromium (Cr³⁺)
- Chromium has the electron configuration: \[ [Ar] 3d^5 4s^1 \]
- For Cr³⁺, it loses 1 electron from 4s and 2 from 3d, leading to: \[ 3d^3 \]
- The distribution of electrons in 3d subshell for Cr³⁺ is:
- ↑ ↑ ↑
- Number of unpaired electrons (n) = 3

### Step 3: Calculate Spin-Only Magnetic Moments
Now, we can calculate the spin-only magnetic moment for each ion:

1. **Cobalt (Co²⁺)**:
\[ \mu_s = \sqrt{3(3 + 2)} = \sqrt{15} \approx 3.87 \text{ Bohr magneton} \]

2. **Chromium (Cr²⁺)**:
\[ \mu_s = \sqrt{4(4 + 2)} = \sqrt{24} \approx 4.90 \text{ Bohr magneton} \]

3. **Manganese (Mn²⁺)**:
\[ \mu_s = \sqrt{5(5 + 2)} = \sqrt{35} \approx 5.92 \text{ Bohr magneton} \]

4. **Chromium (Cr³⁺)**:
\[ \mu_s = \sqrt{3(3 + 2)} = \sqrt{15} \approx 3.87 \text{ Bohr magneton} \]

### Step 4: Identify the Ions with the Same Magnetic Moment
From the calculations:
- Co²⁺ and Cr³⁺ both have the same number of unpaired electrons (3) and thus the same spin-only magnetic moment of approximately 3.87 Bohr magneton.

### Conclusion
The metallic ions that have almost the same spin-only magnetic moment are **Cobalt (Co²⁺)** and **Chromium (Cr³⁺)**.

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