Which of the following species have same magnetic moment?

Options

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

Answer

To determine which of the given species have the same magnetic moment, we need to analyze the electronic configurations and the number of unpaired electrons for each species. The magnetic moment (µ) can be calculated using the formula:

\[ \mu = \sqrt{n(n + 2)} \]

where \(n\) is the number of unpaired electrons.

### Step-by-Step Solution:

1. **Identify the electronic configurations:**
- **Chromium (Cr)**: Atomic number 24, electronic configuration is \( [Ar] 3d^5 4s^1 \).
- **Chromium 3+ (Cr³⁺)**: Loses 3 electrons, configuration becomes \( [Ar] 3d^3 \).
- **Iron (Fe)**: Atomic number 26, electronic configuration is \( [Ar] 3d^6 4s^2 \).
- **Iron 3+ (Fe³⁺)**: Loses 3 electrons, configuration becomes \( [Ar] 3d^5 \).
- **Cobalt (Co)**: Atomic number 27, electronic configuration is \( [Ar] 3d^7 4s^2 \).
- **Cobalt 2+ (Co²⁺)**: Loses 2 electrons, configuration becomes \( [Ar] 3d^7 \).
- **Nickel (Ni)**: Atomic number 28, electronic configuration is \( [Ar] 3d^8 4s^2 \).
- **Nickel 2+ (Ni²⁺)**: Loses 2 electrons, configuration becomes \( [Ar] 3d^8 \).

2. **Count the number of unpaired electrons:**
- **Cr³⁺ (3d³)**: Configuration is \( \uparrow \uparrow \uparrow \) (3 unpaired electrons).
- **Fe³⁺ (3d⁵)**: Configuration is \( \uparrow \uparrow \uparrow \uparrow \uparrow \) (5 unpaired electrons).
- **Co²⁺ (3d⁷)**: Configuration is \( \uparrow \uparrow \uparrow \uparrow \uparrow \downarrow \downarrow \) (3 unpaired electrons).
- **Ni²⁺ (3d⁸)**: Configuration is \( \uparrow \uparrow \uparrow \uparrow \uparrow \downarrow \downarrow \downarrow \) (2 unpaired electrons).

3. **Calculate the magnetic moments:**
- For **Cr³⁺**:
\[ \mu = \sqrt{3(3 + 2)} = \sqrt{15} \]
- For **Fe³⁺**:
\[ \mu = \sqrt{5(5 + 2)} = \sqrt{35} \]
- For **Co²⁺**:
\[ \mu = \sqrt{3(3 + 2)} = \sqrt{15} \]
- For **Ni²⁺**:
\[ \mu = \sqrt{2(2 + 2)} = \sqrt{8} = 2\sqrt{2} \]

4. **Compare the magnetic moments:**
- **Cr³⁺** and **Co²⁺** both have 3 unpaired electrons, resulting in the same magnetic moment of \( \sqrt{15} \).
- **Fe³⁺** has 5 unpaired electrons, resulting in a different magnetic moment of \( \sqrt{35} \).
- **Ni²⁺** has 2 unpaired electrons, resulting in a different magnetic moment of \( 2\sqrt{2} \).

### Conclusion:
The species that have the same magnetic moment are **Cr³⁺** and **Co²⁺**.

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