Which of the following pairs will show the same magnetic moment ('spin only') ?

Options

A. `[Cr(H_(2)O)_(6)]^(3+) amd [Fe(H_(2)O)_(5)NO]^(2+)`
B. `[Mn(CN)_(6)]^(4-) and [Fe(CN)_(6)]^(3-)`
C. `[Ni(CO)_(4)] and [Zn(NH_(3))_(4)]^(2+)`
D. All of these

Explanation

To determine which of the given pairs will show the same magnetic moment ('spin only'), we will follow these steps:

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

### Step 2: Identify the Oxidation States
For each complex in the pairs provided, we need to determine the oxidation state of the central metal ion.

### Step 3: Determine the Number of Unpaired Electrons
Once we have the oxidation state, we can determine the electronic configuration of the metal ion and find out how many unpaired electrons are present.

### Step 4: Calculate the Magnetic Moment
Using the number of unpaired electrons, we can calculate the magnetic moment for each complex using the formula from Step 1.

### Step 5: Compare the Magnetic Moments
Finally, we will compare the calculated magnetic moments of the pairs to see which ones are the same.

### Example Calculation
Let's assume we have the first pair: [Cr(H2O)6]^{3+} and [Mn(H2O)6]^{2+}.

1. **Determine Oxidation States:**
- For [Cr(H2O)6]^{3+}: Let oxidation state of Cr be \( x \).
\[ x + 0 = +3 \implies x = +3 \]
- For [Mn(H2O)6]^{2+}: Let oxidation state of Mn be \( y \).
\[ y + 0 = +2 \implies y = +2 \]

2. **Electronic Configurations:**
- Cr^{3+} (Atomic number 24): Configuration is [Ar] 3d^3.
- Mn^{2+} (Atomic number 25): Configuration is [Ar] 3d^5.

3. **Count Unpaired Electrons:**
- For Cr^{3+}: 3 unpaired electrons (3d^3).
- For Mn^{2+}: 5 unpaired electrons (3d^5).

4. **Calculate Magnetic Moments:**
- For Cr^{3+}:
\[ \mu = \sqrt{3(3 + 2)} = \sqrt{15} \approx 3.87 \, \mu_B \]
- For Mn^{2+}:
\[ \mu = \sqrt{5(5 + 2)} = \sqrt{35} \approx 5.92 \, \mu_B \]

5. **Comparison:**
Since \( \mu \) values are different, this pair does not show the same magnetic moment.

### Conclusion
By repeating this process for each of the pairs, we can determine which pairs have the same magnetic moment.