Which of the following complex ion has a magnetic moment same as `[Cr(H_(2)O)_(6)]^(3+)`?

Options

A. `[Mn(H_(2)O)_(6)]^(4+)`
B. `[Mn(H_(2)O)_(6)]^(3+)`
C. `[Fe(H_(2)O)_(6)]^(3+)`
D. `[Cu(H_(3)N)_(4)]^(2+)`

Answer

To determine which of the following complex ions has a magnetic moment the same as \([Cr(H_2O)_6]^{3+}\), we will follow these steps:

### Step 1: Determine the oxidation state of chromium in \([Cr(H_2O)_6]^{3+}\)
- Let \(x\) be the oxidation state of chromium.
- Since water is a neutral ligand, the equation is:
\[ x + 6(0) = +3 \implies x = +3 \]
- Therefore, the oxidation state of chromium in this complex is +3.

### Step 2: Write the electronic configuration of chromium
- The atomic number of chromium (Cr) is 24.
- The electronic configuration of Cr is:
\[ [Ar] 4s^1 3d^5 \]

### Step 3: Determine the electronic configuration of \(Cr^{3+}\)
- For \(Cr^{3+}\), we remove three electrons:
- The first electron is removed from the \(4s\) orbital, and the next two are removed from the \(3d\) orbital.
- Thus, the configuration becomes:
\[ 4s^0 3d^3 \quad \text{or simply} \quad d^3 \]

### Step 4: Count the number of unpaired electrons
- In the \(d^3\ configuration, the three electrons will occupy three different \(d\) orbitals, resulting in:
- 3 unpaired electrons.

### Step 5: Calculate the magnetic moment
- The magnetic moment (\(\mu\)) can be calculated using the formula:
\[ \mu = \sqrt{n(n + 2)} \]
where \(n\) is the number of unpaired electrons.
- Substituting \(n = 3\):
\[ \mu = \sqrt{3(3 + 2)} = \sqrt{15} \text{ Bohr magneton} \]

### Step 6: Identify the complex ion with the same magnetic moment
- We need to find a complex ion that also has 3 unpaired electrons (i.e., a \(d^3\) configuration).
- Let's analyze the options provided (not specified in the question but assumed to be similar to the video transcript):
- For example, consider \([Mn(H_2O)_6]^{4+}\):
- The oxidation state of manganese is +4.
- The electronic configuration of Mn is \(4s^2 3d^5\).
- For \(Mn^{4+}\), we remove 4 electrons (2 from \(4s\) and 2 from \(3d\)):
\[ 4s^0 3d^3 \quad \text{or simply} \quad d^3 \]
- This also has 3 unpaired electrons.

### Conclusion
- Therefore, the complex ion \([Mn(H_2O)_6]^{4+}\) has the same magnetic moment as \([Cr(H_2O)_6]^{3+}\).