There are three elements A, B and C. their atomic number are `Z_(1), Z_(2)` and `Z_(3)` respectively. If `Z_(1)-Z_(2)=2 and (Z_(1)+Z_(2))/(2)=Z_(3)-2` and the electronic configuration of element A is `[Ar]3d^(6)4s^(2)`, then correct order of magnetic momentum is/are:

Options

A. `B^(+) gt A^(2+) gt C^(2+)`
B. `A^(3+) gt B^(2+) gt C`
C. `B gt A gt C^(2+)`
D. `B=A^(3+) gt C^(3+)`

Answer

To solve the problem step-by-step, we need to determine the atomic numbers of elements A, B, and C based on the given conditions and then find their electronic configurations to evaluate their magnetic moments.

### Step 1: Determine Atomic Number of Element A (Z1)
The electronic configuration of element A is given as \([Ar] 3d^6 4s^2\).
- The atomic number of Argon (Ar) is 18.
- Therefore, the atomic number \(Z_1\) can be calculated as:
\[ Z_1 = 18 + 6 + 2 = 26 \]

### Step 2: Determine Atomic Number of Element B (Z2)
We are given the equation:
\[ Z_1 - Z_2 = 2 \]
Substituting \(Z_1 = 26\):
\[ 26 - Z_2 = 2 \implies Z_2 = 26 - 2 = 24 \]

### Step 3: Determine Atomic Number of Element C (Z3)
We are also given:
\[ \frac{Z_1 + Z_2}{2} = Z_3 - 2 \]
Substituting the values of \(Z_1\) and \(Z_2\):
\[ \frac{26 + 24}{2} = Z_3 - 2 \implies \frac{50}{2} = Z_3 - 2 \implies 25 = Z_3 - 2 \implies Z_3 = 25 + 2 = 27 \]

### Step 4: Identify Elements A, B, and C
- \(Z_1 = 26\) corresponds to Iron (Fe).
- \(Z_2 = 24\) corresponds to Chromium (Cr).
- \(Z_3 = 27\) corresponds to Cobalt (Co).

### Step 5: Determine Electronic Configurations
1. **Element A (Fe, Z1 = 26)**:
- Configuration: \([Ar] 3d^6 4s^2\)
- Unpaired electrons: 4 (from \(3d^6\))

2. **Element B (Cr, Z2 = 24)**:
- Configuration: \([Ar] 3d^5 4s^1\)
- Unpaired electrons: 6 (from \(3d^5\) and \(4s^1\))

3. **Element C (Co, Z3 = 27)**:
- Configuration: \([Ar] 3d^7 4s^2\)
- Unpaired electrons: 3 (from \(3d^7\))

### Step 6: Determine Magnetic Moments
The magnetic moment (\(\mu\)) is calculated based on the number of unpaired electrons (\(n\)) using the formula:
\[ \mu = \sqrt{n(n + 2)} \]
- For **Element A (Fe)**: \(n = 4\)
\[ \mu_A = \sqrt{4(4 + 2)} = \sqrt{24} \approx 4.9 \]

- For **Element B (Cr)**: \(n = 6\)
\[ \mu_B = \sqrt{6(6 + 2)} = \sqrt{48} \approx 6.93 \]

- For **Element C (Co)**: \(n = 3\)
\[ \mu_C = \sqrt{3(3 + 2)} = \sqrt{15} \approx 3.87 \]

### Step 7: Order of Magnetic Moments
Based on the number of unpaired electrons:
- B (Cr) has the highest magnetic moment.
- A (Fe) has a moderate magnetic moment.
- C (Co) has the lowest magnetic moment.

### Final Order of Magnetic Moments
The correct order of magnetic moments is:
\[ \text{B (Cr)} > \text{A (Fe)} > \text{C (Co)} \]