Magnetic moment value for a d-block ion is 4.90 BM determine value of total spin for it (consider clock wise spin).

Answer

To determine the total spin for a d-block ion with a magnetic moment value of 4.90 BM, we can follow these steps:

### Step 1: Understand the formula for magnetic moment
The magnetic moment (\( \mu \)) is given by the formula:
\[ \mu = \sqrt{n(n + 2)} \]
where \( n \) is the number of unpaired electrons.

### Step 2: Substitute the given magnetic moment value
We know from the problem that:
\[ \mu = 4.90 \text{ BM} \]
Thus, we can set up the equation:
\[ 4.90 = \sqrt{n(n + 2)} \]

### Step 3: Square both sides to eliminate the square root
Squaring both sides gives:
\[ (4.90)^2 = n(n + 2) \]
Calculating \( (4.90)^2 \):
\[ 24.01 = n(n + 2) \]

### Step 4: Rearrange the equation
Rearranging gives us:
\[ n^2 + 2n - 24.01 = 0 \]

### Step 5: Use the quadratic formula to solve for \( n \)
The quadratic formula is given by:
\[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our case, \( a = 1 \), \( b = 2 \), and \( c = -24.01 \).
Calculating the discriminant:
\[ b^2 - 4ac = 2^2 - 4 \cdot 1 \cdot (-24.01) = 4 + 96.04 = 100.04 \]
Now substituting into the quadratic formula:
\[ n = \frac{-2 \pm \sqrt{100.04}}{2 \cdot 1} \]
Calculating \( \sqrt{100.04} \approx 10 \):
\[ n = \frac{-2 \pm 10}{2} \]

### Step 6: Calculate the possible values for \( n \)
Calculating the two possible values:
1. \( n = \frac{8}{2} = 4 \)
2. \( n = \frac{-12}{2} = -6 \) (not possible since \( n \) cannot be negative)

Thus, \( n = 4 \).

### Step 7: Determine the total spin
The total spin (\( S \)) is given by the formula:
\[ S = \frac{n}{2} \]
Substituting \( n = 4 \):
\[ S = \frac{4}{2} = 2 \]

### Final Answer
The total spin for the d-block ion is \( S = 2 \).

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