Spin only magnetic moment of the compound `Hg[Co(SCN)_(4)]` is `sqrt(x)` B.M. The value of 'x' is

Answer

To find the value of 'x' in the spin-only magnetic moment of the compound `Hg[Co(SCN)_(4)]`, we will follow these steps:

### Step 1: Determine the oxidation state of cobalt in the complex
The overall charge of the complex is -2. The thiocyanate ion (SCN) has a charge of -1, and there are 4 thiocyanate ions.

Let the oxidation state of cobalt be \( x \).

\[ x + 4(-1) = -2 \]

This simplifies to:

\[ x - 4 = -2 \]

Solving for \( x \):

\[ x = -2 + 4 = +2 \]

### Step 2: Determine the electronic configuration of Co²⁺
Cobalt (Co) has an atomic number of 27. The electronic configuration of neutral cobalt is:

\[ [Ar] 3d^7 4s^2 \]

For Co²⁺, we remove two electrons, which will be removed from the 4s orbital first:

\[ Co^{2+} : [Ar] 3d^7 \]

### Step 3: Determine the number of unpaired electrons
In the \( 3d \) subshell, there are 7 electrons. The distribution of these electrons in the five \( d \) orbitals will be:

1. Fill each orbital with one electron first (Hund's rule):
- 1 electron in each of the 5 orbitals: \( \uparrow \uparrow \uparrow \uparrow \uparrow \)
2. Then start pairing:
- The next two electrons will pair up in two of the orbitals.

The arrangement will look like this:

- Orbital 1: \( \uparrow \)
- Orbital 2: \( \uparrow \)
- Orbital 3: \( \uparrow \)
- Orbital 4: \( \uparrow \downarrow \)
- Orbital 5: \( \uparrow \downarrow \)

This results in **3 unpaired electrons**.

### Step 4: Calculate the spin-only magnetic moment
The formula for the spin-only magnetic moment (\( \mu \)) is given by:

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

Where \( n \) is the number of unpaired electrons. Here, \( n = 3 \):

\[ \mu = \sqrt{3(3 + 2)} = \sqrt{3 \times 5} = \sqrt{15} \]

### Step 5: Determine the value of 'x'
From the magnetic moment calculated, we have:

\[ \mu = \sqrt{x} \text{ B.M.} \]

Since we found that \( \mu = \sqrt{15} \text{ B.M.} \), we can equate:

\[ \sqrt{x} = \sqrt{15} \]

Squaring both sides gives:

\[ x = 15 \]

### Final Answer:
The value of \( x \) is **15**.

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