Complex X of composition `Cr(H_(2)O)_(6)Cl_(n)` has a spin only mangnetic moment of 3.83 BM. It reacts with `AgNO_(3)` and shows geometrical isomerism. The IUPAC nomenclature of X is

Options

A. Tetraaquadichlorido chromium (IV) chloride dihydrate
B. Tetraaquadichlorido chromium (III) chloride dihydrate
C. Hexaaqua chromium (III) chloride
D. Dichloridotetraaque chromium (IV) chloride dihydrate

Answer

To solve the question, we need to determine the IUPAC nomenclature of the complex \( \text{Cr(H}_2\text{O)}_6\text{Cl}_n \) based on the given information. Let's break down the steps:

### Step 1: Determine the Spin-Only Magnetic Moment
The spin-only magnetic moment (\( \mu \)) is given as 3.83 BM. We can use the formula:
\[ \mu = \sqrt{n(n + 2)} \]
where \( n \) is the number of unpaired electrons.

### Step 2: Calculate the Number of Unpaired Electrons
Setting \( \mu = 3.83 \):
\[ 3.83 = \sqrt{n(n + 2)} \]
Squaring both sides:
\[ 14.6689 = n(n + 2) \]
This simplifies to:
\[ n^2 + 2n - 14.6689 = 0 \]
Using the quadratic formula \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ n = \frac{-2 \pm \sqrt{4 + 58.6756}}{2} \]
Calculating the discriminant:
\[ n = \frac{-2 \pm \sqrt{62.6756}}{2} \]
\[ n \approx \frac{-2 \pm 7.91}{2} \]
Taking the positive root:
\[ n \approx 3 \]

### Step 3: Determine the Oxidation State of Chromium
The electronic configuration of chromium (Cr) is \( [Ar] 3d^5 4s^1 \). In the +3 oxidation state, the configuration becomes \( 3d^3 \) (removing 2 electrons from \( 4s \) and 1 from \( 3d \)).

### Step 4: Determine the Composition of the Complex
The complex is \( \text{Cr(H}_2\text{O)}_6\text{Cl}_n \) with a total charge of 0. The charge from \( \text{Cr}^{3+} \) is +3, and \( \text{Cl}^- \) contributes -1 for each Cl. Thus:
\[ 3 - n = 0 \implies n = 3 \]
So, the complex is \( \text{Cr(H}_2\text{O)}_6\text{Cl}_3 \).

### Step 5: Identify Geometrical Isomerism
The complex can show geometrical isomerism due to the presence of both water and chloride ligands. The possible structures are:
1. **Cis Isomer**: Two Cl ligands adjacent to each other.
2. **Trans Isomer**: Two Cl ligands opposite each other.

### Step 6: Write the IUPAC Name
The IUPAC name is derived from the ligands and the central metal ion:
- The ligand \( \text{H}_2\text{O} \) is called "aqua".
- The ligand \( \text{Cl}^- \) is called "chlorido".
- The oxidation state of Cr is +3.

Thus, the name is:
\[ \text{Tetra-aqua-dichlorido-chromium(III) chloride} \]

### Final Answer
The IUPAC nomenclature of complex \( X \) is:
\[ \text{Tetra-aqua-dichlorido-chromium(III) chloride} \]