Barium ion , `CN^(-)` and `Co^(2+)` form an ionic complex . If that complex is supposed to be `75%` ionised in water with van't Hoff factor `'i'` equal to four , then the coordination number of `Co^(2+)` in the complex can be :

Options

A. 6
B. 4
C. `6 & 4` both possible
D. 5

Explanation

To solve the problem, we need to analyze the ionic complex formed by barium ion (Ba²⁺), cyanide ion (CN⁻), and cobalt ion (Co²⁺). Given that the complex is 75% ionized in water and the van't Hoff factor (i) is equal to 4, we will determine the coordination number of Co²⁺ in the complex.

### Step-by-Step Solution:

1. **Understanding the Complex:**
Let's assume the formula of the complex is \( \text{Ba}_x \text{Co}_y \text{CN}_z \). The barium ion has a charge of +2, and the cyanide ion has a charge of -1. The cobalt ion has a charge of +2.

2. **Charge Balance:**
The overall charge of the complex must be neutral. Therefore, we can set up the charge balance equation:
\[ 2x + 2y - z = 0 \]

3. **Dissociation of the Complex:**
When the complex dissociates in water, it produces ions. The dissociation can be represented as:
\[ \text{Ba}_x \text{Co}_y \text{CN}_z \rightleftharpoons x \text{Ba}^{2+} + y \text{Co}^{2+} + z \text{CN}^- \]
The total number of particles after dissociation will be \( x + y + z \).

4. **Van't Hoff Factor (i):**
The van't Hoff factor \( i \) is given as 4. This means:
\[ i = \frac{\text{Total number of particles after dissociation}}{\text{Total number of particles before dissociation}} = \frac{1 - \alpha + x\alpha + z\alpha}{1} \]
Given that the complex is 75% ionized, \( \alpha = 0.75 \). Thus:
\[ i = 1 - 0.75 + x(0.75) + z(0.75) = 4 \]
Simplifying this gives:
\[ 0.25 + 0.75x + 0.75z = 4 \]
\[ 0.75x + 0.75z = 4 - 0.25 = 3.75 \]
Dividing through by 0.75:
\[ x + z = 5 \]

5. **Substituting for z:**
From the charge balance equation \( 2x + 2y - z = 0 \), we can express \( z \) in terms of \( x \) and \( y \):
\[ z = 2x + 2y \]
Substituting this into \( x + z = 5 \):
\[ x + (2x + 2y) = 5 \]
\[ 3x + 2y = 5 \]

6. **Solving the Equations:**
Now we have two equations:
- \( 3x + 2y = 5 \)
- \( x + z = 5 \)

We can express \( y \) in terms of \( x \):
\[ 2y = 5 - 3x \implies y = \frac{5 - 3x}{2} \]

7. **Finding Integer Values:**
Since \( x \) and \( y \) must be integers, we can try different values of \( x \):
- If \( x = 1 \): \( y = \frac{5 - 3(1)}{2} = 1 \) (valid)
- If \( x = 2 \): \( y = \frac{5 - 3(2)}{2} = -0.5 \) (invalid)
- If \( x = 0 \): \( y = \frac{5 - 3(0)}{2} = 2.5 \) (invalid)

Therefore, \( x = 1 \) and \( y = 1 \) is the only valid solution.

8. **Finding z:**
Using \( x + z = 5 \):
\[ 1 + z = 5 \implies z = 4 \]

9. **Coordination Number of Co²⁺:**
The coordination number of Co²⁺ in the complex is equal to \( y \), which we found to be 1.

### Final Answer:
The coordination number of Co²⁺ in the complex is **1**.