A metallic carbonyl `M (CO)_(x)` is in gaseous state. The rate of diffusion of `CH_(4)` is 3.31 time faster than this gaseous carbonyl under identical conditions. If atomic mass of metal is 63.29, the closest integer value of X is___________.

Explanation

To solve the problem, we will use Graham's law of effusion, which states that the rate of diffusion of a gas is inversely proportional to the square root of its molar mass.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We know that the rate of diffusion of methane (CH₄) is 3.31 times faster than that of the metallic carbonyl \( M(CO)_x \). We need to find the value of \( x \) given that the atomic mass of the metal \( M \) is 63.29 g/mol.

2. **Using Graham's Law**:
According to Graham's law:
\[ \frac{\text{Rate of } CH_4}{\text{Rate of } M(CO)_x} = \sqrt{\frac{M_{M(CO)_x}}{M_{CH_4}}} \]
Given that:
\[ \frac{\text{Rate of } CH_4}{\text{Rate of } M(CO)_x} = 3.31 \]

3. **Calculating Molar Mass of CH₄**:
The molar mass of methane (CH₄) is calculated as follows:
- Carbon (C) = 12 g/mol
- Hydrogen (H) = 1 g/mol × 4 = 4 g/mol
- Total molar mass of CH₄ = 12 + 4 = 16 g/mol

4. **Calculating Molar Mass of \( M(CO)_x \)**:
The molar mass of the metallic carbonyl \( M(CO)_x \) can be expressed as:
\[ M_{M(CO)_x} = M + x \times M_{CO} \]
where \( M_{CO} \) (molar mass of CO) = 12 (C) + 16 (O) = 28 g/mol.
Therefore:
\[ M_{M(CO)_x} = 63.29 + 28x \]

5. **Setting Up the Equation**:
Substitute the values into Graham's law:
\[ 3.31 = \sqrt{\frac{63.29 + 28x}{16}} \]

6. **Squaring Both Sides**:
To eliminate the square root, square both sides:
\[ (3.31)^2 = \frac{63.29 + 28x}{16} \]
Calculate \( (3.31)^2 \):
\[ 3.31^2 = 10.9561 \]
Therefore:
\[ 10.9561 = \frac{63.29 + 28x}{16} \]

7. **Multiplying Both Sides by 16**:
Multiply both sides by 16 to isolate the numerator:
\[ 10.9561 \times 16 = 63.29 + 28x \]
Calculate:
\[ 175.2976 = 63.29 + 28x \]

8. **Solving for \( x \)**:
Rearranging the equation gives:
\[ 28x = 175.2976 - 63.29 \]
Calculate:
\[ 28x = 112.0076 \]
Dividing by 28:
\[ x = \frac{112.0076}{28} \approx 4 \]

9. **Conclusion**:
The closest integer value of \( x \) is 4.

### Final Answer:
The closest integer value of \( x \) is **4**.