The critical constants `P_(C) & T_(C)` for methane are 45 atm and 180 K. The correct statement is -

Options

A. `V_(C)=2.4L`
B. `b=0.04 L//mol`
C. `V_(C)=0.8L`
D. `b=0.8L//mol`

Answer

To solve the problem, we need to analyze the critical constants given for methane and derive the necessary relationships to find the correct statement.

### Step-by-Step Solution:

1. **Identify the Critical Constants**:
- Given:
- Critical Pressure, \( P_C = 45 \, \text{atm} \)
- Critical Temperature, \( T_C = 180 \, \text{K} \)

2. **Use the Critical Constants Formulas**:
- The formulas for the critical constants are:
\[ P_C = \frac{A}{27B^2} \]
\[ T_C = \frac{8A}{27Rb} \]
- Where \( A \) and \( B \) are constants related to the Van der Waals equation of state, and \( R \) is the universal gas constant.

3. **Set Up the Equations**:
- From the first equation:
\[ A = 27P_C B^2 \]
- From the second equation:
\[ A = \frac{27RT_C}{8B} \]

4. **Equate the Two Expressions for A**:
- Setting the two expressions for \( A \) equal gives:
\[ 27P_C B^2 = \frac{27RT_C}{8B} \]

5. **Simplify the Equation**:
- Cancel \( 27 \) from both sides:
\[ P_C B^2 = \frac{RT_C}{8B} \]
- Multiply both sides by \( 8B \):
\[ 8P_C B^3 = RT_C \]

6. **Solve for B**:
- Rearranging gives:
\[ B^3 = \frac{RT_C}{8P_C} \]
- Taking the cube root:
\[ B = \left(\frac{RT_C}{8P_C}\right)^{1/3} \]

7. **Substitute the Values**:
- Using \( R = 0.0821 \, \text{L atm K}^{-1} \text{mol}^{-1} \):
\[ B = \left(\frac{0.0821 \times 180}{8 \times 45}\right)^{1/3} \]

8. **Calculate B**:
- Calculate the value inside the parentheses:
\[ B = \left(\frac{14.778}{360}\right)^{1/3} = \left(0.0416\right)^{1/3} \approx 0.34 \, \text{L/mol} \]

9. **Determine Critical Volume**:
- The critical volume \( V_C \) is given by:
\[ V_C = 3B \]
- Therefore:
\[ V_C = 3 \times 0.34 \approx 1.02 \, \text{L/mol} \]

10. **Identify the Correct Statement**:
- Compare calculated values with the options provided in the question to find the correct statement.