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 regarding the critical constants \( P_C \) and \( T_C \) for methane, we will follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Critical Pressure \( P_C = 45 \) atm
- Critical Temperature \( T_C = 180 \) K

2. **Use the Critical Constants Formulas**:
- The critical pressure is given by the formula:
\[ P_C = \frac{A}{27B^2} \]
- The critical temperature is given by:
\[ T_C = \frac{8A}{27RB} \]
Here, \( A \) and \( B \) are constants specific to the substance, and \( R \) is the gas constant.

3. **Set Up the Equations**:
- From the first equation, we can express \( A \):
\[ A = 27B^2 P_C \]
- From the second equation, we can express \( A \) as well:
\[ A = \frac{27RBT_C}{8} \]

4. **Equate the Two Expressions for A**:
\[ 27B^2 P_C = \frac{27RBT_C}{8} \]

5. **Simplify the Equation**:
- Cancel \( 27B \) from both sides (assuming \( B \neq 0 \)):
\[ B P_C = \frac{RT_C}{8} \]

6. **Substitute the Known Values**:
\[ B \cdot 45 = \frac{R \cdot 180}{8} \]

7. **Solve for B**:
- Rearranging gives:
\[ B = \frac{R \cdot 180}{8 \cdot 45} \]
- Simplifying further:
\[ B = \frac{R \cdot 180}{360} = \frac{R}{2} \]

8. **Determine the Value of R**:
- The gas constant \( R \) can be taken as \( 0.0821 \, \text{L atm K}^{-1} \text{mol}^{-1} \).
- Thus:
\[ B = \frac{0.0821}{2} = 0.04105 \, \text{L/mol} \]

9. **Calculate the Critical Volume**:
- The critical volume \( V_C \) is given by:
\[ V_C = 3B \]
- Therefore:
\[ V_C = 3 \times 0.04105 = 0.12315 \, \text{L} \]

10. **Conclusion**:
- The correct statement is that the value of \( B \) is approximately \( 0.04 \, \text{L/mol} \).