Calculate van der Waals constansts a and b if critical temperature and critical pressure are `30^(@)C` and 72 atm respectively .

Answer

To calculate the van der Waals constants \( a \) and \( b \) using the given critical temperature and critical pressure, we can follow these steps:

### Step-by-Step Solution:

1. **Convert Critical Temperature to Kelvin**:
- Given critical temperature \( T_c = 30^\circ C \).
- Convert to Kelvin:
\[ T_c = 30 + 273 = 303 \, K \]

2. **Identify Critical Pressure**:
- Given critical pressure \( P_c = 72 \, atm \).

3. **Use the van der Waals equations for critical constants**:
- The formulas for the van der Waals constants \( a \) and \( b \) in terms of critical temperature and pressure are:
\[ T_c = \frac{8a}{27R} \quad \text{and} \quad P_c = \frac{a}{27b^2} \]
- Here, \( R \) is the ideal gas constant, which is \( 0.0821 \, L \cdot atm/(K \cdot mol) \).

4. **Calculate Constant \( b \)**:
- Rearranging the equation for \( b \):
\[ b = \frac{RT_c}{8P_c} \]
- Substituting the values:
\[ b = \frac{(0.0821)(303)}{8 \times 72} \]
- Calculate \( b \):
\[ b = \frac{24.9063}{576} \approx 0.0433 \, L/mol \]

5. **Calculate Constant \( a \)**:
- Rearranging the equation for \( a \):
\[ a = \frac{27P_cb^2}{1} \]
- First, calculate \( b^2 \):
\[ b^2 = (0.0433)^2 \approx 0.00187689 \, L^2/mol^2 \]
- Now substitute \( P_c \) and \( b^2 \):
\[ a = 27 \times 72 \times 0.00187689 \]
- Calculate \( a \):
\[ a \approx 3.59 \, L^2 \cdot atm/mol^2 \]

### Final Results:
- \( a \approx 3.59 \, L^2 \cdot atm/mol^2 \)
- \( b \approx 0.0433 \, L/mol \)