Find the temperature at which 5 moles of `SO_2` will occupy a volume of 10 litre at a pressure of 15 atm. a=6.71` atm litre^2mol^(−2)` ;b=0.0564 `litre mol^(−1)` .

Answer

To find the temperature at which 5 moles of \( SO_2 \) will occupy a volume of 10 liters at a pressure of 15 atm, we will use the Van der Waals equation for real gases. The Van der Waals equation is given by:

\[ \left( P + \frac{a n^2}{V^2} \right) (V - nb) = nRT \]

Where:
- \( P \) = pressure (15 atm)
- \( n \) = number of moles (5 moles)
- \( V \) = volume (10 liters)
- \( R \) = gas constant (0.0821 L·atm/(K·mol))
- \( a \) = Van der Waals constant (6.71 atm·L²/mol²)
- \( b \) = Van der Waals constant (0.0564 L/mol)

### Step-by-step Solution:

1. **Identify the given values:**
- \( P = 15 \, \text{atm} \)
- \( n = 5 \, \text{moles} \)
- \( V = 10 \, \text{liters} \)
- \( a = 6.71 \, \text{atm·L}^2/\text{mol}^2 \)
- \( b = 0.0564 \, \text{L/mol} \)

2. **Calculate \( \frac{a n^2}{V^2} \):**
\[ \frac{a n^2}{V^2} = \frac{6.71 \times (5^2)}{(10^2)} = \frac{6.71 \times 25}{100} = \frac{167.75}{100} = 1.6775 \, \text{atm} \]

3. **Calculate \( n \cdot b \):**
\[ n \cdot b = 5 \times 0.0564 = 0.282 \, \text{L} \]

4. **Substitute values into the Van der Waals equation:**
\[ \left( 15 + 1.6775 \right) \left( 10 - 0.282 \right) = 5RT \]

5. **Simplify the left side:**
\[ 16.6775 \times 9.718 = 5RT \]

6. **Calculate \( 16.6775 \times 9.718 \):**
\[ 16.6775 \times 9.718 \approx 162.36 \]

7. **Set up the equation:**
\[ 162.36 = 5RT \]

8. **Solve for \( T \):**
\[ T = \frac{162.36}{5R} \]
where \( R = 0.0821 \, \text{L·atm/(K·mol)} \).

9. **Calculate \( 5R \):**
\[ 5R = 5 \times 0.0821 = 0.4105 \]

10. **Substitute \( 5R \) back into the equation for \( T \):**
\[ T = \frac{162.36}{0.4105} \approx 395.2 \, \text{K} \]

### Final Answer:
The temperature at which 5 moles of \( SO_2 \) will occupy a volume of 10 liters at a pressure of 15 atm is approximately **395 K**.