Find the temperature at which 7 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 7 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:

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

Where:
- \( P \) = pressure (15 atm)
- \( n \) = number of moles (7 moles)
- \( V \) = volume (10 liters)
- \( a \) = Van der Waals constant for \( SO_2 \) (6.71 atm liter\(^2\) mol\(^{-2}\))
- \( b \) = Van der Waals constant for \( SO_2 \) (0.0564 liter mol\(^{-1}\))
- \( R \) = ideal gas constant (0.0821 atm liter K\(^{-1}\) mol\(^{-1}\))
- \( T \) = temperature in Kelvin (what we need to find)

### Step 1: Substitute the values into the Van der Waals equation

First, we rearrange the equation to isolate \( T \):

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

Now substituting the values:

\[ 15 + \frac{6.71 \times 7^2}{10^2} = \frac{7 \times 0.0821 \times T}{10 - 7 \times 0.0564} \]

### Step 2: Calculate \( \frac{a n^2}{V^2} \)

Calculating \( n^2 \):

\[ n^2 = 7^2 = 49 \]

Now calculate \( \frac{6.71 \times 49}{100} \):

\[ \frac{6.71 \times 49}{100} = \frac{328.39}{100} = 3.2839 \]

So,

\[ P + \frac{a n^2}{V^2} = 15 + 3.2839 = 18.2839 \]

### Step 3: Calculate \( V - nb \)

Calculating \( nb \):

\[ nb = 7 \times 0.0564 = 0.3948 \]

Now calculate \( V - nb \):

\[ 10 - 0.3948 = 9.6052 \]

### Step 4: Substitute into the equation

Now we substitute these values back into the equation:

\[ 18.2839 = \frac{7 \times 0.0821 \times T}{9.6052} \]

### Step 5: Rearranging to find \( T \)

Rearranging gives:

\[ T = \frac{18.2839 \times 9.6052}{7 \times 0.0821} \]

Calculating \( 7 \times 0.0821 \):

\[ 7 \times 0.0821 = 0.5747 \]

Now calculate \( 18.2839 \times 9.6052 \):

\[ 18.2839 \times 9.6052 = 175.565 \]

Now substituting back:

\[ T = \frac{175.565}{0.5747} \approx 305.48 \text{ K} \]

### Final Answer

The temperature at which 7 moles of \( SO_2 \) will occupy a volume of 10 liters at a pressure of 15 atm is approximately **305.48 K**.