Find the temperature at which 4 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)` .

Options

A. 478
B. 450
C. 600
D. 179

Explanation

To find the temperature at which 4 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:

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

Where:
- \( P \) = pressure (15 atm)
- \( n \) = number of moles (4 moles)
- \( V \) = volume (10 liters)
- \( a \) = Van der Waals constant (6.71 atm L\(^2\) mol\(^{-2}\))
- \( b \) = Van der Waals constant (0.0564 L mol\(^{-1}\))
- \( R \) = gas constant (0.0821 atm L K\(^{-1}\) mol\(^{-1}\))
- \( T \) = temperature (unknown)

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

Substituting the known values into the equation:

\[ \left( 15 + \frac{6.71 \times 4^2}{10^2} \right) (10 - 4 \times 0.0564) = 4 \times 0.0821 \times T \]

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

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

\[ \frac{6.71 \times 16}{100} = \frac{107.36}{100} = 1.0736 \]

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

Calculate \( (10 - 4 \times 0.0564) \):

\[ 10 - 0.2256 = 9.7744 \]

### Step 4: Substitute back into the equation

Now substituting these values back into the equation:

\[ (15 + 1.0736)(9.7744) = 4 \times 0.0821 \times T \]

Calculating \( (15 + 1.0736) \):

\[ 16.0736 \times 9.7744 = 4 \times 0.0821 \times T \]

### Step 5: Calculate the left side

Calculating the left side:

\[ 16.0736 \times 9.7744 \approx 157.52 \]

### Step 6: Calculate the right side

Calculating the right side:

\[ 4 \times 0.0821 = 0.3284 \]

Thus, we have:

\[ 157.52 = 0.3284T \]

### Step 7: Solve for \( T \)

Now, solving for \( T \):

\[ T = \frac{157.52}{0.3284} \approx 478.2 \text{ K} \]

### Final Answer

The temperature at which 4 moles of \( SO_2 \) will occupy a volume of 10 liters at a pressure of 15 atm is approximately **478 K**.

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