2 moles of `H_2` at 5 atm expands isothermally and reversibly at `57^@C` to 1 atm. Work done is

Options

A. `-2.302 xx 8.314 xx 330 xx log 5 J`
B. `-2.303 xx 8.314 xx 660 xx log 5 J`
C. `-2.303 xx 8.314 xx 660 xx log 1/5 J`
D. `-2.303 xx 8.314 xx 330 xx log 1/5 J `

Explanation

To solve the problem of calculating the work done during the isothermal and reversible expansion of 2 moles of H₂ gas from 5 atm to 1 atm at a temperature of 57°C, we can follow these steps:

### Step 1: Convert the temperature to Kelvin
To convert the temperature from Celsius to Kelvin, we use the formula:
\[ T(K) = T(°C) + 273 \]
Given \( T = 57°C \):
\[ T = 57 + 273 = 330 \, K \]

### Step 2: Identify the given values
- Number of moles (n) = 2 moles
- Initial pressure (P₁) = 5 atm
- Final pressure (P₂) = 1 atm
- Temperature (T) = 330 K
- Universal gas constant (R) = 8.314 J/(mol·K)

### Step 3: Use the formula for work done in isothermal expansion
The formula for work done (W) during an isothermal and reversible expansion is given by:
\[ W = -nRT \ln\left(\frac{P_1}{P_2}\right) \]
Since we need to use logarithm base 10, we can convert it using:
\[ W = -2.303 \cdot nRT \log\left(\frac{P_1}{P_2}\right) \]

### Step 4: Substitute the values into the formula
Substituting the known values into the work done formula:
\[ W = -2.303 \cdot (2) \cdot (8.314) \cdot (330) \cdot \log\left(\frac{5}{1}\right) \]

### Step 5: Calculate the logarithm
Calculate \( \log(5) \):
\[ \log(5) \approx 0.699 \]

### Step 6: Substitute the logarithm value back into the equation
Now substituting the logarithm value back into the equation:
\[ W = -2.303 \cdot (2) \cdot (8.314) \cdot (330) \cdot (0.699) \]

### Step 7: Perform the calculations
Calculating the product:
1. \( 2 \cdot 8.314 \cdot 330 \approx 5499.24 \)
2. \( 5499.24 \cdot 0.699 \approx 3848.99 \)
3. Finally, \( W = -2.303 \cdot 3848.99 \approx -8842.36 \, J \)

### Step 8: Conclusion
The work done during the isothermal and reversible expansion of 2 moles of H₂ gas from 5 atm to 1 atm at 57°C is approximately:
\[ W \approx -8842.36 \, J \]