Calculate the work done when 1 mol of an ideal gas is compressed reversibly from 1 bar to 4 bar at a constant temperature of 300 K

Options

A. 4.01 kJ
B. 3.458 kJ
C. 18.02 kJ
D. `14.01 kJ`

Explanation

To calculate the work done when 1 mole of an ideal gas is compressed reversibly from 1 bar to 4 bar at a constant temperature of 300 K, we can follow these steps:

### Step 1: Identify the given values
- Initial pressure (P1) = 1 bar
- Final pressure (P2) = 4 bar
- Number of moles (n) = 1 mol
- Temperature (T) = 300 K

### Step 2: Use the formula for work done in isothermal processes
For a reversible isothermal process, the work done (W) can be calculated using the formula:
\[ W = -2.303 \, nRT \log \left( \frac{P_1}{P_2} \right) \]

### Step 3: Substitute the values into the formula
- The gas constant (R) = 8.314 J/(mol·K)
- Substitute the values into the formula:
\[ W = -2.303 \times 1 \, \text{mol} \times 8.314 \, \text{J/(mol·K)} \times 300 \, \text{K} \times \log \left( \frac{1 \, \text{bar}}{4 \, \text{bar}} \right) \]

### Step 4: Calculate the logarithm
Calculate the logarithm:
\[ \log \left( \frac{1}{4} \right) = \log(1) - \log(4) = 0 - 0.6021 = -0.6021 \]

### Step 5: Substitute the logarithm back into the equation
Now substitute this value back into the equation:
\[ W = -2.303 \times 1 \times 8.314 \times 300 \times (-0.6021) \]

### Step 6: Perform the calculations
Calculate the work done:
1. First, calculate the product:
\[ 2.303 \times 1 \times 8.314 \times 300 \times 0.6021 \]
2. This gives:
\[ W = 2.303 \times 8.314 \times 300 \times 0.6021 \approx 3458.3 \, \text{J} \]

### Step 7: Convert to kilojoules
Convert the work done from joules to kilojoules:
\[ W = \frac{3458.3 \, \text{J}}{1000} = 3.4583 \, \text{kJ} \]

### Final Answer
The work done when 1 mole of an ideal gas is compressed from 1 bar to 4 bar at 300 K is approximately:
\[ W \approx 3.458 \, \text{kJ} \]

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