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. `-8.02kJ `
C. 18.02kJ
D. 3.45kJ

Answer

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 use the formula for work done in a reversible isothermal process:

### Step-by-Step Solution

1. **Identify the Given Values:**
- Number of moles (n) = 1 mol
- Initial pressure (P1) = 1 bar
- Final pressure (P2) = 4 bar
- Temperature (T) = 300 K
- Gas constant (R) = 8.314 J/(mol·K)

2. **Use the Formula for Work Done:**
The work done (W) during a reversible isothermal process can be calculated using the formula:
\[ W = -nRT \ln\left(\frac{P_2}{P_1}\right) \]
Here, since we are compressing the gas, we will take the logarithm of the ratio of the final pressure to the initial pressure.

3. **Substitute the Values into the Formula:**
\[ W = -1 \times 8.314 \times 300 \times \ln\left(\frac{4}{1}\right) \]

4. **Calculate the Natural Logarithm:**
\[ \ln(4) = 1.3863 \]
Thus, we can rewrite the equation:
\[ W = -1 \times 8.314 \times 300 \times 1.3863 \]

5. **Perform the Calculation:**
\[ W = -1 \times 8.314 \times 300 \times 1.3863 \approx -3457.69 \text{ J} \]

6. **Convert the Work Done to Kilojoules:**
Since the answer is required in kilojoules, we convert joules to kilojoules:
\[ W = -3457.69 \text{ J} \times \frac{1 \text{ kJ}}{1000 \text{ J}} = -3.45769 \text{ kJ} \]
Rounding to two decimal places, we get:
\[ W \approx -3.46 \text{ kJ} \]

7. **Final Answer:**
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 is approximately **-3.46 kJ**.