One mole of an ideal gas expands isothermally to double its volume at `27^(@)C` . The work done by the gas is nearly

Options

A. `2760 cal`
B. `414 cal`
C. `1380 cal`
D. `600 cal`

Explanation

To solve the problem of calculating the work done by one mole of an ideal gas that expands isothermally to double its volume at 27°C, we can follow these steps:

### Step 1: Understand the Process
The gas expands isothermally, which means the temperature remains constant throughout the process. The formula for work done (W) during an isothermal expansion of an ideal gas is given by:

\[ W = nRT \ln\left(\frac{V_2}{V_1}\right) \]

Where:
- \( n \) = number of moles of gas
- \( R \) = universal gas constant
- \( T \) = absolute temperature in Kelvin
- \( V_2 \) = final volume
- \( V_1 \) = initial volume

### Step 2: Convert Temperature to Kelvin
The temperature given is 27°C. To convert this to Kelvin:

\[ T = 27 + 273 = 300 \, K \]

### Step 3: Determine the Volume Ratio
Since the gas expands to double its volume, we have:

\[ V_2 = 2V_1 \]
Thus, the volume ratio \( \frac{V_2}{V_1} = 2 \).

### Step 4: Substitute Values into the Work Formula
Now, we can substitute the known values into the work formula. We need to use the value of \( R \) in appropriate units. The value of \( R \) in Joules is approximately 8.31 J/(mol·K). To convert this to calories, we use the conversion factor \( 1 \, \text{cal} = 4.2 \, \text{J} \):

\[ R = \frac{8.31}{4.2} \approx 1.976 \, \text{cal/(mol·K)} \]

Now substituting \( n = 1 \) mole, \( R \approx 1.976 \, \text{cal/(mol·K)} \), \( T = 300 \, K \), and \( \ln(2) \approx 0.693 \):

\[ W = 1 \times 1.976 \times 300 \times \ln(2) \]

### Step 5: Calculate the Work Done
Now we can calculate the work done:

\[ W \approx 1.976 \times 300 \times 0.693 \]

Calculating this step-by-step:

1. Calculate \( 1.976 \times 300 \):
\[ 1.976 \times 300 \approx 592.8 \]

2. Now multiply by \( 0.693 \):
\[ 592.8 \times 0.693 \approx 411.6 \]

Thus, the work done \( W \approx 412 \, \text{calories} \).

### Final Answer
The work done by the gas is nearly **414 calories**.

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