Calculate the work done during isothermal and reversible expansion of 2 moles of an ideal gas from 10 L to 100 L at 300 K.

Answer

To calculate the work done during the isothermal and reversible expansion of an ideal gas, we can use the formula for work done in an isothermal process:

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

Where:
- \( W \) = work done
- \( n \) = number of moles of gas
- \( R \) = universal gas constant (8.314 J/(mol·K))
- \( T \) = temperature in Kelvin
- \( V_1 \) = initial volume
- \( V_2 \) = final volume

### Step 1: Identify the given values
- Number of moles, \( n = 2 \) moles
- Initial volume, \( V_1 = 10 \) L
- Final volume, \( V_2 = 100 \) L
- Temperature, \( T = 300 \) K
- Universal gas constant, \( R = 8.314 \) J/(mol·K)

### Step 2: Calculate the natural logarithm of the volume ratio
\[ \frac{V_2}{V_1} = \frac{100 \text{ L}}{10 \text{ L}} = 10 \]

Now, calculate the natural logarithm:
\[ \ln(10) \approx 2.303 \]

### Step 3: Substitute the values into the work formula
Now, substituting the values into the work done formula:
\[ W = -nRT \ln\left(\frac{V_2}{V_1}\right) \]
\[ W = -2 \times 8.314 \, \text{J/(mol·K)} \times 300 \, \text{K} \times 2.303 \]

### Step 4: Calculate the work done
Calculating the above expression:
\[ W = -2 \times 8.314 \times 300 \times 2.303 \]
\[ W = -2 \times 2494.2 \]
\[ W = -4988.4 \, \text{J} \]

### Step 5: Convert work done from Joules to kilocalories
To convert Joules to kilocalories, we use the conversion factor \( 1 \, \text{kcal} = 4184 \, \text{J} \):
\[ W = \frac{-4988.4 \, \text{J}}{4184 \, \text{J/kcal}} \approx -1.192 \, \text{kcal} \]

### Final Answer
The work done during the isothermal and reversible expansion of the gas is approximately:
\[ W \approx -1.192 \, \text{kcal} \]