One mole of oxygen of volume 1 litre at 4 atm pressure to attain 1 atm pressure by result of isothermal expansion. Find work done by the gas

Options

A. ~155 KJ
B. ~206 KJ
C. ~355 KJ
D. ~554 KJ

Answer

To solve the problem of calculating the work done by one mole of oxygen gas during isothermal expansion from 4 atm to 1 atm, we will follow these steps:

### Step 1: Understand the Isothermal Process
In an isothermal process, the temperature remains constant. The work done by the gas during isothermal expansion can be calculated using the formula:

\[ W = nRT \ln\left(\frac{P_1}{P_2}\right) \]

where:
- \( W \) is the work done,
- \( n \) is the number of moles,
- \( R \) is the universal gas constant,
- \( T \) is the absolute temperature,
- \( P_1 \) is the initial pressure,
- \( P_2 \) is the final pressure.

### Step 2: Convert Units
We need to convert the pressures from atm to pascals (Pa) for consistency in SI units:
- \( P_1 = 4 \, \text{atm} = 4 \times 10^5 \, \text{Pa} \)
- \( P_2 = 1 \, \text{atm} = 1 \times 10^5 \, \text{Pa} \)

### Step 3: Calculate Temperature
We can calculate the temperature using the ideal gas law:

\[ PV = nRT \implies T = \frac{PV}{nR} \]

Given:
- \( P = 4 \times 10^5 \, \text{Pa} \)
- \( V = 1 \, \text{L} = 1 \times 10^{-3} \, \text{m}^3 \)
- \( n = 1 \, \text{mol} \)
- \( R = 8.314 \, \text{J/(mol K)} \)

Substituting the values:

\[ T = \frac{(4 \times 10^5 \, \text{Pa})(1 \times 10^{-3} \, \text{m}^3)}{(1)(8.314)} = \frac{400}{8.314} \approx 48.1 \, \text{K} \]

### Step 4: Calculate Work Done
Now we can substitute \( n \), \( R \), \( T \), \( P_1 \), and \( P_2 \) into the work formula:

\[ W = nRT \ln\left(\frac{P_1}{P_2}\right) \]

Substituting the values:

\[ W = (1)(8.314)(48.1) \ln\left(\frac{4 \times 10^5}{1 \times 10^5}\right) \]

Calculating the logarithm:

\[ \ln\left(\frac{4 \times 10^5}{1 \times 10^5}\right) = \ln(4) \approx 1.386 \]

Now substituting everything back into the work formula:

\[ W = 8.314 \times 48.1 \times 1.386 \approx 498.5 \, \text{J} \]

### Step 5: Convert to Kilojoules
To convert the work done into kilojoules:

\[ W \approx 0.4985 \, \text{kJ} \approx 0.5 \, \text{kJ} \]

### Final Answer
The work done by the gas during the isothermal expansion is approximately **0.5 kJ**.

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