Starting with the same initial conditions, an ideal gas expands from volume `V_1` to `V_2` in three different ways, the work done by the gas is `W_1` if the process is purely isothermal, `W_2` if purely isobaric and `W_3` if purely adiabatic, then

Options

A. (a) `W_2 gt W_1 gt W_3`
B. (b) `W_2 gt W_3 gt W_1`
C. (c) `W_1 gt W_2 gt W_3`
D. (d) `W_1gt W_3gtW_2`

Explanation

To solve the problem of comparing the work done by an ideal gas during three different expansion processes (isothermal, isobaric, and adiabatic), we will analyze each process step-by-step.

### Step 1: Understanding the Processes
1. **Isothermal Process**: The temperature remains constant. For an ideal gas, the work done during an isothermal expansion from volume \( V_1 \) to \( V_2 \) can be calculated using the formula:
\[ W_1 = nRT \ln\left(\frac{V_2}{V_1}\right) \]
where \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the absolute temperature.

2. **Isobaric Process**: The pressure remains constant. The work done during an isobaric expansion can be calculated using the formula:
\[ W_2 = P \Delta V = P(V_2 - V_1) \]
where \( P \) is the constant pressure during the process.

3. **Adiabatic Process**: No heat is exchanged with the surroundings. The work done in an adiabatic process is given by:
\[ W_3 = \frac{P_1 V_1 - P_2 V_2}{\gamma - 1} \]
where \( \gamma \) is the heat capacity ratio \( C_p/C_v \), and \( P_1 \) and \( P_2 \) are the pressures at volumes \( V_1 \) and \( V_2 \) respectively.

### Step 2: Comparing the Work Done
To compare \( W_1 \), \( W_2 \), and \( W_3 \), we need to consider the nature of the processes:
- In the **isobaric process**, the gas does work against a constant pressure, which generally results in a larger area under the curve in a PV diagram.
- In the **isothermal process**, while the temperature is constant, the work done is dependent on the logarithm of the volume ratio, which is less than the linear relationship in the isobaric case.
- In the **adiabatic process**, the work done is typically the least because the gas expands without heat exchange, leading to a drop in temperature and pressure.

### Step 3: Conclusion
From the analysis, we can conclude that:
\[ W_2 > W_1 > W_3 \]
Thus, the correct order of work done is:
- \( W_2 \) (isobaric) > \( W_1 \) (isothermal) > \( W_3 \) (adiabatic).

### Final Answer
The correct option is \( W_2 > W_1 > W_3 \).