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 find the relationship between the work done by an ideal gas during expansion in three different processes (isothermal, isobaric, and adiabatic), we can analyze the work done in each process using the area under the pressure-volume (P-V) curves.

### Step-by-Step Solution:

1. **Understanding the Processes**:
- **Isothermal Process (W1)**: The gas expands at a constant temperature. The P-V curve is a hyperbola. The work done can be calculated using the formula:
\[ W_1 = nRT \ln\left(\frac{V_2}{V_1}\right) \]
- **Isobaric Process (W2)**: The gas expands at a constant pressure. The P-V curve is a horizontal line. The work done is given by:
\[ W_2 = P(V_2 - V_1) \]
- **Adiabatic Process (W3)**: The gas expands without heat exchange. The P-V curve is steeper than the isothermal curve. The work done can be calculated using the formula:
\[ W_3 = \frac{P_1 V_1 - P_2 V_2}{\gamma - 1} \]
where \(\gamma\) is the heat capacity ratio (C_p/C_v).

2. **Analyzing the Areas Under the Curves**:
- The work done by the gas in each process corresponds to the area under the respective P-V curve.
- For the isobaric process (W2), the area is the largest because it represents a constant pressure over a change in volume.
- The isothermal process (W1) has a smaller area than the isobaric process but larger than the adiabatic process (W3).
- The adiabatic process (W3) has the smallest area under the curve, indicating the least work done.

3. **Establishing the Relationship**:
- From the analysis of the areas under the curves, we can conclude:
\[ W_2 > W_1 > W_3 \]
- This means that the work done during the isobaric process is greater than that done during the isothermal process, which in turn is greater than that done during the adiabatic process.

### Final Relation:
Thus, the relationship between the work done in the three processes is:
\[ W_2 > W_1 > W_3 \]