A sample of ideal gas undergoes isothermal expansion in a reversible manner from volume `V_(1)` to volume `V_(2)`. The initial pressure is `P_(1)` and the final pressure is `P_(2)`. The same sample is then allowed to undergoes reversible expansion under adiabatic conditions from volume `V_(1) to V_(2)`. The initial pressure being same but final pressure is `P_(3)`.
The work of expansion in adiabatic process `(w_(adi))` is related to work of expansion in isothermal process `(w_(iso))` is

Options

A. `w_(adi) = w_(iso)`
B. `w_(adi) lt w_(iso)`
C. `w_(adi) = 2w_(iso)`
D. `w_(adi) gt w_(iso)`

Explanation

To find the relationship between the work done in the adiabatic process \((w_{adi})\) and the work done in the isothermal process \((w_{iso})\), we can analyze the processes graphically and mathematically.

### Step 1: Understand the Processes
- **Isothermal Expansion**: The gas expands at a constant temperature. The work done during this process can be expressed as:
\[ w_{iso} = nRT \ln\left(\frac{V_2}{V_1}\right) \]
where \(n\) is the number of moles of the gas, \(R\) is the ideal gas constant, and \(T\) is the absolute temperature.
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