Starting with the same initial Conditions, an ideal gas expands from Volume \(\mathrm{V}_{1}\) to \(\mathrm{V}_{2}\) in three different ways. The Work done by the gas is \(\mathrm{W}_{1}\) if the process is purely isothermal, \(\mathrm{W}_{2}\) if purely isobaric and \(\mathrm{W}_{3}\) if purely adiabatic Then (A) \(\mathrm{W}_{2}>\mathrm{W}_{1}>\mathrm{W}_{3}\) (B) \(\mathrm{W}_{2}>\mathrm{W}_{3}>\mathrm{W}_{1}\) (C) \(\mathrm{W}_{1}>\mathrm{W}_{2}>\mathrm{W}_{3}\) (D) \(\mathrm{W}_{1}>\mathrm{W}_{3}>\mathrm{W}_{2}\)

Here's a brief explanation of what happens and the work done in each process: 1. Isothermal Process: An isothermal process is a change of a system, in which the temperature remains constant: \(ΔT = 0\). The work done in an isothermal process is given by \(W_1 = nRT \ln{\frac{V_2}{V_1}}\) where \(n\) is the number of moles, \(R\) is the gas constant, \(T\) is the absolute temperature and \(V_2\) and \(V_1\) are the final and initial volumes respectively. 2. Isobaric Process: The change of a system in which the pressure remains constant is called an isobaric process. The work done in this process is given by \(W_2 = p(V_2-V_1)\), i.e., the difference in volume multiplied by the constant pressure. 3. Adiabatic Process: It is a process in which there is no heat transfer into or out of the system and can be described by \(W_3 = \frac{1}{γ-1}(p_{2}V_{2}-p_{1}V_{1})\), where \(γ = \frac{C_p}{C_v}\) is the heat capacity ratio, \(p_2\) and \(p_1\) are the final and initial pressures, and \(V_{2}\) and \(V_{1}\) are the final and initial volumes.

The work done \(W\) on a system is related to the change in internal energy \(ΔU\) and heat transferred \(Q\) as \( W = ΔU - Q \). Therefore, the work done is dependent on the amount of heat transferred. 1. In an isothermal process, the change in internal energy is zero because the temperature is constant, so the entire heat transfer goes to do work, hence more work \(W_1\) is done. 2. In an isobaric process, some part of the heat transfer is used to increase the internal energy (to maintain constant pressure) and the rest does work. So \(W_2 < W_1\). 3. In an adiabatic process, there is no heat transfer, hence no external work is done. Therefore \(W_3 < W_2\). Based on these comparisons, the correct order is: \(W_1 > W_2 > W_3\) Thus, the correct answer is Option (C).