Order Isothermal Processes by Magnitude of Work Involved

To find the correct order of the magnitude of work involved in the given isothermal processes, let us consider two states: State 1 with $(P_1, V_1)$ and State 2 with $(P_2, V_2)$.

1. Expansion Process (from State 1 to State 2, where $V_2 > V_1$ and $P_1 > P_2$):

• Single-stage expansion ($w_A$): The gas expands in one step against a constant external pressure $P_{\text{ext}} = P_2$. $\text{Magnitude of work: } |w_A| = P_2(V_2 - V_1)$

• Multi-stage expansion ($w_B$): The expansion occurs in multiple intermediate steps. As the number of stages increases, the area under the $P-V$ curve increases toward the reversible path. $\text{Magnitude of work: } |w_B| > |w_A|$

• Reversible expansion ($w_{\text{rev}}$): Represented by the area directly under the isothermal $P-V$ curve, serving as the upper bound for expansion work: $|w_{\text{rev}}| = nRT \ln\left(\frac{V_2}{V_1}\right)$ Thus, for expansion: $|w_{\text{rev}}| > |w_B| > |w_A|$

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2. Compression Process (from State 2 to State 1, where $V_1 < V_2$ and $P_1 > P_2$):

• Single-stage compression ($w_C$): The gas is compressed in one step against a constant external pressure $P_{\text{ext}} = P_1$. $\text{Magnitude of work: } |w_C| = P_1(V_2 - V_1)$

• Multi-stage compression ($w_D$): The gas is compressed in multiple intermediate steps. As the number of stages increases, the work required to compress the gas decreases toward the reversible work magnitude. $\text{Magnitude of work: } |w_C| > |w_D|$

• Reversible compression ($w_{\text{rev}}$): Serves as the lower bound for compression work: $|w_C| > |w_D| > |w_{\text{rev}}|$

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3. Comparison of all magnitudes:

Since $P_1 > P_2$, the area of the rectangle corresponding to single-stage compression ($P_1 \Delta V$) is larger than the area under the reversible curve, which in turn is larger than the area corresponding to single-stage expansion ($P_2 \Delta V$).

Combining all the inequalities: $|w_C| > |w_D| > |w_{\text{rev}}| > |w_B| > |w_A|$

Thus, the order of magnitude of work is: $|w_C| > |w_D| > |w_B| > |w_A|$

This corresponds to Option C.