Question

Arrange the following isothermal processes in order of the magnitude of the work (p - V) involved between states 1 and 2.

A. Expansion in single stage $w_A$
B. Expansion in multi stages $w_B$
C. Compression in single stage $w_C$
D. Compression in multi stages $w_D$

Choose the correct option.

• A. $|w_B|>|w_A|>|w_C|>|w_D|$
• B. $|w_C|>|w_D|>|w_A|>|w_B|$
• C. $|w_C|>|w_D|>|w_B|>|w_A|$
• D. $|w_B|>|w_A|>|w_D|>|w_C|$

Hint:

Remember that for expansion, work magnitude increases with the number of stages (closer to reversible), while for compression, work magnitude decreases with the number of stages (closer to reversible). Compression work is always greater than expansion work between the same limits.

Answer 1

The Correct Option is C

In thermodynamics, the work done during a p-V process is represented by the area under the curve on a P-V diagram. For an isothermal process between two fixed states (1 and 2), we compare expansion and compression work.

Step 1: Compare Expansion Work ($w_A$ vs $w_B$).
For expansion (from volume $V_1$ to $V_2$ where $V_2>V_1$), the maximum work is obtained during a reversible process. In an irreversible expansion, work done depends on the external pressure. A multi-stage expansion ($w_B$) approximates the reversible path more closely than a single-stage expansion ($w_A$). Thus, the magnitude of work done in multi-stages is greater than in a single stage:
$$ |w_B|>|w_A| $$

Step 2: Compare Compression Work ($w_C$ vs $w_D$).
For compression (from volume $V_2$ to $V_1$), the minimum work is required during a reversible process. Irreversible compression requires more work than reversible compression because the external pressure is always higher than the internal pressure. A single-stage compression ($w_C$) requires more work than a multi-stage compression ($w_D$) because the constant external pressure applied is the final high pressure throughout the process:
$$ |w_C|>|w_D| $$

Step 3: Compare Compression vs Expansion.
For the same volume change between two states, the area under the compression curve (where $P_{ext} \ge P_{internal}$) is always greater than the area under the expansion curve (where $P_{ext} \le P_{internal}$). Therefore, even the smallest compression work is greater than the largest expansion work:
$$ |w_{compression}|>|w_{expansion}| $$

Combining these results, the final order of magnitudes is:
$$ |w_C|>|w_D|>|w_B|>|w_A| $$