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

Work done in an irreversible isothermal process is given by $w = -P_{ext}\Delta V$. We compare the magnitudes $|w|$.

1. For Expansion ($V_1 \to V_2$):
Single stage ($w_A$): $P_{ext} = P_2$. Magnitude $|w_A| = P_2(V_2 - V_1)$.
Multi-stage ($w_B$): $P_{ext}$ decreases in steps. The area under the path is larger than the single rectangle of $w_A$. Thus, $|w_B|>|w_A|$.

2. For Compression ($V_2 \to V_1$):
Single stage ($w_C$): $P_{ext} = P_1$. Magnitude $|w_C| = P_1(V_2 - V_1)$. Since $P_1>P_2$, $|w_C|$ is very large.
Multi-stage ($w_D$): $P_{ext}$ increases in steps. The area required is less than the single large rectangle of $w_C$ but still more than the reversible work. Thus, $|w_C|>|w_D|$.

3. General relationship:
Work done on the system (compression) is always of greater magnitude than work done by the system (expansion) for the same state change. Specifically:
$$ |w_{comp, single}|>|w_{comp, multi}|>|w_{rev}|>|w_{exp, multi}|>|w_{exp, single}| $$
Assigning the given variables:
$$ |w_C|>|w_D|>|w_B|>|w_A| $$
This matches option 3.