Arrange the following in order of magnitude of work done by the system / on the system at constant temperature :
(a) $$|w_{\text{reversible}}|$$ for expansion in infinite stage.
(b) $$|w_{\text{irreversible}}|$$ for expansion in single stage.
(c) $$|w_{\text{reversible}}|$$ for compression in infinite stage.
(d) $$|w_{\text{irreversible}}|$$ for compression in single stage.

Choose the correct answer from the options given below :

For an isothermal process of an ideal gas, the work is obtained from the integral $$w = -\int P_{\text{ext}}\,dV$$, where the sign is negative for work done by the system (expansion) and positive for work done on the system (compression). We shall compare only the magnitudes $$|w|$$.

Step 1 : Reversible expansion or compression (infinite stages)
At every step $$P_{\text{ext}} = P_{\text{int}}$$, therefore
$$|w_{\text{reversible}}| = nRT\,\ln\!\left(\frac{V_2}{V_1}\right)\quad-(1)$$
For an expansion, $$V_2 \gt V_1$$, the logarithm is positive, and for a compression, $$V_2 \lt V_1$$, the logarithm is negative. Taking absolute value removes the sign, so the magnitude is the same for expansion and compression:
$$|w_{\text{rev,exp}}| = |w_{\text{rev,comp}}|$$.

Step 2 : Single-step (free) expansion
In a one-step irreversible expansion, the external pressure is suddenly lowered to the final pressure $$P_2$$. Hence
$$|w_{\text{irrev,exp}}| = P_2\,(V_2 - V_1)\quad-(2)$$
Because during a reversible expansion the pressure continuously matches higher internal values throughout the path, $$P_{\text{avg,reversible}} \gt P_2$$. Therefore
$$|w_{\text{irrev,exp}}| \lt |w_{\text{reversible}}|$$.

Step 3 : Single-step compression
For a one-step irreversible compression the external pressure is suddenly raised to the initial internal pressure $$P_1$$. Now
$$|w_{\text{irrev,comp}}| = P_1\,(V_1 - V_2)\quad-(3)$$
Throughout a reversible compression the gas would face lower intermediate pressures than $$P_1$$, so $$P_1 \gt P_{\text{avg,reversible}}$$. Consequently
$$|w_{\text{irrev,comp}}| \gt |w_{\text{reversible}}|$$.

Step 4 : Collecting the magnitudes
(a) $$|w_{\text{reversible}}|$$, expansion  ≡  $$|w_{\text{rev,exp}}|$$
(b) $$|w_{\text{irreversible}}|$$, expansion  ≡  $$|w_{\text{irrev,exp}}|$$
(c) $$|w_{\text{reversible}}|$$, compression  ≡  $$|w_{\text{rev,comp}}|$$
(d) $$|w_{\text{irreversible}}|$$, compression  ≡  $$|w_{\text{irrev,comp}}|$$

From Steps 1-3:
$$|w_{\text{irrev,comp}}| \gt |w_{\text{rev,comp}}| = |w_{\text{rev,exp}}| \gt |w_{\text{irrev,exp}}|,$$
that is
$$d \gt c = a \gt b.$$

Thus the correct sequence is Option B.