Q.20 Consider two different paths in which the volume of an ideal gas doubles isothermally:
- Reversible expansion (work done = Wrev)
- Irreversible expansion, with the external pressure equal to the final pressure of the gas (work done = Wirrev)
Here,
Wrev / Wirrev = ?
(A) 2 ln 2
(B) (1/2) ln 2
(C) (1/2)(1/2)
(D) 2(1/2)
W_rev / W_irrev equals 2 ln 2 for an ideal gas undergoing isothermal expansion where volume doubles. This ratio arises because reversible work maximizes due to continuous equilibrium, while irreversible work uses constant final pressure. The correct answer is option (A).
Problem Breakdown
The query involves an ideal gas expanding isothermally from volume V₁ to 2V₁. Initial pressure P₁ = nRT/V₁, final pressure P_f = P₁/2. Work done by the gas (positive magnitude) compares between paths.
Reversible Work Calculation
For reversible isothermal expansion, P_ext = P_gas at each step. Using P = nRT/V,
W_rev = ∫_{V₁}^{2V₁} P dV = nRT ln(2V₁/V₁) = nRT ln 2
This represents maximum work, as the area under the PV isotherm curve.
Irreversible Work Calculation
Irreversible case sets constant P_ext = P_f = P₁/2 = nRT/(2V₁). Work is
W_irrev = P_ext (2V₁ – V₁) = (nRT/(2V₁)) ⋅ V₁ = nRT/2
Less work occurs since P_ext stays low throughout.
Ratio Derivation
W_rev / W_irrev = (nRT ln 2) / (nRT/2) = 2 ln 2 ≈ 1.386
Option Analysis
| Option | Value | Matches? | Reason |
|---|---|---|---|
| (A) | 2 ln 2 | Yes | Exact ratio from derivation |
| (B) | (1/2) ln 2 | No | Inverse of half reversible work |
| (C) | (1/2) × (1/2) | No | Equals 0.25, unrelated to ln 2 |
| (D) | 2 × (1/2) | No | Equals 1, ignores logarithmic nature |
Introduction to Isothermal Expansion Work
Ideal gas isothermal expansion work reversible irreversible volume doubles scenarios compare maximum versus actual work output. Reversible paths yield nRT ln 2 for doubling volume; irreversible with P_ext = P_f gives half that value.
Core Derivations and PV Diagrams
- Reversible: Hyperbolic PV curve, work = shaded area under isotherm: W_rev = nRT ln 2.
- Irreversible: Vertical drop to P_f, then horizontal: W_irrev = nRT/2.
- Ratio 2 ln 2 shows reversible superiority.
CSIR NET Exam Insights
Common trap: confusing signs or P_ext choice. Always use magnitudes for “work done”; verify P_f = P_i/2 for isothermal doubling.
Practical Applications
Thermodynamic efficiency in engines, refrigeration relies on reversible approximations for maximum work extraction.
