--- title: "Second law statements; Kelvin-Planck and Clausius equivalence" subject: "Fundamental of Thermodynamics & Heat Transfer" chapter: 5 key: "semester_2/fundamental_of_thermodynamics_and_heat_transfer:second-law-classical-kelvin-clausius" source: https://www.bctengineer.com/q/semester_2/fundamental_of_thermodynamics_and_heat_transfer/second-law-classical-kelvin-clausius site: BCT Engineering --- # Second law statements; Kelvin-Planck and Clausius equivalence Fundamental of Thermodynamics & Heat Transfer · Chapter 5: Second Law of Thermodynamics - Asked in: 2068 Bhadra, 6 marks - Video: [Equivalence of Kelvin Plank And Clausius Statements](https://www.youtube.com/watch?v=XPVE0rs-jCY) — BEST MECHANICAL ENGINEERING - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_2/fundamental_of_thermodynamics_and_heat_transfer/second-law-classical-kelvin-clausius - Whole chapter: https://www.bctengineer.com/md/semester_2/fundamental_of_thermodynamics_and_heat_transfer/chapter-5 ## Question Write down classical statements of second law of thermodynamics. Derive the equivalence between Kelvin-Planck's and Clausius's statement of 2nd law of thermodynamics. ## Short answer Kelvin-Planck: No cycle can convert heat from a single reservoir entirely into work. Clausius: Heat cannot pass from a colder to a hotter body without work input. Violating one enables a combined device that violates the other, proving equivalence. ## Answer #### Classical statements of the second law **Kelvin-Planck statement:** It is impossible for a device operating in a cycle to receive heat from a single reservoir and produce a net amount of work. (No heat engine can have a thermal efficiency of 100 %; some heat must be rejected to a sink.) **Clausius statement:** It is impossible to construct a device operating in a cycle whose sole effect is the transfer of heat from a colder body to a hotter body. (A refrigerator needs work input.) #### Equivalence The two statements are equivalent: violating one means violating the other. We prove it by contradiction. **(a) Violation of Clausius implies violation of Kelvin-Planck.** Suppose a device C transfers heat Q_L from a cold reservoir (T_L) to a hot reservoir (T_H) without any work (violates Clausius). Now run an ordinary heat engine E between the same reservoirs. It takes Q_H from the hot reservoir, produces work W = Q_H − Q_L, and rejects Q_L to the cold reservoir. ```text Hot T_H <--Q_L-- [C] --- Q_L ---+ | Q_H | v | [E] --> W = Q_H - Q_L | | Q_L | v | Cold T_L ---------------------------+ ``` The cold reservoir gives Q_L to C and receives Q_L from E, so its net change is zero. The combined device (C + E) takes net heat Q_H − Q_L from the hot reservoir alone and produces work W = Q_H − Q_L in a cycle. This violates Kelvin-Planck. **(b) Violation of Kelvin-Planck implies violation of Clausius.** Suppose an engine K takes heat Q_H from the hot reservoir and converts it completely to work W = Q_H (violates Kelvin-Planck). Use this work to drive a refrigerator R between the same reservoirs: R takes Q_L from the cold reservoir and rejects Q_H + Q_L to the hot reservoir. The hot reservoir gives Q_H to K and receives Q_H + Q_L from R, a net gain of Q_L. The combined device (K + R) therefore moves Q_L from the cold to the hot reservoir with no net work input. This violates Clausius. Since a violation of either statement leads to a violation of the other, the two statements are equivalent. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which statement is the Kelvin-Planck statement? A. No cyclic device can transfer heat from cold to hot with no other effect B. Every engine has 100% efficiency at infinite temperature C. No cyclic device can receive heat from a single reservoir and produce net work D. Entropy of an isolated system never decreases Answer: C. Kelvin-Planck says a cyclic device cannot produce net work while exchanging heat with only one reservoir. (Key point: Kelvin-Planck statement) 2. The Clausius statement says that: A. Heat flows only from hot to cold in any device B. Heat cannot be converted fully to work in a cycle C. Work cannot be converted to heat D. Heat cannot flow from cold to hot as the sole effect of a cyclic device Answer: D. A refrigerator needs work input; heat cannot move from a colder to a hotter body as the only effect. (Key point: Clausius statement) 3. In proving that violating Clausius violates Kelvin-Planck, what is added to the Clausius-violating device? A. An ordinary heat engine between the same two reservoirs B. A refrigerator driven by external work C. A throttling valve D. A third reservoir at intermediate temperature Answer: A. An engine takes Q_H and rejects Q_L; the Q_L returned by the violating device cancels, so the combination takes heat from the hot reservoir only and produces work. (Key point: Violating Clausius) 4. When a Kelvin-Planck-violating engine drives a refrigerator, what is the net effect? A. The hot reservoir loses Q_H + Q_L B. Entropy of the cold reservoir rises C. Work is produced from one reservoir D. Q_L is moved from cold to hot with no net work input Answer: D. The hot reservoir gains net Q_L, which came from the cold reservoir, with no external work, violating Clausius. 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