Second law statements; Kelvin-Planck and Clausius equivalence

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.

   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.