Mathematical Kelvin Formulation of the Second Law of Thermodynamics

A mathematical Kelvin formulation of the second law of thermodynamics in the form of a limit is proposed: If the efficiency of a heat engine approaches unity, then the rejected work vanishes. This limit allows deriving the behavior of a Carnot cycle near absolute zero of temperature. Also, the unattainability of absolute zero can be shown. In turn, these results allow deriving the behavior of the entropy near absolute zero, as has already been shown previously. The point of view is the phenomenological, macroscopic, and non-statistical one of classical thermodynamics.