Helmholtz-Kelvin statement of the first and second laws as conservation and dissipation of energy enables us to apply these principles in the broadest and most philosophical way. Lord Kelvin extended the application of the second law to cosmic physics and, with Boltzmann, to predictions as to the ultimate thermal death of the earth. Meanwhile Clausius, Maxwell and Boltzmann began to apply the second law to the kinetic theory of gases, a phase of the subject which belongs essentially to the last stage of its development. Maxwell in particular emphasized the important point that since the heat of a body is the kinetic energy of its molecular motions, the second law is in reality not a mathematical but a statistical truth. It can not, says Maxwell, be reduced to a form as axiomatic as that of the first law, but stands upon a lower plane of probability, because it depends upon the motions of millions of molecules of which we can not get hold of a single one. Could we reduce ourselves to molecular dimensions, and with the gift of molecular vision trace the movements of individual molecules, the distinction between work and heat would
and for a reversible (Carnot) cycle the mathematical statement of the second law is the "Carnot-Clausius equation":
This means that the positive and negative entropies of the system in passing from to and in reversing backwards from to must balance each other. Or as Gibbs has expressed it, "The second law requires (for a reversible cycle) that the algebraic sum of all the heat received from external bodies, divided, each portion thereof, by the absolute temperature at which it is received shall be zero." The criterion of irreversible processes is the "inequality of Clausius"
which implies that the phenomenon will proceed irrevocably or irreversibly in a definite direction, entropy increasing or available energy dissipating to a maximum until a final state of rest or equilibrium (uniformly distributed temperature) is attained. Reversible thermodynamics deals, then, with equations; irreversible thermodynamics with inequalities, because in reversible processes the total entropy of a system remains unchanged while in irreversible processes it continually increases.
- Maxwell, Nature, London, 1877-8, XVII., 279.