Thus since no actual engine can attain the efficiency of the ideal reversible engine, it follows that in all actual conversions of heat into mechanical energy there is necessarily incurred a waste of energy; and the change of entropy in such conversions may be taken as the measure of this unavoidable waste, or it may be defined as that entity which multiplied by the lowest available temperature gives the magnitude of the necessarily lost energy.
It is difficult to obtain a clear mental picture of entropy, mainly because we have no sensory response to it as we have to other characteristics such as temperature or volume. But it is one of the most useful concepts ever introduced into physics. It permits, for instance, a simplification of the statement of Carnot's principle as great as that resulting from the idea of "absolute" temperature. Thus, as can be seen from the above alternative expression of the principle, the result can be stated simply that for the system of the two reservoirs and the working substance the change in entropy vanishes; or in reversible cycles entropy is conserved. Another illustration of the simplification introduced by this concept lies in the conclusion derived by Clausius (and seen most clearly perhaps from the second of the above definitions) that in all nonreversible processes, that is in all natural changes, the entropy must increase. This result which he embodied in his statement of the second law, Die Entropie der Welt strebt einem Maximum zu,'' together with the equally challenging form he gave to the first law," Die Energie der Welt ist constant," was placed by Gibbs at the head of his great monograph. They fitly summarize the foundations upon which he built. A final (example of the usefulness of the idea of entropy is afforded by the compactness and simplicity it leads to the combination of the two laws of thermodynamics into what may be styled the "prime" fundamental equation of the science.
For reversible processes where the work performed can be expressed as the product of the pressure by the resulting volume change, that is for fluids (gasses, vapors, and liquids) and for noncrystalline and unstrained solids, it reads,
Change in intrinsic energy=
(a) Change in heat energy -Change in mechanical energy
(b) Temperature x Change in entropy - Pressure x Change in volume
This equation is fundamental, in the sense that if from experiment we know for any substance how the change in energy depends on the entropy and volume changes, then the equation suffices to determine all of its thermal and mechanical properties. The equation is of course valid only for bodies uniform in composition.
This fundamental equation formed the starting point for Gibbs' development and extension of thermodynamics. In a sense it may be said to embody the whole of his indebtedness to his predecessors. No one had in the slightest degree anticipated the line of his further development of the subject. Prior to him no one had realized that the equation could be generalized to include non-homogeneous bodies, or had seen that when so expanded it would hold the key to the great domain of chemical equilibrium.
The story of how Gibbs was led step by step with inexorable logic to his great generalization and the completeness with which he explored its consequences and implications form a narrative almost unique in the history of science. "On the Equilibrium of Heterogeneous Substances" appeared upon the scientific horizon in the 1870's as unheralded as had Carnot's Reflexions in the 1820's; but whereas Carnot's work required that of Kelvin and Clausius to bring it to fruition, Gibbs' work forms a completed whole in whose framework the developments of the succeeding threequarters of a century in the fields it covers appear for the most part as necessary and inevitable consequences. Like Sir Isaac Newton's Principia, this work of Willard Gibbs stands out in the history of man's intellectual progress as an imperishable monument to the power of abstract thought and logical reasoning.
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