Thermodynamic Definition and Quantum-Theoretic Pictorial Illustration of Entropy

Carnot analyzed an engine operating between two reservoirs. Through a peculiar mode of reasoning, he found the correct optimum shaft work done during a cyclic change of state of the engine. Clausius justified Carnot's result by enunciating two laws of thermodynamics, and introducing the concept of entropy as a ratio of heat and temperature of a thermodynamic equilibrium state. In this paper, we accomplish five purposes: (i) We consider a Carnot engine. By appropriate algebraic manipulations we express Carnot's optimum shaft work in terms of available energies or exergies of the end states of one reservoir with respect to the other, and Clausius' entropy S in terms of the energies and available energies of the same end states. (ii) We consider the optimum shaft work done during a cyclic change of state of an engine operating between a reservoir, and a system with fixed amounts of constituents and fixed volume, but variable temperature. We express the optimum shaft work in terms of the available energies of the end states of the system, and Clausius' entropy in terms of the energies and available energies of the same end states. Formally, the entropy expression is identical to that found for the Carnot engine, except that here the change of state of the system is not isothermal. (iii) We consider the optimum shaft work done during a cyclic change of state of a general engine operating between a reservoir R and system A which initially is in any state A 1 , stable or thermodynamic equilibrium or not stable equilibrium, In state A 1 , the values of the amounts of constituents are n 1 , and the value of the volume is V 1 whereas, in the final state A 0 , no ¬= n 1 and V 0 ¬= V 1 . Using the laws of thermodynamics presented by Gyftopoulos and Beretta, we prove that such an optimum exists, call it generalized available energy with respect to R, and use it together with the energy to define a new property Σ 1 . We note that the expression for Σ is formally identical to and satisfies the same criteria as Clausius' entropy S. The only difference is that Σ applies to all states, whereas Clausius' S applies only to stable equilibrium states. So we call Σ entropy and denote it by S. (iv) We use the unified quantum theory of mechanics and thermodynamics developed by Hatsopoulos and Gyftopoulos, and find a quantum theoretic expression for S in terms of the density operator ρ that yields all the probabilities associated with measurement results. (v) We note that the quantum-theoretic expression for S can be interpreted as a measure of the shape of an atom, molecule, or other system because ρ can be thought of as such a shape, and provide pictorial illustrations of this interpretation. For given values of energy E, amounts of constituents n, and volume V, the value of the measure is zero for all shapes that correspond to projectors (wave functions), positive for density operators that are not projectors, and the largest for the ρ that corresponds to the unique stable equilibrium state determined by the given E, n, and V. Accordingly, spontaneous entropy generation occurs as a system adapts its shape to conform to the internal and external forces. Beginning with an arbitrary initial ρ, this adaptation continues only until no further spontaneous change of shape can occur, that is, only until a stable equilibrium state is reached.