Cannot 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 and 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 A1, stable or thermodynamic equilibrium or not stable equilibrium. In state A1, the values of the amounts of constituents are n1, and the value of the volume is V1 whereas, in the final state A0, n0n1 and V0 ≠ V1 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 Σ1 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 quantumtheoritic expression for S can be interpreted as a measure of the shape of an atom, molecule, or other system because ρ can be though 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.

1.
Beretta
G. P.
,
Gyftopoulos
E. P.
,
Park
J. L.
, and
Hatsopoulos
G. N.
,
1984
, “
Quantum Thermodynamics. A New Equation of Motion for a Single Constituent of Matter
,”
Il Nuovo Cimento
, Vol.
82B
,
2
, pp.
169
191
.
2.
Beretta
G. P.
,
Gyftopoulos
E. P.
, and
Park
J. L.
,
1985
, “
Quantum Thermodynamics. A New Equation of Motion for a General Quantum System
,”
Il Nuovo Cimento
, Vol.
87B
,
1
, pp.
77
97
.
3.
Brandt, S., and Dahmen, H. D., 1995, The Picture Book of Quantum Mechanics, Springer-Verlag, New York, NY.
4.
Callen, H. B., 1985, Thermodynamics and an Introduction to Thermostatistics, J. Wiley, New York, NY, pp. 27–32.
5.
Carnot, S., 1824, Reflections on the Motive Power of Fire, ed., E. Mendoza, Dover Publications, New York, NY.
6.
Clausius, R., 1867, “On Different Forms of the Fundamental Equations of the Mechanical Theory of Heat and their Convenience for Application,” The Second Law of Thermodynamics, ed., J. Kestin, Dowden, Hutchinson and Ross, pp. 162–193.
7.
C¸ubukc¸u
E.
, and
Gyftopoulos
E. P.
,
1994
, “
The Quantum Expression for Entropy
,”
ASME
AES-Vol.
33
, ASME, pp.
313
321
.
8.
Gyftopoulos, E. P., and Beretta, G. P., 1991a, Thermodynamics: Foundations and Applications, Macmillan, New York, NY.
9.
Gyftopoulos, E. P., and Beretta, G. P., 1991b, op. cit., Chap. 6, pp. 84–99.
10.
Gyftopoulos, E. P., and Beretta, G. P., 1991c, op. cit., pp. 108–112, and 136–137.
11.
Gyftopoulos, E. P., and Beretta, G. P., 1991d, op. cit., pp. 34–35, and pp. 98–99.
12.
Gyftopoulos, E. P., and Beretta, G. P., 1991e, op. cit., pp. 137–138.
13.
Gyftopoulos, E. P., and Beretta, G. P., 1991f, op. cit., pp. 132–133.
14.
Gyftopoulos, E. P., and Beretta, G. P., 1991g, op. cit., Chaps. 9, 10, and 11.
15.
Gyftopoulos
E. P.
, and
C¸ubukc¸u
E.
,
1997
, “
Entropy: Thermodynamic Definition and Quantum Expression
,”
Physical Review E
, Vol.
55
, 4, pp.
3851
3858
.
1.
Hatsopoulos
G. N.
, and
Gyftopoulos
E. P.
, “
A Unified Quantum Theory of Mechanics and Thermodynamics. Part I. Postulates
,”,
1976
a,
Foundations of Physics
Vol.
6
,
1
, pp.
15
31
;
2.
A Unified Quantum Theory of Mechanics and Thermodynamics. Part IIa. Available Energy
,”
Foundations of Physics
, Vol.
6
,
2
, pp.
127
141
; and
3.
A Unified Quantum Theory of Mechanics and Thermodynamics. Part IIb. Stable Equilibrium States
,”
Foundations of Physics
, Vol.
6
,
4
, pp.
439
454
.
1.
Hatsopoulos
G. N.
, and
Gyftopoulos
E. P.
,
1976
b, “
A Unified Quantum Theory of Mechanics and Thermodynamics. Part III. Irreducible Quantal Dispersions
,”
Foundations of Physics
, Vol.
6
,
5
, pp.
561
570
.
2.
Jauch, J. M., 1973, Foundations of Quantum Mechanics, Addison-Wesley, Reading, MA, p. 132.
3.
Kroemer, H., 1994, Quantum Mechanics for Engineering, Materials Science, and Applied Physics, Prentice Hall, Englewood Cliffs, NJ.
4.
Leighton, R. B., 1959, Principles of Modern Physics, pp. 178–179.
5.
Louisell, W. H., 1973, Quantum Statistical Properties of Radiation, J. Wiley, New York, NY.
6.
Park
J. L.
,
1968
, “
Nature of Quantum States
,”
American Journal of Physics
, Vol.
36
, pp.
211
226
.
7.
Park
J. L.
,
1988
, “
Thermodynamic Aspects of Schroedinger’s Probability Relations
,”
Foundations of Physics
,
18
,
2
, pp.
225
244
.
8.
Schroedinger
E.
,
1936
,
Proceedings, Cambridge Philosophical Society
, Vol.
32
, p.
446
446
.
9.
Slater, J. C., 1963, Quantum Theory of Molecules and Solids, Vol. 1, McGraw-Hill, New York, NY.
This content is only available via PDF.
You do not currently have access to this content.