Abstract
It is only when mixing two or more pure substances along a reversible path that the entropy of the mixing can be made physically manifest. It is not, in this case, a mere mathematical artifact. This mixing requires a process of successive stages. In any finite number of stages, the external manifestation of the entropy change, as a definite and measurable quantity of heat, isa fully continuous function of the relevant variables. It is only at an infinite and unattainable limit thata non-uniform convergence occurs. And this occurs when considered in terms of the number of stages together with a ‘distinguishability parameter’ appropriate to the particular device which is used to achieve reversibility. These considerations, which are of technological interest to chemical engineers, resolve a paradox derived in chemical theory called Gibbs' Paradox.
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References
Boyer, T. H.: 1970, ‘Sharpening Bridgman's Resolution of the Gibbs' Paradox’,American Journal of Physics 38, 771–73.
Bridgman, P. W.: 1941,The Nature of Thermodynamics, Harvard University Press, Cambridge.
Denbigh, K. G.: 1988,The Principles of Chemical Equilibrium, 4th ed., Cambridge University Press, Cambridge.
Denbigh, K. G. and Denbigh, J. S.: 1985,Entropy in Relation to Incomplete Knowledge Cambridge University Press, Cambridge.
Gibbs, J. W.: 1876,The Scientific Papers, vol. 1, Dover, New York, pp. 166–67.
Grad, H.: 1961, ‘The Many Faces of Entropy’,Communications on Pure and Applied Mathematics 14, 323–54.
Guggenheim, E. A.: 1957,Thermodynamics, 3rd ed., North-Holland, Amsterdam.
Hestenes, D.: 1970, ‘Entropy and Indistinguishability,American Journal of Physics 38, 840–45.
Hobson, A.: 1971,Concepts in Statistical Mechanics, Gordon & Breach, New York.
Klein, M. J.: 1959, ‘Remarks on the Gibbs' Paradox’,Nederlands Tizdschrift voor Natuurkunde 25, 73–76.
Kubo, R.: 1965,Statistical Mechanics, North-Holland, Amsterdam.
Landé, A.: 1955,Foundations of Quantum Theory, Yale University Press, New Haven, CT.
Landé, A.: 1965,Foundations of Quantum Mechanics, Cambridge University Press, Cambridge.
Landolt-Börnstein Tabellen.: 1931,Zweiter Erq. Bd., Springer-Verlag, Berlin.
Landsberg, P. T. and D. Tranah: 1978, ‘The Gibbs Paradox and Quantum Gases’,American Journal of Physics 46, 228–30.
Lesk, A. H.: 1980, ‘On the Gibbs’ Paradox: What Does Indistinguishability Really Mean?’Journal of Physics A. Mathematical and General,13, L111-L114.
Mandl, F.: 1974,Statistical Physics, Wiley, New York.
Planck, M.: 1927,Treatise on Thermodynamics, 3rd English ed., Longmans, Green & Co., London.
Popper, K.: 1959,The Logic of Scientific Discovery, Hutchinson, London.
Reif, F.: 1965,Fundamentals of Statistical and Thermal Physics, McGraw-Hill, New York.
Rudin, W.: 1976,Principles of Mathematical Analysis, 3rd ed., McGraw-Hill, Kogakusha, Tokyo.
Schrödinger, E.: 1948,Statistical Thermodynamics Cambridge University Press, Cambridge.
Von Neumann, J.: 1955,Mathematical Foundations of Quantum Mechanics, English ed., Princeton University Press, Princeton. New Jersey.
Yourgrau, W., A. van der Merwe, and G. Raw: 1966,Treatise on Irreversible and Statistical Thermophysics, MacMillan, New York.
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Denbigh, K.G., Redhead, M.L.G. Gibbs' paradox and non-uniform convergence. Synthese 81, 283–312 (1989). https://doi.org/10.1007/BF00869318
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DOI: https://doi.org/10.1007/BF00869318