{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,29]],"date-time":"2025-11-29T16:10:34Z","timestamp":1764432634010,"version":"build-2065373602"},"reference-count":14,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2008,3,20]],"date-time":"2008-03-20T00:00:00Z","timestamp":1205971200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Gibbs\u2019 Paradox is shown to arise from an incorrect traditional definition of the entropy that has unfortunately become entrenched in physics textbooks. Among its flaws, the traditional definition predicts a violation of the second law of thermodynamics when applied to colloids. By adopting Boltzmann\u2019s definition of the entropy, the violation of the second law is eliminated, the properties of colloids are correctly predicted, and Gibbs\u2019 Paradox vanishes.<\/jats:p>","DOI":"10.3390\/entropy-e10010015","type":"journal-article","created":{"date-parts":[[2008,3,20]],"date-time":"2008-03-20T10:02:41Z","timestamp":1206007361000},"page":"15-18","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":40,"title":["Gibbs\u2019 Paradox and the Definition of Entropy"],"prefix":"10.3390","volume":"10","author":[{"given":"Robert H.","family":"Swendsen","sequence":"first","affiliation":[{"name":"Physics Department, Carnegie Mellon University, Pittsburgh, PA 15213, USA"}]}],"member":"1968","published-online":{"date-parts":[[2008,3,20]]},"reference":[{"key":"ref_1","first-page":"108","article-title":"On the Equilibrium of Heterogeneous Substances","volume":"3","author":"Gibbs","year":"1873","journal-title":"Transactions of the Connecticut Academy"},{"key":"ref_2","unstructured":"Gibbs, J. W. (1948). The Collected Works of J. W. Gibbs, Yale University Press."},{"key":"ref_3","unstructured":"Gibbs, J. W. (1902). Elementary Principles of Statistical Mechanics, Yale University Press. Reprinted by (Dover, New York, 1960)."},{"key":"ref_4","unstructured":"Pathria, R. K. (1996). Statistical Mechanics, Butterworth-Heinemann. [2nd ed.]. See especially Eq. (1a), which agrees with Eq. 1 in this paper if the temperature is replaced by the energy as a variable."},{"key":"ref_5","unstructured":"Landau, L. D., and Lifshitz, E. M. (1980). Statistical Physics, Pergamon Press."},{"key":"ref_6","unstructured":"Reif, F. (1965). Fundamentals of Statistical and Thermal Physics, McGraw-Hill. This book and some others use a definition of the partition function that also leads to the traditional expression for the entropy."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Everett, D. H. (1988). Basic Principles of Colloid Science, Royal Society of Chemistry.","DOI":"10.1039\/9781847550200-00202"},{"key":"ref_8","first-page":"373","article-title":"\u00dcber die Beziehung zwischen dem zweiten Hauptsatze der mechanischen W\u00e4rmetheorie und der Wahrscheinlichkeitsrechnung respektive den S\u00e4tzen \u00fcber das W\u00e4rmegleichgewicht","volume":"76","author":"Boltzmann","year":"1877","journal-title":"Wien. Ber."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1119\/1.2174962","article-title":"Statistical mechanics of colloids and Boltzmann\u2019s definition of the entropy","volume":"74","author":"Swendsen","year":"2006","journal-title":"Am. J. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1143","DOI":"10.1023\/A:1015161825292","article-title":"Statistical mechanics of classical systems of distinguishable particles","volume":"107","author":"Swendsen","year":"2002","journal-title":"J. Stat. Phys."},{"key":"ref_11","unstructured":"Planck, M. (1901). \u00dcber das Gesetz der Energieverteilung im Normalspektrum. Drudes Annalen, 553\u2013562. Reprinted in Ostwalds Klassiker der exakten Wissenschaften, Band 206, Die Ableitung der Strahlungsgesetze, pp.65\u201374. The equation appears with an arbitrary additive constant on p. 68 of the reprinted text."},{"key":"ref_12","unstructured":"Planck, M. (1906). Theorie der W\u00e4rmestrahlung, J. A. Barth. Translated into English by Morton Masius in M. Planck, The Theory of Heat Radiation; Dover: New York, 1991; p. 119."},{"key":"ref_13","unstructured":"Erickson, G., Neudorfer, P., and Smith, C. R. (1992). Maximum-Entropy and Bayesian Methods, Kluwer."},{"key":"ref_14","first-page":"133","article-title":"On the So-Called Gibbs Paradox, and on the Real Paradox","volume":"9","year":"2007","journal-title":"Entropy"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/10\/1\/15\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T22:20:28Z","timestamp":1760221228000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/10\/1\/15"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,3,20]]},"references-count":14,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2008,3]]}},"alternative-id":["e10010015"],"URL":"https:\/\/doi.org\/10.3390\/entropy-e10010015","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2008,3,20]]}}}