Abstract
From 1873 to 1878, the American physicist Josiah Willard Gibbs offered to the scientific community three great articles that proved to be a milestone for the science of thermodynamics. On the other hand, between 1886 and 1896, the French physicist Pierre Maurice Marie Duhem translated thermodynamics into the language of Lagrange’s analytical mechanics. At the same time, he expanded its scope to include thermal phenomena, electromagnetic phenomena, and all kinds of irreversible processes. Duhem formulated a version of thermodynamics characterized by the conceptual unification of mechanics, physics, and chemistry. Overall, the work of both physicists on thermodynamics is tremendous, full of axioms, theorems, corollaries, proofs, and hundreds of equations. Therefore, it would be a utopian aim to provide a short analysis of their work. Instead, the present study will attempt to give a brief outline of the main tools and concepts used by the two physicists. I will argue that each scientist approaches thermodynamics in a new and unique way, which reveals their scientific styles as reflected in their personalities, the writing styles, their behavior toward publicity, and their inclination for publication. Finally, I will examine the influence of their theories on the development of chemical thermodynamics.
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Notes
Saurel defended his dissertation in 1900 on a subject related to Gibbs’ and Duhem’s work on the equilibrium of chemical systems (Jaki 1984, pp. 136–137).
Wilson used the notes of another Gibbs’ student, because he lost his notes (Wilson 1936, p. 19).
It has been published as a book: Bordoni (2012). Taming complexity: Duhem's third pathway to thermodynamics. Urbino: Editrice Montefeltro.
However, August Heinrich Horstmann approached the dissociation reactions in the gas phase, in the solid and in solution through the entropy, and the concept of disgregation. He used the disgregation because he thought that it was the only factor contributing to the variation of entropy. From 1868 to 1873, Horstmann published three papers where he developed his theory of dissociation. He proposed that the degree of dissociation has to take a value for which the entropy of the system is at a maximum. He derived an expression for the equilibrium constant in terms of disgregation (Horstmann 1873; Jensen 2009; Darrigol 2018b, pp. 6–7).
Martin Klein conjectures that the historical Tait–Clausius entropy dispute on the pages of the Philosophical Magazine arouse Gibbs’ interest in thermodynamics (Klein 1989, p. 6).
In 1876, Andrews published a second paper with a higher degree of accuracy, in which the temperature range extended by three more series of measurements at 6.5 °C, 64 °C, and 100 °C. He did not offer any theoretical explanation for the critical point, except perhaps, some vague statements about the action of internal attractive and resistive molecular forces (Andrews 1876, pp. 448–449). It seems that Andrews was unaware of Gibbs’ graphical papers.
Gibbs cited both Andrews’ and Thomson’s studies in his Graphical Method.
Clausius put forward a detailed account of the reasons that make van der Waals’ equation to deviate from Andrews’ experiments. He proposed modification of van der Waals equation, which contained four constants (van der Waals’ equation contains two constants) and tested his equation with Andrews’ experimental data taken from his pressure–volume diagram of carbonic acid. This modified equation reproduced the experiments at low densities of the gas but failed at high densities near the point where pressure increases rapidly. The modified Clausius’ equation failed to reproduce Andrews’ straight-line segment. However, it predicted the existence of the metastable states of the supersaturated vapor or the superheated liquid (Clausius 1880, pp. 398–400).
Gibbs divided the thermodynamic surface into two parts, of which one represents the homogeneous phases, and the other mixtures of heterogeneous phases. Gibbs called the first part primitive surface, and the second part derived surface (GSP 1906, pp. 35–36). Gibbs pointed out that none of these surfaces exists when the system is not in a state of thermodynamic equilibrium.
A developable surface is the type of surface that a tangent plane creates. The surface generated by a straight line or by a straight line tangential to a space curve is a ruled surface. A developable surface could be a ruled surface, but the opposite does not hold. Gibbs described the surfaces obtained from the rolling planes as developable surfaces. However, these surfaces are in fact ruled surfaces spanned by lines joining corresponding points of common tangent planes.
This triangle is similar to that presented in Gibbs’ first graphical paper and discussed above.
A diathermal wall between two thermodynamic systems allows heat transfer from one system to the other, but does not allow transfer of matter across it. A diathermal envelope encloses a closed system.
William Thomson introduced the concept of dissipated energy in his attempt to formulate the second law of thermodynamics. In the last section of his essay, Gibbs discussed problems related to the surface of the dissipated energy.
Gibbs did not use the terminology bimodal and spinodal. Maxwell describing Gibbs’ thermodynamic curve made use of the terms convexo-convex curve and spinode curve for the bimodal and the spinodal surfaces, respectively.
This figure is similar to Fig. 2 of Gibbs’ article (GSP 1906, p. 44), except the identification of the various sections of the diagram for clarification. Maxwell presented a more detailed diagram in his textbook the Theory of heat (1902, p. 207).
Saurel studied the condition for the tangential relationship between the bimodal and spinodal curves (1902, pp. 483–484).
Duhem, who considered Gibbs an algebraist, alleged that the geometrical demonstrations of these papers did not play almost any role in the study of thermodynamic properties. They rather constitute the algebraic analysis in the form of Cartesian algebra (1908, p. 17).
According to Professor Walters of Yale, who commented on Gibbs’ dissertation, Gibbs showed his predilection for mechanics and skill in the use of the geometrical approach (Wheeler 1962, p. 28).
Gibbs contrasted Watt's indicator to his graphical representations to emphasize the usefulness of the proposed entropy–temperature diagram. “The method in which the coordinates represent volume and pressure has a certain advantage in the simple and elementary character of the notions upon which it is based, and its analogy with Watt's indicator has doubtless contributed to render it popular” (GSP 1906, p. 11).
Nevertheless, Gibbs used as well geometrical representations in the heterogeneous substances to interpret the thermodynamic behavior of pure and multi-component systems. He recognized that the three independent variables of the first two papers were unsuitable when the system has more than three independent variables. He thought that he could overcome this obstacle upon reducing the number of variables into two or three. He used the temperature, the pressure, and the ζ function (see below) by making constant the total mass of the system equal to unity. Through this line of reasoning, he studied the conditions of equilibrium of homogeneous substances, as well as binary and ternary systems. For ternary systems, he adopted a different method of representation changing the Cartesian coordinate system into a two-dimensional equilateral triangle, or a three-dimensional prism. (GSP 1906, pp. 115–129). The first diagram is currently in use by physicists and engineers to illustrate phase transformations in multi-component systems.
An isolated system is a thermodynamic system enclosed by rigid immovable walls (as for instance a Dewar vessel) through which neither mass nor energy (heat) can pass. An isolated system obeys the conservation law, that is, its total energy and mass stay constant. The isolated system is actually an idealized model system away from common experience.
Gibbs never used the term chemical potential in the heterogeneous substances. Instead, he defined the plain word potential in his writings. The term chemical potential has been attributed to the American chemist Wilder Dwight Bancroft, who renamed Gibbs’ potential in a letter sent to Gibbs in 1899 (Baierlein 2001, p. 431). It appears that Gibbs accepted the new terminology tacitly, as can be deduced in his reply letter to Bancroft in May 1899 (GSP 1906, p. 425). In the current textbooks of physical chemistry, this state function refers to as chemical potential.
Müller (2006) has discussed several applications of the chemical potential (Gibbs paradox, mixtures and solutions, phase rule, law of mass action, semipermeable membranes, osmosis, Raoult’s law, etc.) in the context of modern chemical thermodynamics, as well as the method for its measurement.
The German chemist Wilhelm Meyerhoffer working with van’t Hoff at the University of Amsterdam used for the first time the name phase rule. He published a paper entitled Die Phasenregel und ihre Anwendungen (The phase rule and its applications) (1893).
For the system of pure water with its vapor, there are two phases (liquid and vapor), then r = 2. If we treat water as a single component, n = 1, the phase rule predicts one degree of freedom. We can change either the temperature under constant pressure or the pressure under constant temperature and maintain the system in equilibrium state without changing the number of phases or the components in each phase. If, however, we consider the dissociation of water, the number of components in the liquid phase is not one, but three: neutral water molecules, hydronium ions, and hydroxyl ions. In this case, r is still two, but the number of components is now three. The phase rule predicts three degrees of freedom, which is absurd. In this case, the conditions of neutrality and the dissociation reaction should be taken into account in the phase rule equation for determining the correct number of degrees of freedom.
Gibbs deduced the phase rule far more concisely by the following reasoning, "A system of r coexistent phases, each of which has the same n independently variable components is capable of n + 2 − r variations of phase. The temperature, the pressure, and the potentials for the actual components have the same values in the different phases, and the variations of these quantities are by [equation] [97] subject to as many conditions as there are different phases. Therefore, the number of independent variations of the phase of the system, will he n + 2 − r" (GSP 1906, p. 96).
Duhem rejected the method of reversible cycles first introduced by Clausius on the ground that it was a lengthy and painful procedure for the study of physical and chemical processes. Alternatively, he proposed the use of the thermodynamic potential as an easier and more effective tool (see below). Contrary to Gibbs, Duhem, and Max Planck, van ‘t Hoff considered the method of reversible cycles as the most suitable to study thermodynamic systems than using abstract physical conceptions and mathematical functions, such as entropy (see below).
Clifford Truesdell (1984, p. 22) stressed that in the whole heterogeneous substances he found one, and only one passage, where Gibbs hinted the state of irreversibility, when he discussed the case of an imperfect electrochemical apparatus in which Clausius’ inequality dη ≥ dQ/t for the irreversible process may hold (GSP 1906, p. 339).
Pierre Duhem, who studied Gibbs' thermodynamics thoroughly, considered this work as a manifestation of thermostatics (statique chimique). Arguing Gibbs’ reference to equilibrium processes, he posed the following question: "suppose the original condition of a body is one that satisfies Gibbs' criterion of equilibrium. If its condition is then forcibly altered, will that body upon release tend to resume or at least remain near its original condition”? Duhem considered that equilibrium in nature is not realizable; it is a virtual equilibrium since any physical process presupposes the departure from the equilibrium state accompanied by dissipation (loss) of energy as in viscous fluids and deformable solids.
Gibbs was slow to publish. He was not easily satisfied with his intellectual product, and some of his publications remained on the shelf for several years before becoming available to the reader. He had completed the book Elementary Principles in Statistical Mechanics since 1892. The book appeared 10 years later, even though his notes on this subject had been distributed to students attending his lectures (Klein 1989, p. 14).
Clausius, and Rankine, contributed to both theoretical trends of thermodynamics. They both based the development of thermodynamics on a molecular interpretation, although using different concepts. Clausius, even when he worked in the more phenomenological register, relied on kinetic molecular intuition (free heat, disgregation, etc.). For Clausius' motivation to develop the molecular theory in thermodynamics, see Garber (1970), Darrigol (2018a, pp. 58–60).
Rational mechanics is the theoretical branch of physics that deals with problems concerning complex movements of bodies, i.e., processes occurring in time, and belongs to the general context of dynamics like equilibrium in the frame of statics. It has its origin in Newton’s physics but developed in the eighteenth century by the French physicists and mathematicians Bernoulli, Euler, D’Alembert, and advanced by Lagrange.
Scrutiny of these theoretical traditions may reveal a finer classification of these theoretical approaches to thermodynamics. Stefano Bordoni discerns five streams sorted according to their conceptual distance from mechanics (2013, pp. 618–619).
Massieu, Professor of Mineralogy and Geology at the University of Rennes, published two founding papers in 1869, where he introduced the two functions (1869). He was able to determine several physical constants of bodies utilizing his characteristic functions. Massieu wrote on the subject in 1876 a more detailed memoir of great clarity (1876). He remained for a long time unknown working away from the scientific community. However, as Horstmann, he kept abreast of the recent developments of physics, since he had relied on the concept of entropy and adhered to the molecular theory of fluids which only began to develop (Balian 2017).
Duhem included the German-speaking physicists Arthur von Oettingen, professor at the Dorpart University in Estonia, among the physicists who viewed thermodynamics in a remarkable generalized context. Stefano Bordoni (2013, pp. 635–640) gives a short account on Oettingen’s thermodynamics.
For a thorough discussion of the events that took place during Duhem’s presentation, see Jaki (1984, pp. 50–53).
Jules Moutier was a professor of physics and chemistry at Collège Stanislas, a state lycėe in Paris, where Duhem continued his schooling after a short period in a private school. Duhem had great respect for Moutier as a teacher and physicist and learned from him the new thermodynamics of Gibbs (Jaki pp. 29–30 and 260–263).
Gibbs cited Massieu’s first paper and his functions in a footnote of the heterogeneous substances (GSP 1906, pp. 86–87).
Duhem never used this terminology. I borrowed it from Gibbs' Elementary principles in statistical mechanics, where he discussed the thermodynamic analogies with rational mechanics. By the mid-twentieth century, rational thermodynamics became a school of thought developed by Clifford Truesdell, Bernard Coleman, and Walter Noll (Truesdell 1984).
Carnot’s fundamental theorem deals with the maximum obtainable motive power in heat engines for a given amount of heat. Carnot provided the proof of the maximum efficiency u for a perfect engine in a lengthy footnote in his single memoir Reflexions sur la puissance motrice du feu et sur les machines propres a développer cette puissance. He demonstrated that the efficiency of a perfect engine working reversibly was at a maximum, but he did not know its value.
Clausius’ theorem states that a system exchanging a quantity of heat Q with external reservoirs and undergoing a cyclic process is one that ultimately returns a system to its original state. The theorem was expressed mathematically by the so-called Clausius inequality for a cyclic process, i.e., \( {\oint }\frac{dQ}{T} \le 0 \). T is the absolute temperature of the external reservoir.
van’t Hoff formulated the principle of mobile equilibrium in his treatise Études de Dynamique Chimique published in (1884). According to this principle, every equilibrium between two different conditions of matter (systems) is displaced by lowering the temperature, at constant volume, toward that system the formation of which evolves heat. He applied this principle to every possible case, both chemical and physical equilibrium (van’t Hoff 1884, pp. 161–176). Duhem discusses van’t Hoff’s law of displacement of equilibrium with temperature that marks the difference between exothermic and endothermic compounds in connection with the effect of temperature changes at the moment of equilibrium. He contrasts once again this thermodynamic law with the doctrine of the maximum work enunciated by Berthelot (Duhem 1893a, pp. 143–144).
L’Évolution de la mécanique was published in seven parts in Revue Générale des Sciences. As a book, it appeared in 1903 by A. Joanin, Paris. The edition of 1905 has been used in this study. There exists an English translation, The evolution of mechanics by Sijthoof and Noordhoff published in 1980, to which I have no access. English translation of portions of chapter VII is available in Maugin’s book (2014, pp. 176–183).
As the reader of l’Évolution may note, Duhem, in writing this book, has used material of several of his previous publications. Therefore, this collection may be taken as an overview of Duhem’s philosophical, historical, and scientific deliberations on rational thermodynamics.
Under the name hysteresis, Duhem refers to a number of systems that suffer permanent alterations, e.g., dielectric, chemical, elastic hysteresis, or alterations imposed on metals by the effect of temperature, such as annealing, hardening, etc. (Duhem 1905, pp. 318–319).
Duhem has described the theoretical analysis of these complex phenomena of deformable solids as a function of one and two variables in seven memoirs published between 1895 and 1902 under the general title Les Déformations Permanentes et l'Hystérésis (1895, 1898, 1902).
Duhem calls this coefficient as the coefficient of hysteresis when he refers the phenomena of hysteresis within the context of his theory on permanent alteration (1905, p. 320).
Gibbs and Duhem were honored by the Scientific Academies of their countries and abroad and by some of their colleagues, at least from those who had an understanding of their work. For Gibbs’ recognition, see Wheeler, pp. 83–93, and 97–99; for Duhem, see Jaki (1984, pp. 141–147). However, personal recognition does not mean necessarily a wide reception of their work.
Gibbs distributed reprints of his heterogeneous substances to a large number of contemporary physicists, chemists, mathematicians, and astronomers that, according to him, might have some interest in his work. The extensive mailing list of the individuals that received reprints is reported in Wheeler 1962, pp. 235–248. However, most of the recipients showed characteristic indifference to his work, probably due to their difficulty in understanding his concise writing style or because Gibbs was unknown to his European colleagues. Among the few, Maxwell indicated an immediate interest in Gibbs’ thermodynamics.
The release of a scientific journal does not necessarily imply that it will have a wide readership. It depends on the quality of the journal and the level of research reflected on its pages.
It published in four large volumes of 1430 pages in total from 1897 to 1899.
Criticism of Maxwell's classical electromagnetic theory was widespread among French physicists in Duhem's time. But it was rather shallow compared to the detailed and rigorous analysis of it given by Duhem. Nevertheless, Duhem's treatise on Maxwell’s electromagnetic theory was omitted from contemporary French reviews, articles, and monographs (Jaki 1984, pp. 283–284).
Duhem, who studied thoroughly Gibbs’ work and exposed his thermodynamics among the contemporaries in France, made once the following ironic comment regarding Gibbs’ dense writing style: “Il semble parfois qu'en publiant ses travaux, Gibl)s eût été possédé du désir de les voir passer inaperçus; s'il en fut ainsi, il fut bien souvent servi à souhait; bien souvent, ses idées demeurèrent ignorées de ceux-là mêmes qui auraient eu le plus grand intérêt à les connaitre.” (It sometimes seems that by publishing his works, Gibbs would have been possessed of the desire to see them go unnoticed; if this was the case, his wishes were often fulfilled; very often, his ideas remained ignored to those who would have had the greatest interest in knowing them) (Duhem 1908, p. 14).
Commenting on Gibbs’ writing style, Duhem concluded: “If therefore, Gibbs has left his discoveries of chemical mechanics in an abstract and purely algebraic form, it is not that he was incapable of presenting them in a language more concrete and more accessible to experimenters, it is because of his intellect.” (Duhem 1908, p. 26).
Maxwell, van der Waals, Helmholtz, J. J. Thomson, and Hertz were among Gibbs’ contemporary physicists who showed an understanding of the heterogeneous substances. However, only Maxwell spoke enthusiastically to his contemporaries in Britain about Gibbs and included Gibbs’ graphical representations in his textbook Theory of Heat. Maxwell’s premature death in 1879, at the age of 48, put an end to his excellent work in physics, while Gibbs lost a keen supporter of his thermodynamics.
For a thorough discussion of the various factors that delayed the transfer of thermodynamics to chemistry and the subsequent formulation of the chemical thermodynamics by the joint efforts of van’t Hoff, Arrhenius and Ostwald, see Dais (2019).
Not only chemists but also the majority of physicists had difficulty in accepting or assimilating the concept of entropy. As noted, William Thomson never mentioned the entropy in his work, and even its creator, Clausius, did not give entropy any prominent place in his research. The founders of the discipline of physical chemistry chose to avoid any reference to entropy. The next generation of chemists was familiar with the concept of entropy, but they preferred to use alternative measures for the chemical affinity and chemical equilibrium. Gibbs made use of the chemical potential and free energy under constant pressure, Helmholtz used the free energy under constant volume, and Duhem the thermodynamic potential. Reconciliation of entropy and chemistry was achieved during the decades after the First World War, especially by the American physical chemists (Lewis, Randall, Trevor, Noyes, and others). The “tortuous” course of entropy to enter research and education is the subject of an excellent paper by Kragh and Weininger (1996).
In a letter to Gibbs in 1887, Ostwald proposed the translation of the heterogeneous substances, admitted that “I cannot deny that at present the study of your work is pretty difficult, particularly for the chemist, who is usually not at home in a mathematical treatment” (quoted in Moore et al. 2002, p. 115).
Characteristic is the statement of the organic chemist Friedrich Wöhler, while comparing mathematics and observation: “My imagination is fairly active, but I am somewhat slow in my thinking. No one is less oriented to be a critic than I. The organ for philosophical thought is entirely missing in me, as you know so well, just as that for mathematics. Only for observation do I imagine that I have a passable facility in my brain, which may be connected with a sort of instinct to be able to predict empirical relationships” (Rocke 1993, p. 34).
Duhem was so impressed by Gibbs’ heterogeneous substances that he sat down and wrote a review on Gibbs’ work as a whole. This review was initially published in the Bulletin des Sciences mathėmatique in 1907. 1 year later, Duhem published an enriched version as a separate book (1908); see Klein (1990, p. 53).
The correspondence between Gibbs and Ostwald when the latter asked Gibbs’ permission to translate the heterogeneous substances reveals some facets of Gibbs’ scientific style. For instance, he refused to write a short introduction in the German translation and delayed to provide a personal portrait for the same publication. The Ostwald–Gibbs correspondence was published by Moore et al. (2002).
Helmholtz derived the equation in his 1882 famous paper Über die Thermodynamik Chemischer Vorgänge (1882) (on the thermodynamics of chemical processes) dealing with the free and the bound energy of chemical processes.
In 1889, the twenty-four-year-old Nernst showed that the free energy of a chemical reaction could be measured by making the reaction the source of a galvanic cell and measuring the electrochemical potential. Nernst showed that there was a simple proportionality between the electrochemical potential and free energy. This relationship is now known as Nernst’s equation. However, this methodology had practical difficulties. For each chemical process, a separate galvanic cell should be implemented, which was not feasible for all cases. Furthermore, very dilute solutions should be used to allow the direct use of concentrations in Nernst’s equation.
Several physicists in Europe and America including Duhem delivered formal proof of the phase rule.
A report for Stassfurt deposits and van’t Hoff experimental findings is issued in the Data in Geochemistry. 1991. Bulletin-United States Geological Survey, issue 491, pp. 210–217. This survey did not include phase diagrams from these studies due to the limited space of the report.
Bancroft research program based on the phase rule had a limited scope and finally collapsed unable to compete with the advancement of physical chemistry. In later years, the phase rule became the central research subject of engineers. Details for the bankruptcy of Bancroft’s research program have been provided by Servos (1982).
Semipermeable membrane was an imaginary device, known as the equilibrium box, invented by van’t Hoff to study reversible changes of the concentrations of gas mixtures or solutions, so that chemical reactions could be performed, at least in principle, in a reversible way. This model could take into account any transformation of individual substances in a mixture or solution (van’t Hoff 1912, p. 54; Kipnis 1991, p. 215).
Nernst adopted Duhem’s terminology for the potential, but he used Gibbs’ notion of the chemical potential symbolized by the same letter μ as in heterogeneous substances.
To the best of my knowledge, it is unknown when and by whom the name of Duhem was added to this equation. Duhem has derived an analogous equation in his first dissertation for two components, and later for the general case, when he considered the effect of the masses of substances on his thermodynamic potential. (Duhem 1886, pp. 32–35 and 140–143). The same topic reappeared in his Traitė Ėlementaire de Mėchanique Chimique Fondėe sur la Thermodynamique (1897b, Vol. 3, pp. 1–4). See Miller 1963). Duhem considered that the thermodynamic potential Φ under constant pressure and temperature is a homogeneous function of the first degree of the masses of the various substances of the system. Using Euler’s theorem, he derived an equation analogous to Eq. (24).
The Duhem–Margules equation, sometimes called Gibbs–Duhem–Margules equation, describes the relationship between the mole fraction Ni (expression of composition) with partial pressure Pi of the irth component in a liquid mixture expressed by \( \mathop \sum \nolimits_{1}^{{\text{n}}}\, {\text{N}}_{{\text{i}} }{\text{dlogP}}_{{\text{i}}} = 0 \). The integration of this equation gives Raoult’s law for the partial pressures of the mixture, whereas Henry's law results when the mole fraction of the solute goes to zero (Ni ⟶ 0). Ni = Pi. In other words, the concentration of the solute dissolved in the solvent is proportional to the partial pressure of the gas above the solution.
Jaki has given a detailed account on the reception of Duhem’s work by the French community and abroad during his lifetime. Few French physicists, including Jules Tannery, Henri Poincaré, and Paul Langevin, gave support or at least were sympathetic to Duhem’s publications. Others, like Le Chatelier, Jean Perrin, Brillouin, Berthelot, and several others of lower caliber avoided citing Duhem’s name in their work (Jaki 1984, pp. 279–302).
Yves-André Rocard, professor at the Sorbonne and director of the physical laboratories at the École Normale Superieure, seems to have been the first French author to refer in print to Gibbs–Duhem equation in his Thermodynamique in 1952 (Jaki 1984, p. 308).
Emil Jouguet was a graduate of the École Polytechnique and from 1920 to 1939 professor at l'École des mines de Paris. He had collaborated with Duhem at Bordeaux, and thus he was able to evaluate Duhem's physics and his overall theoretical achievements. Jouguet has created the theory of detonation waves with application to high explosives continuing his mentor's efforts on this subject that considered a side effect of the false equilibrium. Jouguet credited Duhem with having laid the theory of explosives on especially solid foundations (Jaki 1984, p. 305).
The rather eccentric expression of the nonsensical branches of mechanics indicates fields of physics, mechanics, and electromagnetism. The list of these fields includes the so-called false equilibria, friction, viscosity, hysteresis phenomena, and electromagnetic theory of materials. These are precisely dissipative phenomena such as thermodynamically irreversible reactions, plasticity, viscoelasticity, and memory effects.
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Dais, P. Impact of Gibbs’ and Duhem’s approaches to thermodynamics on the development of chemical thermodynamics. Arch. Hist. Exact Sci. 75, 175–248 (2021). https://doi.org/10.1007/s00407-020-00259-8
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DOI: https://doi.org/10.1007/s00407-020-00259-8