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Celebrating the birth of De Donder’s chemical affinity (1922–2022): from the uncompensated heat to his Ave Maria

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Un chimiste qui ne s’intéresserait pas à l’affinité

Serait comme un poète qui ne s’interésserait pas… à l’affinité!

(A chemist who is not interested in affinity is like a poet who is not interested… in affinity!)

Aphorisms of Théophile De Donder (recollected around 1927‒1930 by Jean Bosquet)

Abstract

Théophile De Donder, a Belgian mathematician born in Brussels, elaborated two important ideas that created a bridge between thermodynamics and chemical kinetics. He invented the concept of the degree of advancement of a reaction, and, in 1922, he provided a precise mathematical form to the already known chemical affinity by translating Clausius’s uncompensated heat into formal language. These concepts merge in an important inequality that was the starting point for the formalization of non-equilibrium thermodynamics. The present article aims to reconstruct how De Donder elaborated his ideas and developed them by exploring his teaching activity and its connection with his scientific production. Furthermore, it emphasizes the role played by the discussions with his disciples who became his collaborators. The paper analyzes De Donder’s efforts in participating in the second Solvay Chemistry Council in 1925 to call the attention of chemists to his mathematical approach. We explain why his work did not receive much attention at the time, and how, despite this, his formalization of chemical affinity became the basis for the birth of the so-called Brussels school of thermodynamics.

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Fig. 1

© Académie royale de Belgique–Luc Schrobiltgen

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Notes

  1. The concept must not be confused with electronic affinity.

  2. See (Sadoun-Goupil 1991) for a detailed history of the concept.

  3. We borrowed this term from the title of Kondepudi and Prigogine 2015).

  4. Most of the existing literature on the history of De Donder’s chemical affinity is in French. All of the quotations have been translated by the author.

  5. The unit of affinity is joule per mole (J mol−1).

  6. Unquoted biographical notes are contained in (Van den Dungen 1958) and (Glansdorff et al. 1987).

  7. Until the end of 1960s, there was only one University in Brussels, which then split into two different Universities, i.e. the Université Libre de Bruxelles (ULB) and the Vrije Universiteit Brussel (VUB).

  8. Private discussion with Prof. Lambert. Poincaré’s motto is “Thought must never submit to dogma, to a party, to a passion, to an interest, to a preconception, or to anything other than facts themselves; for if thought were to submit, it would cease to be” (Poincaré 1909).

  9. The concept of affinity appeared explicitly in chemistry at the beginning of the seventeenth century. The history of this concept is beyond the scope of our paper. For some landmarks see (Jensen 2015, p. 105), while for a more extended treatment see (Sadoun-Goupil 1991).

  10. This anecdote on De Donder is contained in a footnote which is present only in the original French version of the book.

  11. In the symbol \({Q}^{n}\) introduced by De Donder, the superscript \(n\) stands for “non-compensée”. We preferred the modern term uncompensated instead of the older term non-compensated, used e.g. in (Lengyels 1989, p. 445).

  12. The history of the development of the extent of reaction has not been reconstructed to date. This analysis is beyond the scope of the present paper.

  13. The pages are unnumbered.

  14. In (De Donder 1920b) De Donder declared that he had revised his lectures: the paper was published on June 5, while, as already said, the preface of Lectures I is dated July 1920.

  15. See the Appendix for a review of their statements.

  16. In (De Donder 1920a), (De Donder 1920b), and (De Donder 1920c), De Donder did not use the letter \(G\), which is the modern symbol. In 1920‒1922 and in the following years, he used different letters for this thermodynamic potentials. A complete list can be found in (Sadoun-Goupil 1991, p. 305).

  17. De Donder considered also the second condition, which would select the minimum among the stationary points. We shall not delve into these technical details.

  18. See the Appendix.

  19. It is a generalization because the equation contains the extent of reaction, an additional parameter in Duhem’s spirit.

  20. The chronology of inclusion in Lectures II is unclear because of the lack of explicit dates in Lectures II.

  21. La gravifique de Weyl-Eddington-Einstein in 1924, Théorie Mathématique de l’électricité in collaboration with van Lerberghe, and Introduction a la Gravifique Einsteinienne at the beginning of 1925. For a brief comment of the books see Glansdorff, Bosquet and Géhéniau (1987).

  22. University of Nancy and Strasbourg (1923); Sorbonne in Paris and University of Nancy (1924).

  23. The other scientists were Jules Boulvin, Giuseppe Cesàro and Auguste Collard.

  24. The jury was composed by François Keelhoff (University of Gand); André Duchesne and Henri Buttgenbach (University of Liége); Paul-Henri Stroobant (Brussels University); Gustave Gillon (University of Louvain).

  25. Armand was the son of Ernest Solvay, who died in 1922.

  26. We compared it with the signature that can be found in (Glansdorff et al. 1987, p. 4).

  27. English translation: “To the Members of the Solvay International Chemistry Congress, April 1925. see page 21”.

  28. English translation: “To Mr Charles Lefébure Acknowledgments and tributes”.

  29. We are aware at least of two of them. One is conserved at the Solvay Archives and the other at the Réserve Précieuse archive of the ULB.

  30. In the digitalized version of this communication on the website of the Solvay Science Project, the footnote is not visible.

  31. Arrhenius equation is van ‘t Hoff’s equation. See (Laidler 1993, p. 114).

  32. As already pointed out, we remember that in van ‘t Hoff view, affinity was still identified with a force and he never defined affinity. He used the concept of “work of affinity” whose dimension is joule.

  33. The equation of the chemical reaction represents Lavoisier’s principle of the conservation of mass.

  34. We can argue that De Donder’s decision was connected with the health of his wife. From Heymans’ letter, we are informed that on February, 17 (1925) De Donder’s wife had some health problems and that her health was not improving as rapidly as De Donder hoped (Fig. 6). Solvay’s dinner was scheduled two months after De Donder’s letter and as it would happen in 1927 De Donder was invited with her. It is plausible that he waited until the day before the dinner at the Taverne Royale to make his final decision.

  35. For example, with this old convention the stoichiometric coefficients of the reaction \({N}_{2}+3{H}_{2}\to 2N{H}_{3}\) are \({\nu }_{1}=1\), \({\nu }_{2}=3\) and \({\nu }_{3}=-2\).

  36. The variable p, which indicates the pressure in the squared formula of the annotations of Fig. 13, can be compared with the letter p on the word page of Fig. 8. The correspondence between the two handwritings confirms that De Donder wrote it.

  37. De Donder tried to generalize the concept of affinity in the context of what he called relativistic thermodynamics. We postpone the discussion of this topic to a future work.

  38. For an explanation of this concept see (Kondepudi and Prigogine 2015, p. 131).

  39. We focused on ULB and leave aside the study of the impact in other Belgian Universities and the rest of Europe’s countries.

  40. For a review of the textbook and a tribute to its authors see (Jensen 2015, p. 129).

  41. Gibbs’ free energy as presented in Lewis-Randall’s book would not have been applicable to non-equilibrium conditions.

  42. The statement quoted is the (official) English translation provided by van Rysselberghe for De Donder’s book and it is the same statement quoted by Géhéniau (in French).

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Acknowledgements

We want to acknowledge the time and effort devoted by the reviewer to improving the quality and readability of this work. We also benefited from the help of Marina Solvay, and his colleagues Yoanna Alexiou and Nicolas Brunmayr in understanding some handwritten French words and from the support of the archivists of the Université Libre de Bruxelles (ULB) for the digitalization of the documents. The author is also grateful to Franklin Lambert for the interesting discussions, Brigitte van Tiggelen, Kenneth Bertram and Jan Danckaert for their contribution to the Solvay Science Project and his colleague Catherine Judson for revising the English language and for her fruitful suggestions.

Funding

This work was supported by the Fund for Natural Sciences in Society at the Vrije Universiteit Brussel (VUB).

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  1. Theoretische Natuurkunde, Applied Physics Research Group (APHY) and International Solvay Institutes, Vrije Universiteit Brussel (VUB), Pleinlaan 2, B-1050, Brussels, Belgium

    Alessio Rocci

  2. University of Padova, Padua, Italy

    Alessio Rocci

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Appendix

Appendix

According to Keith Laidler, “The story of how chemical thermodynamics developed is a somewhat tangled one, since several investigators worked along different lines and quite independently of one another” (Laidler 1993, p. 107). In this appendix, we briefly present some key facts related to chemical affinity by reviewing the history of chemical thermodynamics. We shall also present some results mentioned in the main text which are connected with De Donder’s work covering the period between the 1850s and 1920s, that is until De Donder began his research program. The original quotations can be found in (Laidler 1993).

Between 1850 and 1865, Rudolf Clausius presented his synthesis of thermodynamics. The publication of his fourth memory in 1864 marked the birth of the theory. Although he briefly considered also chemical processes, the recognized father of chemical thermodynamics is Josiah Willard Gibbs. (Prigogine and Defay 1944, p. xi). In Europe, the dissemination of Gibbs’ work is mainly due to Henry Le Chatelier’s French translation dated 1867. Gibbs’ theoretical approach has brought many benefits to physical chemistry (Prigogine and Defay 1944, p. xi). Gibbs, Françoise J. D. Massieu, Hermann von Helmholtz and Pierre Duhem shifted the focus of chemistry to thermodynamic potentials. Between the end of the nineteenth century and the beginning of the twentieth century, chemists were very wary of entropy (Kragh and Weininger 1996). De Donder contributed to the development of this concept for irreversible processes and to formalising the connection with his chemical affinity. His work opened the road to the modern concepts of entropy flow and entropy production, related to the usual concept of heat and uncompensated heat respectively, developed in the context of his theory by Ilya Prigogine after the Second World War.

Gibbs investigated the problem of the conditions for physical and chemical equilibrium and invented the chemical potential which is a function of the amounts of all components (Laidler 1993, p. 111). This procedure led him to his famous Phase Rule. Two phases are different physical states of the same chemical species. Gibbs’ rule relates the number of degrees of freedom of the system, phases, constituents and the number of real chemical reactions.

Besides Gibbs’ approach, a second school of chemical thermodynamics was developed by Jacobus Henricus van ‘t Hoff and Walter Nernst (Prigogine and Defay 1944, p. xii). “The thermodynamic work of van’t Hoff […] was in marked contrast to that of Gibbs and Helmholtz” (Laidler 1993, p. 114). Van ‘t Hoff focused on chemical reactions investigating the role of the heat of reaction and the concept of maximum work. According to Prigogine and Defay, their approach was limited by the use of equilibrium states and “a thermodynamics theory of chemical reactions is necessarily a thermodynamics of irreversible phenomena” (Prigogine and Defay 1944, p. xiii). Furthermore, neither of the quantities used by van ‘t Hoff-Nernst school was a function of state because their values are not measurable by measurements made only on the system itself, regardless of the nature of the transformation experienced by the system between two different times.

Around 1865, Marcelin Berthelot developed the hypothesis that the heat exchanged in a chemical reaction should give a measure of the affinity of the reactants (Sadoun-Goupil 1991, p. 277; Cattani 2010, p. 138). “The occurrence of a number of endothermic reactions showed that this could not be the case” (Laidler 1993, p. 112) and Helmholtz tried to overcome this difficulty (Laidler 1993, p. 114). Van’t Hoff criticized Berthelot’s ideas and improved his approach. He conceived the chemical affinity as a force, but he never defined it explicitly. Instead, he focused on “the work done by affinity” (Sadoun-Goupil 1991, p. 314), which in van ‘t Hoff thermodynamics is related to the heat exchanged in the reaction and is a measurable quantity. The heat of reaction \(q\) appears in van ‘t Hoff equation, which expresses the variation with temperature \(T\) of the equilibrium constant\(k\).

Discussing his equation, van ‘t Hoff “presented an important qualitative discussion of how \(k\) is affected by temperature. If heat is evolved when a reaction occurs from left to right (\(q\) is negative) the equilibrium constant will decrease as the temperature is raised. Conversely, if \(q\) is positive (heat is absorbed) a rise in temperature will increase\(k\). Later in 1884, […] Le Chatelier quoted these conclusions of van’t Hoff, which have come to be referred to as the Le Chatelier principle.

The idea of the equilibrium constant, accepted by van ‘t Hoff, was already introduced. The experiments of Bethelot and Léon Péan de Saint-Gilles on reactions of esterification with acids had grounded the basis for the collaboration between the mathematician Cato Maximilien Guldberg and the chemist Peter Waage. They introduced the equilibrium constant \(k\). Their work inserts in the history of chemical kinetics: these two authors devoted a series of papers to the investigation of chemical affinity. Their names are related to the law of mass action or the Guldberg-Waage law (Laidler 1993, p. 115). The two authors argued in terms of forces that the equilibrium of a reaction, represented by \(k\), is proportional to a ratio where both the numerator and denominator are a product of the concentrations of reactants or products to the power of their stoichiometric coefficients.

De Donder showed that Berthelot’s idea was not completely wrong. He realized that it is the heat of reaction associated with the uncompensated heat introduced by Clausius that correctly gives a measurement of affinity. He emphasized this fact by enunciating the “correct form of Berthelot principle” (De Donder and Rysselberghe 1936, p. 27). His disciples called his extension De Donder-Berthelot theorem (Prigogine and Defay 1944, p. 47). De Donder overcame the problems of van ‘t Hoff-Nernst school by defining chemical affinity as a function of state which allowed him to describe non-equilibrium phenomena. As explained in the main text, he connected it with the other functions of state by inventing the extent of reaction. By adding this parameter and formalizing the concept of affinity, De Donder extended the Guldberg-Waage law, van ‘t Hoff equation and Le Chatelier principle (De Donder and van Rysselberghe 1936, p. 63 and p. 84). In De Donder’s approach, the first two results are connected and inserted in a wider context (Prigogine and Defay 1944, p. 105–107).

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Rocci, A. Celebrating the birth of De Donder’s chemical affinity (1922–2022): from the uncompensated heat to his Ave Maria. Found Chem 26, 37–73 (2024). https://doi.org/10.1007/s10698-023-09488-5

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