E. W. Lund University of Oslo
Oslo, Norway
Guldberg and Waage and the Law of Mass Action
Last year, the hundredth anniversary of the first publication by Guldberg and Waage dealing with "The Law of Mass Action" was celebrated in their native country (1). While the importance of this and their later publication of 1867 is generally appreciated by writers on the history of chemistry, the usual presentation of Guldberg and Waage's work, commonly found in textbooks, was criticized in an article in the Textin 1956 (2). In book Errors series of THB JOURNAL that article, however, important parts of Guldberg and Waage's work were not given an adequate treatment, in the present author's opinion. I therefore feel that readers might profit from a further examination of excerpts from the original papers, in assessing the significance of Guldberg and Waage's contributions to knowledge of both equilibrium and reaction rates. In their first publication (S), presented to the Christiania (now Oslo) Academy of Science and Letters on March 11, 1864, Guldberg and Waage introduced the concept of a reaction sequence. They distinguished clearly between simple and composite reactions and considered a composite reaction to consist of several simple ones in a sequence. The remainder of their treatment is limited to simple substitution reactions. Throughout their work, Guldberg and Waage applied the concept of "chemical force" which at that time was considered to be the clue to the solution of the problem of chemical affinity. Briefly, their arguments run as follows: When a compound is formed in a chemical reaction, a strong chemical force is assumed to be acting. Very often, however, one must conclude that two opposing forces are active at the same time under given conditions. This was particularly assumed to be the
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case in reactions for which the direction may be reversed by changing temperature or relative concentrations. By careful selection of experimental conditions a state of equilibrium can be attained a t which the two opposing forces balance. It is evident that Guldberg and Waage's work was influenced by the investigations of Berthelot and St. Gilles in 1861-63 (4) dealing with the formation and ddecomposition of esters. Guldberg and Waage's own experimental work was primarily concerned with heterogeneous equilibria in systems containing sulfates and carbonates of barium and potassium. In such cases they find the same kind of situation as reported by Berthelot and St. Gilles in their experiments, namely that in a reaction of the kind A B = A' B' there is a limiting situation where all four substances are present a t the same time. Guldberg and Waage considered this to be a state of equilibrium at which two opposing forces were of equal magnitude. A few direct quotations indicate their starting point and their conclusion in the form of a general law.
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In order to study the chemical force we have chosen the conditions where the formation and decomposition are taking place at the same time. In this case, the forces which produce the formation of the two molecules and those which decompose these molecules in regenerating the original molecules, are acting s t the same time and give rise to a. state of equilibrium. When two molecules A and B with masses M and N tend to form the two new molecules A' and B' by a substitution of their elements, the total volume being V , the force which tends to produce this substitution may be expressed by u ( M / V ) " ( N / V ) b where a , a, and b are specific coefficients far the species Let us suppose that one has 8. system of four substances A, B, A', and B' with masses p, q, p', and q' respeotively. When the state
...
of equilibrium is attained, the quantity z of A and of B is transformed into A' and B' and the masses are p - I , q - z, p',+ z , and q' z . The force which tends to form A' B' w111 he a [ ( p - z ) / V I S [ ( q- z ) / V I b , and that which tends to form A B will be a ' [ ( p ' z)/VI"[(qf z ) / V ] b ' . These two forces are equal and one ohtsina the equation
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Thus with this statement a mathematical formulation of the condition for chemical equilihrium was given for the first time. It is clear however, that the authors failed to realize that the powers to which a concentration had to be raised should be integers deducible from the chemical equation. They considered the powers to be parameters which had to be determined by experiment. The quotations given above are taken from the puhlication in the French weekly periodical Les Mondes in the issue of May 19, 1864 (6). I n their first publication of March, Guldberg and Waage formulated two laws for the force, one of Mass Action and one of Volume Action. As shown above, the two laws were combined into one of concentration action two mouths later. I n the summer of 1864 Guldberg and Waage also presented papers dealing with the influence of time on chemical reactions. They proposed the idea that the driving force is proportional to the rate of the reaction. Again they considered the substitution reaction A B = A' B' and treated two cases mathematically.
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Let Us assume that the new bodies A'and B' do not react. Let I, and o be the number of molecules of A and B. v the veloeitv. t
where k is a constant depending on the nature of the bodies, the volume, the temperature, and the solvent. Then, let us consider the more general case where the new substances A' and B' react and give the origind bodies A and B. The force which tends to produce A B ia equal to oza'9' and, b' = n, one has the velocity pntting a'
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v = d x / d t = k [ ( p - x)]'[(q
- x)]b - a x "
I n this case s state of equilibrium is attained far a. certain value of 2 .
For this value of x, the net velocity is equal to zero. The authors then showed how it is possible to carry out an approximate integration of their rate equation. They applied the formula to a few reactions, i.e., some of the esterificatious reported by Berthelot and St. Gilles and the reaction between barium peroxide and hydrochloric acid for which experimental values had been given by Brodie in 1863 (7). I n the case of the ester formation they found that an increase of the amount of acid makes the reaction go faster, whereas an increase in the amount of alcohol slows it down. They applied their formulae to various cases (diierent kinds of alcohols and of acids and different temperatures). Very probably, this different influence of acid and alcohol on the rate of reaction was the main reason why they introduced the powers to he determined experimentally in their equilibrium and rate equations. I n a second paper (8)written in French and presented as an official publication of Oslo University in 1867, Guldberg and Waage made a change in the formulation. They gave a detailed discussion of their view on the
forces acting in a chemical system. From this discussion it follows that they considered forces to he acting between all pairs of molecules present. Most important were the "affinity forces" between the chemically reacting species. I n addition, weaker "secondary forces" were assumed to he acting between any other pnir of molecules present. These latter forces might rr:tard or accelerate the reaction. Again they considered a system containing four substances A, B, A', and B' in a dynamical chemical equilibrium with the equation A B = A' B'. This time the concentrations (which they called "active masses") were denoted by the letters p, q, p', and q', respectively. In order to simplify matters the authors introduced the concept af an ideal reaction-in which the secondary forces might be neglected. From their experiments they considered themselves in the position to conclude that the affinity force between A and B was proportional to the product of the concentrations, i.e., equal to kpq, where k is a constant ("aflinity coefficient"). The opposite force, causing regeneration of A and B, will be k'p'q' where k' is the affinity coefficient for that reaction. This force is in equilibrium with the first force, and consequently kpq = k'p'q'. A few direct quotations may be appropriate to show their further treatment.
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After hwing determined the active masses of p, q, p', and q' one can find the ratio between the coefficients k and k'. On the other hand. havine found the ouatient k'lk. , , one can ealeulate in advance the result of the rertctians for an arbitrary initial state of the four substances. Let us denote by P , Q, P', and Q' the absolute quantities of the four bodies A, B, A', and B' before the reaction starts and let z be the number of atoms of A and B which are transformed into A' and B'. Let us sumose that the total volume is constant during the reaction and eq%S to V . One then obtains p = (P - z)/V z)/V p' = ( P '
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q = (Q - z ) / V q' = (Q' z)/V
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After having introduced these values into the equation and multiplied by Veone finds (P
- z ) ( Q - 2) = ( k ' / k ) ( P ' + XI(&' + z )
With the aid of this equation, one easily determines the value of z . When the two bodies A and A' have constant active masses during the reaction and when the values of these masses are equal, the formula reduces t o
This is the situation, a t least approximately, when A and A' are solids while B and B' artre liquids.
They applied this formula to the reaction between barium sulfate and potassium carbonate and found very good agreement between observed and calculated values. (Today we know this occurred because the amount of solutes chosen were such that the solutions had approximately the same ionic strength.) Extensive applications were also made to esterification reactions. For these reactions they found that the secondary forces could not be neglected. Therefore in the case of the reaction between acetic acid and ethyl alcohol, the equilihrium equation is pq = 0.2372 p'q'
- 0.01843 pp' - 0.00626 pp'
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0.0372 p'q
+ 0.00723 9.9'
Here p, q, p', and q' denote the concentrations of acetic Volume 42, Number 10, Odober 1965
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acid, ethyl alcohol, ethyl acetate, and water respectively. The last four terms on the right hand side express the action of secondary forces. They clearly are of minor importance. In the same paper of 1867, Guldberg and Waage also considered reaction rate in the new formulation and gave observed and calculated values in a number of cases. We know today that some of the ideas of Guldberg and Waage have been abandoned, for instance the vague concept of "chemical force." This was, however, at the time a frequently used idea in the theory of chemical affinity. Furthermore, it was perhaps unfortunate that they based the theories so much on heterogeneous equilibria. Their treatment of this case was not quite correct because they considered the active masses of BaSOPand of BaC03 to be equal. On the other hand, they introduced useful ideas about the concept of a reaction sequence and the influence of "foreign" substances on reaction rate and equilibrium. Of far greater importance, however, was their formulation of a general equilibrium equation, namely kpq = k'p'q'. Their motivation for this equation was based on general arguments in terms of forces and they checked their hypotheses by experimental data, both on reaction rates and on equilibrium. The essential content of their work was made generally known due to the application of the equilibrium equation by the Danish thermochemist Julius Thomsen in a paper of 1869 (9). August Horstmann quoted the same formula in 1873 and considered it to be in correspondence with his own theory of dissociation (10). Horstmann also used Guldberg and Waage's experimental results in the case of the equilihrium between BaSOa, BaC03, K2SOa, and K2C03. He found the ratio between the concentration of KsSOa and K&O, to he constant and independent of the relative amounts of BaSOa and BaCOa present. This result, he admits, was already suggested by Guldberg and Waage. As a supplement to this account of Guldherg and Waage's work, it seems appropriate to make a brief note on other papers which represent two important steps in the further development of the field. In 1877 van't Hoff gave a treatment (11) of the kinetics of ester formation, based on the data of Berthelot and St. Gilles. He assumed that the rate of a reaction is proportional to the product of the numbers (P and Q ) of the interacting molecules and inversely proportional to the total volume, i.e. From this assumption he arrived at the simple equilibrium equation pq = '/rp'q' (here written in Guldberg and Waage's notation). In spite of a somewhat incorrect rate law the equilihrium equation is correct because there are two reactants on both sides.
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In the same year Horstmann published a paper (12) dealing with thermal dissociation of a solid into two gaseous products. From thermodynamic arguments he derived an equilihrium equation of the form pzm X pan = d. Here p, and pa are the partial pressures of the gaseous products, m and n are the number of moles formed per mole of solid, and d is a function of temperature only. As far as it has been possible to determine, this is the first appearance of the powers in an equilibrium equation as being equal to the coefficients in the chemical equations. Some of the publications in the 1870's must have given Guldberg and Waage the impression that their papers of 1864 and 1867 had not been generally known. They must have felt the need to write about their ideas in a more widely circulated journal, because such an article appeared in Journal fur Praktische Chenzie in 1879 (15). In this last paper, molecular kinetics and energy considerations were taken into account. The authors stated that the molecules of a certain compound may be in different states of energy and that only a certain fraction of the molecules is in such a state that a collision can initiate a reaction. The reaction rate is proportional not only to these fractions of the active masses, hut also to the active masses themselves. Ideas along the same line had, however, been puhlished by Pfaundler in 1867 and 1874 (14, 15). This last paper by Pfaundler and that by Horstmann in 1873 are especially interesting because they represent the first applications of the second law of thermodynamics to chemical equilibrium. As is well known, this approach in the hands of J. Willard Gihhs has provided the most powerful tool in solving the problem of chemical affinity. Literature Cited (1) "The Law of Mass Action" (a. centenary volume), Det Norske Videnskaps-Akademi i Oslo, Oslo, 1964. E. A,, J. CHEM.EDUC.,33,544 (1956). (2) GUGGENAEIM, (3) G U ~ B E R GC., M., AND WAAGE,P., "Avhandl. Nomke Videnskws-Akademi Oslo." Mat. Natorv. Kl., 1864, p. 35. (4) BERTHELOT, M., AND PEAN DE SAINT-GILLES,L., Ann. Chim. Phys. (3), 65, 385, (1862); 66, 5 (1862); 68, 225 ilRRR>. GULDBERQ, C. M., AND WAAGE,P., Les Mondes, 12, 107113 (1864). Ibid., 12,627-32 (1864). BBODIE,B.C.,Ann. Phya. Chem. 120,294(1863). C. M., AND WAAGE,P., etude^ S L I 1.8 ~ Affinitks GULDBERG, ehimique," Christiania, 1867; Ostwald's Klassiker, 104. J., Ann. Phys. Chem., 138,65 (1869). THOMSEN, H O R S ~ A N A,, N , Ann. Chem. & Phann., 170, 192 (1873); Ostwald'sKh~iker,137. J. H., Ber., 10,669 (1877). YAN'THOFF, HORSTMANN, A,, Vwhand. NaturhistMed. Ver. Heidelberg, 1 (5), 465 (1877); Ostwald's Klassiker, 137. GULDBERG, C. M., AND WAAGE,P., J . Prakt. Chm., 127, 69 (1879); Ostwald's Klassiker, 104. PFAUNDLER, L., Ann. Phys. Chem., 131,55 (1867). L., Ann. Phys. Chem., Jubelband, 182 (1874). PFAUNDLER, \----,-
