Journal of Chemical Education 42(10): 548 (1965)

E. W. Lund Guldberg and Waage and the University of Oslo Oslo, Norway Law of Mass Action

Last year, the hundredth anniversary of case in reactions for which the direction may be re- the first publication by Guldberg and Waage dealing versed by changing temperature or relative concen- with "The Law of Mass Action" was celebrated in their trations. By careful selection of experimental con- native country (1). While the importance of this and ditions a state of equilibrium can be attained at which their later publication of 1867 is generally appreciated the two opposing forces balance. by writers on the history of chemistry, the usual presen- It is evident that Guldberg and Waage's work was in- tation of Guldberg and Waage's work, commonly found fluenced by the investigations of Berthelot and St. Gilles in textbooks, was criticized in an article in the Text- in 1861-63 (4) dealing with the formation and dde- book Errors series of THB JOURNALin 1956 (2). In composition of esters. Guldberg and Waage's own that article, however, important parts of Guldberg experimental work was primarily concerned with hetero- and Waage's work were not given an adequate treat- geneous equilibria in systems containing sulfates and ment, in the present author's opinion. I therefore carbonates of barium and potassium. In such cases feel that readers might profit from a further examination they find the same kind of situation as reported by of excerpts from the original papers, in assessing the Berthelot and St. Gilles in their experiments, namely significance of Guldberg and Waage's contributions to that in a reaction of the kind A + B = A' + B' there knowledge of both equilibrium and reaction rates. is a limiting situation where all four substances are In their first publication (S), presented to the present at the same time. Guldberg and Waage con- Christiania (now Oslo) Academy of Science and Letters sidered this to be a state of equilibrium at which two on March 11, 1864, Guldberg and Waage introduced opposing forces were of equal magnitude. A few direct the concept of a reaction sequence. They distinguished quotations indicate their starting point and their clearly between simple and composite reactions and conclusion in the form of a general law. considered a composite reaction to consist of several simple ones in a sequence. The remainder of their In order to study the chemical force we have chosen the condi- treatment is limited to simple substitution reactions. tions where the formation and decomposition are taking place Throughout their work, Guldberg and Waage applied at the same time. In this case, the forces which produce the formation of the two molecules and those which decompose these the concept of "chemical force" which at that time was molecules in regenerating the original molecules, are acting st the considered to be the clue to the solution of the problem same time and give rise to a. state of equilibrium. of chemical affinity. Briefly, their arguments run as When two molecules A and B with masses M and N tend to follows: When a compound is formed in a chemical re- form the two new molecules A' and B' by a substitution of their action, a strong chemical force is assumed to be acting. elements, the total volume being V, the force which tends to Very often, however, one must conclude that two 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 opposing forces are active at the same time under given us suppose that one has 8. system of four substances A, B, A', conditions. This was particularly assumed to be the and B' with masses p, q, p', and q' respeotively. When the state

548 / Journal of Chemical Education of equilibrium is attained, the quantity z of A and of B is trans- forces acting in a chemical system. From this dis- formed into A' and B' and the masses are p - I, q - z, p',+ z, cussion it follows that they considered forces to he and q' + z. The force which tends to form A' + B' w111 he a[(p- z)/VIS[(q- z)/VIb, and that which tends to form A + acting between all pairs of molecules present. Most B will be a'[(p' + z)/VI"[(qf + z)/V]b'. These two forces important were the "affinity forces" between the chem- are equal and one ohtsina the equation ically reacting species. In addition, weaker "second- ary forces" were assumed to he acting between any other pnir of molecules present. These latter forces Thus with this statement a mathematical formulation might rr:tard or accelerate the reaction. of the condition for chemical equilihrium was given for Again they considered a system containing four sub- the first time. It is clear however, that the authors stances A, B, A', and B' in a dynamical chemical failed to realize that the powers to which a concentra- equilibrium with the equation A + B = A' + B'. tion had to be raised should be integers deducible from This time the concentrations (which they called "active the chemical equation. They considered the powers masses") were denoted by the letters p, q, p', and q', to be parameters which had to be determined by ex- respectively. periment. In order to simplify matters the authors introduced The quotations given above are taken from the puhli- the concept af an ideal reaction-in which the secondary cation in the French weekly periodical Les Mondes in forces might be neglected. From their experiments the issue of May 19, 1864 (6). In their first publication they considered themselves in the position to conclude of March, Guldberg and Waage formulated two laws that the affinity force between A and B was propor- for the force, one of Mass Action and one of Volume tional to the product of the concentrations, i.e., equal Action. As shown above, the two laws were combined to kpq, where k is a constant ("aflinity coefficient"). into one of concentration action two mouths later. The opposite force, causing regeneration of A and B, In the summer of 1864 Guldberg and Waage also will be k'p'q' where k' is the affinity coefficient for that presented papers dealing with the influence of time on reaction. This force is in equilibrium with the first chemical reactions. They proposed the idea that the force, and consequently kpq = k'p'q'. driving force is proportional to the rate of the reaction. A few direct quotations may be appropriate to show Again they considered the substitution reaction A + their further treatment. B = A' + B' and treated two cases mathematically. After hwing determined the active masses of p, q, p', and q' Let Us assume that the new bodies A'and B' do not react. Let one can find the ratio between the coefficients k and k'. On the I, and o be the number of molecules of A and B. v the veloeitv. t 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 where k is a constant depending on the nature of the bodies, the into A' and B'. Let us sumose that the total volume is constant volume, the temperature, and the solvent. during the reaction and eq%S to V. One then obtains Then, let us consider the more general case where the new sub- stances A' and B' react and give the origind bodies A and B. p = (P - z)/V q = (Q - z)/V The force which tends to produce A + B ia equal to oza'9' and, p' = (P' + z)/V q' = (Q' + z)/V pntting a' b' = n, one has the velocity + After having introduced these values into the equation and v = dx/dt = k[(p - x)]'[(q - x)]b - ax" multiplied by Veone finds In this case s state of equilibrium is attained far a. certain value (P - z)(Q - 2) = (k'/k)(P' + XI(&' + z) of 2. With the aid of this equation, one easily determines the value of z. For this value of x, the net velocity is equal to zero. When the two bodies A and A' have constant active masses during the reaction and when the values of these masses are The authors then showed how it is possible to carry equal, the formula reduces to 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. This is the situation, at least approximately, when A and A' Gilles and the reaction between barium peroxide and are solids while B and B' artre liquids. hydrochloric acid for which experimental values had been given by Brodie in 1863 (7). They applied this formula to the reaction between In the case of the ester formation they found that an barium sulfate and potassium carbonate and found increase of the amount of acid makes the reaction go very good agreement between observed and calculated faster, whereas an increase in the amount of alcohol values. (Today we know this occurred because the slows it down. They applied their formulae to various amount of solutes chosen were such that the solutions cases (diierent kinds of alcohols and of acids and differ- had approximately the same ionic strength.) ent temperatures). Very probably, this different Extensive applications were also made to esterifica- influence of acid and alcohol on the rate of reaction was tion reactions. For these reactions they found that the the main reason why they introduced the powers to he secondary forces could not be neglected. Therefore in determined experimentally in their equilibrium and the case of the reaction between acetic acid and ethyl rate equations. alcohol, the equilihrium equation is In a second paper (8)written in French and presented pq = 0.2372 p'q' - 0.01843 pp' - 0.00626 pp' as an official publication of Oslo University in 1867, + 0.0372 p'q + 0.00723 9.9' Guldberg and Waage made a change in the formulation. They gave a detailed discussion of their view on the Here p, q, p', and q' denote the concentrations of acetic

Volume 42, Number 10, Odober 1965 / 549 acid, ethyl alcohol, ethyl acetate, and water respec- In the same year Horstmann published a paper (12) tively. The last four terms on the right hand side dealing with thermal dissociation of a solid into two express the action of secondary forces. They clearly gaseous products. From thermodynamic arguments he are of minor importance. derived an equilihrium equation of the form pzm X In the same paper of 1867, Guldberg and Waage also pan = d. Here p, and pa are the partial pressures of the considered reaction rate in the new formulation and gaseous products, m and n are the number of moles gave observed and calculated values in a number of formed per mole of solid, and d is a function of tempera- cases. ture only. As far as it has been possible to determine, We know today that some of the ideas of Guldberg this is the first appearance of the powers in an equilib- and Waage have been abandoned, for instance the rium equation as being equal to the coefficients in the vague concept of "chemical force." This was, how- chemical equations. ever, at the time a frequently used idea in the theory Some of the publications in the 1870's must have of chemical affinity. Furthermore, it was perhaps given Guldberg and Waage the impression that their unfortunate that they based the theories so much on papers of 1864 and 1867 had not been generally known. heterogeneous equilibria. Their treatment of this They must have felt the need to write about their ideas case was not quite correct because they considered the in a more widely circulated journal, because such an active masses of BaSOPand of BaC03to be equal. article appeared in Journal fur Praktische Chenzie in On the other hand, they introduced useful ideas about 1879 (15). In this last paper, molecular kinetics and the concept of a reaction sequence and the influence of energy considerations were taken into account. The "foreign" substances on reaction rate and equilibrium. authors stated that the molecules of a certain compound Of far greater importance, however, was their formula- may be in different states of energy and that only a tion of a general equilibrium equation, namely kpq = certain fraction of the molecules is in such a state that a k'p'q'. Their motivation for this equation was based collision can initiate a reaction. The reaction rate is on general arguments in terms of forces and they proportional not only to these fractions of the active checked their hypotheses by experimental data, both masses, hut also to the active masses themselves. on reaction rates and on equilibrium. Ideas along the same line had, however, been puh- lished by Pfaundler in 1867 and 1874 (14, 15). This The essential content of their work was made gen- last paper by Pfaundler and that by Horstmann in erally known due to the application of the equilibrium 1873 are especially interesting because they represent equation by the Danish thermochemist Julius Thomsen the first applications of the second law of thermody- in a paper of 1869 (9). August Horstmann quoted the namics to chemical equilibrium. As is well known, this same formula in 1873 and considered it to be in cor- approach in the hands of Willard Gihhs has provided respondence with his own theory of dissociation (10). J. the most powerful tool in solving the problem of chem- Horstmann also used Guldberg and Waage's experi- ical affinity. mental results in the case of the equilihrium between BaSOa, BaC03, K2SOa, and K2C03. He found the Literature Cited ratio between the concentration of KsSOa and K&O, (1) "The Law of Mass Action" (a. centenary volume), Det to he constant and independent of the relative amounts Norske Videnskaps-Akademi i Oslo, Oslo, 1964. of BaSOa and BaCOa present. This result, he admits, (2) GUGGENAEIM,E. A,, J. CHEM.EDUC., 33,544 (1956). was already suggested by Guldberg and Waage. (3) GU~BERG,C. M., AND WAAGE,P., "Avhandl. Nomke As a supplement to this account of Guldherg and Videnskws-Akademi Oslo." Mat. Natorv. Kl., 1864, Waage's work, it seems appropriate to make a brief p. 35. (4) BERTHELOT,M., AND PEAN DE SAINT-GILLES,L., Ann. note on other papers which represent two important Chim. Phys. (3), 65, 385, (1862); 66, 5 (1862); 68, 225 steps in the further development of the field. In 1877 ilRRR>. \----,- van't Hoff gave a treatment (11) of the kinetics of GULDBERQ,C. M., AND WAAGE,P., Les Mondes, 12, 107- ester formation, based on the data of Berthelot and 113 (1864). Ibid., 12,627-32 (1864). St. Gilles. He assumed that the rate of a reaction is BBODIE,B.C.,Ann. Phya. Chem. 120,294(1863). proportional to the product of the numbers (P and Q) GULDBERG,C. M., AND WAAGE,P., etude^ SLI~1.8 Affinitks of the interacting molecules and inversely proportional ehimique," Christiania, 1867; Ostwald's Klassiker, 104. to the total volume, i.e. THOMSEN,J., Ann. Phys. Chem., 138,65 (1869). HORS~ANN,A,, Ann. Chem. & Phann., 170, 192 (1873); Ostwald'sKh~iker,137. YAN'THOFF,J. H., Ber., 10,669 (1877). From this assumption he arrived at the simple equi- HORSTMANN,A,, Vwhand. NaturhistMed. Ver. Heidelberg, librium equation pq = '/rp'q' (here written in Guldberg 1 (5), 465 (1877); Ostwald's Klassiker, 137. GULDBERG,C. M., AND WAAGE,P., J. Prakt. Chm., 127, and Waage's notation). In spite of a somewhat in- 69 (1879); Ostwald's Klassiker, 104. correct rate law the equilihrium equation is correct PFAUNDLER,L., Ann. Phys. Chem., 131,55 (1867). because there are two reactants on both sides. PFAUNDLER,L., Ann. Phys. Chem., Jubelband, 182 (1874).

550 / Journal of Chemical Educafion