A very important epoch in the history of Berthollet's doctrine was attained when, in 1867, the Norwegian chemists, Guldberg and Waage, expressed it as an algebraical formula. They defined the active mass as the number of molecules contained in a given volume, and assumed, as follows from the spirit of Berthollet's teaching, that the action between the substances was equal to the product of the masses of the reacting substances.
Hence if the salts MX and NY be taken in equivalent quantities (m = 1 and n = 1) and the salts MY and NX are not added to the mixture but proceed from it, then if k represent the coefficient of the rate of the action of MX on NY and if k′ represent the same coefficient for the pair MY and NX, then we shall have at the moment when the decomposition equals x a measure of action for the first pair: k(1 - x)(1 - x) and for the second pair k′xx, and a state of equilibrium or limit will be reached when k(1 - x)2 = k′x2, whence the ratio k/k′ = [x/(1 - x)]2.
Therefore in the case of the action of alcohol on an acid, when x = ⅔, the magnitude k/k′ = 4, that is, the reaction of the alcohol on the acid is four times as fast as that of the ethereal salt on water. If the ratio k/k′ be known, then the influence of mass may be easily determined from it. Thus if instead of one molecule of alcohol two be taken, then the equation will be k(2 - x)(1 - x) = k′xx, whence x = 0·85 or 85 percent., which is close to the result of experiment. If 300 molecules of alcohol be taken, then x proves to be approximately 100 per cent., which is also found to be the case by experiment.31
But it is impossible to subject the formation of salts to any process directly analogous to that which is so conveniently effected in etherification. Many efforts have, however, been made to solve the problem of the measure of reaction in this case also. Thus, for example, Khichinsky (1866), Petrieff (1885), and many others investigated the distribution of metals and haloid groups in the case of one metal and several haloids taken in excess, as acids; or conversely with an excess of bases, the distribution of these bases with relation to an acid; in cases where a portion of the substances forms a precipitate and a portion remains in solution. But such complex cases, although they in general confirm Berthollet's teaching (for instance, a solution of silver nitrate gives some silver oxide with lead oxide, and a solution of nitrate of lead precipitates some lead oxide under the action of silver oxide, as Petrieff demonstrated), still, owing to the complexity of the phenomena (for instance, the formation of basic and double salts), they cannot give simple results. But much more instructive and complete are researches like those made by Pattison Muir (1876), who took the simple case of the precipitation of calcium carbonate, CaCO 3 , from the mixture of solutions of calcium chloride and sodium or potassium carbonate, and found in this case that not only was the rate of action (for example, in the case of CaCl 2 + Na 2 CO 3 , 75 per cent. of CaCO 3 was precipitated in five minutes, 85 per cent. in thirty minutes, and 94 per cent. in two days) determined by the temperature, relative mass, and amount of water (a large mass of water decreases the rate), but that the limit of decomposition was also dependent on these influences. However, even in researches of this kind the conditions of reaction are complicated by the non-uniformity of the media, inasmuch as a portion of the substance is obtained or remains in the form of a precipitate, so that the system is heterogeneous. The investigation of double saline decompositions offers many difficulties which cannot be considered as yet entirely overcome. Although many efforts have long since been made, the majority of the researches were carried on in aqueous solutions, and as water is itself a saline compound and able to combine with salts and