The ideas of Berthollet have been found to be sound in their essentials, and they have served as the foundation for the modern theory of chemical action, though their full develop ment cannot be explained without the use of the differential calculus. The basis of the theory of mass-action, so far as solutions are concerned at all events, appears to be substantially as fol lows : Two substances in solution cannot com bine with each other, except when a molecule, or ion (see SOLUTIONS) of the one, in its wan dering through the solution, chances to encoun ter a molecule or ion of the other. Now while we do not know the actual number of encoun ters that take place in a given time between molecules of different kinds, we do ialOW that in a homeogeneous solution the chance that any one given molecule of the first kind will encoun ter some molecule of the second kind within (say) the next second, is strictly proportional to the number of molecules of the second kind that are present in the solution; and conversely, the chance that any given molecule of the second kind will encounter some molecule of the first kind within the next second is strictly propor tional to the number of molecules of the first kind that are present Since the number of molecules of each kind that are present in an actual solution is practically infinite, this amounts to saying that the actual number of encounters between molecules of different kinds, in one second, is proportional to the product of the ntunber of molecules of the first and second kinds that are present. As an illus tration of the usefulness of this principle, we may consider the equilibrium of a mixture of acetic acid and ethyl alcohol. Some of the acid combines with some of the alcohol to form water and ethyl-acetic ester (see Esms), but the reaction is never complete, since a state of equilibrium is attained after a time, in which the inverse combination takes place just as fast as the direct one. The molecular weight of acetic acid (CHICOO.H) is 60, that of ethyl alcohol (C.H.OH) is 46, that of ethyl-acetic ester or ethyl acetate (CH.COO.GH.) is 88, and that of water (H.0) is 18. A mass of any substance which contains as many grams as there are units in the molecular weight of the substance is lcnown as a egram-molecule of the substance. This name is rather unhappily chosen, but the idea itself is a useful one, and is commonly employed in modern writings upon theoretical chemistry. Let us suppose that one gram-molecule of acetic acid (60 grams) is originally mixed with M gram-molecules of ethyl alcohol (46M grams), and with N gram molecules (18M grams) of water, and let us inquire what the composition of the mixture will be when the state of final chemical equi librium has been attained. The advantage of taldng the grain-molecule as a unit of mass is, that when this unit is used the ntunber of grams of acetic acid, alcohol and water that are originally present will be proportional to 1. M and N, and we may speak of M and N and write them in our equations precisely as though they were really the number of actual molecules present. The acetic acid and alcohol act upon each other as indicated by the equation CH*C00.H+CJI:OH = CH.COO.G1-10,+H.O. Now let us assume that when the state of equi librium has been attained, X gram-molecules of the alcohol have been tlecomposed. This implies that X gram-molecules of the acetic acid have also been decomposed, and that X gram-mole cules, each of water and of ethyl acetate have been formed. The total numbers of gram-mole cules of the various substances that are present when the final state of equilibrium is attained are therefore as follows: Acetic acid, 1—X ; alcohol, M—X; water, X; ethyl acetate, N+X. The number of molecular collisions per second, in which a molecule of acetic acid encounters a molecule of alcohol, is therefore (in the final state) proportional to (1—X) (M—X) ; and since the chemical action is itself proportional to the number of such collisions, we may assume that the number of gram-molecules of ethyl acetate formed per second, in the state of equilibrium, is A(1—X) (M—X), where A is a constant whose value we do not know. The same line of reasoning shows that the number of gram-molecules of ethyl acetate that are lost from the solution in the same time, through combining with water to reproduce acetic acid and alcohol, is B(N+X)X, where B is another constant, whose value is also unknown. Since
the existence of equilibrium requires that the quantity of ethyl acetate present shall be con stant, we have A (1—X) (M—X)=13(N+X)X. Now it is known by experiment that when the original mixture is free from water, and con tains chemically equivalent amounts of acetic acid and alcohol, so that M=.1 and N=0, the state of final equilibrium is attained when X=M. If these values of M and X are sub stituted in the foregoing equation, we find that A and B are connected by the necessary relation A=4B. If we replace A by 4B and then divide through by B the foregoing equation reduces to 4 (1—X) (M—X )=( N+X) X, or +N) X+4M=0, a quadratic equation from which the value of X (that is, the number of gram-molecules of acetic acid decomposed) may be inferred, in the final state of equilibrium, for any desired initial mixture of acetic acid, alco hol and water. This example has been given at some length, both because it illustrates clearly the principles of chemical equilibrium and the law of mass-action, and because reactions of this very kind, in which esters are formed by the direct action of an acid upon an alcohol, have a special historic interest, since their study has contributed in no small measure toward placing the modern theory of chemical equi librium upon a firm foundation.
When it is desired to determine the state of a chemical system after the lapse of a definite interval from an initial instant for which its state is given, we must form a differential equation in which the condition is expressed that the chemical change, per unit of time, is proportional (as above) to the product of the number of gram-molecules of the reacting sub stances that are present at the instant consid ered; and having formed this equation and inte grated it, we obtain an expression in which the composition of the system is expressed as a function of the time. When several substances that may react upon one another are present, the differential equation is more complicated in form, as might be expected; but for details of this sort reference must be made to works on theoretical chemistry. When the system con tains several acids and one or more bases, the distribution of the bases among the various acids may be investigated in accordance with similar principles, and by comparing the nu merical results that are thus obtained with the facts of observation, estimates of the true rela tive '
When, as is often the case, the course of a reaction depends upon the temperature, the principles of mass-action apply as before, but regard must also be had for the laws of ther modynamics (q.v.), which usually impose cer tain limitations upon the equations. The full theory of chemical changes in which thermody namical considerations play an important part was given by J. Willard Gibbs, in a paper of great power and originality, entitled 'On the Equilibrium of Heterogeneous Substances,' pub lished in the 'Transactions of the Connecticut Academy of Arts and Sciences' for 1875. Gibbs' basic phase law, or "phase rule"' as it is commonly called, is as follows: n different bodies (chemical substances, either simple or compound) can form n+2 phases, and these can co-exist at one single point only; that is, at a definite temperature and pressure.
The great importance of a full understand ing of the laws of chemical equilibrium rests in the fact that by far the larger part of all chemi cal processes, both in nature and in the indus trial arts, result not in complete reactions, but in a condition of chemical equilibrium, with meas urable quantities present of every possible compound.
Bibliography. —Arrhenius, S. C., 'Theories of Chemistry' (London 1907) ; Bigelow, S. L., 'Theoretical and Physical Chemistry' (New York 1912) ; Comey, A. M., of Chemical Solubilities' (London 1896) ; Findlay, A., 'The Phase Rule' (London 1911) ; Hoff, J. H. van't, 'Lectures on Theoretical and Physical Chemistry' (London 1899) ; Nernst, W., 'Theoretical Chemistry' (London 1895).