the action of these forces? This question is of fundamental importance, since it leads directly to those laws which regulate the chemical process. Besides the already mentioned fundamental law of chemical combination, that of constant and multiple proportions, there is the law of chemical mass-action, discovered by Guldberg and Waage in 1867, which we will now develop from a kinetic standpoint.

*Kinetic Basis of the Law of Chemical Mass-action.*—We will assume that the molecular species A_{1}, A_{2}, ... A′_{1}, A′_{2}, ... are present in a homogeneous system, where they can react on each other only according to the scheme

A_{1} + A_{2} + ... ↔ A′_{1} + A′_{2} + ...;

this is a special case of the general equation

*n*_{1}A_{1} + *n*_{2}A_{2} + ... ↔ *n*′_{1}A′_{1} + *n*′_{2}A′_{2} + ...,

in which only one molecule of each substance takes part in the reaction. The reacting substances may be either gaseous or form a liquid mixture, or be dissolved in some selected solvent; but in each case we may state the following considerations regarding the course of the reaction. For a transformation to take place from left to right in the sense of the reaction equation, all the molecules A_{1}, A_{2}, ... must clearly collide at one point; otherwise no reaction is possible, since we shall not consider side-reactions. Such a collision need not of course bring about that transposition of the atoms of the single molecules which constitutes the above reaction. Much rather must it be of such a kind as is favourable to that loosening of the bonds that bind the atoms in the separate molecules, which must precede this transposition. Of a large number of such collisions, therefore, only a certain smaller number will involve a transposition from left to right in the sense of the equation. But this number will be the same under the same external conditions, and the greater the more numerous the collisions; in fact a direct ratio must exist between the two. Bearing in mind now, that the number of collisions must be proportional to each of the concentrations of the bodies A_{1}, A_{2}, ..., and therefore, on the whole, to the product of all these concentrations, we arrive at the conclusion that the velocity *v* of the transposition from left to right in the sense of the reaction equation is *v* = *kc*_{1}*c*_{2} ..., in which *c*_{1}, *c*_{2}, ... represent the spatial concentrations, *i.e.* the number of gram-molecules of the substances A_{1}, A_{2}, ... present in one litre, and k is, at a given temperature, a constant which may be called the velocity-coefficient.

Exactly the same consideration applies to the molecules A′_{1}, A′_{2}.... Here the velocity of the change from right to left in the sense of the reaction-equation increases with the number of collisions of all these molecules at one point, and this is proportional to the product of all the concentrations. If *k*′ denotes the corresponding proportionality-factor, then the velocity *v*′ of the change from right to left in the sense of the reaction-equation is *v*′ = *k*′*c*′_{1}*c*′_{2}.... These spatial concentrations are often called the “active masses” of the reacting components. Hence the reaction-velocity in the sense of the reaction-equation from left to right, or the reverse, is proportional to the product of the “active-masses” of the left-hand or right-hand components respectively.

Neither *v* nor *v*′ can be separately investigated, and the measurements of the course of a reaction always furnish only the difference of these two quantities. The reaction-velocity Law of chemical statics.actually observed represents the difference of these two partial reaction-velocities, whilst the amount of change observed during any period of time is equal to the change in the one direction, minus the change in the opposite direction. It must not be assumed, however, that on the attainment of equilibrium all action has ceased, but rather that the velocity of change in one direction has become equal to that in the opposite direction, with the result that no further total change can be observed, *i.e.* the system has reached equilibrium, for which the relation *v* - *v*′ = 0 must therefore hold, or what is the same thing

*kc*_{1}*c*_{2} ... = *k*′*c*′_{1}*c*′_{2} ...,

this is the fundamental law of chemical statics.

The conception that the equilibrium is not to be attributed to absolute indifference between the reacting bodies, but that these continue to exert their mutual actions undiminished and the opposing changes now balance, is of fundamental significance in the interpretation of changes of matter in general. This is generally expressed in the form: *the equilibrium in this and other analogous cases is not static but dynamic.* This conception was a direct result of the kinetic-molecular considerations, and was applied with special success to the development of the kinetic theory of gases. Thus with Clausius, we conceive the equilibrium of water-vapour with water, not as if neither water vaporized nor vapour condensed, but rather as though the two processes went on unhindered in the equilibrium state, *i.e.* during contact of saturated vapour with water, in a given time, as many water molecules passed through the water surface in one direction as in the opposite direction. This view, as applied to chemical changes, was first advanced by A. W. Williamson (1851), and further developed by C. M. Guldberg and P. Waage and others.

From the previous considerations it follows that the reaction-velocity at every moment, *i.e.* the velocity with which the chemical process advances towards the
Law of chemical kinetics.
equilibrium state, is given by the equation

V = *v* - *v*′ = *kc*_{1}*c*_{2} ... - *k*′*c*′_{1}*c*′_{2} ...;

this states the fundamental law of chemical kinetics.

The equilibrium equation is simply a special case of this more general one, and results when the total velocity is written zero, just as in analytical mechanics the equilibrium conditions follow at once by specialization of the general equations of motion.

No difficulty presents itself in the generalization of the previous equations for the reaction which proceeds after the scheme

*n*_{1}A_{1} + *n*_{2}A_{2} + ... = *n*′_{1}A′_{1} + *n*′_{2}A′_{2} + ...,

where *n*_{1}, *n*_{2}, ..., *n*′_{1}, *n*′_{2}, ... denote the numbers of molecules of the separate substances which take part in the reaction, and are therefore whole, mostly small, numbers (generally one or two, seldom three or more). Here as before, *v* and *v*′ are to be regarded as proportional to the number of collisions at one point of all molecules necessary to the respective reaction, but now *n*_{1} molecules of A_{1}, *n*_{2} molecules of A_{2}, &c., must collide for the reaction to advance from left to right in the sense of the equation; and similarly *n*′_{1} molecules of A′_{1}, *n*′_{2} molecules of A′_{2}, &c., must collide for the reaction to proceed in the opposite direction. If we consider the path of a single, arbitrarily chosen molecule over a certain time, then the number of its collisions with other similar molecules will be proportional to the concentration C of that kind of molecule to which it belongs. The number of encounters between two molecules of the kind in question, during the same time, will be in general C times as many, *i.e.* the number of encounters of two of the same molecules is proportional to the square of the concentration C; and generally, the number of encounters of n molecules of one kind must be regarded as proportional to the nth power of C, *i.e.* C^{n}.

The number of collisions of *n*_{1} molecules of A_{1}, *n*_{2} molecules of A_{2} ... is accordingly proportional to C_{1}^{n1}C_{2}^{n2} ..., and the reaction-velocity corresponding to it is therefore

*v* = *k*C_{1}^{n1}C_{2}^{n2} ...,

and similarly the opposed reaction-velocity is

*v*′ = *k*′C′_{1}^{n′1}C′_{2}^{n′2} ...;

the resultant reaction-velocity, being the difference of these two partial velocities, is therefore

V = *v* - *v*′ = *k*C_{1}^{n1}C_{2}^{n2} ... - *k*′C′_{1}^{n′1}C′_{2}^{n′2} ...

This is the most general expression of the law of chemical mass-action, for the case of homogeneous systems.

Equating V to zero, we obtain the equation for the equilibrium state, viz.

C_{1}^{n1}C_{2}^{n2} ... / C′_{1}^{n′1}C′_{2}^{n′2} ... = *k* / *k*′ = K;

K is called the “equilibrium-constant.”