Law of mass action

In chemistry, the law of mass action is the proposition that the rate of the chemical reaction is directly proportional to the product of the activities or concentrations of the reactants. It explains and predicts behaviors of solutions in dynamic equilibrium. Specifically, it implies that for a chemical reaction mixture that is in equilibrium, the ratio between the concentration of reactants and products is constant.
Two aspects are involved in the initial formulation of the law: 1) the equilibrium aspect, concerning the composition of a reaction mixture at equilibrium and 2) the kinetic aspect concerning the rate equations for elementary reactions. Both aspects stem from the research performed by Cato M. Guldberg and Peter Waage between 1864 and 1879 in which equilibrium constants were derived by using kinetic data and the rate equation which they had proposed. Guldberg and Waage also recognized that chemical equilibrium is a dynamic process in which rates of reaction for the forward and backward reactions must be equal at chemical equilibrium. In order to derive the expression of the equilibrium constant appealing to kinetics, the expression of the rate equation must be used. The expression of the rate equations was rediscovered later independently by Jacobus Henricus van 't Hoff.
The law is a statement about equilibrium and gives an expression for the equilibrium constant, a quantity characterizing chemical equilibrium. In modern chemistry this is derived using equilibrium thermodynamics. It can also be derived with the concept of chemical potential.
History.
Two chemists generally expressed the composition of a mixture in terms of numerical values relating the amount of the product to describe the equilibrium state.
Cato Maximilian Guldberg and Peter Waage, building on Claude Louis Berthollet's ideas about reversible chemical reactions, proposed the law of mass action in 1864. These papers, in Danish, went largely unnoticed, as did the later publication (in French) of 1867 which contained a modified law and the experimental data on which that law was based.
In 1877 van 't Hoff independently came to similar conclusions, but was unaware of the earlier work, which prompted Guldberg and Waage to give a fuller and further developed account of their work, in German, in 1879. Van 't Hoff then accepted their priority.
1864.
The equilibrium state (composition).
In their first paper, Guldberg and Waage suggested that in a reaction such as 
the "chemical affinity" or "reaction force" between A and B did not just depend on the chemical nature of the reactants, as had previously been supposed, but also depended on the amount of each reactant in a reaction mixture. Thus the Law of Mass Action was first stated as follows:
In this context a substitution reaction was one such as <chem>{alcohol} + acid <=> {ester} + water</chem>. Active mass was defined in the 1879 paper as "the amount of substance in the sphere of action". For species in solution active mass is equal to concentration. For solids, active mass is taken as a constant. formula_2, a and b were regarded as empirical constants, to be determined by experiment.
At equilibrium, the chemical force driving the forward reaction must be equal to the chemical force driving the reverse reaction. Writing the initial active masses of A,B, A' and B' as p, q, p' and q' and the dissociated active mass at equilibrium as formula_3, this equality is represented by
formula_3 represents the amount of reagents A and B that has been converted into A' and B'. Calculations based on this equation are reported in the second paper.
Dynamic approach to the equilibrium state.
The third paper of 1864 was concerned with the kinetics of the same equilibrium system. Writing the dissociated active mass at some point in time as x, the rate of reaction was given as
Likewise the reverse reaction of A' with B' proceeded at a rate given by
The overall rate of conversion is the difference between these rates, so at equilibrium (when the composition stops changing) the two rates of reaction must be equal. Hence
1867.
The rate expressions given in Guldberg and Waage's 1864 paper could not be differentiated, so they were simplified as follows. The chemical force was assumed to be directly proportional to the product of the active masses of the reactants.
This is equivalent to setting the exponents a and b of the earlier theory to one. The proportionality constant was called an affinity constant, k. The equilibrium condition for an "ideal" reaction was thus given the simplified form
[A]eq, [B]eq etc. are the active masses at equilibrium. In terms of the initial amounts reagents p,q etc. this becomes
The ratio of the affinity coefficients, k'/k, can be recognized as an equilibrium constant. Turning to the kinetic aspect, it was suggested that the velocity of reaction, v, is proportional to the sum of chemical affinities (forces). In its simplest form this results in the expression
where formula_13 is the proportionality constant. Actually, Guldberg and Waage used a more complicated expression which allowed for interaction between A and A', etc. By making certain simplifying approximations to those more complicated expressions, the rate equation could be integrated and hence the equilibrium quantity formula_3 could be calculated. The extensive calculations in the 1867 paper gave support to the simplified concept, namely,
This is an alternative statement of the Law of Mass Action.
1879.
In the 1879 paper the assumption that reaction rate was proportional to the product of concentrations was justified microscopically in terms of the frequency of independent collisions, as had been developed for gas kinetics by Boltzmann in 1872 (Boltzmann equation). It was also proposed that the original theory of the equilibrium condition could be generalised to apply to any arbitrary chemical equilibrium.
The exponents α, β etc. are explicitly identified for the first time as the stoichiometric coefficients for the reaction.
Modern statement of the law.
The affinity constants, k+ and k−, of the 1879 paper can now be recognised as rate constants. The equilibrium constant, K, was derived by setting the rates of forward and backward reactions to be equal. This also meant that the chemical affinities for the forward and backward reactions are equal. The resultant expression
Diffusion in condensed matter.
Yakov Frenkel represented diffusion process in condensed matter as an ensemble of elementary jumps and quasichemical interactions of particles and defects. Henry Eyring applied his theory of absolute reaction rates to this quasichemical representation of diffusion. Mass action law for diffusion leads to various nonlinear versions of Fick's law.
In mathematical ecology.
The Lotka–Volterra equations describe dynamics of the predator-prey systems. The rate of predation upon the prey is assumed to be proportional to the rate at which the predators and the prey meet; this rate is evaluated as "xy", where "x" is the number of prey, "y" is the number of predator. This is a typical example of the law of mass action.
In mathematical epidemiology.
The law of mass action forms the basis of the compartmental model of disease spread in mathematical epidemiology, in which a population of humans, animals or other individuals is divided into categories of susceptible, infected, and recovered (immune). The SIR model is a useful abstraction of disease dynamics which applies well to many disease systems and provides useful outcomes in many circumstances when the Mass Action Principle applies. Individuals in human or animal populations unlike molecules in an ideal solution do not mix homogeneously. There are some disease examples in which this non-homogeneity is great enough such that the outputs of the SIR model are invalid. For these situations in which the assumptions of mass action do not apply, more sophisticated graph theory models may be useful. For more information, see Compartmental models in epidemiology.
In sociophysics.
Sociophysics uses tools and concepts from physics and physical chemistry to describe some aspects of social and political behavior. It attempts to explain why and how humans behave much like atoms, at least in some aspects of their collective lives. The law of mass action (generalized if it is necessary) is the main tool to produce the equation of interactions of humans in sociophysics.