Law of mass action

In chemistry, the law of mass action is the proposition that the rate of a 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. The law was formulated by Cato Maximilian Guldberg and Peter Waage in 1864, who derived equilibrium constants using kinetic data and the rate equation they proposed. It was later clarified independently by the Dutch chemist Jacobus Henricus van 't Hoff in 1877.

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, For species in solution active mass is equal to concentration. For solids, active mass is taken as a constant. <math>\alpha</math>, 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 <math>\xi</math>, this equality is represented by

:<math>\alpha(p-\xi)^a(q-\xi)^b=\alpha'(p'+\xi)^{a'}(q'+\xi)^{b'}\!</math>

<math>\xi</math> represents the extent of reaction; 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. Thus, today the "law of mass action" sometimes refers to the (correct) equilibrium constant formula,

and at other times to the (usually incorrect) <math>r_f</math> rate formula.

Applications to other fields

In plasma physics

In a plasma, the ionization of the atoms can be understood as a chemical equilibrium between each ionization state with the next ionization state and a freed electron:

:<chem>A{} <=> A+ + e-</chem>

:<chem>A+ <=> A^2+ + e-</chem>

:<chem>A^2+ <=> A^3+ + e-</chem>

:etc.

and accordingly a law of mass action arises for each reaction, which in the ideally dilute limit is the Saha ionization equation.

In semiconductor physics

The law of mass action also has implications in semiconductor physics. Regardless of doping, the product of electron and hole densities is a constant at equilibrium. This constant depends on the thermal energy of the system (i.e. the product of the Boltzmann constant, <math>k_\text{B}</math>, and temperature, <math>T</math>), as well as the band gap (the energy separation between conduction and valence bands, <math>E_g \equiv E_C-E_V</math>) and effective density of states in the valence <math>(N_V(T))</math> and conduction <math>(N_C(T))</math> bands. When the equilibrium electron <math>(n_o)</math> and hole <math>(p_o)</math> densities are equal, their density is called the intrinsic carrier density <math>(n_i)</math> as this would be the value of <math>n_o</math> and <math>p_o</math> in a perfect crystal. Note that the final product is independent of the Fermi level <math>(E_F)</math>:

: <math>n_o p_o = \left(N_C e^{-\frac{E_C-E_F}{k_\text{B} T\right)\left(N_V e^{-\frac{E_F-E_V}{k_\text{B} T\right)=N_C N_V e^{-\frac{E_g}{k_\text{B} T = n_i^2</math>

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 principle of mass action is at the heart of the transmission term of compartmental models in epidemiology, which provide a useful abstraction of disease dynamics. The law of mass action formulation of the SIR model corresponds to the following "quasichemical" system of elementary reactions:

: The list of components is S (susceptible individuals), I (infected individuals), and R (removed individuals, or just recovered ones if we neglect lethality);

: The list of elementary reactions is

:: <chem>S + I -> 2I</chem>

:: <chem>I -> R</chem>.

: If the immunity is unstable then the transition from R to S should be added that closes the cycle (SIRS model):

:: <chem>R -> S</chem>.

A rich system of law of mass action models was developed in mathematical epidemiology by adding components and elementary reactions.

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 classical SIR model and their simple generalizations like SIS or SEIR, are invalid. For these situations, more sophisticated compartmental models or distributed reaction-diffusion models may be useful.