Law of mass action

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. \alpha, 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 \xi, this equality is represented by :\alpha(p-\xi)^a(q-\xi)^b=\alpha'(p'+\xi)^{a'}(q'+\xi)^{b'}\! \xi 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. 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 : \left(\frac{dx}{dt}\right)_\text{forward}=k(p-x)^a(q-x)^b Likewise the reverse reaction of A' with B' proceeded at a rate given by : \left(\frac{dx}{dt}\right)_\text{reverse}=k'(p'+x)^{a'}(q'+x)^{b'} 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 : (p-x)^{a}(q-x)^{b}=\frac{k'}{k} (p'+x)^{a'}(q'+x)^{b'}... 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. : \mbox{affinity} = \alpha[A][B]\! 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 : k[A]_\text{eq}[B]_\text{eq}=k'[A']_\text{eq}[B']_\text{eq} [A]eq, [B]eq etc. are the active masses at equilibrium. In terms of the initial amounts reagents p,q etc. this becomes : (p-\xi)(q-\xi)=\frac{k'}{k}(p'+\xi)(q'+\xi) 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 : v = \psi (k(p-x)(q-x)-k'(p'+x)(q'+x))\! where \psi 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 \xi could be calculated. The extensive calculations in the 1867 paper gave support to the simplified concept, namely, :The rate of a reaction is proportional to the product of the active masses of the reagents involved. 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. : \text{affinity}=k[\ce A]^{\alpha}[\ce B]^{\beta}\dots 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 : K=\frac{{\left [ A' \right ]}^{\alpha'}{\left [ B' \right ]}^{\beta'} \dots } {[A]^\alpha [B]^\beta \dots} is correct Thus, today the "law of mass action" sometimes refers to the (correct) equilibrium constant formula, and at other times to the (usually incorrect) r_f 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: :A{} A+ + e- :A+ A^2+ + e- :A^2+ A^3+ + e- :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, k_\text{B}, and temperature, T), as well as the band gap (the energy separation between conduction and valence bands, E_g \equiv E_C-E_V) and effective density of states in the valence (N_V(T)) and conduction (N_C(T)) bands. When the equilibrium electron (n_o) and hole (p_o) densities are equal, their density is called the intrinsic carrier density (n_i) as this would be the value of n_o and p_o in a perfect crystal. Note that the final product is independent of the Fermi level (E_F): : 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 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 :: S + I -> 2I :: I -> R. : If the immunity is unstable then the transition from R to S should be added that closes the cycle (SIRS model): :: R -> S. 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. == See also ==