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{{Other uses|Mass action (disambiguation)}}

In chemistry, the '''law of mass action''' is a mathematical model that explains and predicts behaviors of [[solutions]] in [[dynamic equilibrium]]. Two aspects are involved in the initial formulation of the law: 1) the equilibrium aspect, concerning the composition of a [[chemical reaction|reaction]] mixture at [[chemical equilibrium|equilibrium]] and 2) the [[chemical kinetics|kinetic]] aspect concerning the [[rate equation]]s for [[elementary reaction]]s. Both aspects stem from the research by [[Cato Maximilian Guldberg|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 [[reaction rate|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 has been 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]].

== History ==

[[Cato Maximilian Guldberg]] and [[Peter Waage]], building on [[Berthollet|Claude Louis Berthollet]]’s ideas<ref name="Levere" >{{cite book | last = Levere | first = Trevor, H. | title = Affinity and Matter – Elements of Chemical Philosophy 1800–1865 | publisher = Gordon and Breach Science Publishers | year = 1971 | isbn = 2-88124-583-8}}</ref> about [[chemical equilibrium|reversible chemical reactions]], proposed the '''Law of Mass Action''' in 1864.<ref name="GW1">C.M. Guldberg and P. Waage,"Studies Concerning Affinity" ''C. M. Forhandlinger: Videnskabs-Selskabet i Christiana'' (1864), 35</ref><ref name="GW2">P. Waage, "Experiments for Determining the Affinity Law" ,''Forhandlinger i Videnskabs-Selskabet i Christiania'', (1864) 92.</ref><ref name="GW3">C.M. Guldberg, "Concerning the Laws of Chemical Affinity", ''C. M. Forhandlinger i Videnskabs-Selskabet i Christiania'' (1864) 111</ref> These papers, in Norwegian, 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.<ref name="GW4">C.M. Guldberg and P. Waage, "Experiments concerning Chemical Affinity"; German translation by Abegg in ''Ostwald's Klassiker der Exacten Wissenschaften'', no. 104, Wilhelm Engleman, Leipzig, 1899, pp 10-125</ref>

In 1877 [[Jacobus Henricus van 't Hoff|van 't Hoff]] independently came to similar conclusions,<ref>J.H. van 't Hoff, ''Berichte der Berliner Chem. Ges''., (1877) '''10'''</ref> 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.<ref name="GW5">C.M. Guldberg and P. Waage, "Concerning Chemical Affinity" ''Erdmann's Journal für Practische Chemie'', (1879), '''127''', 69-114. Reprinted, with comments by Abegg in ''Ostwald's Klassiker der Exacten Wissenschaften'', no. 104, Wilhelm Engleman, Leipzig, 1899, pp 126-171</ref> Van 't Hoff then accepted their priority.

=== 1864 ===
==== The equilibrium state (composition)====
In their first paper,<ref name="GW1"/> Guldberg and Waage suggested that in a reaction such as
:A + B <math>\leftrightharpoons</math> A' + B'
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:

:When two reactants, A and B, react together at a given temperature in a "substitution reaction," the affinity, or chemical force between them, is proportional to the active masses, [A] and [B], each raised to a particular power
::<math>\mbox{affinity} = \alpha[A]^a[B]^b\!</math>.

In this context a substitution reaction was one such as alcohol + acid <math>\rightleftharpoons</math> ester + water. Active mass was defined in the 1879 paper as "the amount of substance in the sphere of action".<ref>It was stated in the 1879 paper that "the sphere of action can be represented by unit volume"</ref> 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 [[Chemical_equilibrium|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 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.<ref name="GW2"/>

==== Dynamic approach to the equilibrium state ====
The third paper of 1864<ref name="GW3"/> 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
:<math>\frac{dx}{dt}=k(p-x)^a(q-x)^b</math>
Likewise the reverse reaction of A' with B' proceeded at a rate given by
:<math>\frac{dx}{dt}=k'(p'+x)^{a'}(q'+x)^{b'}</math>
At equilibrium the two rates of reaction must be equal. This follows from the fact that the composition of a mixture at equilibrium does not change with time. Hence
:<math>(p-x)^{a}(q-x)^{b}=\frac{k'}{k} (p'+x)^{a'}(q'+x)^{b'}</math>...

=== 1867 ===
The rate expressions given in the 1864 paper could not be integrated, so they were simplified as follows.<ref name ="GW4"/> The chemical force was assumed to be directly proportional to the product of the active masses of the reactants.
:<math>\mbox{affinity} = \alpha[A][B]\!</math>
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

:<math>k[A]_\text{eq}[B]_\text{eq}=k'[A']_\text{eq}[B']_\text{eq}\!</math>

[A]eq, [B]eq etc. are the active masses at equilibrium. In terms of the initial amounts reagents p,q etc. this becomes
:<math>(p-\xi)(q-\xi)=\frac{k'}{k}(p'+\xi)(q'+\xi)</math>
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
:<math>v = \psi (k(p-x)(q-x)-k'(p'+x)(q'+x))\!</math>
where <math>\psi</math> 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 <math>\xi</math> 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<ref name="GW5"/> the assumption that reaction rate was proportional to the product of concentrations was justified microscopically in terms of collision theory, as had been developed for gas reactions. It was also proposed that the original theory of the equilibrium condition could be generalised to apply to any arbitrary chemical equilibrium.

: <math>\mbox{affinity}=k[A]^{\alpha}[B]^{\beta}\dots\!</math>

The exponents α, β etc. are explicitly identified for the first time as the [[stoichiometry|stoichiometric coefficients]] for the reaction. Since reaction rate was considered to be proportional to chemical affinity, it follows that for a general reaction of the type

:<math> \alpha A + \beta B \dots \rightleftharpoons \sigma S + \tau T \dots</math>
:<math>\mbox{forward reaction rate} = k_+ [A]^\alpha[B]^\beta \dots \,\!</math>
:<math>\mbox{backward reaction rate} = k_{-} [S]^\sigma[T]^\tau \dots \,\!</math>

where [A], [B], [S] and [T] are active masses and ''k''+ and ''k''− are affinity constants. Since at equilibrium the affinities and reaction rates for forward and backward reactions are equal, it follows that

:<math>K=\frac{k_+}{k_-}=\frac{[S]^\sigma [T]^\tau \dots } {[A]^\alpha [B]^\beta \dots}</math>

== Contemporary statement of the law==
The affinity constants, k+ and k-, of the 1879 paper can now be recognised as [[rate constant]]s. 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

:<math>K=\frac{[S]^\sigma [T]^\tau \dots } {[A]^\alpha [B]^\beta \dots}\,</math>

is correct <ref>[http://www.science.uwaterloo.ca/~cchieh/cact/c123/massacti.html Uwaterloo.ca], stating "The law of mass action is universal, applicable under any circumstance."</ref> even from the modern perspective, apart from the use of concentrations instead of activities (the concept of chemical activity was developed by [[Josiah Willard Gibbs]], in the 1870s, but was not [[Josiah Willard Gibbs#Scientific recognition|widely known]] in Europe until the 1890s). The derivation from the reaction rate expressions is no longer considered to be valid. Nevertheless, Guldberg and Waage were on the right track when they suggested that the driving force for both forward and backward reactions is equal when the mixture is at equilibrium. The term they used for this force was chemical affinity. Today the expression for the equilibrium constant is derived by setting the [[chemical potential]] of forward and backward reactions to be equal. The generalisation of the Law of Mass Action, in terms of affinity, to equilibria of arbitrary stoichiometry was a bold and correct conjecture.

The hypothesis that reaction rate is proportional to reactant concentrations is, strictly speaking, only true for [[elementary reaction]]s (reactions with a single mechanistic step), but the empirical rate expression

:<math>r_f=k_f[A][B]\,</math>

is also applicable to [[rate equation|second order]] reactions that may not be concerted reactions. Guldberg and Waage were fortunate in that reactions such as ester formation and hydrolysis, on which they originally based their theory, do indeed follow this rate expression.

In general many reactions occur with the formation of reactive intermediates, and/or through parallel reaction pathways. However, all reactions can be represented as a series of elementary reactions and, if the mechanism is known in detail, the rate equation for each individual step is given by the <math>r_f</math> expression so that the overall rate equation can be derived from the individual steps. When this is done the equilibrium constant is obtained correctly from the rate equations for forward and backward reaction rates.

In biochemistry, there has been significant interest in the appropriate mathematical model for chemical reactions occurring in the intracellular medium. This is in contrast to the initial work done on chemical kinetics, which was in simplified systems where reactants were in a relatively dilute, pH-buffered, aqueous solution. In more complex environments, where bound particles may be prevented from disassociation by their surroundings, or diffusion is slow or anomalous, the model of mass action does not always describe the behavior of the reaction kinetics accurately. Several attempts have been made to modify the mass action model, but consensus has yet to be reached. Popular modifications replace the rate constants with functions of time and concentration. As an alternative to these mathematical constructs, one school of thought is that the mass action model can be valid in intracellular environments under certain conditions, but with different rates than would be found in a dilute, simple environment {{Citation needed|date=July 2007}}.

The fact that Guldberg and Waage developed their concepts in steps from 1864 to 1867 and 1879 has resulted in much confusion in the literature as to which equation the Law of Mass Action refers. It has been a source of some textbook errors.<ref>"Textbook errors IX: More About the Laws of Reaction Rates and of Equilibrium", E.A. Guggenheim, ''J. Chem. Ed''., (1956) '''33''', 544-545</ref> Thus, today the "law of mass action" sometimes refers to the (correct) equilibrium constant formula,
<ref>[http://www.chem.purdue.edu/gchelp/gloss/lawmaction.html Law of Mass Action]</ref>
<ref>[http://www.chem.sc.edu/goode/C112Web/CH14NF/sld007.htm SC.edu]</ref>
<ref>[http://www.luc.edu/faculty/afitch/chap13/sld011.htm The Law of Mass Action]</ref>
<ref>[http://funnel.sfsu.edu:16080/courses/geol480/lectures/lecture4.pdf SFSU.edu]</ref>
<ref>[http://www.chemistry.wustl.edu/~courses/genchem/Tutorials/Buffers/fundamental.htm Recap of FundamentRecap of Fundamental Acid-Base Concepts]</ref>
<ref>[http://www.chm.davidson.edu/ChemistryApplets/equilibria/BasicConcepts.html Chemical Equilibria: Basic Concepts]</ref>
<ref>[http://www.science.uwaterloo.ca/~cchieh/cact/c123/massacti.html Chemical equilibrium - The Law of Mass Action]</ref>
<ref>[http://www.chem.indiana.edu/academics/ugrad/Courses/s117/documents/kr_110606.pdf Indiana.edu]</ref>
<ref>[http://www.nuc.berkeley.edu/courses/classes/E-115/Slides/Sect_10_thermo_chemical_reactions.ppt Berkeley.edu]</ref>
<ref>[http://antoine.frostburg.edu/chem/senese/101/acidbase/faq/equivalence-point-HF-NaOH.shtml General Chemistry Online: FAQ: Acids and bases: What is the pH at the equivalence point an HF/NaOH titration?]</ref>
and at other times to the (usually incorrect) <math>r_f</math> rate formula.
<ref>[http://www.biochem.northwestern.edu/holmgren/Glossary/Definitions/Def-L/law_of_mass_action.html law of mass action definition]</ref>
<ref>[http://www.caam.rice.edu/~bpeercy/math468/lab4.html Lab 4 - Slow Manifolds]</ref>

== Applications to other fields ==

===In semiconductor physics===

The law of mass action [[Mass action law (electronics)|is also applied in semiconductor physics]] to describe the carrier density in a semiconductor. Regardless of doping, the product of electron and hole densities is a material property dependent on temperature.

===Diffusion in condensed matter===

[[Yakov Frenkel]] represented [[diffusion]] process in [[condensed matter]] as an [[Diffusion#Jumps on the surface and in solids|ensemble of elementary jumps]] and quasichemical interactions of particles and defects. [[Henry Eyring]] applied his theory of [[Transition state theory|absolute reaction rates]] to this quasichemical representation of diffusion. Mass action law for diffusion leads to various nonlinear versions of [[Fick's law]].<ref name=GorbanMMNP2011>A.N. Gorban, H.P. Sargsyan and H.A. Wahab (2011), [http://arxiv.org/pdf/1012.2908v4.pdf Quasichemical Models of Multicomponent Nonlinear Diffusion], [http://journals.cambridge.org/action/displayJournal?jid=MNP Mathematical Modelling of Natural Phenomena], Volume 6 / Issue 05, 184−262.</ref>

===In mathematical ecology===

The [[Lotka–Volterra equation]]s 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 (typically human) population is divided into categories of susceptible, infected, and recovered. However, the so-called [[SIR model]] is being replaced by more sophisticated models using graph theory, since individuals in a human population - unlike molecules in an ideal solution - do not mix homogeneously. For more information, see [[Compartmental models in epidemiology]].

===In sociophysics===

Sociophysics<ref>S. Galam, Sociophysics. A Physicist's Modeling of Psycho-political Phenomena, Springer, 2012, DOI: 10.1007/978-1-4614-2032-3, ISBN 978-1-4614-2032-3</ref> 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.

== See also ==
* [[Chemical equilibrium]]
* [[Chemical potential]]
* [[Equilibrium constant]]
* [[Reaction quotient]]

== References ==
{{reflist|colwidth=35em}}

==Further reading==
*[http://chimie.scola.ac-paris.fr/sitedechimie/hist_chi/text_origin/guldberg_waage/Concerning-Affinity.htm Studies Concerning Affinity]. P. Waage and C.M. Guldberg; Henry I. Abrash, Translator.
*"Guldberg and Waage and the Law of Mass Action", E.W. Lund, ''J. Chem. Ed''., (1965), '''42''', 548-550.
* [http://www.graphpad.com/curvefit/law_of_mass_action.htm A simple explanation of the mass action law]. H. Motulsky.

{{DEFAULTSORT:Law Of Mass Action}}
[[Category:History of chemistry]]
[[Category:Equilibrium chemistry]]
[[Category:Chemical kinetics]]

Game complexity - Revision history

197.155.4.118: /* Example: tic-tac-toe */ Added English name with link to article

en>Monkbot: /* Complexities of some well-known games */Fix CS1 deprecated date parameter errors (test)

en>Monkbot: /* Complexities of some well-known games */Fix CS1 deprecated date parameter errors