Mass action
Encyclopedia
In Chemistry, the law of mass action is a mathematical model that explains and predicts behaviors of solutions in dynamic equilibrium
. It can be described with two aspects: 1) the equilibrium aspect, concerning the composition of a reaction
mixture at equilibrium
and 2) the kinetic
aspect concerning the rate equation
s for elementary reaction
s. Both aspects stem from the research by Guldberg and Waage (1864–1879) 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.
Taken as a statement about kinetics, the law states that the rate of an elementary reaction (a reaction that proceeds through only one transition state, that is one mechanistic step) is proportional to the product of the concentrations of the participating molecules. In modern chemistry this is derived using statistical mechanics
.
Taken as a statement about equilibrium, that law gives an expression for the equilibrium constant, a quantity characterizing chemical equilibrium
. In modern chemistry this is derived using equilibrium thermodynamics
.
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 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.
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.
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 alcohol + acid
ester + water. 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. 
, 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
, this equality is represented by
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.
Likewise the reverse reaction of A' with B' proceeded at a rate given by
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
...
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
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 
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.
The exponents α, β etc. are explicitly identified for the first time as the 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
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
is correct 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 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
is also applicable to 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
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 .
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. Thus, today the "law of mass action" sometimes refers to the (correct) equilibrium constant formula,
and at other times to the (usually incorrect)
rate formula.
The law of mass action 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.
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
.
Dynamic equilibrium
A dynamic equilibrium exists once a reversible reaction ceases to change its ratio of reactants/products, but substances move between the chemicals at an equal rate, meaning there is no net change. It is a particular example of a system in a steady state...
. It can be described with two aspects: 1) the equilibrium aspect, concerning the composition of a reaction
Chemical reaction
A chemical reaction is a process that leads to the transformation of one set of chemical substances to another. Chemical reactions can be either spontaneous, requiring no input of energy, or non-spontaneous, typically following the input of some type of energy, such as heat, light or electricity...
mixture at equilibrium
Chemical equilibrium
In a chemical reaction, chemical equilibrium is the state in which the concentrations of the reactants and products have not yet changed with time. It occurs only in reversible reactions, and not in irreversible reactions. Usually, this state results when the forward reaction proceeds at the same...
and 2) the kinetic
Chemical kinetics
Chemical kinetics, also known as reaction kinetics, is the study of rates of chemical processes. Chemical kinetics includes investigations of how different experimental conditions can influence the speed of a chemical reaction and yield information about the reaction's mechanism and transition...
aspect concerning the rate equation
Rate equation
The rate law or rate equation for a chemical reaction is an equation that links the reaction rate with concentrations or pressures of reactants and constant parameters . To determine the rate equation for a particular system one combines the reaction rate with a mass balance for the system...
s for elementary reaction
Elementary reaction
An elementary reaction is a chemical reaction in which one or more of the chemical species react directly to form products in a single reaction step and with a single transition state....
s. Both aspects stem from the research by Guldberg and Waage (1864–1879) 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
Reaction rate
The reaction rate or speed of reaction for a reactant or product in a particular reaction is intuitively defined as how fast or slow a reaction takes place...
for the forward and backward reactions must be equal.
Taken as a statement about kinetics, the law states that the rate of an elementary reaction (a reaction that proceeds through only one transition state, that is one mechanistic step) is proportional to the product of the concentrations of the participating molecules. In modern chemistry this is derived using statistical mechanics
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
.
Taken as a statement about equilibrium, that law gives an expression for the equilibrium constant, a quantity characterizing chemical equilibrium
Chemical equilibrium
In a chemical reaction, chemical equilibrium is the state in which the concentrations of the reactants and products have not yet changed with time. It occurs only in reversible reactions, and not in irreversible reactions. Usually, this state results when the forward reaction proceeds at the same...
. In modern chemistry this is derived using equilibrium thermodynamics
Equilibrium thermodynamics
Equilibrium Thermodynamics is the systematic study of transformations of matter and energy in systems as they approach equilibrium. The word equilibrium implies a state of balance. Equilibrium thermodynamics, in origins, derives from analysis of the Carnot cycle. Here, typically a system, as...
.
History
Cato Maximilian GuldbergCato Maximilian Guldberg
Cato Maximilian Guldberg was a Norwegian mathematician and chemist.-Career:Guldberg worked at the Royal Frederick University. Together with his brother-in-law, Peter Waage, he proposed the law of mass action...
and Peter Waage
Peter Waage
Peter Waage , the son of a ship's captain, was a significant Norwegian chemist and professor at the Royal Frederick University. Along with his brother-in-law Cato Maximilian Guldberg, he co-discovered and developed the law of mass action between 1864 and 1879.He grew up in Hidra...
, building on Claude Louis Berthollet’s ideas about reversible chemical reactions
Chemical equilibrium
In a chemical reaction, chemical equilibrium is the state in which the concentrations of the reactants and products have not yet changed with time. It occurs only in reversible reactions, and not in irreversible reactions. Usually, this state results when the forward reaction proceeds at the same...
, proposed the Law of Mass Action in 1864. 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.
In 1877 van 't Hoff
Jacobus Henricus van 't Hoff
Jacobus Henricus van 't Hoff, Jr. was a Dutch physical and organic chemist and the first winner of the Nobel Prize in chemistry. He is best known for his discoveries in chemical kinetics, chemical equilibrium, osmotic pressure, and stereochemistry...
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.
The equilibrium state (composition)
In their first paper, Guldberg and Waage suggested that in a reaction such as- A + B
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
.
In this context a substitution reaction was one such as alcohol + acid
ester + water. 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. , 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
, this equality is represented by 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 asLikewise the reverse reaction of A' with B' proceeded at a rate given by
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
...1867
The rate expressions given in the 1864 paper could not be integrated, 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
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 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 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.
The exponents α, β etc. are explicitly identified for the first time as the stoichiometric coefficients
Stoichiometry
Stoichiometry is a branch of chemistry that deals with the relative quantities of reactants and products in chemical reactions. In a balanced chemical reaction, the relations among quantities of reactants and products typically form a ratio of whole numbers...
for the reaction. Since reaction rate was considered to be proportional to chemical affinity, it follows that for a general reaction of the type
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
Contemporary view
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 expressionis correct 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
Josiah Willard Gibbs
Josiah Willard Gibbs was an American theoretical physicist, chemist, and mathematician. He devised much of the theoretical foundation for chemical thermodynamics as well as physical chemistry. As a mathematician, he invented vector analysis . Yale University awarded Gibbs the first American Ph.D...
, in the 1870s, but was not 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
Chemical potential
Chemical potential, symbolized by μ, is a measure first described by the American engineer, chemist and mathematical physicist Josiah Willard Gibbs. It is the potential that a substance has to produce in order to alter a system...
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
Elementary reaction
An elementary reaction is a chemical reaction in which one or more of the chemical species react directly to form products in a single reaction step and with a single transition state....
s (reactions with a single mechanistic step), but the empirical rate expression
is also applicable to second order
Rate equation
The rate law or rate equation for a chemical reaction is an equation that links the reaction rate with concentrations or pressures of reactants and constant parameters . To determine the rate equation for a particular system one combines the reaction rate with a mass balance for the system...
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
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 .
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. Thus, today the "law of mass action" sometimes refers to the (correct) equilibrium constant formula,
and at other times to the (usually incorrect)
rate formula.Applications to Other Fields
In semiconductor physicsThe law of mass action 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.
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
Compartmental models in epidemiology
In order to model the progress of an epidemic in a large population, comprising many different individuals in various fields, the population diversity must be reduced to a few key characteristics which are relevant to the infection under consideration...
.
Further reading
- 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.
- A simple explanation of the mass action law. H. Motulsky.