Condition in the course of a reversible chemical reaction in which no net change in the amounts of reactants and products occurs: Products are reverting to reactants at the same rate as reactants are forming products. For practical purposes, the reaction under those conditions is completed. Expressed in terms of the law of mass action, the reaction rate to form products is equal to the reaction rate to re-form reactants. The ratio of the reaction rate constants (i.e., of the amounts of reactants and products, each raised to the proper power), defines the equilibrium constant. Changing the conditions of temperature or pressure changes the reaction's equilibrium; a high temperature or pressure may be used to “push” a reaction that at ordinary conditions makes little product. See also H.-L. Le Châtelier.
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In a chemical reaction, when reactants are mixed together in a reaction vessel (and heated if needed), the whole of reactants do not get converted into the products. After some time (which may be shorter than millionths of a second or longer than the age of the universe), there will come a point when a fixed amount of reactants will exist in harmony with a fixed amount of products, the amounts of neither changing anymore. This is called chemical equilibrium.
to be at equilibrium the rates of the forward and backward (reverse) reactions have to be equal. In this chemical equation with harpoon arrows pointing both ways to indicate equilibrium, A and B are reactant chemical species, S and T are product species, and α, β, σ, and τ are the stoichiometric coefficients of the respective reactants and products. The equilibrium position of a reaction is said to lie far to the right if, at equilibrium, nearly all the reactants are used up and far to the left if hardly any product is formed from the reactants.
and the ratio of the rate constants is also a constant, now known as an equilibrium constant.
By convention the products form the numerator. However, the law of mass action is valid only for concerted one-step reactions that proceed through a single transition state and is not valid in general because rate equations do not, in general, follow the stoichiometry of the reaction as Guldberg and Waage had proposed (see, for example, nucleophilic aliphatic substitution by SN1 or reaction of hydrogen and bromine to form hydrogen bromide). Equality of forward and backward reaction rates, however, is a necessary condition for chemical equilibrium, though it is not sufficient to explain why equilibrium occurs.
Despite the failure of this derivation, the equilibrium constant for a reaction is indeed a constant, independent of the activities of the various species involved, though it does depend on temperature as observed by the van 't Hoff equation. Adding a catalyst will affect both the forward reaction and the reverse reaction in the same way and will not have an effect on the equilibrium constant. The catalyst will speed up both reactions thereby increasing the speed at which equilibrium is reached.
Although the macroscopic equilibrium concentrations are constant in time reactions do occur at the molecular level. For example, in the case of ethanoic acid dissolved in water and forming ethanoate and hydronium ions,
Le Chatelier's principle (1884) is a useful principle that gives a qualitative idea of an equilibrium system's response to changes in reaction conditions. If a dynamic equilibrium is disturbed by changing the conditions, the position of equilibrium moves to counteract the change. For example, adding more S from the outside will cause an excess of products, and the system will try to counteract this by increasing the reverse reaction and pushing the equilibrium point backward (though the equilibrium constant will stay the same).
If mineral acid is added to the ethanoic acid mixture, increasing the concentration of hydronium ion, the amount of dissociation must decrease as the reaction is driven to the left in accordance with this principle. This can also be deduced from the equilibrium constant expression for the reaction:
A quantitative version is given by the reaction quotient.
J.W. Gibbs suggested in 1873 that equilibrium is attained when the Gibbs energy of the system is at its minimum value (assuming the reaction is carried out under constant pressure). What this means is that the derivative of the Gibbs energy with respect to reaction coordinate (a measure of the extent of reaction that has occurred, ranging from zero for all reactants to a maximum for all products) vanishes, signalling a stationary point. This derivative is usually called, for certain technical reasons, the Gibbs energy change. This criterion is both necessary and sufficient. If a mixture is not at equilibrium, the liberation of the excess Gibbs energy (or Helmholtz energy at constant volume reactions) is the “driving force” for the composition of the mixture to change until equilibrium is reached. The equilibrium constant can be related to the standard Gibbs energy change for the reaction by the equation
The relationship between the Gibbs energy and the equilibrium constant can be found by considering chemical potentials. The thermodynamic condition for chemical equilibrium is
In this article only the constant pressure case is considered. The constant volume case is important in geochemistry and atmospheric chemistry where pressure variations are significant. Note that, if reactants and products were in standard state (completely pure), then there would be no reversibility and no equilibrium. The mixing of the products and reactants contributes a large entropy (known as entropy of mixing) to states containing equal mixture of products and reactants. The combination of the standard Gibbs energy change and the Gibbs energy of mixing determines the equilibrium state.
In general an equilibrium system is defined by writing an equilibrium equation for the reaction
In order to meet the thermodynamic condition for equilibrium, the Gibbs energy must be stationary, meaning that the derivative of G with respect to reaction coordinate (ΔG) must be zero. It can be shown that ΔG is, in fact, equal to the difference between the chemical potentials of the products and those of the reactants. Therefore, the sum of the Gibbs energies of the reactants must be the equal to the sum of the Gibbs energies of the products.
Substituting expressions like this into the Gibbs energy equation:
which at constant pressure and temperature becomes:
By substituting the chemical potentials:
the relationship becomes:
At equilibrium and therefore
O is the standard molar Gibbs energy change for the reaction and K is the equilibrium constant. Note that activities and equilibrium constants are dimensionless numbers.
The expression for the equilibrium constant can be re-written as the product of a concentration quotient, Kc and an activity coefficient quotient, Γ.
For reactions in the gas phase partial pressure is used in place of concentration and fugacity coefficient in place of activity coefficient. In the real world, for example, when making ammonia in industry, fugacity coefficients must be taken into account. Fugacity, f, is the product of partial pressure and fugacity coefficient. The chemical potential of a species in the gas phase is given by
In aqueous solution, equilibrium constants are usually determined in the presence of an "inert" electrolyte such as sodium nitrate NaNO3 or Potassium perchlorate KClO4. The ionic strength, I, of a solution containing a dissolved salt, X+Y-, is given by
To use a published value of an equilibrium constant in conditions of ionic strength different from the conditions used in its determination, the value should be adjustedSoftware (below).
A mixture may be appear to have no tendency to change, though it is not at equilibrium. For example, a mixture of SO2 and O2 is metastable as there is a kinetic barrier to formation of the product, SO3.
Applying the general formula for an equilibrium constant to the specific case of ethanoic acid one obtains
a constant factor is incorporated into the equilibrium constant.
A particular case is the self-ionization of water itself
The self-ionization constant of water is defined as
It is perfectly legitimate to write [H+] for the hydronium ion concentration, since the state of solvation of the proton is constant (in dilute solutions) and so does not affect the equilibrium concentrations. Kw varies with variation in ionic strength and/or temperature.
The concentrations of H+ and OH- are not independent quantities. Most commonly [OH-] is replaced by Kw[H+]-1 in equilibrium constant expressions which would otherwise hydroxide.
Solids also do not appear in the equilibrium equation. An example is the Boudouard reaction :
for which the equation (without solid carbon) is written as:
Consider the case of a dibasic acid H2A. When dissolved in water, the mixture will contain H2A, HA- and A2-. This equilibrium can be split into two steps in each of which one proton is liberated.
The effect of changing temperature on an equilibrium constant is given by the van 't Hoff equation
In these applications, terms such as stability constant, formation constant, binding constant, affinity constant, association/dissociation constant are used. In biochemistry, it is common to give units for binding constants, which serve to define the concentration units used when the constant’s value was determined.
When the only equilibrium is that of the formation of a 1:1 adduct as the composition of a mixture, there are any number of ways that the composition of a mixture can be calculated. For example, see ICE table for a traditional method of calculating the pH of a solution of a weak acid.
There are three approaches to the general calculation of the composition of a mixture at equilibrium.
In general, the calculations are rather complicated. For instance, in the case of a dibasic acid, H2A dissolved in water the two reactants can be specified as the conjugate base, A2-, and the proton, H+. The following equations of mass-balance could apply equally well to a base such as 1,2-diaminoethane, in which case the base itself is designated as the reactant A:
With TA the total concentration of species A. Note that it is customary to omit the ionic charges when writing and using these equations.
When the equilibrium constants are known and the total concentrations are specified there are two equations in two unknown "free concentrations" [A] and [H]. This follows from the fact that [HA]= β1[A][H], [H2A]= β2[A][H]2 and [OH] = Kw[H]-1
so the concentrations of the "complexes" are calculated from the free concentrations and the equilibrium constants. General expressions applicable to all systems with two reagents, A and B would be
This diagram, for the hydrolysis of the aluminium Lewis acid Al3+aq shows the species concentrations for a 5×10-6M solution of an aluminium salt as a function of pH. Each concentration is shown as a percentage of the total aluminium.
Another common instance where precipitation occurs is when a metal cation interacts with an anionic ligand to form an electrically-neutral complex. If the complex is hydrophopbic, it will precipitate out of water. This occurs with the nickel ion Ni2+ and dimethylglyoxime, (dmgH2): in this case the lattice energy of the solid is not particularly large, but it greatly exceeds the energy of solvation of the molecule Ni(dmgH)2.
At equilibrium, G is at a minimum:
For a closed system, no particles may enter or leave, although they may combine in various ways. The total number of atoms of each element will remain constant. This means that the minimization above must be subjected to the constraints:
where is the number of atoms of element i in molecule j and bi0 is the total number of atoms of element i, which is a constant, since the system is closed. If there are a total of k types of atoms in the system, then there will be k such equations.
This is a standard problem in optimisation, known as constrained minimisation. The most common method of solving it is using the method of Lagrange multipliers, also known as undetermined multipliers (though other methods may be used).
where the are the Lagrange multipliers, one for each element. This allows each of the to be treated independently, and it can be shown using the tools of multivariate calculus that the equilibrium condition is given by
(For proof see Lagrange multipliers)
This is a set of (m+k) equations in (m+k) unknowns (the and the ) and may, therefore, be solved for the equilibrium concentrations as long as the chemical potentials are known as functions of the concentrations at the given temperature and pressure. (See Thermodynamic databases for pure substances).
This method of calculating equilibrium chemical concentrations is useful for systems with a large number of different molecules. The use of k atomic element conservation equations for the mass constraint is straightforward, and replaces the use of the stoichiometric coefficient equations.
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