Equilibrium chemistry is concerned with systems inchemical equilibrium. The unifying principle is that thefree energy of a system at equilibrium is the minimum possible, so that the slope of the free energy with respect to thereaction coordinate is zero.^[1][2] This principle, applied to mixtures at equilibrium provides a definition of anequilibrium constant. Applications includeacid–base,host–guest,metal–complex,solubility,partition,chromatography andredox equilibria.

A chemical system is said to be in equilibrium when the quantities of the chemical entities involved do not andcannot change in time without the application of an external influence. In this sense a system in chemical equilibrium is in astable state. The system atchemical equilibrium will be at a constant temperature, pressure or volume and a composition. It will be insulated from exchange of heat with the surroundings, that is, it is aclosed system. A change of temperature, pressure (or volume) constitutes an external influence and the equilibrium quantities will change as a result of such a change. If there is a possibility that the composition might change, but the rate of change is negligibly slow, the system is said to be in ametastable state. The equation of chemical equilibrium can be expressed symbolically as

reactant(s) ⇌ product(s)

The sign ⇌ means "are in equilibrium with". This definition refers tomacroscopic properties. Changes do occur at the microscopic level of atoms and molecules, but to such a minute extent that they are not measurable and in a balanced way so that the macroscopic quantities do not change. Chemical equilibrium is a dynamic state in which forward and backward reactions proceed at such rates that the macroscopic composition of the mixture is constant. Thus, equilibrium sign ⇌ symbolizes the fact that reactions occur in both forward${\displaystyle \rightharpoonup }$ and backward${\displaystyle \leftharpoondown }$ directions.

Asteady state, on the other hand, is not necessarily an equilibrium state in the chemical sense. For example, in a radioactivedecay chain the concentrations of intermediate isotopes are constant because the rate of production is equal to the rate of decay. It is not a chemical equilibrium because the decay process occurs in one direction only.

Thermodynamic equilibrium is characterized by the free energy for the whole (closed) system being a minimum. For systems at constant volume theHelmholtz free energy is minimum and for systems at constant pressure theGibbs free energy is minimum.^[3] Thus a metastable state is one for which the free energy change between reactants and products is not minimal even though the composition does not change in time.^[4]

The existence of this minimum is due to the free energy of mixing of reactants and products being always negative.^[5] Forideal solutions theenthalpy of mixing is zero, so the minimum exists because theentropy of mixing is always positive.^[6][7] The slope of the reaction free energy, δG_r with respect to thereaction coordinate,ξ, is zero when the free energy is at its minimum value.

${\displaystyle \delta G_{\mathrm{r}}={\left(\frac{\mathrm{\partial }G}{\mathrm{\partial }\xi }\right)}_{T,P};\delta G_{\mathrm{r}}(\mathrm{E}\mathrm{q})=0}$

Chemical potential is the partial molar free energy. The potential,μ_i, of theith species in a chemical reaction is the partial derivative of the free energy with respect to the number of moles of that species,N_i:

${\displaystyle {\mu }_i={\left(\frac{\mathrm{\partial }G}{\mathrm{\partial }{N}_i}\right)}_{T,P}}$

A general chemical equilibrium can be written as^{[note 1]}

${\displaystyle \sum _j{n}_j{\mathrm{R}\mathrm{e}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}}_j\rightleftharpoons \sum _k{m}_k{\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{t}}_k}$

n_j are thestoichiometric coefficients of the reactants in the equilibrium equation, andm_j are the coefficients of the products. The value of δG_r for these reactions is a function of the chemical potentials of all the species.

${\displaystyle \delta G_{\mathrm{r}}=\sum _k{m}_k{\mu }_k-\sum _j{n}_j{\mu }_j}$

The chemical potential,μ_i, of theith species can be calculated in terms of itsactivity,a_i.

${\displaystyle {\mu }_i={\mu }_i^{\ominus }+RT\ln a_i}$

μ^o
_i is the standard chemical potential of the species,R is thegas constant andT is the temperature. Setting the sum for the reactantsj to be equal to the sum for the products,k, so that δG_r(Eq) = 0:

${\displaystyle \sum _j{n}_j\left({\mu }_j^{\ominus }+RT\ln a_j\right)=\sum _k{m}_k\left({\mu }_k^{\ominus }+RT\ln a_k\right)}$

Rearranging the terms,

${\displaystyle \sum _k{m}_k{\mu }_k^{\ominus }-\sum _j{n}_j{\mu }_j^{\ominus }=-RT\left(\sum _k\ln {a_k}^{m_k}-\sum _j\ln {a_j}^{n_j}\right)}$

${\displaystyle \mathrm{\Delta }{G}^{\ominus }=-RT\ln K.}$

This relates thestandard Gibbs free energy change, ΔG^o to anequilibrium constant,K, thereaction quotient of activity values at equilibrium.

${\displaystyle \mathrm{\Delta }{G}^{\ominus }=\sum _k{m}_k{\mu }_k^{\ominus }-\sum _j{n}_j{\mu }_j^{\ominus }}$

${\displaystyle \ln K=\sum _k\ln {a_k}^{m_k}-\sum _j\ln {a_j}^{n_j};K=\frac{\prod _k{a_k}^{m_k}}{\prod _j{a_j}^{n_j}}}$

It follows that any equilibrium of this kind can be characterized either by the standard free energy change or by the equilibrium constant. In practice concentrations are more useful than activities. Activities can be calculated from concentrations if theactivity coefficient are known, but this is rarely the case. Sometimes activity coefficients can be calculated using, for example,Pitzer equations orSpecific ion interaction theory. Otherwise conditions must be adjusted so that activity coefficients do not vary much. For ionic solutions this is achieved by using a background ionic medium at a high concentration relative to the concentrations of the species in equilibrium.

If activity coefficients are unknown they may be subsumed into the equilibrium constant, which becomes a concentration quotient.^[8] Each activitya_i is assumed to be the product of a concentration, [A_i], and an activity coefficient,γ_i:

${\displaystyle a_i=[{\mathrm{A}}_i]{\gamma }_i}$

This expression for activity is placed in the expression defining the equilibrium constant.^[9]

${\displaystyle K=\frac{\prod _k{a_k}^{m_k}}{\prod _j{a_j}^{n_j}}=\frac{\prod _k{\left([{\mathrm{A}}_k]{\gamma }_k\right)}^{m_k}}{\prod _j{\left([{\mathrm{A}}_j]{\gamma }_j\right)}^{n_j}}=\frac{\prod _k[{\mathrm{A}}_k{]}^{m_k}}{\prod _j[{\mathrm{A}}_j{]}^{n_j}}\times \frac{\prod _k{{\gamma }_k}^{m_k}}{\prod _j{{\gamma }_j}^{n_j}}=\frac{\prod _k[{\mathrm{A}}_k{]}^{m_k}}{\prod _j[{\mathrm{A}}_j{]}^{n_j}}\times \mathrm{\Gamma }}$

By setting the quotient of activity coefficients,Γ, equal to one,^{[note 2]} the equilibrium constant is defined as a quotient of concentrations.

${\displaystyle K=\frac{\prod _k[{\mathrm{A}}_k{]}^{m_k}}{\prod _j[{\mathrm{A}}_j{]}^{n_j}}}$

In more familiar notation, for a general equilibrium

α A +β B ... ⇌σ S +τ T ...

${\displaystyle K=\frac{[\mathrm{S}{]}^{\sigma }[\mathrm{T}{]}^{\tau }...}{[\mathrm{A}{]}^{\alpha }[\mathrm{B}{]}^{\beta }...}}$

This definition is much more practical, but an equilibrium constant defined in terms of concentrations is dependent on conditions. In particular, equilibrium constants for species in aqueous solution are dependent onionic strength, as the quotient of activity coefficients varies with the ionic strength of the solution.

The values of the standard free energy change and of the equilibrium constant are temperature dependent. To a first approximation, thevan 't Hoff equation may be used.

${\displaystyle \frac{d\ln K}{dT}=\frac{\mathrm{\Delta }{H}^{\ominus }}{R{T}^2}\text{ or }\frac{d\ln K}{d\frac{1}{T}}=-\frac{\mathrm{\Delta }{H}^{\ominus }}{R}}$

This shows that when the reaction is exothermic (ΔH^o, the standardenthalpy change, is negative), thenK decreases with increasing temperature, in accordance withLe Châtelier's principle. The approximation involved is that the standard enthalpy change, ΔH^o, is independent of temperature, which is a good approximation only over a small temperature range. Thermodynamic arguments can be used to show that

${\displaystyle {\left(\frac{\mathrm{\partial }H}{\mathrm{\partial }T}\right)}_p=C_p}$

whereC_p is theheat capacity at constant pressure.^[10]

When dealing with gases,fugacity,f, is used rather than activity. However, whereas activity isdimensionless, fugacity has the dimension ofpressure. A consequence is that chemical potential has to be defined in terms of a standard pressure,p^o:^[11]

${\displaystyle \mu ={\mu }^{\ominus }+RT\ln \frac{f}{p^{\ominus }}}$

By conventionp^o is usually taken to be 1bar.Fugacity can be expressed as the product ofpartial pressure,p, and a fugacity coefficient,Φ:

${\displaystyle f=p\mathrm{\Phi }}$

Fugacity coefficients are dimensionless and can be obtained experimentally at specific temperature and pressure, from measurements of deviations fromideal gas behaviour. Equilibrium constants are defined in terms of fugacity. If the gases are at sufficiently low pressure that they behave as ideal gases, the equilibrium constant can be defined as a quotient of partial pressures.

An example of gas-phase equilibrium is provided by theHaber–Bosch process ofammonia synthesis.

N₂ + 3 H₂ ⇌ 2 NH₃;     ${\displaystyle K=\frac{{f_{\mathrm{N}{\mathrm{H}}_3}}^2}{f_{{\mathrm{N}}_2}{f_{{\mathrm{H}}_2}}^3}}$

This reaction is stronglyexothermic, so the equilibrium constant decreases with temperature. However, a temperature of around 400 °C is required in order to achieve a reasonable rate of reaction with currently availablecatalysts. Formation of ammonia is also favoured by high pressure, as the volume decreases when the reaction takes place. The same reaction,nitrogen fixation, occurs at ambient temperatures in nature, when the catalyst is anenzyme such asnitrogenase. Much energy is needed initially to break the nitrogen–nitrogen triple bond even though the overall reaction is exothermic.

Gas-phase equilibria occur duringcombustion and were studied as early as 1943 in connection with the development of theV2rocket engine.^[12]

The calculation of composition for a gaseous equilibrium at constant pressure is often carried out using ΔG values, rather than equilibrium constants.^[13][14]

Two or more equilibria can exist at the same time. When this is so, equilibrium constants can be ascribed to individual equilibria, but they are not always unique. For example, three equilibrium constants can be defined for adibasicacid, H₂A.^{[15][note 3]}

A²⁻ + H⁺ ⇌ HA⁻;     ${\displaystyle K_1=\frac{[{\text{HA}}^{-}]}{[{\text{H}}^{+}][{\text{A}}^{2-}]}}$

HA⁻ + H⁺ ⇌ H₂A;     ${\displaystyle K_2=\frac{[{\text{H}}_2^{}\text{A}]}{[{\text{H}}^{+}][{\text{HA}}^{-}]}}$

A²⁻ + 2 H⁺ ⇌ H₂A;     ${\displaystyle {\beta }_2=\frac{[{\text{H}}_2^{}\text{A}]}{[{\text{H}}^{+}{]}^2[{\text{A}}^{2-}]}}$

The three constants are not independent of each other and it is easy to see thatβ₂ =K₁K₂. The constantsK₁ andK₂ are stepwise constants andβ is an example of an overall constant.

The concentrations of species in equilibrium are usually calculated under the assumption that activity coefficients are either known or can be ignored. In this case, each equilibrium constant for the formation of a complex in a set of multiple equilibria can be defined as follows

α A +β B ... ⇌ A_αB_β...;     ${\displaystyle K_{\alpha \beta \dots }=\frac{[{\mathrm{A}}_{\alpha }{\mathrm{B}}_{\beta }\dots ]}{[\mathrm{A}{]}^{\alpha }[\mathrm{B}{]}^{\beta }\dots }}$

The concentrations of species containing reagent A are constrained by a condition ofmass-balance, that is, the total (or analytical) concentration, which is the sum of all species' concentrations, must be constant. There is one mass-balance equation for each reagent of the type

${\displaystyle T_{\mathrm{A}}=[\mathrm{A}]+\sum [{\mathrm{A}}_{\alpha }{\mathrm{B}}_{\beta }\dots ]=[\mathrm{A}]+\sum \left(\alpha K_{\alpha \beta }\dots [\mathrm{A}{]}^{\alpha }[\mathrm{B}{]}^{\beta }\dots \right)}$

There are as many mass-balance equations as there are reagents, A, B..., so if the equilibrium constant values are known, there aren mass-balance equations inn unknowns, [A], [B]..., the so-called free reagent concentrations. Solution of these equations gives all the information needed to calculate the concentrations of all the species.^[16]

Thus, the importance of equilibrium constants lies in the fact that, once their values have been determined by experiment, they can be used to calculate the concentrations, known as thespeciation, of mixtures that contain the relevant species.

There are five main types of experimental data that are used for the determination of solution equilibrium constants. Potentiometric data obtained with aglass electrode are the most widely used with aqueous solutions. The others areSpectrophotometric,Fluorescence (luminescence) measurements andNMRchemical shift measurements;^[8][17] simultaneous measurement ofK and ΔH for 1:1 adducts in biological systems is routinely carried out usingIsothermal Titration Calorimetry.

The experimental data will comprise a set of data points. At the i'th data point, the analytical concentrations of the reactants,T_A(i),T_B(i) etc. will be experimentally known quantities and there will be one or more measured quantities,y_i, that depend in some way on the analytical concentrations and equilibrium constants. A general computational procedure has three main components.

1. Definition of a chemical model of the equilibria. The model consists of a list of reagents, A, B, etc. and the complexes formed from them, with stoichiometries A_pB_q... Known or estimated values of the equilibrium constants for the formation of all complexes must be supplied.
2. Calculation of the concentrations of all the chemical species in each solution. The free concentrations are calculated by solving the equations of mass-balance, and the concentrations of the complexes are calculated using the equilibrium constant definitions. A quantity corresponding to the observed quantity can then be calculated using physical principles such as theNernst potential orBeer-Lambert law which relate the calculated quantity to the concentrations of the species.
3. Refinement of the equilibrium constants. Usually aNon-linear least squares procedure is used. A weighted sum of squares,U, is minimized.$${\displaystyle U=\sum _{i=np}^{i=1}{w}_i{\left(y_i^{\mathrm{o}\mathrm{b}\mathrm{s}\mathrm{e}\mathrm{r}\mathrm{v}\mathrm{e}\mathrm{d}}-y_i^{\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{c}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d}}\right)}^2}$$ The weights,w_i and quantitiesy may be vectors. Values of the equilibrium constants are refined in an iterative procedure.^[16]

Brønsted and Lowry characterized an acid–base equilibrium as involving a proton exchange reaction:^[18][19][20]

acid + base ⇌ conjugate base + conjugate acid.

An acid is a proton donor; the proton is transferred to the base, a proton acceptor, creating a conjugate acid. For aqueous solutions of an acid HA, the base is water; the conjugate base is A⁻ and the conjugate acid is the solvated hydrogen ion. In solution chemistry, it is usual to use H⁺ as an abbreviation for the solvated hydrogen ion, regardless of the solvent. In aqueous solution H⁺ denotes asolvated hydronium ion.^{[21][22][note 4]}

The Brønsted–Lowry definition applies to other solvents, such asdimethyl sulfoxide: the solvent S acts as a base, accepting a proton and forming the conjugate acid SH⁺. A broader definition of acid dissociation includeshydrolysis, in which protons are produced by the splitting of water molecules. For example,boric acid,B(OH)
₃, acts as a weak acid, even though it is not a proton donor, because of the hydrolysis equilibrium

B(OH)
₃ +H
₂O ⇌B(OH)⁻
₄ + H⁺.

Similarly,metal ion hydrolysis causes ions such as[Al(H
₂O)
₆]³⁺
to behave as weak acids:^[23]

[Al(H
₂O)
₆]³⁺
⇌[Al(H
₂O)
₅(OH)]²⁺
+H⁺
.

Acid–base equilibria are important in a very widerange of applications, such asacid–base homeostasis,ocean acidification,pharmacology andanalytical chemistry.

A host–guest complex, also known as a donor–acceptor complex, may be formed from aLewis base, B, and aLewis acid, A. The host may be either a donor or an acceptor. Inbiochemistry host–guest complexes are known asreceptor-ligand complexes; they are formed primarily bynon-covalent bonding. Many host–guest complexes has 1:1 stoichiometry, but many others have more complex structures. The general equilibrium can be written as

p A +q B ⇌ A_pB_q

The study of these complexes is important forsupramolecular chemistry^[24][25] andmolecular recognition. The objective of these studies is often to find systems with a highbinding selectivity of a host (receptor) for a particular target molecule or ion, the guest or ligand. An application is the development ofchemical sensors.^[26] Finding a drug which either blocks a receptor, anantagonist which forms a strong complex the receptor, or activate it, anagonist, is an important pathway todrug discovery.^[27]

The formation of a complex between a metal ion, M, and a ligand, L, is in fact usually a substitution reaction. For example, Inaqueous solutions, metal ions will be present asaquo ions, so the reaction for the formation of the first complex could be written as^{[note 5]}

[M(H₂O)_n] + L ⇌ [M(H₂O)_n−1L] + H₂O

However, since water is in vast excess, the concentration of water is usually assumed to be constant and is omitted from equilibrium constant expressions. Often, the metal and the ligand are in competition for protons.^{[note 4]} For the equilibrium

p M +q L +r H ⇌ M_pL_qH_r

a stability constant can be defined as follows:^[28][29]

${\displaystyle {\beta }_{pqr}=\frac{[{\mathrm{M}}_p{\mathrm{L}}_q{\mathrm{H}}_r]}{[\mathrm{M}{]}^p[\mathrm{L}{]}^q[\mathrm{H}{]}^r}}$

The definition can easily be extended to include any number of reagents. It includeshydroxide complexes because the concentration of the hydroxide ions is related to the concentration of hydrogen ions by theself-ionization of water

${\displaystyle [{\text{OH}}^{-}]=\frac{K_{\mathrm{w}}}{[{\text{H}}^{+}]}}$

Stability constants defined in this way, areassociation constants. This can lead to some confusion aspK_a values aredissociation constants. In general purpose computer programs it is customary to define all constants as association constants. The relationship between the two types of constant is given inassociation and dissociation constants.

Inbiochemistry, an oxygen molecule can bind to an iron(II) atom in ahemeprosthetic group inhemoglobin. The equilibrium is usually written, denoting hemoglobin by Hb, as

Hb + O₂ ⇌ HbO₂

but this representation is incomplete as theBohr effect shows that the equilibrium concentrations are pH-dependent. A better representation would be

[HbH]⁺ + O₂ ⇌ HbO₂ + H⁺

as this shows that when hydrogen ion concentration increases the equilibrium is shifted to the left in accordance withLe Châtelier's principle. Hydrogen ion concentration can be increased by the presence of carbon dioxide, which behaves as a weak acid.

H₂O + CO₂ ⇌HCO⁻
₃ + H⁺

The iron atom can also bind to other molecules such ascarbon monoxide. Cigarette smoke contains some carbon monoxide so the equilibrium

HbO₂ + CO ⇌Hb(CO) + O₂

is established in the blood of cigarette smokers.

Chelation therapy is based on the principle of usingchelating ligands with a highbinding selectivity for a particular metal to remove that metal from the human body.

Complexes withpolyamino carboxylic acids find a wide range of applications.EDTA in particular is used extensively.

A reduction–oxidation (redox) equilibrium can be handled in exactly the same way as any other chemical equilibrium. For example,

Fe²⁺ + Ce⁴⁺ ⇌ Fe³⁺ + Ce³⁺;     ${\displaystyle K=\frac{[{\text{Fe}}^{3+}][{\text{Ce}}^{3+}]}{[{\text{Fe}}^{2+}][{\text{Ce}}^{4+}]}}$

However, in the case of redox reactions it is convenient to split the overall reaction into two half-reactions. In this example

Fe³⁺ + e⁻ ⇌ Fe²⁺

Ce⁴⁺ + e⁻ ⇌ Ce³⁺

The standard free energy change, which is related to the equilibrium constant by

${\displaystyle \mathrm{\Delta }{G}^{\ominus }=-RT\ln K}$

can be split into two components,

${\displaystyle \mathrm{\Delta }{G}^{\ominus }=\mathrm{\Delta }{G}_{\text{Fe}}^{\ominus }+\mathrm{\Delta }{G}_{\text{Ce}}^{\ominus }}$

The concentration of free electrons is effectively zero as the electrons are transferred directly from the reductant to the oxidant. Thestandard electrode potential,E⁰ for the each half-reaction is related to the standard free energy change by^[30]

${\displaystyle \mathrm{\Delta }{G}_{\text{Fe}}^{\ominus }=-nF{E}_{\text{Fe}}^0;\mathrm{\Delta }{G}_{\text{Ce}}^{\ominus }=-nF{E}_{\text{Ce}}^0}$

wheren is the number of electrons transferred andF is theFaraday constant. Now, the free energy for an actual reaction is given by

${\displaystyle \mathrm{\Delta }G=\mathrm{\Delta }{G}^{\ominus }+RT\ln Q}$

whereR is thegas constant andQ areaction quotient. Strictly speakingQ is a quotient of activities, but it is common practice to use concentrations instead of activities. Therefore:

${\displaystyle E_{\text{Fe}}=E_{\text{Fe}}^0+\frac{RT}{nF}\ln \frac{[{\text{Fe}}^{3+}]}{[{\text{Fe}}^{2+}]}}$

For any half-reaction, the redox potential of an actual mixture is given by the generalized expression^{[note 6]}

${\displaystyle E=E^0+\frac{RT}{nF}\ln \frac{[\text{oxidized species}]}{[\text{reduced species}]}}$

This is an example of theNernst equation. The potential is known as a reduction potential. Standard electrode potentials are available in atable of values. Using these values, the actual electrode potential for a redox couple can be calculated as a function of the ratio of concentrations.

The equilibrium potential for a general redox half-reaction (See#Equilibrium constant above for an explanation of the symbols)

α A +β B... +n e⁻ ⇌σ S +τ T...

is given by^[31]

${\displaystyle E=E^{\ominus }+\frac{RT}{nF}\ln \frac{{\{\text{S}\}}^{\sigma }{\{\text{T}\}}^{\tau }...}{{\{\text{A}\}}^{\alpha }{\{\text{B}\}}^{\beta }...}}$

Use of this expression allows the effect of a species not involved in the redox reaction, such as the hydrogen ion in a half-reaction such as

MnO⁻
₄ + 8 H⁺ + 5 e⁻ ⇌ Mn²⁺ + 4 H₂O

to be taken into account.

The equilibrium constant for a full redox reaction can be obtained from the standard redox potentials of the constituent half-reactions. At equilibrium the potential for the two half-reactions must be equal to each other and, of course, the number of electrons exchanged must be the same in the two half reactions.^[32]

Redox equilibrium play an important role in theelectron transport chain. The variouscytochromes in the chain have different standard redox potentials, each one adapted for a specific redox reaction. This allows, for example, atmosphericoxygen to be reduced inphotosynthesis. A distinct family of cytochromes, thecytochrome P450 oxidases, are involved insteroidogenesis anddetoxification.

When asolute forms asaturated solution in asolvent, the concentration of the solute, at a given temperature, is determined by the equilibrium constant at that temperature.^[33]

${\displaystyle \ln K=-RT\ln \left(\frac{\sum _k{a_k}^{m_k}(\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n})}{a(\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{d})}\right)}$

The activity of a pure substance in the solid state is one, by definition, so the expression simplifies to

${\displaystyle \ln K=-RT\ln \left(\sum _k{a_k}^{m_k}(\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n})\right)}$

If the solute does not dissociate the summation is replaced by a single term, but if dissociation occurs, as with ionic substances

${\displaystyle K_{\mathrm{S}\mathrm{P}}=\prod _k{a_k}^{m_k}}$

For example, with Na₂SO₄,m₁ = 2 andm₂ = 1 so the solubility product is written as

${\displaystyle K_{\mathrm{S}\mathrm{P}}=[\mathrm{N}{\mathrm{a}}^{+}{]}^2[\mathrm{S}{\mathrm{O}}_4^{2-}]}$

Concentrations, indicated by [...], are usually used in place of activities, but activity must be taken into account of the presence of another salt with no ions in common, the so-called salt effect. When another salt is present that has an ion in common, thecommon-ion effect comes into play, reducing the solubility of the primary solute.^[34]

When a solution of a substance in one solvent is brought into equilibrium with a second solvent that is immiscible with the first solvent, the dissolved substance may be partitioned between the two solvents. The ratio of concentrations in the two solvents is known as apartition coefficient ordistribution coefficient.^{[note 7]} The partition coefficient is defined as the ratio of theanalytical concentrations of the solute in the two phases. By convention the value is reported in logarithmic form.

${\displaystyle \log p=\log \frac{[\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{e}{]}_{\mathrm{o}\mathrm{r}\mathrm{g}\mathrm{a}\mathrm{n}\mathrm{i}\mathrm{c}\mathrm{p}\mathrm{h}\mathrm{a}\mathrm{s}\mathrm{e}}}{[\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{e}{]}_{\mathrm{a}\mathrm{q}\mathrm{u}\mathrm{e}\mathrm{o}\mathrm{u}\mathrm{s}\mathrm{p}\mathrm{h}\mathrm{a}\mathrm{s}\mathrm{e}}}}$

The partition coefficient is defined at a specified temperature and, if applicable, pH of the aqueous phase. Partition coefficients are very important inpharmacology because they determine the extent to which a substance can pass from the blood (an aqueous solution) through a cell wall which is like an organic solvent. They are usually measured using water andoctanol as the two solvents, yielding the so-calledoctanol-water partition coefficient. Many pharmaceutical compounds areweak acids orweak bases. Such a compound may exist with a different extent of protonation depending onpH and theacid dissociation constant. Because the organic phase has a lowdielectric constant the species with no electrical charge will be the most likely one to pass from the aqueous phase to the organic phase. Even at pH 7–7.2, the range of biological pH values, the aqueous phase may support an equilibrium between more than one protonated form. log p is determined from the analytical concentration of the substance in the aqueous phase, that is, the sum of the concentration of the different species in equilibrium.

Solvent extraction is used extensively in separation and purification processes. In its simplest form a reaction is performed in an organic solvent and unwanted by-products are removed by extraction into water at a particular pH.

A metal ion may be extracted from an aqueous phase into an organic phase in which the salt is not soluble, by adding aligand. The ligand, L^a−, forms a complex with the metal ion, M^b+, [ML_x]^(b−ax)+ which has a stronglyhydrophobic outer surface. If the complex has no electrical charge it will be extracted relatively easily into the organic phase. If the complex is charged, it is extracted as anion pair. The additional ligand is not always required. For example,uranyl nitrate, UO₂(NO₃)₂, is soluble indiethyl ether because the solvent itself acts as a ligand. This property was used in the past for separating uranium from other metals whose salts are not soluble in ether. Currently extraction intokerosene is preferred, using a ligand such astri-n-butyl phosphate, TBP. In thePUREX process, which is commonly used innuclear reprocessing, uranium(VI) is extracted from strong nitric acid as the electrically neutral complex [UO₂(TBP)₂(NO₃)₂]. The strong nitric acid provides a high concentration of nitrate ions which pushes the equilibrium in favour of the weak nitrato complex. Uranium is recovered by back-extraction (stripping) into weak nitric acid. Plutonium(IV) forms a similar complex, [PuO₂(TBP)₂(NO₃)₂] and the plutonium in this complex can be reduced to separate it from uranium.

Another important application of solvent extraction is in the separation of thelanthanoids. This process also uses TBP and the complexes are extracted into kerosene. Separation is achieved because thestability constant for the formation of the TBP complex increases as the size of the lanthanoid ion decreases.

An instance of ion-pair extraction is in the use of a ligand to enable oxidation bypotassium permanganate, KMnO₄, in an organic solvent. KMnO₄ is not soluble in organic solvents. When a ligand, such as acrown ether is added to an aqueous solution of KMnO₄, it forms a hydrophobic complex with the potassium cation which allows the uncharged ion pair [KL]⁺[MnO₄]⁻ to be extracted into the organic solvent. See also:phase-transfer catalysis.

More complex partitioning problems (i.e. 3 or more phases present) can sometimes be handled with afugacity capacity approach.

In chromatography substances are separated by partition between a stationary phase and a mobile phase. The analyte is dissolved in the mobile phase, and passes over the stationary phase. Separation occurs because of differing affinities of theanalytes for the stationary phase. A distribution constant,K_d can be defined as

${\displaystyle K_{\mathrm{d}}=\frac{a_{\mathrm{s}}}{a_{\mathrm{m}}}}$

wherea_s anda_m are the equilibrium activities in the stationary and mobile phases respectively. It can be shown that the rate of migration,ν, is related to the distribution constant by

${\displaystyle \overline{\nu }\propto \frac{1}{1+f{K}_{\mathrm{d}}}.}$

f is a factor which depends on the volumes of the two phases.^[35] Thus, the higher the affinity of the solute for the stationary phase, the slower the migration rate.

There is a wide variety of chromatographic techniques, depending on the nature of the stationary and mobile phases. When the stationary phase is solid, the analyte may form a complex with it. Awater softener functions by selective complexation with asulfonateion exchange resin. Sodium ions form relatively weak complexes with the resin. Whenhard water is passed through the resin, the divalent ions of magnesium and calcium displace the sodium ions and are retained on the resin, R.

RNa + M²⁺ ⇌ RM⁺ + Na⁺

The water coming out of the column is relatively rich in sodium ions^{[note 8]} and poor in calcium and magnesium which are retained on the column. The column is regenerated by passing a strong solution of sodium chloride through it, so that the resin–sodium complex is again formed on the column.Ion-exchange chromatography utilizes a resin such aschelex 100 in whichiminodiacetate residues, attached to a polymer backbone, formchelate complexes of differing strengths with different metal ions, allowing the ions such as Cu²⁺ and Ni²⁺ to be separated chromatographically.

Another example of complex formation is inchiral chromatography in which is used to separateenantiomers from each other. The stationary phase is itself chiral and forms complexes selectively with the enantiomers. In other types of chromatography with a solid stationary phase, such asthin-layer chromatography the analyte is selectivelyadsorbed onto the solid.

Ingas–liquid chromatography (GLC) the stationary phase is a liquid such aspolydimethylsiloxane, coated on a glass tube. Separation is achieved because the various components in the gas have different solubility in the stationary phase. GLC can be used to separate literally hundreds of components in a gas mixture such ascigarette smoke oressential oils, such aslavender oil.

• Thermodynamic databases for pure substances

• Chemical EquilibriumDownloadable book

• Atkins, P.W.; De Paula, J. (2006).Physical Chemistry (8th. ed.). Oxford University Press.ISBN 0-19-870072-5.
• Denbeigh, K. (1981).The principles of chemical equilibrium (4th. ed.). Cambridge, U.K.: Cambridge University Press.ISBN 0-521-28150-4. A classic book, last reprinted in 1997.
• Mendham, J.; Denney, R. C.; Barnes, J. D.; Thomas, M. J. K. (2000),Vogel's Quantitative Chemical Analysis (6th ed.), New York: Prentice Hall,ISBN 0-582-22628-7

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