In achemical reaction,chemical equilibrium is the state in which both thereactants andproducts are present inconcentrations which have no further tendency to change with time, so that there is no observable change in the properties of thesystem.[1] This state results when the forward reaction proceeds at the same rate as thereverse reaction. Thereaction rates of the forward and backward reactions are generally not zero, but they are equal. Thus, there are no net changes in the concentrations of the reactants and products. Such a state is known asdynamic equilibrium.[2][3]It is the subject of study ofequilibrium chemistry.
Theconcept of chemical equilibrium was developed in 1803, afterBerthollet found that somechemical reactions arereversible.[4] For any reaction mixture to exist at equilibrium, therates of the forward and backward (reverse) reactions must be equal. In the followingchemical equation, arrows point both ways to indicate equilibrium.[5] A and B arereactant chemical species, S and T are product species, andα,β,σ, andτ are thestoichiometric coefficients of the respective reactants and products:
α A +β B ⇌σ S +τ T
The equilibrium concentration position of a reaction is said to lie "far to the right" if, at equilibrium, nearly all the reactants are consumed. Conversely the equilibrium position is said to be "far to the left" if hardly any product is formed from the reactants.
Despite the limitations 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 thevan 't Hoff equation. Adding acatalyst 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.[2][6]
Although themacroscopic equilibrium concentrations are constant in time, reactions do occur at the molecular level. For example, in the case ofacetic acid dissolved in water and formingacetate andhydronium ions,
CH3CO2H + H2O ⇌ CH3CO−2 + H3O+
a proton may hop from one molecule of acetic acid onto a water molecule and then onto an acetate anion to form another molecule of acetic acid and leaving the number of acetic acid molecules unchanged. This is an example ofdynamic equilibrium. Equilibria, like the rest of thermodynamics, are statistical phenomena, averages of microscopic behavior.
Le Châtelier's principle (1884) predicts the behavior of an equilibrium system when changes to its reaction conditions occur.If a dynamic equilibrium is disturbed by changing the conditions, the position of equilibrium moves to partially reverse the change. For example, adding more S (to the chemical reaction above) 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).
Ifmineral acid is added to the acetic 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:
If {H3O+} increases {CH3CO2H} must increase andCH3CO−2 must decrease. The H2O is left out, as it is the solvent and its concentration remains high and nearly constant.
J. W. Gibbs suggested in 1873 that equilibrium is attained when the "available energy" (now known asGibbs free energy or Gibbs energy) of the system is at its minimum value, assuming the reaction is carried out at a constant temperature and pressure. What this means is that the derivative of the Gibbs energy with respect toreaction coordinate (a measure of theextent of reaction that has occurred, ranging fromzero for all reactants to a maximum for all products) vanishes (because dG = 0), signaling astationary point. This derivative is called the reaction Gibbs energy (or energy change) and corresponds to the difference between thechemical potentials of reactants and products at the composition of the reaction mixture.[1] This criterion is both necessary and sufficient. If a mixture is not at equilibrium, the liberation of the excess Gibbs energy (orHelmholtz 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 standardGibbs free energy change for the reaction by the equation
where [A] is theconcentration of A, etc., is independent of theanalytical concentration of the reactants. For this reason, equilibrium constants forsolutions are usuallydetermined in media of high ionic strength.Kc varies withionic strength, temperature and pressure (or volume). LikewiseKp for gases depends onpartial pressure. These constants are easier to measure and encountered in high-school chemistry courses.
At constant temperature and pressure, one must consider theGibbs free energy,G, while at constant temperature and volume, one must consider theHelmholtz free energy,A, for the reaction; and at constant internal energy and volume, one must consider the entropy,S, for the reaction.
The constant volume case is important ingeochemistry andatmospheric chemistry where pressure variations are significant. Note that, if reactants and products were instandard state (completely pure), then there would be no reversibility and no equilibrium. Indeed, they would necessarily occupy disjoint volumes of space. The mixing of the products and reactants contributes a large entropy increase (known asentropy of mixing) to states containing equal mixture of products and reactants and gives rise to a distinctive minimum in the Gibbs energy as a function of the extent of reaction.[7] The standard Gibbs energy change, together with the Gibbs energy of mixing, determine the equilibrium state.[8][9]
In this article only theconstant pressure case is considered. The relation between the Gibbs free energy and the equilibrium constant can be found by consideringchemical potentials.[1]
At constant temperature and pressure in the absence of an applied voltage, theGibbs free energy,G, for the reaction depends only on theextent of reaction:ξ (Greek letterxi), and can only decrease according to thesecond law of thermodynamics. It means that the derivative ofG with respect toξ must be negative if the reaction happens; at the equilibrium this derivative is equal to zero.
:equilibrium
In order to meet the thermodynamic condition for equilibrium, the Gibbs energy must be stationary, meaning that the derivative ofG with respect to the extent of reaction,ξ, must be zero. It can be shown that in this case, the sum ofchemical potentials times the stoichiometric coefficients of the products is equal to the sum of those corresponding to the reactants.[10] Therefore, the sum of the Gibbs energies of the reactants must be the equal to the sum of the Gibbs energies of the products.
whereμ is in this case a partial molar Gibbs energy, achemical potential. The chemical potential of a reagent A is a function of theactivity, {A} of that reagent.
InsertingdNi =νi dξ into the above equation gives astoichiometric coefficient () and a differential that denotes the reaction occurring to an infinitesimal extent (dξ). At constant pressure and temperature the above equations can be written as
which is the Gibbs free energy change for the reaction. This results in:
.
By substituting the chemical potentials:
,
the relationship becomes:
:
which is thestandard Gibbs energy change for the reaction that can be calculated using thermodynamical tables.Thereaction quotient is defined as:
Therefore,
At equilibrium:
leading to:
and
Obtaining the value of the standard Gibbs energy change, allows the calculation of the equilibrium constant.
For a reactional system at equilibrium:Qr = Keq;ξ = ξeq.
If the activities of constituents are modified, the value of the reaction quotient changes and becomes different from the equilibrium constant:Qr ≠ Keq and then
In simplifications where the change in reaction quotient is solely due to the concentration changes,Qr is referred to as themass-action ratio, and the ratioQr/Keq is referred to as the disequilibrium ratio.[citation needed]
If activity of a reagenti increases the reaction quotient decreases. Then and The reaction will shift to the right (i.e. in the forward direction, and thus more products will form).
If activity of a productj increases, then and The reaction will shift to the left (i.e. in the reverse direction, and thus less products will form).
Note that activities and equilibrium constants are dimensionless numbers.
The expression for the equilibrium constant can be rewritten as the product of a concentration quotient,Kc and anactivity coefficient quotient,Γ.
[A] is the concentration of reagent A, etc. It is possible in principle to obtain values of the activity coefficients, γ. For solutions, equations such as theDebye–Hückel equation or extensions such asDavies equation[11]Specific ion interaction theory orPitzer equations[12] may be used. However this is not always possible. It is common practice to assume thatΓ is a constant, and to use the concentration quotient in place of the thermodynamic equilibrium constant. It is also general practice to use the termequilibrium constant instead of the more accurateconcentration quotient. This practice will be followed here.
For reactions in the gas phasepartial pressure is used in place of concentration andfugacity coefficient in place of activity coefficient. In the real world, for example, when makingammonia 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 thereal gas phase is given by
so the general expression defining an equilibrium constant is valid for both solution and gas phases.[citation needed]
In aqueous solution, equilibrium constants are usually determined in the presence of an "inert" electrolyte such assodium nitrate, NaNO3, orpotassium perchlorate, KClO4. Theionic strength of a solution is given by
whereci andzi stand for the concentration and ionic charge of ion typei, and the sum is taken over all theN types of charged species in solution. When the concentration of dissolved salt is much higher than the analytical concentrations of the reagents, the ions originating from the dissolved salt determine the ionic strength, and the ionic strength is effectively constant. Since activity coefficients depend on ionic strength, the activity coefficients of the species are effectively independent of concentration. Thus, the assumption thatΓ is constant is justified. The concentration quotient is a simple multiple of the equilibrium constant.[13]
However,Kc will vary with ionic strength. If it is measured at a series of different ionic strengths, the value can be extrapolated to zero ionic strength.[12] The concentration quotient obtained in this manner is known, paradoxically, as a thermodynamic equilibrium constant.
Before using a published value of an equilibrium constant in conditions of ionic strength different from the conditions used in its determination, the value should be adjusted.
A mixture may appear to have no tendency to change, though it is not at equilibrium. For example, a mixture ofSO2 andO2 ismetastable as there is akinetic barrier to formation of the product,SO3.
2 SO2 + O2 ⇌ 2 SO3
The barrier can be overcome when acatalyst is also present in the mixture as in thecontact process, but the catalyst does not affect the equilibrium concentrations.
When pure substances (liquids or solids) are involved in equilibria their activities do not appear in the equilibrium constant[14] because their numerical values are considered one.
Applying the general formula for an equilibrium constant to the specific case of a dilute solution of acetic acid in water one obtains
CH3CO2H + H2O ⇌ CH3CO2− + H3O+
For all but very concentrated solutions, the water can be considered a "pure" liquid, and therefore it has an activity of one. The equilibrium constant expression is therefore usually written as
Because water is the solvent, and has an activity of one, the self-ionization constant of water is defined as
It is perfectly legitimate to write [H+] for thehydronium ion concentration, since the state ofsolvation 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 byKw[H+]−1 in equilibrium constant expressions which would otherwise includehydroxide ion.
Solids also do not appear in the equilibrium constant expression, if they are considered to be pure and thus their activities taken to be one. An example is theBoudouard reaction:[14]
2 CO ⇌ CO2 + C
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.
K1 and K2 are examples ofstepwise equilibrium constants. Theoverall equilibrium constant,βD, is product of the stepwise constants.
Note that these constants aredissociation constants because the products on the right hand side of the equilibrium expression are dissociation products. In many systems, it is preferable to use association constants.
β1 andβ2 are examples of association constants. Clearlyβ1 = 1/K2 andβ2 = 1/βD;log β1 = pK2 andlog β2 = pK2 + pK1[15]For multiple equilibrium systems, also see: theory ofResponse reactions.
The effect of changing temperature on an equilibrium constant is given by thevan 't Hoff equation
Thus, forexothermic reactions (ΔH is negative),K decreases with an increase in temperature, but, forendothermic reactions, (ΔH is positive)K increases with an increase in temperature.[16] An alternative formulation is
At first sight this appears to offer a means of obtaining the standard molar enthalpy of the reaction by studying the variation ofK with temperature. In practice, however, the method is unreliable because error propagation almost always gives very large errors on the values calculated in this way.
Equilibrium can be broadly classified as heterogeneous and homogeneous equilibrium.[19] Homogeneous equilibrium consists of reactants and products belonging in the same phase whereas heterogeneous equilibrium comes into play for reactants and products in different phases.
The industrial synthesis such asammonia in theHaber–Bosch process (depicted right) takes place through a succession of equilibrium steps includingadsorption processes
In certainoscillating reactions, the approach to equilibrium is not asymptotically but in the form of a damped oscillation .[14]
The relatedNernst equation in electrochemistry gives the difference in electrode potential as a function of redox concentrations.
When molecules on each side of the equilibrium are able to further react irreversibly in secondary reactions, the final product ratio is determined according to theCurtin–Hammett principle.
In these applications, terms such as stability constant, formation constant, binding constant, affinity constant, association constant and 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 many ways that the composition of a mixture can be calculated. For example, seeICE 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.
The most basic approach is to manipulate the various equilibrium constants until the desired concentrations are expressed in terms of measured equilibrium constants (equivalent to measuring chemical potentials) and initial conditions.
Satisfy the equation ofmass balance. The equations of mass balance are simply statements that demonstrate that the total concentration of each reactant must be constant by the law ofconservation of mass.
In general, the calculations are rather complicated or complex. For instance, in the case of a dibasic acid, H2A dissolved in water the two reactants can be specified as theconjugate base, A2−, and theproton, H+. The following equations of mass-balance could apply equally well to a base such as1,2-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
It is easy to see how this can be extended to three or more reagents.
The composition of solutions containing reactants A and H is easy to calculate as a function ofp[H]. When [H] is known, the free concentration [A] is calculated from the mass-balance equation in A.
The diagram alongside, shows an example of the hydrolysis of thealuminiumLewis acid Al3+(aq)[23] shows the species concentrations for a 5 × 10−6 M solution of analuminium salt as a function of pH. Each concentration is shown as a percentage of the total aluminium.
The diagram above illustrates the point that aprecipitate that is not one of the main species in the solution equilibrium may be formed. At pH just below 5.5 the main species present in a 5 μM solution of Al3+ arealuminium hydroxides Al(OH)2+,AlOH+2 andAl13(OH)7+32, but on raising the pHAl(OH)3 precipitates from the solution. This occurs because Al(OH)3 has a very largelattice energy. As the pH rises more and more Al(OH)3 comes out of solution. This is an example ofLe Châtelier's principle in action: Increasing the concentration of the hydroxide ion causes more aluminium hydroxide to precipitate, which removes hydroxide from the solution. When the hydroxide concentration becomes sufficiently high the soluble aluminate,Al(OH)−4, is formed.
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 ishydrophobic, it will precipitate out of water. This occurs with thenickel ion Ni2+ anddimethylglyoxime, (dmgH2): in this case the lattice energy of the solid is not particularly large, but it greatly exceeds the energy ofsolvation of the molecule Ni(dmgH)2.
where is the chemical potential in the standard state,R is thegas constantT is the absolute temperature, andAj is the activity.
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:
whereaij is the number of atoms of elementi in moleculej andb0 i is the total number of atoms of elementi, which is a constant, since the system is closed. If there are a total ofk types of atoms in the system, then there will bek such equations. If ions are involved, an additional row is added to the aij matrix specifying the respective charge on each molecule which will sum to zero.
where theλi are the Lagrange multipliers, one for each element. This allows each of theNj andλj to be treated independently, and it can be shown using the tools ofmultivariate calculus that the equilibrium condition is given by
(For proof seeLagrange multipliers.) This is a set of (m + k) equations in (m + k) unknowns (theNj and theλi) and may, therefore, be solved for the equilibrium concentrationsNj as long as the chemical activities are known as functions of the concentrations at the given temperature and pressure. (In the ideal case,activities are proportional to concentrations.) (SeeThermodynamic databases for pure substances.) Note that the second equation is just the initial constraints for minimization.
This method of calculating equilibrium chemical concentrations is useful for systems with a large number of different molecules. The use ofk atomic element conservation equations for the mass constraint is straightforward, and replaces the use of the stoichiometric coefficient equations.[20] The results are consistent with those specified by chemical equations. For example, if equilibrium is specified by a single chemical equation:,[25]
where νj is the stoichiometric coefficient for thej th molecule (negative for reactants, positive for products) andRj is the symbol for thej th molecule, a properly balanced equation will obey:
Multiplying the first equilibrium condition by νj and using the above equation yields:
^Berthollet, C.L. (1803).Essai de statique chimique [Essay on chemical statics] (in French). Paris, France: Firmin Didot. On pp. 404–407, Berthellot mentions that when he accompanied Napoleon on his expedition to Egypt, he (Berthellot) visited Lake Natron and found sodium carbonate along its shores. He realized that this was a product of the reverse of the usual reaction Na2CO3 + CaCl2 → 2NaCl + CaCO3↓ and therefore that the final state of a reaction was a state of equilibrium between two opposing processes. From p. 405:" ... la décomposition du muriate de soude continue donc jusqu'à ce qu'il se soit formé assez de muriate de chaux, parce que l'acide muriatique devant se partager entre les deux bases en raison de leur action, il arrive un terme où leurs forces se balancent." ( ... thedecomposition of the sodium chloride thus continues until enough calcium chloride is formed, because the hydrochloric acid must be shared between the two bases in the ratio of their action [i.e., capacity to react]; it reaches an end [point] at which their forces are balanced.)
^The notation ⇌ was proposed in 1884 by the Dutch chemistJacobus Henricus van 't Hoff. See:van 't Hoff, J.H. (1884).Études de Dynamique Chemique [Studies of chemical dynamics] (in French). Amsterdam, Netherlands: Frederik Muller & Co. pp. 4–5. Van 't Hoff called reactions that didn't proceed to completion "limited reactions". From pp. 4–5:"Or M. Pfaundler a relié ces deux phénomênes ... s'accomplit en même temps dans deux sens opposés." (Now Mr. Pfaundler has joined these two phenomena in a single concept by considering the observed limit as the result of two opposing reactions, driving the one in the example cited to the formation of sea salt [i.e., NaCl] and nitric acid, [and] the other to hydrochloric acid and sodium nitrate. This consideration, which experiment validates, justifies the expression "chemical equilibrium", which is used to characterize the final state of limited reactions. I would propose to translate this expression by the following symbol:
HCl + NO3 Na ⇌ NO3 H + Cl Na .
I thus replace, in this case, the = sign in the chemical equation by the sign ⇌, which in reality doesn't express just equality but shows also the direction of the reaction. This clearly expresses that a chemical action occurs simultaneously in two opposing directions.)
^Brady, James E. (2004-02-04).Chemistry: Matter and Its Changes (4th ed.). Fred Senese.ISBN0-471-21517-1.
^Atkins, P.; de Paula, J.; Friedman, R. (2014).Physical Chemistry – Quanta, Matter and Change, 2nd ed., Fig. 73.2. Freeman.
^Clugston, Michael J. (1990). "A mathematical verification of the second law of thermodynamics from the entropy of mixing".Journal of Chemical Education.67 (3): 203.Bibcode:1990JChEd..67Q.203C.doi:10.1021/ed067p203.
^ Mortimer, R. G.Physical Chemistry, 3rd ed., p. 305, Academic Press, 2008.
^Davies, C. W. (1962).Ion Association. Butterworths.
^Eigen, Manfred (December 11, 1967)."Immeasurably fast reactions"(PDF).Nobel Prize.Archived(PDF) from the original on 2022-10-09. RetrievedNovember 2, 2019.
