Theequilibrium constant of a chemical reaction is the value of itsreaction quotient atchemical equilibrium, a state approached by a dynamic chemical system after sufficient time has elapsed at which its composition has no measurable tendency towards further change. For a given set of reaction conditions, the equilibrium constant is independent of the initial analytical concentrations of the reactant and product species in the mixture. Thus, given the initial composition of a system, known equilibrium constant values can be used to determine thecomposition of the system at equilibrium. However, reaction parameters like temperature, solvent, andionic strength may all influence the value of the equilibrium constant.
A knowledge of equilibrium constants is essential for the understanding of many chemical systems, as well as the biochemical processes such as oxygen transport byhemoglobin in blood andacid–base homeostasis in the human body.
Stability constants, formation constants,binding constants, association constants anddissociation constants are all types ofequilibrium constants.
For a system undergoing areversible reaction described by the generalchemical equation
a thermodynamic equilibrium constant, denoted by, is defined to be the value of thereaction quotientQt when forward and reverse reactions occur at the same rate. Atchemical equilibrium, the chemical composition of the mixture does not change with time, and theGibbs free energy change for the reaction is zero. If the composition of a mixture at equilibrium is changed by addition of some reagent, a new equilibrium position will be reached, given enough time. An equilibrium constant is related to the composition of the mixture at equilibrium by[1][2]
where {X} denotes thethermodynamic activity of reagent X at equilibrium, [X] the numerical value[3] of the correspondingconcentration in moles per liter, and γ the correspondingactivity coefficient. If X is a gas, instead of [X] the numerical value of thepartial pressure in bar is used.[3] If it can be assumed that the quotient of activity coefficients,, is constant over a range of experimental conditions, such as pH, then an equilibrium constant can be derived as a quotient of concentrations.
An equilibrium constant is related to the standardGibbs free energy change of reaction by
whereR is theuniversal gas constant,T is theabsolute temperature (inkelvins), andln is thenatural logarithm. This expression implies that must be a pure number and cannot have a dimension, sincelogarithms can only be taken of pure numbers. must also be a pure number. On the other hand, thereaction quotient at equilibrium
does have the dimension of concentration raised to some power (see§ Dimensionality, below). Such reaction quotients are often referred to, in the biochemical literature, as equilibrium constants.
For an equilibrium mixture of gases, an equilibrium constant can be defined in terms ofpartial pressure orfugacity.
An equilibrium constant is related to the forward and backwardrate constants,kf andkr of the reactions involved in reaching equilibrium:
A cumulative or overall constant, given the symbolβ, is the constant for the formation of a complex from reagents. For example, the cumulative constant for the formation of ML2 is given by
The stepwise constant,K, for the formation of the same complex from ML and L is given by
It follows that
A cumulative constant can always be expressed as the product of stepwise constants. There is no agreed notation for stepwise constants, though a symbol such asKL
ML is sometimes found in the literature. It is best always to define each stability constant by reference to an equilibrium expression.
A particular use of a stepwise constant is in the determination of stability constant values outside the normal range for a given method. For example,EDTA complexes of many metals are outside the range for the potentiometric method. The stability constants for those complexes were determined by competition with a weaker ligand.
The formation constant of[Pd(CN)4]2− was determined by the competition method.
In organic chemistry and biochemistry it is customary to use pKa values foracid dissociation equilibria.
wherelog denotes a logarithm to base 10 orcommon logarithm, andKdiss is a stepwiseacid dissociation constant. For bases, thebase association constant, pKb is used. For any given acid or base the two constants are related bypKa + pKb = pKw, so pKa can always be used in calculations.
On the other hand, stability constants formetal complexes, and binding constants forhost–guest complexes are generally expressed as association constants. When considering equilibria such as
it is customary to use association constants for both ML and HL. Also, in generalized computer programs dealing with equilibrium constants it is general practice to use cumulative constants rather than stepwise constants and to omit ionic charges from equilibrium expressions. For example, if NTA,nitrilotriacetic acid, N(CH2CO2H)3 is designated as H3L and forms complexes ML and MHL with a metal ion M, the following expressions would apply for the dissociation constants.
The cumulative association constants can be expressed as
Note how the subscripts define the stoichiometry of the equilibrium product.
When two or more sites in an asymmetrical molecule may be involved in an equilibrium reaction there are more than one possible equilibrium constants. For example, the moleculeL-DOPA has two non-equivalent hydroxyl groups which may be deprotonated. DenotingL-DOPA as LH2, the following diagram shows all the species that may be formed (X =CH
2CH(NH
2)CO
2H).
The concentration of the species LH is equal to the sum of the concentrations of the two micro-species with the same chemical formula, labelled L1H and L2H. The constantK2 is for a reaction with these two micro-species as products, so that [LH] = [L1H] + [L2H] appears in the numerator, and it follows that thismacro-constant is equal to the sum of the twomicro-constants for the component reactions.
However, the constantK1 is for a reaction with these two micro-species as reactants, and [LH] = [L1H] + [L2H] in the denominator, so that in this case[4]
and thereforeK1 =k11k12 / (k11 +k12).Thus, in this example there are four micro-constants whose values are subject to two constraints; in consequence, only the two macro-constant values, for K1 and K2 can be derived from experimental data.
Micro-constant values can, in principle, be determined using a spectroscopic technique, such asinfrared spectroscopy, where each micro-species gives a different signal. Methods which have been used to estimate micro-constant values include
Although the value of a micro-constant cannot be determined from experimental data, site occupancy, which is proportional to the micro-constant value, can be very important for biological activity. Therefore, various methods have been developed for estimating micro-constant values. For example, the isomerization constant forL-DOPA has been estimated to have a value of 0.9, so the micro-species L1H and L2H have almost equal concentrations at allpH values.
pH is defined in terms of theactivity of the hydrogen ion
In the approximation of ideal behaviour, activity is replaced by concentration. pH is measured by means of a glass electrode, a mixed equilibrium constant, also known as a Brønsted constant, may result.
It all depends on whether the electrode is calibrated by reference to solutions of known activity or known concentration. In the latter case the equilibrium constant would be a concentration quotient. If the electrode is calibrated in terms of known hydrogen ion concentrations it would be better to write p[H] rather than pH, but this suggestion is not generally adopted.
In aqueous solution the concentration of the hydroxide ion is related to the concentration of the hydrogen ion by
The first step in metal ionhydrolysis[7] can be expressed in two different ways
It follows thatβ* =KKW. Hydrolysis constants are usually reported in theβ* form and therefore often have values much less than 1. For example, iflogK = 4 andlog KW = −14,logβ* = 4 + (−14) = −10 so thatβ* = 10−10. In general when the hydrolysis product containsn hydroxide groupslogβ* = logK +n logKW
Conditional constants, also known as apparent constants, are concentration quotients which are not true equilibrium constants but can be derived from them.[8] A very common instance is where pH is fixed at a particular value. For example, in the case ofiron(III) interacting withEDTA, a conditional constant could be defined by
This conditional constant will vary with pH. It has a maximum at a certain pH. That is the pH where the ligand sequesters the metal most effectively.
In biochemistry equilibrium constants are often measured at a pH fixed by means of abuffer solution. Such constants are, by definition, conditional and different values may be obtained when using different buffers.
For equilibria in agas phase,fugacity,f, is used in place of activity. However, fugacity has thedimension ofpressure, so it must be divided by a standard pressure, usually 1 bar, in order to produce a dimensionless quantity,f/po. An equilibrium constant is expressed in terms of the dimensionless quantity. For example, for the equilibrium 2NO2 ⇌ N2O4,
Fugacity is related topartial pressure,, by a dimensionless fugacity coefficientϕ:. Thus, for the example,
Usually the standard pressure is omitted from such expressions. Expressions for equilibrium constants in the gas phase then resemble the expression for solution equilibria with fugacity coefficient in place of activity coefficient and partial pressure in place of concentration.
Thermodynamic equilibrium is characterized by the free energy for the whole (closed) system being a minimum. For systems at constant temperature and pressure theGibbs free energy is minimum.[9] The slope of the reaction free energy with respect to theextent of reaction,ξ, is zero when the free energy is at its minimum value.
The free energy change, dGr, can be expressed as a weighted sum of change in amount times thechemical potential, the partial molar free energy of the species. The chemical potential,μi, 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,Ni
A general chemical equilibrium can be written as
wherenj are thestoichiometric coefficients of the reactants in the equilibrium equation, andmj are the coefficients of the products. At equilibrium
The chemical potential,μi, of theith species can be calculated in terms of itsactivity,ai.
μo
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 thatδGr(Eq) = 0
Rearranging the terms,
This relates thestandard Gibbs free energy change, ΔGo to an equilibrium constant,K, thereaction quotient of activity values at equilibrium.
At equilibrium the rate of the forward reaction is equal to the backward reaction rate. A simple reaction, such asester hydrolysis
has reaction rates given by expressions
According toGuldberg andWaage, equilibrium is attained when the forward and backward reaction rates are equal to each other. In these circumstances, an equilibrium constant is defined to be equal to the ratio of the forward and backward reaction rate constants
The concentration of water may be taken to be constant, resulting in the simpler expression
This particular concentration quotient,, has the dimension of concentration, but the thermodynamic equilibrium constant,K, is always dimensionless.
It is very rare for activity coefficient values to have been determined experimentally for a system at equilibrium. There are three options for dealing with the situation where activity coefficient values are not known from experimental measurements.
An equilibrium constant is related to the standardGibbs free energy of reaction change,,[12] for the reaction by the expression
Therefore,K, must be adimensionless number from which a logarithm can be derived. In the case of a simple equilibrium
the thermodynamic equilibrium constant is defined in terms of theactivities, {AB}, {A} and {B}, of the species in equilibrium with each other:
Now, each activity term can be expressed as a product of a concentration and a correspondingactivity coefficient,. Therefore,
When, the quotient of activity coefficients, is set equal to 1, we get
K then appears to have the dimension of 1/concentration. This is what usually happens in practice when an equilibrium constant is calculated as a quotient of concentration values. This can be avoided by dividing each concentration by its standard-state value (usually mol/L or bar), which is standard practice in chemistry.[3]
The assumption underlying this practice is that the quotient of activities is constant under the conditions in which the equilibrium constant value is determined. These conditions are usually achieved by keeping the reaction temperature constant and by using a medium of relatively highionic strength as the solvent. It is not unusual, particularly in texts relating to biochemical equilibria, to see an equilibrium constant value quoted with a dimension. The justification for this practice is that the concentration scale used may be either mol dm−3 or mmol dm−3, so that the concentration unit has to be stated in order to avoid there being any ambiguity.
Note. When the concentration values are measured on themole fraction scale all concentrations and activity coefficients are dimensionless quantities.
In general equilibria between two reagents can be expressed as
in which case the equilibrium constant is defined, in terms of numerical concentration values, as
The apparent dimension of thisK value is concentration1−p−q; this may be written as M(1−p−q) or mM(1−p−q), where the symbol M signifies amolar concentration (1M = 1 mol dm−3). The apparent dimension of adissociation constant is the reciprocal of the apparent dimension of the correspondingassociation constant, andvice versa.
When discussing thethermodynamics of chemical equilibria it is necessary to take dimensionality into account. There are two possible approaches.
In both approaches the numerical value of the stability constant is unchanged. The first is more useful for practical purposes; in fact, the unit of the concentration quotient is often attached to a published stability constant value in the biochemical literature. The second approach is consistent with the standard exposition ofDebye–Hückel theory, where,etc. are taken to be pure numbers.
For reactions in aqueous solution, such as an acid dissociation reaction
the concentration of water may be taken as being constant and the formation of thehydronium ion is implicit.
Water concentration is omitted from expressions defining equilibrium constants, except when solutions are very concentrated.
Similar considerations apply tometal ion hydrolysis reactions.
If both the equilibrium constant, and thestandard enthalpy change,, for a reaction have been determined experimentally, thestandard entropy change for the reaction is easily derived. Since and
To a first approximation the standard enthalpy change is independent of temperature. Using this approximation,definite integration of thevan 't Hoff equation
gives[13]
This equation can be used to calculate the value of log K at a temperature, T2, knowing the value at temperature T1.
The van 't Hoff equation also shows that, for anexothermic reaction (), when temperature increasesK decreases and when temperature decreasesK increases, in accordance withLe Chatelier's principle. The reverse applies when the reaction isendothermic.
WhenK has been determined at more than two temperatures, a straight line fitting procedure may be applied to a plot of against to obtain a value for.Error propagation theory can be used to show that, with this procedure, the error on the calculated value is much greater than the error on individual log K values. Consequently, K needs to be determined to high precision when using this method. For example, with a silverion-selective electrode each log K value was determined with a precision of ca. 0.001 and the method was applied successfully.[14]
Standard thermodynamic arguments can be used to show that, more generally, enthalpy will change with temperature.
whereCp is theheat capacity at constant pressure.
The calculation ofK at a particular temperature from a knownK at another given temperature can be approached as follows if standard thermodynamic properties are available. The effect of temperature on equilibrium constant is equivalent to the effect of temperature on Gibbs energy because:
where ΔrGo is the reaction standard Gibbs energy, which is the sum of the standard Gibbs energies of the reaction products minus the sum of standard Gibbs energies of reactants.
Here, the term "standard" denotes the ideal behaviour (i.e., an infinite dilution) and a hypothetical standard concentration (typically 1 mol/kg). It does not imply any particular temperature or pressure because, although contrary to IUPAC recommendation, it is more convenient when describing aqueous systems over wide temperature and pressure ranges.[15]
The standard Gibbs energy (for each species or for the entire reaction) can be represented (from the basic definitions) as:
In the above equation, the effect of temperature on Gibbs energy (and thus on the equilibrium constant) is ascribed entirely to heat capacity. To evaluate the integrals in this equation, the form of the dependence of heat capacity on temperature needs to be known.
If the standard molar heat capacityCo
p can be approximated by some analytic function of temperature (e.g. theShomate equation), then the integrals involved in calculating other parameters may be solved to yield analytic expressions for them. For example, using approximations of the following forms:[16]
then the integrals can be evaluated and the following final form is obtained:
The constantsA,B,C,a,b and the absolute entropy,S̆ o
298 K, required for evaluation ofCo
p(T), as well as the values ofG298 K andS298 K for many species are tabulated in the literature.
The pressure dependence of the equilibrium constant is usually weak in the range of pressures normally encountered in industry, and therefore, it is usually neglected in practice. This is true forcondensed reactant/products (i.e., when reactants and products are solids or liquid) as well as gaseous ones.
For a gaseous-reaction example, one may consider the well-studied reaction of hydrogen with nitrogen to produce ammonia:
If the pressure is increased by the addition of an inert gas, then neither the composition at equilibrium nor the equilibrium constant are appreciably affected (because the partial pressures remain constant, assuming an ideal-gas behaviour of all gases involved). However, the composition at equilibrium will depend appreciably on pressure when:
In the example reaction above, the number of moles changes from 4 to 2, and an increase of pressure by system compression will result in appreciably more ammonia in the equilibrium mixture. In the general case of a gaseous reaction:
the change of mixture composition with pressure can be quantified using:
wherep denote the partial pressures andX the mole fractions of the components,P is the total system pressure,Kp is the equilibrium constant expressed in terms of partial pressures andKX is the equilibrium constant expressed in terms of mole fractions.
The above change in composition is in accordance withLe Chatelier's principle and does not involve any change of the equilibrium constant with the total system pressure. Indeed, for ideal-gas reactionsKp is independent of pressure.[17]
In a condensed phase, the pressure dependence of the equilibrium constant is associated with the reaction volume.[18] For reaction:
the reaction volume is:
whereV̄ denotes apartial molar volume of a reactant or a product.
For the above reaction, one can expect the change of the reaction equilibrium constant (based either on mole-fraction or molal-concentration scale) with pressure at constant temperature to be:
The matter is complicated as partial molar volume is itself dependent on pressure.
Isotopic substitution can lead to changes in the values of equilibrium constants, especially if hydrogen is replaced bydeuterium (ortritium).[19] Thisequilibrium isotope effect is analogous to thekinetic isotope effect on rate constants, and is primarily due to the change inzero-point vibrational energy of H–X bonds due to the change in mass upon isotopic substitution.[19] The zero-point energy is inversely proportional to the square root of the mass of the vibrating hydrogen atom, and will therefore be smaller for a D–X bond that for an H–X bond.
An example is ahydrogen atom abstraction reaction R' + H–R ⇌ R'–H + R with equilibrium constant KH, where R' and R are organic radicals such that R' forms a stronger bond to hydrogen than does R. The decrease in zero-point energy due to deuterium substitution will then be more important for R'–H than for R–H, and R'–D will be stabilized more than R–D, so that the equilibrium constant KD for R' + D–R ⇌ R'–D + R is greater than KH. This is summarized in the rulethe heavier atom favors the stronger bond.[19]
Similar effects occur in solution foracid dissociation constants (Ka) which describe the transfer of H+ or D+ from a weak aqueous acid to a solvent molecule: HA + H2O = H3O+ + A− or DA + D2O ⇌ D3O+ + A−. The deuterated acid is studied inheavy water, since if it were dissolved in ordinary water the deuterium would rapidly exchange with hydrogen in the solvent.[19]
The product species H3O+ (or D3O+) is a stronger acid than the solute acid, so that it dissociates more easily, and its H–O (or D–O) bond is weaker than the H–A (or D–A) bond of the solute acid. The decrease in zero-point energy due to isotopic substitution is therefore less important in D3O+ than in DA so that KD < KH, and the deuterated acid in D2O is weaker than the non-deuterated acid in H2O. In many cases the difference of logarithmic constants pKD – pKH is about 0.6,[19] so that the pD corresponding to 50% dissociation of the deuterated acid is about 0.6 units higher than the pH for 50% dissociation of the non-deuterated acid.
For similar reasons theself-ionization of heavy water is less than that of ordinary water at the same temperature.
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