Equilibrium constant

Types of equilibrium constants

Cumulative and stepwise formation constants

A cumulative or overall constant, given the symbol β, is the constant for the formation of a complex from reagents. For example, the cumulative constant for the formation of ML₂ is given by

M + 2 L ⇌ ML₂; [ML₂] = β₁₂[M][L]²

The stepwise constant, K, for the formation of the same complex from ML and L is given by

ML + L ⇌ ML₂; [ML₂] = K[ML][L] = Kβ₁₁[M][L]²

It follows that

β₁₂ = Kβ₁₁

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 as K^L
_ML is sometimes found in the literature. It is best always to define each stability constant by reference to an equilibrium expression.

Competition method

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.

ML + L′ ⇌ ML′ + L ${\displaystyle [{\mathrm{M}\mathrm{L}}^{\prime }]=K\frac{[\mathrm{M}\mathrm{L}][{\mathrm{L}}^{\prime }]}{[\mathrm{L}]}=K\frac{{\beta }_{\mathrm{M}\mathrm{L}}[\mathrm{M}][\mathrm{L}][{\mathrm{L}}^{\prime }]}{[\mathrm{L}]}=K{\beta }_{\mathrm{M}\mathrm{L}}[\mathrm{M}][{\mathrm{L}}^{\prime }];{\beta }_{{\mathrm{M}\mathrm{L}}^{\prime }}=K{\beta }_{\mathrm{M}\mathrm{L}}}$

The formation constant of [Pd(CN)₄]²⁻ was determined by the competition method.

Association and dissociation constants

In organic chemistry and biochemistry it is customary to use pK_a values for acid dissociation equilibria.

${\displaystyle \mathrm{p}{K}_{\mathrm{a}}=-\log K_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{s}}=\log \left(\frac{1}{K_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{s}}}\right)}$

where log denotes a logarithm to base 10 or common logarithm, and K_diss is a stepwise acid dissociation constant. For bases, the base association constant, pK_b is used. For any given acid or base the two constants are related by pK_a + pK_b = pK_w, so pK_a can always be used in calculations.

On the other hand, stability constants for metal complexes, and binding constants for host–guest complexes are generally expressed as association constants. When considering equilibria such as

M + HL ⇌ ML + H

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.

${\displaystyle \begin{array}{ll}{\text{H}}_3^{}\text{L}\stackrel{-\rightharpoonup }{\leftharpoondown -}{\text{H}}_2^{}\text{L}+\text{H};& \text{p}{K}_1=-\log \left(\frac{[{\text{H}}_2^{}\text{L}][\text{H}]}{[{\text{H}}_3^{}\text{L}]}\right)\\ {\text{H}}_2^{}\text{L}\stackrel{-\rightharpoonup }{\leftharpoondown -}\text{HL}+\text{H};& \text{p}{K}_2=-\log \left(\frac{[\text{HL}][\text{H}]}{[{\text{H}}_2^{}\text{L}]}\right)\\ \text{HL}\stackrel{-\rightharpoonup }{\leftharpoondown -}\text{L}+\text{H};& \text{p}{K}_3=-\log \left(\frac{[\text{L}][\text{H}]}{[\text{HL}]}\right)\end{array}}$

The cumulative association constants can be expressed as

${\displaystyle \begin{array}{ll}\text{L}+\text{H}\stackrel{-\rightharpoonup }{\leftharpoondown -}\text{HL};& \log {\beta }_{011}=\log \left(\frac{[\text{HL}]}{[\text{L}][\text{H}]}\right)=\text{p}{K}_3\\ \text{L}+2\text{H}\stackrel{-\rightharpoonup }{\leftharpoondown -}{\text{H}}_2^{}\text{L};& \log {\beta }_{012}=\log \left(\frac{[{\text{H}}_2^{}\text{L}]}{[\text{L}][\text{H}{]}^2}\right)=\text{p}{K}_3+\text{p}{K}_2\\ \text{L}+3\text{H}\stackrel{-\rightharpoonup }{\leftharpoondown -}{\text{H}}_3^{}\text{L};& \log {\beta }_{013}=\log \left(\frac{[{\text{H}}_3^{}\text{L}]}{[\text{L}][\text{H}{]}^3}\right)=\text{p}{K}_3+\text{p}{K}_2+\text{p}{K}_1\\ \text{M}+\text{L}\stackrel{-\rightharpoonup }{\leftharpoondown -}\text{ML};& \log {\beta }_{110}=\log \left(\frac{[\text{ML}]}{[\text{M}][\text{L}]}\right)\\ \text{M}+\text{L}+\text{H}\stackrel{-\rightharpoonup }{\leftharpoondown -}\text{MLH};& \log {\beta }_{111}=\log \left(\frac{[\text{MLH}]}{[\text{M}][\text{L}][\text{H}]}\right)\end{array}}$

Note how the subscripts define the stoichiometry of the equilibrium product.

Micro-constants

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 molecule L-DOPA has two non-equivalent hydroxyl groups which may be deprotonated. Denoting L-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 constant K₂ is for a reaction with these two micro-species as products, so that [LH] = [L¹H] + [L²H] appears in the numerator, and it follows that this macro-constant is equal to the sum of the two micro-constants for the component reactions.

K₂ = k₂₁ + k₂₂

However, the constant K₁ is for a reaction with these two micro-species as reactants, and [LH] = [L¹H] + [L²H] in the denominator, so that in this case

1/K₁ =1/ k₁₁ + 1/k₁₂,

and therefore K₁ =k₁₁ k₁₂ / (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 as infrared spectroscopy, where each micro-species gives a different signal. Methods which have been used to estimate micro-constant values include

• Chemical: blocking one of the sites, for example by methylation of a hydroxyl group, followed by determination of the equilibrium constant of the related molecule, from which the micro-constant value for the "parent" molecule may be estimated.
• Mathematical: applying numerical procedures to ¹³C NMR data.

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 for L-DOPA has been estimated to have a value of 0.9, so the micro-species L¹H and L²H have almost equal concentrations at all pH values.

pH considerations (Brønsted constants)

pH is defined in terms of the activity of the hydrogen ion

pH = −log₁₀ {H⁺}

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.

HL ⇌ L + H; ${\displaystyle \mathrm{p}K=-\log \left(\frac{[\mathrm{L}]\{\mathrm{H}\}}{[\mathrm{H}\mathrm{L}]}\right)}$

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.

Hydrolysis constants

In aqueous solution the concentration of the hydroxide ion is related to the concentration of the hydrogen ion by

${\displaystyle {\mathit{\text{K}}}_{\text{W}}^{}=[\text{H}][\text{OH}]}$

${\displaystyle [\text{OH}]={\mathit{\text{K}}}_{\text{W}}^{}{[\text{H}]}^{-1}}$

The first step in metal ion hydrolysis can be expressed in two different ways

${\displaystyle \{\begin{array}{ll}\text{M}({\text{H}}_2^{}\text{O})\stackrel{-\rightharpoonup }{\leftharpoondown -}\text{M}(\text{OH})+\text{H};& [\text{M}(\text{OH})]={\beta }^{\ast }[\text{M}][\text{H}{]}^{-1}\\ \text{M}+\text{OH}\stackrel{-\rightharpoonup }{\leftharpoondown -}\text{M}(\text{OH});& [\text{M}(\text{OH})]=K[\text{M}][\text{OH}]=K{K}_{\text{W}}[\text{M}][\text{H}{]}^{-1}\end{array}}$

It follows that β^* = KK_W. Hydrolysis constants are usually reported in the β^* form and therefore often have values much less than 1. For example, if log K = 4 and log K_W = −14, log β^* = 4 + (−14) = −10 so that β^* = 10⁻¹⁰. In general when the hydrolysis product contains n hydroxide groups log β^* = log K + n log K_W

Conditional constants

Conditional constants, also known as apparent constants, are concentration quotients which are not true equilibrium constants but can be derived from them. A very common instance is where pH is fixed at a particular value. For example, in the case of iron(III) interacting with EDTA, a conditional constant could be defined by

${\displaystyle K_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}=\frac{[\text{Total Fe bound to EDTA}]}{[\text{Total Fe not bound to EDTA}]\times [\text{Total EDTA not bound to Fe}]}}$

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 a buffer solution. Such constants are, by definition, conditional and different values may be obtained when using different buffers.

Gas-phase equilibria

For equilibria in a gas phase, fugacity, f, is used in place of activity. However, fugacity has the dimension of pressure, so it must be divided by a standard pressure, usually 1 bar, in order to produce a dimensionless quantity, ⁠f/p^o⁠. An equilibrium constant is expressed in terms of the dimensionless quantity. For example, for the equilibrium 2NO₂ ⇌ N₂O₄,

${\displaystyle \frac{f_{{\mathrm{N}}_2{\mathrm{O}}_4}}{p^{\ominus }}=K{\left(\frac{f_{\mathrm{N}{\mathrm{O}}_2}}{p^{\ominus }}\right)}^2}$

Fugacity is related to partial pressure, ${\displaystyle p_X}$, by a dimensionless fugacity coefficient ϕ: ${\displaystyle f_X={\varphi }_X{p}_X}$. Thus, for the example,

${\displaystyle K=\frac{{\varphi }_{{\mathrm{N}}_2{\mathrm{O}}_4}{p}_{{\mathrm{N}}_2{\mathrm{O}}_4}/p^{\ominus }}{{\left({\varphi }_{\mathrm{N}{\mathrm{O}}_2}{p}_{\mathrm{N}{\mathrm{O}}_2}/p^{\ominus }\right)}^2}}$

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.

${\displaystyle K=\frac{{\varphi }_{{\mathrm{N}}_2{\mathrm{O}}_4}{p}_{{\mathrm{N}}_2{\mathrm{O}}_4}}{{\left({\varphi }_{\mathrm{N}{\mathrm{O}}_2}{p}_{\mathrm{N}{\mathrm{O}}_2}\right)}^2}}$

Thermodynamic basis for equilibrium constant expressions

Thermodynamic equilibrium is characterized by the free energy for the whole (closed) system being a minimum. For systems at constant temperature and pressure the Gibbs free energy is minimum. The slope of the reaction free energy with respect to the extent of reaction, ξ, is zero when the free energy is at its minimum value.

${\displaystyle {\left(\frac{\mathrm{\partial }G}{\mathrm{\partial }\xi }\right)}_{T,P}=0}$

The free energy change, dG_r, can be expressed as a weighted sum of change in amount times the chemical potential, the partial molar free energy of the species. The chemical potential, μ_i, of the ith 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

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

where n_j are the stoichiometric coefficients of the reactants in the equilibrium equation, and m_j are the coefficients of the products. At equilibrium

${\displaystyle \sum _k{m}_k{\mu }_k=\sum _j{n}_j{\mu }_j}$

The chemical potential, μ_i, of the ith species can be calculated in terms of its activity, 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 the gas constant and T is the temperature. Setting the sum for the reactants j to be equal to the sum for the products, k, so that δG_r(Eq) = 0

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

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 the standard Gibbs free energy change, ΔG^o to an equilibrium constant, K, the reaction quotient of activity values at equilibrium.

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

${\displaystyle \ln K=\sum _k\ln {a_k}^{m_k}-\sum _j\ln {a_j}^{n_j};K=\frac{\prod _k{a_k}^{m_k}}{\prod _j{a_j}^{n_j}}\equiv \frac{{\{\mathrm{R}\}}^{\rho }{\{\mathrm{S}\}}^{\sigma }...}{{\{\mathrm{A}\}}^{\alpha }{\{\mathrm{B}\}}^{\beta }...}}$

Equivalence of thermodynamic and kinetic expressions for equilibrium constants

At equilibrium the rate of the forward reaction is equal to the backward reaction rate. A simple reaction, such as ester hydrolysis

${\displaystyle \text{AB}+{\text{H}}_2^{}\text{O}\stackrel{-\rightharpoonup }{\leftharpoondown -}\text{AH}+\text{B}(\text{OH})}$

has reaction rates given by expressions

${\displaystyle \text{forward rate}=k_f[\text{AB}][{\text{H}}_2^{}\text{O}]}$

${\displaystyle \text{backward rate}=k_b[\text{AH}][\text{B}(\text{OH})]}$

According to Guldberg and Waage, 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

${\displaystyle K=\frac{k_f}{k_b}=\frac{[\text{AH}][\text{B}(\text{OH})]}{[\text{AB}][{\text{H}}_2^{}\text{O}]}}$.

The concentration of water may be taken to be constant, resulting in the simpler expression

${\displaystyle K^c=\frac{[\text{AH}][\text{B}(\text{OH})]}{[\text{AB}]}}$.

This particular concentration quotient, ${\displaystyle K^c}$, has the dimension of concentration, but the thermodynamic equilibrium constant, K, is always dimensionless.

Unknown activity coefficient values

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.

1. Use calculated activity coefficients, together with concentrations of reactants. For equilibria in solution estimates of the activity coefficients of charged species can be obtained using Debye–Hückel theory, an extended version, or SIT theory. For uncharged species, the activity coefficient γ₀ mostly follows a "salting-out" model: log₁₀ γ₀ = bI where I stands for ionic strength.
2. Assume that the activity coefficients are all equal to 1. This is acceptable when all concentrations are very low.
3. For equilibria in solution use a medium of high ionic strength. In effect this redefines the standard state as referring to the medium. Activity coefficients in the standard state are, by definition, equal to 1. The value of an equilibrium constant determined in this manner is dependent on the ionic strength. When published constants refer to an ionic strength other than the one required for a particular application, they may be adjusted by means of specific ion theory (SIT) and other theories.

Dimensionality

An equilibrium constant is related to the standard Gibbs free energy of reaction change, ${\displaystyle {\mathrm{\Delta }}_R{G}^{\ominus }}$, for the reaction by the expression

${\displaystyle {\mathrm{\Delta }}_R{G}^{\ominus }={\left(\frac{\mathrm{\partial }G}{\mathrm{\partial }\xi }\right)}_{P,T}=-RT\ln K.}$

Therefore, K, must be a dimensionless number from which a logarithm can be derived. In the case of a simple equilibrium

${\displaystyle \text{A}+\text{B}\stackrel{-\rightharpoonup }{\leftharpoondown -}\text{AB},}$

the thermodynamic equilibrium constant is defined in terms of the activities, {AB}, {A} and {B}, of the species in equilibrium with each other:

${\displaystyle K=\frac{\{AB\}}{\{A\}\{B\}}.}$

Now, each activity term can be expressed as a product of a concentration ${\displaystyle [X]}$ and a corresponding activity coefficient, ${\displaystyle \gamma (X)}$. Therefore,

${\displaystyle K=\frac{[AB]}{[A][B]}\times \frac{\gamma (AB)}{\gamma (A)\gamma (B)}=\frac{[AB]}{[A][B]}\times \mathrm{\Gamma }.}$

When ${\displaystyle \mathrm{\Gamma }}$, the quotient of activity coefficients, is set equal to 1, we get

${\displaystyle K=\frac{[AB]}{[A][B]}.}$

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.

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 high ionic 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 the mole fraction scale all concentrations and activity coefficients are dimensionless quantities.

In general equilibria between two reagents can be expressed as

${\displaystyle \mathit{\text{p}}\text{A}+\mathit{\text{q}}\text{B}\stackrel{-\rightharpoonup }{\leftharpoondown -}{\text{A}}_{\mathit{\text{p}}}^{}{\text{B}}_{\mathit{\text{q}}}^{},}$

in which case the equilibrium constant is defined, in terms of numerical concentration values, as

${\displaystyle K=\frac{[{\text{A}}_p{\text{B}}_q]}{[\text{A}{]}^p[\text{B}{]}^q}.}$

The apparent dimension of this K value is concentration^1−p−q; this may be written as M^(1−p−q) or mM^(1−p−q), where the symbol M signifies a molar concentration (1M = 1 mol dm⁻³). The apparent dimension of a dissociation constant is the reciprocal of the apparent dimension of the corresponding association constant, and vice versa.

When discussing the thermodynamics of chemical equilibria it is necessary to take dimensionality into account. There are two possible approaches.

1. Set the dimension of Γ to be the reciprocal of the dimension of the concentration quotient. This is almost universal practice in the field of stability constant determinations. The "equilibrium constant" ${\displaystyle \frac{K}{\mathrm{\Gamma }}}$, is dimensionless. It will be a function of the ionic strength of the medium used for the determination. Setting the numerical value of Γ to be 1 is equivalent to re-defining the standard states.
2. Replace each concentration term ${\displaystyle [X]}$ by the dimensionless quotient ${\displaystyle \frac{[X]}{[X^0]}}$, where ${\displaystyle [X^0]}$ is the concentration of reagent X in its standard state (usually 1 mol/L or 1 bar). By definition the numerical value of ${\displaystyle \gamma (X^0)}$ is 1, so Γ also has a numerical value of 1.

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 of Debye–Hückel theory, where ${\displaystyle \gamma (AB)}$, etc. are taken to be pure numbers.

Water as both reactant and solvent

For reactions in aqueous solution, such as an acid dissociation reaction

AH + H₂O ⇌ A⁻ + H₃O⁺

the concentration of water may be taken as being constant and the formation of the hydronium ion is implicit.

AH ⇌ A⁻ + H⁺

Water concentration is omitted from expressions defining equilibrium constants, except when solutions are very concentrated.

${\displaystyle K=\frac{[A][H]}{[AH]}}$ (K defined as a dissociation constant)

Similar considerations apply to metal ion hydrolysis reactions.

Enthalpy and entropy: temperature dependence

If both the equilibrium constant, ${\displaystyle K}$ and the standard enthalpy change, ${\displaystyle \mathrm{\Delta }{H}^{\ominus }}$, for a reaction have been determined experimentally, the standard entropy change for the reaction is easily derived. Since ${\displaystyle \mathrm{\Delta }G=\mathrm{\Delta }H-T\mathrm{\Delta }S}$ and ${\displaystyle \mathrm{\Delta }G=-RT\ln K}$

${\displaystyle \mathrm{\Delta }{S}^{\ominus }=\frac{\mathrm{\Delta }{H}^{\ominus }+RT\ln K}{T}}$

To a first approximation the standard enthalpy change is independent of temperature. Using this approximation, definite integration of the van 't Hoff equation

${\displaystyle \mathrm{\Delta }{H}^{\ominus }=-R\frac{d\ln K}{d(1/T)}}$

gives

${\displaystyle \ln K_2=\ln K_1-\frac{\mathrm{\Delta }{H}^{\ominus }}{R}\left(\frac{1}{T_2}-\frac{1}{T_1}\right)}$

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 an exothermic reaction (${\displaystyle \mathrm{\Delta }H<0}$), when temperature increases K decreases and when temperature decreases K increases, in accordance with Le Chatelier's principle. The reverse applies when the reaction is endothermic.

When K has been determined at more than two temperatures, a straight line fitting procedure may be applied to a plot of ${\displaystyle \ln K}$ against ${\displaystyle 1/T}$ to obtain a value for ${\displaystyle \mathrm{\Delta }{H}^{\ominus }}$. Error propagation theory can be used to show that, with this procedure, the error on the calculated ${\displaystyle \mathrm{\Delta }{H}^{\ominus }}$ 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 silver ion-selective electrode each log K value was determined with a precision of ca. 0.001 and the method was applied successfully.

Standard thermodynamic arguments can be used to show that, more generally, enthalpy will change with temperature.

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

where C_p is the heat capacity at constant pressure.

A more complex formulation

The calculation of K at a particular temperature from a known K 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:

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

where Δ_rG^o 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.

The standard Gibbs energy (for each species or for the entire reaction) can be represented (from the basic definitions) as:

${\displaystyle G_{T_2}^{\ominus }=G_{T_1}^{\ominus }-S_{T_1}^{\ominus }(T_2-T_1)-T_2{\int }_{T_1}^{T_2}\frac{C_p^{\ominus }}{T}dT+{\int }_{T_1}^{T_2}{C}_p^{\ominus }dT}$

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 capacity C^o
_p can be approximated by some analytic function of temperature (e.g. the Shomate 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:

• For pure substances (solids, gas, liquid): $${\displaystyle C_p^{\ominus }\approx A+BT+C{T}^{-2}}$$
• For ionic species at T < 200 °C: $${\displaystyle C_p^{\ominus }\approx (4.186a+b{\breve{S}}_{T_1}^{\ominus })\frac{(T_2-T_1)}{\ln \left(\frac{T_2}{T_1}\right)}}$$

then the integrals can be evaluated and the following final form is obtained:

${\displaystyle G_{T_2}^{\ominus }\approx G_{T_1}^{\ominus }+(C_p^{\ominus }-S_{T_1}^{\ominus })(T_2-T_1)-T_2\ln \left(\frac{T_2}{T_1}\right)C_p^{\ominus }}$

The constants A, B, C, a, b and the absolute entropy, S̆ ^o
_{298 K}, required for evaluation of C^o
_p(T), as well as the values of G_{298 K} and S_{298 K} for many species are tabulated in the literature.

Pressure dependence

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 for condensed 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:

N₂ + 3 H₂ ⇌ 2 NH₃

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:

• the pressure is changed by compression or expansion of the gaseous reacting system, and
• the reaction results in the change of the number of moles of gas in the system.

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:

α A + β B ⇌ σ S + τ T

the change of mixture composition with pressure can be quantified using:

${\displaystyle K_p=\frac{{p_{\mathrm{S}}}^{\sigma }{p_{\mathrm{T}}}^{\tau }}{{p_{\mathrm{A}}}^{\alpha }{p_{\mathrm{B}}}^{\beta }}=\frac{{X_{\mathrm{S}}}^{\sigma }{X_{\mathrm{T}}}^{\tau }}{{X_{\mathrm{A}}}^{\alpha }{X_{\mathrm{B}}}^{\beta }}{P}^{\sigma +\tau -\alpha -\beta }=K_X{P}^{\sigma +\tau -\alpha -\beta }}$

where p denote the partial pressures and X the mole fractions of the components, P is the total system pressure, K_p is the equilibrium constant expressed in terms of partial pressures and K_X is the equilibrium constant expressed in terms of mole fractions.

The above change in composition is in accordance with Le Chatelier's principle and does not involve any change of the equilibrium constant with the total system pressure. Indeed, for ideal-gas reactions K_p is independent of pressure.

In a condensed phase, the pressure dependence of the equilibrium constant is associated with the reaction volume. For reaction:

α A + β B ⇌ σ S + τ T

the reaction volume is:

${\displaystyle \mathrm{\Delta }\overline{V}=\sigma {\overline{V}}_{\mathrm{S}}+\tau {\overline{V}}_{\mathrm{T}}-\alpha {\overline{V}}_{\mathrm{A}}-\beta {\overline{V}}_{\mathrm{B}}}$

where V̄ denotes a partial 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:

${\displaystyle {\left(\frac{\mathrm{\partial }\ln K_X}{\mathrm{\partial }P}\right)}_T=\frac{-\mathrm{\Delta }\overline{V}}{RT}.}$

The matter is complicated as partial molar volume is itself dependent on pressure.

Effect of isotopic substitution

Isotopic substitution can lead to changes in the values of equilibrium constants, especially if hydrogen is replaced by deuterium (or tritium). This equilibrium isotope effect is analogous to the kinetic isotope effect on rate constants, and is primarily due to the change in zero-point vibrational energy of H–X bonds due to the change in mass upon isotopic substitution. 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 a hydrogen atom abstraction reaction R' + H–R ⇌ R'–H + R with equilibrium constant K_H, 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 K_D for R' + D–R ⇌ R'–D + R is greater than K_H. This is summarized in the rule the heavier atom favors the stronger bond.

Similar effects occur in solution for acid dissociation constants (K_a) 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 in heavy water, since if it were dissolved in ordinary water the deuterium would rapidly exchange with hydrogen in the solvent.

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 K_D < K_H, and the deuterated acid in D₂O is weaker than the non-deuterated acid in H₂O. In many cases the difference of logarithmic constants pK_D – pK_H is about 0.6, 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 the self-ionization of heavy water is less than that of ordinary water at the same temperature.