Equilibrium constant

The equilibrium constant of a chemical reaction is the value of its reaction quotient at chemical 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 the composition of the system at equilibrium. However, reaction parameters like temperature, solvent, and ionic 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 by hemoglobin in blood and acid–base homeostasis in the human body.
Stability constants, formation constants, binding constants, association constants and dissociation constants are all types of equilibrium constants.

Basic definitions and properties

For a system undergoing a reversible reaction described by the general chemical equation
a thermodynamic equilibrium constant, denoted by, is defined to be the value of the reaction quotient Q_t when forward and reverse reactions occur at the same rate. At chemical equilibrium, the chemical composition of the mixture does not change with time, and the Gibbs 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
where denotes the thermodynamic activity of reagent X at equilibrium, the numerical value of the corresponding concentration in moles per liter, and γ the corresponding activity coefficient. If X is a gas, instead of the numerical value of the partial pressure in bar is used. 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 standard Gibbs free energy change of reaction by
where R is the universal gas constant, T is the absolute temperature, and is the natural logarithm. This expression implies that must be a pure number and cannot have a dimension, since logarithms can only be taken of pure numbers. must also be a pure number. On the other hand, the reaction quotient at equilibrium

does have the dimension of concentration raised to some power. 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 of partial pressure or fugacity.
An equilibrium constant is related to the forward and backward rate constants, k_f and k_r of the elementary reactions involved in reaching equilibrium:

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
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 as K 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.
The formation constant of Palladium cyanide|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.
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, 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
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, N₃ 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.

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 -DOPA has two non-equivalent hydroxyl groups which may be deprotonated. Denoting -DOPA as LH₂, the following diagram shows all the species that may be formed.
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 = + 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.
However, the constant K₁ is for a reaction with these two micro-species as reactants, and = + in the denominator, so that in this case
and therefore K₁ =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 -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)

is defined in terms of the activity 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 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
The first step in metal ion hydrolysis can be expressed in two different ways
It follows that. Hydrolysis constants are usually reported in the β^* form and therefore often have values much less than 1. For example, if and so that β^* = 10⁻¹⁰. In general when the hydrolysis product contains n hydroxide groups

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