EQUILIBRIUM CONSTANT

Stability constants, formation constants, binding constants, association constants and dissociation constants are all types of 'equilibrium constant'. See also Determination of equilibrium constants for experimental and computational methods.
For a general chemical reaction
:αA + βB … $$
ightleftharpoons σS + τT …
the equilibrium constant can be defined by^[1]
:
where {A} is the activity of the chemical species A etc (activity is a dimensionless quantity). It is conventional to put the activities of the products in the numerator and those of the reactants in the denominator. See Chemical equilibrium for a derivation of this expression.
For equilibria in a gas phase, the activity of a gaseous component is the product of the component's partial pressure (made dimensionless with the utility of standard pressure) and the fugacity coefficient for this component.
For equilibria in solution activity is the product of concentration and activity coefficient. It is common practice to determine equilibrium constants in a medium of high ionic strength. In those circumstances the quotient of activity coefficients is effectively constant and the equilibrium constant is taken to be a concentration quotient.
:
However, the value of K_c will depend on the ionic strength.
All equilibrium constants depend on temperature and pressure (or volume).
A knowledge of equilibrium constants is essential for the understanding of many natural processes such as oxygen transport by haemoglobin in blood and acid-base homeostasis in the human body.

Contents

Types of equilibrium constants

Association and dissociation constants

Stepwise formation constants

Competition method

Micro-constants

pH considerations (Brønsted constants)

Conditional constants

Temperature dependence

Data sources

References

Types of equilibrium constants

Association and dissociation constants

In organic chemistry and biochemistry it is customary to use pK_a values for acid dissociation equilibria.
:$pK\_a=-lg\; K\_\{diss\}\; =\; lg\; (1/K\_\{diss\})$
where ''K''_diss is a stepwise acid dissociation constant (lg stands for log₁₀). 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
:$M\; +\; HL$
ightleftharpoons ML + H
it is customary to use association constants for both ML and HL. Also, in generalised computer programs dealing with equilibrium constants it is general practice to use overall constants rather than stepwise constants and to omit ionic charges from equilibrium expressions. For example, if NTA, nitrilotriacetic acid, HC(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.
:$H\_3L$
ightleftharpoons H_2L+H:pK_1=-lg left(rac{[H_2L][H]} {[H_3L]}
ight)
:$H\_2L$
ightleftharpoons HL+H:pK_2=-lg left(rac{[HL][H]} {[H_2L]}
ight)
:$HL$
ightleftharpoons L+H:pK_3=-lg left(rac{[L][H]} {[HL]}
ight)
The overall association constants can be expressed as
:$L+H$
ightleftharpoons HL:lg eta_{011} =lg left(rac{[HL]}{[L][H]}
ight)=pK_3
:$L+2H$
ightleftharpoons H_2L:lg eta_{012} =lg left(rac{[H_2L]}{[L][H]^2}
ight)=pK_3+pK_2
:$L+3H$
ightleftharpoons H_3L:lg eta_{013} =lg left(rac{[H_3L]}{[L][H]^3}
ight)=pK_3+pK_2+pK_1
:$M+L$
ightleftharpoons ML:lg eta_{110} =lg left(rac{[ML]}{[M][L]}
ight)
:$M+L+H$
ightleftharpoons MLH:lg eta_{111} =lg left(rac{[MLH]}{[M][L][H]}
ight)
Note how the subscripts define the stoichiometry of the equilibrium product.

Stepwise formation constants

The stepwise constant for protonation of ML can be easily derived as follows.
:$ML+H$
ightleftharpoons MLH: [MLH]=K[ML][H]=K eta_{110}[M]
The first protonation constants are
:[L¹H] = k₁₁[L][H], [L²H] = k₁₂[L][H]
The concentration of LH^- is the sum of the concentrations of the two micro-species. Therefore, the equilibrium constant for the reaction, the 'macro-constant', is the sum of the 'micro-constants'.
:K₁ = k₁₁ + k₁₂
In the same way,
:K₂ = k₂₁ + k₂₂
Lastly, the overall constant is
:β₂=K₁K₂=k₁₁k₂₁=k₁₂k₂₂
Thus, althought there are six micro-and macro-constants, only three of them are mutually independent. Moreovever, the isomerization constant, K_i, is equal to the ratio of the microconstants.
:K_i=k₁₁/k₁₂
In L-Dopa the isomeriztion constant is 0.9, so the micro-species L¹H and L²H have almost equal concentrations ''at all pH values''.
In general a macro-constant is equal to the sum of all the micro-constants and the occupancy of each site is proportional to the micro-constant. The site of protonation can be very important, for example, for biological activity.
Micro-constants cannot be determined individually by the usual methods, which give macro-constants. Methods which have been used to determine micro-constants include:

★ blocking one of the sites, for example by methylation of a hydroxyl group, to determine one of the micro-constants

★ using a spectroscopic technique, such as infrared spectroscopy, where the different micro-species give different signals.

★ applying mathematical procedures to ¹³C NMR data.^[2]

pH considerations (Brønsted constants)

pH is defined in terms of the activity of the hydrogen ion
:$pH\; =\; -lg\; \{H^+\}$
If, when determining an equilibrium constant, pH is measured by means of a glass electrode, a mixed equilibrium constant, also known as a Brønsted constant, may result.
:$HL$
ightleftharpoons L+H:pK =-lg left(rac{[L]{H}}{[HL]}
ight)
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
:$K\_W=[H][OH]:\; [OH]=K\_W[H]^\{-1\}$
The first step in metal ion hydrolysis ^[3] can be expressed in two different ways
#$M(H\_2O)$
ightleftharpoons M(OH) +H:[M(OH)]=eta^
★ [M][H]^{-1}
#$M+OH$
ightleftharpoons M(OH):[M(OH)]=K[M][OH]=K K_W[M][H]^{-1}
It follows that
★ =K K_W. Hydrolysis constants are usually reported in the
★ form and this leads to them appearing to have strange values. For example, if lg''K''=4 and lg K_W=-14, lg
★ = 4 -14 = -10. In general when the hydrolysis product contains ''n'' hydroxide groups lg
★ = lg K + ''n'' lg ''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.^[4] 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
:
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.

Temperature dependence

It can be shown that the equilibrium constant is related to the standard Gibbs energy change of reaction as:
:,
where Δ''G''° is the standard Gibbs energy change of reaction, ''R'' is the gas constant, and ''T'' the absolute temperature.
This relationship is also written as:
:$Delta\; G^circ\; =\; -RT\; ln\; K$
A direct consequence of this important relation is the Van't Hoff equation, which relates the change in temperature to the change in the equilibrium constant given the enthalpy change.

Data sources

[IUPAC SC-Database] A comprehensive database of published data on equilibrium constants of metal complexes and ligands
[NIST Standard Reference Database 46] Critically Selected Stability Constants of Metal Complexes
[Inorganic and organic acids and bases] pKa data in water and DMSO

References

1. F.J,C. Rossotti and H. Rossotti, The Determination of Stability Constants, McGraw-Hill, 1961.
2.
D.N. Hague and A.D. Moreton,'' J. Chem. Soc. Perkin Trans.2,'' 265-270, 1994; M. Borkovec and G.J.M. Koper, Anal. ''Chem,,'', '72', 3272-3279, 2000.
3. C.F. Baes and R.E. Mesmer, ''The Hydrolysis of Cations'', Wiley, 1976
4. G. Schwarzenbach and H. Flaschka, ''Complexometric titrations'', Methuen, 1969