Constants that describe stability of coordination complexes
In coordination chemistry, a stability constant (also called formation constant or binding constant) is an equilibrium constant for the formation of a complex in solution. It is a measure of the strength of the interaction between the reagents that come together to form the complex. There are two main kinds of complex: compounds formed by the interaction of a metal ion with a ligand and supramolecular complexes, such as host–guest complexes and complexes of anions. The stability constant(s) provide(s) the information required to calculate the concentration(s) of the complex(es) in solution. There are many areas of application in chemistry, biology and medicine.
History.
Jannik Bjerrum (son of Niels Bjerrum) developed the first general method for the determination of stability constants of metal-ammine complexes in 1941. The reasons why this occurred at such a late date, nearly 50 years after Alfred Werner had proposed the correct structures for coordination complexes, have been summarised by Beck and Nagypál. The key to Bjerrum's method was the use of the then recently developed glass electrode and pH meter to determine the concentration of hydrogen ions in solution. Bjerrum recognised that the formation of a metal complex with a ligand was a kind of acid–base equilibrium: there is competition for the ligand, L, between the metal ion, Mn+, and the hydrogen ion, H+. This means that there are two simultaneous equilibria that have to be considered. In what follows electrical charges are omitted for the sake of generality. The two equilibria are
formula_1
formula_2
Hence by following the hydrogen ion concentration during a titration of a mixture of M and HL with base, and knowing the acid dissociation constant of HL, the stability constant for the formation of ML could be determined. Bjerrum went on to determine the stability constants for systems in which many complexes may be formed.
formula_3
The following twenty years saw a veritable explosion in the number of stability constants that were determined. Relationships, such as the Irving-Williams series were discovered. The calculations were done by hand using the so-called graphical methods. The mathematics underlying the methods used in this period are summarised by Rossotti and Rossotti. The next key development was the use of a computer program, LETAGROP to do the calculations. This permitted the examination of systems too complicated to be evaluated by means of hand-calculations. Subsequently, computer programs capable of handling complex equilibria in general, such as SCOGS and MINIQUAD were developed so that today the determination of stability constants has almost become a "routine" operation. Values of thousands of stability constants can be found in two commercial databases.
Theory.
The formation of a complex between a metal ion, M, and a ligand, L, is in fact usually a substitution reaction. For example, in aqueous solutions, metal ions will be present as aqua ions, so the reaction for the formation of the first complex could be written as
formula_4
The equilibrium constant for this reaction is given by
formula_5
[L] should be read as "the concentration of L" and likewise for the other terms in square brackets. The expression can be greatly simplified by removing those terms which are constant. The number of water molecules attached to each metal ion is constant. In dilute solutions the concentration of water is effectively constant. The expression becomes
formula_6
Following this simplification a general definition can be given, for the general equilibrium
formula_7
formula_8
The definition can easily be extended to include any number of reagents. The reagents need not always be a metal and a ligand but can be any species which form a complex. Stability constants defined in this way, are "association" constants. This can lead to some confusion as p"K"a values are "dissociation" constants. In general purpose computer programs it is customary to define all constants as association constants. The relationship between the two types of constant is given in association and dissociation constants.
Stepwise and cumulative 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
 ;chem ;{M} + 2L ;= ; ML2 ;/chem ;;formula_9
The stepwise constants, "K"1 and "K"2 refer to the formation of the complexes one step at a time.
 ;chem ;{M} + L ;= ; ML ;/chem ;;formula_10
 ;chem ;{ML} + L ;= ; ML2 ;/chem ;;formula_11
It follows that
formula_12
A cumulative constant can always be expressed as the product of stepwise constants. Conversely, any stepwise constant can be expressed as a quotient of two or more overall constants. There is no agreed notation for stepwise constants, though a symbol such as "K" is sometimes found in the literature. It is good practice to specify each stability constant explicitly, as illustrated above.
Hydrolysis products.
The formation of a hydroxo complex is a typical example of a hydrolysis reaction. A hydrolysis reaction is one in which a substrate reacts with water, splitting a water molecule into hydroxide and hydrogen ions. In this case the hydroxide ion then forms a complex with the substrate.
formula_13; formula_14
In water the concentration of hydroxide is related to the concentration of hydrogen ions by the self-ionization constant, "K"w.
formula_15
The expression for hydroxide concentration is substituted into the formation constant expression
formula_16
formula_17
In general, for the reaction 
formula_18 formula_19
In the older literature the value of log K is usually cited for an hydrolysis constant. The log "β"* value is usually cited for an hydrolysed complex with the generic chemical formula MpLq(OH)r.
Acid–base complexes.
A Lewis acid, A, and a Lewis base, B, can be considered to form a complex AB.
formula_20formula_21
There are three major theories relating to the strength of Lewis acids and bases and the interactions between them.
For more details see: acid–base reaction, acid catalysis, Extraction (chemistry)
Thermodynamics.
The thermodynamics of metal ion complex formation provides much significant information. In particular it is useful in distinguishing between enthalpic and entropic effects. Enthalpic effects depend on bond strengths and entropic effects have to do with changes in the order/disorder of the solution as a whole. The chelate effect, below, is best explained in terms of thermodynamics.
An equilibrium constant is related to the standard Gibbs free energy change for the reaction
formula_22
"R" is the gas constant and "T" is the absolute temperature. At 25 °C, Δ"G"⊖ = (−5.708 kJ mol−1) ⋅ log "β". Free energy is made up of an enthalpy term and an entropy term.
formula_23
The standard enthalpy change can be determined by calorimetry or by using the Van 't Hoff equation, though the calorimetric method is preferable. When both the standard enthalpy change and stability constant have been determined, the standard entropy change is easily calculated from the equation above.
The fact that stepwise formation constants of complexes of the type ML"n" decrease in magnitude as "n" increases may be partly explained in terms of the entropy factor. Take the case of the formation of octahedral complexes.
formula_24
For the first step "m" = 6, "n" = 1 and the ligand can go into one of 6 sites. For the second step "m" = 5 and the second ligand can go into one of only 5 sites. This means that there is more randomness in the first step than the second one; Δ"S"⊖ is more positive, so Δ"G"⊖ is more negative and formula_25. The ratio of the stepwise stability constants can be calculated on this basis, but experimental ratios are not exactly the same because Δ"H"⊖ is not necessarily the same for each step. Exceptions to this rule are discussed below, in #chelate effect and #Geometrical factors.
Ionic strength dependence.
The thermodynamic equilibrium constant, "K"⊖, for the equilibrium
formula_26
can be defined as
formula_27
where {ML} is the activity of the chemical species ML etc. "K"⊖ is dimensionless since activity is dimensionless. Activities of the products are placed in the numerator, activities of the reactants are placed in the denominator. See activity coefficient for a derivation of this expression.
Since activity is the product of concentration and activity coefficient ("γ") the definition could also be written as
formula_28
where [ML] represents the concentration of ML and "Γ" is a quotient of activity coefficients. This expression can be generalized as
formula_29
To avoid the complications involved in using activities, stability constants are determined, where possible, in a medium consisting of a solution of a background electrolyte at high ionic strength, that is, under conditions in which "Γ" can be assumed to be always constant. For example, the medium might be a solution of 0.1 mol dm−3 sodium nitrate or 3 mol dm−3 sodium perchlorate. When "Γ" is constant it may be ignored and the general expression in theory, above, is obtained.
All published stability constant values refer to the specific ionic medium used in their determination and different values are obtained with different conditions, as illustrated for the complex CuL (L = glycinate). Furthermore, stability constant values depend on the specific electrolyte used as the value of "Γ" is different for different electrolytes, even at the same ionic strength. There does not need to be any chemical interaction between the species in equilibrium and the background electrolyte, but such interactions might occur in particular cases. For example, phosphates form weak complexes with alkali metals, so, when determining stability constants involving phosphates, such as ATP, the background electrolyte used will be, for example, a tetralkylammonium salt. Another example involves iron(III) which forms weak complexes with halide and other anions, but not with perchlorate ions.
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.
Temperature dependence.
All equilibrium constants vary with temperature according to the Van 't Hoff equation 
formula_30
Alternatively
formula_31
"R" is the gas constant and "T" is the thermodynamic temperature. Thus, for exothermic reactions, where the standard enthalpy change, Δ"H"⊖, is negative, "K" decreases with temperature, but for endothermic reactions, where Δ"H"⊖ is positive, "K" increases with temperature.
Factors affecting the stability constants of complexes.
The chelate effect.
Consider the two equilibria, in aqueous solution, between the copper(II) ion, Cu2+ and ethylenediamine (en) on the one hand and methylamine, MeNH2 on the other.
formula_32
formula_33
In the first reaction the bidentate ligand ethylene diamine forms a chelate complex with the copper ion. Chelation results in the formation of a five-membered ring. In the second reaction the bidentate ligand is replaced by two monodentate methylamine ligands of approximately the same donor power, meaning that the enthalpy of formation of Cu–N bonds is approximately the same in the two reactions. Under conditions of equal copper concentrations and when the concentration of methylamine is twice the concentration of ethylenediamine, the concentration of the bidentate complex will be greater than the concentration of the complex with 2 monodentate ligands. The effect increases with the number of chelate rings so the concentration of the EDTA complex, which has six chelate rings, is much higher than a corresponding complex with two monodentate nitrogen donor ligands and four monodentate carboxylate ligands. Thus, the phenomenon of the chelate effect is a firmly established empirical fact: under comparable conditions, the concentration of a chelate complex will be higher than the concentration of an analogous complex with monodentate ligands.
The thermodynamic approach to explaining the chelate effect considers the equilibrium constant for the reaction: the larger the equilibrium constant, the higher the concentration of the complex.
formula_34
formula_35
When the analytical concentration of methylamine is twice that of ethylenediamine and the concentration of copper is the same in both reactions, the concentration [Cu(en)]2+ is much higher than the concentration [Cu(MeNH2)2]2+ because "β"11 ≫ "β"12.
The difference between the two stability constants is mainly due to the difference in the standard entropy change, Δ"S"⊖. In the reaction with the chelating ligand there are two particles on the left and one on the right, whereas in equation with the monodentate ligand there are three particles on the left and one on the right. This means that less entropy of disorder is lost when the chelate complex is formed than when the complex with monodentate ligands is formed. This is one of the factors contributing to the entropy difference. Other factors include solvation changes and ring formation. Some experimental data to illustrate the effect are shown in the following table.
These data show that the standard enthalpy changes are indeed approximately equal for the two reactions and that the main reason why the chelate complex is so much more stable is that the standard entropy term is much less unfavourable, indeed, it is favourable in this instance. In general it is difficult to account precisely for thermodynamic values in terms of changes in solution at the molecular level,