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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.^[1] 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.^[2] 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

${\displaystyle \mathrm {H+L} \leftrightharpoons \mathrm {HL} }$

${\displaystyle \mathrm {M+L} \leftrightharpoons \mathrm {ML} }$

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.

${\displaystyle \mathrm {M} +q\mathrm {L} \leftrightharpoons \mathrm {ML} _{q}}$

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.^[3] The next key development was the use of a computer program, LETAGROP^[4][5] 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^[6] and MINIQUAD^[7] 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.^[8][9]

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

${\displaystyle \mathrm {+\mathrm {L} \leftrightharpoons \mathrm {+\mathrm {H_{2}O} }$

The equilibrium constant for this reaction is given by

${\displaystyle \beta ^{'}={\frac {}{}}}$

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

${\displaystyle \beta =\mathrm {\frac {ML}{ML}} .}$

Following this simplification a general definition can be given, for the general equilibrium

${\displaystyle p\mathrm {M} +q\mathrm {L} \leftrightharpoons \mathrm {M} _{p}\mathrm {L} _{q}}$

${\displaystyle \beta _{pq...}={\frac {\mathrm {M} _{p}\mathrm {L} _{q}...}{\mathrm {M} ^{p}\mathrm {L} ^{q}...}}}$

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 pK_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 constantsedit

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

${\displaystyle {\ce {{M}+ 2L <=> ML2}}}$;     ${\displaystyle \beta _{1,2}=\mathrm {\frac {ML_{2}}{ML^{2}}} }$

The stepwise constants, K₁ and K₂ refer to the formation of the complexes one step at a time.

${\displaystyle {\ce {{M}+ L <=> ML}}}$;     ${\displaystyle K_{1}=\mathrm {\frac {ML}{ML}} }$

${\displaystyle {\ce {{ML}+ L <=> ML2}}}$;     ${\displaystyle K_{2}=\mathrm {\frac {ML_{2}}{MLL}} }$

It follows that

${\displaystyle \beta _{1,2}=K_{1}K_{2}\,}$

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^L
_ML is sometimes found in the literature. It is good practice to specify each stability constant explicitly, as illustrated above.

Hydrolysis productsedit

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.

${\displaystyle \mathrm {M+OH} \leftrightharpoons \mathrm {M(OH)} }$;      ${\displaystyle K=\mathrm {\frac {M(OH)}{MOH}} }$

In water the concentration of hydroxide is related to the concentration of hydrogen ions by the self-ionization constant, K_w.

${\displaystyle K_{w}=\mathrm {H} ^{+}\mathrm {OH} ^{-1}}$

The expression for hydroxide concentration is substituted into the formation constant expression

${\displaystyle K={\frac {\mathrm {M} (\mathrm {OH} )}{\mathrm {M} K_{\mathrm {w} }\mathrm {H} ^{-1}}}}$

${\displaystyle \beta _{1,-1}^{*}=KK_{\mathrm {w} }={\frac {\mathrm {M} (\mathrm {OH} )}{\mathrm {M} \mathrm {H} ^{-1}}}}$

In general, for the reaction

${\displaystyle \mathrm {M} +n\mathrm {OH} \leftrightharpoons \mathrm {M(OH} )_{n}}$      ${\displaystyle \log \beta _{1,-n}^{*}\approx \log K-14n}$

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 M_pL_q(OH)_r.

Acid–base complexesedit

A Lewis acid, A, and a Lewis base, B, can be considered to form a complex AB.

${\displaystyle \mathrm {A} +\mathrm {B} \leftrightharpoons \mathrm {AB} }$    ${\displaystyle ,K=\mathrm {\frac {AB}{AB}} }$

There are three major theories relating to the strength of Lewis acids and bases and the interactions between them.

1. Hard and soft acid–base theory (HSAB).^[10] This is used mainly for qualitative purposes.
2. Drago and Wayland proposed a two-parameter equation which predicts the standard enthalpy of formation of a very large number of adducts quite accurately. −ΔH^⊖ (A − B) = E_AE_B + C_AC_B. Values of the E and C parameters are available.^[11]
3. Guttmann donor numbers: for bases the number is derived from the enthalpy of reaction of the base with antimony pentachloride in 1,2-Dichloroethane as solvent. For acids, an acceptor number is derived from the enthalpy of reaction of the acid with triphenylphosphine oxide.^[12]

For more details see: acid–base reaction, acid catalysis, Extraction (chemistry)

Thermodynamicsedit

The thermodynamics of metal ion complex formation provides much significant information.^[13] 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

${\displaystyle \Delta G^{\theta }=-2.303RTlog\beta }$

R is the gas constant and T is the absolute temperature. At 25 °C, ΔG^⊖ = (−5.708 kJ mol⁻¹) ⋅ log β. Free energy is made up of an enthalpy term and an entropy term.

${\displaystyle \Delta G^{\theta }=\Delta H^{\theta }-T\Delta S^{\theta }}$

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.

${\displaystyle \mathrm {M(H_{2}O)} _{m}\mathrm {L_{n-1}} +\mathrm {L} \leftrightharpoons \mathrm {M(H_{2}O)} _{m-1}\mathrm {L_{n}} }$

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; ΔS^⊖ is more positive, so ΔG^⊖ is more negative and ${\displaystyle K_{1}>K_{2}}$. The ratio of the stepwise stability constants can be calculated on this basis, but experimental ratios are not exactly the same because ΔH^⊖ is not necessarily the same for each step.^[14] Exceptions to this rule are discussed below, in #chelate effect and #Geometrical factors.

Ionic strength dependenceedit

The thermodynamic equilibrium constant, K^⊖, for the equilibrium

${\displaystyle \mathrm {M+L} \leftrightharpoons \mathrm {ML} }$

can be defined^[15] as

${\displaystyle K^{\ominus }=\mathrm {\frac {\{ML\}}{\{M\}\{L\}}} }$

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

${\displaystyle K^{\ominus <!--\; \ominus \; -->}=\frac{\mathrm{M}\mathrm{L}}{\mathrm{M}\mathrm{L}}\times <!--\; \times \; -->\frac{{\mathrm{\gamma <!--\; \gamma \; -->}}_{\mathrm{M}\mathrm{L}}}{{}_{}}}$Zdroj:https://en.wikipedia.org?pojem=Stability_constants_of_complexes
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