View source for Stability constants of complexes

{{Short description|Constants that describe stability of coordination complexes}}
{{context|date=February 2015}}
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 (chemistry)|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 complex|metal-ammine complexes]] in 1941.<ref>{{cite book|last=Bjerrum|first=J.|title=Metal-ammine formation in aqueous solution|publisher=Haase|location=Copenhagen|year=1941}}</ref> 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.<ref>{{cite book
 |title=Chemistry of Complex Equilibria
 |last1=Beck |first1=M. T.
 |last2=Nagypál|first2= I.
 |year=1990
 |publisher=Horwood
 |isbn=0-85312-143-5
|chapter=Chapter 1}}</ref> 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, M<sup>n+</sup>, and the hydrogen ion, H<sup>+</sup>. 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
:<math>\mathrm{H + L} \leftrightharpoons \mathrm{HL}</math>
:<math>\mathrm{M + L} \leftrightharpoons \mathrm{ML}</math>
Hence by following the hydrogen ion concentration during a [[titration]] of a mixture of M and HL with [[Base (chemistry)|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.
:<math>\mathrm{M} + q \mathrm{L} \leftrightharpoons \mathrm{ML}_q</math>
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.<ref>
{{cite book
 |title=The Determination of Stability Constants
 |last1=Rossotti |first1=F. J. C.
 |last2=Rossotti|first2= H.
  |year=1961
 |publisher=McGraw–Hill}}</ref> The next key development was the use of a computer program, LETAGROP<ref>{{cite journal|last1=Dyrssen|first1=D.|last2=Ingri|first2= N. |last3=Sillen|first3= L. G. |year=1961|title=Pit-mapping - A general approach to Computer refinement of stability constants.|journal=Acta Chem. Scand.|volume=15|pages=694–696|doi=10.3891/acta.chem.scand.15-0694|doi-access=free}}</ref><ref>{{cite journal|last1=Ingri|first1=N.|last2=Sillen|first2= L. G. |year=1964|title=High-speed computers as a supplement to graphical methods, IV. An ALGOL version of LETAGROP-VRID|journal=Arkiv för Kemi|volume=23|pages=97–121}}</ref> 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<ref>{{cite journal|last=Sayce|first=I. G.|year=1968|title=Computer calculations of equilibrium constantsof species present in mixtures of metal ions and complexing reagents|journal=Talanta|volume=15|pages=1397–1421|pmid=18960446|issue=12|doi=10.1016/0039-9140(68)80200-0}}</ref> and MINIQUAD<ref>{{cite journal|last1=Sabatini|first1=A.|last2=Vacca |first2=A. |last3=Gans|first3= P. |year=1974|title=MINIQUAD - A general computer program for the computation of Stability constants|journal=Talanta|volume=21|pages=53–77|doi=10.1016/0039-9140(74)80063-9|pmid=18961420|issue=1}}</ref> 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.<ref name=LDP>[http://www.acadsoft.co.uk/scdbase/scdbase.htm IUPAC SC-Database] A comprehensive database of published data on equilibrium constants of metal complexes and ligands</ref><ref name="NIST">[https://www.nist.gov/srd/nist46 NIST Standard Reference Database 46] NIST Critically Selected Stability Constants of Metal Complexes: Version 8.0 (This database has been discontinued.)</ref>

==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 solution]]s, metal ions will be present as [[Metal ions in aqueous solution|aqua ions]], so the reaction for the formation of the first complex could be written as
:<math>\mathrm{[M(H_2O)}_n] + \mathrm{L} \leftrightharpoons \mathrm{[M(H_2O)}_{n-1}\mathrm{L}] + \mathrm{H_2O}</math>
The [[equilibrium constant]] for this reaction is given by
:<math>\beta^'=\frac{[\mathrm{M(H_2O)}_{n-1}\mathrm{L}] [\mathrm{H_2O}]} {[\mathrm{M(H_2O)}_n] [\mathrm{L}] }</math>
[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
:<math>\beta =\mathrm{\frac{[ML] } {[M] [L] }}.</math>
Following this simplification a general definition can be given, for the general equilibrium
:<math>p\mathrm M + q\mathrm L \leftrightharpoons \mathrm M_p \mathrm L_q </math>
:<math>\beta_{pq...}=\frac{[\mathrm{M}_p\mathrm{L}_q...] } {[\mathrm{M}]^p [\mathrm{L}]^q ... }</math>
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 [[acid dissociation constant|p''K''<sub>a</sub> 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 [[equilibrium constant#Association and dissociation constants|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 ML<sub>2</sub> is given by
:<chem>{M} + 2L <=> ML2</chem>;{{spaces|5}}<math>\beta_{1,2}=\mathrm{\frac{[ML_2]}{[M][L]^2}}</math>
The stepwise constants, ''K''<sub>1</sub> and ''K''<sub>2</sub> refer to the formation of the complexes one step at a time.
:<chem>{M} + L <=> ML</chem>;{{spaces|5}}<math>K_{1}=\mathrm{\frac{[ML]}{[M][L]}}</math>
:<chem>{ML} + L <=> ML2</chem>;{{spaces|5}}<math>K_{2}=\mathrm{\frac{[ML_2]}{[ML][L]}}</math>
It follows that
:<math>\beta_{1,2}=K_1 K_2\,</math>
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''{{su|b=ML|p=L}} 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.
:<math>\mathrm{M + OH} \leftrightharpoons \mathrm{M(OH)}</math>; {{spaces|4}} <math>  K =\mathrm{\frac{[M(OH)] } {[M] [OH] }}</math>
In water the concentration of [[hydroxide]] is related to the concentration of hydrogen ions by the [[Self-ionization of water|self-ionization constant]], ''K''<sub>w</sub>.
:<math>K_w =  \mathrm{[H]}^ + \mathrm{[OH]}^{-}</math>
The expression for hydroxide concentration is substituted into the formation constant expression
:<math>K=\frac{[\mathrm{M}(\mathrm{OH})] } {[\mathrm{M}] K_\mathrm{w} [\mathrm{H}]^{-1} }</math>
:<math>\beta^*_{1,-1}= K K_\mathrm{w} =\frac{[\mathrm{M}(\mathrm{OH})] } {[\mathrm{M}] [\mathrm{H}]^{-1} }</math>
In general, for the reaction 
:<math>\mathrm{M} + n\mathrm{OH} \leftrightharpoons \mathrm{M(OH})_n</math> {{spaces|4}} <math> \log \beta^*_{1,-n} \approx \log K - 14n </math>
In the older literature the value of log K is usually cited for an hydrolysis constant. The log ''β''<sup>*</sup> value is usually cited for an hydrolysed complex with the generic chemical formula M<sub>p</sub>L<sub>q</sub>(OH)<sub>r</sub>.

===Acid–base complexes===
{{Main|acid–base equilibrium}}
A [[Lewis acid]], A, and a [[Lewis base]], B, can be considered to form a complex AB.
:<math>\mathrm A + \mathrm B \leftrightharpoons \mathrm {AB} </math>{{spaces|4}}<math> , K =\mathrm{\frac{[AB] } {[A] [B] }}</math>

There are three major theories relating to the strength of Lewis acids and bases and the interactions between them.
#Hard and soft acid–base theory ([[HSAB]]).<ref>{{cite book|last=Pearson|first=R. G.|title=Chemical Hardness: Applications from Molecules to Solids |publisher=Springer-VCH|year=1997|isbn=3-527-29482-1 }}</ref> This is used mainly for qualitative purposes.
#Drago and Wayland proposed a two-parameter equation which predicts the standard enthalpy of formation of a very large number of adducts quite accurately. {{nowrap begin}}−Δ''H''<sup>⊖</sup> (A − B) = ''E''<sub>A</sub>''E''<sub>B</sub> + ''C''<sub>A</sub>''C''<sub>B</sub>.{{nowrap end}} Values of the ''E'' and ''C'' parameters are available.<ref>{{cite journal|last1=Drago| first1=R. S. |last2=Wong |first2=N. |last3=Bilgrien |first3=C. |last4= Vogel|first4= C. |title=''E'' and ''C'' parameters from Hammett substituent constants and use of ''E'' and ''C'' to understand cobalt–carbon bond energies|year=1987|journal=Inorg. Chem.|volume=26|issue=1|pages=9–14|doi=10.1021/ic00248a003}}</ref>
#Guttmann [[donor number]]s: 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]].<ref>{{cite book|last=Gutmann|first=V.|title=The Donor–Acceptor Approach to Molecular Interactions|publisher=Springer|year=1978|isbn=0-306-31064-3|url-access=registration|url=https://archive.org/details/donoracceptorapp0000gutm}}</ref>

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

==Thermodynamics==
The thermodynamics of metal ion complex formation provides much significant information.<ref>{{cite book|first=F. J. C. |last=Rossotti|title=Modern coordination chemistry|editor1-last=Lewis|editor1-first= J.|editor2-first= Wilkins|editor2-last= R. G.|publisher=Interscience|location=New York|year=1960|chapter=The thermodynamics of metal ion complex formation in solution}}</ref> In particular it is useful in distinguishing between [[enthalpy|enthalpic]] and [[entropy|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
:<math>\Delta G^ \theta = -2.303 RT \log \beta  </math>
''R'' is the [[gas constant]] and ''T'' is the [[kelvin|absolute temperature]]. At 25&nbsp;°C, {{nowrap|1=Δ''G''<sup>⊖</sup> = (−5.708 kJ&nbsp;mol<sup>−1</sup>) ⋅ log ''β''}}. Free energy is made up of an enthalpy term and an entropy term.
:<math>\Delta G^ \theta = \Delta H^ \theta -T\Delta S^ \theta  </math>
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<sub>''n''</sub> 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.
:<math>\mathrm{M(H_2 O)}_m \mathrm {L_{n-1}}+ \mathrm{ L} \leftrightharpoons \mathrm{M(H_2 O)}_{m-1} \mathrm {L_{n}}</math>
For the first step ''m''&nbsp;=&nbsp;6, ''n''&nbsp;=&nbsp;1 and the ligand can go into one of 6 sites. For the second step ''m''&nbsp;=&nbsp;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''<sup>⊖</sup> is more positive, so Δ''G''<sup>⊖</sup> is more negative and <math>K_1 >K_2</math>. The ratio of the stepwise stability constants can be calculated on this basis, but experimental ratios are not exactly the same because Δ''H''<sup>⊖</sup> is not necessarily the same for each step.<ref>{{cite book |title=Chemistry of Complex Equilibria |last1=Beck |first1=M. T. |last2=Nagypál|first2= I. |year=1990 |publisher=Horwood |isbn=0-85312-143-5}} sections 3.5.1.2, 6.6.1 and 6.6.2</ref> Exceptions to this rule are discussed below, in [[#chelate effect]] and [[#Geometrical factors]].

=== Ionic strength dependence ===
The thermodynamic equilibrium constant, ''K''<sup>⊖</sup>, for the equilibrium
:<math>\mathrm {M+L} \leftrightharpoons \mathrm{ML} </math>
can be defined<ref name=rr>{{cite book |title=The Determination of Stability Constants |last1=Rossotti |first1=F. J. C. |last2=Rossotti|first2= H.  |year=1961 |publisher=McGraw–Hill
|chapter= Chapter 2: Activity and Concentration Quotients}}</ref> as
:<math> K^{\ominus} =\mathrm{\frac{\{ML\}}  {\{M\}\{L\}} }</math>
where {ML} is the [[activity (chemistry)|activity]] of the chemical species ML etc. ''K''<sup>⊖</sup> 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
:<math>K^{\ominus} = \mathrm{\frac{[ML]}{[M][L]}}\times \frac{\gamma_\mathrm{ML}}{\gamma_\mathrm{M}\gamma_\mathrm{L}} =\mathrm{\frac{[ML]}{[M][L]}}\times\Gamma</math>
where [ML] represents the concentration of ML and ''Γ'' is a quotient of activity coefficients. This expression can be generalized as
:<math>\beta_{pq...}^\ominus=\frac{[\mathrm{M}_p\mathrm{L}_q...] } {[\mathrm{M}]^p [\mathrm{L}]^q ... }\times \Gamma</math>

[[File:Cu gly ionic strength.png|thumb|Dependence of the stability constant for formation of [Cu(glycinate)]<sup>+</sup> on ionic strength (NaClO<sub>4</sub>)<ref>{{cite journal|last1=Gergely|first1=A.|last2=Nagypál|first2=I. |first3=Farkas|last3= E. |year=1974|title=A réz(II)-aminosav törzskomplexek vizes oldatában lejátszodó protoncsere-reakciók kinetikájának NMR-vizsgálata |trans-title=NMR study of the proton exchange process in aqueous solutions of copper(II)-aminoacid parent complexes
|journal=Magyar Kémiai Folyóirat|volume=80|pages=545–549 }}</ref>]]
To avoid the complications involved in using activities, stability constants are [[Determination of equilibrium constants|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.<ref name=rr/> For example, the medium might be a solution of 0.1&nbsp;mol&nbsp;dm<sup>−3</sup> [[sodium nitrate]] or 3&nbsp;mol&nbsp;dm<sup>−3</sup> [[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, [[phosphate]]s form weak complexes with [[alkali metals]], so, when determining stability constants involving phosphates, such as [[adenosine triphosphate|ATP]], the background electrolyte used will be, for example, a [[Quaternary ammonium cation|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 interaction theory|specific ion theory]] (SIT) and other theories.<ref>
{{cite web
 |url         = http://www.iupac.org/web/ins/2000-003-1-500
 |title       = Project: Ionic Strength Corrections for Stability Constants
 |publisher   = [[IUPAC]]
 |access-date  = 2008-11-23
 |url-status     = dead
 |archive-url  = https://web.archive.org/web/20081029193538/http://www.iupac.org/web/ins/2000-003-1-500
 |archive-date = 2008-10-29
 }}</ref>

=== Temperature dependence ===
All equilibrium constants vary with [[temperature]] according to the [[Van 't Hoff equation]]<ref>
{{cite book |title=Physical Chemistry |last1=Atkins |first1=P. W. |last2=De Paula|first2= J. |year=2006 |publisher=Oxford University Press |isbn=0-19-870072-5|chapter= Section 7.4: The Response of Equilibria to Temperature}}</ref> 
:<math alt="The derivative of the natural logarithm of any equilibrium constant K with respect to the temperature T in kelvins equals the standard enthalpy change for the reaction divided by the product R times T squared. Here R represents the gas constant, which equals the thermal energy per mole per kelvin. The standard enthalpy is written as Delta H with a superscript plimsol mark represented by a circled minus. This equation follows from the definition of the Gibbs free energy Delta G equals R times T times the natural logarithm of K." >\frac {d (\ln K)} {dT} = \frac{\Delta \mathit{H}_\mathrm{m}^\ominus} {RT^2}</math>
Alternatively
:<math>\frac{d \ln K}{d(1/T)}=-\frac{{\Delta H^\ominus}}{{R}} </math>
''R'' is the [[gas constant]] and ''T'' is the thermodynamic temperature. Thus, for [[exothermic]] reactions, where the standard [[enthalpy change]], Δ''H''<sup>⊖</sup>, is negative, ''K'' decreases with temperature, but for [[endothermic]] reactions, where Δ''H''<sup>⊖</sup> is positive, ''K'' increases with temperature.

== Factors affecting the stability constants of complexes ==

=== The chelate effect ===
[[File:Cu chelate.svg|thumb|Cu<sup>2+</sup> complexes with methylamine (left) end ethylene diamine (right)]]
Consider the two equilibria, in aqueous solution, between the [[copper]](II) ion, Cu<sup>2+</sup> and [[ethylenediamine]] (en) on the one hand and [[methylamine]], MeNH<sub>2</sub> on the other.
:<math>\mathrm{ Cu^{2+} + en \rightleftharpoons [Cu(en)]^{2+}}</math>
:<math>Cu^{2+} + 2 MeNH_2 \rightleftharpoons [Cu(MeNH_2)_2]^{2+}</math>

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 [[equilibrium thermodynamics|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.
:<math>\mathrm{[Cu(en)] = \beta_{11}[Cu][en]|}</math>
:<math>\mathrm{[Cu(MeNH_2)_2]=  \beta _{12}[Cu][MeNH_2]}^2</math>
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)]<sup>2+</sup> is much higher than the concentration [Cu(MeNH<sub>2</sub>)<sub>2</sub>]<sup>2+</sup> because {{math|''β''<sub>11</sub>&nbsp;≫&nbsp;''β''<sub>12</sub>.}}

The difference between the two stability constants is mainly due to the difference in the standard entropy change, Δ''S''<sup>⊖</sup>. 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 (order and disorder)|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.<ref name=GE>{{Greenwood&Earnshaw}} p. 910</ref>
:{| class="wikitable"
! Equilibrium !! {{math|log&nbsp;''β''}} !! {{math|Δ''G''<sup>⊖</sup> /kJ&nbsp;mol<sup>−1</sup>}} !! {{math|Δ''H''<sup>⊖</sup> /kJ&nbsp;mol<sup>−1</sup>}} !! {{math|−''T''Δ''S''<sup>⊖</sup> /kJ&nbsp;mol<sup>−1</sup>}}
|-
| {{math|Cd<sup>2+</sup> + 4&nbsp;MeNH<sub>2</sub> {{eqm}} {{chem|Cd(MeNH|2|)|4|2+}}}}
||6.55|| −37.4 || −57.3||19.9 
|-
| {{math|Cd<sup>2+</sup> + 2&nbsp;en {{eqm}} {{chem|Cd(en)|2|2+}}}}
||10.62|| −60.67 || −56.48||−4.19
|}
[[File:Metal-EDTA.svg|thumb|100px| An EDTA complex]]
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, but it is clear that the chelate effect is predominantly an effect of entropy. Other explanations, including that of Schwarzenbach,<ref>{{cite journal |last=Schwarzenbach |first=G. |year=1952 |title=Der Chelateffekt |trans-title=The Chelate Effect|journal=Helv. Chim. Acta|volume=35 |issue=7|pages=2344–2359|doi=10.1002/hlca.19520350721}}</ref> are discussed in Greenwood and Earnshaw.<ref name=GE />
[[File:Diethylene triamine.png|thumb|left|150px|Diethylenetriamine]]
The chelate effect increases as the number of chelate rings increases. For example, the complex [Ni(dien)<sub>2</sub>)]<sup>2+</sup> is more stable than the complex [Ni(en)<sub>3</sub>)]<sup>2+</sup>; both complexes are octahedral with six nitrogen atoms around the nickel ion, but dien ([[diethylenetriamine]], 1,4,7-triazaheptane) is a [[denticity|tridentate]] ligand and [[ethylenediamine|en]] is bidentate. The number of chelate rings is one less than the number of donor atoms in the ligand. [[EDTA]] (ethylenediaminetetracetic acid) has six donor atoms so it forms very strong complexes with five chelate rings. Ligands such as [[DTPA]], which have eight donor atoms are used to form complexes with large metal ions such as [[lanthanide]] or [[actinide]] ions which usually form 8- or 9-coordinate complexes.
5-membered and 6-membered chelate rings give the most stable complexes. 4-membered rings are subject to internal strain because of the small inter-bond angle is the ring. The chelate effect is also reduced with 7- and 8- membered rings, because the larger rings are less rigid, so less entropy is lost in forming them.

====Deprotonation of aliphatic –OH groups====
[[file:Ethanolamine.png|thumb|left|125px|2-aminoethanol]]
[[file:2-amino-2-(hydroxymethyl)propane-1,3-diol 200.svg|thumb|125px|[[Tris]]]]
Removal of a [[Hydron (chemistry)|proton]] from an [[aliphatic]] [[Hydroxyl|–OH]] group is difficult to achieve in aqueous solution because the energy required for this process is rather large. Thus, ionization of aliphatic –OH groups occurs in aqueous solution only in special circumstances. One such circumstance is found with compounds containing the H<sub>2</sub>N–C–C–OH substructure. For example, compounds containing the [[ethanolamine|2-aminoethanol]] substructure can form metal–chelate complexes with the deprotonated form, H<sub>2</sub>N–C–C–O<sup>−</sup>. The chelate effect supplies the extra energy needed to break the O–H bond.

An important example occurs with the molecule [[tris]]. This molecule should be used with caution as a [[buffer solution|buffering agent]] as it will form chelate complexes with ions such as Fe<sup>3+</sup> and Cu<sup>2+</sup>.

===The macrocyclic effect{{anchor|The macrocyclic effect}}<!-- This section is linked from [[Macrocyclic effect]] -->===
It was found that the stability of the complex of copper(II) with the [[macrocyclic]] ligand [[cyclam]] (1,4,8,11-tetraazacyclotetradecane) was much greater than expected in comparison to the stability of the complex with the corresponding open-chain [[amine]].<ref>{{cite journal|last1=Cabinness|first1=D. K.|last2=Margerum|first2= D. W.|year=1969|title=Macrocyclic effect on the stability of copper(II) tetramine complexes|journal=J. Am. Chem. Soc.|volume=91|issue=23|pages=6540–6541|doi=10.1021/ja01051a091}}</ref>
This phenomenon was named the '''macrocyclic effect''' and it was also interpreted as an entropy effect. However, later studies suggested that both enthalpy and entropy factors were involved.<ref>{{cite book|last=Lindoy|first=L. F.|authorlink=Leonard Francis Lindoy|title=The Chemistry of Macrocyclic Ligand Complexes|publisher=Cambridge University Press|year=1990|isbn=0-521-40985-3|chapter= Chapter 6: Thermodynamic considerations}}</ref>

An important difference between macrocyclic ligands and open-chain (chelating) ligands is that they have selectivity for metal ions, based on the size of the cavity into which the metal ion is inserted when a complex is formed. For example, the [[crown ether]] [[18-crown-6]] forms much stronger complexes with the [[potassium]] ion, K<sup>+</sup> than with the smaller [[sodium]] ion, Na<sup>+</sup>.<ref>{{cite journal|last=Pedersen|first=C. J. |year=1967|journal=J. Am. Chem. Soc.|volume=89|title=Cyclic polyethers and their complexes with metal salts|issue=26|pages= 7017–7036|doi=10.1021/ja01002a035}}</ref>

In [[hemoglobin]] an iron(II) ion is complexed by a macrocyclic [[porphyrin]] ring. The article [[hemoglobin]] incorrectly states that oxyhemoglogin contains iron(III). It is now known that the iron(II) in hemoglobin is a [[crystal field theory#High-spin and low-spin|low-spin complex]], whereas in oxyhemoglobin it is a high-spin complex. The low-spin Fe<sup>2+</sup> ion fits snugly into the cavity of the porphyrin ring, but high-spin iron(II) is significantly larger and the iron atom is forced out of the plane of the macrocyclic ligand.<ref>{{Greenwood&Earnshaw}} p. 1100, Figure 25.7</ref> This effect contributes the ability of hemoglobin to bind oxygen reversibly under biological conditions. In [[Vitamin B12]] a cobalt(II) ion is held in a [[corrin ring]]. [[Chlorophyll]] is a macrocyclic complex of magnesium(II).
{|
|[[File:cyclam.svg|130px]]
|[[File:Porphyrin.svg|150px]]
|-
|align="center"|[[Cyclam]]
|[[Porphine]], the simplest [[porphyrin]].
|}

{|
|[[File:Various crown ethers (molecular diagrams).png|700px]]
|-
|Structures of common [[crown ethers]]: [[12-crown-4]], [[15-crown-5]], [[18-crown-6]], [[dibenzo-18-crown-6]], and [[aza-crown ether|an aza-crown ether]]
|}

=== Geometrical factors ===
Successive stepwise formation constants ''K<sub>n</sub>'' in a series such as ML<sub>''n''</sub> (''n'' = 1, 2, ...) usually decrease as ''n'' increases. Exceptions to this rule occur when the geometry of the ML<sub>''n''</sub> complexes is not the same for all members of the series. The classic example is the formation of the diamminesilver(I) complex [Ag(NH<sub>3</sub>)<sub>2</sub>]<sup>+</sup> in aqueous solution.
:<math>\mathrm{Ag^+ + NH_3 \rightleftharpoons [Ag(NH_3)]^+ ; K_1 =  \frac{{[Ag(NH_3)]}}{{[Ag][NH_3]}}} </math> 
:<math>\mathrm{Ag(NH_3)^+ + NH_3 \rightleftharpoons [Ag(NH_3)_2]^+ ; K_2 =  \frac{{[Ag(NH_3)_2]}}{{[Ag(NH_3][NH_3]}}} </math> 
In this case, ''K''<sub>2</sub>&nbsp;> ''K''<sub>1</sub>. The reason for this is that, in aqueous solution, the ion written as Ag<sup>+</sup> actually exists as the four-coordinate tetrahedral aqua species [Ag(H<sub>2</sub>O)<sub>4</sub>]<sup>+</sup>. The first step is then a substitution reaction involving the displacement of a bound water molecule by ammonia forming the tetrahedral complex [Ag(NH<sub>3</sub>)(H<sub>2</sub>O)<sub>3</sub>]<sup>+</sup>. In the second step, all the aqua ligands are lost and a linear, two-coordinate product [H<sub>3</sub>N–Ag–NH<sub>3</sub>]<sup>+</sup> is formed. Examination of the thermodynamic data<ref>{{cite journal|last1=Lundeen|first1=M.|last2=Hugus|first2=Z. Z.|year=1992|title=A calorimetric study of some metal ion complexing equilibria |journal=Thermochim. Acta|volume=196|issue=1|pages=93–103 |doi=10.1016/0040-6031(92)85009-K}}</ref> shows that the difference in entropy change is the main contributor to the difference in stability constants for the two complexation reactions.
{|class="wikitable"
!equilibrium
!{{math|Δ''H''<sup>⊖</sup> /kJ&nbsp;mol<sup>−1</sup>}}
!{{math|Δ''S''<sup>⊖</sup> /J&nbsp;K<sup>−1</sup>&nbsp;mol<sup>−1</sup>}}
|-
|align="center"|Ag<sup>+</sup> + NH<sub>3</sub> {{eqm}} [Ag(NH<sub>3</sub>)]<sup>+</sup>
|align="center"|−21.4||align="center"|8.66
|-
|align="center"|[Ag(NH<sub>3</sub>)]<sup>+</sup> + NH<sub>3</sub> {{eqm}} [Ag(NH<sub>3</sub>)<sub>2</sub>]<sup>+</sup>
|align="center"|−35.2||align="center"|−61.26
|}
Other examples exist where the change is from octahedral to tetrahedral, as in the formation of [CoCl<sub>4</sub>]<sup>2−</sup> from [Co(H<sub>2</sub>O)<sub>6</sub>]<sup>2+</sup>.

=== Classification of metal ions ===
Ahrland, Chatt and Davies proposed that metal ions could be described as class A if they formed stronger complexes with ligands whose donor atoms are [[nitrogen]], [[oxygen]] or [[fluorine]] than with ligands whose donor atoms are [[phosphorus]], [[sulfur]] or [[chlorine]] and class B if the reverse is true.<ref>{{cite journal
|last1=Ahrland
|first1=S.
|last2=Chatt|first2 =J. |last3=Davies|first3= N. R.
 |year=1958
|title=The relative affinities of ligand atoms for acceptor molecules and ions
|journal=Quart. Rev.
|volume=12
|issue=3|pages=265–276
|doi=10.1039/QR9581200265
}}</ref> For example, Ni<sup>2+</sup> forms stronger complexes with [[amine]]s than with [[phosphines]], but Pd<sup>2+</sup> forms stronger complexes with phosphines than with amines. Later, Pearson proposed the theory of [[HSAB theory|hard and soft acids and bases]] (HSAB theory).<ref>{{cite journal|title=Hard and Soft Acids and Bases|last=Pearson|first= R. G.|journal= J. Am. Chem. Soc. |year=1963| volume= 85 |issue=22|pages=3533–3539|doi=10.1021/ja00905a001}}</ref> In this classification, class A metals are hard acids and class B metals are soft acids. Some ions, such as copper(I), are classed as borderline. Hard acids form stronger complexes with hard bases than with soft bases. In general terms hard–hard interactions are predominantly electrostatic in nature whereas soft–soft interactions are predominantly covalent in nature. The HSAB theory, though useful, is only semi-quantitative.<ref>{{cite book
 |title=Chemistry of Complex Equilibria
 |last1=Beck |first1=M. T.
 |last2=Nagypál|first2= I.
 |year=1990
 |publisher=Horwood
 |isbn=0-85312-143-5
 |page=354}}</ref>

The hardness of a metal ion increases with oxidation state. An example of this effect is given by the fact that Fe<sup>2+</sup> tends to form stronger complexes with ''N''-donor ligands than with ''O''-donor ligands, but the opposite is true for Fe<sup>3+</sup>.

===Effect of ionic radius===
The [[Irving–Williams series]] refers to high-spin, octahedral, divalent metal ion of the first transition series. It places the stabilities of complexes in the order
:Mn < Fe < Co < Ni < Cu > Zn
This order was found to hold for a wide variety of ligands.<ref>{{cite journal |last1=Irving | first1 = H. M. N. H. |last2=Williams|first2= R. J. P.  |year=1953|title=The stability of transition-metal complexes|journal=J. Chem. Soc.|pages=3192–3210|doi=10.1039/JR9530003192 }}</ref> There are three strands to the explanation of the series.
#The [[ionic radius]] is expected to decrease regularly for Mn<sup>2+</sup> to Zn<sup>2+</sup>. This would be the normal periodic trend and would account for the general increase in stability.
#The [[Crystal field theory#Crystal field stabilization energy|crystal field stabilisation energy]] (CFSE) increases from zero for manganese(II) to a maximum at nickel(II). This makes the complexes increasingly stable. CFSE returns to zero for zinc(II).
#Although the CFSE for copper(II) is less than for nickel(II), octahedral copper(II) complexes are subject to the [[Jahn–Teller effect]] which results in a complex having extra stability.

Another example of the effect of ionic radius the steady increase in stability of complexes with a given ligand along the series of trivalent lanthanide ions, an effect of the well-known [[lanthanide contraction]].

== Applications ==
Stability constant values are exploited in a wide variety of applications. [[Chelation therapy]] is used in the treatment of various metal-related illnesses, such as iron overload in β-[[thalassemia]] sufferers who have been given blood transfusions. The ideal ligand binds to the target metal ion and not to others, but this degree of selectivity is very hard to achieve. The synthetic drug [[deferiprone]] achieves selectivity by having two oxygen donor atoms so that it binds to Fe<sup>3+</sup> in preference to any of the other divalent ions that are present in the human body, such as Mg<sup>2+</sup>, Ca<sup>2+</sup> and Zn<sup>2+</sup>. Treatment of poisoning by ions such as Pb<sup>2+</sup> and Cd<sup>2+</sup> is much more difficult since these are both divalent ions and selectivity is harder to accomplish.<ref>{{cite journal|last1=Arena|first1=G.|last2=Contino|first2=A.|last3=Longo |first3=E.|last4=Sciotto|first4=D. |last5=Spoto|first5=G. |year=2001|title=Selective complexation of soft Pb<sup>2+</sup> and Hg<sup>2+</sup> by a novel allyl functionalized thioamide calix[4]arene in 1,3-alternate conformation: a UV-visible and H-1 NMR spectroscopic investigation|journal=J. Chem. Soc., Perkin Trans.|volume= 2|issue=12|pages=2287–2291|doi=10.1039/b107025h}}</ref> Excess copper in [[Wilson's disease]] can be removed by [[penicillamine]] or [[Triethylene tetramine]] (TETA). [[DTPA]] has been approved by the [[U.S. Food and Drug Administration]] for treatment of [[plutonium]] poisoning.

DTPA is also used as a complexing agent for [[gadolinium]] in MRI [[MRI contrast agent|contrast enhancement]]. The requirement in this case is that the complex be very strong, as Gd<sup>3+</sup> is very toxic. The large stability constant of the octadentate ligand ensures that the concentration of free Gd<sup>3+</sup> is almost negligible, certainly well below toxicity threshold.<ref>{{cite book|last1=Runge|first1=V. M.|last2=Scott|first2=S. |title=Contrast-enhanced Clinical Magnetic Resonance Imaging |publisher=University Press of Kentucky|year=1998|isbn=0-8131-1944-8}}</ref> In addition the ligand occupies only 8 of the 9 coordination sites on the gadolinium ion. The ninth site is occupied by a water molecule which exchanges rapidly with the fluid surrounding it and it is this mechanism that makes the [[paramagnetic]] complex into a contrast reagent.

[[EDTA]] forms such strong complexes with most divalent cations that it finds many [[edta#Uses|uses]]. For example, it is often present in washing powder to act as a water softener by sequestering calcium and magnesium ions.

The selectivity of macrocyclic ligands can be used as a basis for the construction of an [[ion selective electrode]]. For example, [[potassium selective electrode]]s are available that make use of the naturally occurring macrocyclic antibiotic [[valinomycin]].

<!-- <gallery widths="100px" heights="100px" perrow="6">
Image:Deferiprone.svg|Deferiprone
Image:Penicillamine structure.png|penicillamine
Image:Triethylene tetramine.png|triethylenetetramine, TETA 
Image:EDTA.svg|Ethylenediamine tetracetic acid, EDTA
Image:Diethylentriaminpentaessigsäure.svg|diethylenetriaminepentacetic acid DTPA
Image:Valinomycin.svg|Valinomycin
</gallery> -->
{|class="wikitable"| border="0"|-
|[[File:Deferiprone.svg|center|75px]]
|[[File:Penicillamine structure.png|center|100px]]
|[[File:Triethylene tetramine.png|center|150px]] 
|[[File:EDTA.svg|center|120px]]
|-
|align="center"|[[Deferiprone]]
|align="center"|[[Penicillamine]]
|align="center"|[[triethylenetetramine]], TETA
|align="center"|[[Ethylenediaminetetraacetic acid]], EDTA
|-
|[[File:Diethylentriaminpentaessigsäure.svg|center|170px]]
|[[File:Valinomycin.svg|center|200px]]
|[[File:Tributyl-phosphate-2D-skeletal.png|center|170px]]
|-
|align="center"|[[Diethylenetriaminepentaacetic acid]], DTPA
|align="center"|[[Valinomycin]]
|align="center"|[[Tributyl phosphate|Tri-''n''-butyl phosphate]]
|}
An [[ion-exchange resin]] such as [[chelex 100]], which contains chelating ligands bound to a [[polymer]], can be used in [[water softener]]s and in [[chromatography|chromatographic]] separation techniques. In [[solvent extraction]] the formation of electrically neutral complexes allows cations to be extracted into organic solvents. For [[Liquid-liquid extraction#Solvation mechanism|example]], in [[nuclear reprocessing|nuclear fuel reprocessing]] [[uranium]](VI) and [[plutonium]](VI) are extracted into kerosene as the complexes [MO<sub>2</sub>(TBP)<sub>2</sub>(NO<sub>3</sub>)<sub>2</sub>] (TBP = [[tributyl phosphate|tri-''n''-butyl phosphate]]). In [[phase-transfer catalysis]], a substance which is insoluble in an organic solvent can be made soluble by addition of a suitable ligand. For example, [[potassium permanganate]] oxidations can be achieved by adding a catalytic quantity of a crown ether and a small amount of organic solvent to the aqueous reaction mixture, so that the oxidation reaction occurs in the organic phase.

In all these examples, the ligand is chosen on the basis of the stability constants of the complexes formed. For example, TBP is used in [[nuclear fuel]] reprocessing because (among other reasons) it forms a complex strong enough for solvent extraction to take place, but weak enough that the complex can be destroyed by [[nitric acid]] to recover the [[uranyl]] cation as [[Nitrate|nitrato]] complexes, such as [UO<sub>2</sub>(NO<sub>3</sub>)<sub>4</sub>]<sup>2−</sup> back in the aqueous phase.

=== Supramolecular complexes ===
[[Supramolecular]] complexes are held together by hydrogen bonding, hydrophobic forces, van der Waals forces, π-π interactions, and electrostatic effects, all of which can be described as [[noncovalent bonding]]. Applications include [[molecular recognition]], [[host–guest chemistry]] and anion [[sensors]].

A typical application in molecular recognition involved the determination of formation constants for complexes formed between a tripodal substituted [[urea]] molecule and various [[saccharide]]s.<ref>{{cite journal|last1=Vacca|first1=A.|last2=Nativi|first2= C. |last3=Cacciarini|first3= M. |last4=Pergoli|first4= R. |last5= Roelens|first5= S. |year=2004|title=A New Tripodal Receptor for Molecular Recognition of Monosaccharides. A Paradigm for Assessing Glycoside Binding Affinities and Selectivities by <sup>1</sup>H NMR Spectroscopy|journal=J. Am. Chem. Soc.|volume=126|pages=16456–16465|doi= 10.1021/ja045813s|pmid=15600348|issue=50}}</ref> The study was carried out using a non-aqueous solvent and [[NMR]] chemical shift measurements. The object was to examine the selectivity with respect to the saccharides.

An example of the use of [[supramolecular chemistry|supramolecular]] complexes in the development of [[Molecular sensor|chemosensor]]s is provided by the use of transition-metal ensembles to sense for [[adenosine triphosphate|ATP]].<ref>{{cite journal|last1=Marcotte|first1=N.|last2=Taglietti |first2=A. |year=2003|title=Transition-metal-based chemosensing ensembles: ATP sensing in physiological conditions|journal= Supramol. Chem.|volume=15|issue=7|pages=617–717|doi=10.1080/10610270310001605205 |s2cid=96018200 }}</ref>

Anion complexation can be achieved by [[Molecular encapsulation|encapsulating]] the anion in a suitable cage. Selectivity can be engineered by designing the shape of the cage. For example, dicarboxylate anions could be encapsulated in the ellipsoidal cavity in a large macrocyclic structure containing two metal ions.<ref>{{cite journal|last1=Boiocchi|first1=M.|last2=Bonizzoni|first2= M.|last3=Fabbrizzi|first3= L. |last4=Piovani|first4= G. |last5= Taglietti|first5= A. |year=2004|title=A dimetallic cage with a long ellipsoidal cavity for the fluorescent detection of dicarboxylate anions in water|journal=Angew. Chem. Int. Ed.|volume=43|pages=3847–3852|doi=10.1002/anie.200460036|pmid=15258953|issue=29 }}</ref>

== Experimental methods ==
{{main|Determination of equilibrium constants}}
The method developed by Bjerrum is still the main method in use today, though the precision of the measurements has greatly increased. Most commonly, a solution containing the metal ion and the ligand in a medium of high [[ionic strength]] is first acidified to the point where the ligand is fully [[Protonation|protonated]]. This solution is then [[titration|titrated]], often by means of a computer-controlled auto-titrator, with a solution of [[carbon dioxide|CO<sub>2</sub>]]-free base. The concentration, or [[Chemical activity|activity]], of the hydrogen ion is monitored by means of a glass electrode. The data set used for the calculation has three components: a statement defining the nature of the chemical species that will be present, called the model of the system, details concerning the concentrations of the reagents used in the titration, and finally the experimental measurements in the form of titre and pH (or [[Electromotive force|emf]]) pairs.

Other [[ion-selective electrode]]s (ISE) may be used. For example, a [[fluoride selective electrode|fluoride electrode]] may be used with the determination of stability complexes of fluoro-complexes of a metal ion.

It is not always possible to use an ISE. If that is the case, the titration can be monitored by other types of measurement. [[Ultraviolet–visible spectroscopy]], [[fluorescence spectroscopy]] and [[NMR spectroscopy]] are the most commonly used alternatives. Current practice is to take absorbance or fluorescence measurements at a range of wavelengths and to fit these data simultaneously. Various NMR [[chemical shift]]s can also be fitted together.

The chemical model will include values of the protonation constants of the ligand, which will have been determined in separate experiments, a value for log&nbsp;''K''<sub>w</sub> and estimates of the unknown stability constants of the complexes formed. These estimates are necessary because the calculation uses a [[least squares#Non-linear least squares|non-linear least-squares]] algorithm. The estimates are usually obtained by reference to a chemically similar system. The stability constant databases<ref name=LDP/><ref name=NIST/> can be very useful in finding published stability constant values for related complexes.

In some simple cases the calculations can be done in a spreadsheet.<ref>{{cite book|last=Billo|first=E. J.|title=Excel for Chemists: A comprehensive guide|publisher=Wiley-VCH|year=1997|edition=2nd|chapter=Chapter 22|isbn=0-471-18896-4|chapter-url-access=registration|chapter-url=https://archive.org/details/unset0000unse_s1y5|url-access=registration|url=https://archive.org/details/unset0000unse_s1y5}}</ref> Otherwise, the calculations are performed with the aid of a general-purpose computer programs. The most frequently used programs are:
* Potentiometric and/or spectrophotometric data: PSEQUAD<ref>{{cite book|last1=Zekany|first1=L.|last2=Nagypál|first2= I. |title=Computational methods for the determination of formation constants|editor=Leggett|publisher=Plenum|year=1985|chapter=Chapter 8: PSEQUAD: A comprehensive program for the evaluation of potentiometric and/or spectrophotometric equilibrium data using analytical derivatives|isbn=0-306-41957-2}}</ref>
*Potentiometric data: [http://www.hyperquad.co.uk/HQ2013.htm HYPERQUAD],<ref>{{cite journal|last1=Gans|first1=P.|last2=Sabatini|first2= A. |last3=Vacca|first3= A. |year=1996|title=Investigation of equilibria in solution. Determination of equilibrium constants with the HYPERQUAD suite of programs|journal=Talanta|volume=43|pages=1739–1753|doi=10.1016/0039-9140(96)01958-3|pmid=18966661|issue=10}}</ref> BEST,<ref>{{cite book|first1=A. E.|last1= Martell |first2= R. J. |last2=Motekaitis|title=The determination and use of stability constants|publisher=Wiley-VCH|date=1992}}</ref> [http://jplusconsulting.com/products/reactlab-equilibria/ ReactLab pH PRO]
* Spectrophotometric data: [http://www.hyperquad.co.uk/HypSpec.htm HypSpec], SQUAD,<ref>{{cite book|last=Leggett|first=D.|title=Computational methods for the determination of formation constants|editor=Leggett|publisher=Plenum|year=1985|chapter=Chapter 6: SQUAD: Stability quotients from absorbance data|isbn=0-306-41957-2}}</ref> SPECFIT,<ref>{{cite journal|last1=Gampp|first1=M.|last2=Maeder|first2= M. |last3=Mayer|first3= C. J. |last4= Zuberbühler|first4= A. D.|title=Calculation of equilibrium constants from multiwavelength spectroscopic data—I : Mathematical considerations |journal=Talanta|volume=32|pages=95–101|doi=10.1016/0039-9140(85)80035-7|year=1985|pmid=18963802|issue=2}}</ref><ref>{{cite journal|last1=Gampp|first1=M.|last2=Maeder|first2= M. |last3=Mayer|first3= C. J. |last4= Zuberbühler|first4= A. D.|title=Calculation of equilibrium constants from multiwavelength spectroscopic data—II1: Specfit: two user-friendly programs in basic and standard fortran 77|journal=Talanta|volume=32|pages=251–264 |doi=10.1016/0039-9140(85)80077-1|year=1985|pmid=18963840|issue=4}}</ref> [http://jplusconsulting.com/products/reactlab-equilibria/ ReactLab EQUILIBRIA].<ref>Jplus Consulting Pty Ltd</ref>
* NMR data [http://www.hyperquad.co.uk/hypnmr.htm HypNMR],<ref>{{cite journal|last1=Frassineti|first1=C.|last2=Alderighi|first2=L. |last3=Gans|first3= P. |last4=Sabatini|first4=A. |last5=Vacca|first5=A. |last6= Ghelli|first6= S. |year=2003|title=Determination of protonation constants of some fluorinated polyamines by means of <sup>13</sup>C NMR data processed by the new computer program HypNMR2000. Protonation sequence in polyamines|journal=Anal. Bioanal. Chem.|volume=376|pages=1041–1052|doi=10.1007/s00216-003-2020-0|pmid=12845401|issue=7|s2cid=14533024 }}</ref> [http://www.nuigalway.ie/chem/Mike/wineqnmr.htm WINEQNMR2] {{Webarchive|url=https://web.archive.org/web/20190714173527/http://www.nuigalway.ie/chem/Mike/wineqnmr.htm |date=2019-07-14 }}<ref>{{cite journal|last=Hynes|first=M. J.|year=1993|title=EQNMR: A computer program for the calculation of stability constants from nuclear magnetic resonance chemical shift data|journal=J. Chem. Soc., Dalton Trans.|issue=2|pages=311–312|doi=10.1039/DT9930000311}}</ref>
In biochemistry, formation constants of adducts may be obtained from [[Isothermal titration calorimetry]] (ITC) measurements. This technique yields both the stability constant and the standard enthalpy change for the equilibrium.<ref>{{cite book |last1=O'Brien |first1=R. |last2=Ladbury|first2= J. E. |last3=Chowdry|first3= B. Z.  |title=Protein-Ligand interactions: hydrodynamics and calorimetry |editor1-last=Harding|editor1-first= S. E.|editor2-last= Chowdry|editor2-first= B. Z. |publisher=Oxford University Press |year=2000 |chapter=Chapter 10 |isbn=0-19-963749-0 }}</ref> It is mostly limited, by availability of software, to complexes of 1:1 stoichiometry.

==Critically evaluated data==

The following references are for critical reviews of published stability constants for various classes of ligands. All these reviews are published by [[IUPAC]] and the full text is available, free of charge, in pdf format.

*[[ethylenediamine]] (en) <ref>{{cite journal|last=Paoletti|first=P.|year=1984|title=Formation of metal complexes with ethylenediamine: a critical survey of equilibrium constants, enthalpy and entropy values|journal=Pure Appl. Chem.|volume= 56|issue=4|pages=491–522|doi=10.1351/pac198456040491|s2cid=5600085 | url=http://www.iupac.org/publications/pac/pdf/1984/pdf/5604x0491.pdf}}</ref>
*[[Nitrilotriacetic acid]] (NTA)<ref>{{cite journal|last=Anderegg|first=G.|year=1982|title=Critical survey of stability constants of NTA complexes|journal=Pure Appl. Chem.|volume=54|issue=12|pages=2693–2758|doi=10.1351/pac198254122693|s2cid=44040841 |url=http://iupac.org/publications/pac/pdf/1982/pdf/5412x2693.pdf}}</ref>
*[[aminopolycarboxylic acid]]s (complexones)<ref>{{cite journal|last1=Anderegg|first1=G |last2=Arnaud-Neu|first2=F. |last3=Delgado|first3=R. |last4=Felcman|first4=J. |last5=Popov|first5= K.|year=2003|title=Critical evaluation of stability constants of metal complexes of complexones for biomedical and environmental applications (IUPAC Technical Report)|journal=Pure Appl. Chem.|volume=77|issue=8|pages=1445–95|doi=10.1351/pac200577081445|s2cid=94957415 |url=http://iupac.org/publications/pac/pdf/2005/pdf/7708x1445.pdf}}</ref>
*[[Alpha hydroxy acid]]s and other hydroxycarboxylic acids<ref>{{cite journal|last1=Lajunen|first1=L. H. J.|last2=Portanova|first2=R. |last3=Piispanen|first3= J. |last4= Tolazzi|first4= M.  |year=1997|title=Critical evaluation of stability constants for alpha-hydroxycarboxylic acid complexes with protons and metal ions and the accompanying enthalpy changes. Part I: Aromatic ortho-hydroxycarboxylic acids (Technical Report)|journal=Pure Appl. Chem.|volume=69|issue=2|pages=329–382|doi=10.1351/pac199769020329|s2cid=100224661 |url=http://iupac.org/publications/pac/pdf/1997/pdf/6902x0329.pdf
}}</ref><ref>{{cite journal |last1=Portanova |first1=R |last2=Lajunen|first2= L. H. J. |last3=Tolazzi|first3= M. |last4= Piispanen|first4= J.  |year=2003|title=Critical evaluation of stability constants for alpha-hydroxycarboxylic acid complexes with protons and metal ions and the accompanying enthalpy changes. Part II. Aliphatic 2-hydroxycarboxylic acids (IUPAC Technical Report)|journal=Pure Appl. Chem.
|volume=75|issue=4|pages=495–540|doi=10.1351/pac200375040495|s2cid=97662244 |url=http://iupac.org/publications/pac/pdf/2003/pdf/7504x0495.pdf}}</ref>

*[[crown ether]]s<ref>{{cite journal
|last1=Arnaud-Neu
|first1=F.
|last2=Delgado|first2= R. |last3=Chaves|first3= S.
 |year=2003
|title=Critical evaluation of stability constants and thermodynamic functions of metal complexes of crown ethers (IUPAC Technical Report)
|journal=Pure Appl. Chem.
|volume=75
|issue=1
|pages=71–102
|doi=10.1351/pac200375010071
|s2cid=97549495
|url=http://iupac.org/publications/pac/pdf/2003/pdf/7501x0071.pdf
}}</ref>

*[[phosphonic acid]]s<ref>{{cite journal
|last1=Popov
|first1=K.
|last2=Rönkkömäki|first2= H. |last3=Lajunen|first3= L. H. J. 
|year=2001
|title=Critical evaluation of stability constants of phosphonic acids (IUPAC Technical Report)|journal=Pure Appl. Chem.
|volume=73
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|doi=10.1351/pac200173101641
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}}</ref><ref>{{cite journal
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*[[imidazole]]s and [[histamine]]s<ref>{{cite journal
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|journal=Pure Appl. Chem.
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|doi=10.1351/pac199769071549
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}}</ref>

*[[amino acid]]s with polar side-chains<ref>{{cite journal
|last=Berthon
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|title=Critical evaluation of the stability constants of metal complexes of amino acids with polar side chains (Technical Report)
|journal=Pure Appl. Chem.
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|doi=10.1351/pac199567071117
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}}</ref>

*[[nucleotides]]<ref>{{cite journal
|last1=Smith
|first1=R. M.
|last2=Martell |first2=A. E. |last3=Chen|first3= Y.
 |year=1991
|title=Critical evaluation of stability constants for nucleotide complexes with protons and metal ions and the accompanying enthalpy changes
|journal=Pure Appl. Chem.
|volume=63
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}}</ref>

*[[acetylacetone]]<ref>{{cite journal
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}}</ref>

*general<ref>{{cite journal
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}}</ref>

*Chemical speciation of environmentally significant heavy metals with inorganic ligands. Part 1: The Hg<sup>2+</sup>–Cl<sup>−</sup>, OH<sup>−</sup>, {{chem|CO|3|2−}}, {{chem|SO|4|2−}}, and {{chem|PO|4|3−}} systems.<ref>{{cite journal|last1=Powell|first1=Kipton, J.|last2=Brown|first2=Paul L. |last3=Byrne|first3=Robert H. |last4=Gajda|first4=Tamás |last5=Hefter|first5=Glenn |last6=Sjöberg|first6= Staffan |last7= Wanner|first7= Hans |year=2005|title=Chemical speciation of environmentally significant heavy metals with inorganic ligands. Part 1: The Hg<sup>2+</sup>–Cl<sup>−</sup>, OH<sup>−</sup>, {{chem|CO|3|2−}}, {{chem|SO|4|2−}}, and {{chem|PO|4|3−}} aqueous systems|journal=Pure Appl. Chem.|volume=77|issue=4|pages=739–800|doi=10.1351/pac200577040739|s2cid=93650787 |url=http://www.iupac.org/publications/pac/pdf/2005/pdf/7704x0739.pdf}}</ref>
*Chemical speciation of environmentally significant metals with inorganic ligands Part 2: The Cu<sup>2+</sup>–OH<sup>−</sup>, Cl<sup><sup>−</sup></sup>, {{chem|CO|3|2−}}, {{chem|SO|4|2−}}, and {{chem|PO|4|3−}} aqueous systems<ref>{{cite journal|last1=Powell|first1=Kipton J.|last2=Brown|first2=Paul L. |last3=Byrne|first3=Robert H. |last4=Gajda|first4=Tamás |last5=Hefter|first5=Glenn |last6=Sjöberg|first6= Staffan |last7= Wanner|first7= Hans |year=2007|title=Chemical speciation of environmentally significant metals with inorganic ligands Part 2: The Cu<sup>2+</sup>–OH<sup>−</sup>, Cl<sup><sup>−</sup></sup>, {{chem|CO|3|2−}}, {{chem|SO|4|2−}}, and {{chem|PO|4|3−}} systems|journal=Pure Appl. Chem.|volume=79|issue=5|pages=895–950|doi=10.1351/pac200779050895|s2cid=195819532 |url=http://www.iupac.org/publications/pac/pdf/2007/pdf/7905x0895.pdf}}</ref>
*Chemical speciation of environmentally significant metals with inorganic ligands Part 3: The Pb<sup>2+</sup>–OH<sup>−</sup>, Cl<sup><sup>−</sup></sup>, {{chem|CO|3|2−}}, {{chem|SO|4|2−}}, and {{chem|PO|4|3−}} systems<ref>{{cite journal|last1=Powell|first1=Kipton J.|last2=Brown|first2=Paul L. |last3=Byrne|first3=Robert H. |last4=Gajda|first4=Tamás |last5=Hefter|first5=Glenn |last6=Leuz|first6= Ann-Kathrin |last7=Sjöberg|first7= Staffan |last8= Wanner|first8= Hans |year=2009|title=Chemical speciation of environmentally significant metals with inorganic ligands Part 3: The Pb<sup>2+</sup>–OH<sup>−</sup>, Cl<sup><sup>−</sup></sup>, {{chem|CO|3|2−}}, {{chem|SO|4|2−}}, and {{chem|PO|4|3−}} systems|journal=Pure Appl. Chem.|volume=81|issue=12|pages=2425–2476 |doi=10.1351/PAC-REP-09-03-05|s2cid=93738375 |url=http://www.iupac.org/publications/pac/pdf/2009/pdf/8112x2425.pdf}}</ref>
*Chemical speciation of environmentally significant metals with inorganic ligands. Part 4: The Cd<sup>2+</sup>–OH<sup>−</sup>, Cl<sup>−</sup>, {{chem|CO|3|2−}}, {{chem|SO|4|2−}}, and {{chem|PO|4|3−}} systems<ref>{{cite journal|last1=Powell|first1=Kipton J.|last2=Brown|first2=Paul L. |last3=Byrne|first3=Robert H. |last4=Gajda|first4=Tamás |last5=Hefter|first5=Glenn |last6=Leuz|first6= Ann-Kathrin |last7=Sjöberg|first7= Staffan |last8= Wanner|first8= Hans |year=2011|title=Chemical speciation of environmentally significant metals with inorganic ligands. Part 4: The Cd<sup>2+</sup>–OH<sup>−</sup>, Cl<sup>−</sup>, {{chem|CO|3|2−}}, {{chem|SO|4|2−}}, and {{chem|PO|4|3−}} systems|journal=Pure Appl. Chem.|volume=83|issue=5|pages=1163–1214|doi=10.1351/PAC-REP-10-08-09|s2cid=195819251 |url=http://www.iupac.org/publications/pac/pdf/2011/pdf/8305x1163.pdf}}</ref>

==Databases==
*The [[Ki Database]] is a public domain database of published binding affinities (Ki) of drugs and chemical compounds for receptors, neurotransmitter transporters, ion channels, and enzymes.
*[[BindingDB]] is a public domain database of measured binding affinities, focusing chiefly on the interactions of protein considered to be drug-targets with small, drug-like molecules

== References ==
{{Reflist|30em}}

== Further reading ==
* {{cite book
|last1=Sigel
|first1=Roland K. O.
|last2=Skilandat
|first2=Miriam
|last3=Sigel
|first3=Astrid
|last4=Operschall
|first4=Bert P.
|last5=Sigel
|first5=Helmut
|editor1-first=Astrid |editor1-last=Sigel|editor2-first= Helmut |editor2-last=Sigel |editor3-first=Roland K. O. |editor3-last=Sigel
|title=Cadmium: From Toxicology to Essentiality
|series=Metal Ions in Life Sciences
|volume=11
|year=2013
|publisher=Springer
|pages=191–274
|chapter=Chapter 8. Complex formation of cadmium with [[Carbohydrate|sugar]] residues, [[nucleobase]]s, [[phosphate]]s, [[nucleotide]]s and [[nucleic acid]]s
|doi=10.1007/978-94-007-5179-8_8|pmid=23430775
|isbn=978-94-007-5178-1
|url=http://www.zora.uzh.ch/id/eprint/79639/2/MILS-11_ch8_sigel_final_re.pdf
}}
* {{cite book
|last1=Sóvágó
|first1=Imre
|last2=Várnagy
|first2=Katalin
|editor1-first=Astrid |editor1-last=Sigel|editor2-first= Helmut |editor2-last=Sigel |editor3-first=Roland K. O. |editor3-last=Sigel
|title=Cadmium: From Toxicology to Essentiality
|series=Metal Ions in Life Sciences
|volume=11
|year=2013
|publisher=Springer
|pages=275–302
|chapter=Chapter 9. Cadmium(II) complexes of amino acids and peptides
|doi=10.1007/978-94-007-5179-8_9|pmid=23430776
|isbn=978-94-007-5178-1
}}
* {{cite book |last1=Yatsimirsky |first1=Konstantin Borisovich |title=Instability Constants of Complex Compounds |year=1960 |publisher=OUP |last2=Vasilyev|first2= Vladimir Pavlovich|translator-first= D. A.|translator-last= Patterson}}

==External links==
*[http://www.hyperquad.co.uk Stability constants website]: Contains information on computer programs, applications, databases and hardware for experimental titrations.
{{Chemical equilibria}}

[[Category:Equilibrium chemistry]]
[[Category:Coordination chemistry]]