Editing Equilibrium constant (section)

== Types of equilibrium constants ==

=== Cumulative and stepwise formation constants ===
A cumulative or overall constant, given the symbol ''β'', is the constant for the formation of a complex from reagents. For example, the cumulative constant for the formation of ML<sub>2</sub> is given by
:M + 2&nbsp;L {{eqm}} ML<sub>2</sub>; {{spaces|5}} [ML<sub>2</sub>] = ''β''<sub>12</sub>[M][L]<sup>2</sup>
The stepwise constant, ''K'', for the formation of the same complex from ML and L is given by
:ML + L {{eqm}} ML<sub>2</sub>; {{spaces|5}} [ML<sub>2</sub>] = ''K''[ML][L] = ''Kβ''<sub>11</sub>[M][L]<sup>2</sup>
It follows that
:''β''<sub>12</sub> = ''Kβ''<sub>11</sub>
A cumulative constant can always be expressed as the product of stepwise constants. There is no agreed notation for stepwise constants, though a symbol such as ''K''{{su|b=ML|p=L}} is sometimes found in the literature. It is best always to define each stability constant by reference to an equilibrium expression.

==== Competition method ====
A particular use of a stepwise constant is in the determination of stability constant values outside the normal range for a given method. For example, [[EDTA]] complexes of many metals are outside the range for the potentiometric method. The stability constants for those complexes were determined by competition with a weaker ligand.
:ML + L′ {{eqm}} ML′ + L {{spaces|5}} <math>[\mathrm{ML}']=K\frac{[\mathrm{ML}][\mathrm{L}']}{[\mathrm{L}]} = K \frac{\beta_\mathrm{ML}[\mathrm{M}][\mathrm{L}][\mathrm{L}']}{[\mathrm{L}]}= K \beta_\mathrm{ML}[\mathrm{M}][\mathrm{L}']; \quad \beta_{\mathrm{ML}'}=K\beta_\mathrm{ML}</math>
The formation constant of [[Palladium(II) cyanide|[Pd(CN)<sub>4</sub>]<sup>2−</sup>]] was determined by the competition method.

=== Association and dissociation constants ===
In organic chemistry and biochemistry it is customary to use p''K''<sub>a</sub> values for [[Acid dissociation constant|acid dissociation]] equilibria.
:<math>\mathrm{p}K_\mathrm{a}=-\log K_{\mathrm{diss}} = \log \left(\frac{1}{K_\mathrm{diss}}\right)\,</math>
where ''log'' denotes a logarithm to base 10 or [[common logarithm]], and ''K''<sub>diss</sub> is a stepwise [[acid dissociation constant]]. For bases, the [[Acid dissociation constant#Bases and basicity|base association constant]], p''K''<sub>b</sub> is used. For any given acid or base the two constants are related by {{math|1=p''K''<sub>a</sub> + p''K''<sub>b</sub> = p''K''<sub>w</sub>}}, so p''K''<sub>a</sub> can always be used in calculations.

On the other hand, stability constants for [[Coordination chemistry|metal complexes]], and binding constants for [[Host–guest chemistry|host–guest]] complexes are generally expressed as association constants. When considering equilibria such as
:M + HL {{eqm}} ML + H
it is customary to use association constants for both ML and HL. Also, in generalized computer programs dealing with equilibrium constants it is general practice to use cumulative constants rather than stepwise constants and to omit ionic charges from equilibrium expressions. For example, if NTA, [[nitrilotriacetic acid]], N(CH<sub>2</sub>CO<sub>2</sub>H)<sub>3</sub> is designated as H<sub>3</sub>L and forms complexes ML and MHL with a metal ion M, the following expressions would apply for the dissociation constants.
:<math chem>\begin{array}{ll}
\ce{H3L <=> {H2L} + H}; & \ce{p}K_1=-\log \left(\frac{[\ce{H2L}][\ce{H}]} {[\ce{H3L}]} \right)\\
\ce{H2L <=> {HL} + H}; & \ce{p}K_2=-\log \left(\frac{[\ce{HL}][\ce{H}]} {[\ce{H2L}]} \right)\\
\ce{HL <=> {L} + H}; & \ce{p}K_3=-\log \left(\frac{[\ce{L}][\ce{H}]} {[\ce{HL}]} \right)
\end{array}</math>
The cumulative association constants can be expressed as
:<math chem>\begin{array}{ll}
\ce{{L} + H <=> HL}; & \log \beta_{011} =\log \left(\frac{[\ce{HL}]}{[\ce{L}][\ce{H}]} \right)=\ce{p}K_3 \\
\ce{{L} + 2H <=> H2L}; & \log \beta_{012} =\log \left(\frac{[\ce{H2L}]}{[\ce{L}][\ce{H}]^2} \right)=\ce{p}K_3+\ce{p}K_2 \\
\ce{{L} + 3H <=> H3L}; & \log \beta_{013} =\log \left(\frac{[\ce{H3L}]}{[\ce{L}][\ce{H}]^3} \right)=\ce{p}K_3+\ce{p}K_2+\ce{p}K_1 \\
\ce{{M} + L <=> ML}; & \log \beta_{110} =\log \left(\frac{[\ce{ML}]}{[\ce{M}][\ce{L}]} \right)\\
\ce{{M} + {L} + H <=> MLH}; & \log \beta_{111} =\log \left(\frac{[\ce{MLH}]}{[\ce{M}][\ce{L}][\ce{H}]} \right)
\end{array}</math>
Note how the subscripts define the stoichiometry of the equilibrium product.

=== Micro-constants ===
When two or more sites in an asymmetrical molecule may be involved in an equilibrium reaction there are more than one possible equilibrium constants. For example, the molecule [[Levodopa|{{small|L}}-DOPA]] has two non-equivalent hydroxyl groups which may be deprotonated. Denoting {{small|L}}-DOPA as LH<sub>2</sub>, the following diagram shows all the species that may be formed (X = {{chem|CH|2|CH(NH|2|)CO|2|H}}).

:[[Image:Micro constants.png|300px]]

The concentration of the species LH is equal to the sum of the concentrations of the two micro-species with the same chemical formula, labelled L<sup>1</sup>H and L<sup>2</sup>H. The constant ''K''<sub>2</sub> is for a reaction with these two micro-species as products, so that [LH] = [L<sup>1</sup>H] + [L<sup>2</sup>H] appears in the numerator, and it follows that this '''macro-constant''' is equal to the sum of the two '''micro-constants''' for the component reactions.
:''K''<sub>2</sub> = ''k''<sub>21</sub> + ''k''<sub>22</sub>
However, the constant ''K''<sub>1</sub> is for a reaction with these two micro-species as reactants, and [LH] = [L<sup>1</sup>H] + [L<sup>2</sup>H] in the denominator, so that in this case<ref>{{cite journal |last1=Splittgerber |first1=A. G. |last2=Chinander |first2=L.L. |title=The spectrum of a dissociation intermediate of cysteine: a biophysical chemistry experiment |journal=Journal of Chemical Education |date=1 February 1988 |volume=65 |issue=2 |page=167 |doi=10.1021/ed065p167 |bibcode=1988JChEd..65..167S }}</ref>
:1/''K''<sub>1</sub> =1/ ''k''<sub>11</sub> + 1/''k''<sub>12</sub>,
and therefore ''K''<sub>1</sub> =''k''<sub>11</sub> ''k''<sub>12</sub> / (''k''<sub>11</sub> + ''k''<sub>12</sub>).
Thus, in this example there are four micro-constants whose values are subject to two constraints; in consequence, only the two macro-constant values, for K<sub>1</sub> and K<sub>2</sub> can be derived from experimental data.

Micro-constant values can, in principle, be determined using a spectroscopic technique, such as [[infrared spectroscopy]], where each micro-species gives a different signal. Methods which have been used to estimate micro-constant values include
* Chemical: blocking one of the sites, for example by methylation of a hydroxyl group, followed by determination of the equilibrium constant of the related molecule, from which the micro-constant value for the "parent" molecule may be estimated.
* Mathematical: applying numerical procedures to <sup>13</sup>C&nbsp;NMR data.<ref>{{cite journal |pages=265–70 |doi=10.1039/P29940000265 |title=Protonation sequence of linear aliphatic polyamines by 13C NMR spectroscopy |year=1994 |last1=Hague |first1=David N. |last2=Moreton |first2=Anthony D. |journal=J. Chem. Soc., Perkin Trans.|volume= 2 |issue=2}}</ref><ref>{{cite journal |pages=3272–9 |doi=10.1021/ac991494p |title=A Cluster Expansion Method for the Complete Resolution of Microscopic Ionization Equilibria from NMR Titrations |year=2000 |last1=Borkovec |first1=Michal |last2=Koper |first2=Ger J. M. |journal=Anal. Chem. |volume=72 |issue=14 |pmid=10939399}}</ref>

Although the value of a micro-constant cannot be determined from experimental data, site occupancy, which is proportional to the micro-constant value, can be very important for biological activity. Therefore, various methods have been developed for estimating micro-constant values. For example, the isomerization constant for {{small|L}}-DOPA has been estimated to have a value of 0.9, so the micro-species L<sup>1</sup>H and L<sup>2</sup>H have almost equal concentrations at all [[pH]] values.

=== pH considerations (Brønsted constants) ===
[[pH]] is defined in terms of the [[Activity (chemistry)|activity]] of the hydrogen ion
:pH = −log<sub>10</sub> {H<sup>+</sup>}
In the approximation of ideal behaviour, activity is replaced by concentration. pH is measured by means of a glass electrode, a mixed equilibrium constant, also known as a Brønsted constant, may result.
:HL {{eqm}} L + H; {{spaces|5}} <math>\mathrm{p}K =-\log \left(\frac{[\mathrm{L}]\{\mathrm{H}\}}{[\mathrm{HL}]} \right) </math>
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
:<chem>\mathit{K}_W =[H][OH]</chem>
:<chem>[OH]=\mathit{K}_W[H]^{-1}</chem>
The first step in metal ion [[hydrolysis]]<ref name=BM>{{cite book|first1 = C. F. |last1 = Baes |first2 = R. E. |last2 = Mesmer |title=The Hydrolysis of Cations|publisher= Wiley|date= 1976|chapter= Chapter 18. Survey of Hydrolysis Behaviour|pages= 397–430}}</ref> can be expressed in two different ways
:<math chem>\begin{cases}
\ce{M(H2O) <=> {M(OH)} + H}; &[\ce{M(OH)}]=\beta^*[\ce{M}][\ce{H}]^{-1} \\
\ce{{M} + OH <=> M(OH)}; &[\ce{M(OH)}]=K[\ce{M}][\ce{OH}]=K K_\ce{W}[\ce{M}][\ce{H}]^{-1}
\end{cases}</math>
It follows that {{math|1=''β''<sup>*</sup> = ''KK''<sub>W</sub>}}. Hydrolysis constants are usually reported in the ''β''<sup>*</sup> form and therefore often have values much less than 1. For example, if {{math|1=log ''K'' = 4}} and {{math|1=log K<sub>W</sub> = −14,}} {{math|1=log ''β''<sup>*</sup> = 4 + (−14) = −10}} so that ''β<sup>*</sup>'' = 10<sup>−10</sup>. In general when the hydrolysis product contains ''n'' hydroxide groups {{math|1=log ''β''<sup>*</sup> = log ''K'' + ''n'' log ''K''<sub>W</sub>}}

=== Conditional constants ===
Conditional constants, also known as apparent constants, are concentration quotients which are not true equilibrium constants but can be derived from them.<ref>{{cite book|first1=G. |last1=Schwarzenbach |first2=H. |last2=Flaschka |title=Complexometric titrations|publisher=Methuen|date=1969}}{{page needed|date=November 2011}}</ref> 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
:<math>K_{\mathrm{cond}}=\frac{[\mbox{Total Fe bound to EDTA}]}{[\mbox{Total Fe not bound to EDTA}]\times [\mbox{Total EDTA not bound to Fe}] }</math>
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.

=== Gas-phase equilibria ===
For equilibria in a [[gas phase]], [[fugacity]], ''f'', is used in place of activity. However, fugacity has the [[dimension]] of [[pressure]], so it must be divided by a standard pressure, usually 1 bar, in order to produce a dimensionless quantity, {{sfrac|''f''|''p''<sup><s>o</s></sup>}}. An equilibrium constant is expressed in terms of the dimensionless quantity. For example, for the equilibrium 2NO<sub>2</sub> {{eqm}} N<sub>2</sub>O<sub>4</sub>,
:<math>\frac{f_\mathrm{N_2O_4}}{p^\ominus} = K \left(\frac{f_\mathrm{NO_2}}{p^\ominus}\right)^2</math>
Fugacity is related to [[partial pressure]], ''<math>p_X</math>'', by a dimensionless fugacity coefficient ''ϕ'':  ''<math>f_X = \phi_X p_X</math>''. Thus, for the example,
:<math>K=\frac{\phi_\mathrm{N_2O_4} p_\mathrm{N_2O_4}/{p^\ominus}}{\left(\phi_\mathrm{NO_2}p_\mathrm{NO_2}/{p^\ominus}\right)^2}</math>
Usually the standard pressure is omitted from such expressions. Expressions for equilibrium constants in the gas phase then resemble the expression for solution equilibria with fugacity coefficient in place of activity coefficient and partial pressure in place of concentration.
:<math>K=\frac{\phi_\mathrm{N_2O_4} p_\mathrm{N_2O_4}}{\left(\phi_\mathrm{NO_2}p_\mathrm{NO_2}\right)^2}</math>