Stability constant

Introduction

 See Chemical equilibrium for definitions and derivations.

Stability constants, formation constants, binding constants, association constants and dissociation constants are all types of equilibrium constant. For a general reaction

αA + βB … ${\displaystyle \rightleftharpoons }$  σS + τT …

the equilibrium constant can be defined by

${\displaystyle K={\frac {{\{S\}}^{\sigma }{\{T\}}^{\tau }...}{{\{A\}}^{\alpha }{\{B\}}^{\beta }...}}}$ 

where {A} is the activity of the chemical species A etc. It is conventional to put the activities of the products in the numerator and those of the reactants in the denominator. Subject to certain conditions, (see Chemical equilibrium) one of more activity terms may be replaced by a concentration term. If all activities are replaced by concentrations a concentration quotient is obtained.

${\displaystyle K={\frac {{[S]}^{\sigma }{[T]}^{\tau }...}{{[A]}^{\alpha }{[B]}^{\beta }...}}}$ 

Types of equilibrium constants

Association and dissociation constants

In organic chemistry and biochemistry it is customary to use pK_a values for acid dissociation equilibria.

${\displaystyle pK_{a}=-\lg K_{diss}=\lg(1/K_{diss})}$ 

where K_diss is a stepwise acid dissociation constant (lg stands for log₁₀). On the other hand stability constants for metal complexes, and binding constants for host-guest complexes are generally expressed as association constants. When considering equilibria such as

${\displaystyle M+HL\rightleftharpoons ML+H}$ 

it is customary to use association constants for both ML and HL. Also, in generalised computer programs dealing with equilibrium constants it is general practice to use overall constants rather then stepwise constants and to omit ionic charges from equilibrium expressions. For example, if NTA, nitrilotriacetic acid, HC(CH₂CO₂H)₃ is designated as H₃L and forms complexes ML and MHL with a metal ion M, the following expressions would apply for the dissociation constants.

${\displaystyle H_{3}L\rightleftharpoons H_{2}L+H:pK_{1}=-\lg \left({\frac {[H_{2}L][H]}{[H_{3}L]}}\right)}$ 

${\displaystyle H_{2}L\rightleftharpoons HL+H:pK_{2}=-\lg \left({\frac {[HL][H]}{[H_{2}L]}}\right)}$ 

${\displaystyle HL\rightleftharpoons L+H:pK_{3}=-\lg \left({\frac {[L][H]}{[HL]}}\right)}$ 

The overall association constants can be expressed as

${\displaystyle L+H\rightleftharpoons HL:\lg \beta _{011}=\lg \left({\frac {[HL]}{[L][H]}}\right)=pK_{3}}$ 

${\displaystyle L+2H\rightleftharpoons H_{2}L:\lg \beta _{012}=\lg \left({\frac {[H_{2}L]}{[L][H]^{2}}}\right)=pK_{3}+pK_{2}}$ 

${\displaystyle L+3H\rightleftharpoons H_{3}L:\lg \beta _{013}=\lg \left({\frac {[H_{3}L]}{[L][H]^{3}}}\right)=pK_{3}+pK_{2}+pK_{1}}$ 

${\displaystyle M+L\rightleftharpoons ML:\lg \beta _{110}=\lg \left({\frac {[ML]}{[M][L]}}\right)}$ 

${\displaystyle M+L+H\rightleftharpoons MLH:\lg \beta _{111}=\lg \left({\frac {[MLH]}{[M][L][H]}}\right)}$ 

Note how the subscripts define the stoichiometry of the equilibrium product.

Stepwise formation constants

The stepwise constant for protonation of ML can be easily derived as follows.

${\displaystyle ML+H\rightleftharpoons MLH:[MLH]=K[ML][H]=K\beta _{110}[M][[L][H]:K=\beta _{111}/\beta _{110}}$ 

There is no agreed notation for stepwise constants of this kind, though a symbol such as ${\displaystyle K_{ML}^{MLH}}$  is sometimes found in the literature. It is best always to define a 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.

${\displaystyle ML+L'\rightleftharpoons ML'+L:[ML']=K{\frac {[ML][L']}{[L]}}:\beta _{ML'}=K\beta _{ML}}$ 

pH considerations

The correct definition of pH is in terms of the activity of the hydrogen ion

${\displaystyle pH=-\lg {H^{+}}}$ 

If, when determining an equilibrium constant, pH is measured, for example by a glass electrode, a mixed equilibrium constant may result.

${\displaystyle HL\rightleftharpoons L+H:pK=-\lg \left({\frac {[L]\{H\}}{[HL]}}\right)}$ 

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 accepted.

Hydrolysis constants

In aqueous solution the concentration of the hydroxide ion is related to the concentration of the hydrogen ion by

${\displaystyle K_{W}=[H][OH]:[OH]=K_{W}[H]^{-1}}$ 

The first step in metal ion hydrolysis can be expressed in two different ways

1. ${\displaystyle M(H_{2}O)\rightleftharpoons M(OH)+H:[M(OH)]=\beta ^{*}[M][H]^{-1}}$ 
2. ${\displaystyle M+OH\rightleftharpoons M(OH):[M(OH)]=K[M][OH]=KK_{W}[M][H]^{-1}}$ 

It follows that ${\displaystyle \beta *=KK_{W}}$ . Hydrolysis constants are usually reported in the ${\displaystyle \beta *}$  form and this leads to them appearing to have strange values. For example, if lgK=4 and lg K_W=-14, lg ${\displaystyle \beta ^{*}}$  = 4 -14 = -10. In general when the hydrolysis product contains n hydroxide groups lg ${\displaystyle \beta ^{*}}$  = lg K + n lg K_W

Conditional constants

Conditional constants, also known as apparent constants, are concentration quotients which are not true equilibrium constants but can be derived from them. 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

${\displaystyle K_{cond}={\frac {[{\mbox{Total Fe bound to EDTA}}]}{[{\mbox{Total Fe not bound to EDTA}}]\times [{\mbox{Total EDTA not bound to Fe}}]}}}$ 

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. Such constants are, by definition, conditional.

Experimental methods

A general equilibrium expression for an equilibrium constant

${\displaystyle K={\frac {{[S]}^{\sigma }{[T]}^{\tau }...}{{[A]}^{\alpha }{[B]}^{\beta }...}}}$ 

shows that it is a function of the concentrations [A], [B] etc. of the chemical species in equilibrium. The equilibrium constant value can be determined if any one of these concentrations can be measured. The general procedure is that the concentration in question is measured for a series of solutions with different compositions. The data are then treated by a more or less complicated mathematical procedure to get the values. There are four main experimental methods.

Potentiometric measurements

A free concentration [A] or activity {A} is measured by means of an ion selective electrode such as the glass electrode. If the electrode is calibrated using activity standards it is assumed that the Nernst equation applies in the form

E=E⁰+RT/nF ln{A}

where E⁰ is the standard electrode potential. When standard buffers solutions of known pH are used for calibration the meter reading will be pH. When the electrode is calibrated with solutions of known concentration a modified Nernst equation is assumed.

E=E⁰+s lg[A]

s an empirical slope factor

Absorbance measurements

It is assumed that the Beer-Lambert law applies.

${\displaystyle A=l\sum {\epsilon c}}$ 

where l is the optical path length, ${\displaystyle \epsilon }$  is a molar absorbance and c is a concentration. More than one of the species may contribute to the absorbance. In principle absorbance may be measured at one wavelength only, but in present-day practice it is common to record complete spectra.

Fluorescence (luminescence) measurements

It is assumed that the scattered light intensity is a linear function of species’ concentrations.

${\displaystyle I=\sum {\phi c}}$ 

where ${\displaystyle \phi }$  is a proportionality constant.

NMR chemical shift measurements

Chemical exchange is assumed to be rapid on the NMR time-scale. An individual chemical shift is the mol-fraction weighted average of the shift of contributing species.

${\displaystyle \Delta ={\frac {\sum c_{i}\delta _{i}}{\sum c_{i}}}}$ 

Calorimetric measurements

Simultaneous measurement of K and ${\displaystyle \Delta }$ H for 1:1 adducts is routinely carried out using Isothermal Titration Calorimetry. Extension to more complex systems is limited by the availability of suitable software.

Range and limitations

1. Potentiometry. The most widely used electrode is the glass electrode which is selective for the hydrogen ion. This is suitable for all acid-base equilibria. Lg  values between about 2 and 11 can be measured by potentiometric titration using a glass electrode. This enormous range is possible because of the logarithmic response of the electrode. The limitations arise because the Nernst equation breaks down at very low or very high pH.
2. Absorbance and Luminescence. An upper limit on lg ${\displaystyle \beta }$  of 4 is usually quoted, corresponding to the precision of the measurements, but it also depends on how intense the effect is. Spectra of contributing species should be clearly distinct from each other
3. NMR. Limited precision of chemical shift measurements also puts an upper limit of about 4 on lg ${\displaystyle \beta }$ . Limited to diamagnetic systems.
4. Calorimetry. Insufficient evidence is currently available.

Data sources

[IUPAC SC-Database] A comprehensive database of published data on equilibrium constants of metal complexes and ligands

[NIST Standard Reference Database 46] Critically Selected Stability Constants of Metal Complexes

[Inorganic and organic acids and bases] pKa Data in water and DMSO