Determination of equilibrium constants

• See Chemical equilibrium and stability constant for definitions and derivations

Experimental methods

A general expression for an equilibrium constant, expressed as a concentration quotient,

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

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 known analytical concentrations of the reactants. Typically, a titration is performed with one or more reactants in the titration vessel and one or more reactants in the burette. Knowing the analytical ooncentrations of reactants initially in the reaction vessel and in the burette, all analytical concentrations can be derived as a function of the volume (or mass) of titrant added.

There are four main experimental methods. For less commonly used methods see Rossotti and Rossotti

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 buffer solutions of known pH are used for calibration the meter reading will be pH. See IUPAC Compendium of Analytical Nomenclature, Table 8.4.2for primary pH standards.

When the electrode is calibrated with solutions of known concentration, by means of a strong acid/strong base titration, for example, a modified Nernst equation is assumed.

E=E⁰+s lg[A]

s an empirical slope factor

Spectrophotometric measurements

Absorbance

It is assumed that the Beer-Lambert law applies.

$A=\ell \sum {\epsilon c}$

where $\ell$ is the optical path length, $\epsilon$ is a molar absorbance at unit path length 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) intensity

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

$I=\sum {\phi c}$

where $\phi$ 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.

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

Calorimetric measurements

Simultaneous measurement of K and $\Delta$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 $\beta$ values between about 2 and 11 can be measured directly 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. The range can be extended by using the competition method.
2. Absorbance and Luminescence. An upper limit on lg $\beta$ 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 $\beta$. Limited to diamagnetic systems.
4. Calorimetry. Insufficient evidence is currently available.

Computational methods

It is assumed that the experimental data which have been collected comprise a set of data points. At each data point, i, the analytical concentrations of the reactants, T_A(i), T_B(i) etc. are known along with a measured quantity, y_i. A general computational procedure has three main components.

1. Definion of a chemical model of the equilibria
2. Calculation of the concentrations of all the chemical species in each solution
3. Refinement of the equilibrium constants

The chemical model

The chemical model consists of a set of reactants and the complexes formed from them. Denoting the reactants by A, B ..., each complex is specified by its stoichiometric coefficients.

$pA+qB...\rightleftharpoons A_{p}B_{q}...:\beta _{pq...}={\frac {[A_{p}B_{q}...]}{[A]^{p}[B]^{q}...}}$

Ionic charges are omitted from these expression. For consistency all the equilibrium constants should be association constants. When using general-purpose computer programs it is usual to use cumulative constants. With aqueous solutions the constant for the self-disscotiation of water should be included.

$K_{W}=[H][OH]$

It is quite usual to omit from the model species whose concentrations are considered to be negligible. For example it is usually assumed then there is no interaction between the reactants and/or complexes and the electrolyte used to maintain constant ionic strength or the buffer used to maintain constant pH. These assumptions may or may not be justified. Also, it is implicitly assumed that there are no other complex species present. When complexes are wrongly ignored a systematic error is introduced into the calculations.

Equilibrium constant values are usually estimated initially by reference to data sources.

Speciation calculations

The speciation calculations consist of solving the equations of mass-balance, at each data point, for the concentrations of free reactant, [A], [B] ...

$T_{A}=[A]+\sum p\beta _{pq...}[A]^{p}[B]^{q}...$

$T_{B}=[B]+\sum q\beta _{pq...}[A]^{p}[B]^{q}...$

$...$

In general solving these non-linear equations presents a formidable challenge because of the huge range over which the free concentrations may vary. Once the free reactant concentrations have been calculated the concentrations of the complexes are derived from them and the equilibrium constants. Note that the free reactant concentrations can be regarded as implicit parameters in the equilibrium constant refinement process.

The speciation calculations are repeated at each refinement step.

Equilibrium constant refinement

The objective of the refinement process it to find equilibrium constant values that give the best fit to the experimental data. This is usually acheived by minimising an objective function, U, by the method of non-linear least-squares.

$U=\sum _{i}\sum _{j}W_{ij}\left(y_{i}^{obs}-y_{i}^{calc}\right)^{2}\left(y_{j}^{obs}-y_{j}^{calc}\right)^{2}$

The matrix of weights, W, should be, ideally, the inverse of the variance-covarance matrix of the observations. It is rare for this to be known. However, when it is the expectation value of U is one, which means that the data are fitted within experimental error. Most often only the diagonal elements are known, in which case the objective function simplifies to

$U=\sum _{i}W_{i}\left(y_{i}^{obs}-y_{i}^{calc}\right)^{2}$

Unit weights, W_i=1, are ofter used, but in that case the expectation value of U is the root mean square of the experimental errors. The minimization may be performed using the Gauss-Newton method with a Levenberg-Marquardt parameter. This method requires the Jacobian, J, to be calculated.

$J_{j,k}={\frac {\partial y_{j}^{calc}}{\partial p_{k}}}$

where p_k is the kth parameter of the refinement.

One or more equilibrium constants may be parameters of the refinement. However, examination of the expressions for the measured quantities (above) reveals that the equilibrium constants do not appear in the expressions, but that the concentrations, c, are implicit functions of these parameters. Therefore the Jacobian must be obtained using implicit differentiation. Parameter increments are calculated by solving the normal equations

$\mathbf {\delta p=\left(J^{T}J+\lambda I\right)^{-1}J^{T}\left(y^{obs}-y^{calc}\right)}$

where $\lambda$ is the Marquardt parameter. The increments are added to the current parameter estimates.

A particular issue arises with NMR and spectrophotometric data. For the latter the observed quantity is absorbance, A, and the Beer-Lambert law can be written as

$A_{i}=\sum \epsilon _{pq...}c_{pq...,i}$

It can be seen that absorbance, A, is a linear function of the molar absorbptivities, $\epsilon$, at the path length used. In matrix notation

$\mathbf {A=\mathrm {E} C}$

There are two approaches to the calculation of the unknown molar absorptivities

1) The $\epsilon$ values are considered to be parameters of the minimization and the Jacobian is constructed on that basis. However, the $\epsilon$ values themselves are calculated at each step of the refinement by linear least-squares:

$\mathbf {\mathrm {E} =\left(C^{T}C\right)^{-1}C^{T}A}$

using the refined values of the equilibrium constants to obtain the speciation. The matrix $\mathbf {\left(C^{T}C\right)^{-1}C^{T}}$ is an example of a pseudo-inverse.

2) The Beer-Lambert law is written as

$\mathbf {A=\left(\left(C^{T}C\right)^{-1}C^{T}A\right)C}$

Golub and Pereyra showed how the pseudo-inverse can be differentiated so that parameter increments for both molar absorptivities and equilibrium constants can be calculated by solving the normal equations.

Implementations

Some simple systems are amenable to spreadsheet calculations. These calculations do not follow the general procedures outlined here and use SOLVER to perform the least-squares minimization.

A large number of computer programs for equilibrium constant calculation have been published. See for bibliography. The most frequently used programs are:

• Potentiometric data: Hyperquad, BEST PSEQUAD
• Spectrophotometric data:Hyperquad, SQUAD,Specfit, Specfit/32 is now available commercially
• NMR data HypNMR, EQNMR