Determination of equilibrium constants

Equilibrium constants are determined in order to quantify chemical equilibria. When an equilibrium constant is expressed as a concentration quotient,

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

it is implied that the activity quotient is constant. In order for this assumption to be valid equilibrium constants should be determined in a medium of relatively high ionic strength. Where this is not possible, consideration should be given to possible activity variation.

The equilibrium expression above 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 concentrations 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.

The equilibrium constants may be derived by best-fitting of the experimental data with a chemical model of the equilibrium system.

Experimental methods

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.

pH=nF/RT (E⁰-E)

At 298K, 1 pH unit is approximately equal to 59 mV.

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 log₁₀[A]

s an empirical slope factor. A solution of known hydrogen ion concentration may be prepared by standardization of a strong acid against borax. Constant-boiling hydrochloric acid may also be used as a primary standard for hydrogen ion concentration.

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 ${\bar {\delta }}$ is the mole-fraction-weighted average of the shifts $\delta$ of nuclei in contributing species.

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

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. Log₁₀ $\beta$ 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 log₁₀ $\beta$ 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 log₁₀ $\beta$. 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 i'th data point, the analytical concentrations of the reactants, $T_{A}(i)$, $T_{B}(i)$ etc. are known along with a measured quantity, $y_{i}$, that depends on one or more of these analytical concentrations. 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 chemical species present in solution, both the reactants added to the reaction mixture and the complex species formed from them. Denoting the reactants by A, B ..., each complex species is specified by the stoichiometric coefficients that relate the particular combination of reactants forming them.

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

When using general-purpose computer programs, it is usual to use cumulative, association constants, as shown above. When reactants and complexes are chemical species ionic charges should be shown explicitly. With aqueous solutions the concentrations of proton (hydronium ion) and hydroxide ion are constrained by the self-dissociation of water.

$H_{2}O\rightleftharpoons H^{+}+OH^{-}:K_{W}'={\frac {[H^{+}][OH^{-}]}{[H_{2}O]}}$

With dilute solutions the concentration of water can be assumed to be constant so the equilibrium expression is written in the familiar form of the ionic product of water.

$K_{W}=[H^{+}][OH^{-}]\,$

When both H⁺ and OH⁻ must be considered as reactants, one of them is eliminated from the model by specifying that its concentration is to be derived from the concentration of the other. Usually the concentration of the hydroxide ion is given by

$[OH^{-}]=K_{W}[H^{+}]^{-1}\,$

In this case the equilibrium constant for the formation of hydroxide has the stoichiometric coefficients -1 in regard to the proton and zero for the other reactants. This has important implications for all protonation equilibria in aqueous solution and for hydrolysis constants in particular.

It is quite usual to omit from the model those 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.

Model selection

This is the process whereby a variety of models are examined in order to find the model that best fits the experimental data, within experimental error. The main difficulty is with the so-called minor species. These are species whose concentration is so low that the effect on the measured quantity is at or below the level of error in the experimental measurement. The constant for a minor species may prove impossible to determine if there is no means to increase the concentration of the species.

Speciation calculations

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

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

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

$...$

To simplify and unify the representation, some authors include the free reactant terms in the sums by declaring identity (unit) $\beta$ constants:

$[A]^{_{}}=\beta _{10...}[A]^{1}[B]^{0}...$ where $\beta _{10...^{}}={[A_{1}B_{0}...]}/{[A]^{1}[B]^{0}...}=1$

$[B]^{_{}}=\beta _{01...}[A]^{0}[B]^{1}...$ where $\beta _{01...^{}}={[A_{0}B_{1}...]}/{[A]^{0}[B]^{1}...}=1$

$...$

In this manner, all chemical species, including the free (unreacted) reactants, are treated in the same manner, i.e. as having been formed from some combination of reactants specified by the coefficients $p,q,...$, as by

$[A_{p^{}}B_{q}...]=\beta _{pq...}[A]^{p}[B]^{q}...$

whence

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

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

$...$

suffices. This unification makes for simpler coding of computer programs during the calculation of species concentrations and derivatives.

At each data point, the $T$ would be known quantities, from the details of the assembly of reaction components used to generate the sample solutions. In a titration experiment, the total concentration of any reactant $R$ at any i'th titration point would be given by

$T_{R}={\frac {v_{0}[R]_{0}+v_{i}[R]_{t}}{v_{0}+v_{i}}}$

where a volume $v_{0}$ of titrate containing a concentration $[R]_{0}$ of a reactant would be supplemented with a volume increment $v_{i}$ of titrant containing a concentration $[R]_{t}$ of the same reactant. Usually, only one reactant is added and the other reactants' $[R]_{t}$ values would be 0.

In general, solving these non-linear equations presents a formidable challenge because of the huge range over which the free concentrations may vary. For this reason, logarithmic expressions are often used because logarithms span a much narrower range.

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 (invariable) in the equilibrium constant refinement process; even though they can vary among the data, for instance in a titration, they are merely computational intermediates obtained from the total reactant concentrations $T$, which are fixed values even in a titration, and the current set of $\beta$ estimates. As such, they are not themselves refined. Neither are the known $\beta$ values (known from previous experiment) nor the identity $\beta$ constants.

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 achieved by minimising an objective function, U, by the method of non-linear least-squares.

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

The matrix of weights, W, should be, ideally, the inverse of the variance-covariance 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

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

with $W_{i,j}=0$ when j≠ i. Unit weights, $W_{i,i}=1$, are often 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 or without a Levenberg-Marquardt parameter. This method requires the elements of the Jacobian matrix, 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, the measured quantities (see above) represented by $y$ are not expressed in terms of the equilibrium constants, but in terms of the species concentrations, which are implicit functions of these parameters. Therefore the Jacobian elements must be obtained using implicit differentiation.

Parameter increments $\delta P$ are calculated by solving the normal equations, the first-order Taylor series gathered in matrix form as

$\mathbf {y^{obs}=y^{calc}+J\delta P}$

which are solved with

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

where $\lambda$ is the optional Marquardt parameter, I is the identity matrix, and the superscript T indicates the transpose of the matrix. The increments $\delta P$ are added to the current parameter estimates to obtain better estimates, the species concentrations and $y^{calc}$ values are recalculated at every data point and the procedure is repeated until no further reduction in U is achieved.

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 a 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