Equilibrium constants are determined in order to quantify chemical equilibria. When an equilibrium constant is expressed as a concentration quotient,
it is implied that the activity quotient is constant. For this assumption to be valid, equilibrium constants must 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. In all cases the range can be extended by using the competition method. An example of the application of this method can be found in palladium(II) cyanide.
Potentiometric measurements A free concentration [A] or activity {A} of a species 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
where is the standard electrode potential. When buffer solutions of known pH are used for calibration the meter reading will be a pH.
At 298 K, 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.
where is 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.
Range and limitations The most widely used electrode is the glass electrode, which is selective for the hydrogen ion. This is suitable for all acid–base equilibria. values between about 2 and 11 can be measured directly by potentiometric titration using a glass electrode. This enormous range of stability constant values (ca. 100 to 1011) 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.
When a glass electrode is used to obtain the measurements on which the calculated equilibrium constants depend, the precision of the calculated parameters is limited by secondary effects such as variation of liquid junction potentials in the electrode. In practice it is virtually impossible to obtain a precision for log β better than ±0.001.
Spectrophotometric measurements
Absorbance It is assumed that the Beer–Lambert law applies.
where is the optical path length, is a molar absorbance at unit path length and 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.
Range and limitations An upper limit on 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
Fluorescence (luminescence) intensity It is assumed that the scattered light intensity is a linear function of species’ concentrations.
where is a proportionality constant.
Range and limitations
The magnitude of the constant may be higher than the value of the molar extinction coefficient, ε, for a species. When this is so, the detection limit for that species will be lower. At high solute concentrations, fluorescence intensity becomes non-linear with respect to concentration due to self-absorption of the scattered radiation.
NMR chemical shift measurements Chemical exchange is assumed to be rapid on the NMR time-scale. An individual chemical shift is the mole-fraction-weighted average of the shifts of nuclei in contributing species.
Example: the pKa of the hydroxyl group in citric acid has been determined from 13C chemical shift data to be 14.4. Neither potentiometry nor ultraviolet–visible spectroscopy could be used for this determination.
Range and limitations Limited precision of chemical shift measurements also puts an upper limit of about 4 on . Limited to diamagnetic systems. 1H NMR cannot be used with solutions of compounds in 1H2O.
Calorimetric measurements Simultaneous measurement of and 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 Insufficient evidence is currently available.
The competition method The competition method may be used when a stability constant value is too large to be determined by a direct method. It was first used by Schwarzenbach in the determination of the stability constants of complexes of EDTA with metal ions.
For simplicity consider the determination of the stability constant of a binary complex, AB, of a reagent A with another reagent B.
where the [X] represents the concentration, at equilibrium, of a species X in a solution of given composition.
A ligand C is chosen which forms a weaker complex with A The stability constant, KAC, is small enough to be determined by a direct method. For example, in the case of EDTA complexes A is a metal ion and C may be a polyamine such as diethylenetriamine.
The stability constant, K for the competition reaction
can be expressed as
It follows that
where K is the stability constant for the competition reaction. Thus, the value of the stability constant may be derived from the experimentally determined values of K and .
Computational methods It is assumed that the collected experimental data comprise a set of data points. At each th data point, the analytical concentrations of the reactants, , etc. are known along with a measured quantity, , that depends on one or more of these analytical concentrations. A general computational procedure has four main components: Definition of a chemical model of the equilibria Calculation of the concentrations of all the chemical species in each solution Refinement of the equilibrium constants Model selection The value of the equilibrium constant for the formation of a 1:1 complex, such as a host-guest species, may be calculated with a dedicated spreadsheet application, Bindfit: In this case step 2 can be performed with a non-iterative procedure and the pre-programmed routine Solver can be used for step 3.
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. {\mathit p A} + \mathit q B \cdots <=> A_\mathit{p}B_\mathit{q} \cdots:
When using general-purpose computer programs, it is usual to use cumulative association constants, as shown above. Electrical charges are not shown in general expressions such as this and are often omitted from specific expressions, for simplicity of notation. In fact, electrical charges have no bearing on the equilibrium processes other that there being a requirement for overall electrical neutrality in all systems.
With aqueous solutions the concentrations of proton (hydronium ion) and hydroxide ion are constrained by the self-dissociation of water. H2O <=> H+ + OH-:
With dilute solutions the concentration of water is assumed constant, so the equilibrium expression is written in the form of the ionic product of water.
When both H+ and OH− must be considered as reactants, one of them is eliminated from the model by specifying that its concentration be derived from the concentration of the other. Usually the concentration of the hydroxide ion is given by
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 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 A speciation calculation is one in which concentrations of all the species in an equilibrium system are calculated, knowing the analytical concentrations, TA, TB etc. of the reactants A, B etc. This means solving a set of nonlinear equations of mass-balance
for the free concentrations [A], [B] etc. When the pH (or equivalent e.m.f., E).is measured, the free concentration of hydrogen ions, [H], is obtained from the measured value as or and only the free concentrations of the other reactants are calculated. The concentrations of the complexes are derived from the free concentrations via the chemical model.
Some authors include the free reactant terms in the sums by declaring identity (unit) constants for which the stoichiometric coefficients are 1 for the reactant concerned and zero for all other reactants. For example, with 2 reagents, the mass-balance equations assume the simpler form.
In this manner, all chemical species, including the free reactants, are treated in the same way, having been formed from the combination of reactants that is specified by the stoichiometric coefficients.
In a titration system the analytical concentrations of the reactants at each titration point are obtained from the initial conditions, the burette concentrations and volumes. The analytical (total) concentration of a reactant R at the th titration point is given by
where R0 is the initial amount of R in the titration vessel, is the initial volume, [R] is the concentration of R in the burette and is the volume added. The burette concentration of a reactant not present in the burette is taken to be zero.
In general, solving these nonlinear equations presents a formidable challenge because of the huge range over which the free concentrations may vary. At the beginning, values for the free concentrations must be estimated. Then, these values are refined, usually by means of Newton–Raphson iterations. The logarithms of the free concentrations may be refined rather than the free concentrations themselves. Refinement of the logarithms of the free concentrations has the added advantage of automatically imposing a non-negativity constraint on the free concentrations. Once the free reactant concentrations have been calculated, the concentrations of the complexes are derived from them and the equilibrium constants.