Espectofotometria

online version of the paper in the July 2000 JCE issue, Vol. 77 No. 7 p. 927

A Straightforward Method to Determine Equilibrium Constants from Spectrophotometric Data
E. Keszei, M. G. Takács, and B. Vizkeleti Eötvös University, H-1518 Budapest 112, P.O.B. 32, keszei@chem.elte.hu Spectrophotometry is an excellent experimental method providing sufficient information on the equilibriumconcentrations of a (usually liquid) reaction mixture. For many decades, chemists have made a great effort to infer equilibrium constants from this rich source of information. A paradox feature of this effort was that they usually did not aim at maximum or optimum information concerning the equilibrium constant but tried to fulfil peculiar conditions to be able to use one of the numerous simplifiedtreatments to evaluate experimental data. The common feature of all these simplifications was to remove the essentially non-linear nature of the related model equations, to select and manipulate experimental data in such a way that finally, to get the equilibrium constant, only linear models had to be treated. Problems arising from this "selection and manipulation" of the experimental data have beendiscussed in detail by Richard Ramette (1) in this journal. He also proposed – among others – a clear and correct treatment of the linear models, described criteria to choose the most appropriate one to handle experimental data, and developed a computer program – as early as 1967 – to infer equilibrium constants from practically all experimental "cases". However, his treatment also remained in theframework of linearized models only. Almost two decades later, Michael Collins extended Ramette's algorithm to use with other methods in addition to spectrophotometry (e. g., solution calorimetry, NMR chemical shift) (2), still keeping the essentially linearized evaluation of the experimental data. Though there have been several applications of non-linear parameter estimation in evaluation ofspectrophotometric and other equilibrium measurements, they were based on the mathematical and numerical treatment of the particular nonlinear model (3,4). A critical overview of early chemical equilibrium applications of non-linear parameter estimation is given in Ref. (3). However, earlier applications usually had to deal with many numerical and computer programming details, being too much tedious toimplement in undergraduate lab courses. The present paper – after another decade – shows a non-linear evaluation method for spectrophotometric data. This method uses the physical description as it is for a model function to infer the equilibrium constant from experimental data. In 1986, Collins mentioned, "in favor of the Ramette approach" the "widespread microcomputer availability". Nowadays,thanks to the widespread availability of sophisticated mathematical software packages, nonlinear parameter estimation became an easy task to make with minimum mathematical and no programmation skill. The availability of the non-linear methods makes it possible to concentrate on optimum information when calculating the equilibrium constant, with the additional convenience of almost no mathematicalmanipulation of the underlying (simple) 1

equations. Details of the numerical techniques of non-linear parameter estimation can be found in current statistical or numerical textbooks (5-7).

Derivation of the model As the aim of this paper is mainly to show the ease and superiority of non-linear methods, we shall only discuss simple cases and show a few examples. Let us consider the prototypedonor-acceptor reaction, with the same notation as in Refs. 1 and 2:

→ D + X ← DX

(1)

We are interested in the equilibrium constant of this reaction, which can be written as

K=

[DX ] ⋅ [D] ⋅ [X]

f DX , fD ⋅ fX

(2)

where brackets stand for equilibrium concentrations, and the symbols ƒ indicate the corresponding activity coefficients. Keeping in mind that activity coefficients...