The Art of Titration. From Classical End Points to Modern Differential and Dynamic Analysis


  • Dr. Ruthild Winkler-Oswatitsch,

    1. Max-Planck-Institut für Biophysikalische Chemie, Abt. Biochemische Kinetik, D-3400 Göttingen (Germany)
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  • Prof. Dr. Manfred Eigen

    Corresponding author
    1. Max-Planck-Institut für Biophysikalische Chemie, Abt. Biochemische Kinetik, D-3400 Göttingen (Germany)
    • Max-Planck-institut für Biophysikalische Chemie, Abt. Biochemische Kinetik, D-3400 Göttingen (Germany)
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  • This work is based on a thesis [1] carrried out at the Max-Planck-Institut für Physikalische Chemie, Göttingen, and submitted at the Technische Hochschule, Wien, im November 1969.

  • Gerold Schwarzenbach in memoriam


Titration represents the quantitative determination of a chemical change in response to the variation in concentration of a standrad. Information on reaction parameters is usually obtained from an analysis of the shape of the titration curve. In this paper the derivatives of titration functions with respect to concentration variables and with respect to equilibrium parameters are compared with one another. The latter can be determined directly from amplitudes and time constants of the dynamic responses to equilibrium perturbations, brought about by fast alterations of temperature, pressure or electirc field.

Three general situations have to be distinguished: 1. If the interaction is very strong (i.e. the complex stability is high: K → ∞), the (average) number of sample praticles can be counted directly by means of their reaction with a calibrated number of standard particles.—2. If, on the contrary, the interaction is weak (i.e. the complex stability is low), the sample will react quantitatively only in the presence of a large excess of standard; the half-way point of which provides a direct measure of the binding constant.—3. Only for moderate interactions can individual curves characteristic of the binding strength be observed. The limiting cases 1 and 2 yield titration curves which reflect only the general class to which the system belongs.

The dynamical procedures are fairly insensitive in case 1, which represents the classical end-point situation and is most suitable for quantitative anlaysis, i.e. for a determination of sample concentrations. For case 2, the classical and dynamical techniques yield comparable information: mass-action parameters are obtained from the extrema of the curves, which occur only in the presence of an excess of the standard. In case 3 the dynamical treatment provides a more direct access to the equilibrium constants, and also to the enthalpies or volumes of reaction as well as to the rate constants. The advantage of the dynamic method is due to the fact that in the two derivatives of the titration function d Ti/d ln q and d Ti/d ln p (where q is the titration variable, i.e. the ratio of standard and sample concentration, and p the mass-action parameter, i.e. a reduced binding constant) the terms resulting from a differentiation with respect to p are more closely related to the reaction parameters than those following from a variation of q.

The dynamic analysis is generalized for applications to multiple-step titration processes, where it allows for the measurement fo equilibrium and rate parameters of individual steps. A uniform representation utilizing trigonometric functions was chosen which expresses clearly the singualr character of the end-point. The relations are summarized intabular form. They provide in conjunction with the illustrations the basis for a comparative discussion.