Two methods are described for calculating, with the aid of a digital computer, the successive formation constants of a complex system, using ligandnumber data. Weighted least-squares criteria were used in both methods for determining the constants.A LARGE number of graphical methods have been described for calculating successive formation constants defining complex equilibria.l These methods tend to be subjective in nature and thus do not give objective estimates of the precision of the calculated constants; moreover, they weight the data in an arbitrary fashion, and are also often tedious to apply. More recently, several reports describe the use of digital computers for computing the constants, where by objective methods tests can be applied, but few (cf. ref.3) give detailed account of the methods for specific problems. The present Paper describes two methods which can be generally applied to one of the most frequently used experimental techniques for determining stability constants, vix. , the measurement of the pH of solutions of the ligands with and without the addition of the central co-ordinating ion. The methods of calculation apply specifically to mononuclear systems with a maximum co-ordination number of 2, but they can be easily extended to systems with a maximum of 3 or 4, and possibly up to 6, although a different numerical technique in which orthogonal polynomials are used would be more suitable for the problems of high order.4
EXPERIMENTALThe present work arose from a study of complex-formation between silver ions and a number of aliphatic amino-compounds. Full experimental details and complete results will be published later, but, briefly, the experimental technique is a refined form of the pH-titration method used by Bjerrum5 to give free-ligand concentrations [A] and the corresponding ligand number 6, at constant ionic strength of 0 . 5 ~. A newly developed cell with free diffusion liquid junction,$ in conjunction with a stable electrometer (E.I.L. Vibron), were used, so that e.m.f. measurements were reproducible to f O . 1 mv.
WEIGHTING PROCEDUREThe system studied involved the following equilibria : andwhere A is an aliphatic amino-compound, it being understood that all species are solvated.Conventionally, two equilibrium constants are used to describe this system, viz. , p1 and p2, where PI refers to the first equilibrium, and P2 = @,, where k , refers to the second equilibrium. The formation function for such a system is given byThere will, therefore, exist a series of equations of the form where i i and [A]: are members of a set of experimentally determined values, and Ri, which should theoretically be zero, are residuals due to experimental errors. The problem was to Rossotti and Rossotti, " The Determination of Stability Constants, " McGraw-Hill, New York, 1961.