Graphical and computational methods for determining the stability constants of mono- and polynuclear complexes with a common intersection point of the family of formation curves

Aqueous polynuclear systems have been analyzed, for which the family of formation curves intersects at a common point. The analyzed graphical and computational method for determining the stability constants can be used as initial values within the iterative calculation process. In some cases, the stability constants are calculated using only the coordinates of the common intersection point. The obtained equations could be of special interest when the experimental data can be interpreted in several models. In these cases, given the large volume of experimental data, the calculation is simple and the model can certainly be chosen with high safety. The obtained equations may also be applied for critical evaluation of tabular data if the coordinates of the intersection point are known. A series of real polynuclear systems have been analyzed and useful conclusions have been made.


INTRODUCTION
At present, for the determination of the chemical equilibrium model, specially developed programs using modern computing techniques are used. However, despite the existing belief that the methods for determining the number, composition, and stability of species formed in the equilibrium system, in which several complex compounds of arbitrary composition are formed simultaneously, are studied well enough, in some cases to solve this task. Certain difficulties arise. First, in the case of the formation of several complex compounds, it is difficult to predict the model in the first approximation. Second, as a result of the computerized search, several models with values close to the minimum deviation squares are obtained [1][2][3]. The plurality of solutions to this problem, e.g. lack of a single solution, occurs when the existing information is insufficiently complete. As examples of the systems with an extremely large number of species formed, "core-link" complexes can serve, where they can vary within wide limits of integer values. For some series of these complexes [4][5][6], the mathematical processing of the experimental data and the computerized calculation of the equilibrium model are impossible without preliminary information on the composition of the complexes. This information can be obtained through functional transformations of experimental data in the form of additional concentration functions and their derivatives [7][8][9]. At the same time, in the presence of a vast volume of experimental data, in case of their uniform distribution, along with the graphic processing, it is necessary to use numerical analysis methods [10,11]. In this paper, the polynuclear systems are analyzed, for which the family of curves intersects at a point. The theoretical analysis of the correlation between the position of the intersection point and the composition of the complexes has been the subject of discussion in a series of papers [12][13][14][15][16][17][18]. The authors [12][13][14] found a criterion for the appearance of the real and the apparent intersection point. It has been shown that there are two conditions for the appearance of the real intersection point: (a) the derivative of the formation function in respect to the total concentration of one of the components under the invariability conditions for the second component must be zero and (b) the sign of these variables change as they pass through the common point of intersection. In the present work, to demonstrate the significance of the derived earlier expressions, a series of real important systems have been taken into consideration.

EXPERIMENTAL PART
It has been examined the systems, in which reactions with the formation of mononuclear complexes MLn and polynuclear complexes MqLp take place according to the general scheme: These two types of reactions are described by the following equilibrium constants The partial molar fractions fn and fpq for mononuclear and polynuclear complexes respectively may be written as In a previous paper [18], the authors showed that the derivative of the formation function Z with respect to the total (analytical) concentration of the metal ion M can be expressed by the partial molar fractions fpq as follows: , ln When a single complex MLN is formed, the following expressions are valid: ) are formed, from Eq. (4), taking into account Eq. (9), the following conclusions can be derived: Therefore, if the coordinates of the common intersection point are known, then from the Eq. (10) the complex stability constant is easily calculated. The stability constants of the polynuclear complexes MqLp and MQLP can be calculated by solving the system from two equations (11) for two values different CM. then their stability constants are calculated using the equation

System 3. A series of mononuclear complexes and a series of polynuclear complexes
In all the analyzed examples, the value [ ] L depends only on the composition of the formed complexes and on the stability of the mononuclear complexes. The Z value is a function of the stability constants of the complexes only for the last considered system.

RESULTS AND DISCUSSION
To elucidate the concrete practice of the above expressions, some real polynuclear systems have been analyzed.

Germanic acid and polygermanates
Previously it was obtained a family of the titration curves pH versus the number OHadded per Ge for different total germanium concentration CGe, which were intersected at the point (0.6; 8.8) [16]. These data were interpreted as evidence for the species H2GeO3, should be formed, was tested. It was found that neither of them could explain the data [16]. By applying Eq. (5)  with the coordinates (1.33; 6.91) (Fig. 2). The presence of the common point is a strong indication for a mixture of polynuclear and mononuclear complexes. In Table 1 These calculated values are close to those from those tabulated (see Table 1). For the model I the inconsistent value of the stability constant for the mononuclear complex ZnL2 was obtained. Therefore, this model must be discarded. Consequently, now it is possible to choose between model II and model III.
Unfortunately, if at the common intersection point two polynuclear complexes of the composition Zn3L4 and Zn2L2are formed, this point is not a real one [12,13] and the derived equations (4)- (14) are not applicable for this model. , does not fall within the class of polynuclear systems examined here, because the common point of intersection is apparent. The stability constants, calculated with the expressions, derived earlier by one of the authors, can be used as initial values in the iterative calculation process, as well as for the verification of existing tabular data. In some cases, it is possible to calculate the stability constants using only the coordinates of the common real intersection point. The obtained equations are of special interest when the experimental data may be interpreted in several models. In these cases, given a large volume of experimental data, the calculation is simple and the model can certainly be chosen large. As it has been shown above, the obtained equations can also be used for critical evaluation if the coordinates of the real intersection point are known. Nevertheless, it should be noted that this approach contains some limitations. Firstly, it is applicable for the determination of stability constants only for chemical species, which are formed under the conditions of the intersection of formation curves. Secondly, the above equations are applicable only for certain models. Thirdly, the experimental data must cover an adequately large range of component concentrations to bypass potential mistakes in interpreting the nature of the common point of intersection.