The Method of Continuous Variation

Suppose that several complexes can form between species P and X  

If one complex (say, PX2) predominates, the method of continuous variation (also called Job s method) allows us to identify the stoichiometry of the predominant complex. 

The classical procedure is to mix P and X and dilute to constant volume so that the total concentration [P] + [X] is constant. For example, 2.50 mM solutions of P and X could be mixed as shown in Table 19-1 to give various X P ratios, but constant total concentration. The absorbance of each solution is measured, typically at max for the complex, and a graph is made showing corrected absorbance (defined in Equation 19-21) versus mole fraction of X. Maximum absorbance is reached at the composition corresponding to the stoichiometry of the predominant complex. 

Corrected absorbance is the measured absorbance minus the absorbance that would be produced by free P and free X alone  

NOTE All solutions are diluted to a total volume of 25.0 mL with a buffer.  

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Of the three methods, the method of continuous variations, also called Job s method, is the most popular. In this method a series of solutions is prepared such that the total moles of metal and ligand, tot> in each solution is the same. Thus, if ( m)( and ( l)( are, respectively, the moles of metal and ligand in the i-th solution, then... 

In essence, the corrected absorbance gives the change in absorbance due to the formation of the metal-ligand complex. An example of the application of the method of continuous variations is shown in Example 10.7. 

Both the method of continuous variations and the mole-ratio method rely on an extrapolation of absorbance data collected under conditions in which a linear relationship exists between absorbance and the relative amounts of metal and ligand. When a metal-ligand complex is very weak, a plot of absorbance versus Ay or n-J m may be curved, making it impossible to determine the stoichiometry by extrapolation. In this case the slope ratio may be used. 

In this experiment the method of continuous variations is used to determine the stoichiometry and equilibrium constant for the organic complex of 3-aminopyridine with picric acid in CHCI3, and the inorganic complex of Fe +with salicylic acid. 

Data from the spectrophotometric titrations of Fe + with SCN , and of Cu + with EDTA are used to determine the stoichiometry of the resulting complexes using the method of continuous variations. 

This experiment describes the use of FIA for determining the stoichiometry of the Fe +-o-phenanthroline complex using the method of continuous variations and the mole-ratio method. Directions are also provided for determining the stoichiometry of the oxidation of ascorbic acid by dichromate and for determining the rate constant for the reaction at different pH levels and different concentration ratios of the reactants. 

Another method for determining the composition of a complex in solution is that known as Job s method or the method of continuous variations. Suppose that the formation of a complex between metal A and m ligands B can be shown as... 

Determination of the stoichiometry of a complex by the method of continuous variations (Job s method)... 

A plotting protocol (also known as the method of continuous variation) that provides useful information about protein-ligand and protein-protein interactions. Mole fractions Xa and Xb of two interacting substances, say A and B, are varied such that the total molarity remains constant. Note that Xa = [A]/([A] + [B]) and Xb = [B]/([A] + [B]), such that (Xa + Xb) = 1. If an enzyme prefers to bind AB as a one-to-one complex, then the enzymatic activity will be maximal at a mole fraction Xa of 0.5 (Le., the point at which A and B are present in a one-to-one stoichiometry). Similarly, if AB2 is the active species, then the enzyme will be most active at a Xa value of 0.33. In this manner, the stoichiometry of binding can be readily determined, and the technique can yield information concerning the affinity of the enzyme for the active species. 

In this case, X + Xb = 1, and this formulation allows one to analyze binding interactions between two such components by the method of continuous variation. See... 

Furthermore, the apparent extinction coefficient for the complex increases steadily with increasing concentration of nickel chloride. On the basis of this evidence an attempt was made to determine the nature of the equilibrium involved and the extent to which dissociation takes place. Such an equilibrium has also been suggested by Jicha and Busch (17) on the basis of the results obtained from the method of continuous variations. The appropriate equilibrium appears to be that given in Equation 22. 

Here arc some precautions for the method of continuous variation ... 

The method of continuous variation can be carried out with many separate solutions, as in Table 19-1. However, a titration is more sensible. Figure 19-9a shows a titration of EDTA with Cu2+. In Figure 19-9b, the abscissa has been transformed into mole fraction of Cu2+(= [moles of Cu2+]/[moles of Cu2+ + moles of EDTA]) instead of volume of Cu2+. The sharp maximum at a mole fraction of 0.5 indicates formation of a 1 1 complex. If the equilibrium constant is not large, the maximum is more curved than in Figure 19-9b. The curvature can be used to estimate the equilibrium constant.7... 

Isosbestic (crossing) points are observed when a solution contains variable proportions of two components with a constant total concentration. A Scatchard plot is used to measure an equilibrium constant, and the method of continuous variation allows us to determine the stoichiometry of a complex. In flow injection analysis,... 

It has been reported112 on the basis of a spectrometric study, using the method of continuous variations, that U02+ ions form a soluble 1 2 complex [U02(PhN202)2] initially, but with excess of cupferron an insoluble product, NH4[U02(PhN202)3], is obtained which has a solubility product of 5.8 2.5 x 10-10. 

Job s method, or the method of continuous variation, was first published in 1928 as a means for determining the nature of metal complexes in solution. This spectrophotometric method is particularly useful for identifying metal complexes that, for stability reasons, may be difficult to isolate as a solid from solution. We will use this method to investigate the solution structure of the [Ni(salpd)] complex. 

The three most common techniques used for complex-ion studies are (1) the method of continuous variations, (2) the mole-ratio method, and (3) the slope-ratio method. 

Spreadsheet Summary In Chapter 12 of Applications of Microsoft Excel in Analytical Chemistry, we investigate the method of continuous variations using the slope and intercept functions and we learn how to produce inset plots. 

Based upon the total zinc content of this preparation of enzyme, 3.5 gram atoms per mole, the lines drawn through the linear portions of the experimental data intersect at 64 mole % of zinc. The point of intersection would be expected to be 50 mole % of zinc if OP were to form a 1 1 complex with each zinc atom of LADH. However, if only 2 of the 3.5 zinc atoms of the enzyme react with OP under these conditions, maximal formation of the 1 1 Zn OP complex occurs when the molar ratio of OP to total zinc bears the relationship 2/3.5—t.e., at the point of 3.5/(3.5 + 2.0) corresponding to 64 mole % of zinc, as here observed. Hence, it appears that the method of continuous variations is capable of signaling the presence of "nonreacting zinc atoms, in accord with the titration data shown previously (20). 

For CyD complexes a number of stoichiometric ratios has been observed [2]. The most commonly reported ratios are H G = 1 1 and H G = 2 1. However, other stoichiometries as well as ternary CyD-containing complexes [47] are known. An example of 2 1 stoichiometry is the camphor-a-CyD complex in which the guest molecule is embedded inside a capsule formed by two host molecules [48]. Fenbu-fen (y-oxo-[l,l -biphenyl]-4-butanoic acid) is an interesting example of a compound which shows stoichiometry dependence on the CyD cavity size. It does not form an inclusion complex with a-CyD, but displays H G = 1 1 stoichiometry with f-CyD and H G = 1 2 stoichiometry with y-CyD [49, 50]. Metoprolol is another such compound which forms 1 1 complexes with a-CyD and f-CyD but with y-CyD it forms an H G = 1 2 complex [51]. A similar phenomenon detected using HPLC for a complex with a first-generation dendrimer is presented in Chapter 5 [52]. On the other hand, 1-adamantanecarboxylic acid and f-CyD form a complex with temperature-dependent stoichiometry, H G = 1 1 at 25 °C and H G = 1 2 at 0 °C [28]. For the complexation of dodecyltrimethylammonium bromide with a-CyD two competing associations with stoichiometries of H G = 1 1 and H G = 2 1 have been reported [53]. Use of the method of continuous variations in such situations becomes questionable and information about the complex stoichiometry is revealed directly from the titration measurement described in Section 9.2.3. 

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