On the Use of the Method of Continuous Variations Victor M. S. Oil and Nuno C. Oiiveira University of Coimbra. Portugal The so-called method of continuous variations (or isomolar solutions) is one of the experimental mixing techniques commonly used in the study of reaction stoichiometries (and extent), namely for the determination of formulas and formation constants of-comnlexes. The method ~ ~ ~ ~ ~is also ~ known as Job's method, for his general development of spectrophotometric measurement techniaues (1) for the fundamentals previously established by othkr authors (2-6). For the study of a reaction ~~
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a series of solutions is prepared where the total A plus B molar concentration, [A], [B], = M, is kept constant while [A]d[B], varies in small steps, usually hy mixing different volumes of the two components A and B such that the volume remains constant along the series (if necessary, dilution to constant volume is carried out). I t can be shown that the concentration of the nroduct A.BD (or the amount of A.Bb if the product is ohtained in a different phase) has a maximum for an A-to-H molar ratioeaual toalb. Thus, a plot of [&Bal, or of any property that is aiinear function of [&Ba], against x = [B],/([A], [B],) yields a curve having a maximum at x = bl(a b) = 1/(1+ alb) and zero values for x = 0 and x = 1. The plot approaches two straight-line segments if the equilibrium constant K, of reaction 1is large, and the apex of the angle yields alb. The smaller K,, the more curved is the plot ohtained. Job (I) has shown that when a physical property of the solution is a linear function of the concentration for each &lute species present, the deviation of this property from that calculated with the assumption of no reaction is also an extremum for the stoichiometric mixture [A]d[B], = alb. This is the case of light absorption by solutions that follow Beer-Lambert law, ;se beingrnade i f the corrected absorbance
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where the quantity in parentheses represents the absorbance calculated assuming that no reaction had occurred. Although the method is certainly open to some objections and accompanied by several limitations, i t currently suffers an undeserved ill repute. In particular, i t is undoubtedly a simple method of large didactical relevance in many cases (7),and most of the limitations to its direct use in a nonsonhiaticated analvsis are common to other methods. In this paper, &e recall and systematize the restrictions to the ~-~~annlication of Job's method. illustrate its use associated to proton magnetic resonance spectral intensities, and present some results of a computer program we have developed for the case of multiple equilibria. ~~
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Limitations of Job's Method First of all, Job's method must he scrutinized in the context of the varietv of exnerimental techniaues used in the %udy of solution Ehemisiry that differ in the composition of the initial samnles that are mixed and in the mechanics of mixing the reagents (8).In particular, the data from a titri~
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metric mixing experiment encompass all the data from the Job's method, and a titration curve (using equimolar solutions) can be readily converted into a Job's plot (and vice versa) by a simple mathematical transformation of the abcissa (9). The main reasons for the poor reputation of Job's method can be summarized as follows: a. Theeaiorence of other equilibria concurrent with reaction 1 (taken herewith as the equation for the formation of complex A.B,I thusaffectinghoth the determination ofolh and nftherquilibriurn constant K.. b. The involvement of species other than A, B, &Ba in the equilihrium being studied. e. The use of a physical property that is not a linear function of the concentration of A,Ba. d. The fact that ordinarily only alb is obtained but not the individual values a, b. e. The fact that the formation constantsdetermined are subject toa large uncertainty, even if a and bare known. f. The poor sensitivity of the method for alb > 3. g. Theoccurrenceof an ill-defined maximum in the Job's plot in the case of a moderately stahle complex. h. The effect of ionic strength in the determination of alb and K.. Some of these reasons are common to other mixing methods. We shall now discuss each separately. a. Although Job (1) has pointed out that the method is not generally applicable when more than one compound is formed, Vosburg et al. (10, 11) and Katzin et al. (12) have investigated the condition in which i t can still help in gaining information as to the formulas for the compounds, if more than one is formed. In narticular, the concentration of compound A.Bb is still a maximumfor the stoichiometric alb ratio if the concentrations of the other concurrent complexes are much smaller for such a ratio. Later on we shall address ourselves to the cases where these conditions are not met. h. Regarding criticism b., three situations have been analvzed bv Jones (13).If the reaction of formation of a complex proceeds with the formation of an arbitrary number of other species, all produced in fixed relative amounts (A' and B' being, respectively, the portions of A and B inthe complex), then the maximum in the concentration of A,Bb still occurs for the alb molar ratio. If the complex produced is a weak monoprotic acid, i t is the total concentration of complex species present, i.e., acid conjugate hase, that is a maximum for the alb molar ratio: thus, in studving cases of -this sort i t is necessary to use a physical properiy
