Selection of Wavelengths for Optimum Precision in Simultaneous 'Spectrophotometric Determinations Michael R. DiTusa and Alfred A. Schilt Northern Illinois University. DeKalb, IL 601 15 Although many texts include a description of simultaneous determinations employing absorption spectrophotometry and treat the mathematics necessary for analyte quantitations, treatment of analytical wavelength selection has been mostly qualitative. A general method for the selection of wavelengths for optimum precision in simultaneous determinations is needed. Such a method is described below. In order to perform reliable simultaneous quantitative determinations by spectrophotometry, two conditions must be satisfied, namely: (1) each species of a multicomponent mixture must absorb in conformance toBeer's law, and (2) no interaction should occur between the components of the mixture that would cause the absorbance of the mixture to be unequal to the sum of the absorhances of the individual components. Equations (1) and (2) express these conditions.
where a,i is the molar absorptivity of species x at wavelength hi. b is the ootical . oath. C, reoresents the molar concentration of species x, A,i, Ayi . . . equals the ahsorbance of components x, y . . . at wavelength Xi, and AT^ equals the total absorbance of all components a t wavelength Xi. For a two-component mixture, the ahsorbance must be measured a t two appropriate wavelengths in order to determine the concentrations of both components. The following set of equations (with b = 1)can then be solvedsimultaneously for the concentrations of x and y: A1 =a&,
+ a,#&
(3)
Az = ax2Cx + a,&
If the absorbance a t hl is monitored as the concentration of species x is changed, and the second term in the above equation is kept constant, the following relationship results: dAlldC, = a,l
~
~
(8)
Therefore, the ratio axlayplotted versus wavelength should give a maximum at wavelength 1 and a minimum a t wavelength 2. This conclusion is identical to that deduced ahove from expressions (4-7). The usefulness of this identity becomes apparent when considering a three-component system. By the properties of matrix algebra, a three-by-three determinant can he reduced to a two-by-two determinant wherein each element is now a two-hy-two d e t e ~ m i n a n t . ~
(4)
Similarly, if the absorbance a t Xz is mhitored as the concentration of species y is changed and the first term is kept constant, then dA2/dC, = =,a
in the sought for quantity C , will he minimal a t that wavelength where a, is at its maximum value. Similarly, eqn. (5) predicts that dCy/dAp will be minimal at a wavelength where a, is a t its maximum value. Expression (6) describes a condition which is favorable for the measurement of a broad range of different concentrations of x in the presence of y, and expression (7) describes a similar condition for the opposite situation. Thus, to find the two wavelengths that best satisfy all of these conditions, one need only plot the ratio of ahsorptivitiei o,/o,. versus ~ a v ~ l e n g tand h , the wavelengths sought will be indicated hy thc mmimum a ~ l dminimum in the plot corresponding to A1 and Xz, respectively. A similar approach may be attempted for a three- (or higher-order) component system, but the relationships become much more difficult to handle mathematically. However, by use of an identity and matrix algebra, a parallel argument may he made. For a two-component system, i t has been shown that the principal diagonal elements of the matrix made up of the molar ahsorptivities appearing in eqns. (3) should be larger than the off diagonal elements such that the determinant of the matrix is a maximum at the optimum wave1engths.l
(5)
should he true. In order to keep the transmitted light measurable (not too strongly absorbed) a t X1 as well as a t Xz for a hroad range of concentrations of both x and y, the following conditions must hold: ay1
