Method for the Choice of Optimal Analytical Positions in Spectrophotometric Analysis of Multicomponent Systems J. Sustek Research Institute of Agrochemical Technology,8 10 04 Bratislava-Predmestie,Czechoslovakia
The relative standard deviations of the absorptivities are suggested as a criterion for the choice of optimal analytical positions in spectrophotometric analysis of multicomponent mixtures. The method was applied to a 5-component model system simulated by means of a computer. The model spectrum was approximated by the linear combination of ten Lorentzian functions. In the case of overdetermined linear absorption equations, three- to fourfold overdetermination offers satisfactory results. A further increase of the number of analytical positions has no practical significance and in some cases may reduce accuracy and precision of the determination.
If an overdetermined system of linear equations is used in the spectrophotometric analysis of multicomponent mixtures (1-3), the problem arises as to what extent the system must be overdetermined in order to obtain sufficiently precise and accurate results. The influence of the choice of wavelengths on the stabilit y of the system of linear absorption equations was studied by Zscheile et al. (41, and it has been confirmed that the highest precision for the determination of the component concentrations may be reached by using all available wavelengths. Most of the authors (5-11) consider the number of 20-50 analytical positions as sufficient. Drozdov-Tikhomirov (12) measured as many as 145 absorbance values on equally spaced wavenumber intervals. With such a procedure, the number of experimental data for processing increases rapidly with increasing number of the analyzed components. Recently, more and more papers dealing with the choice of the optimal analytical positions have appeared. The authors start from the presumption that for each component, and therefore also for each system, there is a limited optimal number of analytical positions, the increasing of which has no essential influence on the precision of the determination of the component concentrations. Gudym and Vorob'eva (13) mention that the increase of the number of analytical positions may lead to a decrease of the analysis precision because errors due to the overlapping of absorption bands are introduced into the system. The problem is to find the suitable criterion for the (1) (2) (3) (4)
(5) (6) (7) (8) (9) (10) (11) (12) (13)
A. F. Vasil'ev, Zavod. Lab., 31, 677 (1965). Z. przybylski, Chem. Anal. (Warsaw). 13, 453 (1968). J. Sustek. M. Livai, and 0. Schiessl. Chem. Lisfy, 66, 168 (1972). F. P. Zscheile, H. C. Murray, G. A. Baker, and R. G. Peddicord, Anal. Chem., 34, 1776 (1962). J. C. Sternberg, H. S. Stillo, and R. H. Schwendeman. Anal. Chem., 32, 84 (1960). I. S. Herschberg and F. L. J. Sixrna. Koninkl. Ned. Akad. Wetenschap. Proc., Ser. B.,65, 244 (1962). H. Cerfontain, H. G. J. Duin, and L. Vollbracht. Anal. Chem., 35, 1005 (1963). I. S. Herschberg, Fresenius' 2. Anal. Chem., 205, 180 (1964). J. M. Arends. H. Cerfontain, I. S. Herschberg, A. J. Prinsen, and A. C. M. Wanders, Anal. Chem., 36, 1802 (1964). K. Doerffel, P. Kugler. and H. J. Binder, Z. Chem., 6, 155 (1966). H. Schmidt, ErdoelKohle, Erdgas, Pefrocbem., 19, 275 (1966). L. N. Drozdov-Tikhornirov, Opf. Spekfrosk., 17, 683 (1964). V. K. Gudym and G. A. Vorob'eva. Zavod. Lab., 38, 778 (1972).
1676
choice of the optimal analytical positions. For two-component systems the concentration determination error may be used as a criterion (5, 14-16). With a greater number of analytical positions and increasing the number of the components, calculation difficulties appear because the respective functions have to be calculated for all possible combinations. As a simple criterion, the ratios of the absorbances and the absorptivities, respectively, of the components being determined to those of the remaining components were used (3,17-19). Other criteria for the choice of the optimal analytical positions for the determination of multicomponent mixtures are mentioned by Kats in his review (20). In the present paper the procedure for the choice of the optimal analytical positions is described for the case where the system of overdetermined linear equations is solved by the method of least squares. The effectiveness of this procedure was verified with a model system.
THEORY A N D CALCULATIONS Assuming the validity of the Lambert-Beer's law and the additivity of the absorbances of the components, the matrix of the component concentrations in the samples can be calculated from the equation
X = QB
(1)
Q = (KTK)-'KT
(2)
K = ACT(CCT)-'
(3 )
where and I t can be seen from Equations 1-3 that the absorptivities matrix K has an influence on the results of the determination of the component concentrations in the samples. Moreover, the elements of the main diagonal of the matrix (KT K)-I appear in the equation for the calculation of the standard deviation of the particular component concentrations ( 3 ) . To be allowed t o eliminate the absorptivities which are subject to measuring errors, it is necessary to know the precision of the calculation of the particular elements of the matrix K, characterized by the standard deviations of the absorptivities. If the number of the analytical positions is greater than the number of components being determined, the system of the linear absorption equations is overdetermined and it is necessary to introduce a further variable, the matrix of the residuals of the mixture absorbances EA:
E,
= KC
-A
(4)
into the relation between absorbance and concentration (14) (15) (16) (17) (18) (19) (20)
C. E. Berry, Ann. Math. Sfat., 16, 398 (1945). J. Swietoslawska, Rocz. Chem., 30, 569 (1956). A . F. Vasil'ev, Zavod. Lab., 31, 1331 (1965). Z. Przybylski. Chem. Anal. (Warsaw). 14, 1047 (1969). M. D. Kats and M. Ya. Rozkin, Zavod. Lab., 38, 688 (1972). M. D. Kats, Zh. Prikl. Spekffosk., 18, 508 (1973). M. D. Kats, Zavod. Lab., 39, 160 (1973).
ANALYTICAL CHEMISTRY, VOL. 46, NO. 12, OCTOBER 1974
( 3 ) .The standard deviation of the absorbance at the pth analytical position is then calculated using the equation /m
zL,lrn
The values e2Agl are elements of the main diagonal of the matrix (EAEAT). The standard deviation of the absorptivity of the j t h component at the gth analytical position is calculated using the equation
where h,, is the j t h element of the main diagonal of the matrix (C CT)-'. The values of the standard deviations of the absorptivities are not suitable as a criterion for the choice of the optimal analytical positions. In fact, the same absolute values S K ~ ]are of different weight for the absorptivities corresponding to the absorption band maxima and for those near the inflection point or in the minima. We used, therefore, the relative standard deviations of the absorptivities
10
5
1
~~
3
EXPERIMENTAL
5
(1965).
30
Table I. Absorptivities and Concentration Range of Component
expressed in per cent., as a criterion for the choice of the optimal analytical positions. The values r K g , were calculated for each component at all the h analytical positions investigated. The analytical position a t which the value r K g l , calculated for the given component, was the least, was considered as the optimum one for this component.
(22) D. Audo, Y. Armand, and P. Arnaud, J. Mol. Struct., 2, 287 (1968).
15
Figure 1. Model spectrum
1
(21) D. Papougek and J. Pliva, Collect. Czech. Chern. Cornrnun.. 30, 3007
10
Numbers at the maxima denote components corresponding to the absorption bands
Component j
Computer System. All computations concerning the preparation of a model system, the computations of the calibration data and of the concentration of the components in the samples, and the statistic evaluation of the results were carried out on a Gier digital computer (Regnecentralen), equipped with a plotter. The programs were written in ALGOL (version GIER-ALGOL 4). Model System. The 5-component model system was simulated by means of a computer. The model spectrum, in which two absorption bands, approximated by Lorentzian functions (21, 221, appertain to each component, is shown in Figure 1. The absorptivities of the band maxima a, and the concentration ranges of the particular components c,i in the calibration mixtures are given in Table I. The concentrations of the components in the calibration mixtures representing the elements of the matrix C ( n = 5, rn = 30) were chosen in a way to give the truest possible picture of the technical products composition. The concentration of the components in the model sample were as follows: e1 = 0.65, e2 = 0.10, c:j = 0.05, c4 = 0.05, and c5 = 0.15. The computed total absorbances at the analytical positions g = 1-32 ithe sum of the absorbances of the components j = 1--5Jwere assembled into the matrix A ( k = 32, rn = 30) and the total absorbances computed for 10 repeat determinations of the model sample were assembled into the matrix B. T h e random errors, the values of which were within the interval from 0 to 0.040, were introduced into the matrices A and B. The distribution of these errors was given by the Gaussian function. The distribution of the errors over the particular analytical positions was carried out by means of a digital computer in a way that their random selection was secured by the program. We considered three variants of the model system: Model System I-without random errors (at the computing only errors due to the rounding off of the absorbances to three decimal places occur, which corresponds to common reading precision); Model System 11-random errors a t selected analytical positions g = 7, 13, 17, 23, 29 (the maxima of the absorption bands of the particular components); and Model System 111-random errors a t all analytical positions, g = 1-32.
1s
Analylicrl p o ~ i l i o n i
Analytical position g
Absorptivity
3 17
0.3 0.5
7
0.6 0.4
0-0.2
9 29 13 26
0.6 0.5
0-0.1
0.6 0.4
0-0.2
14
0.5
0-0.3
23
0.3
2
a0
19
4
Concentration range cJ
1
0.6-1 .O
Table 11. Sequence of Added Analytical Positions No. of
Model system
I
positions, k
5 10 15 20
25 32
I1
5 10
15 20
25 32
I11
5 10 15 20 25 32
Added analytical positions for the components
-
1
2
3
4
3
13(& 1 4 a
17a 7,& 9 , ' 3" 1 9 ~2gfA 16 6 10 4 20 30 2 18 26 1 5 8
12 25
22 24
27
21
11
15
31
16 18
1 g f L 9
