July, 1960
ANALYSISOF
ABSORPTION SPECTRA OF MULTI-COMPONENT SYSTEMS
899
AKALYSIS O F ABSORPTIOY SPECTRA OF MULTICOMPOSEST SYSTEJIS BY RICHARD AT. WALLACE E. I . du Pont de Xeniours & Company, Savannah River Laboratory, Aiken, South Carolina Received January $8, 1960
A method was developed to find the number of components that contribute to the absorption spectrum of a multiconipoiient system. A complementary procedure was also developed to test for the presence of a non-absorbing species. The only assumption involved is that Beer’s law is valid for each component.
Introduction -4bsorption spectra hare been used to study a number of systems in which sei-era1 components exist in solution simultaneously. The stepwise formation of inorganic complexes aiid the formation of different specie. of organic dyes are examples. The analysis of the data in such studies usually in1-olres two assuniptions: (1) that Beer’s law is \ralid for all components independently ; and ( 2 ) that a relation is knon-n, from equilibrium coiisiderations, between the concentrations of the \-arious coinponerits and some other parameter, such as the ligand concentration or the pH. The most complete analyis of this sort has been given by Nem-man aiid Hume2 for the formation of inorganic complexes. The number of components, as well as inforniation concerning their nature and the equilibrium constants for reactions between them, can be obtained from wch an analysis. Frequently, the conditions required for the validity of the second assumption are difficult to achieve cxperimentally. It is possible, however, t o find the number of coniponciits from the first assumption alone. Theoretical Beer’s law for :Lmulticomponent system may be expressed a i m
b=l
nlwre a h is thr. atmrbancc at wa\ e leiigth A, EXh is the cstinctioii cocficient of the kth coniponent at wa\-e length A, fk is the concentration of the ktli coinponciit. and ?n is the total number of coni-
k=l
A
=
EC
(3)
where A is a p X n matrix, E is a p x m matrix, and C a m X n matrix. p is the number of different wave lengths a t which one chooses to measure the absorbance. Since the rank of A is equal t o the rank of E or C, whichever is smaller, and since the rank of both E and C can be no larger than m, the rank of A can be no larger than m. The rank of both E and C will usually be m, provided p and n are equal to or greater than m, and it will only be necessary t o determine the rank of -4to find the number of components in the system. The only conditions that could cause the rank of C to be less than m are: (1) the concentration of one component is zero in all experiments, (2) the concentrations of all components are zero in more than m experiments, and (3) the concentration of one or more components can be expressed as a linear combination of the other components in all experiments. The first of these conditions vio1at)es the hypothesis of m components, the second is trivial, and the third is so unlikely it need not bc given serious consideration. The rank of C will therefore always be m . The rank of E will be less than V L only if (1) thc extinction coefficients of all components are zero at more than m different wave lengths, ( 2 ) the extinction coefficients of one or more components arc zero at all wave lengths, or (3) the spectra of one or more of the coinpoiieiits can be expressed as a lincar combination of the spectra of the other componrnts. The first of these can be cliiiiiiiated by the propcr choice of wave lengths. Tlic second iniplics the csistence of a non-absorbing component. aiid together nith the third (which is powible :tlthough unlikely) coiistitutcs n limitatiou O I I thc method. It is possible under certain circuiiistaiiccs to cleterniine if the rank of E is equal to or lebs than nz and consequently to denionstrate the presence of a lion-absorbing species or one whose spectrum is a linear combination of the spectra of the other coniponents. Let the sum of the concentrations of the conipoiicnts lie constant in a11 experiments.
where ax; is the absorbance a t wave length A in the jth experiment, and ck; is the concentration of coinponent k in the jth experiment. These chaiigcs in the Ck’S may be accomplished: for example, by (>hangingthe ligand concent.ratioii in the formatioil Ckj :f for ,211 j ( $1 of iiiorgaiiic complexes or the pII of solut,ions w i i k=l t aiiiiiig orga#iiicindicat’ors. Equation 3 is the defiuit i o i i of matris iiiultiplic*n~- ‘l’tiis i~oiiditioncw11 he arranged, f o r ~\;iniple, 1)) Iioldiiig the (eoncentration of the central ion contion and nmy he writt,en in the more rnlnpacf forin stant and \.arying the ligand concantratinns i n t h e il! T h e information contained i n t h i s article vas 4e.ieloperi during formation of inorganic complexes. the roiirse of work iindrr contract A‘1’(07-?)-1 with the IJ. S. Atomic Let a n r w matrix A* he formed by suhtractiiig Energy Commission. one of the columiis in A , say the itli, from every (2) L. Newnian and I). N. Huine, J. A m . C / ~ e m SOC. . 79, 4571 coluinii (1957).
2
RICHARD h/l. WALLACE
900 A*
A
=
- A(’)
(5)
d =
A * can then be represented as the product A*
=
EC*
=
[E
uaXjZ(U.hJ)2]1’z
1
(9)
Xj
(6)
c*
AI
[C
(O~A-~)Z]’”
where C* is formed by subtracting the ith column of C from every column of C. Due to the restriction imposed by equatioll the rank Of and therefore A* can be no greater than m - 1. If rank Of A is to m1 the number Of con’ponents~ the Of A * Wil’ be - ; Of A is less than m, the rank Of A if the and A * a-i11 be the same* Thisprocedure is not applicable where two of the components have identical spectra* In that case the rank Of A* ’ d l be one less than that of A, even though the rank of A is less than the number of components. Statistical Criterion for Determining Rank.Since the elements of both A and A* are experimental quantities, it is highly unlikely that any square submatrix in A or A* will be singular in a strictly mathematical sense. h statistical criterion is thprpfore necessary to find \\Then the determinant of one of these subn,atrices may be ered to have vanished, in order to determine the rank by finding the largest non-singular submatrix. The standard deviation g A of a determinant IA ran be in terms of the standard deviations of the individual measurements aaxj and the cofactors of the elements ah-’. u
Vol. 64
(7)
The summation is carried out over all elements of the matrix. Equation 7 follows from the general equation for tlie propagation of errors and the fact that the partial derivative of a determinant with respect to an element is equal to the cofactor of that element. If the of the individual measurements are known, the probability that a matrix is sillgular can be determined from the ratio of the absolute value of its determinant to its deviation iAl/a,A, by referring to a statistical table such as the one found in the “Handbook of Chemistry and Physiw. ”3 It is generally more conyenient, hon.ever, to detcrminc the quantity
where U A - ~ is an element of the inverse A-I of A. The summation is no^^ carried out oyer the square of all elemmts in A-i. Qihile these procedures are applicable to A * as m-ell as A, it must be remembered that the errors in the elements of L4* are larger than those in A , since A*, will be sensitire not ollly to errors in the absorbance measurements but also to those in and the concentration. If errors in the corlcentrations are known, the errors in the elementsof A4*call be estimated from the equation ua*xj = [ u q J z
+
uaA12
+ p:
n*xj
+
a2A1]”*
where C and uCare the concentration and its standard error, respectively. Experimental All absorption measurements were made in the visible region with a Cary Model 11hf recording spectrophotometer with a tungsten lamp as a light source. The spectra were first scanned rapidly to determine the region in which absorption occurred. lfore precise absorption measurements were then made a t 25-mg intervals over this region by measuring the absorbance for about 15 seconds a t each of these points. This procedure was nrcessary in order to obtain accurate values in the region where the absorbance was changing rapidly with wave length. The individual methyl orange and methyl red solutions were prepared by adding 100 microliters of a 0.1% solution of the indicator to 10 ml. of a Clark and Lubs buffer solution of the appropriate pH. Mixed solutions of the indicators were prepared by adding 100 microliters of each of the 0 1% indicators to 10 ml. Of
~ $ ~ ~ ~Both gg2ind ~~ fmethyl & red !$ were$ chosen of these compounds
to test the method.
spectra in the visible region that change as the pH is varied between 2.0 and 7.0. I t seemed probable that each of these compounds existed in two forms and that the rank of each of their A matrices would be two The method was also applied to a mixture of methyl orange and methyl red whose A matrix should then have the rank of four. The concentrations of the dyes were kept laonstant in each solrltion so that the condition expressed by rquation 4 was valid and the ranks of the A* matrices should be I and 3 for each of the dves and mixture, respectively The following proredurr x a s used in applying the method. The spectra of a number of differcnt solutions were measnred in which the same amount of dye was present but in which the pIl was varied between 2.0 and 7.0. The .4 matrix was constructed by wmpling the spectrum of each of these 1.41 (8) coliitions at a number of different, wive lengths and placing the absorbance mensiired for the same solutions in thc same columns and those a t the same ~vavrIrngths in the samP of the mntiix. The A * mirtriccls were ronstructrrl tile tietermillallt of a sing,llar matrix will be rowe by siibtracting th(, elements in the lnet colrinin in each .I fqual to its o m stsandarddeviation, it follows from tnatrix froill corresnonding in column cquation 7 that a’ will be equal to the standard deThe rank of each matiix was determined by first choosing \Tiation, a, of the individual elements if tile errors a 2 X 2 snbmatiiu and calculating d. If the v a l w of (1 assumed to be constantin all measurements. was close to the standard deviation. U, the submatrix w a s (*onsideredto be singular A11 2 X 2 s1ibrnatrices that corild iZ reliable estimate of can be obtained by examha- hr coiistructed from an element of tile fjr:qt and one sdtiition of enough singular matrices. If matrix 1s tiond row or column from the original matrix were examineti non-singular d will be larger than a, and the proba- similarly. If the d value of each of these RBS close to of the rntttriu was taken to be one. If one of the biiity that it is singular can be estimated from the 2U, Xthe2 rank submatrices was found to be non-singular, all 3 X 3 ratio d / a . Rubmatrices that could be formed from that 2 X 2 subFor piirposes of calculations in which computers matrix and one additional T O N and column were examined are used, it is preferable to put equation 8 in its until a non-singular 3 X 3 WAS found or all of them were shown to be singular This procedure \-.-as continued until equivalent form the largest non-singular submatrix RraF found (3) “Handbook of Chemintry and Physics,” Thirty-fourth Edition, The results of the messiirements and calculations are shown in Tables 1 through 111. The rl matrices are shown C. D. Hodgman, Editor, Chemical Rubber Publishing Co., Cleveland a t the top of each of these tables, whilc sections A and B 7, Ohio, p . 228.
sillce
AXALSSIS OF ABSORPTION SPECTILA OF MULTI-COMPONEST SYSTEXS
July, 1960
contain the d values of the submatrices of A and A*, respectively, necessary to establish their rank. The first column in each table gives the order of the submatrix, thr second and third t,he rows and columns, respectively, of the original matrix from which they were taken, while the last column contains the values of d . Since the A* matrices mere formed by a simple subtraction they are not shown. Their indexing however is the same as that for the corresponding A matrix. Values of d were calculated using equation 8 in Tables I and I1 while equation 9 was used with the larger submatrices in Table 111. An IBM 650 computer was used to invert the matrices when equation 9 %.aa employed. The values of d for all the 3 X 3 submatrices in Tables IA and IIA are very close to the smallest value of the absorbance that can be estimated, which is 0.001 absorbance unit. These matrices can then be assumed to be singular, and with this assumption the standard deviation, u, of the individual measurements can be calculated from the values of d to be zt0.003. None of the values of d for the 3 X 8 submat,rices is as large as 20 while the values of d for t.he 2 X 2 submatrices are about 450. The rank of each A matrix is therefore two. Similarly the rank of each of the A* matrices may be taken to be one since the largest value of d for any of the 2 X 2 submatrices in Tables IB and I I B is only slight~lvlarger than 2 a with u equal to =t0.004. The larger value of u arises from the necessity of holding the concentrations constant.
TABLE I A MATRIXFOR METHYLRED w 575 525 475 425
X,
PH
3.4
4.6
0.460 993 .400 ,060
0.342 ,742 .365 .152
Rows
Columns
A.
TABLE I1 METHYLORASGE
~ ~ A T R FOR IX
550 500 475 425
5.4
6.2
0.108 .282 .3 10 .320
0.018 ,077 .288 .400
d
2J 0.145 2 2,3 1,2,3 ,004I 3 1,2,3 2,3,4 .0029 3 1,2.3 1,293 .0020 3 2: 3 , 4 2,3,4 ,0003 3 2,3,4 B. Eva1uat)ion of the rank of the A* matrix 2 4J 172 0.0036 4,2 1,2 .0008 2 2 4,I 1,2 ,0012 2 4>3 1,3 .0040 2 4J 123 ,0036 2 4, 1 1,3 .0084
It may be concluded from these results that both these Bystems contain but two components that contribute to the absorption, and no non-absorbing components. The largest value of d for a 4 X 4 submatrix in Table IIIA is 5 . 5 ~ . The probability of this having occurred by chance is about 1 X 10-6. This 4 X 4 submatrix must therefore be non-singular. The 5 X 5 matrix itself is singular with a d value less than u. Its rank is therefore four. One would have predicted the rank of t,he A* matrix for the mixture of dyes to be three. Actually the value of d for the 3 X 3 submatrix is larger than those for the two 4 X 4 submatrices. The difference, however, is not large enough to state with B great deal of assurance that the rank is three. The small value of d in the 3 X 3 submatrix
r
:io 0 . 670 1.100 0 .i 9 0 0 . "0
2.2
Ill@
0 ,S25 1.250 0.810 0.150
--
3.8
6.2
0 , 3 1.5
0.050 ,515 ,745
..-
.I ,o 770 .4g0
,
Ii")
A. Evaluation of the rank of the tl matrix Order of determinant
Rows
Columns
d
2 3 3 3 3
2,3 2J 1,2, 3 2,3, 4 2,394
2,3 1,2,3 2, 31 1,2,3 7,3,4
0.137 0007 0007 ,0054 ,0026
B. Evahiation of tlie rtlnk of the A * matrix 1,2 I, 2 1,2
172 173
2 2 2 2 2 2
114
1,2 1,3 174
THEA MATRIXFOR
A. Evaluation of t,he rank of the A matrix Order of determinant
A
90 1
'1
193
L3
TABLE 111 MIXTUREOF METHYLORANGE AXD METHYLRED
A
A, m#
2.2
3.4
575 525 475 425 375
0.430 2.325
0.540 1.835 1.245 0.427 0.159
1.395 0.223 0.052
0.0017 ,0054 004 1 0025 ,0027 0006
PH
4.6
0.390 1.070 1.156 0.742 0.319
5.4
6.F
0.022 0.262 1.060 1.055 0.512
0.137 0.508 1.075 0.938 0.438
A. Evaluation of the rank of the A matrix Order of determinant
2 3 4 4 4 4 5
2 3 4 4
Rows
175 1,3,5 1,2,3,5 l, 37 4J l J2! 3, 5 '1 3J 4,
.....
Columns
1,5 L3,5 1,3,4; 5 1,2,3,5 1, 2 , 3 , 5 3, 4J 5
.....
d
0 326 ,136 .0051 ,0077 ,0164 ,0031 ,0009
B. Evaluation of the rank of the A * matrix 112 0.159 L2 17213 .0225 172, 3 I, 2 , 3 , 4 ,0125 1 9 % 3,4 1,2,3,4 .0n9 l J 2, 3,
probably arises from the similarity in the spectra of the two dyes, while the comparatively large values of d for the two 4 X 4 submatrices were due t o the failure to hold the concentration of the two dyes sufficiently constant in all the experiments.
Acknowledgments.-I wish to acknowledge the helpful suggestions and criticisms of W. E. Shuler and also the assistance of J. C. English in inverting the matrices.
