2954
2 2. HUGUS, JR.,AND A. A. EL-AWADY
The Determination of the Number of Species Present in a System:
a New
Matrix Rank Treatment of Spectrophotometric Datal
by Z Z. Hugus, Jr., Department of Chemistry, North Carolina State University, Raleigh, North Carolina 37607
and Abbas A. El-Awady* Department of Chemistry, Western Illinois University, Macomb, Illinois 61466
(Received March 8, 1971)
Publication costs borne completely by The Journal of Physical Chemistry
A method is described for the determination of the number of independent variables (e.g., number of different species) in a chemical system. For the application of the method one measures a set of properties Pi,(such as absorbancesat different wavelengths) in several different mixtures, with the requirement that Pi, = ZkPi,C,,. These are then arranged in a rectangular matrix and the number of independent variables is determined from the rank of the matrix, obtained by finding the number of nonzero eigenvalues. In addition, a statistical criterion for the vanishing of an eigenvalue is proposed.
Introduction In connection with a study of the hydrolytic depolymerization of certain binuclear cobalt(II1) complexes2it became desirable to know the number of species present in the reacting solution in order to fit experimental spectrophotometric data to a detailed kinetic model. Published methods given by Wallacela A i n s ~ o r t h , ~ and Wallace and Katz6 and more recently by Katakis6 and Coleman, Varga, and Mastin’ involve excessive computation, even when high speed digital computing machinery is used. Consequently, we felt that a method of matrix rank analysis that could be used with large rectangular matrices would be of considerable advantage. Although our intention was to apply the procedure only to the determination of the rank of a matrix of spectrophotometric absorbances, the method is of far wider applicability and may be found of utility by workers in the fields of ORD (optical rotatory dispersion), CD (circular dichroism), fluorescence spectroscopy, and so on. For the application of this method, one measures a set of properties, the ith property being denoted by Pi, in several different mixtures, the subscript j denoting the j t h mixture, and there is the requirement that the properties be constitutive, that is, that
I n this equation, Pi, is the value that would be measured for Pi if only the kth species were present, and that a t unit concentration; c k j is the concentratidn of the kth species in the j t h mixture. It is evident that Beer’s law is an example of a relationship of the type of eq 1. The Journal of Physical Chemistry, Vol. 76, N o . 19, 1972
Method of Rank Determination It is a theorem in the theory of matrices that the rank of a matrix which can be written as the product of two other matrices is equal to the smaller of the ranks of the two latter matricesS8 Based on this theorem and on the assumption of Beer’s law for each species present in a mixture, the following analysis results. The absorbance per unit cell legnth, DXi,at the Xth wavelength in the ith solution, is given by m DX$
=
CBEXkCki
1
(2)
where EXk is the extinction coefficient of the lcth species at the Xth wavelength, and C k i is the concentration of the kth species in the ith solution. m is the total number of species that contribute to the absorption of light at any wavelength. Evidently, eq 2 follows the definition of matrix multiplication and may be written in the form
D = EC
(3)
Since the number of wavelengths at which measurements are made will not equal either the number of species present, nor the number of solutions measured, (1) Abstracted in part from the Ph.D. dissertation of A. A. El-Awady, University of Minnesota, 1965. (2) A. A. El-Awady and Z 2. Hugus, Jr., Inorg. Chem., 10, 1415 (1971). (3) R. M . Wallace, J . Phys. Chem., 64, 899 (1960). (4) S. Ainsworth, ibid., 65, 1968 (1961). (5) R. M. Wallace and S. M. Kata, ibid., 68, 3890 (1964). (6) D. Katakis, Anal. Chem., 37, 876 (1965). (7) J. 5. Coleman, L.P. Varga, and S.H. Mastin, Inorg. Chem., 9, 105 (1970) (8) A. C. Aitken, “Determinants and Matrices,” Oliver qnd Boyd, Edinburgh and London, 9th ed, 1956, p 96. I
DETERMINATION OF THE NUMBER OF SPECIES PRESENT IN A SYSTEM all three matrices are rectangular rather than square. Clearly, there should be more experiments than wavelengths, and more wavelengths than the suspected number of species. Since the rank of D is equal to the rank of E or C, whichever is smaller, and since the rank of either C or E can be no larger than m, the rank of D can be no larger than mI9provided the number of wavelengths and the number of experiments are both greater than m. Consider now the matrix, A , formed from the product of D with its transpose,
A AAA‘
=
=
DD
(44
CiDAtDtA’
(4b)
A is symmetric (square) matrix of order n, the number of wavelengths, of rank m. Now, since A is symmetric and real, it can be expressed in terms of its
matrix, v, whose elements are given by equations of the form
=
VIZ
= Cr.kSrjAjkSkr
VZZ
=
=
CJ,kSjlSkZAik
(5)
and this will contain as many terms as there are nonzero eigenvalues of A . Here PI*, P2*, . are the column eigenvectors of A , and *PI,*Pz, * are the row eigenThe at are the vectors (transposes of PI*,P2*, corresponding eigenvalues, and by Pl**Pl,etc., is meant the dyad or outer product. Viewed in this way, the number of nonzero eigenvalues is the mathematical rank of the matrix A . Any nonsingular procedure that serves to diagonalize A would then provide an immediate indication of its rank. To obtain both the eigenvalues and eigenvectors of the matrix A , one needs to utilize the method of orthogonal transformation. lo When the absorbance matrix, D , is derived from experiment, however, each element in D , and therefore in A , is subject to error. In general, on this account one finds the mathematical rank of A to be the same as its order. What is necessary, therefore, is a statistical criterion for the “vanjshing” of an eigenvalue, and one can attack the problem in the following way. The matrix A was defined in eq 4c as I
(9)
Using eq 9, and t’hen eq 7, it follows that GYiI
alP1**Pi+ 02P2**Pz+ . . .
Cj,kSjISklAjk
(8)
Here, the elements of the matrix, S, correspond to the principal axis transformation that diagonalizes A . Moreover, an element, Sjl, is the eigenvector of the matrix A corresponding to the Ith eigenvalue, v z. The matrix S could be obtained using a variety of standard procedures. I n this work an iterative procedure based on the classical method of Jacobi was developed.16 The standard deviation of an element of the diagonal matrix, v say, is given by
eigenvalues and eigenvectors as
A
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The variance of an element in the diagonal matrix will then be given by uu1,2=
i+,,,’zu,,
a ) .
Regarding the standard deviation in A j k as a vector quantityJ11-13 based on the not necessarily orthogonal vectorial standard deviations in the elements of D ,for examples, ZD,,,we then havel4~l5
cuii2
Cj,k(Sj1SkI)2{Ct(Dkt2zDji2t
The quantity P j k is a correlation coefficient between the standard deviations in Dkt and in D j t . We now make the assumption that P j k = 0, unless j = IC, and (9) Wallace3 pointed out that the rank of E or C will be less than m only if (a) the concentration of one or more species is zero in all
experiments, (b) the concentrations of all species are zero in more than m experiments, and (c) the concentrations of one or more species can be expressed as a linear combination of the other species in all experiments. The first two of these conditions are trivial, while the third can be violated by choosing different concentration levels in some experiments. (10) H . Margenau and G. M. Murphy, “The Mathematics of Physics and Chemistry,” Van Nostrand, Princeton, N. J., 1956, p 324 ff. (11) The standard deviation in Ajk is not viewed as a vector in the sense of the row or column vector interpretation of matrices as is generally accepted in matrix calculus. We are introducing geometrical concepts here on top of the usual matrix algebra. One may interpret the errors in the individual measurements as the length of the vectors from the mean of the observations in an N dimensional space. This description is useful when considering correlated errors where the correlations vary from one pair of measurements to the other. (12) W. C. Hamilton, “Statistics in Physical Science,” Ronald Press, New York, N. Y., 1964, pp 132-134, 174 ff. (13) K. J. Holzinger and H. H. Harman, “Factor Analysis,” The University of Chicago Press, Chicago, Ill., 1941, p 56 ff. (14) E. R. Cohen and J. W. M. DuMond, Rev. Mod. Phys., 37, 546 (1965). (15) E. R . Cohen, K. M. Crowe, and J. W. M . DuMond, “The
zA,k
=
Et{
DkIzDli
+ Dfr2Dii)
(7)
If we diagonalize the matrix A , we obtain the
Fundamental Constants of Physics,” Interscience, New York, N. Y., 1957, p 222 ff. (16) D. K. Faddeev and V. N. Faddeeva, “Computational Methods for Linear Algebra,” W. H . Freeman and Co., San Francisco, Calif., 1963, p 482 ff. The Journal of Physical Chemistry, Vol. 76, No. 19,1971
Z Z. HUGUS, JR.,AND A. A. EL-AWADY
2956 = 1, which implies that there is no correlation between uDIiand GDki, for i # i’. This yields the result that for j # IC the quantity in brackets is
{ c1(Dk.t2gD,i2 while for j
=
+
D/i2uDk12)]
(13)
k , the quantity in brackets has the value
{ c24D,22~D112]
(14)
Using the symbol g d i k 2 for quantity 13 and the symbol uA,i2for quantity 14, we then have for the variance in the Zth eigenvalue, v z l gqt2
=
c j , k s j 12Sk ?gA,b2
(15)
Then by comparing each eigenvalue with the square root of its variance as computed from (15), we have a criterion for its differing from zero to a statistically significant extent. However, because our assumptions concerning the correlations in D j , and Dktr may be questionable, there is yet another method that we decided to use to establish whether m species are sufficient to fit the experimental data satisfactorily. The matrix A was defined in eq 4a as the product of the optical density matrix D and its transpose D. Upon diagonalization of A we get
S“AS = N (diagonal) =
I (the identity or unit matrix)
Substitution of ( 4 4 into (16) gives
SDDS FT
=
=
N
N
where
T=DS
Table I: Data from 38 Solutionsa a t Nine Wavelengths (342 Experimental Points)
Eigenvalue
error in eigenvalue
No. of residuals more than 3xvest
1637.301311 2.642417 0,140060 0 * 000091 0.000057 0.000035 0.000027 0.000022 0.00001 5
0.231 0.092 0.111 0.057 0.063 0.071 0.056 0 073 0.052
311 155 0 0 0 0 0 0 0
Std
I
Xmz
222227.0 8741.9 34.67 21.04 14.40 9.88 5.15 2.83 0.00
a The solutions contain [ (en)&o(OH)&o(en)2] 4 + and its hydrolysis products.
On multiplication of (19) by S” on the right we get
TL!?= DSS” = D
A typical treatment of a set of our data, 38 experiments at nine wavelengths, led to the results summarized in Table I. It is evident that only the first three eigenvalues are larger than their standard errors, but perhaps more significantly, the other six eigenvalues are so much smaller than their standard errors that they should surely be regarded as being statistically equivalent to zero. Whether a third species should be assumed at any high level of confidence is open to some question on the first two columns of Table I. However, the results given in the last two columns leave no
(16)
where S”S
By comparing I B z P with g D , ; we then can find, for each value of m, how many residuals in absorbance compared to their estimated errors lie in various ranges. I n addition, we can compute the value of x 2 and use well-known statistical tests to determine whether the fit is satisfactory. Here
(20) ~
If we now arrange the eigenvalues in order of decreasing value-since the matrix A is real and symmetric, the eigenvalues are nonnegative definite-then an approximation to the measured absorbance at the lth wavelength in the ith experiment, D d l ,using only the first m eigenvectors will be given by m
B.ti(m) = CjTijgjz
(22)
where (23)
I n these equations 8 k j and fljz are the j t h components of, respectively, the kth or lth eigenvector, and are represented by row and column matrices, respectively. The Journal of Physical Chemistry, VoL 76,No. 19, 1971
Table 11: Data from Eight Solutionsa at Eight Wavelengths (64 Experimental Points) No. of
Eigenvalue
Std error in eigenvalue
residuals more than 3xusst
Xm2
11.579207 1.167141 0,003276 0.000323 0.000053 0.000016 0.000005 0.000002
0.0173 0.0013 0.0046 0.0058 0.0046 0,0054 0,0036 0.0064
53 9 0 0 0 0 0 0
130091.O 408.3 44.27 8.40 2.55 0.76 0.20 0.00
a The solutions contain methyl red at different pH values (see ref 5).
SHORT-LIVED RADICALS GENERATED BY FAST FLOW TECHNIQUES doubt that a third species must be present, since the fit with only two species (eigenvectors) is completely unsatisfactory. Actually the expectation ~ a l u e of’ x32 ~ ~is~339, ~ and our unduly pessimistic estimates of error-too large by a factor of about 3-cause the value, 34.67, to be as small as it is. An additional set of data taken from the paper of Wallace and Katz6 on methyl red solutions at varying pH’s was treated using our program and gave the results summarized in Table 11. These measurements were made at eight wavelengths on solutions at eight different pH’s. The questionable fourth component mentioned by Wallace and Katz is not supported by the value of xa2which has an expectation value of 61. I n addition, the probability of obtaining a value of x42as
2957
low as 8.4is much smaller than 0.001, although an overestimate of the measurement errors could cause such a low value of x2. In our treatment each error was set at 0.003, the same value used by Wallace and Katz, which seems reasonable. The data were treated by means of a FORTRAN program, $PTIC$N, using a Control Data Corp. 1604 computer. The program as written could be used for up to 200 experiments at ten wavelengths, but adaptation to larger numbers of experiments or of wavelengths would entail only changes in DIMENSION statements. (17) 0. L. Davies, Ed., “Statistical Methods of Research and Production,” Oliver and Boyd, London, 1961,p 283. (18) L. G.Parratt, “Probability and Experimental Errors in Science, An Elementary Survey,” Wiley, New York, N. Y.,1961, pp 188191.
Electron Spin Resonance Studies of Short-Lived Radicals Generated by Fast Flow Techniques in Aqueous Solutions1 by Gideon Czapski Radiation Research Laboratories and Department of Chemistry, Mellon Institute of Science, Carnegie-Mellon University, Pittsburgh, Pennsylvania 16813 (Received April 16, 1971) Publication costa aasiated by Carnegie-Mellon University and the U.S. Atomic Energy Commission
The application of fast reaction flow systems with esr detection to studies of free radicals in aqueous solution is analyaed in detail. It is shown that for short-lived radicals it is essential that the chemical reactions which produce the intermediates proceed within the cavity and that few radicals will be observed if the reactions responsible for the radical formation are complete immediately after mixing. The assumption frequently made in these studies, that this reaction is complete before the solution enters the cavity, is shown to be incorrect. It is believed that many of the controversies in the literature can be readily cleared up by analysis of the kinetics as is done here.
Introduction I n most cases the study of free radical intermediates in aqueous solutions demands fast detection techniques since these radicals normally have very short lifetimes. Most recent studies of the kinetics of free radical reactions in aqueous solutions have been carried out with flash photolysis or pulse radiolysis techniques using optical detection. With these techniques, the absorption spectra and decay of the radicals can be directly observed. The esr spectra of many radicals have been studied in aqueous systems in experiments which mainly employed flow techniques where the radicals were generated chemically. I n most of these studies the emphasis was on the qualitative identification of the radicals and
a determination of the esr parameters with little work being directed toward obtaining kinetic information. Until recently there were difficulties in fast recording of esr spectra. Fessenden, 2a using the pulse radiolysis technique, developed a sampling device for esr measurements by which spectra and kinetics of radicals living longer than several milliseconds could be recorded. Even this method is rather slow for most radicals in ,~ aqueous solution. Recently Smaller, et u Z . , ~ ~ combined pulse radiolysis with esr detection to follow the decay of (1) Supported in part by the U.S. Atomic Energy Commission. (2) (a) R. W. Fessenden, J . Phys. Chem., 68,1508 (1964); (b) E . C. Avery, J. R. Remko, and B. Smaller, J . Chem. Phys., 49,951 (1968). (3) B.Smaller, J. R. Remko, and E. C. Avery, ibid.,48,5174 (1968).
The Journal of Physical Chemistry, Vol. 76, No. f9, 1971
