Simultaneous Determination of Cobalt, Copper, and Nickel by Multivariate Linear Regression Greg Dado and Jeffrey Rosenthal University of Wisconsin-River Falls, River Falls. WI 54022
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One of the exneriments that our students have nerformed -~~ fur several years in the instrumental laboratory is a simultaneousdererminationofthree metal ion.; inasulution by I ' V -~ vis spectroscopy. This experiment has evolved from a need to reinforce concentration and dilution calculations while exposing our students to a more sophisticated experimental nroblem and the associated statistics. The traditional apbrnach, however, required that each studrnt prepare a set standards for e ~ c of h the three metals and record the absorb a n c e ~of each solution a t three wavelengths (as will be shown later, a t least five wavelengths should be used for a system of three components). From the absorbance data, a standard curve was prepared for each metal at each wavelength. The molar absorptivities determined from the standard curves and the absorbances measured from an unknown were arraneed as a set of three eauations and three unknowns from wkch the concentration bf the three metal ions were calculated. Although the students have received a lot of dilution practice, theprocedure is tedious, and the precision of their results is difficult to determine. In order to reduce the number of absorbance measurements and provide a means for evaluating the precision of the results, multivariate linear regression was investigated. ~~
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Theory A good statistics book' should be consulted for a rigorous treatment of multivariate linear regression (MLR); the following is a overview. MLR is similar to linear regression in that the squared deviations between the data and a model areminimized. In the case of MLR the model is a surface in a multidimensional space; when reduced to two dimensions, the model becomes the line of the linear regression. In order to apply MLR to a spectroscopic analysis, we begin with the familiar Beer-Lambert law relating the absorbance of a sample (A) to the optical path length through the sample (1) and the molar absorptivity (6) and concentration (C) of a chromophore, which may be represented as
where the final term, a, is the intercept of a standard curve. For a multicomponent sample where there are no intercomponent interactions, the sample absorbance may be expressed as a linear sum of the individual component absorbance~,as is shown in eq 2 for a sample containingp chromophores. where rj = the molar absorptivity of component i and C, = the concentration of component i. I n a homogeneous solution theoptical path length for each chromophore is the same and in UV-vis spectroscopy generally has a value of 1 cm. Therefore, the 1 term may be incorporated into the molar absorptivity terms:
' Weisberg, Sanford.AppliedLinearRegression; Wiley: New York, 1985.
where mi = cil. Matrix notation may be used to simplify eq 3; the concentrations become a single row matrix (a row vector):
c,.
=
[I, C,.C2, C3'
.. .
.
CJ
(4)
and the molar absorptivities, a single column matrix (a column vector):
The long dimension of both vectors is one greater than the number of components in the sample (p' = p 1),in order to accommodate the intercept, a, which is now represented as mo.Note that bold face is used to denote a vector or matrix and that its dimension appears as the subscript. Equation 3 rewritten in matrix form is given in eq 6.
+
The product of a 1 by p' matrix and a p' by 1 matrix is a scaler, the sample absorbance. In order to analyze a multicomponent sample, the absorbance a t more than one wavelength must be measured. Extending the matrices already defined, the sample absorbance becomes a vector composed of the sample absorbances a t each wavelength, the concentration vector remains the same, and the molar absorptivity vector becomes a matrix of dimensionp' by m (the number of wavelengths) as is shown in eq 7.
F..
...
1
mo,, mo2, , man, m12,. . . mim
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In matrix shorthand eq 7 becomes Each column of the molar absorptivity matrix represents the set of molar absorptivities a t a particular wavelength for each component, while each row (except the first) represents a set of molar absorptivities of one of the components a t every wavelength. The first row of the molar absorptivity matrix contains the intercepts on the absorbance axis at each wavelength. Thus, manis the molar absorptivity of the third component a t the second wavelength. The individual values of the molar absorptivity matrix may be determined by analyzing a set of standards for each Volume 67
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September 1990
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component. Instead of independently determining the molarabsorptivity of eachcomponent a t each wavelength; however, the whole matrix of values may be determined simultaneously by using a set of standards that incorporates all of the components in each of the standard solutions. For a set of n standard solutions eq 8 becomes where n = the number of standard solutions, m = the number of wavelengths, p = the number of components,
vities, the variance of 101must he evaluated. The variance of the estimator of molar absorptivity matrix is given by var(M) = 02(CTC)-'
n-P
As RSS(M) = ETE, the standard error of the estimator of the molar absorptivity matrix may be determined by substitution into eq 12 to yield "&@)
and E is the error matrix. As an example, Agz is the absorbance of standard solution 3 at the second wavelength, CQZis the concentration of the second component in the third standard solution, and ma2is the molar absorptivity of the third component at the second wavelength. The estimator of the molar absorptivity matrix is obtained by minimizing the sum of the squared deviations (ETE): The caret over the symbol of the molar absorptivity matrix denotes an expectation matrix, an experimental estimate of the true matrix. Note that these equations may not be rearranged by the usual algebraic manipulation as matrices are not simple variables. Compare eq 10 to a linear regression analysis of a set of one-component standards a t one wavelength; eq 8 would become
(121
The standard error of regression (c2) is not known but may be estimated from the residual sum of squares of fi by RSS(M1 $2 ss (131
=n-p'
This is a matrix of variances, one for each value of the molar absorptivity matrix. The square root of each value of the variance matrix yields the standard deviation of the corresnondina molar absorntivitv. - I n order to determine the concentration of each component in an unknown samnle. the absorbance of the unknown must be measured at each wavelength and the values arranged as an absorbance matrix. Equation 9 must be rearranged so that the concentration matrix may be evaluated instead of the molar absorntivit~matrix. In order to rearrangeeq 9,allof the miltrir& muit he transposed rrecnll that tor matrires, AR is not equal 10 MA). The transnused form of eq 9 is (151 ATmxn = fiTmxp.CTp,xn + ETmrn where all of the symbols retain their original definitions except n, which is now the number of unknown solutions. For one unknown sample solution, there would one column of absorhances and concentrations (ATmxland CT,.,I). In eq 15 the estimator of M is used, as the unknowns will be determined after an analysis of the standards. The estimator of the unknown concentration matrix is CT = (MMT)-l(M~T)
(16)
and the variance of the unknown concentration estimator is
A comment about the number of standard $olutions and wavelengths used in an experimental analysis is in order. As with any analysis the minimum number of observations is determined by the numher of degrees of freedom. For linear rearession. the number of demees of freedom is determined h
