Anal. Chem. 1981, 53, 2207-2212
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Generalized Standard Addition Method for Multicomponent Inst rument Characterization and Elimination of I nt erf erences in Inductively Coupled Plasma Spectrometry J. H. Kalivas and B. R. Kowalski" Laboratory for Chemomei'rics, Department of Chemistry BG- IO, University of Washington, Seattle, Washington 98 795
Various types of interftrrences are known to exist when using inductively coupled plasma (ICP) emlsslon spectrometry. They have been classified as stray light, direct spectral overlap, overlap of severely broadened lines of high intensity, and conitlnuum emlssion from ion-electron recomblnation. The presence of these interferences often requires alternate (less sensitive) wavelength selection. The generalized standard addition method (GSAM) is a method of multicomponent analysis which provides a means of detecting the Interference effects, quantifying the magnitude of the interferences,,allowing the use of the most sensitive wavelengths for all analytes, and simultaneously determinlng analyte concentrations. In addition, the GSAM can be used to completely Characterize all analytlical wavelengths of a particular ICP instrument. This procedure is quite general and can be applied to most multicomponent methods of analysis.
Inductively coupled plasma (ICP) atomic emission spectrometry (AES), one of the most important new tools for chemical analysis, has been shown to suffer from various forms of interferences resulting from the chemical composition of the samplle matrix (1-4). These multiple interferences have been categorized as stray light, direct spectral overlap, overlap of severly broadened lines of high intensity and continuum emission from ion-electron recombination. Various procedures have been developed to correct for these interference effects (1, 3-6). However, none of these procedures allows for the simultaneous correction of all these interference effects. Also, many of the techniques require prior knowledge of the structure of the spectrum a t and near the analyte emission lines. For example, if line-broadened interference is present, background correction at a wavelength very near the analyte line may result in an over compensation when the correction wavelength is near the broadened interfering emission line. Likewise, a similar problem occurs if background correction is applied when spectral overlap interferences exist. ICP-AIES has also been shown to experience relatively few matrix effects when the proper operating parameters are being used (7,8). However, iit has been demonstrated that viscosity changes caused by variations in the total acid content produce large deviations in analytical sensitivities (Sll).Variations in the total salt content may also produce the same effect (11). Clearly, a method is needed that can detect, characterize, and correct for these interferences and matrix effects without preliminary examination of the spectrum for the sample matrix. Recently our laboratory developed an alternate experimentrd design and a mathematical data analysis procedure that can accomplish these tasks. The method is called the generalized standard addition method (GSAM) (12,13) and is based on the important method of standard additions. In its simplest form, the GSAM is multivariate (more than one analyte) standard addition. After a brief introduction to the GSAM, this paper will deal with the applicability of the GSAM to ICP-AES for multicomponent analysis in ithe presence of direct spectral overlap
Table I. Interferences Studied interfering element, wave- interference length (nm) type
element, wavelength (nm)" Zn, 213.856 (I)
Ni, 213.858 (11) SO Cu, 213.851 ( I ) SO Cu, 213.598 (11) BL Cd, 228.802 (II)c As, 228.812 ( I ) so As, 193.696 (I) A1 CE SO = spectral overlap; BL a I = atomic, I1 = ionic. Second broadened line; CE = continuum emission. order.
=
interference, continuum emission interference, and broadened line interference. It will also be shown that with the GSAM, no longer are direct reading ICPs handicapped with the fixed array of exit slits and compromise analytical wavelengths (Le., the selection of less sensitive emission lines instead of the most sensitive lines). With samples containing broad ranges of elements at various concentrations, direct reading ICPs with the preselected emission lines, were restricted by the exit slit positions for the sensors in that they may exhibit interference from other analytes in the sample. This either requires correction for the interferences or selection of alternate interference-free wavelengths to monitor the analytes. In Table I are listed the elements studied, along with interfering elements and the type of interferences involved. These representative analytes were chosen because of the variety of interferences present, thus allowing for the full potential use of the GSAM to correct for multiple linear interferences. This paper will also show how to use the GSAM for a complete characterization of all analytical wavelengths of a particular ICP instrument. This procedure is quite general and can be applied to most multicomponent analysis methods. T h e Generalized S t a n d a r d Addition Method. The background and theory of the GSAM have been discussed in full detail in previous papers (12,13). Therefore, only the key equations are presented here. The GSAM requires that when there are r analytes to be determined, the responses from p sensors (analytical wavelengths, electrodes, etc.) ( p 2 r ) be recorded before and after n standard additions are made ( n 2 r). The underlying model in matrix notation is
R = CK (1) where R is the ( n + 1) X p matrix of measured responses (initial readings plus readings after standard additions) with n equal to the number of multiple standard additions made and p equal t o the number of sensors. C is the (n + 1) x r concentration (initial plus additions) matrix of the r analytes, and K represents the r x p matrix of linear response constants with each column showing the contribution of each of the r analytes to the pth sensor (r = p assumed in this study). Since concentrations are not always additive, using volume corrected changes for the responses (13), eq 1 becomes AQ = 0 1981 American Chemical Society
ANR
(2)
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ANALYTICAL CHEMISTRY, VOL. 53, NO. 14, DECEMBER 1981
where AQ is now the n X p volume corrected response change matrix and AN is the n X r absolute quantity matrix (e.g., moles, pg) of standards added to the sample. With the method of multiple linear least squares to solve for K (14),the generalized inverse solution is obtained
K =(WAN-1WAQ
____^-__-_I___
element Ni Cu Zn A1
(3)
where T signifies the transpose matrix. After calculation of the K matrix, the vector of initial analyte quantities, no, is recovered by solving 40
= noK
l18noll and llnollsignify the Euclidian norm of the total absolute error in no and the Euclidian norm of the estimated analyte quantities, respectively. Briefly, the Euclidian norm of a vector is the square root of the sum of the squares of the entries in the vector. The Euclidian norm also represents the length of the vector. The ratio represents the total relative errors in all estimated analyte amounts. Similar meaning is given to I16qoll/llqolland ~ ~ 6 K ~ The ~ / [error ~ K magnification ~~. term in eq 5 is the condition number of the K matrix which is defined as cond ( K ) =
AS
Cd Ag Be Ca
(4
where go is the volume corrected initial response vector. For example, for r analytes with multiple interferences, p sensors are selected with p 1 r- The responses of all the p sensors are then recorded before and after the n additions of standards (singly or multiply) corresponding to the r analytes. The restrictions on the number, composition, and order of the n multiple standard additions are (1)n 2 r, (2) the addition must span the r dimensional concentration space ( N A i V must have rank r), end (3) each analyte must effect a response from a t least one sensor (AQTAQmust have rank Z r ) . If n < r , AhrrAN would have rank n rather then rank r thereby not complying with restriction (2) and invalidating eq 3. When n = r, the analyst has obtained just sufficient data to satisfy the above conditions allowing determination of K and no. When n > r, the analyst has an overdetermined system yielding more stable results. As with any analytical calibration system, there is a possibility for error magnification resulting in the relative errors in the estimated analyte concentrations being larger than the relative response errors. This error magnification can be represented by the condition number of the matrix (13). For example, the following equation gives the effect of the error magnification in the estimated analyte quantities for the GSAM model.
11~11~11~~111
(6) for a nonsingular matrix K. IlKll and llK-lll signify the Euclidian norm of the respective matrix. These norms involve eignvalue determinations. The reader is referred to ref 13 and 15 for further discussion of these concepts. Therefore, a method of analysis with the smallest cond ( K ) will have the smallest error magnification in the relative error of no. Likewise, there is error amplification in the estimated K matrix which is given by
(7) The usual meaning is applied to the norms and the relative error of k is the error in one sensor over all analytes (one column of K ) . The error amplification term in eq 7 is cond (AN), the generalized condition number of AN,which may be computed for the rectangular matrix AN by cond (AN) = (cond (wAiVl)'/' (8)
_ _ _ _ l l l
Table 11. Elements with Corresponding Wavelengths
co
a
wavelength,a nm 231.604 (11) 324.754 (I) 213.866 396.153 (1) 193.696 (I) 228.802 ( I ) b 328.068 (I) 234.861 (I) 396.847 (11) 228.616 (11)
I = atomic; I1 = ionic.
element Fe Mg P Pb
wavelength,a nm 259.940 (11) 279.553 (11) 214.914 (I) 220,353 (11) 217.581 (I) 196.026 (I) 251.611 (I) 190.864 (11) 257.610 (11)
Sb
Se Si T1 Mn
Second order,
Table 111. Operating Conditions Plasma incident rf power reflected rf power coolant argon flow rate sample argon flow rate solution uptake rate
1.1kW
