Anal. Chem. 1985, 57, 952-954
952
PORPHYRlNlTHlOGLYCEROL ClDlMlKES SPECTRUM ( x l ) 18 SCANS AVERAGED
I I
GLYCEROL - PORPHYRlNlTHlOGLYCEROL ClDlMlKES SPECTRUM ( x 2 ) 8 SCANS AVERAGED
PORPHYRlNlTHlOGLYCEROL PHOTODISSOCIATION SPECTRUM ( x 1) 1 6 SCANS AVERAGED
W
2 c
2
#I
\-
-
GLYCEROL PORPHYRlNlTHlOGLYCEROL PHOTODISSOCIATION SPECTRUM ( x 2 ) 2 5 SCANS AVERAGED
! 1.OE
tractable molecules of biological importance ionized by the new desorption ionization techniques. The use of a tunable photon source will enhance selectivity and sensitivity because the photodissociation can be performed at a wavelength specific for the compound of interest and having an optimal photon absorption cross section, thus leading to enhanced photofragmentation. Further investigation of the ion photodissociation spectra of the porphyrins over a continuous range of wavelengths would allow the direct comparison of the ion spectra and the electronic absorption spectra. This would provide insight into the ion structure and the dynamics of ion/photon interactions and the electronic structure of the molecular ions.
ACKNOWLEDGMENT The authors thank Michael L. Gross (University of Nebraska) for suggesting the porphyrins for study, Leon Kurlansik (Naval Medical Research Institute) for providing the novel synthetic porphyrins, and Steven Schneider for technical assistance. Registry No. 1, 32407-78-6; 2, 94636-67-6;3, 85771-00-2;4, 94905-12-1; 5, 94636-66-5; 6, 94643-17-1; 7, 52199-35-6; 8, 67595-98-6; 9, 55253-62-8; 10, 83198-46-3; 11, 55022-76-9; 12, 36674-90-5. LITERATURE CITED (1) Kurlansik, L.; Williams, T. J.; Campana, J. E.; Green, 8. N.; Anderson, L. W.; Strong, J. M. Biochem. Biophys. Res. Commun. 1983, 111,
,i
478-483. (2) Kurlansik, L.; Williams, T. J.; Strong, J. M.; Anderson, L. W.; Campana, J. E. Blomed. Mass Spectrom. 1984, 1 1 , 475-481. ION KINETIC ENERGY
0.6E
Flgure 2. The CIDlMIKES and photodissociation/MIKESspectra of the molecular ion ( m l z 737) of compound 8, zinc meso-tetra@aminophenyl)porphyrin, in thioglycerol and of the m l z 737 species in a glycerollthioglycerol mixture: (A) CIDlMIKES spectrum of the porphyrin in a thicgiycerol matrix: (B) CIDlMIKES spectrum of the m l z 737 species. A small percentage of the porphyrin/thiogiyceral sample was admixed with glycerol. The glycerol [(C3H8),H]+ ( m l z 737) cluster ion is isobaric with the porphyrin molecular ion. The glycerol cluster daughter ions are also isobaric with some of the porphyrin daughter ions, which could confuse interpretation of the mass spectrum; (C) PhotodissociationlMIKESspectrum of the porphyrin in a thioglycerol matrix: (D)PhotodissociationlMIKES spectrum of the m / z 737 species from the glycerol-porphyrin/thioglycerol mixture, which is due only to the photodissociation of the porphyrin. ( I n a separate experiment, we were not able to detect any ionic photofragments from the glycerol cluster ion at m l z 737).
(3) McLafferty, F. W., Ed. "Tandem Mass Spectrometry"; Wlley: New York, 1983. (4) Cooks, R . G., Ed. "Collislon Spectroscopy"; Plenum Press: New York.
1978. (5) Kim, M. S.; McLafferty, F. W. J . Am. Chem. Soc. 1978, 100, 3279-3282. (6) Griffiths, I.W.; Mukhtar, E. S.;March, R. E.; Harris, F. M.; Beynon, J. H. Int. J . Mass Spectrom. Ion Phys. 1981, 39, 125-132. (7) Mukhtar, E. S.;Grlffiths, I. W.; Harris, F. M.; Beynon, J. H. Org Mass
Spectrom. 1981, 16, 51. (8) Kim, M. S . ; Morgan, T. G.: Kingston, E. E.; Harris, F. M. Org. Mass Spectrom. 1983, 18, 582-586. (9) Cook, K. D.; Chan, K. W. S . Int. J . Mass Spectrom. Ion Proc. 1983,
54, 135-149. (10) Morgan, R . P.; Beynon, J. H.; Bateman, R. H.; Green, 6. N. Int. J . Mass Spectrom. Ion Phys. 1979, 28, 171-191. (11) Mukhtar, E. S . ; Griffiths, I.W.; Harris, F. M.; Beynon, J. H. Int. J . Mass Spectrom. Ion Phys. 1981, 37, 159-166. (12) Pantell, R. H.; Puthoff, H. E. "Fundamentals of Quantum Electronics"; Wlley: New York, 1969. (13) Bowers, M. T., Ed. "Gas-Phase Ion Chemistry, Vol. 3, Ions and Light"; Academic Press: New York, 1984. (14) Almog, J.; Baldwin, J. E.; Dyer, R. L.; Peters, M. J . Am. Chem. Soc. 1975, 97, 226-227.
(2.41-2.71 eV) used in these studies. Therefore, similar daughter ions and kinetic energy releases from the porphyrin in the CID and photodissociation spectra are observed. However, the collisional activation energy can exceed the 2.71-eV photoexcitation maximum, which implies that the onset for the fragmentation of the glycerol cluster ion lies above 2.7 eV.
Naval Research Laboratory Chemistry Division Washington, D.C. 20375-5000
CONCLUSIONS This rather spectroscopically limited laser/mass spectrometric study demonstrates the concept and potential of ion photodissociation to characterize selectively complex and in-
RECEIVED for review September 21,1984. Accepted January 4, 1985. This work was supported by the Office of Naval Research. E.K.F. thanks the National Research Council for support as an NRC/NRL Cooperative Research Associate.
'Present address: Celanese Research Go., Summit, NJ 07901.
Elaine K. Fukuda' Joseph E. Campana*
Unbiased Generalized Standard Addition Method Sir: During the last decade there has been a lot of interest in the quantitative analysis of multicomponent data. This is due to our increased ability to collect multiwavelength data and to handle the required matrix transformations on com-
puters which are now interfaced to analytical instruments. Recently, a method for multicomponent analysis using standard additions of more than one analyte, component of interest, was developed. This method is called the generalized
0003-2700/85/0357-0952$01.50/0@ 1985 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 57, NO. 4, APRIL 1985
standard addition method (GSAM) (1-4),which is able to detect and correct for spectral interferences, matrix effects, and drift. The computational procedure of the GSAM is composed of two steps. A matrix of linear response constants showing the contribution of analytes to sensor responses is computed at the first step. To compute this matrix, the initial response of the unknown sample is subtracted from the response data matrix. The determined matrix is used a t the second step to compute the concentrations in the sample. Unfortunately, the subtraction of initial response from the response matrix results in biased estimates of the concentrations in the unknown sample. This point is well illustrated by referring to the quantitation of an unknown, using one-dimensional (regular) standard addition. T o estimate the unknown concentration, the response of the analytical sensor is plotted against the number of moles added to the unknown. The best straight line is fitted to these points and extended back to intersect the abscissa. The point where the intersection occurs is the initial number of moles of unknown. The best straight line fit to measured responses will yield the “best linear unbiased estimate” (BLUE) of the unknown. Unlike the onedimensional standard addition, the multidimensional GSAM’s quantitation results are not BLUE. This follows from the subtraction of initial response from the response matrix. In the one-dimensional method, this is equal to enforcing the straight lint t b pass through the initial sample response. In the rest of this note, this argument will be clarified and a technique that allows the achievement of BLUE for the multidimensional GSAM is presented.
THEORY Probem Formulation and Notation. Boldface capital letters are used for matrices, e.g., X, superscript T for transposed matrices, e.g., XT, boldface small characters for vectors, e.g., x, and xo,and italic lower case characters for scalers, e.g., x . Superscript + denotes the pseudoinverse, e.g., X+. The symbol (xlX)designates a partitioned matrix. There are m (i = 1,..., m) sensors, n = 1,...,n) standard additions, and p (k = 1, ..., p ) analytes. The matrix Q which is of the order m X (n + 1) is the matrix of instrumental responses multiplied by the total volume. C is the matrix of number of moles of analytes and is of order p X (n + 1). The matrix A of order m X p connects the two matrices by
f::: 2: :::
...
suggested separating the matrix Q into two terms, as follows: Q = AQ qojToIn doing this separation, one presupposes that qois equal to Q which is the BLUE of the vector of initial sample response. When this assumption is made, eq 2 may be separated into two equations
+
(34
AQ = A AC 90
= Aco
(3b)
Equation 3a is solved first to find A which in turn is used by eq 3b to quantify the unknown number of moles. It is obvious that both A and co are biased due to the assumption Qo = go. Solution of the Unbiased GSAM. In order to get a BLUE and yet avoid the time-consuming solution of a nonlinear matrix equation, it is suggested here to use a nonzero intercept model. The matrices involved will be redefined as A* = (%lA), C* = (jlACT)T,and the matrix Q remains unaltered. Now, eq 1 may bae rewritten as a1 ,o
...
a1,1
a1.p az,p
a2.1
.“
am,,
... a m , ) ’
Q=tl:o
1 1,............... 1
O
ACl,,
(i
Acp,l
...
ACi,,
...
Acp,)(4)
or
Q = A*C* The benefit of using the nonzero intercept model is removing cofrom eq 4 which allows quantitation of A by linear methods. It is also obvious that a. is equal to Go which is now inserted into
qo = a. = Aco
(5)
to obtain a BLUE of the analyte amounts. It may be noted that the solution of a nonzero intercept model does not require a solution of matrices of order greater by one than the zero intercept model, and thus an additional sensor is needed (in contrast to what is claimed in ref 5). Rather, a transformation that allows the solution of matrices with the same order as the zero intercept model can be used (6). The solution of eq 5 is presented in the Appendix. Frank et al. (2) applied the partial least-squares method (PLS) to the GSAM. They found that the results obtained by PLS were superior in terms of noise rejection to those obtained by conventional least squares. The superiority of quantifying by the PLS is well explained by the fact that in the PLS method the initial responses are also used in estimating the constants matrix. Jochum et al. ( 3 ) and Moran and Kowalski (4) derived expressions to describe the effect of random experimental error on the GSAM. The derivation of the error estimates was complicated by the fact that the original computation is made by two steps. By use of the unbiased GSAM this difficulty is avoided as the second step does not introduce error. Thus error estimate by either of the methods (condition number (3) or variance matrix (4)) is much simpler.
:::y:: a2,2 a1,2
‘.‘ ”’
Q2,p) a1.p
X
4m,o 4m,1
953
4m,n
am, I ( 1C . 02 . o
am>2
...
am,p
...
C Zi ,, 1I
...C Cz,n 1,n)
CP, I
...
(1) CP,O
Cp9n
or
Q = AC The first columns qo and co are the vectors of initial sample concentration and response. The matrix C may be written as C = AC + cJTwhere j is a vector with n 1 elements all of which are units. AC is the added concentrations matrix. Now eq 1 is read as
+
Q = A(AC
+ cOjT)
APPENDIX Solution of the Unbiased GSAM. According to the least-squares approach, eq 4 is right multiplied by (jlACT)to obtain the normal equation
(2)
which is a nonlinear equation (because of the multiple, Aco, of two unknown quantities) and its solution requires the use of iterative methods. In order to overcome this difficulty, Saxberg and Kowalski ( I ) , in their formulation of the GSAM,
After the multiplications are performed, eq 6 results in two matrix equations
Qj = ao(n + 1) + A ACj
(7)
Anal. Chem. 1985, 57, 954-955
954
Q = aojT+ A AC
(8)
The mean of the responses for each sensor is defined by
A = Q AC'
n+l
40
4, = C4iJ ]=1
Thus, the vector of mean responses, q, is computed by
Qj n + l The mean of added analyte concentrations is also given by e = -AC j n + l Now, it is observed that eq 7 may be written as a. = q - AE (11) q=-
When this value of a. is inserted into eq 8, the following equation is obtained:
Q
- qjT = AtAC - cjT)
Q=AAC
=A C ~
(13)
whose solution is
(15)
Since qo = ao,the value of a. from eq 11 is inserted into eq 15, to get
+
q = A ( c ~ E)
(16)
and finally CO
= A'ij - E
(17)
LITERATURE CITED (1) Saxberg, 8. E. H.; Kowalski, B. R. Anal. Chem. 1979, 57. 1031 (2) Frank, I.E.; Kalivas, J. H.; Kowalski, 8. R. Anal. Chem. 1983, 5 5 , 1800. (3) Jochum, C.; Jochum, P.; Kowalski, B. R. Anal. Chem. 1981, 5 3 , 85. (4) Moran, M. G.; Kowalski, B. R. Anal. Chem. 1984, 56, 562. (5) Brown, C. W.; Lynch, P. F.; Obremski, R . J.; Lavery, D. S. Anal. Chem. 1982, 5 4 , 1472. (6) Campbell, S. L.; Meyer, C. D. "Generalized Inverses of Linear Transformations"; Pitman: London, 1979; pp 32-41,
(12)
Defining Q = Q - GTand AC = AC - EjT which are zero mean matrices of responses and concentrations, respectively, eq 1 2 is now written
(14)
Quantification of the sample is performed by
Avraham Lorber Nuclear Research Centre-Negev P.O. Box 9001 Beer-Sheva 84190, Israel RECEIVED for review November 2, 1984. Accepted January 2, 1985.
Square Wave Voltammetry at Electrodes Having a Small Dimension Sir: Square wave voltammetry is a large amplitude differential technique in which a wave form composed of a square wave superimposed on a staircase is applied to the working electrode ( I , 2). Current measurements are made near the end of the pulse in each square wave half cycle; the difference of these two currents, when plotted vs. staircase potential, yields a symmetrical, peak-shaped net current voltammogam. The simplicity and fidelity of this net current response as an indicator of analyte properties under conditions which complicate conventional voltammetric methods are the subjects of this paper. The trend towards smaller sample sizes, in situ analysis, and detection in flowing streams makes it harder to use analytical voltammetry under the customary boundary conditions of planar semiinfinite diffusion. This in turn severely limits the ability of traditional voltammetric techniques (Le., differential pulse and cyclic voltammetry) of electroanalysis to quantitate and characterize analytes. The time scale of differential pulse voltammetry (minutes) makes it impractical for many applications, especially those which require contemporaneous, rather than retrospective information. Cyclic voltammetry is limited at low analyte concentrations by charging currents, and the shape of the voltammogram depends on the mode of diffusion. Under conditions of nonplanar diffusion the well-known peaked response tends toward an S-shaped wave (3) while for restricted diffusion it tends toward a symmetrical peak ( 4 ) . Ideally what is desired is a method which preserves quantitative information in a simple peak-shaped response invarient in shape regardless of the electrode geometry, time scale of the experiment, or mechanism of mass transport. Our experiences with pulse methods in a variety of experimental situations suggest that square wave voltammetry possesses these attributes and hence will 0003-2700/85/0357-0954$0 1.50/0
be very useful in many modern applications of voltammetry to quantitative analysis. Figure 1 shows a direct comparison of cyclic staircase and square wave voltammetry used a t identical scan rates with a planar reticulated vitreous carbon electrode and a reversible redox couple. The results for staircase voltammetry are very nearly the same as those for cyclic voltammetry and become identical as the staircase amplitude approaches zero at constant scan rate. Nonplanar diffusion is known to make a significant contribution to the total current measured under these conditions with this type of electrode (5). Note that the staircase voltammogram is severely distorted from the reversible shape afforded by ordinary linear diffusion and offers no easily quantifiable features (6). On the other hand the square wave voltammogram offers a prominent symmetrical peak with a characteristic position, width, and height. Unlike the result for staircase voltammetry, the peak shape is independent of the quantity r/(Dt)'tz where r is a characteristic dimension of the electrode, D is the diffusion coefficient of the reacting species, and t is a characteristic time of the experiment. (When r / ( D t ) l l Z>> 1 only planar semiinfinite diffusion is observed.) For practical purposes the peak potential coincides with the half wave potential of the redox couple, the peak width indicates the effective number of electrons transferred, and the peak height is a measure of analyte concentration. Hence, voltammetric analyses and measurements are greatly simplified by the use of the square wave wave form. Its usefulness for rapid surveys of electrochemical properties should also be obvious. Many other types of electrodes which exhibit nonplanar or restricted diffusion on the time scale of the experiment yield simple peak-shaped voltammograms. Included in this group are an embedded thin wire cross section (microdisk), exposed 0 1985 American Chemical Society
