Acknowledgment The Rohert A. Welch Foundation is gratefully acknowledged for their exemplary support of the Welch Summer Research Program and for their support of this research. Literature Clted 1. Cu1len.J. F. J. Chem.Edue. 1989,66.1043-1045. 2. Kae. M. Probobdiry ond Related Topics in Physieol Science; Interscience: New York. 1959: p 72ff. 3. Kac, M.: Rots. 0.-C.: Schwsrtz. J. T. In Dbcrata Thoughla. Essoys on Malhsmalirs. Science, and Philosophy: Newman, H., Ed.: Birkhauaer: Boston, 1985:p 42. 4. Kac, M. Enigmos o, Chonrs: Univ. of California: Berkeley, 1985: p 117; Kse. M. Am. Moth. Monlhiy 1947.51.263-291;Kse, M. Sci. Am. 1964,211.97A08. 5. Engel. A. In The Teaching o/ Pmbabiity and Statistics, Ride, L., Ed.; Wiley-lnterscimce: New YorklAlmqvist & Wiksell: Stockholm. 1970: p 145. C -. Ilhl~nh~rkGr'.:Fnd.C.W.LacfurrsinStali~ticolM~chhhiiiiA~~~iiiiM~thhmhtical Society: Providence, 1963:p 5. 7. Marx,G.;Gajzago,E.: Gnadig, P.EW. J . P h W 1982.3.39-43. 8. ~ h ~ e n f e sPt , and T. ~ h conceptual r ~oundotionao/ the Stotiaricai ~ p p m o c hin Mechoniea: Moraucsik. M. J.,Tran~l.;Cornell Univ.: Ithaca, 1959;p 30.
Determinationof Concentrations of Species Whose Absorption Bands Overlap Extensively An Instrumental Analysis Laboratory Experiment C. Cappas, N. Holtman, J. Jones, and University of South Alabama. Mobile. AL 36688 Figure 4. Histogram ofmean occbnenceof lhe four states tor the l h r e w a s p l twwoom m0481. Thedlgll-sum 1 an0 2 states are squil'bro~mstates. 0 git-sum 0 and 3 states are nanequilibrium states. Histogram confirms preponderance of eq~ilibriumstates
Appendlx The surprisinge~~mplexity ofall hut thequalitative featuresofthe "simplr"thrre-wasp two-roomweight-vertex beetle model is much roo advanced for the average beginning college student hut shuuld serve 10 con\,inre interestrd r~ndemthat mixing procertse+,and approach toequdibrium, arc another of Nature's wonders. Engel (5) showed that thp average rim? fur repeated trials of n heetle walking the route. 10001 1111 .can he ohtained from a truncated derision tree ia prohahilily tree). This decisiun tree is shown in Figure 3. 11 the expectntwn fur thr lifetime i a e = EtX), then e can be obtained by considering thc pruhnbility of going r u a specific vertex mncroatate or branch of the tree, so that:
-
If this equation is solved for e, then e = 10. Thus, the expectation value for three wasps starting in the configuration (000)and ending in the configuration 1111)is e = 10. The derivation of this equation from the decision tree is quite subtle. The first term in the equation is for the event correspondingtogoing from the initial state {O,O,Olof digit sum 0 to the final state Il,l,l)ofdigit sum 3 involving the three necessary steps,O-1-2-thef first step,O-1 hesaprobability of 111,asa movement from 0-1 must occur. Whenin a digit sum state of 1, three movements to adjacent verticesare possible, one ofwhich can go hack to 0, but two others go to cube vertices with digit sums of 2. Thus, the probability is 213 for the second step, 1-2. Once in a digit sum state of 2 only one movement to the three eligible vertices goes from digit sum 2-3, the final state of (1,1,1].The total probabilitv for these three steps is their product, (111 X 213 X 1131.3. he second term involves the steps, 0-1-2-1, with respective probabilities of 1/1,2/3,213 giving a total probability of 419. However. is onlv - the - ,heetle ~ ~ ~ at ~a vertex with a dieit sum of 1 for which the to the final state isnow, e - 1. It took three probability of steps to get there, so the overall probability is 419(3 + e - 1). The last term involves the steps, 0-1-0, with respective probabilities of 111and 113, giving a total probability of 113. However, the beetle now is only at a vertex with a digit sum of 0 for which the probability of getting to the final state is e itself. It took two steps to get there, so the overall probability is 1/30 + el. ~~~
300
~
Journal of Chemical Education
S. Young
In mixtures of structurally similar compounds, determination of concentrations of components by absorption apectrophotometry often fails to produce accurate results due to considerable soectral overlao of the comoonents. I n such a case, the traditional techniqie of choosing N wavelengths to determine concentrations of Ncomponents results in an illconditioned matrix of molar ahsor&vities. In the theory of linear eauarions, use of an ill-conditioned matrix, defined as a matrix which has a large condition number (vide infra), leads to solutions which are extremely sensitive to errors in the input data ( I ) . Toovercome this problem, measurements are taken a t M wavelengths, where M > N; the technique of multivariate linear least squares is then used to determine both the concentrations and the estimated uncertainties of the comoonents. We have used this aooroach for auantita.. tive analysis of mixtures of rhodium(1) carhonyl complexes whose carhonvl stretchine bands over la^ sianificantlv ( 2 . 3 ) . Due to the iarge numb& of absorhan&vavelengih pairs required for this method, it is impractical to take the data either from a chart recorder or from an analogue or digital readout; instead, computer interfacing is required. Various methods are available, e.g., direct interfacing to a P C from an IEEE-488 or RS-232 interface bus or indirect interfacing by connecting output normally sent to the chart recorder to an AID interface board. By whatever method, the raw data can be sent to data files on the P C for analysis by the multivariate least-squares method. This experiment, appropriate for an instrumental analysis laboratory experiment, involves determination of the cons a mixture of ketones utilizcentrations of the c o m ~ o n e n tof ing 1R spectrophotometry. The data are transferred to a PC throueh an RS-232 interface. and the resulting data set is analyzed using the BASIC algorithm describedielow. Experlmental Mixtures of 2-hexanone, 3-hexanone, 3,3-dimethyl-2-hutanone, and 3-methyl-2-pentanone of various concentrations in carbon tetrachloride were prepared volumetrically using literature values of density. Special care was taken in handling the reagents due to their volatilitv. Table 1 summarizes the concentrations of all mixtures used fo; the six runs. Carbon tetrachloride was chosen as solvent because it does not absorb in the spectral wavelength region of interest and is of only moderate volatility. [Caution: Carbon tetrachloride is a suspected carcinogen, so sample preparation should be carried out in an efficient fume hood.] Samples of the individual components were prepared as 10.0%solutions in carbon tetrachlo-
Table 1. Least-Squares Melhoda Concernrations (M) RUN
COMPONENT
EXP
CALC
UNCERTAINTY
% ERROR
Fiwe I. Absorbance spectra of threecomponent mixture and pure w r n p nenls.
B
0.271 0.2705 0.267 0.2694 0.090 0.0902
C
D A
0.00017 0.000043 0.00012
0.20 0.88 0.25
= Phexanme. B = M e r a n o r e C 3 3.3dimsthyl-2-butanone, and D = 3-methyl-
ride. Spectra of the mixtures and pure components were recorded every em-' from 1730 to 1891 cm-I (region covering the carbonyl stretches of all four ketones, in an 0.25-mm-pathleneth KXr mlu. tion cell on a PE-1430 IR ~pectrophotorneter:~hedata were transferred to an IBM-PC from a PE Model 3600 Data Station usine an RS-232 rnterface huscontrolled by BITCOM softwareand anal;?ed using a BASIC program to calculate concentrations of the compunents Data Analysls The absorbance of a mixture is given by
where A; = absorbance at wavelengthj co = molar absorptivity of component i at wavelengthj b = path length of cell, and C/ = concentration of component i. Using the technique of multivariate least squares ( 4 ) , minimization of the sum of squares of deviation of fit
Figla, 2. Abewbance apecba of nenls.
fourcomponern mixture and pure comp
where crjt2= Zi[(Z; Ayx, - B;)?/(M - N), M = number of wavelength used, and N = number of components. leads t o the following expression for the best fit of the parameters: Ax = B, where A, =
1e,cjkb2; Bj A
=
*
ejkA,b; xi = Cj
Then x = A-'B. One ~ o s s i h l emethod t o obtain A-' is to calculate the matrix in column-by-column fashion using Gaussian elimination (5). Solution of the problem AY = z bv Gaussian elimination where z = (1,0,6,. .) yields y = first column of the matrix A-'. T o obtain column k of A-', solve Ay = z, where z contains all zeros except for the kth element, which is numerically equal to 1. In this way the matrix A-' can be generated column by column. This method is preferable to direct evaluation of matrix elements of A-', in which determinants of arbitrary size must be evaluated. The uncertainty (standard deviation) of the calculated concentrations based upon least-squares error is given by
The ahove algorithm is implemented in a BASIC program used to analyze the data. This program can run on any PC and requires no support software such as canned numerical routines. A copy of the program is available upon request. .Results and Dlscusslon The results of some typical runs are presented in Table 1. In spite of the strongly overlapping nature of the components, as shown in Figure 1, all concentrations were calculated to within 3% error. the average error being 0.76%. The error wan likely due t o manipulation of the somewhat volatile chemical species. This is confirmed bv the small uncertaintiesdue toiheleast-squares fit, which inall cases (except run 1) were on the order of the resolution of the A D converter board (relative uncertainty = 0.0001). The relatively large duruncertainties in run 1suggest . . an instrumental "glitch" ing data collection. Figures 1and 2 almost seem to indicate the presence of an isosbestic point (6).In this case the presence of two primary Volume 68 Number 4 April 1991
30 1
Table 2. Comparlson of "Kln-H' wlth Least-Squarer Method RUN
COMP.
WAVELENGTHS
"N-in-N' "LEAST Sa" COND. % ERR % ERR NUM.
Table 3. Effect d WavelengIh ShM RUN
9 10
6. C, D B. C. D B, C. D 6,C. D 6,C, D
11 12 13 14 15 16
A. B, C. D A.B,C.D A. B. C, D A,B.C,D A,B.C.D A. B. C. D
7
8 1 Figure 3. Thre%mmponent mlx?we-experimental specha.
vs. calculated absorbanoe
COMPONENTS
WAVELENGTHS
1700-17061718 1701-1707-1719 1702-1708-1720 1703-1709-1721 1704-1710-1722 1691-1894-1697-1700 1694-1697-1700-1703
1700-1703-1708-1709 1708-1709-1712-1715
1715-1718-1721-1724 1721-1724-1727-1730
% ERROR
41 21 11 19
37 16
8.9 97*
0.90
4.1 51
'Matrix is ill-xnditioned; cond(A) = 108.800.
components of the mixtures (see Table 1) were determined bv usineabsorbancedata at 1706.1712.1718. and 1724 cm ' f i r thecase of four components i r u n s ' & 6 ) . ~ o rthe case of three components, absorbance data a t 1702,1708, and 1720 cm-' weriused for runs 1and 3, and 1706,. 1718; and 1724 cm-' were used for run 2. Table 2 displays the results of the calculations and allows their comparison with the leastsquares results. For each of the four components, the least squares method gave a better prediction than the "N-in-Ar' method.
Figure 4. Fovsornponent mlxture-experlrnenmlM. calculated absorbance sDeCtra.
carbons bound to the carbonyl leads to nearly complete overlap of spectra of 2- and 3-hexanone, with only a slight shift in 3-methyl-2-pentanone due to the secondary carbon. The presence of the tertiary group bound to the carbonyl in 3,3-dimethyl-2-butanone, however, shifts its spectrum away from the other three, so that the three nearly overlapping species almost seem to form an isosbestic point with the fourth. In general, one would not expect four species that overlap to have a unique isosbestic point. Figures 3 and 4 give plots of the calculated absorbances of the mixtures based upon the concentrations of the species predicted by the least-squares program, along with plots of the experimental absorbances, for runs 1and 4. The agreement is quite good over the entire spectral range; typical deviation between calculated and experimental absorbance is 0.4%. The success of the least-squares method is based in part on the size of the data set taken. To obtain a sufficientlv largedataset, we recommend that thespectrophotometer de interfaced to a PC utilizine either an AA) board or an interface bus. Once stored in &ASCII data file, the raw data can be used for further analysis. T o compare the least squares method with the "N-equations-in-N-unknowns" method, the concentrations of the 302
Journal of Chernlcal Education
Comparlson of "Kin-H' Method wlth Least-Squares Method Even though the "N-in-Wmethod is the method suggested by many analytical textbooks for solving multicomponent systems using spectrometric analysis (7-lo), i t has some pitfalls demonstrated by the data collected in this experiment. These problems are overcome by the multivariate least-squares method. 1. The predicted results of the "N-in-iV" method are sensitive to the choice of wavelength. If the results of run 1 are analyzed using different wavelength positions, widely variate results are obtained. Runs 7-10 in Table 3 show the effect of shift of observation wavelengths by unit increments. Whereas the method for four components seems to work better than for three components, the success of this method depends upon the choice of wavelengths, as shown by runs 11-16 in Table 2. In general, it is not obvious which points to choose to obtain the best prediction of concentrations. Perhaps the best method would be t o monitor the condition number (defined below) of the matrix of absorption coefficients choosine the set of wavelengths which minimize the condition number. 2. The predicted results of the "N-in-IT' method are sensitive to ihstrumental fluctuations and to round-off error. Define a "condition number" for the matrix Aii as follows (I): Let
be the norm of matrix A. Then cond (A) = (11A113(11A-'11).
The condition number ranges from 1 for a unit matrix to extremely large for ill-conditioned matrices t o infinity for a singular matrix. (An ill-conditioned matrix L has a determinant close to zero, so that, although the inverse matrix L-' exists, the values of the elements of L-1 are sensitive to tiny error-induced perturbations of the elements of L.) The importance of the condition number is found in the following theorem ( I ) : Let Ax = b be a series of linear simultaneous equations, with solution x = A-'b. If 6b is the norm of a vector describingerrorsin b, then: bxlllllxll = cond(A) Il6blllllbl~. The condition number thus a d s as an "error amplifier", placing an upper limit of relative concentration error due to relative absorbance error. For runs 1-6, the conditionnumber of the matrices used in the "N-in-N" method are given in Table 2. These matrices are extremely sensitive to error in absorbance. For example, if in run 4 the mixture ahsorhance a t 1718 em-' were changed by 0.0002 absorbance units, the predicted concentrations would be in error by 8%. An extreme case of "ultrasensitivity" is given by run 13. The matrix is so ill-conditioned (condition number = 108,000) that a change in the absorbance of the mixture of 0.0001 absorbanceunits a t 1700 cm-I leads to a770% error in calculated concentrations. Here the "N-in-N" method fails even with four-place accuracy available when using a computer interface. The least-squares analysis is less affected by this prohlem since the total relative error due to a small fluctuation at one wavelength is greatly reduced by the collection and summation of many absorbance data. The above sensitivity also leads to a round-off problem. T o observe the effect of having only three significant digits for absorption data (many instruments read out absorbances digitally to only three places), the "N-in-N" method for run
4 was recalculated in which all absorbances were rounded to 0.001 absorbance unit. The calculated concentrations changed from (0.203,0.201,0.200,0.209) using four places to (0.203,0.252, 0.223,0.134) for three places. This magnitude of error is generally unacceptable for analytical work. The best solution to this prohlem is to utilize the 12-bit accuracy available on most A D or digital hoards for data collection, as described in the experiment, to obtain the necessary fourdigit accuracy. Summary The multivariate least-squares method can be used to obtain concentrations of components of mixtures using ahsorption spectrophotometry even when the absorption bands overlap extensively, as demonstrated by this experiment. The experiment is easy to perform once the software and hardware are set in place. This method can also be a .~.p l i e dto any in which a function of output is linear . system . with concentration, such as overlapping chromatographic peaks. The method has the added admntage that statistical Rnn~vsis be oerformed. so that ~redicteduncertainties , - can -~ are given along with the prkdicted ckcentrations.
.
~~~
Literature Cited 1. Mooro,R. MalhemaiicolElemsnfr o/Scimtifir Computing; Holt-RineharGWinrton: New York, 1980. 2. Branan, D.: Hoffman,N.: MeEIroy, E.: Miller, N.: Ramage, D.; Schatt, A,; Young. S. lnorg
Chrm. 1387.26.2915-2917.
3. Araghiradeh. F.: Brensn, D.: Hoffman,N . ; Jones, J.; McEIroy, E.; Miller. N.; Ramsge, D.:Sslarar,A.;Young, S.Inorg. Chrm. 1388.27.3752-3755. 1. Montenmerv. D.: Peck. E. Introduction to Lineor Reaiession Analysis; Wiloy: New
y";k,
1982
5. Wilkinson, J. Tho Algebraic Eigmwdue Problem; Clarendon: Oxford, 1965. 6. Dmgo, R. Phyairoi Methods in Chemistry; Saunders: New York. 1977. 7. Fritz, ,l;Schenk. G. Quonriratiue Analytical ChemisLry, 2nd ed.:Allvn and Bacon: Needhsm Heighk, MA. 1987. 8. Harris, D. Quonlitoliw C h ~ m i c oA l n o l ~ i s 2nd ; ed.; Freeman:New York, 1987. 9. Kennedv. J. Annlvliral Chemistry Principlar; HarcourGBracoJovsnovich: Sen Dle-
90. 1984. in. Skooe. D.: W e s t , D. Fundamentals of Anolyticol Chamistry.4fhed.: Saundars:Phila-
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