An Experiment on Sequential Simplex Optimization of an Atomic Absorption Analysis Procedure S. Sangsila, G. Labimz, J. S. Poland, and G. W. vanloon Queen's University, Kingston, ON K7L 3N6, Canada Acommon situation encountered by analytical chemists is the need to apply or adapt a particular analytical procedure to a new problem. Often this involves determining optimum instrument settings (i.e., those that produce maximum resoonsel and other conditions for the analvsis. Ootimizatim may be done using the univariate procedure in which each parameter is systematically and individually varied while all other parameters are held constant. This procedure is usuallv time-consumine" and i t does not take account of subtle synergistic effects involving two or more experimental variahles. An alternative to the univariate method is the sequential simplex procedure. By simultaneously varying, according to defined rules, all parameters that affect the magnitude of the analytical signal, the optimum conditions may be estimated. T o illustrate and test simolex ontimization in a student laboratory is time consuming unless an experiment is chosen in which the variables affectine the resoonse mav be readily and rapidly changed. This describes a hydride generation atomic ahsorption experiment using simplex optimization procedures; i t is to be carried out in the laboratorv with a oersonal comouter near the instrument and requires app;oximately t g e e hours for completion. The procedures described in this paper have been used to optimize conditions for the analysis of As, Sb, Se, and Bi, but only data for As is reported here. Hydrlde Generation Atomlc Absorption Spectroscopy In recent vears. hvdride generation atomic ahsorntion s p e c t r o s c o p y i ~ ~ has ~ A ieen i ~ ~ widely used for the h a l y sis of elements caoablc of formine volatile hvdrides (As. Sb. Bi, Ge, Pb, Se, &, Te, In, and Fl). The method has high sensitivitv and is simole.. raoid. - . and relativelv free from matrix interferences. Recent reviews ( I ) summarize some of the developments and applications related t o the technique. ~ydridegenerationaiomic absorption spectroscopy consists of three separate steps. 1. Hydride production. The analyte of interest is converted to its volatile hvdride bv means of a suitable reducine aeent. the most common of which-is NaBHa added as an aaueoksolution. Two chemical reartrona occur rimulrnneously-hydridr tbrmation and decomposition of the exrpss reducing agent (2.3). 2. Troncpart.Thr hydridr is introduced into the atomiler either by direct t r s n a p m with N ~ g m a s ararrieror after collrcrion. In the collection m d e . the hydride is trapped in liquid N9 and subsequently re-evaporated into the atomizer. 3. Atornizotion. Atomization usually occurs in a quarfs cell that is heated electrically or by flame. The atomization mechanism is complex and may occur partially by direct thermal dissociation, hut it also may involve a H free radical mechanism (4,5). In doing analysis by HG-AAS there are a number of experimental variables that will affect the magnitude of the ahsorbance signals. Four such variables that can be readily changed and that may have considerahly different optimum values for each element are acid concentration in the reaction solution, reaction time prior t o sweeping the product into the atomization cell, carrier eas flow rate. and amount of NaBHd reductant.
Simplex Optimlzatlon The sequential simplex algorithm has been described in a number of review articles ( 6 - 8 ) Some . useful definitions are as follows: factor. Experimental variables upon which the response de-
pends. factor value. The setting or experimental value for a particular
factor in a particular experiment. vertex. Set of n faetor values defining a particular experiment.
simplex. An n-dimensional geometric figure composed of n + 1 vertexes.
At the start of a simplex optimization procedure, the the lower boundarvvalues for the n factors (for examnle.~, ~- and upper feasible limits for experimentation) and response to 1 be ootimized must be defined. An initial series of n experiments is then carried out by choosing n 1vertexes that adequatelv span the factor soace. There are svstematic methodsto aid ii the choice of initial vertexes (7,"9).Using results from these ex~eriments,vector aleehra is emoloved to replace the rejected vertex (usually thahielding the poorest response) with a new vertex obtained bv reflection of the rejected vertex through the centroid of the;emaining byperface. A new simplex is thus generated and the process repeated until t h e optimum response is obtained: Modifications to this fixed-size simplex procedure have been made to allow the reflection process t o expand or contract (7, a), to cope with selected vertexes which contain values outside the boundaries of a oarticular exoeriment (7). and t o define criteria for the termination of the experide& (7). If a factor value or response lies outside its predefined allowable region, a very poor response is arbitrarily assigned by the user, and the simplex is forced t o remain within its defined feasible region. In order to terminate the optimization process, the user also defines a percentage of domain of eachfactor level such that, if the length of the reflection for all vertexes is less than the length assigned by these criteria, then the simplexing process stops. Other termination criteria may be applied to indicate that a point of diminishing returns bas been reached. A~
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Apparatus The hydride generation system consists of a three-neck, 125-mL separatory funnel, l-L polyethylene battle as an acid reservoir, and an adjustable gas flow meter '1s in. with stainless steel float (LabCrest Div. F & P Co., Ltd.). These are connected to a heated quartz tube, 1.4-cm i.d. and 14.0-cm Length with optical quartz end windows. The cell is held about 1.5 em above the air-acetylene burner head by an aluminum holder. In this experiment, a Perkin-Elmer 2380 instrument fitted with the appropriate hollow cathode lamps is used. A schematic of the apparatus is shown in Figure 1. The computer is a Zenith 2-151 PC and the Simplex program is written in TurhoPasealusing the Turbo CraohixToolboxfor screen di.iplays and the ~ewlett-~scksrd HP 7 4 % ~Crnphirr Plotter for plot*. Information ronceming this program is availalrle on request to the authors. Volume 66
Number 4
April 1989
351
Reagents . All chemicals used in this experiment are analytical reagent grade or equivalent. A 1000-pgmL-I As(II1)standard solution is prepared hv dissolvinr the aooraoriate amount of NBS Aslo? standard in 5 ; L of 10%& O H doiutibn, then adding 50 mL of waier and making up to 250 mL with 10%HCI. Working standards are prepared daily by serial dilution of this standard. Three percent NaBH4 solution is prepared in 0.5% NaOH solution, filtered (Whatmau #41 paper) and stored at 4 *C. Procedure Boundary conditions are chosen far each experimental variable (Table 1).Five vertexes are then chosen at random from within these boundaries (Vertexes 1-5. Table 2). For analvsis under each specific set of conditions, 30 ngof As(II1) (300 pL 100 ppb solution). is iniected ,~ into the reaction vessel eontainine 10 mL of aoueous HCI of ~elertedconcentration. Nitrogen gas is then pasred info the vessel for about 10 s. After the N1 gas is turned off, the measured amount of NaBH4 is injected and allowed to react for the required time. At the end of this interval NZis again turned on at the selected rate to flush the hydride into the quartz cell, and the maximum absorbance is read using a 0.5-s integration time. Each analysis is repeated, and the average value is used in the optimization program. Preliminarv exoeriments had shown no sienificant difference between resuits ahained using peak height aGd peak area. After resoonnes for the first five vertexes have been measured. the ~~proaram raleulatw ~ubreyuentfactor values, and these nre used to determine a new response. This proress is repeated until the termination criteria are met, at which time the estimate of optimum conditions is reported.
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Results and Dlscusslon Tahle 2 presents a representative set of factor values and responses obtained from a simplex optimization experiment. T h e initial factor values were chosen over t h e entire allowable ranges. Boundary violations occurred for vertexes 6, 8, 12, with t h e arbitrary poor response of -0.1 causing movement of t h e new vertex into the acceotahle factor reeion. 1) Vertexes 3 and 4 were retained for five simplexes (i.e., and thus r e w i r e d t h a t the exoeriment be reoeated usine these factor ialues and t h e average of the old-and new response assigned as t h e response for the vertexes. This operation, resulting from t h e "n 1rule," is invoked in order to ensure t h a t t h e simplex does not retain an experimental
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Table 1. Boundan Condltlons Factor Name
HCl cone
Units %
NZFlow mea
meter units
mg NaBH, mass S Reaction lime *Actual flow rate ranges from 400 rm min-'
LOW
value
High Value
5 5
50 12 40 80
5
5
I5 meter unlo) b 1475 mL min-' 112
meter units).
Table 2.
Vertex
number
Simplex number
14 7 15 8 16 9 176 10 Optimum by univariate memod
Vertwxes
retainede
(%I
4. 13, 11.3 4. 13, 14. 11 4. 13, 15. 14
37.1 15.8 23.0 25.1 20
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352
Slrnplex Opllrnlzatlon tor As Analysis HCI conc
'Vwtexss retained. ranked highest to loveat response. ' s o m y can0 f on r"a l,m l 'Ropoat oip~romn!tvcncx ma bean retainedtor 51n I ) a mpferesl #La% u a c x ,centrod ol a s t o u vonexer reramdl
Journal of Chemical Education
Figure 1.Wdrlde generation apparatus. A. N2 gas inlet; B, adjustable gas-flow meter: C, twc-way. slo~cock: D. threewav E. acid reservoir: F. . . stoocock: . reBcIlm vessel: 0. syrlnge for NaBH. Inpctim. H, sample pon (gro~ndgrass ConnectlOnj:1.10 quartr cell. Dashed line. 1-mmi.d.polyethylene tubing: solid ne, 8-mm1.d. polyetnylene h o ng.
N. flow rate (meter units)
7.7
9.2 7.4 8.5 10
FactM VaIUBS NaBH, mass (mg)
26.1 28.5 23.1 23.1 20
Reaction time ($1
31.2 8.3 31.4 24.4 30
Res~onse (absorbance)
0.080 0.0835 0.084 0.084 0.088
point with an incorrectly measured "high" response (10). From Tahle 2, it is seen that the agreement between the original and remeasured response was good. The termination criterion (20%for each factor) was satisfied after 16 vertexes, with vertex 17 heing calculated by finding the centroid of the hyperface formed by the final four retained vertexes. This final set of factor values was taken to he the optimum.' Using a 15% termination criterion, an additional 18 experiments were required to estimate the ootimum. ~ i & e 2a-d plots factor value versus simplex number, with Fieure 2e olottine - resoonse . versus simolex number. The fir% five points correspond to the initial vertexes. Branches in the graphs indicate that two vertexes were suggested to replace the rejected vertex. The first vertex corresponds to the reflected vertex, the second to the expanded or contracted vertex. The continuous line corresponds to the factor value or response that was retained during the simplex process. The "spikes" reaching the x axis on the plot of response versus simplex number indicate houndary violations, with the response heing assigned a value of -0.1. Tahle 2 also reports estimated optimum conditions for analysis as determined by the univariate approach. A good agreement between these values and those obtained by the simplex procedure is obtained, and there is similar agreement in corresponding HG-AAS experiments on systems involvine Sh. Se. and Bi. Each univariate ootimization required approximately 30-50 individual experiments compared with 10-20 in the case of simplex.
Safety
Each measurement involves 30 ng of As. The use of an efficient extraction hood over the quartz cell should ensure that the TLV of 10 pg m-3in air is not exceeded. Acknowledgment
Financial support from the Canadian International Development Agency for SS and from the Faculty of Applied Science of Queen's University for GL is gratefully acknowledged. Literature Cned 1. Godden, R. G.; Th0mmn.D. R. Analyst 1980,105.1137-1156. 2 Brwks,R. R.;Ryao,D. E.;Zhang,H.Anol.Chim.Arlo 1981,I31,1-16. 3. Agterdenbas, J.; Bau, D. Fmsenius 2.A i d Chem. 1986,323,783-787. 4. Oedina, J.: R u k k a , I. Sptmchim. A d a (1980), 358,119-128. 5. Welr, B.: Meleber, M. Analwt 1983,108,2U224. 6. Morg8n.S. L.;Dcming.S.N. J. Chromofogr. 1975,112,267-285. 7. Deming,S.N.;Margsn,S.L.Anal.Chom. 1974.46,117&1181. 8. Leggett,D.J.J Chem.Edur. 1983.60.707-710. 9. Yarbro,L.A.:Deming.S.N.Ano1. Chim. Acfo 1971,73,391-398. 10. Spendley, W.; Hert, G.R.: Himsworth, F. 8 . T~rhnarnefrics1962.1.441
' Additional methods could be used in order to mao the reoion of the optimum and to determine factor value toleran'ces needed to ensure that the desired high response is maintained (7).
SIMPLEX NUMBER
Figure 2. Oraphs of factor value or response vs. simplex number; a, HCi concentrstlon:b. N2flow rate; C, NaBH,
mass: d, reaction tihe: e, absorbance.
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April 1989
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