Evaluation of a predictive curve-fitting method for processing data from flow systems

125

Analyttca Chuntca Acta, 272 (1993) 125-134 Elsevler Science Publishers B V , Amsterdam

Evaluation of a predictive curve-fitting method for processing data from flow systems Part 1. Flow system with a mixing chamber James M Jordan ‘, Michael D Love and Harry L Pardue Department of Chemistry, Purdue Umverw’y, West Lafayette, IN 47907-1393 (USA) (Received 15th July 1992, revised manuscript recensed 24th September 1992)

Ahstmet

This paper describes the evaluation of a curve-fitting predlctnre approach to processing data from a flow system Hrltha well-stirred mwng chamber The method utdlzes data from the leading edges of response peaks to predict the response that would be measured If sufficient sample were used to give a steady-state response Advantages of the method under optunal condltlons include a lO-fold reduction of dependency on sample volume relative to a peak-height method, a 20- to 65-fold reduction of dependency on flow-rate relative to a peak-area method, and extended linear ranges m situations mvolvmg nomdeal detector response and situations mvolvmg slow reaction kmetlcs Limltatlons include a requirement for larger sample volumes and degraded error coefficients for flow-rate relative to a peak-height method and chamber volume relative to a peak-area method The new approach ISJudged to offer complementary features relative to peak-height and peak-area methods Keyword Flow system, Curve fitting, Data processmg

Quantltatlve methods based on transient responses tend to be less rugged than then eqmhbrrum counterparts because transient responses usually depend more on expenmental variables than eqmhbnum conditions Accordmgly, it has been suggested that kmetlc-based methods should not be used if eqmhbrmm-based methods are available [l] Unfortunately, there are many sltuatlons m which other cnterla strongly favor the use of transient responses In such situations, the approach usually used to compensate for the larger Correspondence to H L Pardue, Department of Chemlstly, 1393 Brown Butldmg, Purdue Umverslty, West Lafayette, IN 47907-1393 (USA) ’ Present address The Procter & Gamble Company, Intematlonal Technology Coordmatlon, Laundry & Cleaning Products, 6060 Center Hill Road, Cmcmnatl, OH 45224 (USA)

dependency on experunental variables 1s to control those vanables mthm very tight tolerances This can be both difficult and expensive for unslolled personnel workmg with nomdeal samples m poorly controlled environments For example, whereas It 1s quite easy for slolled personnel m controlled laboratory envlromnents to control vanables such as sample volume, flow-rate and sample matrices wlthm narrow tolerances, It may not be so easy for unslolled personnel processing large numbers of nomdeal samples m poorly controlled environments to do so A better solution would be to develop and use measurement and data-processing methods that are less dependent on experimental variables Several such methods have been developed [2-51 By usmg such methods with chemical hnetlc processes, It has been possible to reduce

0003-2670/93/$06 00 0 1993 - Elsevler Science Publishers B V All rights reserved

126

effects of expenmentai varzables by factors of loto lOO-fold Although the same pnnclples shouId apply to transient responses from physical and physlco-chemical systems, there have been few analogous studies of such systems This study was undertaken to evaluate the hypothesis that the same types of error-compensatmg methods that have been developed for chemical kmetlc pracesses can be used to Improve the ruggedness of methods based on transient responses of physical and physi~~hemica1 processes As an mltlal test of this hypothesis we chose a flow system with a well-s&red mlxlng chamber [6,7] This system was chosen for the m&al study because the transient response 1s reasonably well understood [S] makmg it possible to compare experimental fmdmgs with theoretical expectations The data-processing approach evaluated IS one we call a predzctzve-kznettc method [9] In this approach, data collected early m the transient response are used to predict the signal that would be measured If the response were monitored to steady state [lO,ll] Frequently the predicted steady-state responses depend much less on experlmental variables than the transient data from which they were predicted Consequently, these predictive methods are frequently more rugged than methods based strictly on transient responses The predlctlve method was implemented by fitting a first-order model [S} to data from the leading edges of peak-shaped responses for both a tracer and a chemical reaction with complex stolchlometry and nomdeal kmetlc behavior Results obtamed with the predIctme method are compared w&h results obtained with peak-height and peak-area methods

Reagents and znstrumerztatwrz Reagents and mstrumentatlon were as described previously 112,131 For tracer studies, the reagent and sample streams contamed 0 15 mol I- ’ potassium mdlde and 0 142 mol l- ’ phosphate buffer (pH 6 0) and the sample stream also contamed vartable concentrations of trnodlde as

J M Jot&n et al /Anal

Chm

Acta 272 (1993) 125-134

the analyte (tracer) For studies mvolvmg a chemtca1 reactlon (lodate + Iodide + hydrochloric acid), the reagent stream contained 0 15 mol I-’ each of potassmm iodide and potassmm lodate m deoxgenated water and the sample stream contamed 0 15 moll-l potassium mdlde and variable concentrations of hydrochlonc acid as the analyte Samples were introduced by the tmme-controlled method [12,13] and trnodlde (both as a tracer and as the product of the reaction among iodide, lodate and hydrochloric acid) was detected ~perometricaIly by pIatlnum electrodes m a thm-layer flow cell Data were acquired on-line with custom-built clrcmtry interfaced to a microcomputer Data were transferred to a supermicrocomputer for long-term storage, processmg and display Data processzng Peak heights were measured m the usual way Data from the leading edges of response peaks were used to predict signals expected If sufficient sample were used to give steady-state responses The equation for the leading edge [8,12,13] IS

(1) where Z,, Z,, and Z, are currents at time t, steady state and zero ttme, respectmeiy, and f and V, are flow-rate and chamber volume, respectively The steady-state current, Z,, IS computed and related to analyte concentration Several approaches used to predict the steady-state current are discussed briefly below The same data range was used for all sample volumes and was selected empu-ically to provide accurate values of steadystate responses for the smallest volume selected It was selected to represent about 85% of the leading edge of the smallest peak processed Iteratzve optws Details of the nonlinear cube-fitting process have been described elsewhere flO,ll] The only change 1s to replace absorbance m the mltlal equations with the tlme-dependent current measured m this study As m previous studies, the Marquardt algorithm 1141 was used and the iterative process was continued until the change m the x2 stat&c between successive approxlmatlons was less than 0 01%

JM Jordan et al /Anal Chzm Acta 272 (1993) 125-134

If the “fixed” parameters, m Eqn 1 are known, then it 1s f, v, and Z,, possible to compute steady-state current directly from one or more values of Zt Use of multlple values of Zt provides the effect of slgnal averaging but use of fiied values of f, V, and Z, obviously removes the posslblllty of compensatmg for changes m these variables To permit the use of this direct-computation optlon and still have the posslblhty of compensatmg for changes m f and V,, we used the Guggenheim [151 and KezdySwmbourne [161 methods to quantify apparent values of f/V, (effectively a first-order rate constant) for each set of tune-dependent responses These experunentally determined values of f/V, were then used m the direct-computation mode (Eqn 1) Modzjied nonlznear regresszon The convergence time for the nonlinear regression program 1s dependent on the mltlal estimates provided, the better the untlal esttmates, the shorter the convergence tune The nonlinear regression program was modified so that results of the dlrect-computatlon method could be used as mltlal estimates of the steady-state slgnal Dzrect computatwrz

RESULTS AND DISCUSSION

All uncertamtles are reported at the level of one standard devlatlon unit ( f S D ) Relative

127

coefflclents [REC(%)I are used to compare effects of expenmental variables on different data-processing approaches As described earher, the relative error coefficient is the percent error m concentration caused by a given change m the variable of interest [121 Unless stated otherwlse all fits to experunental data are made with the unmodified nonlmear regresslon method error

Response curves Figure 1A represents a set of experimental response curves for four different volumes of a fixed concentration of truodlde used as a tracer m the flow system The three smaller peaks represent sample volumes (V, = 10 and 15 ml> too small to give steady-state responses whereas the largest peak represents a sample volume large enough to give a well-defined steady-state response Figure 1B shows the same response curves with a predicted response (solid curve) obtained by flttmg Eqn 1 to the leading edge of the smallest peak The predlcted curve 1s not only superunposed upon large fractions of the leadmg edges of all the peaks but also 1s nearly superunposed upon the steady-state response of the largest volume Thus, by usmg the leadmg edges of such peaks for small volumes it 1s possible to predict the signal expected If sufflclent sample were used to give a steady-state response Because the leading edges of experlmental and fitted responses m Fig 1B are so completely

1 00

s

4

0 60

z3 E

E 0‘

0 60

s 2 .u c 8 s

c,

c2

0

E

40

21 0

20

0

00

u

; I

a0

160

240

Time

(s)

320

400

Time

(s)

Fig 1 Response curves for dtierent volumes of sample Parameters f= 0 0333 ml s-l, V8= 0 7017 ml, Cp = 0 15 mol I-’ (carrier and sample streams), C& = 10 mm01 I-‘, V, (ml, bottom to top) = 0 5, 10, 15 and 5 0, polarrzmg voltage = 200 mV (A) Expenmental data (B) Expenmental data ( ) wth fit ( -1 to the leadmg edge of smallest peak

128

J M Jordan et al /Anal

Chtm Acta 272 (1993) 125-134

As expected, analogous plots for peak heights vs concentration were linear but sensltlvltles (slopes) increased with increased sEes of sample volume Accuracy /prectslon

00

02

04

05

Conccintrotlon

08

10

CmM)

Fig 2 Cahbratlon plots for predIctwe method with different

InJectIon volumes Condltlons as m Fig 1 except V, (ml) = 0 5 (01, 10 co), 15 (A), 3 0 (*) and 5 0 (0)

supenmposed, it could appear that all data contam the same mformatlon This could lead one to conclude that the response at any time could be used to compute the steady-state response and to wonder about the advantage of the predictive method For example, if one knew the true values of the flow parameters, then one could use these parameters and the current at any time along the leading edge to compute the steady-state response However, if the flow parameters were to change without one knowmg rt, then this approach would give incorrect values of the steadystate response whereas the curve-fitting method IS expected to give the correct values This 1s the prmclpal hypothesis to be tested m this study Cahbratlon plots

Figure 2 represents plots of predicted values of steady-state current vs truodlde concentration for five different volumes of SIXdifferent concentrations (mcludmg zero) The most Important observations from these results are that all plots are linear and plots for all but the smallest volume (V, = 0 5 ml) are virtually superimposed upon one another The predictive method virtually nullifies the effects of rather large differences m sample volume for all but the smallest volume for which there 1s msufflclent mformatlon m the leadmg edge of the peak to permit accurate prediction of the steady-state response

To evaluate the potential accuracy of the peak-height and predictive options, data were obtained for SIXsamples containing truodlde concentratlons between 0 and 2 mm01 I-’ with all other condltlons bemg the same
129

JM Jordan et al /AnaL Chwn Acta 272 (1993) 125-134

the dependence on sample volume 1s sufficiently low that it 1s not necessary to control this variable wlthm such tight tolerances to achieve the same levels of rehablhty

60 250 E

$40 x3 c,

0

220 0 A El0 A! 00 00

10

20

InJection

30 Volume

40

5

(mL>

Fig 3 Effects of sample volume on cahbratlon sensitivity Condltlons as m Fig 1 peak height (01, predictive method (0)

Error coefimts To quantify variable dependencies such as those illustrated m Fig 3, we use relative error coefficients (REC) The relative error coefficient 1s the percent error m concentration produced by a unit change m the variable of interest (e g , sample volume) In these studies measured currents varied linearly with concentration with very small background currents for both peak-height and predicted values For such situations, it 1s easily shown that the error coefficients can be calculated from the equation REC(%)

We conclude that m the absence of mterferences both the peak-height and the predlctlve methods can be used to quantify analyte with high degrees of rehablhty The predictive method has the advantage of reduced dependencies on variables To evaluate the rmpreclslon of the results, ten runs were made on a solution contammg 1015 mm01 1-l truodlde with the same condttlons mentioned earlier m this sectlon Average values and standard devlatlons were 0 759 f 0 0052 mm01 1-l for the peak concentrations and 1033 f 0 030 mm01 l- ’ for steady-state concentrations These results are consistent with the general observation that peak-heights tend to be more precise than predictive results Even so, the lmpreclslon of the predictive method 1s not unreasonably large Error compmsatwn

Figure 3 1s a plot of sensltlvlty (slopes of cahbratlon plots) vs sample volume It 1s clear from this plot that the predictwe method depends less on sample volume than the peak-height method for all but the largest volume used which was sufficient to gve a steady-state response With the peak-height method, one must control sample volume urlthm narrow tolerances to achieve high degrees of relrabtilty, with the predictive method

= lOO(dZ/dV)/Z

(2) where REC 1s the relative error coefficient, dZ/dV 1s the slope of a plot of current vs the vanable of interest and Z 1s the measured current at the concentration value where the relative error 1s calculated In this study, Z IS either peak height, ZP, or steady-state current, Z, All relative error coefficients reported below are at 10 mm01 1-l so that absolute error coefficients can be obtained by drvldmg reported values by 100 and relative error coefficients at any other concentration can be obtained by dlvldmg reported values by that concentration (mm01 1-l) We evaluated error coefficients for sample volume, flow-rate and chamber volume Effects of sample volume are illustrated m Fig 1 and relative error coefficients at two levels of sample volume are summarized m Table 1 Because the modified and unmodlfled iterative options gave virtually identical results except that the modified option required, fewer iterations, results are reported here only for the unmodified option As expected, the error coefficients for both peakheight and predictive methods are much lower for larger sample volumes which are closer to that required to give a steady-state response Also, for this situation m which f and V, did not vary, the direct-computation method based on known (nommal) values of f and V8 gives error coefficients about one order of magnitude below

130

JM Jordan et al /Anal

Chtm Acta 272 (1993) 125-134

TABLE 1

TABLE 2

Relative error coeffictents (% $-‘) ’ for sample volume for tracer studies and various predtctlve options (Condmons [I; ] = 10 mm01 l-‘, 5 = 0 7017 ml)

Relative error coefflclents (% p1-l s) B for flow-rate for tracer studies and various predlctlve options (Condltlons as m Table 1)

Iteratlve

Flow-

rate (ml s-l)

Direct computation Nommal b

Guggenhelm

KezdySwmbome

Sample volume (0 33 - 0 37 ml) 0 0333 0010 0 18 0 0250 0 22 0010 0 0167 0010 0 14 Average ’ 0 18 0010

-

-

Sample volume 0 0333 00250 0 0167 Average ’

00045 00011 00044 0 0034

0 0054 00011 00048 0 0038

(0 5- 15 00059 00029 00071 00053

ml) OOCO36 000054 oOcO55 000048

Average b

oou 0

15 Average b

1

so

100

150

Time

200

250

040

Large sample uolume 05 025 10 001

those of the lteratlve method and the du-ect-computatlon method wth determmed values of f and V, These differences probably reflect the uncertamty m the curve-fitting processes Effects of flow-rate for a fixed sample volume, V,, and analyte concentration, C,O,,are illustrated m Fig 4 Despite the differences m curve shapes, the curve-flttmg method predicts the same value

0

Iteratwe

Small sample volume 0 330 064 0 350 052 0 370 011

a Computed at V, = 0 035 and 10 ml for small and large sample volumes, respectwely b Based on known values of f and Vs c Based on 3 to 4 slgmficant figures

I’

Sample volume (ml)

300

(s)

Fig 4 Response curves at different flow-rates Condltlons as m Fig 1 except V, = 0 5 ml, f = 0 0333 (a), 0 0250 (b) and 0 0167 ml s-l (cl

014 0 13

Duect computation Nommal

Guggenhelm

KezdySwmborne

32 32

-

33 33

-

30

028

029

22 18 23

0 034 022 0 18

0 034 023 023

a Computed at f = 0 0250 ml s- 1 b Based on 3 to 4 slgmfk cant figures

of steady-state current for all three data sets Error coeffklents for flow-rate by the different predictive options are summarrzed m Table 2 The error coefficient for flow-rate 1s significantly larger than that for sample volume for all dataprocessmg options The error coefficients for the direct-computation option using nommal volumes of f and V, are quite large because the computation was done assuming that flow-rate remamed constant at 0 0250 ml s-l By usmg either the iterative Guggenheun [15] or Kezdy-Swmbome [16] methods to quantify the effective rate constant (f/V,), the error coefficient was reduced by an order of magnitude relative to the dlrect-computatlon method with nommal values of f and V, Error coefficients for chamber volume were determmed at three sample volumes, results are summarized m Table 3 On average, the error coefflclents for chamber volume are between those for sample volume and flow-rate As expected, direct-computation methods based on determined values of the apparent rate constant are somewhat better than those based on the nominal value of V, Compartsons

wrth conventwnal method.s Error coefficients obtamed by the predlctlve methods are compared m Table 4 Hrlth results by

JM Jordan et al /Anal

131

Chrm Acta 272 (1993) 125-134

option and the direct-computation optlon with determined values of the apparent rate constant offer about 60-fold improvement over the peakarea method but are degraded by about 3-fold relative to the peak-height method This latter result 1s not surprlsmg because peak height 1s expected to be independent of flow-rate [8] Regarding chamber volume, the iterative and direct-computation predictive methods can offer slight (2-fold) improvements relative to the peakheight method but are degraded by a snmlar amount relative to the peak-area method Although both the steady-state current and peak area [18] are expected to be independent of the chamber volume, predicted steady-state responses are somewhat less effective than peak areas m compensating for this variable, probably as a result of maccuracles associated with the predictive process

TABLE 3 Relative error coefficients (o/oAl-‘) a for chamber volume for tracer studies and various predictive options (Conditions [1;]=10mmol1-‘,f=00333m1s-‘) Sample volume (ml)

Iteratlve

Direct computation Nominal

Guggenhelm

KezdySwmborne

0 330 0 350 0 370 Average b

0 055 0054 0 028 0046

0 10 0052 011 0087

0053 0041 0 017 0 037

0041 0 050 0 052 0041

a At Vs = 0 719 ml b Based on 3 to 4 significant figures

peak-height and peak-area methods [13,17] Because results based on the use of the Guggenhelm and Kezdy-Swmboume methods for determuung the apparent rate constant are very sumlar, only the former 1s included m this companson For changes m sample volume, the predictive methods offer only shght improvements over peak-height and peak area at small sample volumes but offer lo- to 20-fold improvements at larger sample volumes Regarding flow-rate at small sample volumes, the iterative predictive method offers about 3-fold improvement over the peak-height method and about 20-fold improvement over the peak-area method At larger sample volumes, the Iterative

Linfxrity

In all studies reported above, truodrde concentrations were limited to the known linear range of the detector In a subsequent study conditions were changed such that truodlde concentrations at peak levels would exceed the linear range of the detector Figure 5 illustrates results obtained by using the iterative predictive option and the peak-height method The predictive method ex-

TABLE 4 Relative error coefficients for tracer studies and different data-processing options (Conditions except when varied [I;] = 10 mmol I-‘, f= 0 0250 ml s-‘, Vs = 0 7017 ml, V, = 0 350 ml a, V, = 10 ml b, Iterative

Predictive

Peak height

Direct computation Nominal

0 0034

022 0 050

Flow-rate (% ~1 -I s) ’ 040 = 33 013 b 23

0 18

14 005

a V,=O33-037ml b V,=O5-15 vs lteratwe method (It)

Improvement ratio e Pk/It

PA/It

Guggenheim

Sample volume (% pl- ‘) 018 a 0 0102 00053 b 00005

Chamber volume (% pl- ‘) d 0046b 0 087

Peak area

0 037

ml ‘f=O0167-00333

0 10

030 0 10

-74 -86

0 017

12

16

94

19

35 04

19 66

22

04

ml s- ’ d Vs = 0 617-O 931 ml ’ Peak height (Pk) and peak area (PA)

132

J M Jordan et al /Anal Chm Acta 272 (1993) 125-134

ct

2

E

E

c,

c,

20.

l5

s k

10.

: 5.

0

: O-O

20

40

Concentrat

60 1 on

80

50

(mM>

Fig 5 Comparwon of cahbratlon plots for peak-height and predictwe options Condltlons as m Fig 1 except V, = 5 ml and polarrzmg voltage = 50 mV

tends the linear range and preserves higher sensltImtIes at higher concentrations relative to the peak-height method because the former method can be implemented urlth data low enough on the edge of the peak that they remam m the linear range of the detector effects Effects of slow reactlons were evaluated by using the reactlon among lodlde, lodate and acid to produce truodlde For low acid concentrations, the reagent concentrations remamed m excess and the error coefficients for sample volume were consistent with those expected from the studies with truodlde as tracer However, for larger sample volumes (5 0 ml) and acid concentration (10 mm01 l-l), reagent 1s depleted and the reaction becomes the rate-hmltmg process The result 1s that the amount of product near the peak-height 1s less than expected Figure 6 shows a typical data set for analyte m excess Although the sample volume 1s large enough to produce the maximum steady-state response, the maximum slgnal 1s less than that expected for a fast reactlon However, by using the predictive method, it 1s possible to compensate for the slow reactlon by extrapolatmg the data along the leading edge to the expected steady-state value finettc

0

10 0

100

150

Time

(s)

200

2 0

Fig 6 Experimental and fitted responses for reachon of lodate, Iodide and hydrochloric acid Conditions as in Fig 1 except V, = 5 0 ml, lodate (reagent stream) = 150 mmol l-‘, hydrochloric acid (sample stream) = 10 mm01 1-l Fxpetlmental ( ), fitted (-1

The effects on cahbratlon plots are dlustrated m Fig 7 The plot of peak height vs concentration curves toward the concentration axis at higher concentrations whereas the predicted values of steady-state signals vary linearly with concentration throughout the range exammed Thus, the predlctlve value can offer extended linearity relative to peak-height methods for sltuatlons mvolvmg slow reactions

3oti

0

4

8

10

7 Cahbratlon m Fig 6 peak-height (01,

predictwe method (0)

IM Jordan et al /And

133

Chtm Acta 272 (1993) 125-134

Concluswns Improvement ratios for error coefficients for selected condltlons are summarized m Table 4 The predlctlve method offers the smaller error coefficients for larger sample volumes This is expected because data for larger samples require less extrapolation to obtain steady-state responses For larger sample sizes, the improvement ratios for sample volume are about lo- and 20-fold relative to peak-height and peak-area methods, respectively Because peak height 1s expected to be independent of flow-rate for detection of tracers and reaction products [8], the predlctlve method either offers only modest (3 5fold) improvement or has a larger error coefflclent than peak height However, the predictive method provided 20- to 70-fold Improvements m the error coefflclents for flow-rate relative to the peak-area method For changes m the volume of the primary dispersing element, the predictive method yielded a 2-fold improvement relative to peak height and was 2 5-fold worse than the peak-area method even though both responses are expected to be independent of the volume of the dispersing element Although it 1s relatively easy to control most of these variables m wellcontrolled laboratory environments, it may not be so easy to do so m poorly-controlled envlronments that exist m manufacturing and field atuatlons These are the situations m which the low variable dependencies of the predictive method will likely be most useful However, even m wellcontrolled laboratory environments, the ablhty to compensate for slow reaction kmetlcs and nonideal nonlinear detector response such as that illustrated for peak heights m Fig 5 may prove very attractive The principal hmltatlon of the predictive method 1s that it requires somewhat larger ratios of sample volume to chamber volume than either the peak-height or peak-area methods For any given flow manifold, the latter options probably can be used successfully with smaller sample volumes than the predictive method as described herem However, there may be other approaches to using the predictive method or other errorcompensative approaches [2-51 that will overcome this hmltatlon In any event, this study has

shown that the predlctlve method 1s complementary to the more conventional options, offering both advantages and hmltatlons under different circumstances The method of choice will depend upon those conditions that are most difficult to control under a given set of circumstances Future studies will focus on more complex flow systems such as single-bead string-reactors and open tubular columns The prmclpal problem with these more complex manifolds will be the selection of appropriate mathematical models for the fitting process Although the responses from these systems are much more complex than that considered m this study, our experience with other types of complex transient responses such as chromatographlc responses lead us to expect that we can apply the approach successfully to these more complex flow manifolds

This research was supported by Grant No GM-13326-24 from the National Institutes of Health

REFERENCES 1 HA Mottola and H B Mark, Jr, Anal Chem , 58 (1986) 264R 2 R C Hams and E Hultman, Clm Chem (Wmston-Salem, NC), 29 (1983) 2079 3 P D Wentzell and S R Crouch, Anal Chem, 58 (1986) 2855 4 H L Pardue, Anal Chum Acta, 216 (1989) 69 5 HA Mottola, Kmetlc Aspects of AnalytIcal ChemlstIy, Wdey, New York, 1988 6 J Ruzlcka, E H Hausen and H Mosbaek, Anal Chum Acta, 92 (1977) 235 7 J F Tyson, Anal Chum Acta, 179 (1986) 131 8 H L Pardue and J M Jordan, Anal Chum Acta, 220 (1989) 23 9 B L Bacon and H L Pardue, Chn Chem , 35 (1989) 360 10 GE Mlehng and H L Pardue, Anal Chem, 50 (1978) 1611 11 G E Mlehng, H L Pardue, J E Thompson and R A Smith, Chn Chem (Wmston-Salem, NC), 25 (1979) 1581 12 J M Jordan and H L Pardue, Anal Chum Acta, 270 (1992) 195 13 J M Jordan, Theoretical treatment of gradient chamber and smgle-bead strmg reaction flow-mJectlon system with emphasis on evaluation of an error-compensating method

134 for data analysis, PhD Thess, Purdue Umverslty, West Lafayette, IN, 1991 14 P R Bevmgton, Data Reduction and Error Analysis for the Physlcal Sciences, McGraw-Hill, New York, NY, 1969, pp 235-242 15 E A Guggenheim, Phil Mag J Scl, 2 (1926) 538

JM Jordan et al /Anal

Chum Acta 272 (1993) 125-134

16 J H Espenson, Chemical Kmetlcs and Reactlon Mechamsms, McGraw Hill, New York, 1981, p 25 17 J M Jordan, S H Hoke and HL Pardue, Anal Chum Acta, 272 (1993) 115 18 J Tyson, Anal Chum Acta, 214 (1988) 57