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Improved ruggedness for flow-based kinetic methods of analysis
Analytica Chimica Acta 360 (1998) 77±87
Improved ruggedness for ¯ow-based kinetic methods of analysis Yunqing Shi, Harry L. Pardue* Department of Chemistry, 1393 BRWN BLDG, Purdue University, West Lafayette, IN 47907-1393, USA Received 29 September 1997; accepted 17 November 1997
Abstract This paper introduces a new approach to improve the ruggedness of ¯ow-based kinetic determinations. After reagents and sample are mixed, the ¯ow is stopped, signal vs. time data are recorded by an on-line computer, and curve-®tting methods are used to compute the signal that would be measured if the reaction were monitored to equilibrium. The new approach is evaluated using the Mo(VI) catalyzed oxidation of iodide to triiodide by hydrogen peroxide. Time-dependent concentrations of triiodide are monitored by changes in absorbance at 360 nm. Effects of changes in pH as well as the concentrations of iodide and Mo(VI) are evaluated using four different data-processing options, two of which were expected to exhibit poor ruggedness to changes in these variables and two of which were expected to exhibit signi®cantly improved ruggedness. All data-processing options yielded linear calibration plots and had similar degrees of precision. As expected, the principal difference among the options was the ability to reject effects of changes in experimental variables. The new options are 10- to 70-fold less dependent on changes in pH and concentrations of iodide and molybdate than the more conventional options. # 1998 Elsevier Science B.V.
1. Introduction Conventional approaches to kinetic-based determinations tend to be less rugged to changes in experimental conditions than equilibrium-based methods based on the same reactions/processes [1]. Although several data-processing options have been developed to improve the ruggedness of kinetic-based methods using batch sample-processing systems [2], similar options have not been developed for or adapted to ¯ow-based methods. Flow-based methods are usually operated in one of two modes, namely continuous ¯ow [3] and stopped¯ow modes [4]. In the continuous-¯ow mode, ¯ow rate *Corresponding author. Fax: 1 765 496 1200; E-mail: [email protected] 0003-2670/98/$19.00 # 1998 Elsevier Science B.V. All rights reserved. PII S0003-2670(97)00686-7
and residence time are controlled carefully so that signals are monitored at a ®xed time after sample and reagents are mixed. For situations in which chemical reactions do not go to completion during the residence time in the ¯ow system, the resulting procedure is equivalent to the ®xed-time method as applied to batch sample-processing systems [5]. In the stopped-¯ow mode, the usual procedure is to stop the ¯ow before reactions have gone to completion, to monitor the time-dependent signal as the reaction continues in the detection cell and to use the slope of the resulting response to determine analyte concentration. This approach is equivalent to the single-rate method as applied to batch sample-processing systems [2]. As with analogous batch-type systems, neither of these ¯ow-based modes includes any inherent ability to compensate for the poorer ruggedness often associated
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with kinetic-based methods [1]. Accordingly, whereas ¯ow-based kinetic methods can offer some of the same advantages associated with batch-based kinetic methods, they also suffer some of the same problems. The purpose of this study was to determine if dataprocessing options similar to those developed to improve the ruggedness of batch-based kinetic methods could offer similar advantages for ¯ow-based methods. The molybdate-catalyzed reaction of hydrogen peroxide and iodide to produce triiodide was chosen as a model system for the study. By using an excess of iodide and a ®xed concentration of molybdate catalyst, it was possible to force the reaction to approach pseudo-®rst-order dependence on hydrogen peroxide concentration. The stopped-¯ow mode was used and signal vs. time data recorded after the ¯ow was stopped were used in a variety of ways to quantify hydrogen peroxide concentration. Data-processing options evaluated include a steady-state option based on the extent of reaction during a ®xed period of time, a single-rate option based on the slope of the response curve, a deconvolution option designed to resolve the ®rstorder component of the response from a zero-order component and an extrapolation option designed to use data from the early part of the reaction to compute the signal expected if the reaction were monitored to completion. Performance characteristics evaluated for the four data-processing options included linearity, imprecision and ruggedness to changes in iodide and molybdate concentrations and pH.
linearity, imprecision and ruggedness to changes in selected variables. 2.3. Phosphate buffer solutions A group of ®ve phosphate buffer solutions was prepared by dissolving 18.5 g of K2HPO43H2O (F.W.228.23, Mallinckrodt) and 15.5, 22.1, 38.5, 54.8 and 76.9 g of KH2PO4 (F.W.136.09, Spectrum Quality Products, Gardena, CA 90248) in ®ve separate volumes of water and diluting each to 500 ml. A separate buffer solution used for iodide studies was prepared by dissolving 151 g (0.555 mol) of KH2PO4 and 72.0 g (0.158 mol) of K2HPO43H2O in water and diluting to 2.0 l. 2.4. Potassium iodide One group of potassium iodide solutions was prepared by dissolving 3.9, 6.2, 9.1, 12.1, and 16.0 g of potassium iodide (F.W.166.00, Mallinckrodt) in separate volumes of water and diluting each to 500 ml. A separate potassium iodide solution (0.068 mol lÿ1) used for pH studies was prepared by dissolving 22.5 g of potassium iodide in water and diluting to 2.0 l. 2.5. Molybdate catalyst
All solutions were prepared in doubly distilled water and all chemicals were used as received.
For all studies except the molybdate dependence, a stock solution of molybdate solution was prepared by dissolving 1.33 g (1.08 mmol) of (NH4)6Mo7O24 4H2O (F.W.1235.86, Fisher Scienti®c, Fair Lawn, NJ 07410) in water and diluting to 100 ml. For the molybdate dependence studies, four molybdate solutions were prepared by dissolving 0.2512, 0.4574, 0.6914 and 0.9092 g of (NH4)6Mo7O244H2O in four separate volumes of water and diluting each to 50 ml.
2.2. Hydrogen peroxide
2.6. Combined solutions
Solutions containing nominal hydrogen peroxide concentrations from 8.82 mmol lÿ1 to 221 mmol lÿ1 were prepared by dilution of a commercially available reagent (3% H2O2, Mallinckrodt Chemical, Paris, KY 40361) with water. Solutions were used without standardization because our primary interests were in
Two groups of carrier solutions were prepared. One group used to study effects of pH contained ®xed concentrations of iodide and molybdate catalyst and variable amounts of the buffer components; the other group used to study effects of iodide concentration contained ®xed concentrations of the buffer compo-
2. Experimental 2.1. Solutions
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nents and molybdate catalyst and variable amounts of potassium iodide. Carrier solutions used to study effects of pH were prepared by mixing 450 ml of potassium iodide solution (0.068 mol lÿ1) and 450 ml of the molybdate solution with 150 ml of each of the ®ve phosphate buffer solutions described above. The ®nal pH values of these solutions were 5.86, 6.04, 6.22, 6.45 and 6.57, respectively. The molybdate and potassium iodide concentrations were ®xed at 8.06 mmol lÿ1 and 0.0507 mol lÿ1, respectively. Several hydrogen peroxide concentrations (8.8±441 mmol/lÿ1) were used. Carrier solutions used to study effects of iodide concentration were prepared by mixing 167 ml of the 0.555/0.158 mol lÿ1 phosphate buffer and 500 ml of the molybdate solution with 500 ml of each of the potassium iodide solutions described above. The hydrogen peroxide and molybdate concentrations was ®xed at 8.07 mmol lÿ1 and the pH was ®xed at 6.1. Several hydrogen peroxide concentrations ((22.1± 882 mmol lÿ1) were used. Carrier solutions used to study effects of molybdate were prepared by mixing 100 ml of potassium iodide solution (0.1085 mol lÿ1) and 1.000 ml of each of the four molybdate solutions described above and 34 ml of phosphate buffer (19.1419 g of KH2PO4 and 9.1726 g of K2HPO43H2O in 200 ml of water). The molybdate concentrations were 0.1090, 0.08289, 0.05483 and 0.03011 mmol lÿ1. The hydrogen peroxide and potassium iodide concentrations were ®xed at 441 mmol lÿ1 and 0.0804 mol lÿ1, respectively and the pH was ®xed at 6.15. 3. Instrumentation Liquid chromatographic instrumentation was used to transport and mix solutions and to monitor timedependent absorbances in a stopped-¯ow mode. Data were collected and processed on line using a personal computer. 3.1. Flow system The liquid chromatograph (Shimadzu Scienti®c Instruments, Columbia, MD 21046) included two pumps (Shimadzu, LC-10AS), a mixing chamber and a variable-wavelength detector (Shimadzu). One
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pump was used to pump sample solutions and the other was used to pump the carrier solutions. Sample and carrier solutions were mixed in the gradient mixing chamber and then passed through a 58 cm length of 0.30 mm i.d. stainless-steel tubing from the mixing chamber to the detection cell (10 mm path length, 8 ml volume). Flow was stopped with sample solution in the detection cell and data for absorbance vs. time were collected and stored in computer memory. The time delay between the mixing chamber and the detection cell was ca. 9 s. 3.2. Data-collection/data-processing system Data were collected on line using a commercially available interface system (National Instruments, 6504 Bridge Point Parkway, Austin, TX 787305039) and computer (Gateway 2000 4DX2-50V, 610 Gateway Drive, P.O. Box 2000, North Sioux City, SD 57049-2000). Data were processed using customdesigned software described elsewhere [6]. 4. Procedures 4.1. Mixing system Flow rates for the sample and carrier solutions were 0.2 and 3.8 ml minÿ1, respectively, corresponding to mixed solutions containing 5% sample solution and 95% carrier solution and a total ¯ow rate of 4.0 ml minÿ1. The 0.5 ml option for the internal volume of the mixing chamber was used in all the studies reported here. Solutions were pumped continuously through the system for ca. 60 s to permit the detector response to reach an initial steady-state value, after which the ¯ow was stopped. 4.2. Data acquisition Data acquisition was initiated immediately after sample and reagent pumps were activated. Absorbances were monitored at 360 nm at a rate of 10 points per second. Data were collected for suf®cient time to provide data over at least 8 half-lives of each reaction, usually 200±400 s depending on reaction rates.
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4.3. Data processing Four data-processing options were evaluated. Two of the options are not expected to permit compensation for changes in experimental conditions whereas two options are expected to be virtually independent of small changes in experimental conditions. One option, identi®ed herein as the ``steady-state option'', involves measurement of the steady-state absorbance obtained just before the ¯ow is stopped. Results obtained in this way are analogous to peakheight results [3] in the sense that they represent a ®xed-time option [5] because the steady-state signal corresponds to the amount of triiodide produced during the time interval from the point of mixing to the measurement cell. A second option, identi®ed herein as the ``single-rate option'', involves the linear leastsquares slope of the ®rst ten data points collected after the ¯ow is stopped. Both of these options are expected to have large dependencies on experimental conditions. Preliminary experiments using a batch mixing approach and a diode-array-based spectrophotometer showed that the iodide/hydrogen peroxide reaction followed ®rst-order kinetics quite well. However, as discussed in more detail later, the stopped-¯ow response curves exhibited more complex behavior, increasing ®rst to a maximum value and then decreased at a near constant rate suggesting a zeroorder component of the response. It was found that a model for combined ®rst-order/zero-order processes ®t the data well. Accordingly, a third data-processing option, identi®ed herein as the ``deconvolution option'', involves the use of a model for simultaneous zero-order/®rst-order processes to resolve the ®rstorder component of absorbance vs. time data from overall responses. The mathematical description for this model is: At A0 A1 ÿ A0 1 ÿ exp ÿk1 t k0 t
(1)
in which At is the time-dependent absorbance, S1 is the absorbance expected when the reaction has gone to completion, A0 is the background absorbance, k1 is the rate constant of the ®rst-order component of the response and k0 is the rate constant of the zero-order component of the response. Fits of Eq. (1) to the data showed that the ®rst-order component of the model ®t data during the ®rst two to
three half lives of the ®rst-order process quite well. Accordingly, a fourth data-processing option, identi®ed herein as the ``extrapolation option'', involved the use of a ®rst-order model to extrapolate data during the ®rst two to three half lives to the value expected if the reaction were to proceed to completion without any complications. The mathematical model for this extrapolation process is: At A0 A1 ÿ A0 1 ÿ exp ÿk1 t
(2)
in which all symbols are de®ned above. Because the deconvolution and extrapolation options are designed to obtain the signal corresponding to complete reaction, these options are also identi®ed herein as pseudo-equilibrium options. Both options are expected to be very rugged to changes in experimental conditions. 4.4. Data interpretation Relative error coef®cients are used to quantify differences in variable dependencies among the different data-processing options. The relative error coef®cient is de®ned as the percent error in concentration produced by a unit change in the variable of interest. Procedures used to quantify relative error coef®cients are described elsewhere [7] and are not repeated here. 5. Results and discussion Unless stated otherwise, all imprecisions are quoted at the level of one standard deviation (1 s.d.). 5.1. Response curves Fig. 1(A) includes a typical set of response curves for different hydrogen peroxide concentrations with ®xed values of iodide and molybdate concentrations and pH. In each case, zero time corresponds to the point at which the pumps and data acquisition were started. The initial increase shortly after pumps are started (Region a) corresponds to any residual triiodide remaining from a previous sample. The more gradual rise toward a steady-state response (Region b) corresponds to entry of the reaction mixture into the measurement cell. The steady-state signal observed in Region b of the responses corresponds to a ®xed
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decreasing pH. It is probable that the decreasing absorbance results from oxidation of the stainlesssteel tubing by triiodide. For this reason, few studies were done at low pH and the system was ¯ushed thoroughly when not in use. 5.2. Data-processing options
Fig. 1. Typical response curves at fixed (A) and variable (B) pH. Frame A: Potassium iodide, 0.0507 mol l ÿ1 ; Molybdate, 8.06 mmol lÿ1; pH, 6.22; Hydrogen peroxide (bottom to top), 8.82, 22.1, 4.1, 88.1, 221, 441 mmol lÿ1. Frame B: Potassium iodide, 0.0507 mol lÿ1; molybdate, 8.06 mmol lÿ1; pH (bottom to top @50 s), 6.57, 6.45, 6.22, 6.04, 5.86; Hydrogen peroxide 82.2 mmol/l.
amount of triiodide formed during the ®xed time period (ca. 9 s) between the mixing and measurement cells. The ¯ow is stopped after this steady-state signal is established and the larger increase following the initial steady-state response corresponds to continued reaction in the solution in the measurement cell. All responses exhibit a gradual decrease in signal as the reaction approaches equilibrium. As shown in Fig. 1(B), this behavior becomes more severe with
The steady-state option is implemented by averaging the last ten data points in Region b just before the ¯ow is stopped. The single-rate option is implemented by computing the least-squares slope of the ®rst ten data points after the ¯ow is stopped. The deconvolution option is implemented by ®tting a model for combined zero-order/®rst-order processes Eq. (1) to data after the ¯ow is stopped and using data from the ®rst-order component of the response. Fig. 2(A) includes data for a ®t of Eq. (1) to a typical set of experimental data as well as the ®rst-order component obtained from the ®t. The model for combined zero-order/®rst-order processes is observed to ®t the data well. Similar ®ts were obtained for other data sets. The maximum value of the ®rst-order component of absorbance, A1;1 , is used to compute hydrogen peroxide concentration. The extrapolation option is implemented by ®tting a model for a ®rst-order process Eq. (2) to data during the early part of the process (usually approximately three half lives of the ®rst-order process) and extrapolating the computed response to t!1. A typical set of experimental and computed data is given in Fig. 2(B). The ®rst-order model is observed to ®t data during the ®rst part of the response well; similar ®ts were obtained for other data sets. The maximum value of the ®rst-order component of absorbance, A1;1 , obtained from the extrapolation process is used to compute hydrogen peroxide concentration. 5.3. Rate constants Apparent values of pseudo-®rst-order rate constants for different values of pH, iodide concentration and molybdate concentration are illustrated in Fig. 3. For conditions used in this study, these rate constants are observed to vary between ca. 0.01±0.1 sÿ1. Rate constants are observed to decrease with increasing pH and to increase with increasing concentrations of iodide and molybdate.
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Fig. 3. Apparent first-order rate constants for different values of pH (a), iodide concentration (a mmol lÿ1, (b)) and Mo(VI) concentration (a mmol lÿ1, (c)). All error bars represent 1 s.d. Plots ((a),(b)) Each value is the average of three runs at each of six hydrogen peroxide concentrations (a8.8±441 mmol lÿ1; b22± 882 mmol lÿ1). Plot (c) Each point is the average of three runs at a single hydrogen peroxide concentration (441 mmol lÿ1). Other conditions as for corresponding frames in Fig. 5.
5.4. Imprecision Fig. 2. Response and fitted curves using combined first-order/zeroorder (A) and first-order (B) models. Conditions (both frames): Potassium iodide, 55.2 mmol lÿ1; Molybdate, 8.07 mmol lÿ1; pH, 6.1; Hydrogen peroxide, 441 mmol/l. Frame A: (a) Experimental data, (b) First-order/zero-order fit, (c) First-order component. Frame B: (a) Experimental data, (b) First-order fit.
Each data point for the different values of pH corresponds to the average of three runs at each of six hydrogen peroxide concentrations between 8.8± 441 mmol lÿ1 and each data point for different iodide concentration corresponds to averages of three runs at each of six hydrogen peroxide concentrations between 22±882 mmol lÿ1. Error bars on the plots represent standard deviations for each of the 18 data sets. These error bars re¯ect the fact that rate constants do not change signi®cantly with hydrogen peroxide concentration.
Table 1 contains pooled relative standard deviations for the different options for different hydrogen peroxide concentrations. There are no apparent trends with different concentrations. When two unusually large values for the single-rate option are ignored, average values of the pooled standard deviations are all similar, with the only signi®cant difference being the smaller value for the steady-state option in the pH study. We conclude that the different options yield similar values of relative standard deviations. 5.5. Linearity Two typical sets of calibration plots obtained at different values of pH are given in Fig. 4(A) and (B). Linearity data for the steady-state option are similar to data in Fig. 4(A) for the single-rate option and data for the extrapolation option are similar to those in Fig. 4(B) for the deconvolution option. Also, data for different values of iodide and molybdate concen-
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Table 1 Summary of imprecision data for different data-processing options Pooled relative standard deviation (%) Concentration (mmol lÿ1)
Steady-state
Initial rate
Deconvolution
Extrapolation
pH dependence b 8.82 22.1 44.1 88.2 221 441 Average Std. Dev.)
3.7 1.8 2.11 1.6 1.4 2.77 2.2 (0.9)
18 e 7.8 4.3 6.1 8.5 6.2 6.6 e (1.6) e
9.4 5.0 3.4 2.4 2.5 3.5 4.3 (2.6)
8.8 3.8 3.3 3.1 4.5 3.3 4.4 (2.2)
Iodide dependence c 22.1 44.1 88.2 221 441 882 Ave. (S.D.)
6.7 2.6 3.4 5.3 3.4 4.6 4.3 (1.5)
16 e 8.1 5.3 6.6 4.4 4.8 5.8 e (1.5) e
5.4 2.0 4.0 5.1 4.1 3.6 4.0 (1.2)
8.3 2.5 3.5 5.2 3.7 4.2 4.6 (2.0)
4.74
6.04
5.88
Molybdate dependence d 0.4412 a
3.31 b
c
d
e
Pooled standard deviations, n5, n5, n4, Value not included in average.
trations are analogous to the pH data in Fig. 4(A) and (B). Least-squares results for all of the calibration plots are summarized in Table 2. Intercepts and standard errors of the estimates are converted to concentration units by dividing by slopes to facilitate comparison among the different options. The large values of the standard deviations of the steady-state and rate options re¯ect the fact that the slopes of the calibration plots change with each of the variables. The standard errors of the estimates re¯ect scatter about the least-squares lines for each set of conditions. For the pH-dependence data, there are no signi®cant differences among the standard errors for the pH-dependence data; however, for the iodide-dependence data, standard errors for the deconvolution and extrapolation options are about three-fold smaller than those for the steady-state and single-rate options. There are no consistent trends in the intercepts among the different options. 5.6. Ruggedness As noted earlier, a major focus of this study was to determine if alternative measurement/data-processing
options could improve the ruggedness of ¯ow-based kinetic methods. As also noted above, changes in pH and concentrations of iodide and molybdate concentrations were used to test the ruggedness of the various options. Data in Fig. 4(A) and (B) illustrate effects of pH on the calibration sensitivities of two of the options. It is apparent from these Fig. that the deconvolution option is much less sensitive to changes in pH than the singlerate option. Analogous behavior was observed for the steady-state and extrapolation options, respectively, as well as for effects of other variables. These differences can be visualized more explicitly using response-ratio plots, i.e. plots of ratios of responses at different values of each variable to the response at a particular value of the variable vs. the different values of the variable. Response-ratio plots for each of the options evaluated in this study are given in Fig. 5(A)±(C) for changes in pH and concentrations of iodide and molybdate. The slopes of such plots re¯ect the different degrees of ruggedness of the various methods; the smaller the slope, the more rugged the option for the variable of interest. It is apparent from these plots that the deconvolution and extrapolation options are
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Fig. 4. Calibration plots for steady-state (A) and deconvolution (B) options for different values of pH. Conditions (both frames): Potassium iodide, 0.0507 mol/l; Molybdate, 8.06 mmol/l; pH (bottom to top, (a)±(e)), 6.57, 6.45, 6.22, 6.04, 5.86.
signi®cantly less dependent on the three variables examined than are the single-rate and steady-state options. Differences in ruggedness visualized with the aid of Fig. 5(A)±(C) are easily quanti®ed using relative error coef®cients [7]. Results are summarized in Table 3. To simplify comparisons, the table contains ratios of numerical values for the extrapolation, single rate and steady-state options to the values for the deconvolution option. The larger the ratio of error coef®cients, the poorer the ruggedness for each variable. Some values which seemed to represent unusual behavior were excluded in computing averages and associated standard deviations. Based on the averages, the extrapolation option seems to be slightly less rugged than the deconvolution option to changes in iodide and molybdate concentrations. However, the differences are small and we conclude that these options exhibit similar degrees of ruggedness for each of the three variables examined. The rate constants measured by the two procedures are summarized in Table 4. Both the deconvolution and extrapolation options are signi®cantly more rugged to changes in these variables than either the single-rate or steady-state options. Average improvement factors of the deconvolution option relative to the single-rate and steady-state options range from about 22- to 40-fold when effects of two unusually small values for the deconvolution option are excluded.
Table 2 Least-squares statistics for calibration plots for different data-processing options Option
Slope a
Std. Error b/Slope b
Average
RSD (%)
pH dependence St. state One rate Deconvolution Extrapolation
0.288 0.0186 1.03 0.960
71 50 3.8 5.7
Iodide dependence St. state One rate Deconvolution Extrapolation
0.388 0.0218 0.91 0.88
34 14 0.9 1.8
a
(mmol lÿ1)ÿ1, b mmol lÿ1.
Average 6.8 5.2 5.1 5.8 13 20 6 5
Intercept b/Slope b Std. Dev.
Average
Std. Dev.
6.0 4.3 2.4 4.1
22 ÿ1.0 5.3 5.1
17 2 1.2 1.9
0.8 5.3 3.3 3.4
0.07 ÿ1.2 39. ÿ6.7
4.6 16 1.3 1.9
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Fig. 5. Effects of pH (A), potassium iodide concentration (B) and molybdate concentration (C) on results obtained by applying four different data-processing methods to the same data sets. All frames: (a) Steady-state option, (b) One rate option, (c) Deconvolution option, (d) Extrapolation option. Frame A: Potassium iodide, 0.0507 mol lÿ1; Molybdate, 8.06 mmol lÿ1; Hydrogen peroxide, 441 mmol lÿ1. Frame B: molybdate, 8.07 mmol lÿ1; pH 6.1; Hydrogen peroxide, 882 mmol lÿ1. Frame C: Potassium iodide, 80.4 mmol lÿ1; pH6.15; Hydrogen peroxide, 441 mmol lÿ1.
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Table 3 Comparison of relative error coefficients (RECS) for different data-processing options Concentration(mmol lÿ1)
(REC%)
Ratio RECS
Deconvolution pH dependence 8.82 22.1 44.1 88.2 221 441
Iodide dependence 22.1 44.1 88.2 221 441 882.
Molybdate dependence 0.08824 a
Extrapolation
ÿ0.0183 a ÿ0.119 ÿ0.103 ÿ0.075 ÿ0.250 ÿ0.0974 Average Std. Dev.
5.76 a 1.63 0.64 0.60 1.35 1.66 1.2 0.5
0.166 ÿ0.548 ÿ0.094 ÿ0.138 0.0275 a ÿ0.199 Average Std. Dev.
1.59 0.41 2.41 2.23 20.85 a 0.03 a 1.7 0.9
ÿ0.29158
1.46
Steady-state 39.2 a 16.3 29.9 50.4 23.1 51.5 34.2 16
Slope 124 a 27.1 31.7 39.0 14.1 29.8 28.3 9
32.3 11.1 69.0 50.3 263 a 34.1 39 21
30.0 7.76 48.3 29.7 152 a 17.4 27 27
31.35
22.56
Value not included in average and standard deviation.
Table 4 Summary of rate constants for different variable studies Deconvolution Average
Extrapolation SD
%SD
Average
SD
%SD
pH
pH dependence
5.86 6.04 6.22 6.45 6.57 KI b
0.0893 a 0.0456 0.0284 0.0131 0.0101 Iodide dependence
0.00460 0.00227 0.000686 0.000741 0.000593
5.15 4.98 2.41 5.67 5.85
0.0882 0.0469 0.0293 0.0142 0.0109
0.00273 0.00364 0.00240 0.000896 0.000923
3.10 7.74 8.17 6.29 8.49
0.1435 0.1076 0.08162 0.05522 0.03519 Mo(VI) c
0.0987 0.00570 0.0706 0.00358 0.0546 0.00202 0.0370 0.00165 0.0240 0.000902 Molybdate dependence
5.77 5.06 3.69 4.47 3.76
0.106 0.0766 0.0592 0.0399 0.0264
0.00451 0.00241 0.00339 0.00153 0.00125
4.24 3.15 5.72 3.83 4.74
0.1090 0.08289 0.05483 0.03011
0.0740 0.0563 0.0378 0.0215
0.94 0.39 1.54 1.80
0.0805 0.0612 0.0416 0.0245
0.000733 0.000577 0.000282 0.00104
0.91 0.94 0.68 4.25
a
0.000696 0.00022 0.000582 0.000386
The units of all rate constants are sÿ1, b mol lÿ1, c mmol lÿ1
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6. Conclusions Results obtained in this study show that alternative measurement/data-processing options can be used to improve the ruggedness of ¯ow-based kinetic methods for noncatalysts used in the stopped-¯ow mode. Ruggedness is improved by at least an order of magnitude, even for the nonideal response curves obtained in this study. It is probable that consistently larger improvements could be obtained with responses that con®rm more closely to ®rst-order behavior. Also, it is probable that other data-processing options developed for batch-type sample processing systems [2] could also be adapted to ¯ow systems. On the one hand, this study did not include direct comparisons with peak-height methods usually used with ¯ow methods. On the other hand, the steady-state option included in this study is a close analogy to the peak-height option because both depend on the amount of analyte reacted during a ®xed time period between points at which reaction is initiated and measurements are made. Accordingly, we can say with some certainty that improvements in ruggedness observed relative to the steady-state option also apply to the more common peak-height option. Also, it is probable that both the deconvolution and extrapola-
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tion options will be more rugged to changes in other variables such as temperature and ¯ow rate than conventional peak-height options. Moreover, for the same reasons that the deconvolution and extrapolation options are more sensitive than the steady-state option, they should also be more sensitive than the peakheight option. Accordingly, for similar levels of measurement error, these pseudo-equilibrium options should also have lower detection limits than the peak-height option. Finally, this study focused solely on the stopped¯ow mode of operation. A future study will attempt to extend these principles to the continuous-¯ow mode. References [1] [2] [3] [4]
H.A. Mottola, H.B. Mark, Anal. Chem. 58 (1986) 264R. M.D. Love, H.L. Pardue, Anal. Chim. Acta 299 (1994) 195. J. Ruzicka, E.H. Hansen, Anal. Chim. Acta 78 (1975) 145. S. Olsen, J. Ruzicka, E.H. Hansen, Anal. Chim. Acta 136 (1982) 101. [5] H.L. Pardue, Clin. Chem. 23 (1977) 2189. [6] J.W. Skoug, W.E. Weiser, I. Cyliax, H.L. Pardue, Trends Anal. Chem. 5 (1986) 32. [7] K.B. Lim, H.L. Pardue, Clin. Chem. 39 (1993) 1850.
Improved ruggedness for ¯ow-based kinetic methods of analysis Yunqing Shi, Harry L. Pardue* Department of Chemistry, 1393 BRWN BLDG, Purdue University, West Lafayette, IN 47907-1393, USA Received 29 September 1997; accepted 17 November 1997
Abstract This paper introduces a new approach to improve the ruggedness of ¯ow-based kinetic determinations. After reagents and sample are mixed, the ¯ow is stopped, signal vs. time data are recorded by an on-line computer, and curve-®tting methods are used to compute the signal that would be measured if the reaction were monitored to equilibrium. The new approach is evaluated using the Mo(VI) catalyzed oxidation of iodide to triiodide by hydrogen peroxide. Time-dependent concentrations of triiodide are monitored by changes in absorbance at 360 nm. Effects of changes in pH as well as the concentrations of iodide and Mo(VI) are evaluated using four different data-processing options, two of which were expected to exhibit poor ruggedness to changes in these variables and two of which were expected to exhibit signi®cantly improved ruggedness. All data-processing options yielded linear calibration plots and had similar degrees of precision. As expected, the principal difference among the options was the ability to reject effects of changes in experimental variables. The new options are 10- to 70-fold less dependent on changes in pH and concentrations of iodide and molybdate than the more conventional options. # 1998 Elsevier Science B.V.
1. Introduction Conventional approaches to kinetic-based determinations tend to be less rugged to changes in experimental conditions than equilibrium-based methods based on the same reactions/processes [1]. Although several data-processing options have been developed to improve the ruggedness of kinetic-based methods using batch sample-processing systems [2], similar options have not been developed for or adapted to ¯ow-based methods. Flow-based methods are usually operated in one of two modes, namely continuous ¯ow [3] and stopped¯ow modes [4]. In the continuous-¯ow mode, ¯ow rate *Corresponding author. Fax: 1 765 496 1200; E-mail: [email protected] 0003-2670/98/$19.00 # 1998 Elsevier Science B.V. All rights reserved. PII S0003-2670(97)00686-7
and residence time are controlled carefully so that signals are monitored at a ®xed time after sample and reagents are mixed. For situations in which chemical reactions do not go to completion during the residence time in the ¯ow system, the resulting procedure is equivalent to the ®xed-time method as applied to batch sample-processing systems [5]. In the stopped-¯ow mode, the usual procedure is to stop the ¯ow before reactions have gone to completion, to monitor the time-dependent signal as the reaction continues in the detection cell and to use the slope of the resulting response to determine analyte concentration. This approach is equivalent to the single-rate method as applied to batch sample-processing systems [2]. As with analogous batch-type systems, neither of these ¯ow-based modes includes any inherent ability to compensate for the poorer ruggedness often associated
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with kinetic-based methods [1]. Accordingly, whereas ¯ow-based kinetic methods can offer some of the same advantages associated with batch-based kinetic methods, they also suffer some of the same problems. The purpose of this study was to determine if dataprocessing options similar to those developed to improve the ruggedness of batch-based kinetic methods could offer similar advantages for ¯ow-based methods. The molybdate-catalyzed reaction of hydrogen peroxide and iodide to produce triiodide was chosen as a model system for the study. By using an excess of iodide and a ®xed concentration of molybdate catalyst, it was possible to force the reaction to approach pseudo-®rst-order dependence on hydrogen peroxide concentration. The stopped-¯ow mode was used and signal vs. time data recorded after the ¯ow was stopped were used in a variety of ways to quantify hydrogen peroxide concentration. Data-processing options evaluated include a steady-state option based on the extent of reaction during a ®xed period of time, a single-rate option based on the slope of the response curve, a deconvolution option designed to resolve the ®rstorder component of the response from a zero-order component and an extrapolation option designed to use data from the early part of the reaction to compute the signal expected if the reaction were monitored to completion. Performance characteristics evaluated for the four data-processing options included linearity, imprecision and ruggedness to changes in iodide and molybdate concentrations and pH.
linearity, imprecision and ruggedness to changes in selected variables. 2.3. Phosphate buffer solutions A group of ®ve phosphate buffer solutions was prepared by dissolving 18.5 g of K2HPO43H2O (F.W.228.23, Mallinckrodt) and 15.5, 22.1, 38.5, 54.8 and 76.9 g of KH2PO4 (F.W.136.09, Spectrum Quality Products, Gardena, CA 90248) in ®ve separate volumes of water and diluting each to 500 ml. A separate buffer solution used for iodide studies was prepared by dissolving 151 g (0.555 mol) of KH2PO4 and 72.0 g (0.158 mol) of K2HPO43H2O in water and diluting to 2.0 l. 2.4. Potassium iodide One group of potassium iodide solutions was prepared by dissolving 3.9, 6.2, 9.1, 12.1, and 16.0 g of potassium iodide (F.W.166.00, Mallinckrodt) in separate volumes of water and diluting each to 500 ml. A separate potassium iodide solution (0.068 mol lÿ1) used for pH studies was prepared by dissolving 22.5 g of potassium iodide in water and diluting to 2.0 l. 2.5. Molybdate catalyst
All solutions were prepared in doubly distilled water and all chemicals were used as received.
For all studies except the molybdate dependence, a stock solution of molybdate solution was prepared by dissolving 1.33 g (1.08 mmol) of (NH4)6Mo7O24 4H2O (F.W.1235.86, Fisher Scienti®c, Fair Lawn, NJ 07410) in water and diluting to 100 ml. For the molybdate dependence studies, four molybdate solutions were prepared by dissolving 0.2512, 0.4574, 0.6914 and 0.9092 g of (NH4)6Mo7O244H2O in four separate volumes of water and diluting each to 50 ml.
2.2. Hydrogen peroxide
2.6. Combined solutions
Solutions containing nominal hydrogen peroxide concentrations from 8.82 mmol lÿ1 to 221 mmol lÿ1 were prepared by dilution of a commercially available reagent (3% H2O2, Mallinckrodt Chemical, Paris, KY 40361) with water. Solutions were used without standardization because our primary interests were in
Two groups of carrier solutions were prepared. One group used to study effects of pH contained ®xed concentrations of iodide and molybdate catalyst and variable amounts of the buffer components; the other group used to study effects of iodide concentration contained ®xed concentrations of the buffer compo-
2. Experimental 2.1. Solutions
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nents and molybdate catalyst and variable amounts of potassium iodide. Carrier solutions used to study effects of pH were prepared by mixing 450 ml of potassium iodide solution (0.068 mol lÿ1) and 450 ml of the molybdate solution with 150 ml of each of the ®ve phosphate buffer solutions described above. The ®nal pH values of these solutions were 5.86, 6.04, 6.22, 6.45 and 6.57, respectively. The molybdate and potassium iodide concentrations were ®xed at 8.06 mmol lÿ1 and 0.0507 mol lÿ1, respectively. Several hydrogen peroxide concentrations (8.8±441 mmol/lÿ1) were used. Carrier solutions used to study effects of iodide concentration were prepared by mixing 167 ml of the 0.555/0.158 mol lÿ1 phosphate buffer and 500 ml of the molybdate solution with 500 ml of each of the potassium iodide solutions described above. The hydrogen peroxide and molybdate concentrations was ®xed at 8.07 mmol lÿ1 and the pH was ®xed at 6.1. Several hydrogen peroxide concentrations ((22.1± 882 mmol lÿ1) were used. Carrier solutions used to study effects of molybdate were prepared by mixing 100 ml of potassium iodide solution (0.1085 mol lÿ1) and 1.000 ml of each of the four molybdate solutions described above and 34 ml of phosphate buffer (19.1419 g of KH2PO4 and 9.1726 g of K2HPO43H2O in 200 ml of water). The molybdate concentrations were 0.1090, 0.08289, 0.05483 and 0.03011 mmol lÿ1. The hydrogen peroxide and potassium iodide concentrations were ®xed at 441 mmol lÿ1 and 0.0804 mol lÿ1, respectively and the pH was ®xed at 6.15. 3. Instrumentation Liquid chromatographic instrumentation was used to transport and mix solutions and to monitor timedependent absorbances in a stopped-¯ow mode. Data were collected and processed on line using a personal computer. 3.1. Flow system The liquid chromatograph (Shimadzu Scienti®c Instruments, Columbia, MD 21046) included two pumps (Shimadzu, LC-10AS), a mixing chamber and a variable-wavelength detector (Shimadzu). One
79
pump was used to pump sample solutions and the other was used to pump the carrier solutions. Sample and carrier solutions were mixed in the gradient mixing chamber and then passed through a 58 cm length of 0.30 mm i.d. stainless-steel tubing from the mixing chamber to the detection cell (10 mm path length, 8 ml volume). Flow was stopped with sample solution in the detection cell and data for absorbance vs. time were collected and stored in computer memory. The time delay between the mixing chamber and the detection cell was ca. 9 s. 3.2. Data-collection/data-processing system Data were collected on line using a commercially available interface system (National Instruments, 6504 Bridge Point Parkway, Austin, TX 787305039) and computer (Gateway 2000 4DX2-50V, 610 Gateway Drive, P.O. Box 2000, North Sioux City, SD 57049-2000). Data were processed using customdesigned software described elsewhere [6]. 4. Procedures 4.1. Mixing system Flow rates for the sample and carrier solutions were 0.2 and 3.8 ml minÿ1, respectively, corresponding to mixed solutions containing 5% sample solution and 95% carrier solution and a total ¯ow rate of 4.0 ml minÿ1. The 0.5 ml option for the internal volume of the mixing chamber was used in all the studies reported here. Solutions were pumped continuously through the system for ca. 60 s to permit the detector response to reach an initial steady-state value, after which the ¯ow was stopped. 4.2. Data acquisition Data acquisition was initiated immediately after sample and reagent pumps were activated. Absorbances were monitored at 360 nm at a rate of 10 points per second. Data were collected for suf®cient time to provide data over at least 8 half-lives of each reaction, usually 200±400 s depending on reaction rates.
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4.3. Data processing Four data-processing options were evaluated. Two of the options are not expected to permit compensation for changes in experimental conditions whereas two options are expected to be virtually independent of small changes in experimental conditions. One option, identi®ed herein as the ``steady-state option'', involves measurement of the steady-state absorbance obtained just before the ¯ow is stopped. Results obtained in this way are analogous to peakheight results [3] in the sense that they represent a ®xed-time option [5] because the steady-state signal corresponds to the amount of triiodide produced during the time interval from the point of mixing to the measurement cell. A second option, identi®ed herein as the ``single-rate option'', involves the linear leastsquares slope of the ®rst ten data points collected after the ¯ow is stopped. Both of these options are expected to have large dependencies on experimental conditions. Preliminary experiments using a batch mixing approach and a diode-array-based spectrophotometer showed that the iodide/hydrogen peroxide reaction followed ®rst-order kinetics quite well. However, as discussed in more detail later, the stopped-¯ow response curves exhibited more complex behavior, increasing ®rst to a maximum value and then decreased at a near constant rate suggesting a zeroorder component of the response. It was found that a model for combined ®rst-order/zero-order processes ®t the data well. Accordingly, a third data-processing option, identi®ed herein as the ``deconvolution option'', involves the use of a model for simultaneous zero-order/®rst-order processes to resolve the ®rstorder component of absorbance vs. time data from overall responses. The mathematical description for this model is: At A0 A1 ÿ A0 1 ÿ exp ÿk1 t k0 t
(1)
in which At is the time-dependent absorbance, S1 is the absorbance expected when the reaction has gone to completion, A0 is the background absorbance, k1 is the rate constant of the ®rst-order component of the response and k0 is the rate constant of the zero-order component of the response. Fits of Eq. (1) to the data showed that the ®rst-order component of the model ®t data during the ®rst two to
three half lives of the ®rst-order process quite well. Accordingly, a fourth data-processing option, identi®ed herein as the ``extrapolation option'', involved the use of a ®rst-order model to extrapolate data during the ®rst two to three half lives to the value expected if the reaction were to proceed to completion without any complications. The mathematical model for this extrapolation process is: At A0 A1 ÿ A0 1 ÿ exp ÿk1 t
(2)
in which all symbols are de®ned above. Because the deconvolution and extrapolation options are designed to obtain the signal corresponding to complete reaction, these options are also identi®ed herein as pseudo-equilibrium options. Both options are expected to be very rugged to changes in experimental conditions. 4.4. Data interpretation Relative error coef®cients are used to quantify differences in variable dependencies among the different data-processing options. The relative error coef®cient is de®ned as the percent error in concentration produced by a unit change in the variable of interest. Procedures used to quantify relative error coef®cients are described elsewhere [7] and are not repeated here. 5. Results and discussion Unless stated otherwise, all imprecisions are quoted at the level of one standard deviation (1 s.d.). 5.1. Response curves Fig. 1(A) includes a typical set of response curves for different hydrogen peroxide concentrations with ®xed values of iodide and molybdate concentrations and pH. In each case, zero time corresponds to the point at which the pumps and data acquisition were started. The initial increase shortly after pumps are started (Region a) corresponds to any residual triiodide remaining from a previous sample. The more gradual rise toward a steady-state response (Region b) corresponds to entry of the reaction mixture into the measurement cell. The steady-state signal observed in Region b of the responses corresponds to a ®xed
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decreasing pH. It is probable that the decreasing absorbance results from oxidation of the stainlesssteel tubing by triiodide. For this reason, few studies were done at low pH and the system was ¯ushed thoroughly when not in use. 5.2. Data-processing options
Fig. 1. Typical response curves at fixed (A) and variable (B) pH. Frame A: Potassium iodide, 0.0507 mol l ÿ1 ; Molybdate, 8.06 mmol lÿ1; pH, 6.22; Hydrogen peroxide (bottom to top), 8.82, 22.1, 4.1, 88.1, 221, 441 mmol lÿ1. Frame B: Potassium iodide, 0.0507 mol lÿ1; molybdate, 8.06 mmol lÿ1; pH (bottom to top @50 s), 6.57, 6.45, 6.22, 6.04, 5.86; Hydrogen peroxide 82.2 mmol/l.
amount of triiodide formed during the ®xed time period (ca. 9 s) between the mixing and measurement cells. The ¯ow is stopped after this steady-state signal is established and the larger increase following the initial steady-state response corresponds to continued reaction in the solution in the measurement cell. All responses exhibit a gradual decrease in signal as the reaction approaches equilibrium. As shown in Fig. 1(B), this behavior becomes more severe with
The steady-state option is implemented by averaging the last ten data points in Region b just before the ¯ow is stopped. The single-rate option is implemented by computing the least-squares slope of the ®rst ten data points after the ¯ow is stopped. The deconvolution option is implemented by ®tting a model for combined zero-order/®rst-order processes Eq. (1) to data after the ¯ow is stopped and using data from the ®rst-order component of the response. Fig. 2(A) includes data for a ®t of Eq. (1) to a typical set of experimental data as well as the ®rst-order component obtained from the ®t. The model for combined zero-order/®rst-order processes is observed to ®t the data well. Similar ®ts were obtained for other data sets. The maximum value of the ®rst-order component of absorbance, A1;1 , is used to compute hydrogen peroxide concentration. The extrapolation option is implemented by ®tting a model for a ®rst-order process Eq. (2) to data during the early part of the process (usually approximately three half lives of the ®rst-order process) and extrapolating the computed response to t!1. A typical set of experimental and computed data is given in Fig. 2(B). The ®rst-order model is observed to ®t data during the ®rst part of the response well; similar ®ts were obtained for other data sets. The maximum value of the ®rst-order component of absorbance, A1;1 , obtained from the extrapolation process is used to compute hydrogen peroxide concentration. 5.3. Rate constants Apparent values of pseudo-®rst-order rate constants for different values of pH, iodide concentration and molybdate concentration are illustrated in Fig. 3. For conditions used in this study, these rate constants are observed to vary between ca. 0.01±0.1 sÿ1. Rate constants are observed to decrease with increasing pH and to increase with increasing concentrations of iodide and molybdate.
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Fig. 3. Apparent first-order rate constants for different values of pH (a), iodide concentration (a mmol lÿ1, (b)) and Mo(VI) concentration (a mmol lÿ1, (c)). All error bars represent 1 s.d. Plots ((a),(b)) Each value is the average of three runs at each of six hydrogen peroxide concentrations (a8.8±441 mmol lÿ1; b22± 882 mmol lÿ1). Plot (c) Each point is the average of three runs at a single hydrogen peroxide concentration (441 mmol lÿ1). Other conditions as for corresponding frames in Fig. 5.
5.4. Imprecision Fig. 2. Response and fitted curves using combined first-order/zeroorder (A) and first-order (B) models. Conditions (both frames): Potassium iodide, 55.2 mmol lÿ1; Molybdate, 8.07 mmol lÿ1; pH, 6.1; Hydrogen peroxide, 441 mmol/l. Frame A: (a) Experimental data, (b) First-order/zero-order fit, (c) First-order component. Frame B: (a) Experimental data, (b) First-order fit.
Each data point for the different values of pH corresponds to the average of three runs at each of six hydrogen peroxide concentrations between 8.8± 441 mmol lÿ1 and each data point for different iodide concentration corresponds to averages of three runs at each of six hydrogen peroxide concentrations between 22±882 mmol lÿ1. Error bars on the plots represent standard deviations for each of the 18 data sets. These error bars re¯ect the fact that rate constants do not change signi®cantly with hydrogen peroxide concentration.
Table 1 contains pooled relative standard deviations for the different options for different hydrogen peroxide concentrations. There are no apparent trends with different concentrations. When two unusually large values for the single-rate option are ignored, average values of the pooled standard deviations are all similar, with the only signi®cant difference being the smaller value for the steady-state option in the pH study. We conclude that the different options yield similar values of relative standard deviations. 5.5. Linearity Two typical sets of calibration plots obtained at different values of pH are given in Fig. 4(A) and (B). Linearity data for the steady-state option are similar to data in Fig. 4(A) for the single-rate option and data for the extrapolation option are similar to those in Fig. 4(B) for the deconvolution option. Also, data for different values of iodide and molybdate concen-
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Table 1 Summary of imprecision data for different data-processing options Pooled relative standard deviation (%) Concentration (mmol lÿ1)
Steady-state
Initial rate
Deconvolution
Extrapolation
pH dependence b 8.82 22.1 44.1 88.2 221 441 Average Std. Dev.)
3.7 1.8 2.11 1.6 1.4 2.77 2.2 (0.9)
18 e 7.8 4.3 6.1 8.5 6.2 6.6 e (1.6) e
9.4 5.0 3.4 2.4 2.5 3.5 4.3 (2.6)
8.8 3.8 3.3 3.1 4.5 3.3 4.4 (2.2)
Iodide dependence c 22.1 44.1 88.2 221 441 882 Ave. (S.D.)
6.7 2.6 3.4 5.3 3.4 4.6 4.3 (1.5)
16 e 8.1 5.3 6.6 4.4 4.8 5.8 e (1.5) e
5.4 2.0 4.0 5.1 4.1 3.6 4.0 (1.2)
8.3 2.5 3.5 5.2 3.7 4.2 4.6 (2.0)
4.74
6.04
5.88
Molybdate dependence d 0.4412 a
3.31 b
c
d
e
Pooled standard deviations, n5, n5, n4, Value not included in average.
trations are analogous to the pH data in Fig. 4(A) and (B). Least-squares results for all of the calibration plots are summarized in Table 2. Intercepts and standard errors of the estimates are converted to concentration units by dividing by slopes to facilitate comparison among the different options. The large values of the standard deviations of the steady-state and rate options re¯ect the fact that the slopes of the calibration plots change with each of the variables. The standard errors of the estimates re¯ect scatter about the least-squares lines for each set of conditions. For the pH-dependence data, there are no signi®cant differences among the standard errors for the pH-dependence data; however, for the iodide-dependence data, standard errors for the deconvolution and extrapolation options are about three-fold smaller than those for the steady-state and single-rate options. There are no consistent trends in the intercepts among the different options. 5.6. Ruggedness As noted earlier, a major focus of this study was to determine if alternative measurement/data-processing
options could improve the ruggedness of ¯ow-based kinetic methods. As also noted above, changes in pH and concentrations of iodide and molybdate concentrations were used to test the ruggedness of the various options. Data in Fig. 4(A) and (B) illustrate effects of pH on the calibration sensitivities of two of the options. It is apparent from these Fig. that the deconvolution option is much less sensitive to changes in pH than the singlerate option. Analogous behavior was observed for the steady-state and extrapolation options, respectively, as well as for effects of other variables. These differences can be visualized more explicitly using response-ratio plots, i.e. plots of ratios of responses at different values of each variable to the response at a particular value of the variable vs. the different values of the variable. Response-ratio plots for each of the options evaluated in this study are given in Fig. 5(A)±(C) for changes in pH and concentrations of iodide and molybdate. The slopes of such plots re¯ect the different degrees of ruggedness of the various methods; the smaller the slope, the more rugged the option for the variable of interest. It is apparent from these plots that the deconvolution and extrapolation options are
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Fig. 4. Calibration plots for steady-state (A) and deconvolution (B) options for different values of pH. Conditions (both frames): Potassium iodide, 0.0507 mol/l; Molybdate, 8.06 mmol/l; pH (bottom to top, (a)±(e)), 6.57, 6.45, 6.22, 6.04, 5.86.
signi®cantly less dependent on the three variables examined than are the single-rate and steady-state options. Differences in ruggedness visualized with the aid of Fig. 5(A)±(C) are easily quanti®ed using relative error coef®cients [7]. Results are summarized in Table 3. To simplify comparisons, the table contains ratios of numerical values for the extrapolation, single rate and steady-state options to the values for the deconvolution option. The larger the ratio of error coef®cients, the poorer the ruggedness for each variable. Some values which seemed to represent unusual behavior were excluded in computing averages and associated standard deviations. Based on the averages, the extrapolation option seems to be slightly less rugged than the deconvolution option to changes in iodide and molybdate concentrations. However, the differences are small and we conclude that these options exhibit similar degrees of ruggedness for each of the three variables examined. The rate constants measured by the two procedures are summarized in Table 4. Both the deconvolution and extrapolation options are signi®cantly more rugged to changes in these variables than either the single-rate or steady-state options. Average improvement factors of the deconvolution option relative to the single-rate and steady-state options range from about 22- to 40-fold when effects of two unusually small values for the deconvolution option are excluded.
Table 2 Least-squares statistics for calibration plots for different data-processing options Option
Slope a
Std. Error b/Slope b
Average
RSD (%)
pH dependence St. state One rate Deconvolution Extrapolation
0.288 0.0186 1.03 0.960
71 50 3.8 5.7
Iodide dependence St. state One rate Deconvolution Extrapolation
0.388 0.0218 0.91 0.88
34 14 0.9 1.8
a
(mmol lÿ1)ÿ1, b mmol lÿ1.
Average 6.8 5.2 5.1 5.8 13 20 6 5
Intercept b/Slope b Std. Dev.
Average
Std. Dev.
6.0 4.3 2.4 4.1
22 ÿ1.0 5.3 5.1
17 2 1.2 1.9
0.8 5.3 3.3 3.4
0.07 ÿ1.2 39. ÿ6.7
4.6 16 1.3 1.9
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Fig. 5. Effects of pH (A), potassium iodide concentration (B) and molybdate concentration (C) on results obtained by applying four different data-processing methods to the same data sets. All frames: (a) Steady-state option, (b) One rate option, (c) Deconvolution option, (d) Extrapolation option. Frame A: Potassium iodide, 0.0507 mol lÿ1; Molybdate, 8.06 mmol lÿ1; Hydrogen peroxide, 441 mmol lÿ1. Frame B: molybdate, 8.07 mmol lÿ1; pH 6.1; Hydrogen peroxide, 882 mmol lÿ1. Frame C: Potassium iodide, 80.4 mmol lÿ1; pH6.15; Hydrogen peroxide, 441 mmol lÿ1.
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Table 3 Comparison of relative error coefficients (RECS) for different data-processing options Concentration(mmol lÿ1)
(REC%)
Ratio RECS
Deconvolution pH dependence 8.82 22.1 44.1 88.2 221 441
Iodide dependence 22.1 44.1 88.2 221 441 882.
Molybdate dependence 0.08824 a
Extrapolation
ÿ0.0183 a ÿ0.119 ÿ0.103 ÿ0.075 ÿ0.250 ÿ0.0974 Average Std. Dev.
5.76 a 1.63 0.64 0.60 1.35 1.66 1.2 0.5
0.166 ÿ0.548 ÿ0.094 ÿ0.138 0.0275 a ÿ0.199 Average Std. Dev.
1.59 0.41 2.41 2.23 20.85 a 0.03 a 1.7 0.9
ÿ0.29158
1.46
Steady-state 39.2 a 16.3 29.9 50.4 23.1 51.5 34.2 16
Slope 124 a 27.1 31.7 39.0 14.1 29.8 28.3 9
32.3 11.1 69.0 50.3 263 a 34.1 39 21
30.0 7.76 48.3 29.7 152 a 17.4 27 27
31.35
22.56
Value not included in average and standard deviation.
Table 4 Summary of rate constants for different variable studies Deconvolution Average
Extrapolation SD
%SD
Average
SD
%SD
pH
pH dependence
5.86 6.04 6.22 6.45 6.57 KI b
0.0893 a 0.0456 0.0284 0.0131 0.0101 Iodide dependence
0.00460 0.00227 0.000686 0.000741 0.000593
5.15 4.98 2.41 5.67 5.85
0.0882 0.0469 0.0293 0.0142 0.0109
0.00273 0.00364 0.00240 0.000896 0.000923
3.10 7.74 8.17 6.29 8.49
0.1435 0.1076 0.08162 0.05522 0.03519 Mo(VI) c
0.0987 0.00570 0.0706 0.00358 0.0546 0.00202 0.0370 0.00165 0.0240 0.000902 Molybdate dependence
5.77 5.06 3.69 4.47 3.76
0.106 0.0766 0.0592 0.0399 0.0264
0.00451 0.00241 0.00339 0.00153 0.00125
4.24 3.15 5.72 3.83 4.74
0.1090 0.08289 0.05483 0.03011
0.0740 0.0563 0.0378 0.0215
0.94 0.39 1.54 1.80
0.0805 0.0612 0.0416 0.0245
0.000733 0.000577 0.000282 0.00104
0.91 0.94 0.68 4.25
a
0.000696 0.00022 0.000582 0.000386
The units of all rate constants are sÿ1, b mol lÿ1, c mmol lÿ1
Y. Shi, H.L. Pardue / Analytica Chimica Acta 360 (1998) 77±87
6. Conclusions Results obtained in this study show that alternative measurement/data-processing options can be used to improve the ruggedness of ¯ow-based kinetic methods for noncatalysts used in the stopped-¯ow mode. Ruggedness is improved by at least an order of magnitude, even for the nonideal response curves obtained in this study. It is probable that consistently larger improvements could be obtained with responses that con®rm more closely to ®rst-order behavior. Also, it is probable that other data-processing options developed for batch-type sample processing systems [2] could also be adapted to ¯ow systems. On the one hand, this study did not include direct comparisons with peak-height methods usually used with ¯ow methods. On the other hand, the steady-state option included in this study is a close analogy to the peak-height option because both depend on the amount of analyte reacted during a ®xed time period between points at which reaction is initiated and measurements are made. Accordingly, we can say with some certainty that improvements in ruggedness observed relative to the steady-state option also apply to the more common peak-height option. Also, it is probable that both the deconvolution and extrapola-
87
tion options will be more rugged to changes in other variables such as temperature and ¯ow rate than conventional peak-height options. Moreover, for the same reasons that the deconvolution and extrapolation options are more sensitive than the steady-state option, they should also be more sensitive than the peakheight option. Accordingly, for similar levels of measurement error, these pseudo-equilibrium options should also have lower detection limits than the peak-height option. Finally, this study focused solely on the stopped¯ow mode of operation. A future study will attempt to extend these principles to the continuous-¯ow mode. References [1] [2] [3] [4]
H.A. Mottola, H.B. Mark, Anal. Chem. 58 (1986) 264R. M.D. Love, H.L. Pardue, Anal. Chim. Acta 299 (1994) 195. J. Ruzicka, E.H. Hansen, Anal. Chim. Acta 78 (1975) 145. S. Olsen, J. Ruzicka, E.H. Hansen, Anal. Chim. Acta 136 (1982) 101. [5] H.L. Pardue, Clin. Chem. 23 (1977) 2189. [6] J.W. Skoug, W.E. Weiser, I. Cyliax, H.L. Pardue, Trends Anal. Chem. 5 (1986) 32. [7] K.B. Lim, H.L. Pardue, Clin. Chem. 39 (1993) 1850.