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Anal. Chem. 1908, 6 0 , 545-548
Improvement of Reaction Rate Measurement Precision Using the Temporally Optimized Fixed-Time Ratemeter Steven A. Engh a n d F. James Holler* Department of Chemistry, University of Kentucky, Lexington, Kentucky 40506
The domain of application of the flxed-time integrating ratemeter method is extended to ail times during first-order and pseudo-first-order reactlons. Propagation of error theory is applied to rate expressions to glve the optimum tlme for rate measurement as a function of Integration wldth. The maximum integration wldth consistent with the optimum measurement time is determined. Experimental data collected with a stopped-flow mixer uslng the iron( I I I)-thlocyanate reaction show excellent agreement with theory. The relative standard deviation of concentration of flve determlnations of a thlocyanate unknown was 0.16% under conditions of random varlabie temperature.
the rate at a single time. Cordos et al. (7) devised an integrating fixed-time method that has these properties and implemented it with a hardware ratemeter. We will call this method the fixed-time ratemeter method to distinguish it from other fixed-time methods and variable time ratemeter methods. The method was devised for conditions of pseudo-zero-order kinetics and provides an approximation of the true rate when applied to a first-order or pseudo-first-order reaction. Several elegant and flexible analog and digital ratemeters have been implemented to take advantage of this method (7-12). THEORY For an irreversible first-order reaction k
It has been shown previously that the precision of reaction rate measurements can be improved by selection of the optimum time at which to make these measurements (1-6). To make best use of an optimum time for measurement, a reliable method of obtaining the rate at a single time is needed. The fixed-time integrating ratemeter method (7) gives an estimate of the rate a t a time at the center of an interval over which the reaction curve can be considered to be linear. In contrast to many methods of rate estimation, this approach is highly resistant to instrumental noise. In this paper we will show that this method is valid under first-order and pseudo-firstorder conditions at any time during the reaction, that the precision of the rate obtained by this method can be temporally optimized, and that there is a maximum integration width consistent with temporal optimization. Finally, we present experimental verification of temporal optimization and signal-to-noise-ratioenhancement resulting from optimum use of the integrating ratemeter. Several strategies have been proposed to temporally optimize reaction rate measurements. Landis et al. (I) recognized that a measurement time o f t = l / k = T is optimum, where k is the first-order or pseudo-first-order rate constant for the reaction, but only used this result for error analysis. Davis and Renoe (2) developed equations to obtain optimum times for wide-interval fixed-time rate measurements. Davis and Pevnick (3) considered coupled enzyme reactions and obtained optimum times for several cases of variations in one or both of the rate constants. Most recently, Wentzell and Crouch presented a two-rate method ( 4 ) and compared the accuracy and precision of their method and several other methods under various conditions (5). In an earlier report, we used propagation of error theory to show that the relative standard deviation of the rate of a first-order reaction is zero at t = l / k = 7 and presented experimental evidence verifying this result (6). Three assumptions were made in that development: (1) the reaction of interest is first order or pseudo first order, (2) sources of error other than variation in the rate constant are negligible, and (3) the rate can be estimated by instrumental or numerical methods at any time during the reaction. To take full advantage of the simplicity of a single-rate measurement at t = 7 it is desirable to use a measurement method that is resistant to instrumental noise and that gives
B-P (1) where k is the first-order rate constant for the reaction, the rate of disappearapce of reactant B is -d[B]/dt = K[B] (2) The integrated form of eq 2 is [B], = [B]oe-kt = [B]oe-t/T (3) where [B], is the concentration measured at any time t, [B], is the initial concentration of B, and 7 is the reciprocal of the first-order rate constant k. From eq 3 the rate of disappearance, R,, of B is 1 -(d[B]/dt), = R, = -[B]oe-t/T (4) where Rt is the rate at t. Figure 1 shows graphically and analytically how the fixed-time ratemeter method is applied at an arbitrary time t to the concentration vs time curve described by eq 3. The method begins with the integration of the curve from t - At to t, and again from t to t + At. The ratemeter rate R ; is given by R', = AI/(At)2 (5) where AI is the difference in the two integrals. If this method is applied to the linear case, i.e., zero-order kinetics, AIg/(At)2 is exactly equal to the true rate. If the method is restricted to integration widths that are short relative to T , then the curve is nearly linear, and AIg/(At)2closely approximates the true rate. At later times, this approximation is not as good, and the analytical expression for the calculated difference in the areas AI, shown in Figure 1 must be used. The analytical expression for the calculated rate is then
When the integrals of eq 6 are evaluated, we obtain eq 7 . 2 [Bloc-,/' R', = [cash (At/T) - 11 = T ( At / T )
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most important consequences of this discussion are that R I, differs from the true rate R, by a constant term for a constant value of At, and that for any fixed value of A t , RIt is linear with concentration. Temporal Optimization. Previous work (6) demonstrated that for first-order or pseudo-first-order reactions the relative standard deviation of the rate is given by
1
0
37
2T
T
Figure 1. Graphical and analytical representations of the integrating ratemeter method. AI, is the graphical approximation of the difference between the integrals and A I c is the analytical difference.
I
0 161
In a similar fashion we can apply propagation of error theory to the ratemeter expression of eq 7. The error in the computed rate due to variations in the rate constant is given by
Differentiation of eq 7 with respect to k yields
i l
[ (i +
A
'3
0 4 0 080 120
Relative absolute error in the ratemeter rate vs integration width relative to T. Figure 2.
Two important consequences result from this equation: (1) the computed rate differs from the true rate by a constant factor that depends only on At/7, and (2) the relationship is valid for any time during the reaction. It is not obvious that eq 7 is in general a good approximation for the true rate R,, however. If the first two terms of the series expansion for the hyperbolic cosine (A~/T)' ( A ~ / T ) ~
++ 2!
~
4!
+ ...
(8)
Rt
=
[(at)coth
(g)
Thus the computed rate is the sum of the true rate and a series error term. When At is chosen to be small relative to 7 , the error term is nearly zero, and the approximation of eq 9 is valid. The relative absolute error in the computed rate Rtt is given by (11)
Figure 2 shows the error computed from eq 11 for values of At/?. It is significant and consistent with the consequences of eq I that eq 11 depends only on the factor Atlr. The two
(t + T ) ] u ~
(15)
(16)
The significance of this time becomes clear when we substitute the series expansion for the hyperbolic cotangent
coth ( A t ) = -27+ - - A -t At 67
(At)3 ... 3 6 0 ~ ~
+
(17)
into eq 16. This gives the time when the relative error is zero as shown in eq 18. In the limit as At approaches zero, this (At)2
which is identically equal to the true rate R, shown in eq 4. When all terms are included in the expression, the computed rate is
-
t = At [coth (At/27)] - T
t=7+--67
[ - c o2s h ( ~ ) - l ] - l (At/7I2
- 11) (14)
Equation 15 can be set to zero and solved for t , the time at which the relative standard deviation of the rate is zero.
are substituted into eq 7, we find that
- = - Rt R', Rt
(e)
Substitution of eq 14 into eq 13 and division of the result by eq 7 yields the relative standard deviation of the rate
160 2 0 0
At/r
cash ( A t / i ) = 1
1) cosh
(At)4 ... 3 6 0 ~ ~
+
time is equal to 7 , and eq 15 reduces to eq 12. At larger values of At, the s u m of the second and higher series terms is positive, which makes the optimum time slightly greater than T . Integration provided by the fixed-time ratemeter and its software equivalent reduces the effect of noise on the signal. The choice of integration width depends on T and the noise power density spectrum of the detection system. The integration width should be short as possible compared with l / f noise and drift. In this way, the measurement is completed before there is significant change in the signal. On the other hand, the integration width should be made wide enough so that the difference in the integrals is sufficiently precise. Usually At is made as large as possible within these limits. Maximum benefit with respect to variations in the rate constant is obtained by making rate measurements at the time t for which eq 15 becomes zero. It is clear from Figure 1 that in order to measure the rate at time t, At can be no larger than t . Thus, by substitution of to = t = A t into eq 16, we have t o = t o coth ( t 0 / 2 7 ) - T (19) which simultaneously gives the maximum integration width and the optimum time at which to make the rate measurement. Equation 19 is a transcendental equation whose nu-
ANALYTICAL CHEMISTRY, VOL. 60, NO. 6, MARCH 15, 1988
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merical solution gives a value of tO/T = 1.25643. Thus an optimum rate measurement can be made by setting At equal to 1.25643~. In order to test the theory, the well-characterized ironthiocyanate reaction was studied by stopped-flow mixing with spectrophotometric detection. EXPERIMENTAL SECTION Instrumentation. A stopped-flow instrument designed and built in our laboratory was used to collect data for these experiments. Instrument operation and data acquisition were under computer control to increase the speed and the precision of the measurements. The computer used was a LSI-11 (Digital Equipment Corp., Maynard, MA) running under the RT-11 operating system (DEC). The data collection software was written in MACRO-11 and the data analysis software was written in FORTRAN IV (DEC). Part of the light beam exiting the monochromator was split off to a reference photomultiplier. The signals from the sample and reference photomultipliers were ratioed to increase the signal-to-noise ratio of the absorbance measurements with respect to variations in the light source. Absorbance curves were stored in disk files for processing by the software ratemeter discussed below. The stopped-flow mixer will be described and characterized in detail in a future paper. Temperature control was provided for the instrument and reagents by a constant temperature bath (Masterline Model 2905, Forma Scientific, Marietta, OH). The reagents were held in jacketed beakers at the inlet of the drive syringes. Water from the constant temperature bath flowed through the jackets and around the instrument. The temperature controller of the water bath was interfaced to the computer to allow the computer to set appropriate temperatures. Software Ratemeter. A software ratemeter was chosen for this work because it could be quickly and easily implemented on the computer interfaced to the stopped-flow instrument. We also wanted to analyze the data in several different ways, and the software ratemeter afforded considerable flexibility in this respect. Because data could be recalled and analyzed repetitively, direct comparison of different integration widths with the same data was possible. From Figure 1 we can see the first rate that either a software or hardware ratemeter can obtain is at t = At. After the first rate is obtained, the hardware ratemeter cannot reuse data from the first two integration periods. Integration for the next rate must proceed from t = 2At, which makes the next rate available at t = 3At. Because data are stored, the software ratemeter can reuse data from previous integration periods, so the next rate obtainable would be at t = At plus a time increment less than At. This allows rates to be obtained at a much higher point density than is possible with a hardware ratemeter. The algorithm to compute integrals makes use of NewtonCoates n point rules where n = 2,3,4, or 5 (13). Intervals longer than 5 points were split into pieces each less than or equal to 5 points and the resultant integrals were summed to provide the totalintegral. All noninteger arithmetic was performed in double precision (64 bits). The algorithm was tested on a variety of exponential functions, and computed values agree with known values to at least seven decimal digits in all cases. Reagents. Reagent grade Fe(N03)3.9Hz0(Merck), KSCN (Baker), and HNO, (Baker) were used as received. All solutions were prepared from distilled deionized water (Barnestead Milli-Q) with class A volumetric glassware. The iron solution was 5.50 X lo-' M in Fe3+and 1 M in HN03. Working thiocyanate solutions were prepared from a 8.87 X loe2 M stock solution and diluted to volume with a dilute HNO, solution immediately before use; the final HN03 concentration was 1 M. Five standard solutions and a pseudounknown were made by dilution of the stock thicyanate solution. The concentrations P M and the pseudounknown ranged from 4.44 X lod to 4.44 X 1 M. Final concentrations of all concentration was 2.44 X reagents except HN03 after mixing were half of those given previously. The final HNO, concentration was 1 M. RESULTS A N D DISCUSSION The rate constant for a reaction depends on parameters such as temperature, ionic strength, and pH. Temperature
0.1
0.2
0.3
0.4
0.5
0.6
TIME , S
Figure 3. Theoretical (a) and experimental (b) curves of relative standard deviation in rate vs time for the Fe(II1)-SCN- reaction.
is probably the easiest of these parameters to vary experimentally and is impossible to maintain absolutely constant. For these reasons, random temperature variations were imposed on the experiments to produce random variations in the rate constant. To obtain random temperatures, random numbers were generated by the central limit theorem (14), and converted to temperatures with a mean of 20.0 "C and a standard deviation of 0.5 "C. This resulted in a mean value of 0.28 s for T with a standard deviation of 0.01 s. The computer automatically adjusted the controller of the temperature bath to the computed random temperature, paused for 15 min to allow equilibration of the reagents to this temperature, and then collected data. Following data collection, the process was repeated with a new random temperature. At each of 20 random temperatures, five data collection pushes were performed and averaged for signal-to-noise-ratio enhancement. An integration width of At/T = 0.25 was chosen as a compromise between a long integration width to minimize the effect of instrumental noise on the measured rate and a short integration width to obtain rates as near as possible to the initiation of the reaction. The first experimental rate was obtained at t = At and the integrals were moved pointwise through the curve until the second integral included the last data point. At each point in time the rates from the 20 rate curves were selected, and the mean, standard deviation, and relative standard deviation of the rate at that time were calculated. The original absorbance curves were fit with a nonlinear least-squares fitting procedure (15) to obtain the mean and standard deviation of the rate constant. Theoretical curves were calculated by substituion of the estimates of T , (Tk, and At in eq 15. The results of this experiment are plotted in Figure 3. Excellent agreement between the theoretical and experimental curves demonstrates the applicability of the theory under varying rate constant conditions due to temperature fluctuations. The difference between the theoretical and experimental curves at the point of minimization can be understood by considering the conditions assumed in the derivation of the theory. Equation 13 represents only the error due to variation in the rate constant. At the point of minimization, the error due to variation in the rate constant vanishes, and the residual 0.17% error is due to instrumental noise. In the experiment above, an average of five pushes of the stopped-flow system a t each temperature were used for signal-to-noise enhancement. To demonstrate the performance of the algorithm under higher noise conditions, only the first of each set of five pushes was used for each temperature. These reaction curves were processed in the same way with integration widths varied from 0.17 to 1.257. These data, plotted in Figure 4, demonstrate some properties of the ratemeter method. Even at the smallest integration widths, the relative standard deviation in the rate is minimized, and most of the advantage of the method is realized by At = 0 . 4 ~ .
548
ANALYTICAL CHEMISTRY, VOL. 60,NO. 6, MARCH 15, 1988
UC
FR', -
C
R:
-
0'1
012
d3
d4
05
06
TIME , S
Figure 4. Relative standard deviation of rate vs time. Integration widths (a) 0.17; (b) 0.27; (C) 0.47; (d) 0.67;(e) 0.87;(f) 7;(9) 1.2587. Same reaction conditions as given for Figure 3.
In addition, the time of minimization increases and the minimum value of the error decreases with an increased integration width. To demonstrate how minimization in the standard deviation of the rate applies to real measurements, a study was performed to determine the between-run precision of the method. Five absorbance versus time curves were averaged for each of five different thiocyanate concentrations and a pseudounknown, under the same random temperature conditions as above. Rates were computed for each point in time after the initial integration width. The rates of the standard thiocyanate concentrations were used to construct calibration curves and the "unknown" concentrations were determined from the standard curves. This process was repeated five times, and the relative standard deviation of the determined pseudounknown concentration vs. time was obtained. The results of this experiment are shown in Figure 5. Clearly, minimization in the error in the rate does lead to minimization of error in measured concentration. The minimum in the curve is a relative standard deviation in concentration of 0.16%. For comparison, the same experiment was performed under temperature invariant conditions. With an integration width of 1.257, the relative standard deviation in concentration was 0.14%. Implications for Rate Methods of Analysis. To make use of the fxed-time ratemeter method, a preliminary estimate of the rate constant for the reaction of interest must be obtained. In Figures 3-5, the relative standard deviation curves round off near the point of minimization. Errors in the value of T , and therefore the estimated optimum time, will have little effect on the precision of the rate measurements. This implies that a relatively unsophisticated method can be used to obtain the rate constant without incurring a large penalty in poorer precision in the measured concentration. Initial rates traditionally have been used for rate methods of analysis. Routine attainment of relative standard deviations of 0.17% show that the temporally optimized ratemeter me-
0
0
01
02
03
04
05
TIME, S
Figure 5. Relative standard deviation of determined SCN- pseudounknown concentration vs time, integration width 0.27. Same reaction conditions as given for Figure 3.
thod gives a significant advantage in precision compared with initial rates. Even if T is not known and the rate is measured at times well removed from the optimum, improvement in precision can be realized compared with initial rates. If the reaction used for analysis exhibits pseudo-first-order behavior from the time of initiation to approximately 2.57, the fixed time ratemeter method can be used with any integration width up to 1.257, and rate measurement will be optimum with respect to integration width and to temporal optimization. Deviations from pseudo-first-order behavior may require smaller integration widths or measurements at times substantially different from the optimum time with smaller integration widths. While the tolerance of this method to deviations from pseudo-first-order behavior has not been investigated, we suspect that this method may yield improvements in reaction rate measurement precision even when there are substantial deviations. Future studies should elucidate the range of applicability of this method beyond strictly pseudo-first-order conditions. LITERATURE CITED Landis, J. B.; Rebec, M.; Pardue, H. L. Anal. Chem. 1977, 4 9 , 785. Davis, J. E.; Renoe, B. Anal. Chem. 1979, 51, 526. Davis, J. E.; Pevnick, J. Anal. Chem. 1979, 57, 529. Wentzell, P. D.; Crouch, S. R. Anal. Chem. 1988, 58, 2851. Wentzell, P. D.; Crouch, S. R. Anal. Chem. 1988, 5 8 , 2855. Holler, F. J.; Calhoun. R. K.; McClanahan. S. F. Anal. Chem. 1982, 54, 755. Cordos, E. M.; Crouch, S. R.; Malmstadt, H. V. Anal. Chem. 1988, 4 0 , 1812. Ingle, J. D., Jr.; Crouch, S. R. Anal. Chem. 1970, 4 2 , 1055. Irackl, E. S.; Malmstadt, H. V. Anal. Chem. 1973, 4 5 , 1766. Wilson, R . L.; Ingle, J. D., Jr. Anal. Chlm. Acta 1976, 83, 203. Bonnell, I. R.; Defreese, J. D. Anal. Chim. Acta 1982, 734, 189. Ryan, M. A.; Ingle, J. D., Jr. Talanta 1981, 28, 539. LaFara, R. L. Computer Methods for Science and Engineering; Hayden: Rochelle Park, NJ, 1973. Dwass, M. Probability and Statistics; W. A. Benjamin: New York, 1970; p 329. Caserta, K. J.; Holler, F. J.; Crouch, S. R.; Enke. C. G. Anal. Chem. 1978, 5 0 , 1534.
RECEIVED for review June 23, 1987. Accepted November 7 , 1987. We are grateful to the National Science Foundation for supporting this work under Grant CHE-8217348.
