Anal. Chem. 1982, 54, 755-761
tributed. Those detected in other series may also be distributed normally although the data were insufficient to rule out other possibilities.
ACKNOWLEDGMENT We thank Q.G. Grind&iff and G. P. Sturm, Jr., Bartlesville Energy Technology Center, for technical assistance in acquiring some of the mass spectra used in this work. P. E. Pulley, Data Processing Center, Texas A&M University, programmed the nonlinear regression and provided other programming assistance. Appreciation is expressed to K. J. Irgolic, Department of Chemistry, Texas A&M University, for acting as graduate advisor to L.R.S. in the absence of R.D.G. LITERATURE CITED Richardson, J. S. Ph.D. Dissertation, Texas A&M Unlversity, College Station, TX, 19'78. Norman, E. J. Ph.D. Dlssertatlon, Texas A&M Unlversity, College Station, TX, 1977. Schronk, L. R. M.S. Thesis, Texas A&M University. College Station, TX, 1978. Schronk, L. R.; Grlgsby, R. D.; Hanks, A. R. Presented at the 27th Annual Conference on Mass Spectrometry and Allled Toplcs, Seattle, WA, June 3-8, 1979; paper No. MAMOAI. Grlndstaff, Q. G.; Hwang, C. S.; Marriott, T. D.; Benson, P. A.: Scheppele, S. E. Presented at the 27th Annual Conference on Mass Spectrometry and Allied Topics, Seattle, WA, June 3-8, 1979; paper No. MAMOA2. Grlgsby, R. D.; Schronk, L. R.; Grindstaff, Q. G.; Scheppele, S. E. Presented at tho 27th Annual Conference on Mass Spectrometry and Allled Topics, Seattle, WA, June 3-8, 1979; paper No. MAMOA3. Grlgsby, R. D.; Schronk. L. R.; Hanks, A. R. Presented at the Texas A&M University Lignite Symposium, College Station, TX, April 17-18, 1980. Hlrsch, D. E.; Hopkins, R. L.; Coleman, H. J.; Cotton, F. 0.: Thompson, C. J. Anal. Chem. 1972, 44, 915-919. Jewell, D. M.; Weber, J. 14.; Bunger. J. W.; Planchet', H.: Latham. D. R. Anal. Chem. 1972, 44, 1391-1395. Coleman, H. J.; Dooley, J. E.; Hlrsch, D. E.; Thompson, C. J. Anal. Chem. 1973, 45, 1724-1737. Dooley, J. E.; Hlrsch. D. E.; Thompson, C. J.; Ward, C. C. Hydrocarbon Process. 1974, 53 (ll), 187-194, and references clted therein. Hirsch, D. E.; Dooley, J. E.; Coleman, H. J.; Thompson, C. J. Rep. Invest.---US.. Bur. Mlnes 1974, No. 7893.
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(13) Scheppele, S. E.; Grindstaff, Q. G.; Pavelka, E. A,; Evans, S.; Banner, E. A.; Tudge, H. U.S.Department of Energy, Bartlesville, OK, and Kratos Sclentiflc Instruments, Ltd., Manchester, Enaland. unoublished result~,1980-1981. (14) Scheppele, S. E.; Grizzle, P. L.; Greenwood, G. J.; Marriott, T. D.: Perreira, N. 0. Anal. Chem. 1976, 48, 2105-2113. (15) Kearns, G. L.; Maranowski, N. C.; Crable, G. F. Anal. Chem. 1959, 31. 1646-1651. (18) Lumpkin,H.E.; Aczel, T. Anal. Chem. 1964, 36, 181-184. (17) Scheppele, S. E.; Benson, P. A.; Greenwood, G. J.; Grindstaff, Q. G.; Aczel, T.; Beier, B. F. A&. Chem. Ser. 1961, No. 195. (18) Grlgsby, R. D.; Hansen, C. 0.; Mannerlng, D. G.; Fox, W. G.; Cole, H. H. Anal. Chem. 1971, 43, 1135-1137. (19) Grlgsby, R. D.; Norman, E. J.; Pulley, P. E. Presented at the 23rd Annual Conference on Mass Spectrometry and Allied Topics, Houston, TX, May 25-30, 1975; paper No. U-9. (20) Grlgsby, R. D. Presented at the 20th Annual Conference on Mass Spectrometry and Allied Topics, Dallas, TX, June 4-9, 1972; paper No. D11. (21) Beynon, J. H. "Mass Spectrometry and its Applications to Organic Chemistry"; Elsevier: New York, 1980; pp 474-479. (22) Thomas, R. H. P. Ph.D. Dissertation, Texas A&M Universltv, Colleae Station, TX, 1978. (23) Burrows, G. "Molecular Dlstillatlon"; Oxford Unlverslty Press: London, 1960. (24) "Characterlzatlon of the Heavy Ends of Petroleum", Annual Report No. 14, Amerlcan Petroleum Institute Research Project 60, 1973, p 19. (25) Lumpkln, H. E.; Elliott, R. M.; Evans, S.; Hazelby, C.; Wolstenholme, W. A. Presented at the 23rd Annual Conference on Mass Spectrometry and Allied Topics, Houston, TX, May 25-30. 1975; Paper No. H-17. (28) Scheppele, S. E. "Characterization of Coal-Derived Liquids and Other Fossil Fuel Related Materials Employing Mass Spectrometry"; US. Department of Energy, Contract No. EX-78-S-01-2537, Quarterly Report, March 30June 29, 1978, pp 119-123.
RECEIVED for review September 17,1981. Accepted December 28, 1981. This work was supported by U S . Department of Energy Contract No. DE-AC19-80BC10171,project 1672 of the Texas Agricultural Experiment Station, and funds from the Center for Energy and Mineral Resources, Texas A&M University. Taken in part from the Ph.D. dissertation of L. R. Schronk and presented in part at the Second Chemical Congress of the North American Continent, Las Vegas, NV, Aug 24-29, 1980.
Minimization of Errors in Fixed-Time Reaction Rate Methods by Optimization of Measurement Time F. J. Holler," R. K. Calhoun, and S. F. McClanahan Department of Chemistry, University of Kentucky, Lexington, Kentucky 40506
A theoretical development of the effect of random and systematic fiuctuatlon in rate constants on the relative precision of reaction rates for first-order and pseudo-first-order reactions is presented. By the use of propagation of error theory, it Is demonstrated that at t = l / k = 7 reaction rates are essentially independent of small fluctuations In the rate constant. Simulated data support the theoretical predictions. The mlnlmization of rate error at a time other than t = 0 leads to a comparison of the relative precision of inltiai concentration of analyte as determlned from lnltlal rates and from rates measured at 7. The comparison, drawn through the use of propagation of error theory and by the generalization of simulated data, shows that much better preclslon may be obtalned when the rate is measured at T . Experimental data collected on the oxidation of iodlde by hydrogen peroxide are in excellent agreement wlth the theory. The relative standard deviation of the rate determined from these data at t = 7 is a factor of 11.3 smaller than that determined for t = 0. 0003-2700/82/0354-0755$01.25/0
Over the past several years there has been steady growth in the literature of kinetic analysis. However, relatively few papers have appeared that explore the fundamental limitations of kinetic analysis or that present comprehensive studies of the methodology of kinetics measurements. Ingle and Crouch ( I , 2) have discussed the relative merits of fixed-time and variable-time met hods and have developed the signalto-noise ratio theory for these methods, particularly with regard to spectrophotometricdetection. Wilson and Ingle have presented a similar treatment in conjunction with the development of a fluorometric reaction rate instrument (3). In a recent work Carr (4)has investigated the effects of random variations in rate constants on the precision of kinetic methods. Because of difficulties inherent in variable-time methods, only fixed-time methods were treated. By assuming that rate measurements are carried out by measuring a change in the concentration of the monitored species, ACM,over the fixed time of the experiment and by applying propagation of Q 1982 Amerlcan Chemical Soclety
756
ANALYTICAL CHEMISTRY, VOL. 54, NO. 4, APRIL 1982
and systematic errors in the rate parameters, pi. The variance of a measured rate at any time t is given by
where k and, therefore, the rate are functions of the rate parameters. From eq 2, we have for the reaction of eq 1
_ aRt - [B],,eWkt(l - kt) ak
(4)
Thus UR?
= [B],2e-2kt(l- kt)'Uk2
(5)
and finally TIME, 5 X 10 Flgure 1. Simulated concentration vs. time and rate vs. time plots for k = 9.0s-'; the hypothetical pseudo-first-order reaction of eq 1: (0) (X) k = 9.5 s-': (A) k = 10.0 s-': (Y) k = 10.5 s-'; (0) k = 11.0 S-1.
error theory, Carr has developed equations for the relative precision of the measured concentration of the analyte under pseudo-firsborder conditions. He presented calculations which demonstrate the dependence of the relative uncertainty of the measured concentration as a function of the extent of reaction and concluded that initial rates provide optimum precision. Landis et al. (5) have taken a slightly different approach and have suggested that for absorption detection, minimum concentration error may be obtained by making rate measurements at times well removed from t = 0, namely, at t = l / k . In this paper we take a general approach and assume that at any point during the course of a first-order or pseudo-first-order reaction it is possible to estimate the rate of the reaction by instrumental and/or numerical methods. By the use of propagation of error theory, we will demonstrate that at t = l / k , a time characteristic of the first-order or pseudo-fiit-order reaction of interest, reaction rates are essentially independent of small fluctuations in parameters affecting the rate constant. As a specific example of such a parameter, we shall explore the effect of temperature on the results of fixed-time kinetic analyses through mathematical development and numerical simulation of experimental data. Finally, we present experimental data to support the results of this discussion and suggest guidelines for minimizing errors in fixed-time kinetic analyses based on the results of this treatment.
GENERAL CONSIDERATIONS For a first-order or pseudo-first-order reaction such as that shown in eq 1,
B
k@i)
products
the rate at any time t can be written as
-(z )
= R , = k(pi)[B]oe-kt
(2)
t
where k ( p i ) is the rate constant, which is dependent upon parameters, pi,such as temperature, pH, ionic strength, etc. The effects of changes in k(pi) are clearly illustrated in Figure 1 for a hypothetical reaction with k(pi) = 10 s-l. Plots of concentration of B vs. time for k(pJ values ranging from 0.9 k(pi) to 1.1 k ( p j ) and corresponding plots of rate disappearance of B are shown. It is particularly interesting to note that at t E 0.1 s the rate vs. time curves cross, suggesting the possibility that at this point in time rate measurements may be relatively independent of systematic variations in k(pi). In the sections which follow, we will explore this possibility through the use of propagation of error theory for both random
This equation suggests that the relative imprecision in the rate measured at time t is zero when t = l / k = T . This somewhat surprising result is consistent with the crossing of the rate curves shown in Figure 1. In a similar fashion we can show that for systematic errors in k
(7) and therefore the effects of such errors may be minimized or eliminated altogether if the rate measurement is carried out at t = T . The implications of eq 6 and 7 are surprisingly simple and potentially powerful, but it is important to recognize the assumptions inherent in this treatment. We have assumed that an estimate of Rt is obtainable at any time and, in particular, at t = T . It is clear that for some chemical systems this may not be possible. We have further assumed zero error in the specification of the measurement time, which is reasonable in view of the availability of modern highly accurate and precise timing devices, and have neglected imprecision resulting from instrumental factors (1-4). In any event, it is possible to estimate uk and Ak by application of propagation of error theory and from the dependence of the rate constant upon the relevant rate parameters as shown in eq 8 and 9. A
more detailed investigation of important parameters and the effects of these upon analytical results occupies the remainder of this paper.
THERMAL EFFECTS Temperature variation is one of the factors affecting the precision and accuracy obtainable in kinetic methods of analysis. Temperature fluctuations may arise from many different sources including mixing of nonisothermal reagents, heat generated by the mixing process, and the enthalpy of the reaction itself (6-10). Generally, investigators have attempted to minimize the effect of temperature changes on kinetics experiments by thermostating all reagents and reaction vessels (6, 7, 11, 12) or by applying numerical corrections to the collected data (13, 14). Carr (4) has shown that the relative uncertainity in the rate constant k( 7') resulting from random temperature fluctuations is given by
ANALYTICAL CHEMISTRY, VOL. 54, NO. 4, APRIL 1982
Flgure 2. Three-dimenslonal plot of eq 11; relative standard deviation of rate (aR/ R , ) as a function of relative time ( t / 7 )and the standard devlatlon 01 temperature (ulr):k = 10 s-' at 300 K E, = 20 kcal/mol.
where E, and R are the Arrhenius activation energy and the universal gas constant, respectively. By combining eq 10 and 6, we obtain
The form of eq 11is illustrated for the hypothetical reaction of eq 1, with an activation energy of 20 kcal/mol, in the three-dimensional plot of Figure 2. In this plot URR,/Rtis presented as time varies1 from 0 to 27 and QT varies from 0 to 1K at a nominal temperature of 300 K. The plot clearly shows the minimum in aR,/Rt at t = 7 and a rather steep increase to a maximum of 11%RSD a t t = 0 and t = 27 for U T = 1 K, a fact which indicates maximum sensitivity to random thermal noise at these times for the time range illustrated and even greater sensitivity a t t > 27. For systematic errors in temperature we evaluate Ak as follows:
Ak = ( $ , ) A T and from the Arrheniue equation
= Ae-Ee/RT
(13)
we obtain, assuming A is not a function of temperature
By combining eq 14 with eq 12 and dividing by eq 13, we have
Ak
-- -EAaT
RT2
757
CBI Simulated data comparlng analytical working curves generated from rates measured at t = 7 (R,) wlth curves generated from initial rates (R,,). Systematic temperature error increases linearly with time from 300 K - A T to 300 K AT during the course of the experiments: (0)A T = 0.1 K; (X) A T = 0.2 K; (A) AT = 0.3 K; (Y) AT = 0.4 K; (0) A T = 0.5 K. Flgure 3.
+
are directly proportional to Ea. In fact, the three-dimensional error surface for aR,/Rtvs. t vs. E, is identical in shape with the plot of Figure 2, and thus, for any given activation energy, minimum error is realized at t = 7.
IMPLICATIONS FOR KINETIC ANALYSIS One might well ask how the results of the previous sections could be used to improve the quality of concentration information obtained from reaction rate methods. The existence of a minimum in the error function at t = 7 suggests that the procedure for the generation of a working curve of rate vs. concentration of analyte can be modified to include a preliminary determination of 7 for the reaction of interest. Once T has been estimated, subsequent rates can be measured a t 7 and plotted as a function of the concentration of analyte. Such a procedure must be critically compared with the widely used practice of using initial rates as a measure of concentration. From the plots of Figures 1 and 2, it should be clear that initial rates in fact exhibit maximum sensitivity to changes in temperature and other rate parameters which occur during the course of a reaction or between successive experiments. Comparison of Rates Measured at t = T with Initial Rates. In order to compare the precision and accuracy of rates measured at t = 7 (R,) with rates measured at t = 0 (Ro),the rates may be evaluated directly as follows. The rate a t any time for the hypothetical reaction of eq 1is given by eq 2. For initial rates t = 0, and eq 2 becomes
which has the form of eq 10. If we finally substitute eq 15 into eq 7, we find that the relative rate error and at t = 7 = l/k(Tl), where Tl is the true mean temperature over the course of the experiment
A three-dimensional plot of eq 16 demonstrates the systematic error function in a manner directly analogous to the plot of Figure 2 except that AT can assume both positive and negative values. This, of course, results in both negative and positive values for the relative rate error. Clearly, ARt/R, = 0 for t = 7 or for Tl = Tzqbut it should also be noted that for any point in the region of the systematic error surface near the point of intersection of the t = 7 and Tl = T2lines, ARt/Rt is negligibly small. As the measurement time approaches t = 0 or when t 2 27, AR,/R, may become as large as several percent for relatively small temperature fluctuations. Inspection of eq 11 and 16 confirqs the qualitative expectation that accuracy and precision of rate measurements
R, =: k(T)[B],,e-k(T)/k(Td
(18)
Figure 3 shows simulated plots of Ro and R, vs. initial concentration of B when the temperature is systematically varied in a linear fashion during the course of the experiment from AT below Tl to AT above T1 for several values of AT as indicated in the figure caption. The chosen thermal gradients produce essentially no effect on the R, working curves while the Ro working curves exhibit a steadily increasing slope as the temperature interval increases to f0.5'. As a demonstration of the effects of indeterminate temperature errors on rates, normal random thermal noise about a mean value of T1 can be used to calculate k ( T ) . The effects on Ro and R , are shown in the simulated curves of Figure 4 for which the
758
ANALYTICAL CHEMISTRY, VOL. 54, NO. 4, APRIL 1982
different thermal noise levels used are indicated in the caption. The improvement in precision achieved by carrying out the measurement at t = T is apparent from a comparison of R, working curves (upper) and Ro working curves (lower). It should also be noted that the family of essentially straight lines of Figure 4 represents an upper limit on the R, vs. [B], curve, which corresponds to zero temperature fluctuation and measurement precisely at 7. Although we have confined our discussion to thermal effects, it should be clear from inspection of eq 11and 16 that similar resulta will be obtained when other variables that affect the rate constant are considered. Effects of Experimental Variables on Measured Concentrations. To this point we have considered only effects upon measured rates; however, the ultimate goal of this discussion is to demonstrate the effects of experimental variables on the accuracy and precision of measured concentrations. Here we adopt the use of the asterisk (*) to denote measured quantities, values derived from measured quantities, and any other experimental parameters operant during the determination of unknown concentrations in the final analytical step. This distinguishes these variables from those operant during calibrations, which bear no asterisk. We shall first consider the simplest imaginable experimental situation for kinetic analysis, i.e., calibration of a reaction rate method by carrying out a single experiment with a known initial concentration, [B]d, of analyte followed by a second experiment to determine an unknown initial concentration [B],. The rate of the reaction for the calibration experiment is given by
Rt = ke-kt[B]std
CBI Flgure 4. Simulated data comparing analytical working curves generated from rates measured at t = T (R,) with curves generated from initial rates (RO). Random temperature fluctuatlons: (0)cT = 0.1 K; (X) = 0.2 K; (A)UT = 0.3 K; (Y) (TT = 0.4 K; (0) UT = 0.5 K.
(19)
from which the calibration factor for subsequent experiments is
Therefore the concentration of an unknown solution calculated from measured quantities is obtained from
The standard deviation of the unknown concentration, assuming ut = 0, is evaluated from the expression
TIME, T/TFIU Flgure 5. Plot of relative standard devlation of measured concentration (LT[~I./[B] ") vs. time ( t h ) : (X) eq 27; (0)simulated data with random fluctuatlons in k resulting from UT = 0.5 K.
In general k* zi k and (Tk* z c k for experiments carried out under nearly identical conditions, and therefore '[Blo'
uk
- fi(1 - kt)k
(27) PI,* Using an analogous procedure for determinate errors we find that
Since
d [B]o* -dk*
Ak A[BlO* -- (1 - k*t)-Ak* - (1- kt)[BIo* k* k
[B]oe-k*t (1 - k*t) ke-kt
--
and
then a[B]o*2 =
[B]02e-2k*t (1 - k*t)2Q*2+ k2e-2kt
and, finally, we obtain
(z)2 ( = (1- k*t)2
[BIo*
$)2
+ (1 - kt)l(
2)
(26)
(28)
Equations 27 and 28 confirm the expectation based on eq 11 and 16 that minimum error in measured concentrations is achieved when rate measurements are carried out at t = T . This point is further illustrated by the simulated data of Figure 5. As in previous simulations, normally distributed temperature fluctuations generated according to the central limit theorem (15)were used to produce realistic random fluctuations in the rate constant of the hypothetical reaction. The plot presents theoretical and simulated values for the relative standard deviations of [Bl0* as functions of the measurement time. The simulation parameters are given in the figure caption. The agreement between the theoretical curve and the simulated points is excellent, and, as expected, minimum error is achieved at t = T . The extension of this somewhat simplified treatment for single point calibration to the widely used procedure of generating working curves of rate vs. concentration and per-
ANALYTICAL CHEMISTRY, VOL. 54, NO. 4, APRIL 1982
forming linear regression analysis on the collected data is straightforward. In thii procedure, measured concentrations are calculated from
Rt* - u [IB],* = -
The systematic error in [B],* is given by m10* Ak Ak* -P I , * - ( k - k*)At - (1- k t ) -k + (1 - k*t)- k*
where a is the intercept and b is the slope of the regression equation. By applying propagation of error techniques to eq 29 as before, we find
A[BIo* [BIo* and for t =
7
cz ( k - k*)At
Plo*
In most instances, such as when [B], is sufficiently removed from the lower range of the calibration curve, a
