Anal. Chem. 1991, 63,1303-1307 (24) &&to,T.; Kato, T. J . RaUMnel. Chmn. 1982, 68, 175. (25) Lutz, G. J. Trans. Am. Nud. Suc. 198S, 44, 27. (26) mtt,A.; Md)owdl, L. S.; Pegg, D. L. Trans. Am. NwI. Suc. 1988, 53, 171. (27) mtt,A.: Daw, H. S.: Fong, B. 8.; Jayawlckrsme, C. K.: McDoweW, L. S.; Pegg, D. L. J . Ra&n”. ”1. chem.A W s 1988, 124, 65. (28) olascock, M. D.; Tlan, W. 2.; Ehmann, W. D. J . Redbanal. Nucl. chem.1985, 92,379. (29) Sboube, W. B.. Jr.; Cunningham, W. C.; Lutz, 0. J. J . Radybenel. NWl. chem.Artlcks 1987, 112, 341. (30) Johansen, 0.; Stelnnes, E. Analyst 1978, 101. 455. (31) Overdjanclc, L.; Kwta, L.; DermelJ,M. J . Redbanal. Chem. 1980, 58, 359. (32) Bymne. A. R.; Demlj, M.; Kosta, L.; Tusek-Znidaric, M. 1cmcroahlm. Acta 1984, 1, 119. (33) -11, M.; SleRovec, 2.; Byme, A. R.; Stegnar, P.; StibllJ, V.; Rossbach, M. Fresenlus J . Anal. chem.1990, 338, 559. (34) . . Ebihara. M.: Salto. N.: Akalwa. H. J. R ” a l . Nw/. C b m . Lett. 1988, 108, 241. (35) Ran, K. W.; Kingston, H. M.; MacCrehan, W. A.; Koch, W. F. Anal. chem.1988, 60, 2024. (36) Ryan, D. E.; Stuart, D. C.; Chattopadhyay, A. Anel. CMm. Acta 1978. 100. .... 87. . (37) Ryan, D. E.; Chatt, A.; Holzbecher, J. Anal. CMn. Acta 1987, 200, 89.
1303
(38) Wuliems, D. R.; Hlelop, J. S. J . Rsdkenal. chsm. 1977, 39, 359. N.; dl casa, M. ” n . mumna~.LM. ion, (39) steua, R.: -a,
30, 65.
(40) Takagl, H.; Klmura, T.; Iwashlma, K.; Yamagata, N. Bun&/ Kapku 198S, 32, 513. (41) Heckman, M. H. J . Assoc. Off. Anal. Chem. 1979, 62, 1045. (42) Greenkng, R. R. Ana/. Chem. 1966, 58, 2511. (43) Gladney, E. S.; O’Malley, B. T.; Roelandts. I.; QlHs,T. E. NBS Spec. h b l . (U.S.) 1987, 260-111. (44) Special Note for Users of IAEA Mllk Powder A-11 and IAEA Animal Muscle H-4. IAEA Communication, 1989, Jan 23. (45) Parr, R. M. J . Redbana/. Nut/. chem.Altlcks 1988, 123, 259. (46) Iyengar, G. V.; Tanner, J. T.; Wolf, W. R.; Zelsler, R. Scl. TotalEnvl141). 1987, 61, 235. (47) Parr, R. M. Humen Del& Dktary Intakes of NuMtknelly I-nt Trace Elements as Maswed by N m b r 8nd Other Technhpes; IAEA: Vienna, 1984-89. (48) Rao, R. R.; Chatt. A. Unpublished data, Dalhousie Unlverstly, Hallfax, iaan (49) TiGI&, L.; F u m . V.; Wyttenbach, A. Bbl. Trace Element Res. 1990, 28-7, 623.
RECEIVED for review November 14,1990. Accepted April 8, 1991.
Error-Compensated Kinetic Determinations by Detecting the Intermediate Product in Successive Reactions, without Prior Knowledge of Reaction Constants Israel Schechter The Fritz Haber Research Center for Molecular Dynamics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
The paper dercrlbes an algorithm for the determination of substrate A and reaction coeffkientr k , and k , in the succ.rrlve reaction A& B A C, by tho klwttc detectton of B. The algorithm k studied by urlng dmuiated data with dttferent levels of suporhposed ndse and different rate constants, inltiai concentrattons, and data denrltles. The algorithm k evaluated malnly for determlnatbn of A but also for findhg tho reaction coclltants. Thk algorithm was found to be suitable for a wide range of concentrations and rate constant values, and In many cases a prior estimate of the order of magnltude of the parameters Is sufficient.
INTRODUCT10N The importance of kinetic approaches to analytical chemistry has been widely recognized (1,2). Many new efforts (3-8) have been encouraged by a recent review (2) evaluating and classifying this field of intereat. Recently, new techniques were developed for kinetic determinations of first- and general-order direct reactions (4.11). In this study we are interested in another common chemical system in analytical chemistry. The substrate, A, that has to be determined, reacts with a reactant to give B. This is a first-order or a pseudo-firstorder reaction when the reactant is in excess. The product B can be eaaily detected (spectroscopically,for instance), but it is not stable and decomposes to C. Our purpose is to evaluate an algorithm that determines A by the kinetic detection of B. In the kinetic process of this system, the concentration of A starts at ita initial value AO, and drops to zero; the concentration of B starts at zero,rises until it achieves a maximum 0003-2700/91/0383-1303$02.50/0
and then drops to zero again. In the simulated experiment, the detector’s signal, which is proportional to the concentration of B, is read at several intervals. The readings include a variable level of noise. The experiment is simulated by the following steps: (a) choosing a set of parameters. (b) calculating the exact signal as a function of time, in the given interval (regularly or randomly). (c) adding Gaussian noise to the data points at a given standard deviation. We evaluate the ability of the algorithm to reconstruct the initial parameters, as a function of the variables’ range, the noise level, the estimates of the parameters, range of data points, and other factors. Such methods have been used earlier for other systems (9,lO). Figure 1 shows some simulated plots, using the exact parameters of Table I, at standard deviations in the range studied in the following.
MATHEMATICAL DESCRIPTION The kinetics of a successive reaction are well-known (12). It consists of two steps: kl
A-B ha
B-C
Let Ao be the initial concentration of A, SO the initial signal (detecting B), S,the signal at time t, and a the sensitivity factor of the detector response. The signal as a function of time is given by
where r = kz/kl. In this function the unknowns are AO, r, and k l . The maximum signal (S-), and the time (t-) when it 0 1991 Amerkan Chemlcal Sockty
1304
ANALYTICAL CHEMISTRY, VOL. 63,NO. 13, JULY 1, 1991
Table I
variable on x axis figure no.
exact Darameters
variable on y axis
matrix size
A
r
kl
2a
no. of pointa
sd
71 X 71
1.0 0.5 0.003 1.25
2b
no. of pointa
sd
71 X 71
1.0 0.5
4a 4c
no. of points no. of points no. of points
At At At
5a
A
A0
51 X 51 X 51 X 51 X
51 51 51 51
1.0 0.5 1.0 0.5 1.0 0.5 0.5
5b
kl
k1°
51 X 51
1.0 0.5
5c
r
ro
51 X 51
1.0
A
sd sd sd
51 X 51 51 X 51 51 X 51
0.5 1.0 0.5 1.0
4b
kl
r
6
ranee " of variables constanta
At = 0-3000, d t = Atlnp 0.003 1.25 0.67 0.002 At = 0-3000, dt = random 0.003 1.25 0.67 0.002 to = 0, sd = 0.0001 0.003 1.25 0.67 0.002 to = 400, sd 0.0001 0.003 1.25 0.67 0.002 to = 1500, sd = 0.0001 0.003 0.67 0.002 np = 100, At = 0-3000, ad = 0.0001 1.25 0.67 np = 100, At = 0-3000, sd = 0.0001 0.003 1.25 0.004 np = 100, At = 0-3000, sd = 0.0001 0.003 1.25A 0.67 0.004 np = 100, At = 0-3000 1.25 0.67 1.25kl np = 100, At = 0-3000 0.003 1.25 1.05r 0.004 n D = 100. At = 0-3000
0.67
0.002
Y
X
10-220
0.01
10-220
0.01
10-210 10-210 10-210 1-5
200-1000 200-1000 200-1000 1-5
0.002-0.01 0.002-0.01 0-2
0-2
0.1-10 0.001-0.1 0-2
0.001 0.001 0.001
If no estimate of AO, k,, and r is available at all, then one can estimate S, and t , from the experimental data points and use eq 3 in order to find the approximate value of r. Then, 12, can be approximated by eq 2 and SO by eq 1. From now on we shall assume that we have estimations of the parametera that do not deviate too far from the true values. In the suggested algorithm eq 1 is fitted to the data by using
0.5
0.4
s
estimated Darameters Ao fl klo
0.3
a modified Levenberg-Marquardt method to solve the nonlinear least-squares problem (13,14). This process gives the desired Ao and also k1 and r. The algorithm requires the Jacobian of S,. it can be dculated numerically, but it is better to supply it analytically:
0.2 0.1
(4)
o'6 0.5
5
0.4
s
0.3 0.2 0.1
EXPERIMENTAL SECTION The data were processed on a supermicrowork station (Silicop Graphics 4D/240), runningunder the UNM operating system. All software was written in FORTRANII. The efficiency of the algorithm depends on many variables and should be presented as a function of all of them. Since a function of two variables can be presented as a three-dimensional plot, the results were accumulatedin matrices and presented in stereoscopic view. A single point in such a matrix represents the error in the determination of A at a given set of input data. Converging to the best possible result on the work station took ca. 0.05 s for a typical input. This implies that it took a few minutes to calculate most of the plots that represent a 51 X 51 matrix. The convergence time is influenced by the initial estimatea. Its absolute value depends also on the computer, the code, and the parameters of the convergence tests. In our case, the longer convergence time was under 0.5 s.
3 O O
400
1200
800
t
1600
2000
Isec.
Flguro 1. Simulated plots of the signal as a function of tlme at two n d s e levels used in the calculations. A o = 1, k , = 0.003, k , = 0.0015.
occurs are also well-known as a function of the parameters (12).
Substitution of , S
in eq 1 gives
and by substitution of t,,
s, =
we get
RESULTS AND DISCUSSION Table I shows the input data used in the calculation of the following plots. First we investigate the influence of the number of data points (namely, the number of points of detection of B, at several times after the reaction began) and the standard deviation of the readings. Then we investigate the influence of the time interval and time range of the readings and find the optimal conditions. We look for the effect of the initial estimates of the parameters, and finally, we study the range of the parameters that enables the use of this algorithm. The following sections refer to the error in
ANALYTICAL CHEMISTRY, VOL. 63, NO. 13, JULY 1, 1991
1305
20 -
220
115
10 0.1 .
Number of Data Points
b
0 0
500
lo00
1500
t
2000
2500
3000
1 sec.
Figure 3. B a s a function of time. Ao = 1, k , = 0.003, r = 0.5. The three time intervals are shown: (a) the buildup of B; (b) the most informative time interval; (c) the tail of the reaction.
n
E 8 t
W
0 220
115
10
Number of Data Points Figure 2. Error in the determination of A as a function of the number of readings of B and the standard deviation of the readings: (a, top) readings at regular time intervals; (b, bottom) readings at random time intervals.
A, but we found that the same conclusions are valid also for the error in determination of the rate constants. Effect of Signal-Noise and Number of Data Points. Figure 2a shows the error in the determination of the substrate A as a function of the number of data points (readings of B) and the standard deviation of the B detection. In this experiment the readings were taken at regular time intervals. The relatively low errors observed are consistent with the low imprecision used in these simulations. The errors are low even at the lowest number of points. However, on the same plot are shown the errors at higher standard deviations, up to 0.01. Even at the highest noise of 0.01, the error is limited to -2%, provided more than 20 points are available. This figure has been recalculated by using the same parameters but by randomizing the time in which the readings were taken (within the same range). The results are shown in Figure 2b. Randomizing the time improves the detection, mainly where a small number of points is available. The modulation of the error as a function of the number of data points, especially at a high noise level, is not significant, since it varies with the parameters. Thus one should get the impression of the average error at a certain number of points and not read the specific one. The excellent results obtained for realistic noise levels demonstrated the efficiency of this algorithm and encouraged us to continue this study. In the following, the effects of other factors are considered. Due to the low effect of the noise, as shown in Figure 2, and in order to isolate the factors, the following calculations were done at a low noise level (lo4). Some of the calculations using several parameters have been repeated with a higher noise level, and the validity of the conclusions has been maintained. Effect of Time Range. The time range in this system is not an absolute quantity. It should be related to the characteristic time scale of the reaction, which can be estimated by the recpirocals of the reaction constants. In Figure 3, B is plotted as a function of time for Ao = 1, kl = 0.003, and k2 = 0.0015. There are three variables that should be investigated The first is the time tothat passes from the beginning of the reaction until the first measurement is made. The second, At, is the time interval between the first and the last
measurement. The third, dt, is the time interval between sequential measurements. We have shown earlier that randomizing dt improves the results. Here we did not randomize it so the errors found in this section are an upper limit to the realistic errors. In the previous plots the measurements were taken in the time range 0-3000 s. Here we investigate three ranges of 1000-s duration. In the first range (a), the measurements start at to = 0 and continue up to t = lo00 s. From the corresponding reaction time scale we realize that this range represents the buildup of B and the beginning of its decay. The second range (b) starts at to = 400 and continues up to t = 1400 s. It represents high signals and the main part of the reaction life time. The third range (c) represents the tail of the reaction. Each range has been studied separately as a function of At (time interval between the first and the last measurement) and the number of points. At a given At the number of points (np) represents a definite dt, as At/np = dt. The results are shown in Figure 4. The main conclusion is that if At is long enough (e.g., longer than 500 s) the recovery of the data is very good (over 99.5%). Even better results were obtained at higher At values. There is no significant dependency on dt. Measurements in the b range give the best resultq the erros in the range c are slightly higher. Measuring at time intervals of range a, at short At may give poor results (5-6% error). Attempts to measure at the very end of the reaction (starting at t > 2500 s) or measuring within a short range at the beginning of the reaction (e.g., t = 0-50 s) causes numerical difficulties in the convergence process. The reason is that at these ranges there is insufficient significant information to reconstruct the function. The conclusions derived in this section should be considered with respect to the time scale of the specific reaction and not to the absolute times mentioned above. For another set of parameters the characteristic time scales should be found, and the conclusions discussed above should be shifted to the new times. Effect of the Initial Estimations. In this section we ask how poor the initial estimate of the various parameters may be and how these affect the accuracy of the determination. Figure 5a shows the error versus A and Ao (the initial estimation of A). It can be seen that for the whole range the error is essentiallyzero. There is a slight rise in the error for smaller values of A, but there is no significant dependence on the difference between A and AO, as far as the algorithm converges. This feature has also been checked and proved to be valid at other ranges of A and AO. In this plot the ratio A o / A changes from 0.2 to 5. The algorithm supports some higher values, and then convergence problems arise. If the initial estimate is too far from the true value, the algorithm will not converge. If the estimates of r and kl are better, more extreme values
1306
ANALYTICAL CHEMISTRY, VOL. 63, NO. 13, JULY 1, 1991 to = 0 sec.
6 -
i
\
Ot 1
5
,'
1
A
Number of Data Points
to = 400 sec. 50-
