Kinetic Studies of Cascade Reactions in High-Throughput Systems

Article abstract of DOI:10.1021/ac034719b

The application of robotic systems to the study of complex reaction kinetics is considered, using the cascade reaction A → B → C as a working example. Practical problems in calculating the rate constants k ₁ and k₂ for the reactions A → B and B → C from concentration measurements of C_A, C_B, or C_C are discussed in the light of the symmetry and invertability of the rate equations. A D-optimal analysis is used to determine the points in time and the species that will give the best (i.e., most accurate) results. When exact data are used, the most robust solution results from measuring the pair of concentrations (C_A, C_C). The system's information function is computed using numeric methods. This function is then used to estimate the amount of information obtainable from a given cascade reaction at any given time. The theoretical findings are compared with experimental results from a set of two-stage cascade experiments monitored using UV-visible spectroscopy. Finally, the pros and cons of using a single reaction sample to estimate both k₁ and k₂ are discussed.

Full text of DOI:10.1021/ac034719b

Anal. Chem. 2003, 75, 6701-6707
Kinetic Studies of Cascade Reactions in
High-Throughput Systems
David Iron,† Hans F. M. Boelens,‡ Johan A. Westerhuis,‡ and Gadi Rothenberg*,‡
Chemical Engineering Department and KdV Institute, University of Amsterdam, Nieuwe Achtergracht 166,
1018 WV Amsterdam, The Netherlands
The determination of reaction rate constants is a primary tool
The application of robotic systems to the study of complex
reaction kinetics is considered, using the cascade reaction
A f B f C as a working example. P ractical problems in
calculating the rate constants k1 and k2 for the reactions
A f B and B f C from concentration measurements of
CA, CB, or CC are discussed in the light of the symmetry
and invertability of the rate equations. A D-optimal
analysis is used to determine the points in time and the
species that will give the best (i.e., most accurate) results.
When exact data are used, the most robust solution
results from measuring the pair of concentrations (CA,
CC). The system’s in for m a tion fu n ction is computed
using numeric methods. This function is then used to
estimate the amount of information obtainable from a
given cascade reaction at any given time. The theoretical
findings are compared with experimental results from a
set of two-stage cascade experiments monitored using
UV-visible spectroscopy. Finally, the pros and cons of
using a single reaction sample to estimate both k1 and k2
are discussed.
in mechanistic studies. Kinetic analysis is more time-consuming
and labor-intensive than simple yes/ no activity tests, because
many samples have to be taken from every reaction to establish
each kinetic profile. When working with high-throughput systems,
such as reactor arrays, this multiple sampling and analysis often
results in a bottleneck.7 One way to prevent this bottleneck is to
optimize the sampling times. Previously, we showed, using a
mathematical model of the reaction rate law, that one can estimate
the amount of future information that can be gained from sampling
each reactor in an array of first-order reactions.8,9 Here, we extend
the application of the information function to the more complex
k
k
1
2
system of cascade reactions of the type A f B f C and discuss
the implications of simultaneous determination of two reaction
rate constants in reaction arrays. A theoretical D-optimal analysis
is compared with the results from a set of cascade experiments.
RESULTS AND DISCUSSION
General Considerations. To find any reaction rate constants
requires measurements of concentrations that will necessarily
contain errors. The effect these errors have on the calculation of
a rate constant depends strongly on the times the concentrations
are sampled and on the value of the reaction rate constants
themselves. The sampling of first-order reactions was considered
elsewhere.10-12 However, most reactions are not governed by first-
order kinetics. In this paper, we will discuss the complications
that arise as the result of a two-stage cascade reaction given by
eq 1,
The last two decades have provided chemists with a variety of
new experimental tools. Among these, the realization of the
concepts of combinatorial synthesis and parallel experimentation
are perhaps the most significant. These methods have revolution-
ized the pharmaceutical industry but, despite a promising start,1-4
have not yet overwhelmed researchers in other fields.5,6 This is
due more to a psychological barrier than to equipment and
infrastructure costs: Working with high-throughput setups neces-
sitates a change in one’s mode of thinking, because the value of
the basic research unit, the scientific experiment, is changed. In
high-throughput systems, “cheap experiments” can be performed
to obtain quickly large amounts of rough-quality data that are then
analyzed to point to the next generation of experiments. This
differs from the conventional “scientific” mode of thinking, where
every experiment must be thoroughly evaluated (reflecting
perhaps the time and labor costs of the work).
k1
k2
A
98 B
98 C
(1)
The introduction of the intermediate B results in considerably
more complex reaction dynamics.13 If the initial concentration of
(7) Lutz, M. W.; Menius, J. A.; Choi, T. D.; Laskody, R. G.; Domanico, P. L.;
Goetz, A. S.; Saussy, D. L. Drug Discovery Today 1 9 9 6 , 1, 277-286.
(8) Boelens, H. F. M.; Iron, D.; Westerhuis, J. A.; Rothenberg, G. Chem. Eur.
J. 2 0 0 3 , 9, 3876-3881.
* Corresponding author. Fax: +31 20 525 5604. E-mail: gadi@science.uva.nl.
† KdV Institute.
(9) This function basically describes the relationship between the errors in the
measured concentrations δ(a) (where
a is a vector that contains the
‡ Chemical Engineering Department.
measured concentrations at the various time points) and the time vector t.
See: Rothenberg, G.; Boelens, H. F. M.; Iron, D.; Westerhuis, J. A. Chim.
Oggi 2 0 0 3 , 21, 80-83.
(1) Jandeleit, B.; Schaefer, D. J.; Powers, T. S.; Turner, H. W.; Weinberg, W.
H. Angew. Chem., Int. Ed. Engl. 1 9 9 9 , 38, 2495-2532.
(2) Lavastre, O. Actual. Chim. 2 0 0 0 , 42-45.
(3) Senkan, S. Angew. Chem., Int. Ed. 2 0 0 1 , 40, 312-329.
(4) Mirodatos, C. Actual. Chim. 2 0 0 0 , 35-39.
(10) Carr, P. W. Anal. Chem. 1 9 7 8 , 50, 1602-1607.
(11) Holler, F. J.; Calhoun, R. K.; McClanahan, S. F. Anal. Chem. 1 9 8 2 , 54,
755-761.
(5) Hoffmann, R. Angew. Chem., Int. Ed. 2 0 0 1 , 40, 3337.
(12) Rothenberg, G.; Boelens, H. F. M.; Iron, D.; Westerhuis, J. A. Catal. Today
(6) Schlo¨ gl, R. Angew. Chem., Int. Ed. Engl. 1 9 9 8 , 37, 2333-2336.
2 0 0 3 , 81, 359-367.
10.1021/ac034719b CCC: $25.00 © 2003 American Chemical Society
Published on Web 10/24/2003
Analytical Chemistry, Vol. 75, No. 23, December 1, 2003 6701

there is a single rate constant k and the reaction profile is simply
given by C(t) ) C0e-kt, where C0 is the concentration at t ) 0.
For such a system, a single measurement is sufficient to determine
k if C0 is known. There is also an explicit expression for k (eq 3)
in terms of the initial concentration C0 and the concentration C1
that is measured at time t1
k ) ln(C0/ C1)/ t1
(3)
For a two-stage reaction, this is not the case. There are two
reaction rate constants, and a minimum of two measurements is
needed to have enough information to determine both of them.
Even when we have sufficient information to determine k1 and k2,
we cannot directly invert eq 2. This means that we have no explicit
expression, such as eq 3, for the rate constants in terms of known
constants and measured data. Instead we must use a numerical
method, such as a generalized Newton root finder, to find
approximate values of k1 and k2 that fit the experimental data.14
We must also choose which concentrations to sample. In the case
of a single first-order reaction, there is only one reaction and we
just need to find an optimal time to sample it. Now we have three
concentrations; this leads to many possible choices. It does not
make sense to solely use data from CA, since this gives us no
information about k2. However, we may attempt to find k1 and k2
solely from measuring CB or CC. We may also measure any two
of the three concentrations at either the same or at different times.
However, as eq 2 is highly nonlinear, the choice of which
concentrations to measure can make it very difficult or even
impossible to make an accurate approximation of the rate
constants.
Let us now consider the difficulties associated with some of
the choices of concentration measurements. The clearest problem
results when we choose to measure the concentration CC at two
different times. Equation 2c is invariant under the transform k1
S k2. This symmetry means that a cascade reaction with k1 ) a
and k2 ) b will have the same CC profile as a reation with k1 ) b
and k2 ) a, for any a and b. In other words, no matter how many
measurements you take, you will never be able to distinguish k1
from k2 if you use only information from CC. Another major
consideration when using information from just the concentration
of C is that it is possible that profiles generated using different
reaction rates will be very similar (see Figure 2).
k1
Figure 1. Concentration profiles for the cascade reaction A f B
k2
f C, with k1 ) 0.2 and k2 ) 0.02.
A is given by A0 and the initial concentration of B and C are both
0, the reaction profiles are given by eqs 2a-2c.
CA(t) ) A0e-k t
(2a)
(2b)
1
k1
(e-k t - e-k t
)
1
2
CB(t) ) A
0k2 - k1
A0
1
2
CC(t) )
(-k2e-k t + k1e-k t) + A0
(2c)
k2 - k1
Here CA(t), CB(t), and CC(t) represent the concentrations, at
time t, of A, B, and C, respectively (see Figure 1). The value of
A0 will have no importance in the remaining analysis, so from
here on, we will scale the concentration values such that A0 ) 1.
The reaction is governed by two rate constants, so it will
require a minimum of two measurements to determine k1 and k2.
However, what should be measured? Is it possible to determine
the rate constants by measuring the concentration of the same
compound at two different times? Or should one measure the
concentration of two different compounds at the same time or at
different times? The following analysis assumes that all of the
compounds can be measured with the same accuracy (in practice,
the accuracy of the measurements may vary for each compound,
and this must also be taken into account).
The reaction profile for CB does not possess the symmetry k1
S k2, so the choice to measure CB at two distinct times may be
reasonable. However, again two measurements will not give
unique values for the rate constants. Figure 3 shows two different
profiles for CB. If we were to take measurements at the points
where these curves intersect, we would have at least two different
solutions for k1 and k2.
These difficulties are generic in the sense that they exist for
any values of k1, k2, t1, and t2. Consider a two-stage reaction with
concentrations of B ) CB1 and CB2 at times t1 and t2, respectively.
If we make an initial guess of the rate constants, k1 and k2 as k10
and k20, we obtain
In this cascade system, even though there is sufficient
information to determine the rate constants, no explicit expression
for k1 and k2 exists. This is true even if we have exact data, because
we cannot directly invert eq 2. Instead, we must use a numerical
solver to find the rate constants. The timing of the measurements
must also be carefully considered. If concentration measurements
are taken at poorly chosen times, small errors in the measure-
ments can be magnified when the rate constants are being
estimated. All of these issues will be discussed in the next section.
Determining Rate Constants from Exact Data. We will now
assume that we have exact concentration data and consider the
problems associated with finding the rate constants, k1 and k2,
that correspond to these data. For a single first-order reaction,
k1 ) k10 + δk1
k2 ) k20 + δk2
(4a)
(4b)
(13) Moore, W. J. Physical Chemistry, 4th ed.; Longman: London, 1976; pp 345-
347.
6702 Analytical Chemistry, Vol. 75, No. 23, December 1, 2003

all values of k1, k2, t1, and t2. This means that the Jacobian is not
invertible and we cannot solve for δk1,2. In general, nonlinear
systems with a Jacobian that cannot be inverted, cannot be
inverted either. Even if we had very good estimates for the rate
constants, any numerical solver will need to invert the Jacobian.
As the Jacobian is not invertible, this will make finding an accurate
estimate impossible. Thus, measuring CB at two different times
is not the best choice.
Measuring CB and CC is, at least in theory, a better option. As
long as we do not choose to sample both concentrations at the
same time (i.e., t1 ) t2) (Note: this means that we cannot use the
appealing choice of taking our measurements at the same time
and thus save a measurement per rate constant calculation.), the
Jacobian will be invertible and the implicit function theorem
guarantees that we will have unique values of k1 and k2 for any
pair of measurements of CB and CC. So at least in theory, this is
an acceptable option. In practice, however, reactions with very
different rate constants can have similar concentration profiles
(see Figure 4).
Figure 2. Two CC profiles with rate constants k1 ) 0.01, k2 ) 0.5
(broken line) and k1 ) 0.01, k2 ) 10 (solid line). Even though the
values of k2 differ by a factor of 20, the curves are almost
indistinguishable.
This problem of instability is solved if we choose the concen-
tration of A as one of the measurements. If CA is measured, then
we immediately have a value for k1 from eq 3. The second
measurement may then be used to obtain the value of k2. Since
we are only solving for one variable now, we may use a scalar
version of Newton’s root finding method to obtain an approximate
value of k2. The scalar version of Newton’s method is much more
robust then the vector version. If we use CA and CB then, as well
as increasing the robustness of the root finder, we also make the
problem of finding k2 well defined by ensuring a unique solution.
Once we have k1 from the measurement of CA, eq 2b becomes a
monotonic function of k2. Thus, by the inverse function theorem,
invertible; given any CB(t2), we can then find a unique value of k2.
Even though multiple solutions may exist if we measure CA and
CC, this is not as much of a problem as it was in the case where
CA was not sampled. In this case, the problem is not generic. There
are only isolated time points where we cannot invert the system.
Note that in this case it is possible to determine the rate constants
uniquely with measurements taken at a single point in time.
To summarize: Considering only invertability, the best options
are to measure the concentrations of A and either B or C.
Measuring the concentrations of B and C is possible, but less
favorable. Determining the rate constants from measurements of
a single compound is not possible. The choice to measure CA is
usually a logical choice for experimental reasons as well. As A is
the starting material, pure A is usually readily available, and it is
much easier to calibrate the equipment for accurate measure-
ments. The chemist is often faced with a problem trying to
calibrate for the intermediate B. Thus, for the general situation,
the best results will be obtained when the concentrations of A
and C are measured.
Figure 3. Broken line curve representing the concentration profile
for B with k1 ) 0.2 and k2 ) 0.018. The solid line curve represents
the concentration for B with k1 ) 0.1 and k2 ) 0.02. Sampling
compound B at points t1 and t2 will render these two profiles
indistinguishable.
where δk1,2 is the error in the guess of k1,2. Linearizing eq 2 about
this guess gives us eq 5,
δCB1
δCB2
δk1
δk2
) J
(5)
(
)
(
)
Here δCB1,B2 are the errors in the concentrations due to δk1,2 and
J is the Jacobian matrix,
∂CB
∂CB
0
0
0
0
(k1 ,k2 ,t1)
(k1 ,k2 ,t1)
∂k1
∂k2
J )
(6)
Minimizing the Effects of the Errors in Concentration
Measurements. We will now consider how errors in the
concentration measurements affect the calculation of k1 and k2.
For now, we will assume that, given two appropriate concentration
measurements, the exact corresponding values of k1 and k2 can
be found. The measurement errors will be composed of systemic
errors, due to faults in the equipment or the calibration model,
and random errors. We will assume the systemic errors are small
∂CB
∂CB
0
0
0
0
(
)
(k1 ,k2 ,t2)
(k1 ,k2 ,t2)
∂k1
∂k2
Substituting eq 2 into J, we find that the determinant of J is 0 for
(14) Press: W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T. Numerical
Recipes in C: The Art of Scientific Computing, 2nd ed.; Cambridge University
Press: Cambridge, U.K., 1993.
Analytical Chemistry, Vol. 75, No. 23, December 1, 2003 6703

Figure 4. Reaction profiles for CB (left) and CC (right). The broken line curves refer to a reaction with rate constants k1 ) 0.5 and k2 ) 0.1,
while the solid line curves represent a reaction with rate constants k1 ) 5.0 and k2 ) 0.1. If measurements are taken after t ) 10 min, then any
small experimental errors will render these profiles indistinguishable.
Here Ji is the Jacobian matrix defined in eq 10:
compared to the random errors and thus may be ignored. The
time at which the measurements are made is crucial. If the
measurements are taken close to the start of the reaction or near
its completion, a small measurement error may result in a large
error in the calculated value of k. Here we will use a D-optimal
approach to find the best times to measure the concentrations.
The criterion for determining D-optimality is to find sampling
times that minimize Var(k1) × Var(k2). Var is the variance that
corresponds to the error in the absence of systemic effects.
We will assume that we are monitoring a cascade reaction such
as in eq 1. We may choose any of the concentration pairs
discussed above, but to simplify the notation, we will use CA and
CB. Each measurement contains errors, so we will write the ith
set of measurements as in eq 7, where δΑi,Βi represent the errors
in the ith set of measurements. Although these measurements
have the same index i, we do not necessarily intend this to mean
that they are taken at the same time, but to mean that this pair of
measurements is used to determine one set of k1,2 values.
∂C
∂CΑ
A(k1,k2,ti1)
(k1,k2,ti1)
∂k1
∂C
∂k2
Ji )
(10)
∂CB
B(k1,k2,ti2)
(k1,k2,ti2)
(
)
∂k1
∂k2
The errors in the concentration measurements are amplified
by the inverse of the Jacobian matrices. Assuming that the
magnitudes of the concentration errors in eq 9 are of the same
order, the D-optimal sampling times are determined by minimizing
the amplification of the errors by the inverse Jacobian or by
maximizing the information function f ) |Det(JTJ)|.9 Larger values
of f correspond to measurements that yield more accurate rate
constant estimates.
We will now consider the D-optimal times for taking the
minimum number of samples. There are five different possible
minimal sample types. Instead of listing all possible formulas, we
will label the concentration sampled at time ti as Ci and the
concentration sampled at tj as Cj. The pair (Ci,Cj) may be one of
(CA,CB), (CA,CC), or (CB,CC) values. With this notation, the function
f is given by eq 11:
CA + δΑi,CB + δBi
(7)
We denote as ti1 and ti2 the sampling times for the ith
measurements of CA and CB, respectively. The measurement
errors will lead to errors in the calculated values of k1,2. We write
the ith set of calculated k values as,
2
2
2
2
∂Ci ∂Cj
∂Ci ∂Cj
∂Ci ∂Cj ∂Ci ∂Cj
∂k1 ∂k2 ∂k2 ∂k1
f ) |
+
- 2
|
(11)
( ) ( ) ( ) ( )
∂k1 ∂k2
∂k2 ∂k1
k1 + δk1i
k2 + δk2i
(8a)
(8b)
The D-optimal sampling times are at the global maximum of
f. For any given values of k1 and k2, we may find this maximum
numerically. Table 1 gives the D-optimal sampling times for a two-
stage reaction for all of the possible sampling combinations. In
both cases where CA is sampled, the results are identical. This is
not a coincidence, as the function f will always be identical for
these two cases (eq 12), and the minimum error in the calculation
of k1 by measuring CA will be at t ) 1/ k1.10 It is not possible to
predict f for a given k1,2 pair a priori, because eq 12 is highly
nonlinear (cf. entries 1-3 with entries 4-6). This means that f
must always be computed (the computation itself is not costly).
where δk1i and δk2i represent the errors, in the calculated values
of k1 and k2, respectively, from the ith set of measurements. Since
we expect the relative error to be small, we may substitute all of
the concentration measurements and calculated rate constants into
eq 2, and then linearize to obtain eq 9.
δΑi
δΒi
δk1i
δk2i
) Ji
(9)
(
)
(
)
6704 Analytical Chemistry, Vol. 75, No. 23, December 1, 2003

Scheme 1
Table 1. Sampling Times That Afford the Maximum f
Values Depending on Compounds Sampleda
concns
sampled
entry
t1/ min
t2/ min
f
1b
2b
3
CA and CB
CA and CC
4.21
4.21
3.89
10.00
10.00
8.85
9.58
9.58
9.58
43.19
43.19
42.74
12.56
12.56
14.80
9.58
475.6
475.6
420.4
14.0
14.0
17.11
22.4
CB and CC
4b
5b
6
CA and CB
CA and CC
CB and CC
7c
8c
9c,d
CA and CB
CA and CC
CB and CC
9.58
9.58
22.4
22.4
a
The rate constants used for these calculations are k1 ) 0.237 and
k2 ) 0.026 (entries 1-3 and 7-9) and k1 ) 0.1 and k2 ) 0.2 (entries
4-6). The first two values match the values of experimental data that
is used to test the validity of the theory, and the second pair is used
to demonstrate the nonlinearity of eq 12. b The first two choices result
in the same values since the function f is the same for these cases.
c The last three entries are the same because when t1 ) t2 the
expressions simplify to the same result in all cases. d Although the
value of f can be calculated in this case, it is difficult to find the rate
constants from data pertaining to CB and CC when t1 approaches t2.
the covariance matrix obtained from the experimental results
(bottom). In both cases, the red areas pertain to sampling times
that afford low error values. As the information function differs
from the covariance matrix (the latter contains an additional
source of variance, in the form of the vector {δA1, δB1.. δAn, δBn},
see eq 9), an outright quantitative comparison cannot be made.
However, much can be inferred from a qualitative comparison of
the two sets of results: There is a clear similarity between the
columns of graphs. This means, essentially, that f values can be
used to determine the best points in time to sample this cascade
reaction (this is advantageous, as f is obtained by simple calcula-
tions, rather than from experiments). Table 2 gives the sampling
times that correspond to the minimum in the covariance. The
model predictions for the pair AB are much better than those for
AC or BC, as can also be seen by comparing the optimal sampling
times in Tables 1 and 2. The reason for this discrepancy may be
that the experimental measurements of CC have the highest noise
in this case (in the theoretical studies, we assumed that all three
compounds were sampled with the same accuracy). The strong
change in the minimum covariance sampling times could also be
due to a flatness of the error surface minima (cf. the rather flat
minimum of the error curve in the case of the single-step reaction
A f B, with the same data8).
Application to High-Throughput Experimentation. In many
high-throughput systems used for kinetic studies, for example,
in catalysis, an array of reactors is interfaced to one (or two)
analysis devices, such as a GC or HPLC. Such setups reflect the
basic chemical requirements (i.e., different chemical reactions are
carried out under different conditions) and the financial limitations
(10-mL-scale autoclave reactors can cost as low as $10, but the
price of an HPLC analyzer can be in the order of $50.000). In
such cases, and especially when using robotic apparatus, the
traditional practice is inverted: the analyzer time becomes the
scarce commodity, but it is relatively inexpensive to discard a
reaction and start a new one in its place.17
Entries 7-9 show the results for the special case when only one
sample is taken, at t1 ) t2. This case is particularly interesting in
high-throughput applications (vide infra).
f ) (t1 exp(-k1t1)2((exp(-k1t1) - k1t2 exp(k2t2))/ (k2 -
k1) - (k1 exp(-k1t2) - k2 exp(k1t2))/ (k2 - k1)2)2 (12)
An Experimental Example. The above mathematical model
was compared with experimental results from a two-stage cascade
reaction. The test reaction should have well-defined cascade
kinetics, and sufficient sample points must be measured to enable
a good statistical analysis. We used the carbon-sulfur coupling
of 3-chlorophenylhydrazonopropane dinitrile A with â-mercapto-
ethanol to give the adduct B that subsequently underwent
unimolecular decomposition to 3-chlorophenylhydrazonocynaoac-
etamide C and ethylene sulfide (Scheme 1).15,16 Pseudo-first-order
conditions were realized for the step A f B by adding an excess
of â-mercaptoethanol.
The reaction was followed using UV-visible spectroscopy. A
total of 32 repetitions of this experiment were performed, the first
of which was used to estimate the spectra of A, B, and C. A total
of 271 UV-visible spectra were recorded for each experiment (one
spectrum every 10 s), and using these spectra, a set of 31
concentration profiles was obtained for A, B, and C. The “true
values” (i.e., best estimates) for k1 and k2 obtained by averaging
all 271 sample points over the 31 profiles. The values were 0.237
and 0.026 min-1, respectively, with standard deviations of 8.5 ×
10-3 and 7.8 × 10-4 min-1. These values were then compared with
the respective k1 and k2 values obtained by sampling the pairs
AB, AC, and BC at various points in time. In each case, two
sample times t1 and t2 were chosen, and the resulting accuracy in
k1 and k2 was calculated.
Figure 5 shows the comparison between the error amplification
calculated by the theoretical model (top) and the determinant of
(17) Another option is to analyze several reactions in parallel, as was recently
demonstrated using gas-phase IR image analysis; see: Hendershot, R. J.;
Fanson, P. T.; Snively, C. M.; Lauterbach, J. A. Angew. Chem., Int. Ed. 20 03 ,
42, 1152-1155.
(15) Bijlsma, S.; Louwerse, D. J.; Smilde, A. K. J. Chemom. 1 9 9 9 , 13, 311-329.
(16) Bijlsma, S.; Boelens, H. F. M.; Hoefsloot, H. C. J.; Smilde, A. K. Anal. Chim.
Acta 2 0 0 0 , 419, 197-207.
Analytical Chemistry, Vol. 75, No. 23, December 1, 2003 6705

Figure 5. Qualitative comparison of f values (mathematical model, top) with the determinant of the covariance matrix for the experimental
results (bottom) when sampling different reactant/product pairs at different times. The red areas pertain to sampling times that afford low error
values. In all cases, and especially when
[Cov(t1,t2)].
A and B are sampled, a direct relationship is observed between f and det-
already passed, two possibilities exist to improve the estimation
of the reaction rate constants. The first is to let the reaction run
and analyze the concentrations at suboptimal sampling times (this
only leads to a suboptimal improvement of the rate constant
estimates). The second possibility is to discard the old reaction
and start the same reaction again at the identical conditions. Then,
it is possible to measure at the optimal sampling time and have
an optimal improvement of the reaction rate estimates. The latter
option is the better alternative as far as the costly sampling time
is concerned. One assumes that the automated robotic system
can reproduce the reaction conditions with high precision.
Finally, it is worthwhile to consider obtaining the so-called
“initial guess estimates” for k1 and k2. If these guesses are good,
it will be easy to reach the optimal sampling times, and vice versa.
However, the methods for obtaining these initial guesses and for
performing the first set of measurements will depend strongly
on the experimental apparatus used and on the amount of
chemical knowledge available. Different approaches should be
used, for example, for parallel reactor arrays and for fast sequential
systems. As we showed earlier for first-order reactions,12 iterative
algorithms may be useful. This problem will be the subject of
future study in our laboratory.
Table 2. Sampling Times Having Lowest Variance
(1-3) and Other Sampling Times
concns
sampled
Det(Cov(k1,k2))
entry
t1/ min
t2/ min
× 10-11
1
2
3
4
5
6
CA and CB
CA and CC
4
9
4
13
13
9
44
24
31
13
13
9
1.26
6.01
4.54
47.8
7.7
CB and CC
CA and CB
CA and CC
CB and CC
57.1
Conventional analysis practices (i.e., following each reaction
for a fixed period of time with periodic sampling) will result in
many cases in sampling at suboptimal times. In practice, this
would mean that the robots would be generating garbage data
and may also miss out on sampling other reactions in the array
at their optimal times. In contrast, if one uses the concept of the
information function, a near-optimal analysis protocol can be
reached.
First, all reactors are sampled twice according to initial guesses
for the rate constants generated by the system. Then, an array of
rate constant estimates is calculated based on these samples, and
from these, the corresponding f values are calculated for each
reaction. From these the optimal sampling time can be obtained.
If according to this information vector, the optimum sampling time
CONCLUSIONS
By combining simple kinetic models and numeric analysis
methods it is possible to formulate for cascade reactions an
6706 Analytical Chemistry, Vol. 75, No. 23, December 1, 2003

information function that can predict the usefulness of sampling
a given reaction at a given time. This approach can be used to
minimize the experimental effort involved in kinetic studies and
calibration experiments. The analysis process has three steps:
first, you must invert the rate equations to find k1 and k2 from
your data; second, you must decide when and which compounds
to sample; finally, you must take into account the individuality of
the experiment; that is, figure out how the chemistry and the
experimental setup itself affect the measured concentrations. In
principle, this method may be applied to reduce the experimental
effort and the amount of “garbage data” in robotic reaction setups.
using MATLAB.18 A detailed description of the sample preparation
methods and the experimental apparatus was published previ-
ously.15
A total of 32 identical experiments were performed and
monitored using UV-visible. A stock solution of 3-chlorophenyl-
hydrazonopropane dinitrile A (1.034 M in 0.1 N NaOH) was
prepared. For each experiment, part of this stock solution was
then diluted to 51.71 µM, buffered to pH 5.4 with KH2PO4, and
mixed in the quartz cuvette with an excess (276:1 mol/ mol) of
â-mercaptoethanol solution (2.5 µL of â-mercaptoethanol in 7.5
µL of KH2PO4 buffer solution). UV-visible spectra of the reaction
mixtures were recorded every 10 s at a wavelength range from
300 to 500 nm.
EXPERIMENTAL SECTION
All chemicals were commercially available (99% pure) and were
used without further purification. KH2PO4 buffers were purchased
from Acros (pro analysis 0.2 M). UV-visible spectra were
recorded using a Hewlett-Packard 8453 spectrophotometer (quartz
cuvettes, 1.00-cm path length). Data processing was performed
Received for review July 1, 2003. Accepted September 12,
2003.
(18) MATLAB Version 6.1, 2001, is commercially available from MathWorks.
AC034719B
Analytical Chemistry, Vol. 75, No. 23, December 1, 2003 6707