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Reaction Progress Kinetic Analysis A Powerful Methodology for Mechanistic Studies of Complex Catalytic Reactions.

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Reviews
D. G. Blackmond
Reaction Kinetics
Reaction Progress Kinetic Analysis: A Powerful
Methodology for Mechanistic Studies of Complex
Catalytic Reactions
Donna G. Blackmond*
Keywords:
catalysis · in situ measurements ·
kinetics · reaction kinetics ·
reaction mechanisms
In memory of Keith J. Laidler (1916–2003)
Angewandte
Chemie
4302
2005 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
DOI: 10.1002/anie.200462544
Angew. Chem. Int. Ed. 2005, 44, 4302 – 4320
Angewandte
Chemie
Reaction Kinetics
Reaction progress kinetic analysis to obtain a comprehensive
picture of complex catalytic reaction behavior is described. This
methodology employs in situ measurements and simple manipulations to construct a series of graphical rate equations that enable
analysis of the reaction to be accomplished from a minimal
number of experiments. Such an analysis helps to describe the
driving forces of a reaction and may be used to help distinguish
between different proposed mechanistic models. This Review
describes the procedure for undertaking such analysis for any new
reaction under study.
1. Introduction
Kinetic investigations of multistep organic reactions form
a critical part of fundamental mechanistic studies as well as
being indispensable to practical applications of organic
synthesis. Such studies are aimed at providing a better
understanding of reaction pathways by giving concentration
dependences of reacting components as well as rate and
equilibrium constants pertaining to elementary steps in the
reaction network. Analysis of experimental data in multistep
catalytic reactions is complicated, however, by the complexity
of the reaction rate laws characteristic of these reactions, and
therefore simplified means of representing the kinetic data
are often sought.
One of the most important examples of such a simplifying
tool is the double-reciprocal plot developed by Lineweaver
and Burk. The original paper outlining this approach[1] has
become the most-cited paper in the history of the Journal of
the American Chemical Society[2] and offers a straightforward
means of linearizing enzymatic reaction rates (Scheme 1)
Scheme 1. Catalytic reaction cycle for a simple enzyme-catalyzed reaction.
given by the Michaelis–Menten Equation [Eq. (1), u = reaction rate (in m min1), umax = maximum reaction rate (in
m min1), [S] = substrate concentration (in m), KMM =
Michaelis constant (in m)].[3] The Lineweaver–Burk Equation
[Eq. (2)] turns the Michaelis–Menten Equation upside down;
umax ½S
K MM þ ½S
ð1Þ
1
1
K
1
¼
þ MM
u umax umax ½S
ð2Þ
u¼
Angew. Chem. Int. Ed. 2005, 44, 4302 – 4320
From the Contents
1. Introduction
4303
2. Reaction Progress Kinetic Analysis:
Definition
4304
3. Getting Started
4304
4. A Simple Example of HighThroughput Kinetics
4307
5. Dealing with Added Complexity
4309
6. Graphical Tools for Reactions with
Two Substrates: Dissecting the
Reaction-Rate Expression
4310
7. Interrogating the Data
4311
8. A Flow Chart for Kinetic Analysis of
Catalytic Reaction Systems
4318
9. Summary
4318
this simple but powerful operation created an enabling
graphical tool for several generations of chemists trying to
make sense of experimental kinetic data in both enzymatic
and nonenzymatic catalytic systems (Figure 1).[4]
Today, 70 years since the Lineweaver–Burk Equation was
developed, dramatic advances both in experimental measurement techniques and in computer analysis tools provide far
more accurate methods for extracting kinetic parameters in
reaction systems. Reaction progress kinetic analysis is a
methodology that makes use of the voluminous data sets that
are now readily obtained from continuous monitoring of the
entire course of a reaction. This analysis has permitted the
development of a new set of enabling graphical tools that we
have termed “graphical rate equations” for a new generation
of chemists. Reaction progress kinetic analysis gives the same
information provided by classical kinetic approaches with
only a fraction of the number of separate experiments. In
addition, monitoring the time evolution of the reaction can
yield significant further clues about issues that can be
problematic in classical experimental approaches, such as
catalyst activation and deactivation, as well as substrate and
product inhibition or acceleration.[5]
In its most comprehensive form, reaction progress kinetic
analysis uses detailed kinetic modeling to assess quantitatively the relative likelihood of different proposed reaction
[*] Prof. D. G. Blackmond
Department of Chemistry
Imperial College
London SW7 2AZ (UK)
Fax: (+ 44) 020-7594-5804
E-mail: d.blackmond@imperial.ac.uk
DOI: 10.1002/anie.200462544
2005 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
4303
Reviews
D. G. Blackmond
measured data, and the time commitment of the analyst. This
Review gives an introducion to reaction progress kinetic
analysis through topical experimental examples from my
research group. Special emphasis will be placed on the
intermediate level of data “interrogation” to help demonstrate the power of modern experimental techniques in rapid
and comprehensive kinetic analysis and to illustrate the
accessibility and intuitive value of this graphical approach.
2. Reaction Progress Kinetic Analysis: Definition
Figure 1. Plots of a) the Michaelis–Menten Equation [Eq. (1)] and
b) the Lineweaver–Burk Equation [Eq. (2)] to describe the reaction
mechanism shown in Scheme 1 for different values of umax and KMM.
mechanisms. At the other end of the spectrum, monitoring
reaction progress can provide a rapid, qualitative “fingerprint” of reaction behavior for streamlining practical exploitation of new chemistry as well as offering general insights
useful for further mechanistic study. In the middle lies a
graphical approach where simple manipulations applied to a
series of seemingly inscrutable experimental kinetic profiles
allow the “interrogation” of the data to yield a wealth of
mechanistic information, while requiring only minimal mathematical prowess on the part of the analyst.
The level at which reaction progress kinetic analysis is
carried out in any particular example will ultimately reflect a
balance between the depth of information desired, the
complexity of the reaction under study, the quality of the
Donna G. Blackmond is a Professor in the
Departments of Chemistry as well as Chemical Engineering and Chemical Technology at
Imperial College (Chair of Catalysis). Previously she held appointments at the University of Hull, the Max-Planck-Institut f.r Kohlenforschung, the University of Pittsburgh,
and Merck & Co, Inc. She was recently
awarded the Arthur C. Cope Scholar Award
from the American Chemical Society. Her
research interests are focussed on the kinetics
of organic reactions with application in pharmaceutical processes.
4304
2005 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
Reaction progress kinetic analysis is faster, requires fewer
experiments, can provide enhanced mechanistic detail, and is
arguably more accurate than classical kinetic techniques.
Why, then, have these methods not yet become common
practice in studies of organic reactions? The answer to this
question lies, I believe, in our general difficulty in relating the
mathematics of a reaction rate law to the molecular story
unfolding within a catalytic reaction system. Kinetics conjures
up images of experimental drudgery, mathematical manipulations, and logarithmic graphs; chemists prefer to spend their
time devising novel transformations in the laboratory. What is
not generally made evident in our training in kinetics is that,
rather than being a rote and rather detached method of
“dotting the i;s and crossing the t;s” undertaken only after a
mechanistic supposition has been all but confirmed by other
work, kinetics and its concomitant mathematics are critical
tools that can help predict how molecules undergo transformations. Furthermore, these tools can be applied each and
every time a reaction is carried out—without altering the
conditions under which the reaction is typically performed.
Kinetics is one of several languages for describing
chemical events. Other possibilities include drawing out a
reaction mechanism or construction of an energy diagram.
The fact that the study of kinetics presents a “language
barrier” to many chemists reflects the different ways reactions
are viewed. To a synthetic chemist, a chemical reaction is
characterized by the singular verdict of its outcome: yield and
selectivity at the end of the reaction event are registered as
numbers in a table. From a kinetic perspective, a chemical
reaction is a journey narrated by the reaction rate law. While
reaction progress is evident in an energy diagram with its
“reaction coordinate” axis, surprisingly few chemists realize
that they may envision kinetic graphs in this way. Where
classical kinetic approaches provide only a bare summary of
the journey that the molecules have undertaken, reaction
progress kinetic analysis allows much more to be learnt by
following the molecules on their journey.
3. Getting Started
The two key components required for carrying out
reaction progress kinetic analysis are: 1) an in situ method
of continuous and accurate experimental data collection and
2) a computational means for manipulating the data (typically
a PC with a spreadsheet program, perhaps supplemented by a
commercial curve-fitting package). Data collection may be
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Angew. Chem. Int. Ed. 2005, 44, 4302 – 4320
Angewandte
Chemie
Reaction Kinetics
carried out by using any method that provides an adequate
and accurate response (both in signal strength and in time
resolution) that is related in a known way to reaction
turnover. Understanding the ways in which we manipulate
the data we obtain requires that a few words are first said
about the different ways in which kinetic data can be
measured.
3.1. Integral Measurements
Kinetic methods that rely on a relationship between the
measurable parameter and species concentration are called
integral measurements because concentration is proportional
to the integral of the reaction rate. For example, FTIR
spectroscopic studies that follow particular vibrations of
reactant or product molecules and also incorporate a chemometric analysis relate the signal intensity to temporal reactant
or product concentrations (Table 1, left column). The reaction
Scheme 2. Examples of reactions monitored using integral (a; see Figure 2 a) and differential (b, see Figure 2 b) measurements of reaction
progress.
Table 1: Integral and differential measurements of reaction progress.
FTIR spectroscopy as an example of an integral
measurement[a]
Reaction calorimetry as an example of a differential
measurement[b]
A=ebc
dc
rate = dt
A
conversion = 1A0
q̇ = DHrxn ? volume ? rate
measured parameter: conversion
processed parameter: reaction rate
measured parameter: reaction rate
processed parameter: conversion
R
R
t¼tðendÞ
t¼t
conversion =
q(t)dt /
t¼0
q(t)dt
t¼0
[a] A = absorbance, c = concentration, e = extinction coefficient, b = cell path length. [b] q̇ = reactant
heat flow, DHrxn = heat of reaction.
method (Figure 2 a).[7] Another
example in which a differential
method (Figure 2 b) was used is
the reaction shown in Scheme 2 b.[8]
The graphs in Figure 2 reflect the
way data are collected, with time as
the x-axis. However, viewing reaction progress as a function of time is
not the best way to extract kinetic
information from a reaction. Con-
rate is obtained by differentiating the experimental concentration versus time data. Thus, with integral methods, rate is
termed a “processed” parameter to the “primary” parameter
of concentration or conversion of the substrate. Typically,
some method of smoothing the concentration data is applied
before calculating the rate to minimize the noise inherent in
the derivative process.[6]
3.2. Differential Measurements
Methods that measure reaction rate directly are termed
differential or derivative measurements (Table 1, right
column). These methods include reaction calorimetry, which
measures the instantaneous heat flow of the reaction. The
reaction rate is related to the heat flow q through the
thermodynamic heat of reaction DHrxn. In this case the rate is
the primary measured parameter, while concentration or
conversion (the processed parameters) is proportional to the
integral of the rate versus the time.
3.3. A “Graphical Rate Equation”
The reaction shown in Scheme 2 a is an example in which
the reaction progress was monitored by using an integral
Angew. Chem. Int. Ed. 2005, 44, 4302 – 4320
Figure 2. a) Integral (see Scheme 2 a) and b) differential (see Scheme 2 b) methods for monitoring reaction progress.
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2005 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
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D. G. Blackmond
sider the Michaelis–Menten rate expression [Eq. (1)] used to
describe the simple mechanism shown in Scheme 1. We can
see that time appears only implicitly in this expression.
Instead, as with all reaction-rate expressions, the Michaelis–
Menten Equation defines a relationship between rate and
substrate concentration. This is where it is useful to combine
the two ways in which reaction progress data are described.
For any given reaction, the differential measurement can be
plotted as the y-axis and the integral measurement as the xaxis, thus leaving time out of the picture. The resulting plot is
termed a “graphical rate equation”. Figure 3 shows such plots
of rate versus substrate concentration for the two examples of
Figure 2.
The graphical rate equation is a powerful tool for probing
concentration dependences. For example, the straight-line
relationship in Figure 3 a shows that the asymmetric transfer
hydrogenation of Scheme 2 a exhibits overall first-order
kinetics in the concentration of the imine substrate. The
curvature in the graphical rate equation in Figure 3 b suggests
a more complex relationship must be used to describe the
enantioselective Heck coupling of Scheme 2 b.
The key to the success of this graphical tool lies in the
accurate construction of the graphical rate equation by
obtaining sufficient high-quality data over the course of the
reaction. This highlights the importance of accurate, rapidresponse in situ measurement tools in reaction progress
kinetic analysis.
3.4. Two (or More) In Situ Techniques are Better than One
Facile access to a range of in situ monitoring techniques
makes it possible in many cases today to employ more than
one such technique simultaneously. Monitoring a reaction by
several orthogonal techniques[9]—methods that rely on different properties of a reacting system to monitor its progress—is
a good way of checking that the desired reaction is indeed
being followed. A multitechnique approach can also provide
valuable mechanistic information that one technique used
alone might miss. The reaction shown in Scheme 3 and the
Scheme 3. Room-temperature hydrogenation of nitrobenzene at
normal pressure using a Pd/carbon catalyst (see Figure 4).
Figure 3. “Graphical rate equations” constructed from the data in Figure 2 a and b.
corresponding analysis shown in Figure 4 are shown as an
example.[10] Hydrogenation of nitro substituents on aromatic
rings using heterogeneous catalysts is a common transforma-
Either concentration or conversion can be used as the
“integral method” x-axis in these plots. The fraction conversion x of the limiting substrate S is related to the
concentration of the substrate by Equation (3).
½S ¼ ½S0 ð1xÞ
u¼
ð3Þ
d½S
dx
¼ ½S0
dt
dt
Although fraction conversion is often preferred, comparisons on this basis can be misleading, because a given value for
fraction converted represents different substrate concentrations for experiments carried out at different initial concentrations of the limiting substrate. Notice also that plotting
concentration as the x-axis means that the reaction progresses
from right to left, because substrate concentration decreases
over the course of the reaction.
4306
2005 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
Figure 4. Reaction progress of the Pd/carbon-catalyzed hydrogenation
of nitrobenzene shown in Scheme 3 as monitored by three different
in situ techniques.
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Angew. Chem. Int. Ed. 2005, 44, 4302 – 4320
Angewandte
Chemie
Reaction Kinetics
tion in the pharmaceutical industry. The conversion of
nitrobenzene, shown in Figure 4, into aniline using Pd/C is
monitored by three separate in situ methods that monitor
different properties of the reaction under study: FTIR
spectroscopy follows the disappearance of the nitro group in
the starting material; hydrogen uptake monitors the stepwise
stoichiometry of the hydrogenation; and reaction calorimetry
measures the thermodynamic heats in all steps of this highly
exothermic reaction. Figure 4 shows clearly that the three
methods are in excellent agreement. This result demonstrates
that the kinetic data are accurately following the reaction in
question and that any potential contribution from side
reactions may be safely neglected.
Note that no discrepancy is observed between a method
following the decay of the starting material and one monitoring the stoichiometry of the hydrogen uptake, although the
hydrogen uptake profile differs for the first and second steps
in Scheme 3. From this information it can be proposed that
intermediate species, such as hydroxylamine, do not build up
in solution over the course of the reaction, since all three
techniques give identical information about the global
reaction stoichiometry. Thus, in this case, the use of these
three methods also offers additional mechanistic insight.
3.5. Calibrating the In Situ Measurement
Whatever the method or methods used to follow the
progress of the reaction, it is vitally important to verify that
the measurement indeed correlates with the turnover of
substrate molecules to product. In most cases it is not feasible
to compare the reaction progress profile with three separate
measurement techniques as in the previous example. More
routinely, calibration of the in situ technique is accomplished
by periodic sampling of the reaction mixture over time and
employing a known analytical method of assessing concentration. Figure 5 shows a comparison of conversion versus
time determined from GC analysis of discrete samples and
from the reaction heat flow for the example shown in
Scheme 4, the copper-free Sonogashira coupling of bromoa-
Scheme 4. Sonogashira coupling of an aryl bromide with phenylacetylene using a
zero-valent Pd phosphine catalyst (see Figure 5).
cetophenone with phenylacetylene using [Pd(PtBu3)2] as
catalyst.[11, 12] Although discrete analytical sampling alone
seldom provides the quantity of data required for reaction
progress kinetic analysis, it serves as a check on the veracity of
the in situ technique used for monitoring the reaction
progress.[13]
3.6. Caveats
The principles and the graphical tools developed here for
reaction progress kinetic analysis owe a large debt to a long
history of graphical approaches in enzyme catalysis that are
beyond the scope of this Review. Curve-fitting and simulation
approaches as well as widely available commercial software
packages that determine rate constants will not be the focus of
this Review. The fact that these approaches are available but
not routinely used by synthetic organic chemists studying
chemical catalytic systems provides a strong rationale for the
development of the approach described here.
The rate equations developed in this Review are based on
the steady-state approximation. Reaction progress kinetic
analysis may also be applied (with certain caveats) to
stoichiometric reactions; however, this extension of the
methodology is beyond the scope of the present Review.
The parallel reaction cycles of asymmetric catalytic reactions
are not treated here, although the simple reaction schemes
given may be easily extended to these systems.
Illustrating the graphical approach described here
required access to data sets for manipulation, and hence all
of the data and examples discussed come from work carried
out in my research group or in collaborative efforts. These
data are used here without attempts at detailed mechanistic
interpetation; interested readers can refer to the published
papers.
4. A Simple Example of “High-Throughput Kinetics”
The asymmetric hydrogenation reaction depicted in
Scheme 5 and carried out in two different solvents is used as
an introductory illustration of kinetic analysis of the progress
of the reaction.[14] Reaction progress, determined by an
integral (Figure 6 a) and by a differential (Figure 6 b)
Figure 5. Fraction converted versus time for the reaction shown in
Scheme 4 as monitored by reaction calorimetry and verified by GC
analysis.
Angew. Chem. Int. Ed. 2005, 44, 4302 – 4320
Scheme 5. Asymmetric hydrogenation of ethyl pyruvate mediated by
cinchona-modified Pt/Al2O3.
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2005 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
4307
Reviews
D. G. Blackmond
Figure 6. Kinetic analysis (by differential and integral methods) of the hydrogenation reaction shown in Scheme 5 carried out in two different solvents: a) Substrate concentration versus time (integral method). b) Reaction rate versus time (differential method). c) Rate versus substrate concentration; a fit to Equation (1) gives umax = 0.008 0.33 % m min1, KMM = 0.684 0.56 % m1 (isopropyl alcohol) and umax = 0.002 0.07 % m min1, KMM = 0.083 0.23 % m1 (toluene). d) Reciprocal rate versus reciprocal substrate (Lineweaver–Burk plot); a fit to Equation (2) gives
umax = 0.007 0.90 % m min1, KMM = 0.600 1.32 % m1 (isopropyl alcohol) and umax = 0.002 0.12 % m min1, KMM = 0.091 0.34 % m1 (toluene).
method, is plotted as a function of time.[15] This example
shows how the progress curves can be used to obtain both a
qualitative comparison of the reaction;s features in the two
solvents as well as a quantitative analysis of kinetic parameters. The important point about monitoring reaction progress
is that, in a single experiment, kinetic data can be obtained
not only at the initial concentration of the reactant but also at
every concentration between this value and that at the
reaction;s end. Thus, continuous monitoring of reaction
progress is akin to carrying out hundreds of separate initial
rate experiments.
The qualitative approach allows some information to be
gained before a reaction mechanism is proposed. The
temporal data of Figure 6 a and 6 b show that the reaction is
slower in toluene and appears to exhibit a greater degree of
“flatness” in the progress curve relative to that in isopropyl
alcohol. This observation suggests that the reaction exhibits
less-positive-order kinetics in toluene than in isopropyl
alcohol. Such a conclusion is even easier to reach by looking
at the graphical rate equation of Figure 6 c, which is constructed by combining Figure 6 a and b. The closer the data
are to a horizontal line, the closer the reaction is to zero-order
kinetics. Thus, the reaction seems to be not-quite zero order
when carried out in toluene, and not-quite first order when
carried out in isopropyl alcohol.
Possible reaction mechanisms for kinetic modeling of the
data need to be considered if a more quantitative approach is
to be used. A good place to start for a constant-pressure
4308
2005 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
hydrogenation reaction is the simple Michaelis–Menten
mechanism shown in Scheme 1, with the rate constant k2
replaced by a pseudo-first-order constant that incorporates
the constant hydrogen concentration. The qualitative observations made above are indeed borne out quantitatively by
the Michaelis–Menten kinetic model of Scheme 1 and Equation (1). Figure 6 c shows that an excellent fit to Equation (1)
is obtained for reactions in both solvents and give statistically
significant values of the kinetic parameters. These parameters
show that the value of KMM in toluene is nearly one order of
magnitude lower than it is in isopropyl alcohol. Since KMM is
related to the inverse of the binding constant,[16] this result
suggests that much stronger catalyst–substrate binding occurs
in toluene, thus confirming the pseudo-zero-order form of
Equation (1) for the reaction in toluene. In isopropyl alcohol,
the Michaelis–Menten Equation cannot be simplified to an
integer order, thus resulting in a reaction order that lies
between 0 and 1, with its value changing with [S].
These same reaction progress data may also be plotted in
the time-honored Lineweaver–Burk double reciprocal plot as
in Figure 6 d, which shows similar values for the kinetic
parameters with similar standard deviations. One advantage
of the “straight-line” approach of Figure 6 d is that slopes and
intercepts can be extracted easily without resorting to a curvefitting program. One drawback is that the relative values of
KMM in the two solvents are intuitively less easy to visualize
from double-reciprocal plots of Figure 6 d than they are from
the direct “graphical rate equation” plots of Figure 6 c.
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Angew. Chem. Int. Ed. 2005, 44, 4302 – 4320
Angewandte
Chemie
Reaction Kinetics
The most important message to take away from this
example is that, in this simple case, one single experiment is
all that is mathematically required to develop a quantitative
kinetic model of the reaction at a given pressure and
temperature. The reason for this is because the experiment
contained in essence hundreds of separate initial rate
measurements at different substrate concentrations. In comparison to the days of labor that the classical kinetic approach
would require, this is very high-throughput kinetic analysis
indeed.
5. Dealing with Added Complexity
The example from the previous section (Scheme 5 and
Figure 6) gives a sense of the ease with which a mechanistic
model can be developed and tested by monitoring the
reaction progress even in a single experiment, and it
demonstrates both qualitative and quantitative approaches
to reaction progress kinetic analysis. Furthermore, it illustrates the value of the “graphical rate equation” (the plot of
rate versus concentration) as a visual aid in assessing
concentration dependences. However, most reactions are
mechanistically and experimentally much more complicated
than this example.
5.1. Reactions with Two Substrates: Beyond the Lineweaver–Burk
Plot
The reaction system in Scheme 1 represents the simplest
case of a catalytic reaction mechanism, which in practice
applies to only a very few cases—including decompositions
and isomerizations, or, as already shown, hydrogenations
under constant pressure. Typically, however, chemists are
more interested in putting molecules together, and thus most
reactions of interest will involve at least two reactant species.
This means that, under synthetically relevant conditions, two
separate reactant concentrations will change simultaneously
over the course of the reaction.
Scheme 6 shows how the simple reaction system of
Scheme 1 is modified for the case where a second substrate
Scheme 6. Mechanism for a simple reaction with two substrates and
one intermediate species.
2 adds to the catalytic intermediate species 5 formed from the
binding of the first substrate 1 to catalyst 4 to form the
product 3. The rate expression for this case can be written in
the same form as the Michaelis–Menten relationship of
Equation (1) to give Equation (4).
Angew. Chem. Int. Ed. 2005, 44, 4302 – 4320
u¼
umax ½1
K MM þ ½1
ð4Þ
umax ¼ k2 ½2½4total , K MM ¼
k1 þ k2 ½2
k1
It can be seen that the kinetic parameters umax and KMM
have now become more complex. They are no longer
constants, but instead they each incorporate a dependence
on the concentration of the second substrate. A Lineweaver–
Burk plot for a two-substrate reaction must be constructed
from a series of experiments in which the concentration of the
substrate [2] is held constant. It is important to emphasize that
the constants obtained from this plot are only valid at that
particular concentration of the second substrate. This limits
the information that can be extracted from these parameters;
for example, the magnitude of KMM will no longer be easily
related to the strength of binding of the substrate to the
catalyst.
KMM can be generalized—that is, the value of KMM can be
related back to values of the elementary step rate constants—
by using a classical kinetic approach in which not one, but a
series of Lineweaver–Burk plots are constructed at different
constant values of [2]. This process will provide the relationship between KMM and [2] that is needed to determine k1, k1,
and k2 individually. Since each data point on each Lineweaver–Burk plot represents a separate experiment, it is clear
that the classical kinetic approach to reactions involving two
substrates involves even more work than it does for experiments with only one substrate.
5.2. Defining the Reaction [“Excess”]
Reaction progress kinetic analysis can help in the streamlining of the approach described above for reactions with two
substrates. If the Michaelis–Menten form of the kinetic
parameters is dismissed and the rate equation written out
instead in terms of the elementary step rate constants, the
relationship in Equation (5 a) and (5 b) is obtained.
u¼
k1 k2 ½1 ½2 ½4total
k1 þ k1 ½1 þ k2 ½2
ð5aÞ
u¼
a ½1 ½2 ½4total
1 þ b ½1 þ c ½2
ð5bÞ
a¼
k1
k
k
k ,b¼ 1 ,c¼ 2
k1 2
k1
k1
Equation (5 b) divides both the numerator and denominator by the constant k1 to give the rate equation in what is
called the “one-plus” form.[17] These equations reveal that for
reactions with two substrates the idea of an integer reaction
order in either substrate will be at best an approximation: [1]
and [2] appear in additive terms in the denominator as well as
being multiplied together in the numerator. Reaction progress data can help make sense of this complex situation,
without constraining one concentration to remain constant;
instead conditions are employed where both [1] and [2]
change over the course of the reaction.
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2005 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
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D. G. Blackmond
Although [1] and [2] both change with time, the manner in
which they change is linked to the reaction stoichiometry. For
the reaction shown in Scheme 6, this stoichiometric “rule”
says that each time one molecule of 1 is converted into one
molecule of product 3, one molecule of 2 is necessarily also
converted. Here we define a parameter called the [“excess”],
which is equal to the difference in the initial concentrations of
the two substrates [Eq. (6)].[18]
,,
½2 ¼ ½20 ½10 þ ½1 ) ½2 ¼ ½ excess þ ½1
ð6Þ
,,
This parameter [“excess”] has units, the same units as [1]
and [2] (typically m or mm). The [“excess”] need not be a large
number—such as the familiar tenfold excess used in classical
kinetic experiments to approximate zero-order conditions in
one substrate. The [“excess”] may be a positive number, a
negative number, or zero ([“excess”] = 0 for equimolar
conditions). A few important points to emphasize about this
parameter [“excess”] include:
*
*
*
[“excess”] gives the difference between the initial concentrations of the two substrates.
[“excess”] does not change as the reaction progresses (for
constant volume reactions).
[“excess”] is not identical to either the number of
equivalents or to a percentage excess concentration,
both of which do change as the reaction progresses.
One of the key features of reaction progress kinetic
analysis is that reactions can be carried out using synthetically
relevant conditions. A synthetically reasonable value for the
excess of one substrate over the other is employed, thus
avoiding the synthetically unreasonable tenfold concentration
differences typical of the “pseudo-zero-order” classical
kinetic approach.
allows the kinetics of reactions with two substrates to be
assessed. And because the concentrations of each reactant
need not be fixed in turn for a series of experiments probing
the other substrate;s concentration dependence, this process
cuts down significantly on the total number of experiments
that must be carried out in order to complete the kinetic
analysis of a reaction.
Equation (7) allows us to illustrate two important ways in
which we use the variable [“excess”] in kinetic analysis, as
outlined in the next two sections.
5.3.1. Experiments Eploying the Same [“Excess”]
First, suppose several experiments are carried out with the
same [“excess”] but different starting concentrations of 1 and
2. Equation (7) illustrates that this is equivalent to carrying
out the same experiment from two different starting points.
This possibility becomes useful to probe reactions for complexities such as product inhibition or catalyst deactivation
(see Section 7.1.2).
5.3.2. Experiments with Different [“Excess”]
Equation (7) shows that there are only two independent
adjustable parameters in this rate equation, a’ and b’, while
the reaction mechanism has three independent rate constants,
k1, k1, and k2. This situation is analogous to having two
simultaneous algebraic equations containing three unknowns,
for which an infinite number of solutions exists. Suppose,
however, that several experiments are carried out with
different [“excess”] values. Collecting data from reactions
with at least two different [“excess”] values gives an extra
equation and allows a unique solution to be obtained for all
three rate constants. This becomes especially important when
different mechanistic models are to be tested (Sections 7.4
and 7.5).
5.3. How [“Excess”] Helps to Streamline Kinetic Analysis
If the stoichiometric relationship of Equation (6) is
substituted into the rate expression Equation (5 b), the
resulting rate equation is given by Equation (7). For a given
set of conditions, the quantities [4]total and [“excess”] are
constant, and k1, k1, and k2 are constants, thus leaving [1] as
the only variable.
ð7Þ
k1 k2
k1 þ k2
,, , b0 ¼
,,
k1 þ k2 ½ excess k1 þ k2 ½ excess ,,
,,
a0 ¼
,,
½ excess ½1 þ ½12
½4total
1 þ b0 ½1
,,
u ¼ a0
Writing the rate expression in this form highlights how the
parameter [“excess”] helps deal with the conundrum of
having two concentrations changing at once. Equation (7)
shows that there is no need to carry out reactions where the
concentration of one substrate is artificially fixed at a
pseudoconstant high value over the course of the reaction.
As long as the [“excess”] is known, then monitoring the
concentration of one substrate as the reaction progresses
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6. Graphical Tools for Reactions with Two
Substrates: Dissecting the Reaction-Rate
Expression
The dependence of the reaction rate on [1] in Equation (7) is complex, with second-order and first-order polynomials in [1] appearing in the numerator and denominator,
respectively. Since [1] and [2] are associated with the
parameter [“excess”], it is often easier in practice to return
to the rate equation in the form of Equation (5 b), where both
[1] and [2] appear explicitly. In this section some limiting cases
of this rate equation are probed.
The denominator of Equation (5 b) is comprised of three
terms: one is a constant (simply the number 1) and the other
two terms contain reactant concentrations [1] and [2], each
multiplied by constants. It is this complexity that makes the
concentration dependences of [1] and [2] nonstraightforward.
The key to relating this equation to the species shown in the
reaction mechanism (Scheme 6) is to probe which, if any, of
the three terms in the denominator dominates the other two.
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Table 2: Limiting cases of the rate expression given in Equation (5 b) for
the catalytic reaction in Scheme 6.
7. “Interrogating” the Data
Dominating species
(Scheme 6)
Dominating term
in denominator
of Eq. (5 b)
Observed
rate equation
5
4
b
c
u = k2[2] [4]total (8)
u = k1[1] [4]total (9)
none
all terms contribute
u=
The [“excess”] relationship allows a set of data pairs of
([2], time) points to be compiled with the data from the
reaction analysis, that is, with ([1], time) and (rate, time) data
pairs. Armed with these data, it is now possible to start
constructing graphical rate equations for reactions with two
substrates. A number of examples below highlight how these
data can give information about catalyst behavior. It should
be noted that all of the kinetic treatment described here is
designed for use in experiments that are carried out under
isothermal conditions where the reaction under study is
dominant (that is, there are minimal side reactions).
k1
k ½1 ½2 ½4total
k1 2
k
k
1þk 1 ½1þk 2 ½2
1
1
(5 b)
The various limiting forms of this equation are discussed
below and are summarized in Table 2.
Note that the ratio given by the constant b in Equation (5 b) is the equilibrium constant for the binding of
substrate 1 and hence gives information about the concentration of the catalyst intermediate species 5. If the term
containing b outweighs the other denominator terms, the
intermediate species 5 dominates and the slow step will be the
reaction consuming this intermediate. This limiting situation
is commonly termed “saturation kinetics” in species 1. The
rate equation in this case simplifies to that of Equation (8) in
Table 2. A case of saturation kinetics for a reaction with only
one substrate was shown in the introductory example of
Section 4.0 and one for a reaction with two substrates will be
shown in Section 7.2.
The constant c gives the ratio of rate constants for the two
reactions of intermediate species 5: forward to yield product 3
versus backward to reform substrate 1. If c is large, then
intermediate 5 does not build up to a significant concentration. In this case the formation of the intermediate 5 will be
rate-limiting, and the unbound catalyst species 4 will be the
“resting state” of the catalyst. Equation (9) in Table 2
describes this case, with an example given in Section 7.3.
It is also possible that none of the terms in the
denominator dominate the rate expression. This situation is
equivalent to saying that the reaction doesn;t have a ratelimiting step—a circumstance more common than is typically
considered; it simply means that the catalyst exists in
significant amounts in both forms 4 and 5 in the system
rather than opting for one particular resting state. In this case
it is necessary to use the full rate Equation (5 b), without
simplification of the denominator, to describe the reaction
(see Section 7.5).
The observed kinetics will be dictated by the reaction
involving the catalytic species that dominates, as summarized
in Table 2. It can be seen from this discussion that observation
of integer reaction orders in catalytic reactions can hide a
wealth of mechanistic information. The next few examples
will demonstrate how reaction progress data can be used to
help discern which, if any, of these limiting cases are
applicable to the reaction under study. While we start from
the form of equation (5), we will see again how the parameter
[“excess”], which defines the relationship between substrates
1 and 2, can help us gather significant mechanistic detail from
a minimal number of experiments.
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7.1. Catalyst Instabilities
7.1.1. Induction Periods
An important consideration that is often neglected in
classical kinetic approaches is the problem of induction
periods in catalytic reactions. Transient behavior at the
beginning of a reaction is common in chemical catalysis
where reactions are often initiated by addition of a catalyst
precursor, rather than a catalyst in its active form. Even when
the catalyst itself is in its active form, there may be other
considerations that cause data obtained at the very outset of
the reaction to be suspect. The example in Scheme 7
Scheme 7. Epoxidation of chalcone by H2O2 in the presence of the catalyst PLL (MeO-poly(ethylene glycol)-poly-(l-leucine); see Figure 7).
DBU = 1,8-diazabicyclo[5.4.0]unde-7-ene.
illustrates this for the JuliJ–Colonna epoxidation of 1,3diphenylpropenone (chalcone).[19] The reaction rate versus
time curve (Figure 7) shows that the rate rises for the first few
minutes of the reaction, before decreasing with time in the
manner expected for a reaction exhibiting positive-order
kinetics in substrate concentration.[20] The inset in Figure 7
shows a plot of the graphical rate equation in the form of
reaction rate versus fraction conversion of the chalcone
substrate and shows that the reaction reaches about 20 %
conversion before the rate profile assumes positive-order
kinetics. During the initial few minutes of reaction, the
measured values of rate will vary dramatically with even small
increments in time. For example, an initial rate measurement
taken at 30 seconds would differ by a factor of 50 % from one
taken at 60 seconds. Such transient behavior might never be
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Scheme 8. a) Heck coupling of aryl bromides. b) Hydrolytic epoxide
ring opening.
Figure 7. Reaction rate versus time profile for the epoxidation of chalcone with H2O2 (see scheme 7).
detected, and misleading interpretation of kinetic data might
result, if reaction-progress analysis was not available.
The potential complications of induction periods can
easily be overcome when kinetic data are obtained using an
in situ method for monitoring reaction progress. Most often it
is the steady-state characteristics of the reaction that are of
interest for our kinetic analysis. Since the reaction rate and
the concentrations of the reactants are known at every point
in the reaction, the data obtained during the induction period
can simply be discarded and the analysis carried out by
selecting only the portion of the reaction that exhibits steadystate behavior. This is what has been done in the examples in
the following sections, both for graphical data “interrogation”
and for detailed kinetic modeling studies.[21]
7.1.2. Catalyst Deactivation and Product Inhibition
A source of additional complexity in catalytic reactions
arises in cases where the catalyst deactivates over time, or
where the concentration of the active catalyst can be altered
over the course of the reaction by substrate or product
inhibition. These issues are often cited in support of the
practice of carrying out initial rate measurements, to remove
time-dependent properties of the reaction from consideration. Such practice may in some cases provide an ideal
picture of the kinetics of the reaction; however, this picture
can be quite misleading in the practical world, where
reactions necessarily are carried out from beginning to end.
One of the most powerful features of reaction progress
kinetic analysis is that it allows us to account for the influence
of such time-dependent processes, rather than simply ignoring
them. Consider the two reactions shown in Scheme 8 a and 8 b,
for which reaction progress data are plotted in the graphical
rate equation form in Figure 8 a[22] and 8 b,[23] respectively.
Palladium-catalyzed Heck couplings of aryl halides (Scheme 8 a) are commercially important carbon–carbon bondforming reactions that are often plagued by catalyst deactivation.[24] Epoxide ring opening catalyzed by a cobalt–salen
complex, as performed by Jacobsen and co-workers, offers a
strikingly efficient method for the kinetic resolution of
racemic epoxides (Scheme 8 b).[25] These reactions are more
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complex than our introductory hydrogenation example,
because in each case two substrate concentrations change
simultaneously over the course of the reaction.
The issue of catalyst deactivation can be probed for the
two reactions by carrying out two separate reactions with
different starting concentrations of species 1 and 2, but
identical values of the [“excess”]. As explained in Section 5.3,
two such experiments represent an identical reaction started
from two different places. Thus, choosing any concentration
value on either of the plots in Figure 8 a and b gives data that
correspond to experiments that are identical except for two
things: 1) the reaction with a higher initial concentration (red
curve) contains more product than that with a lower initial
concentration (blue curve); and 2) the catalyst for the
reaction with a higher initial concentration has completed
more turnovers. By comparing the curves for the reaction
progress under these conditions, it is possible to address both
the influence of the presence of product and the possibility of
catalyst deactivation slowing the reaction over time.
Figure 8 a shows that the two curves (reaction rate versus
[ArX]) fall exactly on top of one another, thus confirming that
neither the presence of product nor the “extra work” done by
the catalyst in the reaction shown in red had any influence on
the reaction kinetics. We will return to this example later to
discuss the interesting “bend” in the curves toward the end of
the reaction. For now, however, it has been shown how two
simple experiments effectively tell us that neither catalyst
deactivation nor product inhibition is a factor in this Heck
reaction.
The reaction shown in Figure 8 b tells a different story. For
a given epoxide concentration, a slightly higher rate was
observed in the reaction indicated by the blue curve (lower
initial concentration) compared to that shown in red (higher
initial concentration). This result suggests that either product
inhibition or catalyst deactivation influences the reaction. In
this case, further experiments are required to establish which
of the two effects is responsible for the lowered rate. For
example, reaction product could be added to the mixture at
the outset of a reaction under the conditions shown by the
blue curve, and the effect on the resulting graphical rate
equation observed: namely, do the new data fall on top of the
blue curve, or do they fall on top of the red curve? If the curve
with addition of product overlays with the blue curve, then it
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consequences of this practice of removing the catalytic
reaction system from its “natural environment” are seldom
addressed. Our goal is to employ the conditions under which
we plan to run a reaction and allow reaction progress kinetics
to help us understand how our reaction works. Kinetic
analysis that studies the reacting system in its natural state
provides a truer account of the journey taken by the reaction.
These two illustrations have shown how reaction progress
kinetic analysis can make use of two simple experiments with
identical [“excess”] to address the questions of catalyst
deactivation and product inhibition. Adding a third experiment to the list (that of product addition at the start of the
reaction) can generally settle these issues definitively for a
given system under a given range of conditions. Initial rate
measurements leave these questions unanswered; however—
especially since the experimental protocol is disarmingly
straightforward—it makes sense not to avoid looking for the
answers.
7.2. Saturation Kinetics in a Reaction with Two Substrates
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ð10Þ
slope ¼ k
,,
yintercept ¼ k ½ excess ,,
can be concluded that catalyst deactivation and not product
inhibition is responsible—this is what was found for the
reaction shown in Scheme 8 b. Further experiments suggested
that such deactivation was significant only at the unusually
high water:epoxide ratios used in these experiments. Since the
hydrolytic kinetic resolution is typically carried out at
0.55 equivalents of water (compared to the 1.3–1.5 equivalents used here), it could be concluded that catalyst deactivation would not be a factor under synthetically relevant
conditions.[26]
The example shown in Figure 8 b brings up an important
consideration in comparing reaction progress kinetic analysis
to classical kinetic approaches. Kinetic behavior may be
altered when reactions are carried out under conditions
significantly removed from those of practical application, as is
often the case in classical kinetic approaches in which
pseudoconstant concentrations of one reactant is used. The
,,
u ¼ k ½2 ¼ k ð½ excess þ ½1Þ
,,
Figure 8. Probing catalyst deactivation by carrying out two reactions at
the same [“excess”] but different initial substrate concentrations.
a) Heck coupling using a C-N palladacycle catalyst (0.0002 m); red
curve: [ArX]0 = 0.16 m, [olefin]0 = 0.24 m; blue curve: [ArX]0 = 0.12 m,
[olefin]0 = 0.20 m; b) epoxide ring opening with 2 mol % [Co(salen)];
red curve: [epoxide]0 = 1.5 m, [H2O]0 = 2 m; blue curve: [epoxide]0 = 1 m,
[H2O]0 = 1.5 m.
Returning to the Heck coupling example of Figure 8 a, we
turn our attention to the unusual bend observed in the rate
curves at high conversion. This unusual shape is also evident
when the data are plotted in the time domain, as in
Figure 9 a,[27] where in addition to the data from Figure 8 a, a
third reaction carried out at a different [“excess”] value is
shown. For most of the reaction, the curve of rate versus
[ArX] in Figure 8 a is a straight line suggestive of first-order
kinetics; however, this line has a non-zero y-intercept (y =
m x + b instead of y = m x). The parameter [“excess”] can be
used to explain this. From the initial reactant concentrations,
it can be seen that the aryl bromide is the limiting reactant in
all three cases in Figure 9 a. If the rate was first order in [ArX]
(or [1]), the graphical rate equation should have had a yintercept of zero. However, if the reaction is first order in
[olefin] (or [2]), then the rate is given by Equation (10):
This result suggests that the Heck reaction under study fits
the limiting condition of Equation (8) in Table 2 and describes
“saturation kinetics” in [ArX]. The resting state of the
catalyst is species 5, that is, the oxidative addition complex
with the substrate. Note that the condition of saturation in a
reaction with two substrates is revealed by a linear rate versus
concentration curve (overall first-order kinetics), while for a
reaction with only one substrate, saturation gives a horizontal
line (overall zero-order kinetics) on the same plot.
This result can be corroborated with another form of the
graphical rate equation. The simplified rate expression of
Equation (8) is divided on both sides by the concentration of
the olefin [2] to give Equation (11), in which the variables that
are functions of time (u and [2]) appear as a quotient on the
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and first-order kinetics in [olefin]. Such efficient mechanistic
analysis from so few experiments was possible only because of
the voluminous reaction progress data available for constructing the graphical rate equations.
7.3. Free Catalyst as the Resting State
Reactions following the simple mechanism in Scheme 6
can look deceptively different from one another when carried
out under different conditions: changing the substrates, the
catalyst, the reaction temperature, or even the relative
substrate concentrations can shift the resting state of the
catalyst in the cycle and thus change the observed substrate
concentration dependences. An example illustrating this
reconsiders this Heck reaction, but now employing the
[Pd(PtBu)3)2] catalyst that Littke and Fu have shown to be
active under mild conditions.[28] Plotting the standard graphical rate equation of rate versus [ArX] for three reactions
carried out at two different [“excess”] values gives a firstorder relationship for rate versus [ArX], with a y-intercept at
zero (Figure 10).[29] This result suggests that the reaction
obeys the limiting form of Equation (9) (Table 2) and that the
unbound catalyst is the resting state in the reaction. This
indicates that the rate-limiting step is the formation of the
oxidative addition Pd complex, while the addition and
insertion of the olefin proceeds much faster. It can also be
seen from the two experiments carried out at the same
[“excess”] value that catalyst deactivation is not a problem for
this catalyst under the conditions employed. Comparing
Figures 10 and 8 a illustrates how differently two different
Pd catalysts can behave even in the same reaction.
Figure 9. a) Primary data of reaction rate versus time for the Heck coupling
in Scheme 8 a (C,N palladacycle catalyst) carried out under three different
sets of initial substrate conditions. b) A variation of the graphical rate equation for the Heck coupling derived from Figure 8 a and 9 a. The reaction rate
divided by the corresponding olefin concentration is plotted versus aryl
halide concentration [see Eq. (11)].
left side of the equation, while the quantities on the right side
are constants.
uðtÞ
¼ k2 ½4total ¼ constant
½2ðtÞ
ð11Þ
A graphical rate equation can be made according to
Equation (11) with our data from the three reactions shown in
Figure 9 a. This is achieved by dividing each row of a column
of rate data points by its corresponding olefin concentration;
Equation (11) says that a horizontal straight line could be
obtained when this function is plotted versus [1] or [2] for
experiments carried out at any [“excess”] value (Figure 9 b).
It is evident from Figures 8, 9 a, and 9 b that three simple
reaction experiments and two graphical rate equations have
given extensive detail concerning the mechanism of this Heck
coupling reaction. These plots reveal that the system suffers
from neither catalyst deactivation nor product inhibition and
it exhibits pseudozero order (saturation) kinetics in [ArX]
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Figure 10. Standard graphical rate of reaction [Eq. (9)] for the Heck
coupling shown in Scheme 8 a using [Pd(PtBu3)2] as the catalyst; the
reaction was carried out at three different sets of initial concentrations
of the reactants.
7.4. An Ambiguous Case of Catalyst Resting State
These examples have highlighted two limiting cases of the
resting state of the catalyst shown in Table 2 for the reaction
system of Scheme 6 with two substrates, demonstrating how
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reaction progress kinetic analysis can help in obtaining a rapid
and comprehensive picture of the reaction;s concentration
driving forces in both cases. Reaction progress kinetic analysis
can also help to sort out mechanistic details efficiently for
more complicated cases that fall in between these two limits.
The reaction shown in Scheme 9,[30] the conjugate addition
of diethylzinc to chalcone mediated by chiral amino alcohols
Scheme 9. Alkylation of chalcone with diethylzinc using Ni(acac)2 and
()-MIB ((2S)-()-3-exo-morpholinoisoborneol) as a chiral ligand (see
Figure 11).
in the presence of a Ni complex, involves a complex mix of
reagents. The active catalyst species itself is formed in situ,
and much conjecture has been put forward about the reaction
mechanism, with complicated dependences of the rate and
selectivity on various reaction variables.[31] This would be a
formidable test case for reaction progress kinetic analysis. The
most important aspects are considered here, a detailed
interpretation will be reported separately.[32]
7.4.1. Failure of the Standard Graphical Rate Equation
In the terminology of Scheme 6, [chalcone] can be
represented by [1] and [Et2Zn] by [2]. Three reaction progress
curves of this reaction at two different [“excess”] values are
shown in Figure 11 as a function of time.[33] Note that the
reaction exhibits an induction period, thus suggesting that
rate data at conversions less than about 20 % will not give a
reliable picture of the steady-state catalytic cycle. At conversions higher than this, the reaction appears to be well-
Figure 11. Primary data (reaction rate versus time) for the dialkylzinc
alkylation of chalcone (see Scheme 9). The reaction was carried out
under three different sets of initial substrate concentrations.
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behaved, and our analysis will focus on this region. Further
details about the relationship between rate and reactant
concentrations cannot easily be discerned from the raw
reaction rate versus time data. Therefore the next step is to
plot graphical rate equations.
The standard graphical rate equation of rate versus [1] is
shown in Figure 12 a. The curves for the two reactions with the
same value of [“excess”] fall on top of each other, which
indicates that product inhibition or catalyst deactivation is not
a factor in this reaction. However, the third reaction, carried
out at different initial concentrations of both chalcone and
diethylzinc, as well as at a different value of [“excess”],
doesn;t fall nicely on the other two. Indeed, it is difficult to
determine any useful relationship between the substrate and
the reaction rate from these plots. In this situation, the best
thing to do is to try plotting other forms of the graphical rate
equation.
7.4.2. Trying Other Graphical Rate Equations: Do the Curves
“Overlay”?
New graphical rate equations can be made by “normalizing” the rate law in Equation (5 b) in different ways. For
instance, the rate can be divided by each substrate concentration in turn to provide a new function to plot against the
other substrate concentration. This procedure gave Figure 9 b
in the example of the palladium-catalyzed Heck coupling,
where all three curves fell on top of each other in a horizontal
straight line which indicated zero-order kinetics (Section 7.2).
In that case, this result was not unexpected, because it was
predicted by Equation (11). In most new reactions, as in the
current example, the shape of curves given by developing new
graphical rate equations will not be known a priori. What is
sought, then, in making these plots is a relationship – whether
a straight line or some other function – that produces curves
that fall on top of one another; the questions is, therefore, do
the curves “overlay”?
Figure 12 b and c show the plots of the graphical rate
equations where the general rate equation has been divided
first by the concentration of chalcone [1] [Eq. (12)], and then
by diethylzinc [2] [Eq. (13)].
u
a ½2 ½4total
¼
½1 1 þ b ½1 þ c ½2
ð12Þ
u
a ½1 ½4total
¼
½2 1 þ b ½1 þ c ½2
ð13Þ
Both equations still appear to be complex. Indeed the plot
in Figure 12 b doesn;t help to deconvolute the picture,
because the curves don;t overlay when plotted. This observation indicates that unlike the previous Heck examples, the
order in [1] is not a simple integer; the b term in the
denominator is not negligible in this case.
However, the curves do overlay when the data is plotted
according to Equation (13). This result indicates that the data
plotted according to Equation (13) versus [1] are independent
of [2], since at any given [1] value one of the three runs has a
value of [2] that is different from the others. This means that
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Figure 12. Graphical rate equations for the alkylation of chalcone with dialkylzinc as in Scheme 9 constructed from the primary data given in
Figure 11. a) Standard graphical rate equation rate versus [1]. b) u/[1] versus [2]. c) u/[2] versus [1]; the solid line represents the curve fit to the
modified Michaelis–Menten Equation [Eq. (15)]. d) Lineweaver–Burk plot showing the experimental data and the fit calculated according to the
inverse form of Equation (15).
the right side of Equation (13) can;t contain [2], and therefore
the term c [2] must be small compared to the other terms.
Thus Equation (13) can be simplified to Equation (14).
u
a ½1 ½4total
¼
½2
1 þ b ½1
form [Eq. (15)] and a simple curve-fitting program applied
u0 ¼
u0max ½1
K0MM þ ½1
ð15Þ
ð14Þ
u
,,
u0 ¼ normalized rate ¼
½2
,,
k2
k
½2 1 ½1
k1
k1
k2
½2 1
k1
u0max ¼ k1 k2 ½4total
K0MM ¼
Equation (14) and its graphical version in Figure 12 c show
that this reaction exhibits overall first-order kinetics in the
concentration of diethylzinc [2] and a saturation kinetics
expression in the concentration of chalcone [1]. Since the
plots in Figure 12 c do not show a horizontal straight line, as
they did in the Heck reaction (Figure 9), it can be deduced
that the condition of saturation in [1] has not been reached
under the concentrations of these conjugate addition reactions.
7.4.3. Quantifying the Results of the Data Interrogation
These graphical conclusions can be quantified just as in
the asymmetric hydrogenation example of Figure 6 (Section 4). Equation (14) can be written in a Michaelis–Menten
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k1
k1
fitting the data in Figure 12 c to Equation (15) yields the
values for umax and KMM that are shown in the Figure.[34] Note
that excellent standard deviations were obtained in these
values based on the data from just three separate experiments.
The straight-line approach could instead be used with the
same data by inverting Equation (15). The Lineweaver–Burk
plot constructed from the same three experiments is shown in
Figure 12 d. The same quantitative information is available
from the plot in this form.
To obtain the information on this plot from a classical
kinetic approach would have required a much greater
experimental effort. It would have been necessary to complete at least two separate Lineweaver–Burk plots in which
data was included from two sets of reactions carried out at two
different (constant) values of [2] while varying [1]. At the very
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least, about half a dozen reactions in each set would have
been needed. Reaction progress kinetic analysis allows the
total number of experiments to be cut by a factor of four.
reactions with the same [“excess”], the key to this plot is not
the shape of the profile but simply whether the curves fall on
top of one another.
7.4.4. Turnover Frequency: Reaction Order in [Catalyst]
7.5. Complex Cases
The reaction mechanism of Scheme 6 predicts that
reaction rate will be proportional to the concentration of
the catalyst. Although there are notable exceptions, most
catalytic reactions do exhibit first-order behavior in the
catalyst concentration. A graphical rate equation can be
developed to confirm the relationship by dividing the general
rate expression of Equation (5 b) (reaction with two substrates) by the total catalyst concentration [4]total [Eq. (16)].
This expression for the rate normalized by catalyst concentration is also referred to as the turnover frequency (TOF).[35]
u
a ½1 ½2
¼ TOF ¼
½4total
1 þ b ½1 þ c ½2
ð16Þ
Any two or more reactions carried out using the same
[“excess”] but different catalyst concentrations may now be
investigated and TOF plotted versus [1] (for example,
chalcone). If the curves fall on top of one another, the
reaction is first order in the catalyst concentration. Note that
catalyst concentration is a constant for the entire reaction,
and TOF is calculated by dividing each rate data point by this
constant number; this is in contrast to the other graphical rate
equations where each data point for the rate was divided by a
quantity ([1] or [2]) that changes over the course of the
reaction.
Figure 13 shows the TOF versus [chalcone] plots for two
reactions shown in Scheme 9 carried out at the same
[“excess”] and different concentrations of the catalyst precursor (Ni salt). Since the curves fall on top of one another,
the reaction is first order in [Ni]. Note that constructing the
graphical rate equation and determining the order in [catalyst] by this method requires no information about the
reaction orders in [1] and [2]. As long as TOF is plotted for
Figure 13. Graphical rate equation for turnover frequency as in Equation (16) for data from the reaction shown in Scheme 9.
Angew. Chem. Int. Ed. 2005, 44, 4302 – 4320
In some reactions with two substrates it may be found that
none of the plots of graphical rate equations result in all the
curves falling on top of one another. What this means is that
neither substrate exhibits an integer order and both substrate
concentration terms in the denominator of the rate expression
Equation (5 b) make a noticeable contribution to the overall
rate expression. The catalyst is almost evenly distributed
between its free state 4 and intermediate 5 so that that the
reaction does not exhibit a definitive rate-limiting step or
resting state.[36] In such cases, the failure of the graphical
approach to produce curves that overlay suggests that
detailed kinetic modeling studies must be carried out to
determine the values of the rate constants in the system.
7.6. Other Reaction Mechanisms
Although the discussion in this section focuses on a simple
reaction with two substrates (Scheme 6), a similar approach
may be applied in constructing graphical rate equations for
more complicated reactions.[37] The method allows the rapid
assessment of whether the concentration dependences in a
particular reaction system obey simple or complex forms.
Indeed, in most cases a simple series of a half-dozen separate
reaction-progress experiments are all that is required to assess
concentration dependences of the two substrates and the
catalyst, while assessing whether the reaction exhibits catalyst
deactivation or product inhibition are also important features
of the reaction.
7.7. Lessons Learned
These examples illustrate several important points about
reaction progress kinetic analysis. 1) Access to powerful
in situ experimental tools enables new experimental protocols
to be devised to collect accurate and plentiful data sets to
streamline the data collection portion of the task. Understanding how the data are going to be manipulated also
contributes to optimizing our experimental design. 2) Kinetics may be used to deconvolute a reaction system and even to
propose new mechanistic possibilities. 3) The straightforward
experimental effort and data manipulation of reaction
progress kinetic analysis allows kinetics to be used at the
outset of a mechanistic investigation rather than simply to
confirm a mechanistic proposal.
The relationships found from the data interrogation do
not depend on the mechanism shown in Scheme 6. The trends
observed can be used to indicate the direction of other
potential schemes, thus carrying out kinetic studies as a first—
instead of a last—resort is good advice for the study of any
new reaction.
www.angewandte.org
2005 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
4317
Reviews
D. G. Blackmond
8. A Flow Chart for Kinetic Analysis of Catalytic
Reaction Systems
A summary of what has been demonstrated in the
examples discussed above can be illustrated by making a
flow chart for the kinetic analysis of a reaction with two
substrates hypothesized to follow the reaction mechanism
shown in Scheme 6. Even with no proposed reaction mechanism in mind, this simple system can be used as the starting
point, and the steps described in Scheme 10 can act as a guide
to tackling kinetic analysis of any new reaction system. The
experiments listed in the flow chart follow those described in
the examples in this Review.
The analysis begins with the choice of a set of reaction
conditions for the first experiment, including initial concentrations of substrates 1 and 2, and catalyst concentration, at a
fixed temperature that will remain constant through the series
of experiments. This experiment will serve as the “standard
conditions”, and the experiments that follow will involve
alterations in one or more of these conditions. Next the
reactions can be carried out at two catalyst concentrations
that are different from the standard conditions, and the
graphical rate equation is plotted as turnover frequency
(TOF) versus [1]. This helps us ascertain the order in the
catalyst concentration. If the reaction is not first order in the
catalyst concentration, then the simple mechanism of
Scheme 1 is an imperfect description of all of
the activities of the catalyst.
Next, at least one further experiment is
carried out with the same value of [“excess”] as
our standard conditions but different values of
starting concentrations of 1 and 2. A plot of these
data in the form of the graphical rate equation of
rate versus [1] will show whether or not the
curves overlay. If they do, then catalyst deactivation or product inhibition is not a feature of these
reactions. If they don;t, further experiments are
carried out with product addition to distinguish
product inhibition from catalyst deactivation.
In the next step further reactions are carried
out at values of [“excess”] different from that of
the standard conditions. With these data, various
graphical rate equations described in Section 7
can be tried. The goal is to find out if the data
overlay in any of the different forms of the
graphical rate equation and helps in determining
whether the reaction exhibits simple integer
order, or if it exhibits saturation kinetics in one
or both of the two substrates.
If none of these plots result in overlay
between the different experiments, then it is
known that either the catalyst does not exhibit a
definitive resting state in the mechanism of
Scheme 6, or that the reaction mechanism itself
is more complex. In this case, other measures
must be used, including detailed quantitative
kinetic modeling of the mechanism and other
proposed mechanisms.
9. Summary
Scheme 10. Flow chart for carrying out kinetic analysis of a reaction with two substrates. The
green boxes indicate experiments to be carried out, black boxes indicate plots to be constructed, magenta boxes are query points for determining whether manipulation of the data
results in overlay between curves from reactions carried out under different conditions.
4318
2005 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
This Review has endeavored to illustrate the
ways in which accurate in situ reaction progress
data coupled with simple data manipulations can
help to illuminate aspects of catalytic behavior.
Particular focus has been placed on a graphical
approach involving simple manipulations that
aims to keep the situation close to the “story” of
the molecules on their reaction pathway, without
straying too far into the mathematical wilderness
of quantitative kinetics. Reaction profiles hold a
www.angewandte.org
Angew. Chem. Int. Ed. 2005, 44, 4302 – 4320
Angewandte
Chemie
Reaction Kinetics
wealth of mechanistic information that we are challenged to
reveal; it is hoped that the ideas discussed here will inspire
chemists to begin to demand more from the data that are now
so easily obtainable through in situ techniques.
Contributions to the experimental work presented here from
co-workers and collaborators cited in the references are
gratefully acknowledged. Valuable discussions with and/or a
critical reading of the manuscript by the following people are
gratefully acknowledged: Prof. Stephen L. Buchwald, Eric
Strieter, Prof. Eric N. Jacobsen, Prof. Andreas Pfaltz, Dr.
Suju P. Mathew, Hiroshi Iwamura, Dr. Joel le Bars, Prof.
Stanley M. Roberts, Dr. Emma Emanuelsson, Dr. Martin
Klussmann, Dr. David H. Wells, Jr., Dr. Frederic G. Buono,
Prof. William Tolman, Dr. Brian G. Cox, Dr. Steve Eyley, Dr.
Marijan Stefinovic, Dr. Yongkui Sun, Dr. Thorsten Rosner,
Lars P. C. Nielsen, Prof. Michael J. Krische, Prof. Robert. G.
Bergman, and Prof. Athel Cornish-Bowden. Research funding
from the EPSRC/DTI (“From Micrograms to Multikilos”), the
Max-Planck-Gesellschaft Sonderprogramm zur FBrderung
hervorragender
Wissenschaftlerrinnen,
Pfizer
Global
Research, Merck Research Laboratories, AstraZeneca, Mitsubishi Pharma, and Mettler-Toledo Autochem is gratefully
acknowledged.
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
Received: November 8, 2004
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
H. Lineweaver, D. Burk, J. Am. Chem. Soc. 1934, 56, 658.
Chem. Eng. News 2003, 81, 25.
L. Michaelis, M. Menten, Biochem. Z. 1913, 49, 1333.
The Michaelis Menten rate law was conceived first by Victor
Henri (V. Henri, C. R. Hebd. Seances Acad. Sci. 1902, 135, 916).
A linearized form of the Michaelis – Menten Equation was
published by Hanes two years before the Lineweaver – Burk
paper[1] (C. S. Hanes, Biochem. J. 1932, 26, 1406). The Lineweaver – Burk and other related graphical methods are covered
in detail in enzyme kinetics textbooks. For an especially lucid
discussion of both the history and the use of graphical methods in
general and the opportunities and limitations of such methods,
see A. Cornish-Bowden, Enzyme Kinetics, 3rd ed., Portland,
London, 2004.
Extensive graphical methods have been developed, primarily for
studying enzyme kinetics, for consideration of reaction complexities such as product inhibition by using a classical kinetic
approach, which entailed a concomitant increase in the number
of required experiments. The textbook cited in Ref. [4] provides
an excellent background to these methods.
Rate may be calculated from (concentration, time) data sets by
fitting the data to an arbitrary function (e.g., a 9th order
polynomial) and then differentiating this function to obtain
(rate, time) data. Care must be taken to insure that the fit is
accurate to obtain an accurate rate profile.
M. Ropic, D. G. Blackmond, unpublished results.
T. Rosner, M. Baumann, A. Togni, D. G. Blackmond, unpublished results.
S. Eyley, personal communication.
J. LeBars, D. G. Blackmond, unpublished results.
J. S. Mathew, D. G. Blackmond, unpublished results.
A. Soheili, J. Albaneze-Walker, J. A. Murry, P. G. Dormer, D. L.
Hughes, Org. Lett. 2003, 5, 4191.
The use of a corroborating technique also helps check the
validity of the method when the question of complications other
Angew. Chem. Int. Ed. 2005, 44, 4302 – 4320
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
[30]
[31]
[32]
[33]
[34]
www.angewandte.org
than side reactions arises. For example, the extent of contribution to the overall signal from physical processes such as heat of
mixing or heat of precipitation may be quantified by using
reaction calorimetry, while by using FTIR spectroscopy the
possibility of an inhomogeneous solution layer adjacent to the
IR window may be explored.
For a review, see: M. Studer, H.-U. Blaser, Adv. Synth. Catal.
2003, 345, 45.
F. Bohnen, A. Gomez, D. G. Blackmond, unpublished results.
If k2 is small compared to k1, the Michaelis parameter KMM in
Equation (1) is identical to the inverse of the binding constant
K1 = k1/k1.
F. G. Helfferich, Kinetics of Homogeneous Multi-Step Reactions,
Elsevier, Amsterdam, 2001, chap 7.2.
The concentration of 2 is strictly given by: [2] = [“excess”] + [1]
+ [5]; however, [5] may be neglected under typical catalytic
conditions where [5] ! [1] and [5] ! [“excess”]. This consideration helps to set a lower limit on [“excess”]. In general,
practical values of [“excess”] should be at least an order of
magnitude greater than the catalyst concentration.
a) S. Colonna, H. Molinari, S. Banfi, S. JuliO, J. Masana, A.
Alvarez, Tetrahedron 1983, 39, 1635; b) S. Banfi, S. Colonna, H.
Molinari, S. JuliO, J. Guixer, Tetrahedron 1984, 40, 5207, and
references therein; c) R. W. Flood, T. P. Geller, S. A. Petty, S. M.
Roberts, J. Skidmore, M. Volk, Org. Lett. 2001, 3, 683.
S. P. Mathew, S. Gunathilagan, D. G. Blackmond, unpublished
results.
The graphical tools described here may only be used where the
steady-state approximation applies. For many catalytic reactions
this rules out data at the very start of the reaction if an induction
period is present (see Section 7.1.1). By definition, the steadystate approximation breaks down for the last few turnovers of
the reaction, where the concentration of the catalytic intermediate species approaches that of the substrates being consumed and the parameter [“excess”] is no longer constant (see
discussion in Ref. [18]). These graphical tools are generally
recommended for use on reaction progress data between 15 and
85 % conversion of the limiting substrate.
Figure 8 a was redrawn in a different form from T. Rosner, J.
LeBars, A. Pfaltz, D. G. Blackmond, J. Am. Chem. Soc. 2001,
123, 1848.
J. LeBars, C. P. Stevenson, E. N. Jacobsen, D. G. Blackmond,
unpublished results.
Reviews: R. F. Heck, Org. React. 1982, 27, 345 – 390; A.
de Meijere, F. E. Meyer, Angew. Chem. 1994, 106, 2473 – 2506;
Angew. Chem. Int. Ed. Engl. 1994, 33, 2379 – 2411; W. Cabri, I.
Candiani, Acc. Chem. Res. 1995, 28, 2 – 7.
a) M. Tokunaga, J. F. Larrow, F. Kakiuchi, E. N. Jacobsen,
Science 1997, 277, 936; b) J. M. Keith, J. F. Larrow, E. N.
Jacobsen, Adv. Synth. Catal. 2001, 343, 109.
For a detailed kinetic analysis of this reaction, see L. P. C. N.
Nielsen, C. P. Stevenson, D. G. Blackmond, E. N. Jacobsen, J.
Am. Chem. Soc. 2004, 126, 1360.
Figure 9 a was constructed from the data in: T. Rosner, J. LeBars,
A. Pfaltz, D. G. Blackmond, J. Am. Chem. Soc. 2001, 123, 1848.
A. F. Littke, G. C. Fu, J.Am. Chem. Soc. 2001, 123, 6989.
C. Ellis, D. G. Blackmond, unpublished results.
The high ee value achieved in this reaction justifies the
simplification of the treatment to include only one catalytic
cycle.
a) A. H. M. deVries, J. F. G. A. Jansen, B. L. Feringa, Tetrahedron 1994, 50, 4479; b) C. Bolm, M. Ewald, M. Felder, Chem.
Ber. 1992, 125, 1205.
H. Iwamura, F. G. Buono, D. G. Blackmond, unpublished results.
H. Iwamura, F. G. Buono, D. G. Blackmond, unpublished results.
The data were fit using the Solver program in Excel. Statistical
analysis was performed using Solvstat.xls, a macro program for
2005 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
4319
Reviews
D. G. Blackmond
use with Excel that is available with E. J. Billo, Excel for
Chemists, Wiley-VCH, New York, 2001.
[35] The left side of Equation (16) has the units of inverse time, and
therefore it appears the turnover frequency should have no
concentration dependence. A closer examination reveals that
the units of TOF are: m(substrate) Q m(catalyst)1 Q time1.
While the units of molarity cancel these are, in reality, molarities
of two different substances. The right hand side of Equation (16)
shows clearly that TOF exhibits a concentration dependence.
4320
2005 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
[36] For a lucid discussion of the concept of the rate-limiting step, see
K. J. Laidler, J. Chem. Educ. 1988, 65, 250.
[37] For examples of kinetic modeling studies of reactions exhibiting
more complex mechanisms, see: a) T. Rosner, P. J. Sears, W. A.
Nugent, D. G. Blackmond, Org. Lett. 2000, 2, 2511; b) D. G.
Blackmond, C. R. McMillan, S. Ramdeehul, A. Schorm, J. M.
Brown, J. Am. Chem. Soc. 2001, 123, 10 103; c) F. G. Buono, P. J.
Walsh, D. G. Blackmond, J. Am. Chem. Soc. 2002, 124, 13 652;
also see Ref. [26].
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