The Rate‐Determining Step is Dead. Long Live the Rate‐Determining State!

Abstract

The concept of a rate-determining step (RD-Step) is central to the kinetics community, and it is basic knowledge even for the undergraduate chemical student. In spite of this, too many different definitions of the RD-Step appear in the literature, all of them with drawbacks. This dilemma has been thoroughly studied by several authors in the attempt to "patch" the drawbacks and bring the RD-Step to a correct physical meaning. Herein we review with simple models the most notable definitions and some challengers of the RD-Step concept, to conclude with the deduction that there are no rate-determining steps, only rate-determining states.

Full text

DOI: 10.1002/cphc.201100137
The Rate-Determining Step is Dead. Long Live the Rate-
Determining State!
Sebastian Kozuch*[a] and Jan M. L. Martin[b]
1. The Rate-Determining Step Dilemma
Contrary to other areas of human intellectual endeavor, in sci-
ence there is no place for ambiguity. A definition of a concept
must be (ideally) clear, unique and unequivocal; building those
definitions is the raison d’Þtre for the existence of the IUPAC
body.[1] When a concept definition loses precision, or grows to
an unmanageable series of ad hoc conditions, we may be look-
ing at a case of an imperfect model of the scientific reality.
Some examples of this are the concepts of Lewis structures,
the formal charge, or the full set of “ideal” systems (ideal
gases, ideal solution, ideal covalent bond, etc). Sometimes, as
in the previous examples, this is only a philosophical matter, as
they are useful models. If we need more accuracy, we can add
a correction term or apply a more complex model. However,
sometimes the ambiguity in a definition can be a symptom of
a very pragmatic problem, which may lead to faulty conclu-
sions.
In chemical kinetics a central concept is the rate-determining
step (RD-Step), also called rate-controlling or rate-limiting
step.[1, 2] Herein we show with simple examples that the defini-
tions of this concept are not clear, unique or unequivocal. The
RD-Step description mutated from a clear but imprecise (and
frankly wrong) wording, to an obscure mathematical terminol-
ogy. These are symptoms of a pragmatic problem:[3–11] there
are no rate-determining steps![3, 4, 12] This statement has pro-
found consequences in homogeneous[3, 4, 13, 14] , heterogeneous
and enzymatic[6, 7, 15, 16] catalysis, kinetic isotope effect stud-
ies,[2,6, 8, 17] and the whole body of kinetic and mechanistic
chemistry.
2. Rate-Determining Step Faulty Definitions
Several definitions for the RD-Step can be found in the litera-
ture. We will cite the ones we consider most famous or promi-
nent, while the reader is invited to review the rest.[5, 18–23]
2.1. The Slowest Step of the Reaction[24–28]
The most typical definition of the RD-Step is “the slowest step
of the reaction”, always regarded as the “bottleneck”. In most
texts this definition goes side by side with the description of
the steady-state approximation (i.e. all steps in a multistep re-
action occur at the same rate when having unstable intermedi-
ates). This is a paradox. If we consider that the steady-state
regime is a very good approximation, then there is no slower
or faster step, and consequently there is no RD-Step ![2, 5, 8, 9, 18, 19]
2.2. The Step with the Smallest Rate Constant[29–31]
Sometimes the RD-Step is more strictly defined as “the step
with the smallest rate constant (k)” (strangely, this definition
was occasionally considered a synonym with the previous
one[29,30]). This definition is equivalent to saying “the step with
the highest activation energy”. It is a measurable designation,
thus more unambiguous and physical. However, unambiguous
and physical does not mean correct. A simple example can
show why.
Let us have a two-step reaction in the steady-state regime,
such as the one in Equation (1), with k1the smallest rate con-
stant:
C1G
k1
!k1
HC2
k2
!!C3ð1Þ
The concept of a rate-determining step (RD-Step) is central to
the kinetics community, and it is basic knowledge even for the
undergraduate chemical student. In spite of this, too many dif-
ferent definitions of the RD-Step appear in the literature, all of
them with drawbacks. This dilemma has been thoroughly stud-
ied by several authors in the attempt to “patch” the drawbacks
and bring the RD-Step to a correct physical meaning. Herein
we review with simple models the most notable definitions
and some challengers of the RD-Step concept, to conclude
with the deduction that there are no rate-determining steps,
only rate-determining states.
[a] Dr. S. Kozuch
Department of Organic Chemistry
The Weizmann Institute of Science
IL-76100 Rehovot (Israel)
Fax: (+972) 8-934-414 2
E-mail: sebastian.kozuch@weizmann.ac.il
[b] Prof. J. M. L. Martin
Center for Advanced Scientific Computing and Modeling (CASCAM)
Department of Chemistry, University of North Texas
Denton, TX 76203-5017 (USA)
ChemPhysChem 2011, 12, 1413 – 1418 "2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim 1413

The rate of reaction for this system is [Eq. (2)]:
r¼k2%k1
k!1þk2
C1
½( ð2Þ
If we consider k!1smaller than k2, then the rate of the reac-
tion will be [Eq. (3)]:
r¼k1C1
½( ð3Þ
Here the RD-Step definition being considered is correct. The
rate of the reaction depends solely on k1(the smallest rate
constant) and neither k2nor k!1influence the kinetics.
On the other hand, if k!1is bigger than k2, the rate equation
is [Eq. (4)]:
r¼k2%k1
k!1
C1
½( ð4Þ
In this case, how can we say that k1determines the rate of
reaction, when all the other rate constants have the same
weight? Furthermore, taking into account that k1/k!1=Keq1,
Equation (2) can be written as Equation (5):
r¼k2Keq1 C1
½( ð5Þ
and k2may be regarded as rate-determining, even though k1is
the smallest rate constant. Clearly the “smallest rate constant”
definition is flawed (especially when having reversible
steps[9, 32, 33]).
2.3. The Step with the Highest-Energy Transition State[34–37]
For the examples of the previous section [Eqs. (3) and (5)] the
“highest-energy transition state” definition will indeed work.
However, this is hardly a universal scenario. In a non-steady
state, two-step reaction of the type given by Equation (6):
C1
k1
!!C2
k2
!!C3ð6Þ
with an energy profile like that in Figure 1, the highest transi-
tion state is T1. However, the first step has smaller activation
energy compared to the second step (T1!I1<T2!I2, or equiva-
lently k1@k2). C1will be swiftly converted to C2, and then it will
slowly “drip” to C3. In this case the rate of reaction is [Eq. (7)]:
r¼k2C2
½( ð7Þ
The previous definition of RD-Step (“The step with the small-
est rate constant”) can work here, but the “highest transition
state” definition is erroneous (especially for irreversible
steps[2, 32]). In Section 4 (catalytic reactions), we show that even
for steady-state reactions this definition can be misleading.
2.4. The Step with the Rate Constant that Exerts the Stron-
ger Effect[1, 2, 6, 8]
IUPAC’s definition says that “a rate-determining step … is an
elementary reaction the rate constant for which exerts a
strong effect—stronger than that of any other rate constant—
on the overall rate”.[1] In other words, a RD-Step is the one
whose rate constant is determining. This definition is a tautolo-
gy (a reiteration of the same idea in different words), but is ac-
tually the most accurate definition. In IUPAC’s gold book[1] a
mathematical description is added: the RD-Step is the one
whose control function (CF,[8] also called degree of rate con-
trol[20,21, 38] , and based on the similar concept of the sensitivity
function[6]) is the highest [Eq. (8)]:
CFi¼ki
z}|{
Normalization
factor
r%@r
@ki
|{z}
Effect of the
rate constant
on the rate
of reaction
¼@ln r
@lnkið8Þ
This differentiation must be computed maintaining all other
rate constants (kj,j¼6i,!i) and the equilibrium constant of that
step (Keq,i
) fixed. It is easier to understand these restrictions if
we pass from the k-representation (based on rate constants) to
the E-representation (based on energies)[3, 4, 7] according to
Equations (9) and (10):
Keq;i¼e!DGr;i=RT ð9Þ
ki¼kBT
he!DG!
i=RT ðTransition State TheoryÞð10Þ
where DGr,i is the Gibbs reaction energy of step i, and DGi
!is
the Gibbs activation energy of the step, as seen in Figure 2.
If we want to calculate the control function [Eq. (8)] for k1in
the two-step reaction of Equation (1), we have to calculate the
response of the global rate of reaction (r) to an infinitesimal
change in k1, keeping k2and Keq1 constant (Figure 2). According
to Equations (9) and (10), this means we must test rwith an in-
finitesimal change on T1(the Gibbs energy of the first transi-
tion state). In other words, the control function that is sup-
posed to test a rate-determining step, is actually examining a
rate-determining transition state.
Let us consider the case of Equation (5), where k2is the “RD-
Step” according to the control factor. In this case, a change in
Figure 1. Energy profile of a model non-steady state reaction with two irre-
versible steps.
1414 www.chemphyschem.org "2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim ChemPhysChem 2011, 12, 1413 – 1418
S. Kozuch and J. M. L. Martin

Keq1 has the same weight as a change in k2. A change in Keq1
keeping constant k2and k!1implies a change in k1, as appears
in Figure 3. So, we can make a change in k1that will affect the
rate of reaction as much as a change in k2, in spite of having a
null control factor for k1. How is this possible? [Eq. (11)]:[39]
r¼k2
z}|{
Step 2
is rate!
determining
Keq C1
½(¼k2%k1
z}|{
But step 1
is equally
important
k!1
C1
½(
ð11Þ
The problem resides in the definition of the control factor
(hence a problem in the definition of the RD-Step). The con-
straint of keeping Keq1 fixed makes this analysis impossible, but
lifting this constraint makes it impossible to define an unequiv-
ocal control factor. The reason is that the control factor can
only analyze the response to a transition-state change, and in
this case we are changing an intermediate (this issue was tack-
led in different ways by the “degree of turn-over frequency
(TOF) control”,[3,40] the “thermodynamic rate control for inter-
mediates”,[41, 42] the “thermodynamic control factor”,[43] the “sen-
sitivity”,[18, 44] and qualitatively by the Curtin–Hammett[45] and
Sabatier classical principles[46]).
Let us recapitulate the deficiencies of IUPAC definition of a
RD-Step:
1) The control factor, expressed as a function of rate con-
stants, has too many constraints (what we called ad hoc condi-
tions in Section 1). As graphically expressed in Figure 2, the
control factor is more a response to a transition state than to a
rate constant.
2) The control factor (and consequently this definition of the
RD-Step) cannot deal with valid cases that fall outside the con-
straint of the formula, as in Equation (11) and Figure 3. This
needs a new function that measures the response of the rate
with a change of intermediate energies.
3. Rate-Determining States: A Physically
Correct Concept
In Section 1 we argued that “when a concept definition loses
precision, or grows to an unmanageable series of ad hoc con-
ditions, we may be looking at a case of a bad model of the sci-
entific reality”. From Section 2 we can conclude that the RD-
Step is indeed a bad model of the scientific reality, as expressly
highlighted or hinted at by several authors. In that case, what
will be a good model for kinetic research ? As we saw in Sec-
tion 2.4, one transition state and one intermediate have the
potential to be the factors that shape the kinetics of the cycle.
In view of this, the correct concept should be the rate-deter-
mining states (RD-States), and not the rate-determining steps.
Can we build an unambiguous, clear, unique and unequivocal
definition for the RD-State? By all means.
Definition: Rate-determining states are the transition state
and intermediate which exert the strongest effect on the over-
all rate with a differential change on their Gibbs energies.
Mathematically, the transition state and intermediate with
the highest degree of rate control (XTS and XI) are the RD-States
[Eq. (12)]:[3,4, 40–42]
XTS;j¼1
r%@r
@!GTS;j
RT
"#
XI;j¼1
r%@r
@GI;j
RT
"#
ð12Þ
where GTS and GIare Gibbs energies of the intermediates and
transition states, and 1/ris a normalization factor. The degree
of rate control can be extended to study the sensitivity to con-
centrations.[12]
What is the reason that makes the RD-State a physically cor-
rect concept? The energy states are points in the reaction pro-
file space, while the steps are the processes that bind two con-
secutive points. In reality, the rate-determining zone[7] may
cover much more than two consecutive states, thus including
several steps (specifically all the steps that are between the
RD-States). This can be seen in Figure 3, where I1and T2are
the RD-States, and all the rate constants that are in between
(k1,k!1and k2) shape the reaction [Eq. (11)].
Figure 2. Energy profile of the two steps reaction of Equation (1). I1,I2and I3
are the Gibbs energies of the C1,C2and C3intermediates ; T1and T2the
Gibbs energies of the transition states. With Equations (9) and (10) we can
convert these energy states into rate and equilibrium constants. The control
function of k1[Eq. (8)] is measured by calculating the change in the rate of
reaction with a change in k1, maintaining k2and Keq1 constant. This is equiva-
lent to a change in T1.
Figure 3. A change in k1with fixed k!1and k2is equivalent to a change in I1,
the Gibbs energy of the first intermediate.
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Long Live the Rate-Determining State !

4. RD-States in Catalysis
4.1. Introduction to Catalytic Cycles
Catalytic reactions are special in that they are cyclic: from the
catalyst’s perspective the initial and final state can be consid-
ered the same state, with the reactants and products as “addi-
tions”. A steady-state regime is easily achieved for most cata-
lytic cycles. In Figure 4 a model three-step catalytic cycle is de-
picted, both in the rate-constant (k) representation and the
energy (E) representation. Both representations are equivalent,
as we can convert each rate constant to an intermediate/tran-
sition state energy pair and vice versa, according to the transi-
tion state theory given by Equation (10).
In catalysis, the rate of reaction is called the TOF, the
number of cycles that occur in a specific time period. The reso-
lution of a three-step catalytic cycle in a steady-state regime is
beyond the scope of this text,[46, 47] and we write here only the
final formula (for simplicity we do not consider the concentra-
tion of reactants and products[4, 12, 47]) [Eq. (13)]:
TOF ¼
k1k2k3!k!1k!2k!3
k2k3þk3k1þk1k2þk!1k3þk1k!2þk!3k2þk!1k!2þk!2k!3þk!3k!1
ð13Þ
We can simplify the TOF equation by neglecting the lesser
terms. First, the product of forward rate constants (first term in
the numerator) is always bigger than the backward product
(second term) for exergonic reactions. Second, typically only
one of the nine terms in the denominator is significant; for a
reaction profile of the type of Figure 4, it will be k!2k1. In this
case, the resulting equation is given by Equation (14):
TOF ¼k1k2k3
k1k!2
¼k3Keq2 ð14Þ
Equation (14) shows that the TOF (the rate of reaction) is
completely independent of k1. However, not only is k1the
smallest forward rate constant, but T1is also the highest
energy state! Clearly, the rules of “smallest rate constant” and
“highest transition state” are not valid to establish the deter-
mining kinetic factors. Is there a simple way to find these de-
termining factors? There is, if we work in the E-representation
and try to find the RD-States instead of the RD-Step.
4.2. Visual Guide to Find the RD-States in Catalysis[3, 4, 7]
The net chemical flow in an exothermic reaction in the steady
state is always in the forward direction ; this fact will be the
first clue to find the RD-States in a catalytic cycle. The second
clue is that from each intermediate we can consider the effec-
tive energy barrier to be crossed as the highest energy transi-
tion state that occurs in the forward direction (this can be
thought as the Curtin–Hammett principle applied to catalytic
cycles[45]). The RD-States are the ones that have the highest ef-
fective energy barrier.
There is one factor that must be taken into account
when dealing with these systems. There is no starting or
ending point in a catalytic cycle, as the catalyst can theo-
retically continue to an infinite number of turn-overs. There-
fore, when we look for the highest TS following each inter-
mediate, we have to look further than one cycle. In Figure 5
we show the methodology to find the RD-States point by
point. Similar methods can be found elsewhere.[6, 15, 16, 32, 33, 43, 48]
Figure 4. A three-step model catalytic cycle, in A) the linear and closed k-
representations (rate-constant representations) and B) the E-representation
(energy representation). The kand Erepresentations are equivalent, and we
can translate one to the other with the transition state theory [Eq. (10)]. DGr
is the reaction Gibbs energy, independent from the catalyst.
Figure 5. Recipe to find the RD-States in a catalytic cycle: 1) Draw the Gibbs
energy profile of two consecutive cycles. 2) From each intermediate, find the
highest energy transition state in the forward chemical flow. 3) The deter-
mining states are the ones with the biggest span between the intermediate
in question and the following highest TS energy. These states are called the
TDI (TOF-determining intermediate) and the TDTS (TOF-determining transi-
tion state). The energy difference between the TDI and the TDTS is called
the energetic span (dE). In this example, I2is the TDI, and T1’ the TDTS. Note
that T1’=T1+DGr.
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S. Kozuch and J. M. L. Martin

4.3. Mathematical Guide to Find the RD-States in Catalysis :
the Energetic Span Model
The catalytic profile of Figure 5 corresponds to the following
TOF equation [Eq. (15)] [a simplification of Eq. (13)]:
TOF ¼k1k2k3
k!2k!3
¼k1Keq2Keq3 ð15Þ
Almost all the rate constants appear in this formulation, thus
speaking of an RD-Step is a poor oversimplification (again).
From the visual technique of Figure 5, we found that I2and T1
are the RD-States. These two determining species are called
the TDI (TOF-determining intermediate, sometimes called the
resting state or the MARI[5, 11,23, 49]) and the TDTS (TOF-determin-
ing transition state, also called … RD-Step !).[3] Can we find the
TDI and the TDTS from Equation (15) ? Yes, if we pass from the
k-representation to the E-representation by applying the transi-
tion state theory [Eq. (10)] to all the rate constants [Eq. (16)]:
TOF¼k1k2k3
k!2k!3
¼kBT
h
e!T1þI1%e!T2þI2%e!T3þI3
e!T2þI3%e!T3þI4
kBT
h
e!T1þI1þI2
eI4¼kBT
he!T1þI2!DGr
ð16Þ
In the last equality we considered that the starting point of
the second cycle is the end of the first one (I4=I1’=I1+DGr).
As with the visual method of Figure 5, writing the rate equa-
tion in the E-representation naturally gave us I2and T1as the
RD-States. In this case DGris added as the acting TDTS is in
the second cycle (in Figure 5 it is shown as T1’=T1+DGr). As a
rule, DGrappears in the TOF equation if the TDTS comes
before the TDI (when considering only one turn-over). After
this example, we can generalize the TOF expression in what is
called the Energetic Span Model [Eq. (17)]:[3, 4, 40]
TOF ¼kBT
he!dE
dE¼TTDTS !ITDI if TDTS appears after TDI
TTDTS !ITDI þDGrif TDTS appears before TDI
(ð17Þ
dEis the energetic span[50] of the cycle, and serves as the ap-
parent activation energy of the full catalytic cycle.[3, 4, 40, 43, 44, 48] It
depends directly on the RD-States (the TDI and TDTS), and has
no need of a cumbersome product of rate constants. Equa-
tion (17) is an excellent approximation when the degree of
TOF control of the TDI and TDTS is close to one (for the com-
plete TOF formula see references [3, 4, 12, 40]).
5. Conclusions
Herein we saw that the rate-determining step is a hard con-
cept to define, with each one of the definitions having defi-
ciencies. These are symptoms of a faulty concept : either we
must choose an RD-Step definition that fits the chemical prob-
lem,[2,39] or we are working in an unsuitable framework
(Figure 6). The literature is full of discussions and “patches” for
these deficiencies, pointing to a new viewpoint : the whole
idea of a RD-Step should be relegated to a historical place in
chemistry, and should make place for a concept with a stron-
ger physical basis, the rate-determining states.
Keywords: catalysis ·kinetics ·rate-determining step ·
reaction mechanisms ·transition-state theory
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Long Live the Rate-Determining State !

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Received: February 18, 2011
Published online on April 27, 2011
1418 www.chemphyschem.org "2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim ChemPhysChem 2011, 12, 1413 – 1418
S. Kozuch and J. M. L. Martin