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In
chemical kinetics Chemical kinetics, also known as reaction kinetics, is the branch of physical chemistry that is concerned with understanding the rates of chemical reactions. It is to be contrasted with chemical thermodynamics, which deals with the direction in ...
, the overall rate of a reaction is often approximately determined by the slowest step, known as the rate-determining step (RDS or RD-step or r/d step) or rate-limiting step. For a given reaction mechanism, the prediction of the corresponding
rate equation In chemistry, the rate law or rate equation for a reaction is an equation that links the initial or forward reaction rate with the concentrations or pressures of the reactants and constant parameters (normally rate coefficients and partial reac ...
(for comparison with the experimental rate law) is often simplified by using this approximation of the rate-determining step. In principle, the time evolution of the reactant and product concentrations can be determined from the set of simultaneous rate equations for the individual steps of the mechanism, one for each step. However, the analytical solution of these
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
s is not always easy, and in some cases
numerical integration In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equatio ...
may even be required. The hypothesis of a single rate-determining step can greatly simplify the mathematics. In the simplest case the initial step is the slowest, and the overall rate is just the rate of the first step. Also, the rate equations for mechanisms with a single rate-determining step are usually in a simple mathematical form, whose relation to the mechanism and choice of rate-determining step is clear. The correct rate-determining step can be identified by predicting the rate law for each possible choice and comparing the different predictions with the experimental law, as for the example of and CO below. The concept of the rate-determining step is very important to the optimization and understanding of many chemical processes such as
catalysis Catalysis () is the process of increasing the rate of a chemical reaction by adding a substance known as a catalyst (). Catalysts are not consumed in the reaction and remain unchanged after it. If the reaction is rapid and the catalyst recyc ...
and
combustion Combustion, or burning, is a high-temperature exothermic redox chemical reaction between a fuel (the reductant) and an oxidant, usually atmospheric oxygen, that produces oxidized, often gaseous products, in a mixture termed as smoke. Combus ...
.


Example reaction: + CO

As an example, consider the gas-phase reaction + CO → NO + . If this reaction occurred in a single step, its
reaction rate The reaction rate or rate of reaction is the speed at which a chemical reaction takes place, defined as proportional to the increase in the concentration of a product per unit time and to the decrease in the concentration of a reactant per uni ...
(''r'') would be proportional to the rate of collisions between and CO molecules: ''r'' = ''k''[][CO], where ''k'' is the reaction rate constant, and square brackets indicate a molar concentration. Another typical example is the Zel'dovich mechanism.


First step rate-determining

In fact, however, the observed reaction rate is second-order in and zero-order in CO, with rate equation ''r'' = ''k''[]2. This suggests that the rate is determined by a step in which two molecules react, with the CO molecule entering at another, faster, step. A possible mechanism in two elementary steps that explains the rate equation is: # + → NO + ''(slow step, rate-determining)'' # + CO → + ''(fast step)'' In this mechanism the
reactive intermediate In chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up matter to the compounds made of atoms, molecules and ions: their composition, s ...
species is formed in the first step with rate ''r''1 and reacts with CO in the second step with rate ''r''2. However, can also react with NO if the first step occurs in the ''reverse direction'' (NO + → 2 ) with rate ''r''−1, where the minus sign indicates the rate of a reverse reaction. The concentration of a reactive intermediate such as [] remains low and almost constant. It may therefore be estimated by the steady state (chemistry), steady-state approximation, which specifies that the rate at which it is formed equals the (total) rate at which it is consumed. In this example is formed in one step and reacts in two, so that : \frac = r_1 - r_2 - r_ \approx 0. The statement that the first step is the slow step actually means that the first step ''in the reverse direction'' is slower than the second step in the forward direction, so that almost all is consumed by reaction with CO and not with NO. That is, ''r''−1 ≪ ''r''2, so that ''r''1 − ''r''2 ≈ 0. But the overall rate of reaction is the rate of formation of final product (here ), so that ''r'' = ''r''2 ≈ ''r''1. That is, the overall rate is determined by the rate of the first step, and (almost) all molecules that react at the first step continue to the fast second step.


Pre-equilibrium: if the second step were rate-determining

The other possible case would be that the second step is slow and rate-determining, meaning that it is slower than the first step in the reverse direction: ''r''2 ≪ ''r''−1. In this hypothesis, ''r''1 − r−1 ≈ 0, so that the first step is (almost) at equilibrium. The overall rate is determined by the second step: ''r'' = ''r''2 ≪ ''r''1, as very few molecules that react at the first step continue to the second step, which is much slower. Such a situation in which an intermediate (here ) forms an equilibrium with reactants ''prior'' to the rate-determining step is described as a ''pre-equilibrium''
Peter Atkins Peter William Atkins (born 10 August 1940) is an English chemist and a Fellow of Lincoln College at the University of Oxford. He retired in 2007. He is a prolific writer of popular chemistry textbooks, including ''Physical Chemistry'', ''I ...
and Julio de Paula, ''Physical Chemistry'' (8th ed., W. H. Freeman 2006) p. 814–815. .
For the reaction of and CO, this hypothesis can be rejected, since it implies a rate equation that disagrees with experiment. # + → NO + ''(fast step)'' # + CO → + ''(slow step, rate-determining)'' If the first step were at equilibrium, then its
equilibrium constant The equilibrium constant of a chemical reaction is the value of its reaction quotient at chemical equilibrium, a state approached by a dynamic chemical system after sufficient time has elapsed at which its composition has no measurable tendency ...
expression permits calculation of the concentration of the intermediate in terms of more stable (and more easily measured) reactant and product species: :K_1 = \frac, : ce= K_1 \frac. The overall reaction rate would then be :r = r_2 = k_2 \ce = k_2 K_1 \frac, which disagrees with the experimental rate law given above, and so disproves the hypothesis that the second step is rate-determining for this reaction. However, some other reactions are believed to involve rapid pre-equilibria prior to the rate-determining step, as shown below.


Nucleophilic substitution

Another example is the unimolecular nucleophilic substitution (SN1) reaction in organic chemistry, where it is the first, rate-determining step that is
unimolecular In chemistry, molecularity is the number of molecules that come together to react in an elementary (single-step) reactionAtkins, P.; de Paula, J. Physical Chemistry. Oxford University Press, 2014 and is equal to the sum of stoichiometric coeffic ...
. A specific case is the
basic BASIC (Beginners' All-purpose Symbolic Instruction Code) is a family of general-purpose, high-level programming languages designed for ease of use. The original version was created by John G. Kemeny and Thomas E. Kurtz at Dartmouth College ...
hydrolysis Hydrolysis (; ) is any chemical reaction in which a molecule of water breaks one or more chemical bonds. The term is used broadly for substitution, elimination, and solvation reactions in which water is the nucleophile. Biological hydrolysi ...
of tert-butyl bromide () by aqueous
sodium hydroxide Sodium hydroxide, also known as lye and caustic soda, is an inorganic compound with the formula NaOH. It is a white solid ionic compound consisting of sodium cations and hydroxide anions . Sodium hydroxide is a highly caustic base and al ...
. The mechanism has two steps (where R denotes the tert-butyl radical ): # Formation of a
carbocation A carbocation is an ion with a positively charged carbon atom. Among the simplest examples are the methenium , methanium and vinyl cations. Occasionally, carbocations that bear more than one positively charged carbon atom are also encount ...
R−Br → + . # Nucleophilic attack by hydroxide ion + → ROH. This reaction is found to be
first-order In mathematics and other formal sciences, first-order or first order most often means either: * "linear" (a polynomial of degree at most one), as in first-order approximation and other calculus uses, where it is contrasted with "polynomials of hig ...
with ''r'' = ''k'' −Br which indicates that the first step is slow and determines the rate. The second step with OH is much faster, so the overall rate is independent of the concentration of OH. In contrast, the alkaline hydrolysis of
methyl bromide Bromomethane, commonly known as methyl bromide, is an organobromine compound with formula C H3 Br. This colorless, odorless, nonflammable gas is produced both industrially and biologically. It has a tetrahedral shape and it is a recognized ozo ...
() is a
bimolecular nucleophilic substitution The SN2 reaction is a type of reaction mechanism that is common in organic chemistry. In this mechanism, one bond is broken and one bond is formed in a concerted way, i.e., in one step. The name SN2 refers to the Hughes-Ingold symbol of the ...
(SN2) reaction in a single
bimolecular In chemistry, molecularity is the number of molecules that come together to react in an elementary (single-step) reactionAtkins, P.; de Paula, J. Physical Chemistry. Oxford University Press, 2014 and is equal to the sum of stoichiometric coeffic ...
step. Its rate law is second-order: ''r'' = ''k'' −Br].


Composition of the transition state

A useful rule in the determination of mechanism is that the concentration factors in the rate law indicate the composition and charge of the
activated complex In chemistry an activated complex is defined by the International Union of Pure and Applied Chemistry (IUPAC) as "that assembly of atoms which corresponds to an arbitrary infinitesimally small region at or near the col (saddle point) of a potential ...
or
transition state In chemistry, the transition state of a chemical reaction is a particular configuration along the reaction coordinate. It is defined as the state corresponding to the highest potential energy along this reaction coordinate. It is often marked ...
. For the –CO reaction above, the rate depends on []2, so that the activated complex has composition , with 2 entering the reaction before the transition state, and CO reacting after the transition state. A multistep example is the reaction between
oxalic acid Oxalic acid is an organic acid with the systematic name ethanedioic acid and formula . It is the simplest dicarboxylic acid. It is a white crystalline solid that forms a colorless solution in water. Its name comes from the fact that early invest ...
and chlorine in aqueous solution: + → 2 + 2 + 2 . The observed rate law is :v = k \frac, which implies an activated complex in which the reactants lose 2 + before the rate-determining step. The formula of the activated complex is + − 2 − +  , or (an unknown number of water molecules are added because the possible dependence of the reaction rate on was not studied, since the data were obtained in water solvent at a large and essentially unvarying concentration). One possible mechanism in which the preliminary steps are assumed to be rapid pre-equilibria occurring prior to the transition state is : + HOCl + + : + :HOCl + → + + 2


Reaction coordinate diagram

In a multistep reaction, the rate-determining step does not necessarily correspond to the highest Gibbs energy on the reaction coordinate diagram.
Keith J. Laidler Keith James Laidler (January 3, 1916 – August 26, 2003), born in England, was notable as a pioneer in chemical kinetics and authority on the physical chemistry of enzymes. Education Laidler received his early education at Liverpool College. H ...
. ''Chemical Kinetics'' (3rd ed., Harper and Row 1987) p. 283–285. .
If there is a
reaction intermediate In chemistry, a reaction intermediate or an intermediate is a molecular entity that is formed from the reactants (or preceding intermediates) but is consumed in further reactions in stepwise chemical reactions that contain multiple elementary s ...
whose energy is lower than the initial reactants, then the activation energy needed to pass through any subsequent
transition state In chemistry, the transition state of a chemical reaction is a particular configuration along the reaction coordinate. It is defined as the state corresponding to the highest potential energy along this reaction coordinate. It is often marked ...
depends on the Gibbs energy of that state relative to the lower-energy intermediate. The rate-determining step is then the step with the largest Gibbs energy difference relative either to the starting material or to any previous intermediate on the diagram. Also, for reaction steps that are not first-order, concentration terms must be considered in choosing the rate-determining step.


Chain reactions

Not all reactions have a single rate-determining step. In particular, the rate of a chain reaction is usually not controlled by any single step.


Diffusion control

In the previous examples the rate determining step was one of the sequential chemical reactions leading to a product. The rate-determining step can also be the transport of reactants to where they can interact and form the product. This case is referred to as
diffusion control Diffusion-controlled (or diffusion-limited) reactions are reactions in which the reaction rate is equal to the rate of transport of the reactants through the reaction medium (usually a solution). The process of chemical reaction can be considered ...
and, in general, occurs when the formation of product from the activated complex is very rapid and thus the provision of the supply of reactants is rate-determining.


See also

* Product-determining step


References

* {{Authority control Chemical kinetics ja:反応速度#律速段階