Inchemical kinetics, the overall rate of a reaction is often approximately determined by the slowest step, known as therate-determining step (RDS orRD-step[1] orr/d step[2][3]) orrate-limiting step. For a givenreaction mechanism, the prediction of the correspondingrate equation (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 thesedifferential equations is not always easy, and in some casesnumerical integration may even be required.[4] 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 ofNO2 and CO below.
The concept of the rate-determining step is very important to the optimization and understanding of many chemical processes such ascatalysis andcombustion.
As an example, consider the gas-phase reactionNO2 + CO → NO + CO2. If this reaction occurred in a single step, itsreaction rate (r) would be proportional to the rate ofcollisions betweenNO2 and CO molecules:r =k[NO2][CO], wherek is the reactionrate constant, and square brackets indicate amolar concentration. Another typical example is theZel'dovich mechanism.
In fact, however, the observed reaction rate issecond-order inNO2 and zero-order in CO,[5] with rate equationr =k[NO2]2. This suggests that the rate is determined by a step in which twoNO2 molecules react, with the CO molecule entering at another, faster, step. A possible mechanism in two elementary steps that explains the rate equation is:
In this mechanism thereactive intermediate speciesNO3 is formed in the first step with rater1 and reacts with CO in the second step with rater2. However,NO3 can also react with NO if the first step occurs in thereverse direction (NO +NO3 → 2NO2) with rater−1, where the minus sign indicates the rate of a reverse reaction.
The concentration of a reactive intermediate such as [NO3] remains low and almost constant. It may therefore be estimated by thesteady-state approximation, which specifies that the rate at which it is formed equals the (total) rate at which it is consumed. In this exampleNO3 is formed in one step and reacts in two, so that
The statement that the first step is the slow step actually means that the first stepin the reverse direction is slower than the second step in the forward direction, so that almost allNO3 is consumed by reaction with CO and not with NO. That is,r−1 ≪r2, so thatr1 −r2 ≈ 0. But the overall rate of reaction is the rate of formation of final product (here CO2), so thatr =r2 ≈r1. 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.
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:r2 ≪r−1. In this hypothesis,r1 − r−1 ≈ 0, so that the first step is (almost) atequilibrium. The overall rate is determined by the second step:r =r2 ≪r1, 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 (hereNO3) forms an equilibrium with reactantsprior to the rate-determining step is described as apre-equilibrium[6] For the reaction ofNO2 and CO, this hypothesis can be rejected, since it implies a rate equation that disagrees with experiment.
If the first step were at equilibrium, then itsequilibrium constant expression permits calculation of the concentration of the intermediateNO3 in terms of more stable (and more easily measured) reactant and product species:
The overall reaction rate would then be
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.
Another example is theunimolecular nucleophilic substitution (SN1) reaction in organic chemistry, where it is the first, rate-determining step that isunimolecular. A specific case is thebasichydrolysis oftert-butyl bromide (t-C
4H
9Br) by aqueoussodium hydroxide. The mechanism has two steps (where R denotes the tert-butyl radicalt-C
4H
9):
This reaction is found to befirst-order withr =k[R−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 ofmethyl bromide (CH
3Br) is abimolecular nucleophilic substitution (SN2) reaction in a singlebimolecular step. Its rate law issecond-order:r =k[R−Br][OH−
].
A useful rule in the determination of mechanism is that the concentration factors in the rate law indicate the composition and charge of theactivated complex ortransition state.[7] For theNO2–CO reaction above, the rate depends on [NO2]2, so that the activated complex has compositionN
2O
4, with 2NO2 entering the reaction before the transition state, and CO reacting after the transition state.
A multistep example is the reaction betweenoxalic acid and chlorine in aqueous solution:H
2C
2O
4 +Cl
2 → 2 CO2 + 2H+
+ 2Cl−
.[7]The observed rate law is
which implies an activated complex in which the reactants lose 2H+
+Cl−
before the rate-determining step. The formula of the activated complex isCl
2 +H
2C
2O
4 − 2H+
−Cl−
+x H2O, orC
2O
4Cl(H
2O)–
x (an unknown number of water molecules are added because the possible dependence of the reaction rate onH2O 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[7]
In a multistep reaction, the rate-determining step does not necessarily correspond to the highestGibbs energy on thereaction coordinate diagram.[8][6] If there is areaction intermediate whose energy is lower than the initial reactants, then the activation energy needed to pass through any subsequenttransition state 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.[8][9]
Also, for reaction steps that are not first-order, concentration terms must be considered in choosing the rate-determining step.[8][6]
Not all reactions have a single rate-determining step. In particular, the rate of achain reaction is usually not controlled by any single step.[8]
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 asdiffusion control 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.