In chemical kinetics, 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 (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 equations is not always easy, and in some cases numerical integration 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 NO2 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 and combustion.
Example reaction: NO2 + CO As an example, consider the gas-phase reaction NO2 + CO → NO + CO2. If this reaction occurred in a single step, its reaction rate (r) would be proportional to the rate of collisions between NO2 and CO molecules: r = k[NO2][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 NO2 and zero-order in CO, with rate equation r = k[NO2]2. This suggests that the rate is determined by a step in which two NO2 molecules react, with the CO molecule entering at another, faster, step. A possible mechanism in two elementary steps that explains the rate equation is:
NO2 + NO2 → NO + NO3 (slow step, rate-determining) NO3 + CO → NO2 + CO2 (fast step) In this mechanism the reactive intermediate species NO3 is formed in the first step with rate r1 and reacts with CO in the second step with rate r2. However, NO3 can also react with NO if the first step occurs in the reverse direction (NO + NO3 → 2 NO2) with rate r−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 the 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 NO3 is formed in one step and reacts in two, so that
d [ NO 3 ] d t = r 1 − r 2 − r − 1 ≈ 0. {\displaystyle {\frac {d{\ce {[NO3]}}}{dt}}=r_{1}-r_{2}-r_{-1}\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 NO3 is consumed by reaction with CO and not with NO. That is, r−1 ≪ r2, so that r1 − r2 ≈ 0. But the overall rate of reaction is the rate of formation of final product (here CO2), so that r = 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.
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: r2 ≪ r−1. In this hypothesis, r1 − r−1 ≈ 0, so that the first step is (almost) at equilibrium. 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 (here NO3) forms an equilibrium with reactants prior to the rate-determining step is described as a pre-equilibrium For the reaction of NO2 and CO, this hypothesis can be rejected, since it implies a rate equation that disagrees with experiment.
NO2 + NO2 → NO + NO3 (fast step) NO3 + CO → NO2 + CO2 (slow step, rate-determining) If the first step were at equilibrium, then its equilibrium constant expression permits calculation of the concentration of the intermediate NO3 in terms of more stable (and more easily measured) reactant and product species:
K 1 = [ NO ] [ NO 3 ] [ NO 2 ] 2 , {\displaystyle K_{1}={\frac {{\ce {[NO][NO3]}}}{{\ce {[NO2]^2}}}},}
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