Rate-determining step

In chemical kinetics, the rate (or velocity) of a reaction mechanism with several steps is often determined by the slowest step, known as the rate-determining step (RDS) or rate-limiting step. The experimental rate equation can help to identify which step is rate-determining.

In a reaction coordinate, the transition state with the highest energy is the rate-determining step of a given reaction.

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.

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 which explains the rate equation is:

1. NO
   ₂ + NO
   ₂ → NO + NO
   ₃ (slow step, rate-determining)
2. NO
   ₃ + CO → NO
   ₂ + CO
   ₂ (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 ${\frac {{d}{[NO_{3}]}}{dt}}=r_{1}-r_{2}-r_{-1}\approx 0$

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 which react by the first step continue to the second step which is just as fast.

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 which react by the first step continue to the second step which is much slower. However this hypothesis can be rejected (for the example reaction) since it implies a rate equation which disagrees with experiment.

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

$K_{1}={\frac {{[NO]}{[NO_{3}]}}{{[NO_{2}]}^{2}}}$

$[NO_{3}]=K_{1}{\frac {[NO_{2}]^{2}}{[NO]}}$

The overall reaction rate would then be

$r=r_{2}=k_{2}[NO_{3}][CO]=k_{2}K_{1}{\frac {[NO_{2}]^{2}[CO]}{[NO]}}$,

which disagrees with the experimental rate law given above, and so disproves the hypothesis that the second step is rate-determining.

Nucleophilic substitution

Another example is the unimolecular nucleophilic substitution (S_N1) reaction in organic chemistry, where it is the first, rate-determining step that is unimolecular. A specific case is the basic hydrolysis of tert-butyl bromide (t-C₄H₉Br) by aqueous sodium hydroxide. The mechanism has two steps (where R denotes the tert-butyl radical t-C₄H₉)

1. Formation of a carbocation R-Br → R⁺ + Br_-
2. Nucleophilic attack by one water molecule R⁺ + OH^- → ROH.

This reaction is found to be first-order with rate = k[RBr], 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 (CH₃Br) is a bimolecular nucleophilic substitution (S_N2) reaction in a single bimolecular step. Its rate law is second-order, rate = k[RBr][OH^-].

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 interact and form the product. This case is referred to as diffusion control and, in general, occurs where the formation of product from the activated complex is very rapid and thus the supply of reactants and their interaction is rate determining.

References

• Zumdahl, Steven S. Chemical Principles. $5^{th}$ Ed. Boston, Houghton Mifflin Company: 2005, pp 727-728.

See also

• Product-determining step