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 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: + CO As an example, consider the gas-phase reaction + CO → NO + . If this reaction occurred in a single step, its reaction rate (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 species is formed in the first step with rate r1 and reacts with CO in the second step with rate r2.