Rate Laws for Complicated Reactions

Author: John Hutchinson

Our collision model in the previous section accounts for the concentration and temperature dependence of the reaction rate, as expressed by the rate law. The concentration dependence arises from calculating the probability of the reactant molecules being in the same vicinity at the same instant. Therefore, we should be able to predict the rate law for any reaction by simply multiplying together the concentrations of all reactant molecules in the balanced stoichiometric equation. The order of the reaction should therefore be simply related to the stoichiometric coefficients in the reaction. However, table 4 shows that this is incorrect for many reactions.

Consider for example the apparently simple reaction

2 ICl(g) + H2(g) 2 HCl(g) + I2(g)

[20]

Based on the collision model, we would assume that the reaction occurs by 2ICl molecules colliding with a single H₂ molecule. The probability for such a collision should be proportional to [ICl]²[H₂]. However, experimentally we observe (see table 4) that the rate law for this reaction is

Rate = k[ICl][H2]

[21]

As a second example, consider the reaction

NO2(g) + CO(g) NO(g) + CO2(g)

[22]

It would seem reasonable to assume that this reaction occurs as a single collision in which an oxygen atom is exchanged between the two molecules. However, the experimentally observed rate law for this reaction is

Rate = k[NO2]2

[23]

In this case, the [CO] concentration does not affect the rate of the reaction at all, and the [NO₂] concentration is squared. These examples demonstrate that the rate law for a reaction cannot be predicted from the stoichiometric coefficients and therefore that the collision model does not account for the rate of the reaction. There must be something seriously incomplete with the collision model.

The key assumption of the collision model is that the reaction occurs by a single collision. Since this assumption leads to incorrect predictions of rate laws in some cases, the assumption must be invalid in at least those cases. It may well be that reactions require more than a single collision to occur, even in reactions involving just two types of molecules as in equation 22. Moreover, if more than two molecules are involved as in equation 20, the chance of a single collision involving all of the reactive molecules becomes very small. We conclude that many reactions, including those in equation 20 and equation 22, must occur as a result of several collisions occurring in sequence, rather than a single collision. The rate of the chemical reaction must be determined by the rates of the individual steps in the reaction.

Each step in a complex reaction is a single collision, often referred to as an elementary process. In single collision process step, our collision model should correctly predict the rate of that step. The sequence of such elementary processes leading to the overall reaction is referred to as the reaction mechanism. Determining the mechanism for a reaction can require gaining substantially more information than simply the rate data we have considered here. However, we can gain some progress just from the rate law.

Consider for example the reaction in equation 22 described by the rate law in equation 23. Since the rate law involved [NO₂]², one step in the reaction mechanism must involve the collision of two NO₂ molecules. Furthermore, this step must determine the rate of the overall reaction. Why would that be? In any multi-step process, if one step is considerably slower than all of the other steps, the rate of the multi-step process is determined entirely by that slowest step, because the overall process cannot go any faster than the slowest step. It does not matter how rapidly the rapid steps occur. Therefore, the slowest step in a multi-step process is thus called the rate determining or rate limiting step.

This argument suggests that the reaction in equation 22 proceeds via a slow step in which two NO₂ molecules collide, followed by at least one other rapid step leading to the products. A possible mechanism is therefore

Step 1

NO2+NO2 NO3+NO

[24]

Step 2

NO3+CO NO2+CO2

[25]

If Step 1 is much slower than Step 2, the rate of the reaction is entirely determined by the rate of Step 1. From our collision model, the rate law for Step 1 must be Rate = k[NO₂]², which is consistent with the experimentally observed rate law for the overall reaction. This suggests that the mechanism in equation 24 and equation 25 is the correct description of the reaction process for equation 22, with the first step as the rate determining step.

There are a few important notes about the mechanism. First, one product of the reaction is produced in the first step, and the other is produced in the second step. Therefore, the mechanism does lead to the overall reaction, consuming the correct amount of reactant and producing the correct amount of reactant. Second, the first reaction produces a new molecule, NO₃, which is neither a reactant nor a product. The second step then consumes that molecule, and NO₃ therefore does not appear in the overall reaction, equation 22. As such, NO₃ is called a reaction intermediate. Intermediates play important roles in the rates of many reactions.

If the first step in a mechanism is rate determining as in this case, it is easy to find the rate law for the overall expression from the mechanism. If the second step or later steps are rate determining, determining the rate law is slightly more involved. The process for finding the rate law in such a case is illustrated in exercise 11.