D30.3 Equilibrium Approximation

When the rate-determining step is not the first step, the rate law of the overall reaction can still be approximated as the rate law for the rate-determining step. However, one or more of the reactants involved in such a rate-determining step would be a reaction intermediate formed from a previous step. Hence, the rate law for the rate-determining step would include the concentration of a species that is rarely measurable. (Even if the concentration of a reaction intermediate can be measured experimentally, the accuracy is usually low because the concentration is very small.)

Experimentally determined rate laws are always expressed in terms of concentrations of reactants and/or products, for which accurate measurements are much easier to obtain. In order to compare the theoretical rate law derived from the mechanism to the experimentally determined rate law, we must express the mechanism rate law only in terms of reactant and product concentrations, that is, we cannot simply use the rate law of the rate-determining step as is (unless the rate-determining step is the first step).

In other words, it is necessary to express an intermediate’s concentration in terms of reactant and/or product concentrations. To do this, some approximation is usually needed. In one situation, where a preceding step that forms the reaction intermediate is at equilibrium, we can make use of the equilibrium approximation.

For an example, consider the following multi-step reaction:

The currently accepted mechanism for this reaction is:

Step 1:	NO(g) + Cl2(g)		NOCl2(g)	fast
Step 2:	NOCl2(g) + NO(g)		2NOCl(g)	slow
Overall:	2NO(g) + Cl2(g)	⟶	2NOCl(g)

In this mechanism, step 1 has a smaller activation energy and hence a larger rate constant than step 2. Therefore step 2 is the rate-determining step, and its rate law is:

However, this rate law involves the concentration of an intermediate, [NOCl₂], so it cannot be directly compared to experimental data. We need to express [NOCl₂] in terms of concentrations of reactants.

When NO and Cl₂ react to form NOCl₂ with rate constant k₁, there are two reactions possible for the NOCl₂:

• it can decompose back to form NO and Cl₂ with rate constant k_-1
  NOCl₂(g) NO(g) + Cl₂(g)
• it can react with NO to form two NOCl with rate constant k₂
  NOCl₂(g) + NO(g) 2 NOCl(g)

Because k₂ << k_-1, NOCl₂ molecules are much more likely to decompose than to react in step 2. When enough NOCl₂ has formed, we can approximate that the rate of its decomposition (rate_-1) is equal to the rate at which it forms (rate₁), meaning that step 1 is at equilibrium (rate₁ = rate_-1). This approximation gives us:

This relationship can be rearranged to solve for the concentration of NOCl₂:

Substituting this expression into the rate law for step 2, we have:

If we make:

then the rate law for the overall reaction becomes:

This rate law can be compared with experimental data to determine whether the proposed mechanism is a plausible one.