D37.1 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 includes the concentration of one or more reaction intermediates.
Experimentally, the concentration of a reaction intermediate is rarely measurable. (Even if it can be measured, the accuracy is usually low.) Therefore, 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).
If a preceding step that forms the reaction intermediate is at equilibrium, then we can make use of the equilibrium approximation to express the concentration of the intermediate in terms of concentrations of reactants.
For example, consider the following multi-step reaction:
The currently accepted mechanism for this reaction is:
| Step 1: | NO(g) + Cl2(g) | ![\xrightleftharpoons[k_{-1}]{k_1}](https://wisc.pb.unizin.org/app/uploads/quicklatex/quicklatex.com-bf5cd59d1013b85671fed15bcdf128ff_l3.png "Rendered by QuickLaTeX.com") |
NOCl2(g) | fast |
| Step 2: | NOCl2(g) + NO(g) |  |
2NOCl(g) | slow |
| Overall: | 2NO(g) + Cl2(g) | ⟶ | 2NOCl(g) |
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 compared to experimental data. We need to express [NOCl2] in terms of concentrations of reactants.
Step 1 in this mechanism has a smaller activation energy and a larger reaction constant than step 2. When NO and Cl2 reacts 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 or react with NO to form two NOCl with rate constant k2. Because k2 << k-1, NOCl2 molecules are 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:
| rate1 | = | rate-1 |
| k1[NO][Cl2] | = | k-1[NOCl2] |
This relationship can be rearranged to solve for the concentration of NOCl2:
![[\text{NOCl}_2] = \left(\dfrac{k_1}{k_{-1}}\right)[\text{NO}][\text{Cl}_2]](https://wisc.pb.unizin.org/app/uploads/quicklatex/quicklatex.com-9ada7312d279b1ba2fd4886f8ba85e5b_l3.png "Rendered by QuickLaTeX.com")
And substituting this into the rate law for step 2, we have:
![\text{rate}_2 = k_2[\text{NOCl}_2][\text{NO}] = k_2\left(\dfrac{k_1}{k_{-1}}\right)[\text{NO}][\text{Cl}_2][\text{NO}] = \left(\dfrac{k_1k_2}{k_{-1}}\right)[\text{NO}]^2[\text{Cl}_2]](https://wisc.pb.unizin.org/app/uploads/quicklatex/quicklatex.com-d221f37525cde34fd6e447206a3fe5e7_l3.png "Rendered by QuickLaTeX.com")
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.
Activity: Rate Law from Mechanism
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