D30.2 First Step is Rate-Determining

When the rate-determining step is the first step in a mechanism, the rate law for the overall reaction can be approximated as the rate law for the first step. The reaction between NO₂ and CO provides an illustrative example:

At temperatures below 225 °C, the experimentally observed rate law is:

This is consistent with a mechanism that involves these two elementary reaction steps:

Step 1:	NO2(g) + NO2(g)		NO3(g) + NO(g)	slow (larger Ea)
Step 2:	NO3(g) + CO(g)		CO2(g) + NO2(g)	fast (smaller Ea)
Overall:	NO2(g) + CO(g)	⟶	CO2(g) + NO(g)

Both steps in this mechanism are bimolecular elementary reactions, and the sum of the two steps agrees with the overall reaction (the NO₃ that is formed in step 1 reacts away in step 2, and one of the NO₂ molecules that reacts in step 1 is reformed in step 2).

The figure below shows the corresponding reaction energy diagram for this mechanism. E_a,1 is the activation energy for step 1, and it reflects the energy difference between the reactants and the first transition state. E_a,2 is the activation energy for step 2, and it reflects the energy difference between the intermediary minimum and the second transition state.

With E_a,1 being much larger than E_a,2, step 1 has a much smaller rate constant than step 2 (k₁ << k₂, assuming that the frequency factor (A) for the two steps are similar). However, step 2 cannot occur until step 1 produces some amount of NO₃ (NO₃ is a reaction intermediate, and therefore its concentration is zero at the beginning of the reaction). Because k₂ is larger, as soon as some NO₃ molecules are formed, they readily react with CO to form CO₂ and NO₂. Hence, the concentration of NO₃ is always very small, and the rate of step 2 cannot go faster than the rate of step 1.

We can also think about this on a molecular-level. At the beginning of the reaction, there are only NO₂ and CO molecules moving around. Collisions between two NO₂ molecules, involving sufficient energy as well as correct orientation, would lead to the step 1 reaction proceeding forward (the rate constant is k₁ = A₁e^-E_a,1/RT). As reactive collisions occur, NO₃ molecules are formed. Because E_a,2 < E_a,1, when a NO₃ molecule collides with a CO molecule, it is much more likely for that collision to have sufficient energy to proceed forward (the rate constant is k₂ = A₂e^-E_a,2/RT). If there were more NO₃ molecules around, the rate of step 2 would be faster, but the concentration of NO₃ is very small and the rate of step 2 is limited.

Step 1 limits the rate of step 2, and is therefore the rate-determining step in this mechanism. Typically, the slowest reaction step is the one with the largest E_a and/or the highest energy transition state, as illustrated in the figure above.

The stoichiometry of step 1, which is an elementary reaction, gives this rate law:

which can be approximated as the rate law for the overall reaction, and it is in agreement with the experimentally observed rate law.