# D36.4 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 of NO2 and CO provides an illustrative example:

NO2(g) + CO(g) ⟶ CO2(g) + NO(g)

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

rate = k[NO2]2

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. Ea,1 is the activation energy for step 1, and it reflects the energy difference between the reactants and the first transition state. Ea,2 is the activation energy for step 2, and it reflects the energy difference between the intermediary minimum and the second transition state.

Figure: Reaction energy diagram for the NO2(g) + CO(g) → CO2(g) + NO(g) reaction. Click on each “i” in the figure for additional information.

With Ea,1 being much larger than Ea,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.

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 Ea 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:

rate1 = k1[NO2]2

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