Write the rate law for the reaction, expressing k(overall) in terms of k1, k2, and k3.
Step 1: UV light + Cl2(g) ⇄ 2Cl(g)
Fast, Endothermic
Step 2: Cl(g) + O3(g) -> OCl(g) + O2(g)
Slow, Endothermic
Step 3: OCl(g) + NO(g) -> NO2(g) + Cl(g)
Fast, Exothermic
To derive the rate law for the reaction, we need to determine the rate-determining step. This is usually the slowest step in the reaction mechanism, which in this case is Step 2.
Based on Step 2, we can write the rate law as follows:
rate = k2[Cl(g)][O3(g)]
Next, let's express the concentration of [O3(g)] in terms of the concentrations of other species. From Step 1, we see that Cl(g) is produced by the reaction of Cl2(g) and UV light. Therefore, [Cl(g)] is equal to [Cl2(g)] at equilibrium.
Substituting [Cl(g)] = [Cl2(g)] and rearranging the rate law, we have:
rate = k2[Cl2(g)][O3(g)]
Now, let's express the concentration of [Cl2(g)] in terms of [Cl(g)]. From Step 1, we see that 2 moles of Cl(g) are produced for every 1 mole of Cl2(g) consumed. Therefore, [Cl2(g)] = 0.5[Cl(g)] at equilibrium.
Substituting [Cl2(g)] = 0.5[Cl(g)] into the rate law, we get:
rate = k2(0.5[Cl(g)])([O3(g)])
Finally, we can rewrite the rate law in terms of k(overall), k1, k2, and k3 by combining the rate constants:
rate = (k(overall))[Cl(g)][O3(g)]
= (k(overall))(0.5[Cl(g)])([O3(g)])
= (k(overall))(k1)(k2)(k3)[Cl(g)][O3(g)]
Therefore, the rate law for the reaction is:
rate = (k(overall))(k1)(k2)(k3)[Cl(g)][O3(g)]
Where k(overall) = k1k2k3.
To determine the rate law for the overall reaction, we need to look at the slowest step, which is Step 2:
Step 2: Cl(g) + O3(g) -> OCl(g) + O2(g)
The rate law for Step 2 can be written as:
Rate = k2[Cl][O3]
Where k2 is the rate constant for Step 2.
Since Step 2 is the rate-determining step, it is also the step that determines the overall rate law. Therefore, the rate law for the overall reaction is also given by:
Rate = k(overall)[Cl][O3]
Now, let's find k(overall) in terms of k1, k2, and k3.
Step 1: UV light + Cl2(g) ⇄ 2Cl(g) (Fast, Endothermic)
Step 1 is a fast step, which means it quickly reaches equilibrium. Therefore, we can assume that the concentration of Cl2 is constant throughout the reaction. Thus, we can write the rate law for Step 1 as:
Rate = k1[UV light][Cl2]
Since Step 1 is not the rate-determining step, the concentration of Cl2 does not affect the overall rate law.
Step 3: OCl(g) + NO(g) -> NO2(g) + Cl(g) (Fast, Exothermic)
Step 3 is also a fast step. However, it does not involve any of the reactants or products in the rate law expression. Therefore, the rate law for Step 3 does not affect the overall rate law.
Since the overall rate law is determined by the rate-determining step (Step 2), we can conclude that:
k(overall) = k2
Therefore, the rate constant k(overall) is equal to k2, and it is not influenced by k1 or k3.