12.6 Reaction mechanisms  (Page 4/13)

Deriving the overall rate law expression for a multistep reaction mechanism

1. $2{\text{NO}}_2\text{Cl}(g)\rightleftharpoons {\text{ClO}}_2(g)+{\text{N}}_2\text{O}(g)+\text{ClO}(g)$ (fast, k ₁ represents the rate constant for the forward reaction and k ₋₁ the rate constant for the reverse reaction)
2. ${\text{N}}_2\text{O}(g)+{\text{ClO}}_2(g)\rightleftharpoons {\text{NO}}_2(g)+\text{NOCl}(g)$ (fast, k ₂ for the forward reaction, k ₋₂ for the reverse reaction)
3. $\text{NOCl}+\text{ClO}⟶{\text{NO}}_2+{\text{Cl}}_2$ (slow, k ₃ the rate constant for the forward reaction)

Determine the overall reaction, write the rate law expression for each elementary reaction, identify any intermediates, and determine the overall rate law expression.

Solution

Next, write the rate law expression for each elementary reaction. Remember that for elementary reactions that are part of a mechanism, the rate law expression can be derived directly from the stoichiometry:

The third step, which is the slow step, is the rate-determining step. Therefore, the overall rate law expression could be written as Rate = k ₃ [NOCl][ClO]. However, both NOCl and ClO are intermediates. Algebraic expressions must be used to represent [NOCl] and [ClO]such that no intermediates remain in the overall rate law expression.

Using elementary reaction 1, $\left[\text{ClO}\right]=\frac{k_1{\left[{\text{NO}}_2\text{Cl}\right]}^2}{k_{-1}\left[{\text{ClO}}_2\right]\left[{\text{N}}_2\text{O}\right]}.$

Using elementary reaction 2, $\left[\text{NOCl}\right]=\frac{k_2\left[{\text{N}}_2\text{O}\right]\left[{\text{ClO}}_2\right]}{k_{\mathrm{-2}}\left[{\text{NO}}_2\right]}.$

Now substitute these algebraic expressions into the overall rate law expression and simplify:

Notice that this rate law shows an inverse dependence on the concentration of one of the product species, consistent with the presence of an equilibrium step in the reaction mechanism.

Check your learning

Determine the overall reaction, write the rate law expression for each elementary reaction, identify any intermediates, and determine the overall rate law expression.

Key concepts and summary

The sequence of individual steps, or elementary reactions, by which reactants are converted into products during the course of a reaction is called the reaction mechanism. The overall rate of a reaction is determined by the rate of the slowest step, called the rate-determining step. Unimolecular elementary reactions have first-order rate laws, while bimolecular elementary reactions have second-order rate laws. By comparing the rate laws derived from a reaction mechanism to that determined experimentally, the mechanism may be deemed either incorrect or plausible.

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