13.39 The reaction of NO with \( \mathrm{O}_{2} \) to give \( \mathrm{NO}_{2} \) is an important process in the formation of smog in any large city: \[ 2 \mathrm{NO}+\mathrm{O}_{2} \rightarrow 2 \mathrm{NO}_{2} \] Experiments show that this reaction is third order overall. The following mechanism has been proposed: \[ \mathrm{NO}+\mathrm{NO} \underset{k_{-1}}{\stackrel{k_{1}}{\rightleftarrows}} \mathrm{~N}_{2} \mathrm{O}_{2} \] \[ \mathrm{~N}_{2} \mathrm{O}_{2}+\mathrm{O}_{2} \xrightarrow{k_{2}} \mathrm{NO}_{2}+\mathrm{NO}_{2} \] (a) If the second step is rate-determining, what is the rate law? (b) Is this rate law consistent with the overall third-order behaviour? Explain. (c) Draw molecular pictures that show different ways the intermediate species might bind together, and identify the one that is most reasonable with respect to the second step of the mechanism.

**(a)** The proposed mechanism is \[ \begin{aligned} \text{Step 1:} \quad & \mathrm{NO} + \mathrm{NO} \underset{k_{-1}}{\stackrel{k_{1}}{\rightleftarrows}} \mathrm{N_2O_2} \quad (\text{fast equilibrium})\\[1mm] \text{Step 2:} \quad & \mathrm{N_2O_2} + \mathrm{O_2} \xrightarrow{k_{2}} 2\,\mathrm{NO_2} \quad (\text{slow, rate-determining}) \end{aligned} \] Since the second step is the rate-determining step, the rate law is given by \[ \text{Rate} = k_{2} [\mathrm{N_2O_2}][\mathrm{O_2}]. \] To express \([\mathrm{N_2O_2}]\) in terms of \([\mathrm{NO}]\), we use the equilibrium established in the first step. The equilibrium constant \(K\) for step 1 is \[ K = \frac{[\mathrm{N_2O_2}]}{[\mathrm{NO}]^2} = \frac{k_1}{k_{-1}}, \] so \[ [\mathrm{N_2O_2}] = K [\mathrm{NO}]^2. \] Substituting this back into the rate law gives \[ \text{Rate} = k_{2} K [\mathrm{NO}]^2 [\mathrm{O_2}]. \] Thus, the rate law is \[ \boxed{\text{Rate} = k_{\text{obs}} [\mathrm{NO}]^2 [\mathrm{O_2}], \quad \text{with } k_{\text{obs}} = k_{2}K.} \] **(b)** The derived rate law is second order in \(\mathrm{NO}\) and first order in \(\mathrm{O_2}\). This means the overall reaction order is \[ 2 + 1 = 3, \] which is consistent with the experimentally observed third-order behavior. **(c)** The first step involves the dimerization of two \(\mathrm{NO}\) molecules to form the intermediate \(\mathrm{N_2O_2}\). There are several possible ways in which the two \(\mathrm{NO}\) molecules might associate. Two common conceptual pictures are: 1. **Head-to-Head Binding via Nitrogen Centers:** Each \(\mathrm{NO}\) molecule has an unpaired electron primarily on the nitrogen atom. The two nitrogen atoms can form a bond, leading to a structure like \[ \overset{\bullet}{\mathrm{N}}=\mathrm{O} \,\,\,\cdots\,\,\, \overset{\bullet}{\mathrm{N}}=\mathrm{O} \quad \longrightarrow \quad \mathrm{N\!-\!N} \text{ bond}, \] where the oxygen atoms remain free. This arrangement is advantageous because when \(\mathrm{O_2}\) approaches in the second step, it can easily access the reactive site on the \(\mathrm{N_2O_2}\) intermediate. 2. **Side-by-Side or End-On Association:** Alternatively, the \(\mathrm{NO}\) molecules might interact via their oxygen atoms or in a more parallel fashion. However, such an arrangement could cause steric hindrance or reduce the accessibility of the active sites needed for the reaction with \(\mathrm{O_2}\). The most reasonable picture, especially with respect to facilitating the second step, is the head-to-head binding via the nitrogen atoms. In this configuration, the intermediate \(\mathrm{N_2O_2}\) has a structure where the two nitrogen atoms are bonded together while the oxygen atoms are still available for further reaction. This arrangement minimizes steric hindrance and allows \(\mathrm{O_2}\) to attack readily, which is consistent with the observed kinetics of the second step. A simple diagram of the favorable binding is: \[ \begin{array}{cccc} \mathrm{O} & & \mathrm{O} \\ \quad | & & | \\ \overset{\bullet}{\mathrm{N}} & - & \overset{\bullet}{\mathrm{N}} \\ \quad & & \\ \end{array} \] Here, the \(\mathrm{N\!-\!N}\) bond is formed between the two nitrogen atoms (each indicated with an unpaired electron), leaving the oxygen atoms free. This orientation facilitates the approach of \(\mathrm{O_2}\) to react with the intermediate in the rate-determining second step. \[ \boxed{\text{The mechanism with head-to-head (nitrogen-to-nitrogen) association is the most reasonable.}} \]