The reaction $2 \mathrm{NO}(g)+\mathrm{Cl}_{2}(g) \longrightarrow 2 \mathrm{NOCl}(g)$ was performed and the following data were obtained under conditions of constant \(\left[\mathrm{Cl}_{2}\right]:\) (a) Is the following mechanism consistent with the data? $$ \begin{aligned} \mathrm{NO}(g)+\mathrm{Cl}_{2}(g) & \rightleftharpoons \mathrm{NOCl}_{2}(g) \quad(\text { fast }) \\ \mathrm{NOCl}_{2}(g)+\mathrm{NO}(g) & \longrightarrow 2 \mathrm{NOCl}(g) &(\text { slow }) \end{aligned} $$ (b) Does the linear plot guarantee that the overall rate law is second order?

The slow step in the proposed mechanism is: $$ \mathrm{NOCl}_{2}(g) + \mathrm{NO}(g) \longrightarrow 2 \mathrm{NOCl}(g) $$ Since the rate of the reaction depends on the slowest step, the rate law for this step can be written as: $$ \text{rate} = k[\mathrm{NOCl}_{2}][\mathrm{NO}] $$ Here, k is the rate constant for this step.

We use the steady-state approximation to assume that the concentration of the intermediate, NOCl₂, remains constant throughout the reaction. This means that the rate of its formation equals the rate of its consumption. The intermediate is formed in the fast step and consumed in the slow step. Therefore, we can write: $$ k_{1}[\mathrm{NO}][\mathrm{Cl}_{2}] = k_{2}[\mathrm{NOCl}_{2}][\mathrm{NO}] $$ Here, \(k_{1}\) and \(k_{2}\) are the rate constants of the fast and slow steps, respectively. Now, we can solve for [\(\mathrm{NOCl}_{2}\)]: $$ [\mathrm{NOCl}_{2}] = \frac{k_{1}}{k_{2}}\frac{[\mathrm{NO}][\mathrm{Cl}_{2}]}{[\mathrm{NO}]} $$ By simplifying, we get: $$ [\mathrm{NOCl}_{2}] = \frac{k_{1}}{k_{2}}[\mathrm{Cl}_{2}] $$ Since Cl₂ has a constant concentration, \(\frac{k_{1}}{k_{2}}[\mathrm{Cl}_{2}]\) is also a constant. Let's call it K: $$ [\mathrm{NOCl}_{2}] = K $$

Now, we can substitute the expression for [\(\mathrm{NOCl}_{2}\)] in the rate law of the slow step: $$ \text{rate} = k[\mathrm{NOCl}_{2}][\mathrm{NO}] = k K [\mathrm{NO}] $$ This rate law indicates a first-order dependence on NO. As the concentration of Cl₂ is held constant, the observed experimental data is consistent with this mechanism.

The linear plot initially given in the problem refers to a plot of the rate versus the concentration of NO. If the plot is linear, it indicates a first-order overall rate law concerning NO. However, this does not guarantee that the reaction is second-order. We can only conclude that the reaction is first-order with respect to NO concentration, as shown by our analysis of the proposed mechanism.