Note that the rate law given at the end of the Information section is second order in \(\mathrm{NO}\) and first order in \(\mathrm{O}_{2}\), as is the experimental rate law. What relationship must exist between \(k_{\exp }, k_{2}\) and \(K\) for the proposed mechanism to be consistent with the experimental rate law?

Start by examining the rate law given. This rate law implies that the rate of reaction is proportionally related to the concentration of NO squared and O2. It could be written mathematically as \(rate = k_{exp}[NO]^{2}[O_{2}]\). Here, \(k_{exp}\) is the experimental second order rate constant for NO and first order for O2.

The proposed mechanism is suggesting that the reaction follows a certain order with respect to NO and O2, and has a certain equilibrium constant K. The mechanism is often represented with a sequence of elementary reactions, each with its own rate constant. As no specific mechanism is given in the problem, let's assume a simple mechanism where the second step is the rate-limiting step (since it matches the experimental rate law). In this hypothetical mechanism, K is typically the ratio of the rate constant for the forward reaction to the rate constant of the reverse reaction in an elementary step, and its meaning would depend on these steps.

The relationship is formulated based on the match between the experimental rate law and the rate law proposed by the mechanism. The experiment rate law suggests that \(rate = k_{exp}[NO]^{2}[O_{2}]\). If we assume as in the previous step that the second elementary reaction is the rate-limiting step, then its rate law would have some form like \(rate = k_{2}[NO][O_{2}][something]\) (the [something] would need to match \(k_{exp}[NO]\) to make the two rate laws align). The equilibrium of the first step, described by K, could provide this [something]. For example, if the first step was \(2NO <=> something\), at equilibrium, \(K = [something]/[NO]^{2}\), hence, \([something] = K[NO]^{2}\). Substituting this into the rate law of the second step would make its form match the experimental rate law. Therefore, \(k_{exp}\) would need to be equal to \(k_{2}K\) to have consistency between the experimental rate law and the proposed mechanism.