In a kinetic study of the reaction \(2 \mathrm{ClO}_{2}(a q)+2 \mathrm{OH}^{-}(a q) \rightarrow\) \(\mathrm{ClO}_{3}^{-}(a q)+\mathrm{ClO}_{2}^{-}(a q)+\mathrm{H}_{2} \mathrm{O}(l)\) the following rate data were obtained. Write a rate law complete with proper values for the orders. What is the overall order of the reaction? $$\begin{array}{cccc} \text { Experiment } & {\left[\mathrm{ClO}_{2}\right]} & {\left[\mathrm{OH}^{-}\right]} & \text {Rate }(\mathbf{M} / \mathrm{s}) \\ \hline 1 & 0.060 \mathrm{M} & 0.030 \mathrm{M} & 0.02484 \\ 2 & 0.020 \mathrm{M} & 0.030 \mathrm{M} & 0.00276 \\ 3 & 0.020 \mathrm{M} & 0.090 \mathrm{M} & 0.00828 \end{array}$$

A rate law equation defines the rate of a reaction with respect to the concentrations of the reactants. It has the general form: Rate = k [A]^m [B]^n Where k is the rate constant, [A] and [B] are the concentrations of the reactants, and m and n are the orders of the reaction with respect to A and B respectively. For the given reaction, our rate law has the form: Rate = k [ClO2]^m [OH-]^n

To determine the orders m and n, we'll analyze how the rate changes with respect to the concentrations of ClO2 and OH-. Let's focus on Experiments 1 and 2 to find m since only the concentration of ClO2 changes between these two experiments: \(\frac{Rate_2}{Rate_1} = \frac{k [ClO2]_2^m [OH^-]_2^n}{k [ClO2]_1^m [OH^-]_1^n}\) Divide the rate expressions and plug in the given concentrations to get: \(\frac{0.00276}{0.02484} = \frac{0.020^m}{0.060^m}\) Dividing the left side of the equation should give \(0.111 = \frac{1}{3^{m}}\) which implies that m = 2. Now, let's look at Experiments 2 and 3 to find the order n, since only the concentration of OH- changes between these two experiments: \(\frac{Rate_3}{Rate_2} = \frac{k [ClO2]_3^m [OH^-]_3^n}{k [ClO2]_2^m [OH^-]_2^n}\) \(\frac{0.00828}{0.00276} = \frac{0.090^n}{0.030^n}\) Dividing the left side of the equation should give \(3 = 3^n\) which implies that n = 1.

Since we have already determined the values of m and n, we can now write the complete rate law: Rate = k [ClO2]^2 [OH-]^1

Finally, we can calculate the overall order of the reaction by adding the orders of individual reactants: Overall order = 2 + 1 = 3 Thus, the overall order of the given reaction is 3.