The iodate ion is reduced by sulfite according to the following reaction: $$\mathrm{IO}_{3}^{-}(a q)+3 \mathrm{SO}_{3}^{2-}(a q) \longrightarrow \mathrm{I}^{-}(a q)+3 \mathrm{SO}_{4}^{2-}(a q)$$ The rate of this reaction is found to be first order in \(\mathrm{IO}_{3}^{-}\) , first order in \(\mathrm{SO}_{3}^{2-}\) , and first order in \(\mathrm{H}^{+}\) . (a) Write the rate law for the reaction. (b) By what factor will the rate of the reaction change if the pH is lowered from 5.00 to 3.50\(?\) Does the reaction proceed more quickly or more slowly at the lower pH? (c) By using the concepts discussed in Section 14.6, ex-plain how the reaction can be pH-dependent even though H' does not appear in the overall reaction.

The rate law expresses the reaction rate as a product of the concentrations of the reactants raised to their respective orders. We are given that the reaction is first order in IO₃⁻, first order in SO₃²⁻, and first order in H⁺. Therefore, the rate law for the reaction can be written as: Rate = k[IO₃⁻][SO₃²⁻][H⁺] where k is the rate constant.

We know that pH is the negative logarithm of the hydrogen ion concentration: pH = -log[H⁺], so [H⁺] = 10^(-pH). Now, let's find the hydrogen ion concentrations at pH 5.00 and 3.50: [H⁺]_5.00 = 10^(-5.00) [H⁺]_3.50 = 10^(-3.50) Now let's find the ratio of the reaction rates at these two pH: Rate ratio = (Rate at pH 3.50) / (Rate at pH 5.00) Rate ratio = (k[IO₃⁻][SO₃²⁻][H⁺]_3.50) / (k[IO₃⁻][SO₃²⁻][H⁺]_5.00) Since [IO₃⁻] and [SO₃²⁻] remain unchanged, they cancel out in the ratio: Rate ratio = [H⁺]_3.50 / [H⁺]_5.00 Now substitute the concentrations of H⁺: Rate ratio = (10^(-3.50)) / (10^(-5.00)) Rate ratio = 10^(1.5) Calculating the exact value: Rate ratio ≈ 31.6 Thus, the rate of the reaction changes by a factor of 31.6 when the pH is lowered from 5.00 to 3.50. Since the reaction is first-order in H⁺, the reaction rate is directly proportional to the H⁺ concentration, which means the reaction proceeds more quickly at lower pH.

Although H⁺ ions are not involved in the overall reaction, they might be involved in one or more elementary steps of the reaction mechanism. For example, a possible elementary step could be the transfer of a proton (H⁺) between reactants, leading to the formation of intermediate species that participate in subsequent steps. The lower the pH, the higher the concentration of H⁺ ions, which increases the rate of proton transfer in the elementary steps. The pH dependency of the given reaction indicates that H⁺ ions play a role in the reaction mechanism, even if they do not appear in the overall balanced equation.