Question

The iodate ion is reduced by sulfite according to the following reaction: $$ \mathrm{IO}_{3}^{-}(a q)+3 \mathrm{SO}_{3}^{2-}(a q) \longrightarrow \Gamma(a q)+3 \mathrm{SO}_{4}^{2-}(a q) $$ 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 \(\mathrm{pH}\) is lowered from \(5.00\) to \(3.50\) ? Does the reaction proceed more quickly or more slowly at the lower \(\mathrm{pH}\) ? (c) By using the concepts discussed in Section 14.6, explain how the reaction can be \(\mathrm{pH}\)-dependent even though \(\mathrm{H}^{+}\)does not appear in the overall reaction.

Short answer

(a) The rate law for the reaction is: Rate = k[\(IO_3^{-}\)][\(SO_3^{2-}\)][\(H^+\)] (b) The rate of the reaction increases by a factor of 31.62 when the pH is lowered from 5.00 to 3.50. The reaction proceeds more quickly at the lower pH. (c) The pH-dependence can be explained by the role that \(H^+\) plays in the intermediate steps of the reaction mechanism, even though it does not appear in the overall reaction. A change in \(H^+\) concentration directly impacts the rate of the reaction, as indicated by the reaction order with respect to \(H^+\).

Step-by-step solution

(a) Writing the rate law

To write the rate law for the reaction, we use the general form: Rate = k[\(A^n\)][\(B^m\)] where k = rate constant n, m = reaction order A, B = reactants in the reaction Since the reaction is first order in \(IO_3^{-}\), first order in \(SO_3^{2-}\), and first order in \(H^+\), the rate law can be written as: Rate = k[\(IO_3^{-}\)][\(SO_3^{2-}\)][\(H^+\)]

(b) Factor of the rate change and the effect of pH change

Let Rate 1 be the rate of reaction at pH 5.00 and Rate 2 be that at pH 3.50. We have: Rate 1 = k[\(IO_3^{-}\)][\(SO_3^{2-}\)][\(H_1^+\)] Rate 2 = k[\(IO_3^{-}\)][\(SO_3^{2-}\)][\(H_2^+\)] Since the concentrations of \(IO_3^{-}\) and \(SO_3^{2-}\) remain constant, their ratios will cancel out: \(\frac{Rate 2}{Rate 1} = \frac{k[\(IO_3^{-}\)][\(SO_3^{2-}\)][\(H_2^+\)]}{k[\(IO_3^{-}\)][\(SO_3^{2-}\)][\(H_1^+\)]} = \frac{H_2^+}{H_1^+}\) To find the change in H+ concentration, we need to find the H+ concentrations at both pH levels: pH = -log(\(H^+\)) At pH 5, \(H_1^+ = 10^{-5}\) At pH 3.5, \(H_2^+ = 10^{-3.5}\) Now, we can find the rate change factor: \(\frac{Rate 2}{Rate 1} = \frac{H_2^+}{H_1^+}\) = \(\frac{10^{-3.5}}{10^{-5}} = 10^{1.5} = 31.62\) (approximately) The rate of the reaction increases by a factor of 31.62 when the pH is lowered from 5.00 to 3.50. The reaction proceeds more quickly at the lower pH.

(c) Explaining the pH-dependence

Although \(H^+\) does not appear in the overall reaction, it plays a crucial role in the reaction mechanism (the sequence of elementary steps). The presence of \(H^+\) influences the reaction by reacting with the reactants during the intermediate steps, thereby affecting their concentrations and the overall rate of the reaction. In this case, the reaction order with respect to \(H^+\) is 1, meaning that a change in the \(H^+\) concentration directly impacts the rate of the reaction. Since the pH of a solution is related to the concentration of \(H^+\) (\(pH = -log(H^+)\)), the rate of the reaction is dependent on the pH. This dependence on the pH, even when \(H^+\) does not appear in the overall reaction, can be explained by the role it plays in the intermediate steps of the reaction mechanism.

Key concepts

Understanding Reaction Rate Law

The reaction rate law expresses the relationship between the rate of a chemical reaction and the concentration of its reactants. For any given reaction, the rate at which it proceeds is directly proportional to the product of the reactants' concentrations raised to a power. This power is known as the reaction order for a given substance. In a general form, the rate law can be written as:
Rate = k[Reactant A]ⁿ[Reactant B]^m...
Here, 'k' represents the rate constant, which can vary with temperature but is independent of the concentration of reactants. The exponents 'n' and 'm' represent the orders of the reaction with respect to its reactants A and B, respectively. These exponents can be determined experimentally and are not necessarily related to the stoichiometry of the overall balanced equation.
Understanding the rate law is crucial because it allows chemists to predict how changes in reactant concentration impact the reaction rate. Regarding the iodate-sulfite reaction, since it is first order in each reactant including the hydrogen ion (H+), we can conclude that a proportional change in the concentration of any one of the reactants will lead to an equivalent change in the rate of the reaction.

pH Dependence in Reactions

The pH of a solution can have a profound influence on the rate of chemical reactions. pH measures the acidity or basicity of an aqueous solution and is inversely related to the hydrogen ion (H+) concentration. In many reactions, such as the one between iodate ion and sulfite, H+ acts as a catalyst, influencing the reaction rate without being consumed in the overall reaction.
When pH is altered, the H+ concentration changes accordingly, and since the rate depends on the H+ concentration, so does the rate of the reaction. The relationship between pH and H+ concentration is logarithmic:Lowering the pH from 5.00 to 3.50 implies an increase in the H+ concentration by a factor of 10^1.5 (or approximately 31.62). As the reaction is first order with respect to H+, the rate of the reaction is directly proportional to the H+ concentration, leading to a significant increase in reaction rate at lower pH values. This is an essential concept in chemical kinetics as it illustrates how environmental conditions can be manipulated to control reaction speeds.

Delving Into Reaction Mechanisms

A reaction mechanism is a series of steps that leads to the transformation of reactants into products. Each step is known as an elementary process, which involves a certain arrangement of molecules coming together, reacting, and then breaking apart to form new molecules. These processes can often include the formation of temporary, unstable intermediates and transition states.
In reactions where the rate is pH-dependent yet H+ does not appear in the overall balanced equation, it's important to understand that the H+ ion can take part in the individual elementary steps of the mechanism. While not directly involved in the final products, H+ can influence the stability of intermediates or the rate at which specific elementary steps occur, thereby altering the overall reaction rate. The iodate-sulfite reaction's pH-dependence, despite the absence of H+ in the net reaction, is a perfect demonstration of how reaction kinetics can't solely be deduced from the equation of the overall reaction but requires an understanding of the underlying mechanism.