Question

A reaction in which \(\mathrm{A}, \mathrm{B},\) and \(\mathrm{C}\) react to form products is zero order in \(\mathrm{A},\) one-half order in \(\mathrm{B},\) and second order in \(\mathrm{C} .\) \begin{equation} \begin{array}{l}{\text { a. Write a rate law for the reaction. }} \\ {\text { b. What is the overall order of the reaction? }} \\ {\text { c. By what factor does the reaction rate change if }[\mathrm{A}] \text { is doubled (and the }} \\ {\text { other reactant concentrations are held constant)? }}\end{array} \end{equation} \begin{equation} \begin{array}{l}{\text { d. By what factor does the reaction rate change if }[\mathrm{B}] \text { is doubled (and the }} \\ {\text { other reactant concentrations are held constant)? }} \\ {\text { e. By what factor does the reaction rate change if }[\mathrm{C}] \text { is doubled (and the }} \\\ {\text { other reactant concentrations are held constant)? }}\\\\{\text { f. By what factor does the reaction rate change if the concentrations of }} \\\ {\text { all three reactants are doubled? }}\end{array} \end{equation}

Short answer

a. Rate = k[A]^{0}[B]^{1/2}[C]^{2}, b. Overall order is 2.5, c. No change in rate, d. Rate increases by √2, e. Rate increases by 4, f. Rate increases by 4√2.

Step-by-step solution

Write the Rate Law

The rate law is determined by the orders of reaction with respect to each reactant. Given that the reaction is zero order in A, one-half order in B, and second order in C, the rate law can be written as: Rate = k[A]^{0}[B]^{1/2}[C]^{2}. The exponents correspond to the reaction order of each reactant.

Determine the Overall Order of the Reaction

The overall order of reaction is the sum of the orders of each reactant: (0) + (1/2) + (2) = 5/2 or 2.5. The reaction is 2.5 order overall.

Effect of Doubling [A] on the Reaction Rate

Since the reaction is zero order in A, changing its concentration has no effect on the reaction rate. Therefore, doubling [A] would result in no change in the reaction rate: Rate Change Factor = [A]^{0} = 1^{0} = 1.

Effect of Doubling [B] on the Reaction Rate

The reaction is one-half order in B. If [B] is doubled, the rate would increase by a factor of 2^{1/2} or √2. Rate Change Factor when [B] is doubled: (2)^{1/2} = √2.

Effect of Doubling [C] on the Reaction Rate

The reaction is second order in C. If [C] is doubled, the rate would increase by a factor of 2^{2} = 4. Rate Change Factor when [C] is doubled: (2)^{2} = 4.

Effect of Doubling All Reactants on the Reaction Rate

When all the reactants are doubled, the rate change factor will be the product of the individual change factors. For [A], the factor is 1; for [B], it is √2; and for [C], it is 4. Hence, the total factor is: 1 * √2 * 4 = 4√2. Note that increasing [A] has no effect as the reaction is zero order with respect to A.

Key concepts

Rate Law

The rate law is a mathematical equation that describes the relationship between the rate of a chemical reaction and the concentration of reactants. It allows us to understand how different factors, such as changes in concentration or the presence of a catalyst, affect the speed at which a reaction takes place. For instance, if we have a reaction where substances A, B, and C combine to form products, the rate law might look something like this:
\( \text{Rate} = k[A]^x[B]^y[C]^z \).
Here, \( k \) is the rate constant, a unique value that depends on the reaction and conditions, while \( x \), \( y \), and \( z \) represent the reaction orders with respect to A, B, and C respectively. These exponents tell us how the rate is affected by changes in concentration of each reactant. If any of the exponents are zero, it indicates that changing the concentration of that reactant has no effect on the rate of reaction.

Overall Reaction Order

Understanding overall reaction order is essential for a comprehensive study of reaction kinetics. The overall reaction order is simply the sum of the exponents in the rate law equation. Taking our previous example, with exponents as the reaction orders of each reactant in the rate law, the overall reaction order is the addition of these values: \( x + y + z \). If we have a reaction where A is zero order, B is one-half order, and C is second order, by adding up these orders (0 for A, 1/2 for B, and 2 for C), we determine that the reaction has an overall order of 2.5. This gives us insight into how the reaction rate will respond to changes in the concentration of all the reactants combined. A reaction's overall order is crucial when comparing different reactions and predicting the impact of concentration changes on the rate of reaction.

Reactant Concentration Effects

Reactant concentration has a pivotal role in chemical kinetics. Each reactant's impact is governed by its reaction order, as depicted in the rate law. A zero-order reaction means that changes in the reactant's concentration don't influence the rate. A reaction with a higher reaction order, like second order, is more sensitive to changes in reactant concentration. For a reaction that's second order in one reactant, doubling the concentration of that particular reactant will quadruple the reaction rate, since the rate change is proportional to the square of its concentration change (\( 2^2 = 4 \)). Understanding how individual and combined changes in reactant concentrations affect the reaction rate can be highly useful in industrial processes and experimental design, allowing chemists to control reaction speeds for desired outcomes.