Question

Consider the following reaction: $${\mathrm{CH}}_3\mathrm{Br}(aq)+{\mathrm{OH}}^{-}(aq)⟶{\mathrm{CH}}_3\mathrm{OH}(aq)+{\mathrm{Br}}^{-}(aq)$$ The rate law for this reaction is first order in ${\mathrm{CH}}_3\mathrm{Br}$ and first order in ${\mathrm{OH}}^{-}$. When $\left[{\mathrm{CH}}_3\mathrm{Br}\right]$ is $5.0\times {10}^{-3}\mathrm{M}$ and $\left[{\mathrm{OH}}^{-}\right]$ is $0.050\mathrm{M},$ the reaction rate at $298\mathrm{K}$ is $0.0432\mathrm{M}/\mathrm{s}$. (a) What is the value of the rate constant? (b) What are the units of the rate constant? (c) What would happen to the rate if the concentration of ${\mathrm{OH}}^{-}$ were tripled? (d) What would happen to the rate if the concentration of both reactants were tripled?

Short answer

(a) The value of the rate constant, k, is approximately 17.28 M^-1s^-1. (b) The units of the rate constant, k, are M^-1s^-1. (c) If the concentration of OH- is tripled, the reaction rate becomes approximately 0.1296 M/s. (d) If the concentrations of both reactants are tripled, the reaction rate becomes approximately 0.3888 M/s.

Step-by-step solution

Write the rate law

The rate law is given as: Rate = k [CH3Br]^1 [OH-]^1, where k is the rate constant that we need to find, and [CH3Br] and [OH-] are the concentrations of the reactants CH3Br and OH-, respectively.

Find the value of the rate constant (k)

Given, the initial concentrations of the reactants are: [CH3Br] = 5.0 × 10^-5 M [OH-] = 0.050 M and the reaction rate is 0.0432 M/s. Substitute these values into the rate law equation: 0.0432 = k (5.0 × 10^-5) (0.050) Solve for k: k = 0.0432 / ((5.0 × 10^-5) × (0.050)) k ≈ 17.28 M^-1s^-1 The value of the rate constant, k, is approximately 17.28 M^-1s^-1.

Identify the units of the rate constant (k)

From the above calculation, we can see that the units of the rate constant, k, are M^-1s^-1.

Calculate the rate if the concentration of OH- is tripled

If the concentration of OH- is tripled, the new concentration will be 3 × 0.050 M = 0.150 M. Use the rate law equation with the new concentration: Rate_new = k [CH3Br] [OH-]_new Rate_new = 17.28 (5.0 × 10^-5) (0.150) Rate_new ≈ 0.1296 M/s If the concentration of OH- is tripled, the reaction rate becomes approximately 0.1296 M/s.

Calculate the rate if the concentrations of both reactants are tripled

If the concentrations of both reactants are tripled, the new concentrations will be: [CH3Br]_new = 3 × (5.0 × 10^-5) = 15.0 × 10^-5 M [OH-]_new = 3 × 0.050 = 0.150 M Use the rate law equation with the new concentrations: Rate_new = k [CH3Br]_new [OH-]_new Rate_new = 17.28 (15.0 × 10^-5) (0.150) Rate_new ≈ 0.3888 M/s If the concentrations of both reactants are tripled, the reaction rate becomes approximately 0.3888 M/s.

Key concepts

Reaction Rate

Understanding the speed at which chemical reactions take place is crucial for both students and scientists. The reaction rate is a measure of the change in concentration of reactants or products over time in a chemical reaction. Think of it as how fast or slow a reactant is consumed or a product is formed.

For example, in the given reaction of ${\mathrm{CH}}_3\mathrm{Br}$ with ${\mathrm{OH}}^{-}$, the reaction rate was given as $0.0432\mathrm{M}/\mathrm{s}$. This means that the concentration of either the reactants or products changes by this amount every second under the specified conditions.

Various factors can affect the reaction rate, such as temperature, concentration of reactants, surface area, and presence of catalysts. A crucial aspect for students to understand is that reaction rates are proportional to the concentration of reactants in many cases, which is reflected in the rate law for the reaction.

Rate Law

The rate law is a mathematical expression that links the reaction rate to the concentration of reactants. It allows us to predict the speed of a chemical reaction under different conditions. The rate law for a reaction is determined experimentally and cannot be deduced from the stoichiometry of the reaction alone.

For the reaction at hand, the rate law was expressed as Rate = k [CH3Br]^1 [OH-]^1. This tells us that the reaction is first order in ${\mathrm{CH}}_3\mathrm{Br}$ and ${\mathrm{OH}}^{-}$, meaning that the rate of reaction is directly proportional to the concentration of each reactant raised to the first power. If the concentration of ${\mathrm{CH}}_3\mathrm{Br}$ or ${\mathrm{OH}}^{-}$ is doubled, the reaction rate will also double.

It is essential to grasp that the rate law provides invaluable insights into the kinetic behavior of a reaction and helps in the study of reaction mechanisms.

Rate Constant

At the heart of the rate law is the rate constant, often symbolized by $k$. This constant is a measure of the intrinsic speed of a chemical reaction, and it varies with temperature but is independent of the reactant concentrations. The rate constant plays a key role in determining how fast a reaction proceeds under certain conditions.

In the solved exercise, the calculated rate constant was $17.28{\mathrm{M}}^{-1}{\mathrm{s}}^{-1}$. The units of the rate constant can vary depending on the overall order of the reaction. Since our reaction is second order (first order with respect to each reactant), the units for the rate constant are ${\mathrm{M}}^{-1}{\mathrm{s}}^{-1}$, revealing the dependence on both concentration and time.

Understanding the rate constant is fundamental as it helps predict the effects of changing conditions on the reaction rate. For instance, tripling the concentration of ${\mathrm{OH}}^{-}$ or both reactants significantly increases the reaction rate, as seen in the provided solution. Such knowledge is particularly valuable in chemical industries and research where controlling the reaction speed is often as critical as the reaction itself.