Question

Consider the reaction $$ \mathrm{A} \longrightarrow \mathrm{B} $$ The rate of the reaction is \(1.6 \times 10^{-2} M / \mathrm{s}\) when the concentration of A is \(0.35 M\). Calculate the rate constant if the reaction is (a) first order in \(\mathrm{A},\) (b) second order in A.

Short answer

The rate constant, \(k\), for (a) when the reaction is first order in A is about \(0.046 M^{-1}s^{-1}\), and for (b) when the reaction is second order in A is about \(0.13 M^{-2}s^{-1}\).

Step-by-step solution

Identify given information

The rate of the reaction is given as \(1.6 \times 10^{-2} M / \mathrm{s}\) and the concentration of A is \(0.35 M\). The order of the reaction for component A is given for each part of the task.

Set up the relevant equations

The rate law for a reaction where the concentration of A affects the rate is defined as: Rate = k[reactants], where 'Rate' is the rate of the reaction, 'k' is the rate constant, and '[reactants]' represents the concentration of the reactants each raised to the power of its order in the reaction. Therefore, for (a), the equation becomes: \(1.6 \times 10^{-2} M/s = k * (0.35M)^1\), and for (b), the equation becomes: \(1.6 \times 10^{-2} M/s = k * (0.35M)^2\).

Solve the equations

In order to find the rate constant, 'k', for each reaction order, rearrange the equations to isolate 'k' and then calculate. For (a): \(k = \frac{1.6 \times 10^{-2} M/s}{(0.35M)^1}\), and for (b): \(k = \frac{1.6 \times 10^{-2} M/s}{(0.35M)^2}\).

Key concepts

Chemical Kinetics

Chemical kinetics is the branch of chemistry that deals with the speed or rate at which a chemical reaction occurs, as well as the factors that affect this rate. Understanding kinetics is crucial in predicting the time course of a reaction, which has significant implications in various fields such as pharmaceuticals, environmental science, and materials engineering.

At the heart of kinetics is the reaction rate, which is a measure of how quickly the concentration of a reactant or product changes over time. Kinetic studies can illustrate the steps of a reaction mechanism, telling us which step is the slowest or rate-determining step and thus influencing the overall rate of reaction. It's fascinating to note how factors such as temperature, pressure, concentration, and the presence of catalysts can change the reaction rate, making chemical kinetics an essential area of study for understanding and controlling chemical processes.

Reaction Order

The concept of reaction order is fundamental in the study of chemical kinetics as it describes the dependence of the reaction rate on the concentration of each reactant. Reaction order can be an integer or fraction, determined experimentally, and it provides insight into the relationship between reactant concentration and how fast a reaction proceeds.

For example, a first-order reaction implies that the rate is directly proportional to the concentration of the reactant. A second-order reaction suggests that the rate is proportional to the square of the reactant's concentration, indicating a more complex relationship. Understanding the reaction order helps us construct the rate law equation for the reaction, which is crucial for calculating the rate constant and for making predictions about reaction behavior under different conditions.

Rate of Reaction

The rate of reaction tells us how fast reactants are converted to products, often expressed as a change in concentration over time. In the exercise, the rate of reaction is given as \(1.6 \times 10^{-2} M / \mathrm{s}\) when the concentration of A is \(0.35 M\). This numerical value gives a snapshot of the reaction speed under specific conditions.

It's vital to recognize that the rate of reaction is not constant over the course of a chemical reaction; it can change as reactants are consumed and products are formed. Rates typically decrease over time as reactants are used up unless there's a compensatory mechanism in place, such as heat or a catalyst, to maintain the rate or even accelerate it.

Rate Law Equation

The rate law equation is a mathematical representation of the relationship between the rate of a chemical reaction and the concentration of its reactants. It incorporates the rate constant (\(k\)) and the concentrations of reactants, each raised to their respective reaction orders, to quantify the rate of reaction.

From the step-by-step solution, we see two instances of rate law equations corresponding to different reaction orders. In case (a), for a first-order reaction, the rate law is \(Rate = k[A]\), and in case (b), for a second-order reaction, the rate law becomes \(Rate = k[A]^2\). These equations are pivotal for finding the rate constant, which is a proportionality constant unique to each reaction at a given temperature, describing its intrinsic reactivity independently of reactant concentrations.