Question

Consider the data showing the initial rate of a reaction (A \(\longrightarrow\) products) at several different concentrations of A. What is the order of the reaction? Write a rate law for the reaction, includ- ing the value of the rate constant, \(k\). $$ \begin{array}{cc} {[\mathrm{A}](\mathrm{M})} & \text { Initial Rate }(\mathrm{M} / \mathrm{s}) \\\ 0.15 & 0.008 \\ \hline 0.30 & 0.016 \\ \hline 0.60 & 0.032 \\ \hline \end{array} $$

Short answer

The reaction is first-order with respect to A. The rate law is \[Rate = k[A]\], and the rate constant is \[k = \frac{1}{0.015} \frac{1}{s} = 66.7 \frac{1}{s}\].

Step-by-step solution

Determining Reaction Order by Rate Changes

In order to determine the order of the reaction with respect to A, examine how the initial rate changes as the concentration of A is changed. If the rate doubles when the concentration of A doubles, the reaction is first-order with respect to A. If the rate increases by a factor of four when the concentration of A doubles, the reaction is second-order with respect to A. If the rate does not change when the concentration of A changes, the reaction is zero-order with respect to A.

Analyzing the Given Data

Look at the initial rates and their corresponding concentrations of A. When the concentration of A increases from 0.15 M to 0.30 M, the initial rate doubles (from 0.008 M/s to 0.016 M/s). Similarly, when the concentration doubles again from 0.30 M to 0.60 M, the initial rate also doubles (from 0.016 M/s to 0.032 M/s). This indicates that every time the concentration of A is doubled, the rate also doubles, suggesting a first-order reaction.

Writing the Rate Law

Since the reaction is first-order with respect to A, the rate law can be written as \[Rate = k[A]\] where \[k\] is the rate constant and \[A\] is the concentration of reactant A.

Calculating the Rate Constant, k

Use one of the given data points to solve for the rate constant \[k\]. For example, using the first data point with \[A = 0.15 M\] and \[Rate = 0.008 M/s\], the rate law becomes \[0.008 M/s = k(0.15 M)\]. Solve for \[k\] by dividing the rate by the concentration: \[k = \frac{0.008 M/s}{0.15 M} = \frac{1}{0.015} \frac{1}{s}\]. Simplify this fraction to find the value of \[k\].

Key concepts

Understanding Reaction Rate

The reaction rate is a measure of how quickly the concentration of a reactant decreases or the concentration of a product increases over time. In our exercise, the reaction rate is examined for reactant A as it transforms into products.

When evaluating reaction rates, it's crucial to note that a linear increase in rate with concentration typically indicates a first-order reaction. For more complex reactions, the relationship could be quadratic or even more intricate, suggesting second-order or higher reaction orders, respectively.

Why Is Reaction Rate Important?

• It gives insights into the speed of chemical processes.
• It helps in the design and optimization of chemical reactions for industrial applications.
• Understanding reaction rates is essential for controlling reaction conditions in synthesis and manufacturing.

The calculated rate will further aid in determining the rate law and the rate constant, which are the next steps in our exploration.

Deciphering the Rate Law

The rate law expresses the relationship between the reaction rate and the concentrations of reactants. It is determined experimentally because it cannot be reliably predicted by merely knowing the balanced chemical equation. The rate law for a reaction is often written in the form of a mathematical equation that links the rate (\(Rate\)) to the concentration of reactants raised to a power, which represents the order of the reaction with respect to each reactant.

In our study, once the order of the reaction with respect to A is established as first-order, the rate law is written as:\[Rate = k[A]\]where:

• \(k\) is the rate constant,
• \([A]\) is the concentration of reactant A.

This law suggests that the rate of the reaction is directly proportional to the concentration of A.

Rate Laws in Practice

By knowing the rate law, chemists can predict how the rate will change when the concentrations of the reactants are varied. This information is invaluable for controlling the speed of a chemical reaction in various practical applications, such as pharmaceutical synthesis and environmental chemistry.

The Role of the Rate Constant

The rate constant, denoted by \(k\), is a proportionality factor that is unique for each chemical reaction at a given temperature. It describes the rate at which the reaction occurs when the reactant concentrations are at standard conditions.

The value of the rate constant provides insights into the reaction kinetics and can be influenced by factors such as temperature, the presence of a catalyst, and the solvent. In the provided exercise, we calculate the rate constant using the formula derived from the rate law and experimental data:\[k = \frac{\text{Rate}}{[A]}\]By substituting the known values from the experiments, we are able to determine the numerical value of the rate constant, thus quantifying the speed of our chemical reaction under the given conditions. The rate constant is crucial in understanding the overall kinetics of a reaction and designing processes that can harness or regulate the rate at which chemical changes occur within a system.

Rate Constant and Temperature

One fascinating aspect of the rate constant is its relationship to temperature, often modeled by the Arrhenius equation. As temperature increases, the rate constant typically does too, leading to a faster reaction rate, which is an important consideration for both natural phenomena and industrial applications.