Question

lwo ditterent molecules, \(\mathrm{A}\) and \(\mathrm{B}\), were reacted to give products according to the equation \(\mathrm{A}+\mathrm{B} \rightarrow\) products The rate of reaction was measured for fixed concentrations of \(\mathrm{A}\) and \(\mathrm{B}\) and the following data was obtained:$$ \begin{aligned} &\begin{array}{|l|l|l|l|} \hline \text { Experiment } & \begin{array}{l} {[\mathrm{A}]} \\ \text { (moles /iter) } \end{array} & \begin{array}{l} \text { [B] } \\ \text { (moles/iter) } \end{array} & \begin{array}{l} \text { Rate } \\ \text { (moles } \left.^{2} \text { /liter }^{2}-\text { sec }\right) \end{array} \\ \hline 1 & 1.0 & 1.0 & 0.05 \\ \hline 2 & 1.0 & 2.0 & 0.10 \\ \hline 3 & 3.0 & 1.0 & 0.45 \\ \hline \end{array}\\\ &\text { Write an expression for the rate of reaction. } \end{aligned} $$

Short answer

The rate of reaction is given by the expression: Rate = \(k [\mathrm{A}]^2 [\mathrm{B}]\), where \(k\) is the proportionality constant, and the orders with respect to A and B are 2 and 1, respectively.

Step-by-step solution

Identify the relationships between the concentrations and the rate of reaction

Examine the changes in concentration and their corresponding changes in the rate of reaction by comparing different experiments. Notice how the rate changes when the concentrations of either A or B change.

Determine the order of the reaction with respect to A and B

Compare the rates of the different experiments and determine the order of the reaction with respect to both A and B. This will help us develop an expression for the rate of reaction. ## Order with respect to A: From experiment 1 to experiment 3, we can see that the concentration of A tripled: \(\frac{[\mathrm{A}]_{3}}{[\mathrm{A}]_{1}} = \frac{3}{1} = 3\) Now, we check the effect on the rate of the reaction: \(\frac{\text{Rate}_3}{\text{Rate}_1} = \frac{0.45}{0.05} = 9\) As \(3^{\text{Order of A}} = 9\), the order of the reaction with respect to A is 2. ## Order with respect to B: From experiment 1 to experiment 2, the concentration of B doubled: \(\frac{[\mathrm{B}]_{2}}{[\mathrm{B}]_{1}} = \frac{2}{1} = 2\) Now, we check the effect on the rate of the reaction: \(\frac{\text{Rate}_2}{\text{Rate}_1} = \frac{0.10}{0.05} = 2\) As \(2^{\text{Order of B}} = 2\), the order of the reaction with respect to B is 1.

Write an expression for the rate of reaction based on the determined order for each molecule

Based on the order of the reaction, we can write an expression for the rate of reaction. The rate of reaction is proportional to the concentration of A to the power of the order with respect to A and the concentration of B to the power of the order with respect to B. Using a proportionality constant, \(k\), we can write: Rate = \(k [\mathrm{A}]^2 [\mathrm{B}]^1\) The final expression for the rate of reaction is: Rate = \(k [\mathrm{A}]^2 [\mathrm{B}]\)

Key concepts

Reaction Order

Understanding the reaction order is essential in determining how the rate of a reaction is influenced by the concentration of reactants. Reaction order tells us the power to which the concentration of a reactant is raised in the rate law. For a reaction involving molecules A and B, as seen in the original exercise, the reaction order with respect to each reactant can be discovered through experiments.

For example, if we observe the concentration of A and notice that tripling its concentration causes the reaction rate to increase by a factor of nine, we can conclude that the reaction is second-order with respect to A. Similarly, doubling the concentration of B and observing a doubling of the reaction rate indicates a first-order reaction with respect to B.

Reaction order helps to elucidate the mechanism of the reaction, providing insight into how reactants interact at the molecular level.

Concentration Effect

The concentration effect refers to the change in the rate of a chemical reaction due to a change in the concentration of reactants. In simple terms, it shows how adjustments in the amounts of substances involved in a reaction affect the speed at which the reaction occurs.

Consider the original exercise where experiments are conducted at different concentrations of A and B. By observing the data, we note how variations in concentrations of A and B lead to changes in the reaction rate. This illustrates the direct relationship between concentration and rate, which allows us to deduce the order of the reaction.

In kinetic studies, understanding the concentration effect is crucial because it serves as the foundation for developing the rate law, enabling chemists to predict reaction behavior under different conditions.

Reaction Kinetics

Reaction kinetics studies the rate at which chemical reactions proceed and the factors that affect these rates. It provides valuable insights into the steps involved in a reaction and how variables such as temperature, concentration, and catalysts influence the speed of the reaction.

The exercise provided demonstrates a practical application of reaction kinetics by helping students to analyze experimental data to determine the reaction order and to develop the rate law expression. By systematic experimentation, such as altering concentrations of reactants, we gain insight into the kinetics of the reaction between A and B.

Mastering reaction kinetics allows us to optimize industrial processes, improve the efficiency of reactions, and understand biological pathways better, making it a cornerstone of both chemistry and biochemical engineering.

Rate Law Expression

The rate law expression is a mathematical equation that links the rate of a reaction to the concentration of its reactants. This expression is derived from experimental data and provides a concise description of how the rate depends on reactant concentrations.

In the original example, the rate law is expressed as: Rate = \(k [\mathrm{A}]^2 [\mathrm{B}]\). Here, \(k\) is the rate constant, which can vary depending on temperature and other conditions. The exponents (2 for A and 1 for B) correspond to the reaction order with respect to each reactant.

Understanding and formulating the rate law is critical for predicting how a reaction will respond to changes in conditions, as it encompasses the basic principles of reaction kinetics and the role of concentration effect in chemical processes.