Question

For the reaction \(\mathrm{a} \mathrm{A} \longrightarrow \mathrm{xP}\) when \([\mathrm{A}]=2.2 \mathrm{mM}\) the rate was found to be \(2.4 \mathrm{~m} \mathrm{M} \mathrm{s}^{-1}\) On reducing concentration of \(\mathrm{A}\) to half, the rate changes to \(0.6 \mathrm{~m} \mathrm{M} \mathrm{s}^{-1}\). The order of reaction with respect to \(\mathrm{A}\) is (a) \(1.5\) (b) \(2.0\) (c) \(2.5\) (d) \(3.0\)

Short answer

The order of reaction with respect to \(A\) is 2.0.

Step-by-step solution

Understanding the Rate Law

For a reaction \( aA \rightarrow xP \), the rate law can be expressed as \( \text{Rate} = k [A]^n \), where \( k \) is the rate constant and \( n \) is the order of the reaction with respect to \([A]\). We need to determine \( n \).

Set Up Equations for Given Data

When \([A] = 2.2 \text{ mM}\), the rate is \(2.4 \text{ mM s}^{-1}\). The rate equation is \( 2.4 = k (2.2)^n \).When \([A]\) is reduced to half, meaning \([A] = 1.1 \text{ mM}\), the rate is \(0.6 \text{ mM s}^{-1}\). The rate equation becomes \( 0.6 = k (1.1)^n \).

Divide the Equations to Find \(n\)

Divide the first equation by the second,\[ \frac{2.4}{0.6} = \frac{k (2.2)^n}{k (1.1)^n} \]This simplifies to \[ 4 = \left(\frac{2.2}{1.1}\right)^n \]

Solve for \(n\)

The fraction \( \frac{2.2}{1.1} \) simplifies to \(2\). Therefore, from \( 4 = 2^n \), we solve for \( n \) as follows:\[ 4 = 2^2 = \left( 2 \right)^n \]Thus, \( n = 2 \).

Conclusion

Based on our calculation, the order of the reaction with respect to \(A\) is \(2.0\).

Key concepts

Rate Law

The rate law is a mathematical equation that expresses the relationship between the concentration of reactants and the rate of a chemical reaction. It is generally written in the form of \( \text{Rate} = k [A]^n \), where:

• \( k \) is the rate constant, a unique value that depends on factors like temperature and the presence of catalysts.
• \( [A] \) is the concentration of the reactant.
• \( n \) is the order of reaction with respect to the reactant \( A \).

Understanding the rate law helps predict how the concentration of reactants affects the rate of a reaction. For instance, in our example, the reaction rate changes based on the concentration of reactant \( [A] \). By manipulating concentrations, one can influence the speed at which a reaction proceeds. The rate law provides a framework to identify these effects and establish how various factors interplay in the dynamics of the reaction.

If you know the initial concentration and the resulting rate, as given in the exercise, you can set up equations to solve for \( n \), the reaction order, to understand its impact on rate changes.

Reaction Kinetics

Reaction kinetics is the study of the factors affecting the speed of chemical reactions. It focuses on understanding how variables like reactant concentrations and temperature influence the rate at which reactions occur. Kinetics provides crucial insights into the mechanisms behind chemical transformations.

One key concept in reaction kinetics is the rate of reaction, which indicates how quickly reactants are converted into products. This can be measured experimentally by observing changes in concentration over time. In our example, we examine how the reaction rate varies with changes in the concentration of \( [A] \).

Reaction kinetics is central to developing efficient chemical processes and can be applied to industries ranging from pharmaceuticals to energy. By understanding the kinetics of a reaction, chemists can optimize conditions to enhance yield and reduce time, making processes more economical and environmentally friendly.

From the exercise, analyzing how the rate responds to changes in concentration helps to elucidate the underlying principles of chemical kinetics.

Determining Reaction Order

Determining the order of a reaction is a critical step in comprehending the rate law. Reaction order refers to the power to which the concentration of a reactant is raised in the rate law expression. It essentially pinpoints how the concentration of reactants influences the rate of reaction.

In the exercise example, the process of determining the reaction order involved multiple steps:

• First, you establish the rate law equation based on the given data.
• Next, you create two separate rate equations using different concentrations and corresponding rate data.
• Then, by dividing these rate equations, you can isolate the reaction order \( n \).

By simplifying the equation, the value of \( n \) becomes evident and indicates the dependency of the reaction's rate on the reactant's concentration. This exercise demonstrates that the order is \( 2 \), which means the reaction rate increases quadratically as the concentration doubles.

Knowing the order of reaction is crucial for predicting how changes in conditions will affect the rate, guiding the optimization of chemical processes.