Problem 115 - A Complete Resource Book in Chemistry for JEE Main

Question

For the reaction \(\mathrm{A} \longrightarrow\) Products, it is found that the rate of reaction increases by a factor of \(6.25\), when the concentration of \(\mathrm{A}\) is increased by a factor of \(2.5 .\) The order of reaction with respect to \(\mathrm{A}\) is (a) \(0.5\) (b) 1 (c) 2 (d) 3

Step-by-Step Solution

Answer

The order of the reaction with respect to \( \mathrm{A} \) is 2.

Step 1: Understand the Rate Law Expression

The rate law for a reaction can be expressed as \( \text{rate} = k[A]^n \) where \( k \) is the rate constant, \([A]\) is the concentration of the reactant \( A \), and \( n \) is the order of the reaction with respect to \( A \). We need to find the value of \( n \).

Step 2: Set Up the Initial and Final Rate Equations

Let the initial concentration of \( A \) be \([A]_1\) and the initial rate be \( \text{rate}_1 = k[A]_1^n \). After the concentration of \( A \) increases by a factor of 2.5, it becomes \( 2.5[A]_1 \). The new rate is \( \text{rate}_2 = k(2.5[A]_1)^n \).

Step 3: Relate the Rate Change to the Concentration Change

We are told that the rate increases by a factor of 6.25, so \( \text{rate}_2 = 6.25 \times \text{rate}_1 \). Substitute the expressions from Step 2 to get:\[ k(2.5[A]_1)^n = 6.25 \times k[A]_1^n \].

Step 4: Simplify the Equation

Cancel \( k \) and \([A]_1^n \) from both sides:\[ (2.5)^n = 6.25 \].

Step 5: Solve for the Order \( n \)

Express 6.25 as a power of 2.5:\( 6.25 = (2.5)^2 \).Thus, the equation becomes:\[ (2.5)^n = (2.5)^2 \].Equating the exponents gives \( n = 2 \).

Key Concepts

Rate Law Expression

The rate law expression is a fundamental concept in chemical kinetics, helping chemists understand how the concentration of reactants affects the rate of a chemical reaction. It typically takes the form \( \text{rate} = k[A]^n \), where \( k \) represents the rate constant, \([A]\) is the concentration of reactant \( A \), and \( n \) denotes the order of the reaction with respect to \( A \). Understanding this equation is crucial because it allows prediction of how different reactant concentrations impact reaction velocity.

• **Rate Constant \( k \)**: A unique value for every reaction at a given temperature, it remains unchanged during the reaction unless the temperature varies.
• **Concentration \([A]\)**: The amount of reactant present in the solution, usually measured in moles per liter.
• **Order of Reaction \( n \)**: Determines how sensitive the rate is to concentration changes.

In our example, by analyzing how changes in \([A]\) affect the rate, we gain insights into the reaction mechanism and dynamics.

Order of Reaction

The order of reaction with respect to a given reactant conveys how variations in reactant concentration influence the rate of a chemical reaction. The order, represented by \( n \) in the rate law equation, can be an integer or a fraction and is not necessarily equal to the stoichiometric coefficient of the reactant in the balanced chemical equation.

• If \( n = 0 \): The reaction rate is independent of the concentration of that reactant.
• If \( n = 1 \): The rate is directly proportional to the concentration of the reactant.
• If \( n = 2 \): The rate is proportional to the square of the concentration, meaning it is highly sensitive to concentration changes.
• If \( n = 2 \), as in our problem: Doubling the concentration of the reactant increases the reaction rate fourfold.

In the exercise, the order calculated was second order, indicating a significant dependency of the rate on the concentration of \( A \). This helps in determining reaction kinetics and providing key data for industrial chemical processes.

Concentration Change Effect

The concentration change effect refers to how the alteration in the concentration of a reactant impacts the speed at which a reaction proceeds. This is a direct consequence of the reaction order, and it illustrates the kinetic nature of chemical processes.
In the given problem, increasing the concentration of reactant \( A \) by a factor of 2.5 resulted in the rate of reaction increasing by a factor of 6.25. Such a relationship is pivotal in demonstrating the reaction's sensitivity to concentration changes and provides a clear example of a second-order reaction.

• **Small Changes, Big Effect**: Even modest increases in concentration can lead to significant rate enhancements, especially in higher-order reactions.
• **Predictive Power**: Understanding these effects is essential for predicting how a reaction will behave under different concentrations, essential for scaling up in industrial processes.
• **Illustrated by Experimentation**: Such effects are often examined through experiments where variables are carefully manipulated and observed.

Studying these effects enriches comprehension of the dynamic behavior of reactants and underpins the efficient design of chemical processes.