Problem 215 - A Complete Resource Book in Chemistry for JEE Main

Question

Consider the reaction, \(2 \mathrm{~A}+\mathrm{B} \longrightarrow\) Products When concentration of alone was doubled, the half life did not change. When the concentration of \(A\) alone was doubled, the rate increased by two times. The unit of rate constant for this reaction is (a) no unit (b) \(\mathrm{mol} \mathrm{L}^{-1} \mathrm{~s}^{-1}\) (c) \(\mathrm{s}^{-1}\) (d) \(\mathrm{L} \mathrm{mol}^{-1} \mathrm{~s}^{-1}\)

Step-by-Step Solution

Answer

The unit of the rate constant is \(\mathrm{s}^{-1}\).

Step 1: Understanding the Reaction Order

The given reaction is: \(2 \mathrm{~A}+\mathrm{B} \longrightarrow\) Products. We need to determine the order of reaction with respect to \(A\) and \(B\).

Step 2: Analyzing the Effect of Reactant \(B\)

The problem states that when the concentration of \(B\) is doubled, the half-life does not change. Typically, the half-life of a reaction is independent of concentration only for a first-order reaction. This suggests that the order with respect to \(B\) is zero.

Step 3: Analyzing the Effect of Reactant \(A\)

When the concentration of \(A\) is doubled, the rate of the reaction is doubled. This indicates that the reaction is first-order with respect to \(A\), because a change in concentration results in a proportional change in rate.

Step 4: Determining the Overall Reaction Order

From the analysis, the reaction is zero-order with respect to \(B\) and first-order with respect to \(A\). Thus, the overall reaction order is \(1 + 0 = 1\).

Step 5: Identifying the Unit of the Rate Constant

For a first-order reaction, the unit of the rate constant is \(\mathrm{s}^{-1}\). This can be verified by the formula \(k = \text{rate} / [\text{concentration}]\). For a first-order reaction where concentration is measured in \(\mathrm{mol/L}\), the unit becomes \(\mathrm{s}^{-1}\).

Key Concepts

First-Order Reaction

In a first-order reaction, the rate at which the reactants transform into products depends on the concentration of only one reactant. This means the reaction rate is directly proportional to the concentration of that single reactant. In our given equation, this was clearly observed with reactant \(A\). When the concentration of \(A\) was doubled, the reaction rate also doubled. This clearly indicates a first-order relationship.

It's important to note that first-order reactions have a distinct property where changes in the concentration of the reactive species lead to equivalent changes in the rate. This linear dependency can be expressed as:

• \( ext{Rate} = k[A]^1 \)

Rate Constant

The rate constant, often denoted by \(k\), is a crucial factor in the rate equation for any chemical reaction. It helps define the reaction speed and is influenced by temperature and the inherent nature of the reactants. In a first-order reaction, the unit for the rate constant is \(\mathrm{s}^{-1}\). This is because the rate of the reaction is expressed in terms of concentration change over time (typically \(\mathrm{mol/L/s}\)). When this changes with respect to concentration (\(\mathrm{mol/L}\)), you are left with \(\mathrm{s}^{-1}\).

The rate constant provides insights into how quickly a reaction proceeds under certain conditions. For example, a higher \(k\) value suggests a faster reaction.

Half-Life

The half-life of a reaction is the time it takes for the concentration of a reactant to reduce to half its initial value. For first-order reactions, half-life is particularly interesting. It remains constant and is independent of the initial concentration of the reactant. This unique property makes the analysis of first-order reactions more straightforward as compared to reactions of other orders.

The half-life \(t_{1/2}\) for a first-order reaction can be calculated using the formula:

• \( t_{1/2} = \frac{0.693}{k} \)

This formula indicates that the half-life is purely dependent on the rate constant \(k\). This lack of dependency on concentration was a key point in the analysis of the given problem.

Reaction Rate

The reaction rate expresses how quickly reactants are converted into products in a chemical reaction. It is usually determined by the changes in concentration of the reactants or products over time.

In the context of the original problem, when the concentration of \(A\) was doubled, the reaction rate doubled as well. This tangible relationship confirmed that the reaction is first-order with respect to \(A\). Meanwhile, \(B\) had no change on the half-life when its concentration was doubled, suggesting it doesn’t influence the reaction rate, emphasizing its zero-order nature here.

• The overall reaction rate in chemistry depends on several factors: concentration of the reactants, presence of a catalyst, and temperature.

Understanding these dependencies is crucial for manipulating conditions to achieve desired reaction speeds.