Problem 36 - A Complete Resource Book in Chemistry for JEE Main

Question

Which of the following is correct for a first order reaction? \(\left(k=\right.\) rate constant \(t_{1 / 2}=\) half-life \()\) (a) \(t_{1 / 2}=0.693 \times k\) (b) \(\mathrm{k} \cdot \mathrm{t}_{1 / 2}=1 / 0.693\) (c) \(\mathrm{k} \cdot \mathrm{t}_{1 / 2}=0.693\) (d) \(6.93 \times k \times t_{1 / 2}=1\)

Step-by-Step Solution

Answer

Option (c) is correct: \( k \cdot t_{1/2} = 0.693.\)

Step 1: Understanding First-order Reactions

For a first-order reaction, the rate of reaction depends on the concentration of one reactant raised to the first power. It is characterized by the differential rate law \( \text{Rate} = k[A] \), where \( k \) is the rate constant and \([A]\) is the concentration of reactant \(A\).

Step 2: Determine the Half-life Expression

For first-order reactions, the half-life \( t_{1/2} \) is the time taken for the concentration of the reactant to reduce to half of its original value. It is given by the formula \( t_{1/2} = \frac{0.693}{k} \).

Step 3: Evaluate Each Option Against the Correct Formula

Compare each given option with the expression derived in Step 2.- Option (a) states \( t_{1/2} = 0.693 \times k \), which is incorrect according to the formula.- Option (b) states \( k \cdot t_{1/2} = \frac{1}{0.693} \), which is incorrect.- Option (c) states \( k \cdot t_{1/2} = 0.693 \), which matches the correct formula rearranged as \( k = \frac{0.693}{t_{1/2}} \), this is the correct option.- Option (d) makes an incorrect assertion about the relationship between \( k \), \( t_{1/2} \), and a constant.

Key Concepts

Rate Constant in First-order Reactions

In the context of first-order reactions, the rate constant, denoted by the symbol \( k \), is a crucial parameter. It quantitatively describes the speed at which the reaction proceeds. For a reaction to be classified as first-order, its rate is directly proportional to the concentration of one reactant, usually expressed as \( \text{Rate} = k[A] \), where \( [A] \) is the concentration of the reactant.
The rate constant \( k \) remains constant throughout the reaction, assuming temperature does not change. This means it is independent of the concentration of the reactants. Typically, the units of the rate constant for a first-order reaction are \( ext{s}^{-1} \). Understanding \( k \) helps chemists determine how quickly a reactant is transformed into products, making it a valuable tool in reaction kinetics.

Half-life Expression for First-order Reactions

In kinetics, the concept of half-life, denoted by \( t_{1/2} \), is significant, especially when analyzing first-order reactions. The half-life is the time required for the concentration of the reactant to decrease to half of its initial value. A unique feature of first-order reactions is that their half-life remains constant throughout the reaction, irrespective of the initial concentration. This is because the rate depends on the concentration raised to the first power.
The relationship between the half-life and the rate constant for a first-order reaction is given by the formula:

• \( t_{1/2} = \frac{0.693}{k} \)

The derivation of this formula comes from integrating the rate law over time and considering that the half-life is the time when the concentration is halved. Recognizing this connection allows students to comprehend how reactions proceed over time, an essential aspect for both theoretical and practical applications.

Differential Rate Law

The differential rate law in chemistry is a mathematical expression that describes how the rate of a reaction depends on the concentration of its reactants. For first-order reactions, the differential rate law is:

• \( \text{Rate} = k[A] \)

Where \([A]\) is the concentration of one reactant and \(k\) is the rate constant. In this law, the rate of reaction is directly proportional to the concentration of \(A\), and the power to which \(A\) is raised is always one, characterizing it as a first-order reaction.
Understanding the differential rate law is essential because it provides a direct relationship between concentration and rate, unlike integrated rate laws which relate concentrations to time. This law is vital for predicting how changes in concentration affect the rate, thus allowing chemists to control reaction speeds in industrial and laboratory settings.