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) \(t_{1 / 2}=0.693 \times \mathrm{k}\) (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) \( k \cdot t_{1/2} = 0.693 \) is correct.

Step 1: Understanding First-Order Reaction Definition

First-order reactions depend on the concentration of one reactant. Their rate equation is given by \( r = k[A] \), where \( k \) is the rate constant and \( [A] \) is the concentration of the reactant.

Step 2: Recall Half-Life Equation for First-Order Reaction

For a first-order reaction, the half-life \( t_{1/2} \) is given by the formula \( t_{1/2} = \frac{0.693}{k} \). This important equation arises because the rate of reaction is proportional to the concentration of a single reactant.

Step 3: Analyze the Given Options

Each provided option represents a mathematical expression involving \( k \) and \( t_{1/2} \). We need to check which one aligns with the known half-life expression.

Step 4: Verify Option (a)

Option (a) states \( t_{1/2} = 0.693 \times k \). To be valid, it would need to be \( t_{1/2} = \frac{0.693}{k} \). Therefore, this is incorrect.

Step 5: Verify Option (b)

Option (b) states \( k \cdot t_{1/2} = \frac{1}{0.693} \). Rearrange the half-life equation to \( k \cdot t_{1/2} = 0.693 \), showing that option (b) is incorrect.

Step 6: Verify Option (c)

Option (c) asserts \( k \cdot t_{1/2} = 0.693 \), which matches the rearranged half-life formula \( k \cdot \frac{0.693}{k} = 0.693 \). This is correct.

Step 7: Verify Option (d)

Examine option (d): \( 6.93 \times k \times t_{1/2} = 1 \). Since \( k \cdot t_{1/2} = 0.693 \), multiplying by 6.93 should give approximately 0.693, not 1. Thus, option (d) is incorrect.

Key Concepts

Rate Constant

The rate constant, often represented by the symbol \( k \), is a critical component in reaction kinetics, particularly in first-order reactions. Its value provides insight into the speed at which a reaction proceeds. The rate constant is unique to each reaction and changes with temperature. For a first-order reaction, the rate equation is expressed as \( r = k[A] \), which shows that the reaction rate depends solely on the concentration \([A]\) of one reactant.

Some key points about the rate constant include:

• It has units of \( ext{s}^{-1} \) for first-order reactions, making it distinctive from other orders.
• Since it varies with temperature, reactions will typically occur faster at higher temperatures due to increased \( k \).
• The larger the rate constant, the faster the reaction, as it indicates how quickly the concentration of the reactant decreases over time.

Understanding the rate constant is essential when studying the kinetics of a reaction, allowing chemists to predict how quickly a product will form under given conditions.

Half-Life

The concept of half-life, denoted as \( t_{1/2} \), refers to the time required for half of the reactant to be consumed in a reaction. It's a particularly useful measurement in first-order reactions because it remains constant regardless of the concentration of the reactant. This constant nature makes it easier to compare different reactions.

For a first-order reaction, the half-life can be calculated using the formula:\[t_{1/2} = \frac{0.693}{k}\]Some important points about half-life are:

• In first-order reactions, the half-life is independent of the initial concentration, distinguishing it from zero- or second-order reactions.
• The constant 0.693 arises from the natural logarithm of 2 (\( \ln(2) \)), representing the concept of exponential decay.
• A shorter half-life indicates a faster reaction, as it means that half of the reactant is consumed more quickly.

Understanding and calculating the half-life can help predict how long a reaction will take or how the concentration of a reactant will diminish over time.

Reaction Kinetics

Reaction kinetics involves studying how rate factors influence the speed of chemical reactions, providing a deeper understanding of the mechanistic pathways involved. For first-order reactions, reaction kinetics focuses on how the concentration of a single reactant affects the reaction rate.

Key concepts in reaction kinetics of first-order reactions include:

• The rate equation is linear, as it depends directly on the concentration of only one reactant.
• Graphs of concentration versus time for first-order reactions show a characteristic exponential decay.
• Temperature, catalysts, and other conditions can affect the reaction rate, altering the rate constant \(k\).

By studying reaction kinetics, particularly in first-order reactions, scientists can develop models to predict reaction behavior, allowing for optimization in industrial and laboratory settings. Understanding how these factors interconnect can also guide the development of new catalyst systems or influence how reactions are scaled from a small to large scale.