Problem 169 - A Complete Resource Book in Chemistry for JEE Main

Question

If a is the initial concentration of reactant and \((a-x) \quad\) (b) \(x=a\left(1-e^{-k_{1}}\right)\) \(\begin{aligned}&\text { is the remaining concentration after time 't' in a first } \\\&\text { order reaction of rate constant } k_{1} \text {, then which of the } & \text { (c) } t_{1 / 2}=\frac{1.414}{k_{1}}\end{aligned}\) following relations is /are correct? (a) \(k_{1}=\frac{2.303}{t} \log \left(\frac{\mathrm{a}}{\mathrm{a}-x}\right)\) (d) \(\mathrm{t}_{\mathrm{av}}=\frac{1}{k_{1}}\)

Step-by-Step Solution

Answer

Expressions (a), (b), and (d) are correct; expression (c) is incorrect.

Step 1: Understand the given expression

We have the expression for the concentration of the reactant remaining after time \(t\) as \(a(1-e^{-k_1t})\). This implies it describes a first-order reaction, where \(a\) is the initial concentration and \(x\) is the concentration change.

Step 2: Verify expression (a)

For a first-order reaction, the relation between rate constant \(k_1\), time \(t\), initial concentration \(a\), and remaining concentration \((a-x)\) is given by the formula: \[ k_1 = \frac{2.303}{t} \log \left(\frac{a}{a-x}\right). \]This is a standard relation for first-order reactions, indicating that expression (a) is correct.

Step 3: Check expression (b)

Expression (b) \(x = a \left(1 - e^{-k_1t}\right)\) is given in the problem as the remaining concentration formula for a first-order reaction. This correctly represents the relation between concentration change and time for first-order kinetics.

Step 4: Evaluate expression (c)

The half-life for a first-order reaction is generally given by: \[ t_{1/2} = \frac{0.693}{k_1}. \]The expression provided in (c), \( t_{1/2} = \frac{1.414}{k_1} \), does not match this standard formula, thus it is incorrect.

Step 5: Validate expression (d)

Expression (d) suggests that the average time \(t_{av}\) is \(\frac{1}{k_1}\), which relates to the general idea of the inverse of the rate constant representing a time scale for the reaction, which aligns with kinetics analysis.

Key Concepts

Rate Constant

The rate constant, often represented as \( k \), is a crucial parameter in the study of chemical kinetics, especially for first-order reactions. It provides insights into the speed at which a reaction proceeds. In a first-order reaction, the rate constant \( k \) is related to how rapidly the concentration of a reactant decreases over time.

For a first-order reaction, the rate of reaction is directly proportional to the concentration of one reactant. The relationship is expressed by the formula:

• \( k_1 = \frac{2.303}{t} \log \left(\frac{a}{a-x}\right) \)

Here, \( a \) represents the initial concentration, \( x \) is the change in concentration, and \( t \) is the time elapsed. The rate constant has units of time^-1, for instance, \( s^{-1} \) if time is measured in seconds.

Understanding the rate constant is essential because it helps predict how long a reaction will take to reach a specific concentration under given conditions. A larger rate constant implies that the reaction proceeds more quickly, while a smaller rate constant indicates a slower reaction.

Half-Life

The concept of half-life, denoted as \( t_{1/2} \), is fundamental in understanding reactions, especially first-order kinetics. The half-life is the time required for the concentration of a reactant to reduce to half of its initial concentration.

For first-order reactions, the half-life is independent of the reactant concentration, making it a handy tool for predicting reaction progress over time. The standard formula for half-life in a first-order reaction is:

• \( t_{1/2} = \frac{0.693}{k_1} \)

This equation shows that the half-life is inversely proportional to the rate constant \( k_1 \). Thus, as the rate constant increases, the half-life decreases, meaning the reactant reaches half its concentration faster.

It is crucial to note that this formula differs from incorrect interpretations, such as \( t_{1/2} = \frac{1.414}{k_1} \). Relying on the accurate formula enables students to better predict and understand reaction timelines in chemical processes.

First-Order Kinetics

First-order kinetics describes a reaction where the rate of reaction is directly proportional to the concentration of a single reactant. It is one of the simplest forms of chemical kinetics and is common in various chemical and biological processes.

For a first-order reaction, the concentration of the reactant changes in a way that can be described by an exponential decay model. The concentration of the reactant at time \( t \) can be calculated using the formula:

• \( x = a \left(1 - e^{-k_1t}\right) \)

Here, \( a \) is the initial concentration, \( x \) is the concentration change, and \( k_1 \) is the rate constant. This equation highlights the exponential nature of first-order reactions, where the concentration decreases at a rate proportional to its current concentration.

Understanding first-order kinetics is essential for predicting how a reaction progresses over time. In real-world applications, it helps chemists and biologists to model and anticipate the behavior of reactions in both controlled and natural environments. This understanding is crucial for everything from industrial manufacturing to medical dosing.