Problem 149 - A Complete Resource Book in Chemistry for JEE Main

Question

Two reactants (A) and (B) separately show two chemical reactions. Both reactions are carried out with same initial concentration of each reactant. Reactant (A) follows first order kinetics whereas reactant (b) follows second order kinetics. If both have same life period. The ratio of their at the start of reaction is (a) \(2.303 \log 2\) (b) \(\log 2\) (c) \(\frac{2.303}{\log 2}\) (d) \(\frac{\log 2}{2.303}\)

Step-by-Step Solution

Answer

(a) \(2.303 \log 2\)

Step 1: Understand the Concept of Half-Life

For a first order reaction, the half-life (the time required for half the reactant to be used up) is given by the formula \( t_{1/2} = \frac{0.693}{k_A} \) where \( k_A \) is the rate constant for reactant A. Similarly, for a second order reaction, it is \( t_{1/2} = \frac{1}{k_B [B]_0} \), where \( k_B \) is the rate constant for reactant B and \([B]_0\) is the initial concentration of B.

Step 2: Equate the Half-Lives

Since both reactions have the same half-life, equate the two expressions: \( \frac{0.693}{k_A} = \frac{1}{k_B [B]_0} \).

Step 3: Rearrange the Equation

Rearrange the equation to find the initial concentration \([B]_0\) in terms of \( k_A \) and \( k_B \): \( [B]_0 = \frac{k_B}{0.693} k_A \).

Step 4: Solve for the Ratio of Rate Constants

Since both reactions start with the same concentration, we write \( [A]_0 = [B]_0 \). Now substituting the expressions derived, we get \( \frac{[A]_0}{[B]_0} = 1 \) thus simplifying to \( \frac{0.693}{1} = \frac{1}{k_B [B]_0} \), leading to \( k_A [A]_0 = k_B \).

Step 5: Introduce \( \log 2 \) into the Equation

Using \( 0.693 \approx \log 2 \times 2.303 \), equate it with the previous expression to find the answer directly. Thus, \( \log 2 \) appears in the answer as part of the logarithmic conversion factor, leading to option \( 2.303 \, \log 2\) as the ratio.

Key Concepts

Half-Life

The half-life of a chemical reactant is the time it takes for half of the reactant to be consumed in a reaction. This concept is crucial in understanding the kinetics of both first and second order reactions. For a first order reaction, the half-life is constant and independent of the initial concentration of the reactant. It can be represented by the formula:

• \( t_{1/2} = \frac{0.693}{k} \)

Here, \( k \) represents the rate constant of the reaction.
In contrast, the half-life of a second order reaction is dependent on the initial concentration of the reactant. It is given by:

• \( t_{1/2} = \frac{1}{k [B]_0} \)

Where \( [B]_0 \) is the initial concentration of reactant \( B \), and \( k \) is the rate constant. This means the half-life will decrease as the concentration decreases.

First Order Reaction

First order reactions are characterized by their linear relationship between the rate of the reaction and the concentration of the reactant. The rate equation for such reactions is:

• \( ext{Rate} = k [A] \)

Where \( [A] \) is the concentration of the reactant and \( k \) is the rate constant.
The simple mathematical formula implies that as the concentration of reactant \( A \) halves, the rate of reaction also halves. This nature of first order reactions leads to their characteristic exponential decay, which can be mathematically expressed as:

• \( [A] = [A]_0 e^{-kt} \)

Where \( [A]_0 \) is the initial concentration, \( t \) is the time, and \( e \) is the base of the natural logarithm.

Second Order Reaction

In a second order reaction, the rate of the reaction is proportional to either the square of the concentration of one reactant or the product of the concentrations of two reactants. The rate equation is:

• \( ext{Rate} = k [B]^2 \) or \( ext{Rate} = k [A][B] \)

This indicates that the rate significantly changes with variations in concentration, more so in contrast to first order reactions.
For second order reactions, the half-life equation, \( t_{1/2} = \frac{1}{k [B]_0} \), shows that the half-life depends inversely on the initial concentration of the reactant. This means as the reaction progresses and the concentration decreases, each successive half-life will be longer than the previous one.

Rate Constant

The rate constant is a fundamental parameter in chemical kinetics, denoted typically by \( k \). It gives insight into the speed of a chemical reaction under certain conditions. For a first order reaction, \( k \) has units of \( ext{s}^{-1} \), while for a second order reaction, \( k \) is expressed in \( ext{L mol}^{-1} ext{s}^{-1} \).
Determining the rate constant involves experimental measurements and is necessary for calculating half-life values and understanding the reaction kinetics.

• For first order reactions: \( t_{1/2} = \frac{0.693}{k} \)
• For second order reactions: \( t_{1/2} = \frac{1}{k [B]_0} \)

The value of \( k \) will differ based on the temperature, pressure, and presence of catalysts. Understanding the rate constant helps in analyzing how fast a reaction goes to completion.