Problem 159 - A Complete Resource Book in Chemistry for JEE Main

Question

The half life \(\left(\mathrm{t}_{1}\right)\) of the first order reaction and half life \(\left(t_{2}\right)\) of the second order reaction are in the ratio \(2: 1\). Hence the ratio of the rates of the above first and second order reactions at the start is (a) \(1: 0.4365\) (b) \(0.3465: 1\) (c) \(2: 1\) (d) \(1: 2\)

Step-by-Step Solution

Answer

(b) \(0.3465: 1\)

Step 1: Understand Half-Life Formulas

For a first-order reaction, the half-life equation is given by \( t_{1/2} = \frac{0.693}{k_1} \), where \( k_1 \) is the rate constant for the first-order reaction. For a second-order reaction, the half-life is given by \( t_{1/2} = \frac{1}{k_2[A]_0} \), where \( k_2 \) is the rate constant for the second-order reaction and \([A]_0\) is the initial concentration of the reactant.

Step 2: Apply Given Ratio to Half-Life Equations

According to the problem, the ratio of half-lives \( t_{1/2}^{(1)} : t_{1/2}^{(2)} = 2:1 \). This can be written as \( \frac{t_{1/2}^{(1)}}{t_{1/2}^{(2)}} = 2 \). Substituting the half-life equations, we get:\[ \frac{\frac{0.693}{k_1}}{\frac{1}{k_2[A]_0}} = 2 \]

Step 3: Simplify the Ratio Equation

This equation can be rearranged:\[ \frac{0.693 \cdot k_2[A]_0}{k_1} = 2 \]

Step 4: Solve for the Ratio of Rate Constants

From the equation \( \frac{0.693 \cdot k_2[A]_0}{k_1} = 2 \), we isolate the constants and rates:- \( k_2[A]_0 = \frac{2k_1}{0.693} \)- The ratio of the rates \( \frac{k_1}{k_2[A]_0} = \frac{0.693}{2} \approx 0.3465 \)Thus, the ratio of the rates of the reactions is roughly \( 0.3465:1 \).

Step 5: Identify the Answer from Options

Comparing the calculated ratio \( 0.3465:1 \) with the given options, the answer that corresponds to our solution is option (b): \( 0.3465: 1 \).

Key Concepts

First-order reaction

A first-order reaction is one where the rate of reaction depends on the concentration of a single reactant to the first power. In simpler terms, if you double the concentration of the reactant, the reaction rate will also double. A unique feature of first-order reactions is that they have a constant half-life. This means that the time it takes for the concentration of the reactant to reduce to half its original amount is independent of how much reactant you started with. For a first-order reaction, the half-life formula is: \[ t_{1/2} = \frac{0.693}{k_1} \]where:

• \( t_{1/2} \) is the half-life,
• \( k_1 \) is the rate constant of the reaction.

This equation illustrates the direct relationship between the rate constant and the half-life. The quicker the reaction (a larger \( k_1 \)), the shorter the half-life. Understanding this relationship helps in predicting how quickly a reaction will proceed, which is especially important in fields ranging from pharmaceuticals to environmental chemistry.

Second-order reaction

Second-order reactions are those where the reaction rate is proportional to either the concentration of a reactant squared or to the product of the concentrations of two reactants. Unlike first-order reactions, the half-life of second-order reactions depends on the initial concentration of the reactant. This variability means that as the concentration of the reactant decreases, the half-life increases. The half-life formula for a second-order reaction is:\[ t_{1/2} = \frac{1}{k_2[A]_0} \]where:

• \( t_{1/2} \) is the half-life,
• \( k_2 \) is the rate constant,
• \([A]_0\) is the initial concentration of the reactant.

This equation shows that if the initial concentration doubles, the half-life shortens significantly. Second-order reactions are often encountered in reactions involving two different molecules or ions that collide, which is crucial in understanding mechanisms in chemical kinetics.

Rate constant

The rate constant, represented as \( k \), is a proportionate factor in the rate law equation that indicates the speed at which a reaction proceeds. It is influenced by various factors, such as temperature and catalysts, but remains independent of reactant concentrations in rate calculations. In the context of our exercise:

• \( k_1 \) is the rate constant for first-order reactions, characterizing how quickly these reactions proceed.
• \( k_2 \) is the rate constant for second-order reactions, determining their specific reaction speeds.

The ratio of these rate constants, when linked with the concentrations of reactants, helps in determining the reaction rate. When given a ratio of half-lives, such as in our exercise, the rate constants can be used to calculate how fast one reaction occurs relative to another. This allows for the determination of reaction speeds and efficiencies in different chemical processes, providing valuable insight for chemists in optimizing and controlling reactions.