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 \(\left(\mathrm{t}_{2}\right)\) 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

The ratio of the rates is (b) \(0.3465: 1\).

Step 1: Understand Half-Life in First and Second Order Reactions

For a first order reaction, the half-life \( t_{1/2} \) is given by \( t_{1/2} = \frac{0.693}{k_1} \). For a second order reaction, the half-life \( t_{1/2} \) is given by \( t_{1/2} = \frac{1}{k_2 [A]_0} \), where \( [A]_0 \) is the initial concentration of reactant. The ratio of their half-lives is given by \( \frac{t_1}{t_2} = 2:1 \).

Step 2: Set Up Equation for Half-life Ratio

Using the half-life formulas, set up the equation: \( \frac{0.693}{k_1} = 2 \times \frac{1}{k_2 [A]_0} \). Simplifying gives \( k_1 = \frac{0.693}{2 t_2} \) and \( k_2 = \frac{1}{t_2 [A]_0} \).

Step 3: Derive Rate Equation for Each Reaction

For the first-order reaction, the rate is \( r_1 = k_1 [A]_0 \). For the second-order reaction, the rate is \( r_2 = k_2 [A]_0^2 \).

Step 4: Express Rates in Terms of Half-life

Substitute \( k_1 \) and \( k_2 \) into the rate equations. For the first-order reaction: \( r_1 = \left(\frac{0.693}{2 t_2}\right) [A]_0 \), and for the second-order reaction: \( r_2 = \left(\frac{1}{t_2 [A]_0}\right) [A]_0^2 \).

Step 5: Find the Ratio of Rates

The ratio of the rates is \( \frac{r_1}{r_2} = \frac{0.693}{2} \times \frac{1}{[A]_0} \div \frac{1}{t_2} = 0.693 \times \frac{t_2}{2 [A]_0} \times \frac{[A]_0}{t_2}\). Simplifying, we get \( \frac{r_1}{r_2} = \frac{0.693}{2} = 0.3465 \). Hence, the ratio of the rates is \( 0.3465:1 \).

Step 6: Select the Answer

The calculated rate ratio \( 0.3465:1 \) matches option (b). Therefore, the correct answer is (b) \( 0.3465: 1\).

Key Concepts

Half-Life in Chemical Reactions

In the world of chemistry, the concept of half-life is crucial for understanding how long a reaction takes to reach half its completion. For reactions, half-life is an indicator of the time required for the concentration of a reactant to reduce to half its initial value. It is a particularly relevant measure for first and second order reactions, each with its unique half-life equations.

For a **first order reaction**, the half-life is constant and is expressed as \( t_{1/2} = \frac{0.693}{k} \), where \( k \) is the rate constant. This means that the time it takes for the reactant to decrease to half is dependent solely on the reaction’s rate constant.

On the other hand, a **second order reaction** has a half-life that depends on the initial concentration of the reactant \( [A]_0 \). The expression is \( t_{1/2} = \frac{1}{k [A]_0} \). Unlike first order reactions, the half-life here changes as the reaction progresses.

Understanding these two formulas allows chemists to predict how quickly a reaction will proceed under certain conditions, critically important in various chemical industries and laboratory settings.

Rate Equations

Rate equations are mathematical representations that link the reaction rate to the concentration of the reactants and the rate constant. These equations provide insight into how fast a reaction progresses. Different types of reactions have specific form rate equations, describing the relationship between the speed of the reaction and the concentrations of the reactants.

For a **first order reaction**, the rate is linearly proportional to the concentration of one reactant. The rate equation is given as \( r = k [A] \), where \( r \) is the rate, \( k \) is the rate constant, and \( [A] \) is the concentration of the reactant.

Conversely, in a **second order reaction**, the rate could involve either:

• Two molecules of a single reactant, expressed as \( r = k [A]^2 \)
• Or one molecule each from two different reactants, shown as \( r = k [A][B] \)

These formulations illustrate how both the concentration and order of reaction influence reaction rates.

Understanding rate equations not only aids in determining how fast reactions occur but also assists in predicting the outcome of reactions under varying concentrations and conditions.

Reaction Kinetics

Reaction kinetics is a part of chemistry that studies the speed or rate of chemical reactions and the factors affecting these rates. It is fundamentally concerned with how reactions happen and how various variables, such as temperature, surface area, concentration, and the presence of catalysts, can alter this.

**Factors Influencing Reaction Rates** include:

• **Concentration of Reactants:** Generally, increased concentration speeds up a reaction because more molecules lead to more frequent collisions.
• **Temperature:** Raising the temperature usually results in faster reaction rates by giving molecules more energy to collide successfully.
• **Catalysts:** These substances increase reaction rates without being consumed in the reaction, by lowering the activation energy needed for the reaction.

Having a deep understanding of reaction kinetics is crucial not only for theoretical analyses but also for practical applications like designing chemical processes, improving industrial operations, and creating innovation in chemical technology.

By mastering reaction kinetics, students can predict how changes in conditions affect reaction rates, enabling better control and optimization of chemical processes.