Problem 184 - A Complete Resource Book in Chemistry for JEE Main

Question

Match the following $$ \begin{array}{ll} \hline \text { Column-I } & \text { Column-II } \\ \hline \begin{array}{ll} \text { (a) } t_{1 / 2}=\frac{0.693}{\mathrm{k}} & \text { (p) Zero order } \\\ \text { (b) } t_{1 / 2}-\frac{\mathrm{a}}{2 \mathrm{k}} & \text { (q) First order } \\ \text { (c) } \mathrm{t}=\frac{1}{\mathrm{k}} & \text { (r) Average life } \\ \text { (d) } \mathrm{t}_{\frac{1}{2}}=\frac{1}{\mathrm{ak}} & \text { (s) Second order } \\ & \end{array} \\ \hline \end{array} $$

Step-by-Step Solution

Answer

(a) matches with (q), (b) matches with (p), (c) matches with (r), (d) matches with (s).

Step 1: Analyze Column-I (a)

The expression \( t_{1/2} = \frac{0.693}{k} \) is a well-known formula for the half-life of a first-order reaction. In first-order reactions, the half-life is a constant that is independent of the initial concentration, which matches the expression given. Thus, (a) matches with (q) First order.

Step 2: Analyze Column-I (b)

The expression \( t_{1/2} = \frac{a}{2k} \) relates to zero-order reactions. In zero-order kinetics, half-life depends on the initial concentration \( a \) and is given by \( t_{1/2} = \frac{a}{2k} \). Therefore, (b) matches with (p) Zero order.

Step 3: Analyze Column-I (c)

The expression \( t = \frac{1}{k} \) is commonly used to represent the average life or mean life of a reaction. This is especially relevant in radioactive decay and is called the mean lifetime. Therefore, (c) matches with (r) Average life.

Step 4: Analyze Column-I (d)

The expression \( t_{1/2} = \frac{1}{ak} \) defines the half-life for a second-order reaction. For second-order reactions, half-life is inversely proportional to both the initial concentration \( a \) and the rate constant \( k \). Hence, (d) matches with (s) Second order.

Key Concepts

Half-life Formulas

Half-life is the time required for half of a reactant to be consumed in a chemical reaction. It is an essential parameter in chemical kinetics, especially in fields such as pharmacology and environmental science. Different types of reactions have distinct half-life formulas.

• First-order reactions: These reactions have a constant half-life, independent of the initial concentration. The formula is \( t_{1/2} = \frac{0.693}{k} \). Here, \( k \) is the rate constant. This simplicity makes first-order half-lives particularly easy to predict and analyze.
• Zero-order reactions: Unlike first-order reactions, the half-life of zero-order reactions depends on the initial concentration \( a \). The formula \( t_{1/2} = \frac{a}{2k} \) shows this dependency, indicating that the half-life will change if \( a \) changes.
• Second-order reactions: For these reactions, the half-life is given by \( t_{1/2} = \frac{1}{ak} \). It is inversely related to both the initial concentration \( a \) and the rate constant \( k \), meaning it varies significantly with changes in these parameters.

Understanding these formulas helps predict how long a reaction will proceed and aids in planning experiments and applications.

Order of Reactions

The order of a reaction provides insight into the relationship between the concentration of reactants and the rate of reaction. It is determined by how the reactant concentration affects the reaction rate.

• Zero-order reactions: The rate of reaction is constant and does not depend on the concentration of reactants. This means \( [A]^0 \) implies the concentration has no effect on the rate.
• First-order reactions: The rate is directly proportional to the concentration of one reactant. A simple example is radioactive decay where reducing the concentration decreases the rate linearly.
• Second-order reactions: The rate depends on the square of the concentration of a single reactant or the product of the concentrations of two reactants, represented mathematically as \( [A]^2 \) or \( [A][B] \).

Understanding the order of a reaction is crucial as it affects the formulation of half-life and indicates how a reaction progresses over time.

Reaction Kinetics

Reaction kinetics is the study of the speed or rate at which chemical reactions occur and the factors that influence these rates. To grasp how quickly products form from reactants or how reactants are consumed, one needs to understand key kinetic concepts.

• Rate constants: These are specific to each reaction and determine how quickly a reaction proceeds. The units of the rate constant vary based on the order of the reaction.
• Activation energy: This is the minimum energy required to start a reaction. The lower the activation energy, the faster the reaction.
• Catalysts: Substances that increase reaction rate without being consumed. They do this by lowering the activation energy.

Grasping reaction kinetics is fundamental for chemists in optimizing conditions, designing reactors, and even in creating new reaction pathways.

Mean Life of Reactions

The mean life of a reaction, often referred to as the average life, provides an insight into the average time it takes for the reactant particles to react. This concept is frequently used in contexts like radioactive decay.

• The mean life is associated with first-order reactions, where it is mathematically defined as \( \tau = \frac{1}{k} \), indicating how long a typical reactant stays unreacted. This constant mean life is crucial in processes such as decay, where each atom independently has a chance to decay.
• Mean life is more comprehensive than half-life as it provides an average time applicable across different fractions of reactive events.

Understanding mean life helps in predicting the duration of reaction processes and is particularly useful in nuclear chemistry for predicting the stability and decay of radioactive substances.