Problem 138 - A Complete Resource Book in Chemistry for JEE Main

Question

If the initial concentration of reactant in certain reaction is doubled, the half life period of the reaction is also doubled. The order of reaction is (a) zero (b) first (c) second (d) \(1.5\)

Step-by-Step Solution

Answer

The order of the reaction is 1.5 (option d).

Step 1: Understanding the half-life concept

The half-life of a reaction (t_{1/2}) is the time required for the concentration of a reactant to decrease to half of its initial concentration. The relationship between half-life and the order of reaction is crucial in determining the order of the reaction.

Step 2: Formulating half-life expressions for different orders

For a zero-order reaction, half-life is given by t_{1/2} = rac{[A]_0}{2k}, where [A]_0 is the initial concentration and k is the rate constant. For a first-order reaction, the half-life is t_{1/2} = rac{ ext{ln}2}{k}, independent of initial concentration. For a second-order reaction, the half-life is t_{1/2} = rac{1}{k[A]_0}.

Step 3: Analyzing the given condition

We know from the problem that doubling the initial concentration also doubles the half-life. Let's evaluate this condition for each order: - For zero-order, if [A]_0 is doubled, t_{1/2} will not double but decrease. - For first-order, t_{1/2} does not depend on [A]_0, so it remains unchanged. - For second-order, doubling [A]_0 halves the half-life, contrary to the given condition.

Step 4: Determining the correct order from options

Based on the evaluation: - Zero-order, first-order, and second-order reactions do not satisfy the condition of doubling half-life when concentration is doubled. - Therefore, the likely order must be one in which half-life is directly proportional to initial concentration, indicating a fractional order that fits would be 1.5. The option that matches is (d).

Key Concepts

Half-Life Concept

To understand how long a reaction will last, we need to learn about "half-life." Half-life is simply the time it takes for half of a reactant to be used up in a chemical reaction. It tells us how fast or slow a reaction happens. For different types of reactions, the half-life can vary.

The half-life also depends on the order of the reaction, which gives us clues on how the concentration of substances varies over time. Let's explore how these concepts interconnect with reaction order for clearer insights.

Zero-Order Reaction

In a zero-order reaction, the rate of the reaction is constant. This means that the rate does not depend on the concentration of reactants. The formula for the half-life of a zero-order reaction is: \[ t_{1/2} = \frac{[A]_0}{2k} \]
where

• \([A]_0\) is the initial concentration and
• \(k\) is the rate constant.

This tells us that if we double the initial concentration

• the half-life gets shorter.

Thus, the reaction speed appears to increase. This is key in understanding that not all reactions behave the same way when concentrations change.

First-Order Reaction

A first-order reaction is quite straightforward when it comes to half-life. The rate of these reactions depends directly on the concentration of one reactant. However, the half-life is uniquely constant and doesn't change with different concentrations.

The half-life for a first-order reaction can be calculated using the formula:\[ t_{1/2} = \frac{\ln 2}{k} \]

• Here, \(k\) is the rate constant, and the \(\ln 2\) factor arises naturally from the mathematics of exponential decay.

This consistent half-life makes first-order reactions particularly easy to analyze over time.

Second-Order Reaction

In second-order reactions, the rate depends on either the concentrations of two reactants or the square of the concentration of a single reactant. These reactions have a more complex half-life expression:\[ t_{1/2} = \frac{1}{k[A]_0} \]

For these reactions, as you double the initial concentration of the reactant

• the half-life is actually halved.

This means that the reaction speeds up with a higher concentration. Such a relationship highlights how small changes in concentration can lead to larger changes in reaction speed for second-order reactions, offering deeper insight into reaction dynamics.