Question

The half-life of a chemical reaction at a particular concentration is \(50 \mathrm{~min}\), when the concentration of reactants is doubled, the half-life becomes \(100 \mathrm{~min}\). Find the order. (a) zero (b) first (c) second (d) third

Short answer

The reaction is second order (c).

Step-by-step solution

Understanding Half-Life and Reaction Order

The half-life of a reaction is the time required for the concentration of a reactant to decrease by half. Reaction order affects how half-life changes with concentration changes. We need to determine the order based on the given changes in half-life at different concentrations.

Observing Half-life Change with Concentration

The problem states that when the concentration of the reactant is doubled, the half-life increases from 50 minutes to 100 minutes. For reactions of different orders, the relationship between half-life and concentration can vary.

Analyzing Zero Order Reaction

In a zero-order reaction, the half-life decreases as concentration increases, according to the formula: \[ t_{1/2} = \frac{[A]_0}{2k} \] where \([A]_0\) is the initial concentration and \(k\) is the rate constant. Here, doubling concentration should lead to a reduced half-life, not an increased half-life.

Analyzing First Order Reaction

In a first-order reaction, the half-life is independent of the concentration, given by the formula: \[ t_{1/2} = \frac{0.693}{k} \]Doubling concentration does not affect the half-life, contradicting our observation.

Analyzing Second Order Reaction

For a second-order reaction, the half-life increases with the increase in concentration. The formula for half-life is: \[ t_{1/2} = \frac{1}{k[A]_0} \]Doubling the concentration results in doubling the half-life, which matches the given change from 50 minutes to 100 minutes.

Determining the Order

Based on the analysis, only the second-order reaction description fits the observed change in half-life with a change in concentration. Thus, the reaction is of second order.

Key concepts

Half-Life

The half-life of a chemical reaction is the time it takes for the concentration of a reactant to reduce to half its original amount. It is a critical concept in chemistry, especially in the study of reaction kinetics.
Understanding how half-life relates to concentration changes is essential to determine the reaction's order.

• For some reactions, half-life is constant, regardless of concentration.
• For others, it may vary depending on how concentration changes.

This relationship helps chemists predict how long it will take for a reaction to reach a certain extent, which is valuable in various industrial and scientific applications.

Zero Order Reaction

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

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

If the initial concentration \([A]_0\) is increased, the half-life becomes shorter because the numerator of the formula grows with concentration, but the entire expression still divides by a constant rate. Conversely, in this problem, the half-life doubled, ruling out a zero-order reaction.

First Order Reaction

In a first-order reaction, the rate depends linearly on the concentration of one reactant. The unique aspect of first-order reactions is that their half-life remains constant regardless of changes in concentration.
The half-life can be calculated with:\[t_{1/2} = \frac{0.693}{k}\]This formula tells us that the half-life

• Does not depend on the initial concentration.

Thus, doubling the concentration will not alter the half-life, which does not align with the problem's condition where the half-life increases. This inconsistency helps eliminate first-order reactions in this context.

Second Order Reaction

A second-order reaction's rate is proportional to either the square of one reactant's concentration or the product of two reactants' concentrations.
The half-life for a second-order reaction is expressed as:\[t_{1/2} = \frac{1}{k[A]_0}\]From this formula:

• The half-life increases as the initial concentration decreases.
• Doubling initial concentration directly doubles the half-life.

When the concentration doubles and the half-life also doubles, it matches what we would expect for a second-order reaction. In this way, examining the half-life's dependency on concentration effectively helps identify second-order reactions, as demonstrated in this exercise.