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) \(1.5\)

Short answer

The order of the reaction is zero (option a).

Step-by-step solution

Understanding the Half-Life Concept

The half-life of a reaction is the time taken for the concentration of a reactant to reduce to half its initial value. The way the half-life changes with initial concentration helps in determining the order of the reaction.

Analyzing the Order-Dependent Half-Life Behavior

For a zero-order reaction, the half-life is given by \( t_{1/2} = \frac{[A]_0}{2k} \), which is directly proportional to the initial concentration \([A]_0\). For a first-order reaction, the half-life \( t_{1/2} = \frac{0.693}{k} \), which is independent of the initial concentration. For a second-order reaction, the half-life \( t_{1/2} = \frac{1}{k[A]_0} \), which is inversely proportional to the initial concentration. We will need to decide which behavior matches the situation described.

Matching the Reaction Order to Given Conditions

In the given problem, the half-life period doubles when the initial concentration of the reactant is doubled. This suggests that the half-life is directly proportional to the initial concentration, typical of a zero-order reaction where doubling \([A]_0\) results in doubling \( t_{1/2} \).

Key concepts

Zero-Order Reactions

Zero-order reactions are fascinating because their rate is independent of the concentration of the reactant. In a zero-order reaction, the rate remains constant over time. This means the rate at which the reactant is used up doesn't change as the concentration goes down. Instead, it relies on the rate constant (k).

Mathematically, the rate of a zero-order reaction is given by:

• Rate = k

In terms of concentration over time for a zero-order reaction, you can use the equation:

• \[ [A] = [A]_0 - kt \]

Here,

• \([A]\) is the concentration at time \(t\)
• \([A]_0\) is the initial concentration
• \(k\) is the rate constant

One key feature of zero-order reactions is how their half-life, the time it takes for half of the reactant to vanish, depends on concentration. The half-life formula in this case is:

• \[ t_{1/2} = \frac{[A]_0}{2k} \]

This makes the half-life directly proportional to the initial concentration, meaning if you double the concentration, the half-life also doubles. This characteristic helped us determine the reaction order in the original problem.

Half-Life

Half-life is a significant concept in chemical kinetics as it tells how long it takes for a reactant's concentration to reduce to half of its original value. Different orders of reactions affect how half-lives change based on reactant concentration.

For zero-order reactions, the half-life depends on the initial concentration. It's calculated using:

• \[ t_{1/2} = \frac{[A]_0}{2k} \]

In first-order reactions, the half-life doesn't change with the concentration of reactants. It remains constant throughout the reaction. The formula for a first-order half-life is:

• \[ t_{1/2} = \frac{0.693}{k} \]

This is because the rate of a first-order reaction depends exponentially on concentration.

For second-order reactions, the half-life is inversely proportional to the initial concentration:

• \[ t_{1/2} = \frac{1}{k[A]_0} \]

In the practice problem, understanding how half-life escalates with the initial concentration helped conclude that it was a zero-order reaction, as it fit the doubling pattern.

Concentration Effects

Concentration effects refer to how alterations in reactant concentration affect the rate of reaction and its kinetics. Different reaction orders exhibit distinct concentration dependencies.

In a zero-order reaction, concentration changes do not affect the reaction rate, which remains fixed. This effect can illustrate operation limits of catalysts or surface reactions where saturation has been reached.

For first-order reactions, the rate is proportional to the concentration of one reactant. This means if you double the concentration, the rate doubles as well.
Second-order reactions have more complex behaviors, being influenced by the concentration of one reactant or two reactants in a bimolecular reaction. Rate here is proportional to the square of the reactant's concentration or the product of concentrations in bimolecular cases.

If a reaction shows that the rate, and in turn the half-life, changes in direct proportion to the concentration, like in the original exercise, it is a distinct marker for it being a zero-order reaction.

Chemical Kinetics

Chemical kinetics is the study of reaction rates and the factors affecting them. It helps us understand the speed at which chemical reactions occur and the steps involved.

Kinetics involves rate laws, which show the relationship between reaction rate and concentrations. The order of reaction is key in kinetics. It tells us how concentration changes impact the rate.

In determining reaction order, chemists often look at the formula derived from the rate law, which usually follows this pattern:

• \[ ext{Rate} = k[A]^x[B]^y \]

Where:

• \(k\) is the rate constant
• \([A]\) and \([B]\) are reactant concentrations
• \(x\) and \(y\) are the reaction orders for the reactants

In the broader context of kinetics, factors such as temperature, catalysts, and reaction conditions can also majorly impact rates.

Understanding these principles of kinetics allows us to predict how changes like concentration doubling will alter the reaction's dynamics as observed in the provided problem.