Question

Provide a conceptual rationale for the differences in the half-lives of zero-, first-, and second-order reactions.

Short answer

The half-lives of zero-, first-, and second-order reactions differ due to their varying dependencies on the rate constants and initial concentrations of the reactants. In a zero-order reaction, the half-life depends on both the initial concentration and the rate constant, causing the reaction to slow down as the reactant concentration decreases. For a first-order reaction, the half-life is independent of the initial concentration and only depends on the rate constant, resulting in a consistent reaction rate. In a second-order reaction, the half-life is inversely proportional to the initial concentration and directly proportional to the rate constant, leading to a slowing down of the reaction as the reactant concentration decreases.

Step-by-step solution

Understand the concept of half-life

Half-life is the time it takes for the concentration of a reactant to decrease to half of its initial concentration. It is an important parameter in understanding the rates of different chemical reactions and depends on the rate constant and the initial concentration of the reactant involved.

Familiarize with different reaction orders

A reaction order is an empirical relationship between the reaction rate and the concentration of the reactant(s) participating in the reaction. The three orders of reactions that we will be discussing are: 1. Zero-order reaction: The rate of reaction is constant and does not depend on the concentration of the reactants. Rate law: \(Rate = k\) 2. First-order reaction: The rate of reaction is directly proportional to the concentration of one reactant. Rate law: \(Rate = k[A]\) 3. Second-order reaction: The rate of reaction is proportional to the square of the concentration of one reactant or the product of the concentrations of two reactants. Rate law: \(Rate = k[A]^2\) or \(Rate = k[A][B]\)

Derive the half-life expressions for each reaction order

The half-life expressions for each reaction order are derived from their respective rate laws. 1. Zero-order reaction half-life: \[t_{1/2} = \frac{[A]_0}{2k}\] 2. First-order reaction half-life: \[t_{1/2} = \frac{0.693}{k}\] 3. Second-order reaction half-life: \[t_{1/2} = \frac{1}{k[A]_0}\]

Compare half-lives of different reaction orders

Now that we have the half-life expressions for each reaction order, we can compare and explain the differences: 1. Zero-order reaction: The half-life depends on both the initial concentration \([A]_0\) and the rate constant \(k\). As the concentration of the reactant decreases, the half-life will decrease as well, meaning the reaction will slow down over time. 2. First-order reaction: The half-life depends only on the rate constant \(k\) and is independent of the initial concentration \([A]_0\). This means the reaction rate will stay consistent throughout the reaction, as the half-life remains constant. 3. Second-order reaction: The half-life is inversely proportional to the initial concentration \([A]_0\) and directly proportional to the rate constant \(k\). As the concentration of the reactant decreases, the half-life increases, indicating that the reaction slows down as the reactant concentration decreases. In conclusion, the half-lives of zero-, first-, and second-order reactions differ primarily because of their dependencies on the rate constants and initial concentrations of the reactants, which affects how their reaction rates change over time.