Question

Derive an expression for the half-life of the reactant A that decays by an \(n\) th-order reaction (with \(n>1\) ) with rate constant \(k\). Reaction Mechanisms

Short answer

The half-life for an nth-order reaction (n>1) depends on the rate constant (k) and the initial concentration ([A]0), and can be derived through integration of the rate law, setting integration limits from the initial concentration to half of that, and solving the resulting equation for the half-life (\(t_{1/2}\)).

Step-by-step solution

Identify Order of Reaction

Recognize that the given reaction is an nth-order reaction with n>1. The rate of this reaction can be described by the rate law: Rate = k[A]^n, where [A] is the concentration of reactant A, and k is the rate constant.

Write Differential Rate Equation

Write the differential rate equation for the decay of reactant A in the form of a differential equation: -\(\frac{d[A]}{dt}\) = k[A]^n.

Integrate the Rate Equation

To find the relationship between the concentration of A and time (t), integrate the rate equation. The integral form will depend on the order n not being equal to 1, since the integration process is different for n=1 (which yields a natural logarithm function).

Set Up Integration Limits

To find the half-life (\(t_{1/2}\)), set the limits of integration from the initial concentration [A]0 to [A]0/2 (where [A]0/2 denotes the concentration of A at its half-life), and the time limits from 0 to \(t_{1/2}\).

Perform Integration

Integrate both sides of the equation with respect to the appropriate variables to obtain a relationship between concentration and time.

Solve for Half-Life

Isolate the half-life (\(t_{1/2}\)) on one side of the equation to derive a formula that relates half-life to the rate constant (k) and the initial concentration ([A]0) specifically for an nth order reaction where n>1.

Key concepts

Chemical Kinetics

Chemical kinetics is the study of how chemical reactions proceed and how to quantify their rates. The speed at which reactants transform into products is crucial in synthesizing new materials, understanding environmental processes, and even in our body's metabolism. Kinetic analyses help scientists and engineers design reactors, control pollution, and understand the stability of substances over time.

At the core of kinetics is the question of how variables like concentration, temperature, and the presence of a catalyst affect the rate of a reaction. It also considers the mechanisms by which reactions occur, including the sequence of elementary steps making up complex reactions.

Rate Laws

Rate laws or rate equations are mathematical expressions that describe the relationship between the rate of a chemical reaction and the concentrations of reactants. These algebraic equations provide insights into the order of reaction with respect to each reactant. For example, a rate law might look like Rate = k[A]ⁿ[B]^m where A and B are reactants, k is the rate constant, and n and m represent the reaction orders with respect to A and B, respectively.

The form of the rate law is determined experimentally and can not be inferred from the chemical equation alone because it reflects the microscopic process of the reaction, including intermediates and activated complexes.

Rate Constant

The rate constant, denoted by k, is the coefficient found in the rate law equation that relates reaction rate to reactant concentration. It is a measure of how quickly a reaction proceeds. This constant is dependent on the temperature and the presence of a catalyst but is independent of reactant concentrations. The units of k vary depending on the overall reaction order and ensure that the rate of the reaction has the correct units of concentration per unit time.

The rate constant can be determined experimentally and once known, it allows the calculation of the reaction rate under different concentration conditions. In the nth-order reaction we are focusing on, k is crucial for determining how the rate and half-life depend on the starting concentration.

Differential Rate Equation

The differential rate equation expresses the rate of change in reactant concentration with time. It is a differential form that directly tells us how fast a reactant concentration decreases or a product concentration increases. For an nth-order reaction where n>1, the differential rate equation is -\(\frac{d[A]}{dt}\) = k[A]ⁿ.

This equation implies that the rate of reaction is proportional to the concentration of reactant A raised to the power of n. In the context of the exercise, this equation is integral(pun intended!) as it is the starting point for deriving the half-life expression.

Reaction Order

Reaction order indicates the dependency of the reaction rate on the concentration of reactants. It is defined as the sum of the powers of the concentration terms in the rate equation. When we talk about an nth-order reaction, we indicate that the rate of the reaction is proportional to the concentration of one reactant raised to the nth power. Simple cases include zero-order (rate is independent of the reactant concentration), first-order (rate is directly proportional to the reactant concentration), or second-order (rate is proportional to the square of the reactant concentration).

Understanding the reaction order is essential as it affects how the rate of reaction changes with varying concentrations and can significantly influence the design of chemical reactors and safety protocols.

Integrated Rate Law

The integrated rate law is derived from the differential rate equation and provides a direct relationship between the concentrations of reactants and time. For different reaction orders, the form of the integrated rate law varies. For example, first-order reactions yield a logarithmic relationship, while second-order reactions result in a reciprocal relationship of concentration over time.

In the case of nth-order reactions with n>1, integration is necessary to find the relationship between concentration of reactant A and the time it takes to reach half its initial concentration, also known as the half-life (\(t_{1/2}\)). The result of this integration is an explicit formula that relates the half-life, the rate constant, and the initial concentration for nth-order reactions, a concept central to the exercise at hand.