Question

The integrated rate law for a zero-order reaction is $$[A]_{t}=-k t+[A]_{0}$$ Derive an equation for the half-life of a zero-order reaction.

Short answer

The half-life of a zero-order reaction is given by \( t_{1/2} = \frac{[A]_0}{2k} \).

Step-by-step solution

Identify Half-Life Conditions

To derive the half-life equation for a zero-order reaction, recognize that the half-life, \( t_{1/2} \), is the time it takes for the concentration of reactant \( [A] \) to decrease to half its original concentration \( [A]_0 \). Therefore, at half-life, \( [A]_{t} = \frac{[A]_0}{2} \).

Set Up the Integrated Rate Law at Half-Life

Substitute the half-life conditions into the integrated rate law. Setting \( [A]_{t} \) as \( \frac{[A]_0}{2} \) and \( t \) as \( t_{1/2} \), the equation becomes \( \frac{[A]_0}{2} = -k t_{1/2} + [A]_0 \).

Isolate the Half-Life \( t_{1/2} \)

Solve for \( t_{1/2} \) by first subtracting \( [A]_0 \) from both sides of the equation: \( \frac{[A]_0}{2} - [A]_0 = -k t_{1/2} \). Simplify to get \( -\frac{[A]_0}{2} = -k t_{1/2} \).

Solve for Half-Life \( t_{1/2} \)

Divide both sides by \( -k \) to isolate \( t_{1/2} \): \( t_{1/2} = \frac{[A]_0}{2k} \). This is the derived equation for the half-life of a zero-order reaction.

Key concepts

Integrated Rate Law

The integrated rate law expresses the concentration of a reactant in a chemical reaction as a function of time. Rather than focusing on the change at a particular moment, this law integrates the rate over time to provide a formula that connects the initial concentration of a reactant with its concentration after a certain period. For zero-order reactions, the concentration of the reactant decreases linearly over time. The integrated rate law for a zero-order reaction can be represented by the equation \[ [A]_{t} = -kt + [A]_0 \] where \( [A]_{t} \) is the concentration of reactant A at time t, \( [A]_0 \) is the initial concentration, and k is the rate constant. When you understand this equation, you can predict how much reactant will remain after any given time, including the half-life, which is the time required for half of the reactant to be consumed.

Zero-Order Kinetics

Zero-order kinetics characterizes reactions where the rate is independent of the concentration of the reactant. This means that no matter how much reactant you start with, the reaction will proceed at a constant rate until the reactant is exhausted. This is often seen in processes where a catalyst is saturated with the reactant, or when the reaction is surface-controlled, such as in certain catalytic reactions. The hallmark of a zero-order reaction is a straight-line plot of reactant concentration versus time, indicating a constant rate of decrease in concentration. The integrated rate law is crucial for calculating various parameters in such reactions, including the half-life, which is derived by setting the concentration to half its original value and solving for time.

Chemical Kinetics

Chemical kinetics is the branch of chemistry concerned with the speed or rate at which a chemical reaction occurs and the factors that affect this rate. It studies the reaction rate, which is the change in concentration of reactants or products over time, and seeks to understand the various steps that occur during a chemical reaction, collectively known as the reaction mechanism. In zero-order reactions, the kinetics are straightforward because the rate does not depend on the concentration of reactants, simplifying the analysis. However, for most reactions, the kinetics are more complex and require detailed study to determine the precise pathway and factors influencing the reaction rate, such as temperature, pressure, and the presence of a catalyst.

Reaction Rate Constants

Reaction rate constants are numerical values that represent the speed of a chemical reaction. They are an intrinsic part of the rate law, which relates the rate of a reaction to the concentrations of reactants, and they differ for every reaction. The constant \( k \) in the zero-order integrated rate law is specific to a particular reaction at a given temperature and indicates the constant rate at which the reactant concentration decreases over time. Understanding \( k \) allows chemists to predict how fast a reaction will proceed and to design processes that optimize reaction conditions for efficiency and yield. For a zero-order reaction, knowing the rate constant is essential to determining the half-life, as it inversely affects the time it takes for the reactant concentration to reduce to half of its original value.