Question

The following expression shows the dependence of the half-life of a reaction \(\left(t_{1 / 2}\right)\) on the initial reactant concentration \([\mathrm{A}]_{0}:\) $$ t_{1 / 2} \propto \frac{1}{[\mathrm{~A}]_{0}^{n-1}} $$ where \(n\) is the order of the reaction. Verify this dependence for zeroth-, first-, and second-order reactions.

Short answer

Zeroth-order: \( t_{1/2} \propto [A]_0 \), First-order: \( t_{1/2} \) is constant, Second-order: \( t_{1/2} \propto \frac{1}{[A]_0} \).

Step-by-step solution

Understand the given expression

We are given that the half-life \( t_{1/2} \) is proportional to \( \frac{1}{[A]_0^{n-1}} \). This expression indicates that the form of the half-life equation depends on the order \( n \) of the reaction.

Zeroth-order reaction analysis

For a zeroth-order reaction, \( n = 0 \). Plug \( n=0 \) into the expression:\[t_{1/2} \propto \frac{1}{[A]_0^{-1}} = [A]_0 \]This shows that for a zeroth-order reaction, the half-life is directly proportional to the initial concentration \([A]_0\).

First-order reaction analysis

For a first-order reaction, \( n = 1 \). Plug \( n = 1 \) into the expression:\[t_{1/2} \propto \frac{1}{[A]_0^{1-1}} = \frac{1}{[A]_0^0} = 1\]This indicates that the half-life for a first-order reaction is independent of the initial concentration \([A]_0\).

Second-order reaction analysis

For a second-order reaction, \( n = 2 \). Plug \( n = 2 \) into the expression:\[t_{1/2} \propto \frac{1}{[A]_0^{2-1}} = \frac{1}{[A]_0}\]This means that for a second-order reaction, the half-life is inversely proportional to the initial concentration \([A]_0\).

Key concepts

Zeroth-Order Reaction

In a zeroth-order reaction, the rate of reaction is independent of the concentration of the reactant. This means that as the reaction progresses, the rate remains constant. A unique feature of zeroth-order reactions is that their half-lives are directly proportional to the initial concentration of the reactant, denoted as - The rate law for a zeroth-order reaction is expressed as: \[ ext{Rate} = k \]- As a result, the concentration of the reactant decreases linearly with time. The half-life formula for a zeroth-order reaction can be described using:\[ t_{1/2} = \frac{[A]_0}{2k} \]Here,

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

With this expression, you can see that as the initial concentration increases, the duration of half-life also increases. This characteristic makes zeroth-order reactions quite predictable in terms of how long they will take to reach a certain point.

First-Order Reaction

First-order reactions depend directly on the concentration of one reactant. This means that the rate of reaction slows down as the reactant is consumed over time.- The rate law for a first-order reaction is: \[ ext{Rate} = k[A] \]- The concentration of the reactant decreases exponentially.For first-order reactions, the half-life is particularly noteworthy because it remains constant throughout the course of the reaction. The formula used is:\[ t_{1/2} = \frac{0.693}{k} \]What's interesting here is that:

• The half-life is independent of the initial concentration \([A]_0\).
• It only depends on the rate constant \(k\).

This consistency makes first-order reactions well-suited for situations where the reaction needs to be predictable over time, such as pharmacokinetics in drug dosage.

Second-Order Reaction

For a second-order reaction, the rate depends on either the concentration of two reactants or the square of the concentration of a single reactant. This results in a rate that changes more significantly as the reactants are consumed.- The rate law for a second-order reaction is given by: \[ ext{Rate} = k[A]^2 \]For second-order reactions, the half-life expression shows that it is inversely proportional to the initial concentration. The formula for half-life is:\[ t_{1/2} = \frac{1}{k[A]_0} \]This indicates:

• When the initial concentration \([A]_0\) is high, the half-life is short.
• Conversely, if the initial concentration is low, the half-life becomes longer.

In real-world applications, second-order reactions are often found in processes where two reactants are needed to interact, such as in bi-molecular reactions in chemistry.

Half-Life

Half-life is a crucial concept as it describes the time required for half of the reactant to be converted in a reaction. This measure provides significant insight into how quickly a reaction proceeds.Different orders of reactions handle half-life in unique ways:- **Zeroth-Order**: The half-life is directly proportional to the initial concentration and is given by \( t_{1/2} = \frac{[A]_0}{2k} \).- **First-Order**: The half-life is independent of the initial concentration, with \( t_{1/2} = \frac{0.693}{k} \).- **Second-Order**: The half-life is inversely proportional to the initial concentration, giving \( t_{1/2} = \frac{1}{k[A]_0} \).Understanding these relationships helps predict how long a reaction will take to reach a certain point, which is valuable in both experimental and industrial settings. Knowing the half-life allows scientists and engineers to control reaction times more accurately.

Initial Concentration

The initial concentration of reactants in a chemical reaction sets the stage for how the reaction will unfold. It is the starting point from which the reaction begins and influences the rate and duration. - In **zeroth-order reactions**, the initial concentration directly affects the reaction's half-life. A higher initial concentration means a longer half-life. - For **first-order reactions**, the initial concentration does not impact the half-life, illustrating a steady rate of reaction over time, regardless of how much reactant you start with. - In **second-order reactions**, the initial concentration and half-life are inversely related. A larger initial concentration results in a shorter half-life. By understanding the initial concentration's role, scientists can design experiments and predict outcomes more effectively by altering this initial condition. This insight is valuable for controlling reaction environments to achieve desired results.