Problem 40 - Chemistry The Central Science In Si Units

Question

(a) Develop an equation for the half-life of a zero-order reaction. (b) Does the half-life of a zero-order reaction increase, decrease, or remain the same as the reaction proceeds?

Step-by-Step Solution

Answer

(a) \( t_{1/2} = \frac{[A]_0}{2k} \); (b) it decreases as the reaction proceeds.

Step 1: Understanding Zero-Order Kinetics

For a zero-order reaction, the rate of reaction is constant and independent of the concentration of the reactant. The rate law is given by the equation \( r = k \), where \( k \) is the rate constant.

Step 2: Defining Half-Life for a Zero-Order Reaction

The half-life \( t_{1/2} \) is the time it takes for the concentration of a reactant to decrease to half its initial concentration. For a zero-order reaction, the equation for concentration over time is \( [A] = [A]_0 - kt \).

Step 3: Deriving the Half-Life Equation

Set \( [A] = \frac{[A]_0}{2} \), where \([A]_0\) is the initial concentration, and solve for \( t = t_{1/2} \). Substitute into the zero-order rate equation: \( \frac{[A]_0}{2} = [A]_0 - kt_{1/2} \).

Step 4: Solving for Half-Life

Rearrange \( \frac{[A]_0}{2} = [A]_0 - kt_{1/2} \) to get \( kt_{1/2} = [A]_0 - \frac{[A]_0}{2} = \frac{[A]_0}{2} \), and solve for \( t_{1/2} \) to get \( t_{1/2} = \frac{[A]_0}{2k} \).

Step 5: Analyzing Half-Life Behavior

The half-life equation \( t_{1/2} = \frac{[A]_0}{2k} \) shows that the half-life of a zero-order reaction depends on the initial concentration \([A]_0\). As the reaction proceeds, \([A]_0\) decreases, thus the half-life decreases.

Key Concepts

Kinetic Equations

Kinetic equations are the mathematical expressions that describe the rate of chemical reactions. They are fundamental to understanding how reactions occur over time. In the context of zero-order reactions, the kinetic equation is particularly simple. In any reaction, the concentration of the reactants and products changes as the reaction progresses, and kinetic equations provide a way to predict these changes.

For zero-order reactions, the kinetic equation can be expressed as:

• The reaction rate (\( r \)) is constant and equals the rate constant (\( k \)).
• The concentration of the reactant (\( [A] \)) decreases linearly over time.
• The equation \( [A] = [A]_0 - kt \) captures this relationship, where \( [A]_0 \) is the initial concentration.

These equations help predict how quickly a reactant decreases over time, allowing chemists to understand and manipulate reaction processes.

Rate Laws

Rate laws are experimental expressions that relate the rate of a reaction to the concentration of the reactants. Unlike kinetic equations, which provide a descriptive view, rate laws allow us to predict the speed of a reaction based on concentration measurements.

In a zero-order reaction, the rate law is quite straightforward:

• Rate (\( r \)) is equal to the rate constant (\( k \)).
• This means that the reaction rate is independent of the concentration of the reactants.

As a result, the reaction proceeds at a constant rate until the reactants are depleted. This characteristic is unique compared to first-order or second-order reactions, where the rate changes as the concentration changes.

Chemical Kinetics

Chemical kinetics involves the study of reaction rates and how different variables influence these rates. It helps scientists understand how different factors such as temperature, pressure, and concentration impact the speed of reactions.

Zero-order reactions present a unique aspect as part of chemical kinetics:

• The rate is invariant with changing concentration levels, meaning the rate constant \( k \) defines the entirety of the rate.
• These reactions can occur when a catalyst is saturated or under conditions where the surface or enzyme active sites are saturated.

Understanding these reactions provides crucial insight into reaction mechanisms and can assist in optimizing industrial processes where control over reaction speed is essential.

Reaction Half-Life

The half-life of a reaction is the time required for half of the reactant to be transformed into product. It's a concept that offers a practical measure of how fast a reaction occurs. For a zero-order reaction, the half-life correlates directly with the initial concentration and the rate constant.

To calculate the half-life (\( t_{1/2} \)) of a zero-order reaction:

• Use the formula \( t_{1/2} = \frac{[A]_0}{2k} \).
• The half-life decreases as the initial concentration decreases, meaning the reaction ends more swiftly as it progresses.

This dependency on initial concentration is a distinguishing feature of zero-order reactions, contrasting with other reaction orders where half-life is typically constant (first-order) or dependent on both concentration and rate (second-order). Understanding these differences is essential for accurately predicting and managing chemical processes.