Problem 45 - Chemistry: An Atoms-Focused Approach

Question

What effect does doubling the initial concentration of a reactant have on the half-life in a reaction that is second order in the reactant?

Step-by-Step Solution

Answer

Answer: Doubling the initial concentration of a reactant in a second-order reaction will cause the half-life to decrease by half.

Step 1: Write the rate law for a second-order reaction

For a second-order reaction, the rate law is given by: Rate = k[A]² where Rate is the reaction rate, k is the rate constant, and [A] is the concentration of the reactant A.

Step 2: Write the integrated rate law for a second-order reaction

To find the integrated rate law, we need to integrate the rate law with respect to time. For second-order reactions, the integrated rate law is: 1/[A] - 1/[A]₀ = kt where [A]₀ is the initial concentration of the reactant A, [A] is the concentration at time t and k is the rate constant.

Step 3: Derive the half-life formula for a second-order reaction

Half-life (t₁/₂) is the time it takes for the concentration of a reactant to fall to half its initial value. Thus, at t = t₁/₂, [A] = [A]₀ / 2. We can substitute this into the integrated rate law equation to derive the half-life formula: 1/([A]₀/2) - 1/[A]₀ = kt₁/₂

Step 4: Simplify the half-life formula and solve for t₁/₂

Now we will simplify the half-life formula: 2/[A]₀ - 1/[A]₀ = kt₁/₂ 1/[A]₀ = kt₁/₂ t₁/₂ = 1/(k[A]₀) The half-life formula is: t₁/₂ = 1/(k[A]₀)

Step 5: Analyze the effect of doubling the initial concentration on half-life

Now, let's see how the half-life changes when we double the initial concentration: Original half-life: t₁/₂ = 1/(k[A]₀) New half-life with doubled initial concentration: t₁/₂' = 1/(k(2[A]₀)) t₁/₂' = 1/(2k[A]₀) Comparing the original half-life with the new half-life after doubling the initial concentration, we see that: t₁/₂' = (1/2)t₁/₂ Thus, doubling the initial concentration of a reactant in a second-order reaction will cause the half-life to decrease by half.

Key Concepts

Rate Law

In chemical kinetics, the rate law is a mathematical equation that describes the rate of a chemical reaction. For second-order reactions, the rate law takes the form:

• Rate = k[A]²

Here, 'Rate' is the speed at which the reaction proceeds, 'k' is the rate constant unique to each reaction, and '[A]' represents the concentration of reactant A. The squared concentration signifies a second-order reaction, where the reaction rate is proportional to the square of the reactant's concentration. Understanding the rate law helps predict how changes in concentration affect reaction speed, laying the foundation for deeper analysis involving integrated rate laws and other kinetics concepts. This would be particularly useful for anyone looking to control reaction speeds in industrial or laboratory settings.

Integrated Rate Law

The integrated rate law combines the information from the rate law to describe how reactant concentrations change with time. For a second-order reaction, it is derived by integrating the rate law equation:

• 1/[A] - 1/[A]₀ = kt

In this formula, [A]₀ is the initial concentration of reactant A, [A] is its concentration at time t, and 'k' is once again the rate constant. The integrated rate law simplifies tasks such as determining the time required for a reactant to reach a specific concentration. It also provides insights into the kinetics of the reaction, showing how concentration decreases over time. Being familiar with this concept is crucial for tasks like predicting reaction progress and estimating time needed for completing a reaction.

Half-Life

The half-life of a reaction helps determine how long it takes for the concentration of a reactant to reduce to half its initial amount. For second-order reactions, the half-life formula is:

• t₁/₂ = 1/(k[A]₀)

This formula reveals that the half-life is inversely proportional to the initial concentration [A]₀. Thus, unlike first-order reactions, the half-life varies with initial concentration in a second-order reaction. When you double the initial concentration, the half-life becomes half of the original half-life, illustrating the dynamic relationship between concentration and reaction time. Understanding this is essential for practical applications such as determining the shelf life of a chemical or managing reactants in a lab setting.

Reaction Kinetics

Reaction kinetics involves the study of the speeds of chemical processes and how they change under different conditions. In second-order reactions, kinetics explores how variations in reactant concentration and other factors affect the rate and overall progress of a reaction.

• Factors such as temperature, concentration, and catalysts play a significant role in reaction kinetics.
• Kinetics provides a detailed map of how a reaction proceeds over time.

This field aids in understanding the mechanisms behind chemical reactions, allowing for the development of more efficient industrial processes and innovations in fields like pharmaceuticals. Learning about reaction kinetics equips students and professionals with skills to manipulate and optimize reactions for desired outcomes.