Problem 54 - Chemistry and Chemical Reactivity

Question

For a first-order reaction, what fraction of reactant remains after five half- lives have elapsed?

Step-by-Step Solution

Answer

After five half-lives, \( \frac{1}{32} \) of the reactant remains.

Step 1: Understand the half-life concept for first-order reactions

In a first-order reaction, the half-life is the time required for half of the reactant to be consumed. This is a constant value and does not depend on the initial concentration of the reactant.

Step 2: Calculate the remaining fraction after one half-life

After one half-life, half of the original reactant remains. Therefore, the remaining fraction is given by \( \frac{1}{2} \).

Step 3: Calculate the remaining fraction after five half-lives

For each half-life, the remaining fraction of the reactant is halved. After five half-lives, the remaining fraction is \( \left( \frac{1}{2} \right)^5 \).

Step 4: Perform the calculation

Calculate \( \left( \frac{1}{2} \right)^5 \), which is \( \frac{1}{32} \). This means that after five half-lives, \( \frac{1}{32} \) of the reactant remains.

Key Concepts

half-life

In chemistry, the concept of half-life is pivotal when discussing reactions, particularly first-order reactions. A half-life is defined as the amount of time it takes for half of a given amount of a reactant to be used up or transformed into a product.
In first-order reactions, this half-life is unique as it remains constant, no matter how much of the original substance you start with.
This makes calculations and predictions about the reaction's progression easier.

• For example, if you start with a full amount of a substance, after one half-life, you'll have 50% of that original amount left.
• The next half-life reduces the amount to 25%, then to 12.5%, and so on.

Utilizing this concept, students can determine how long a reaction will take to reach a certain amount of completion. It also helps in understanding the speed or rate at which reactions proceed.

reaction rate

The reaction rate in chemistry refers to how quickly or slowly reactants are converted into products. For first-order reactions, like others, the rate is influenced by the concentration of the reactants.
The rate of a first-order reaction can be expressed using the formula:\[ r = k[A] \]where:- \( r \) is the reaction rate- \( k \) is the rate constant- \( [A] \) is the concentration of the reactant
Since first-order reactions depend linearly on the reactant's concentration, as the reaction progresses, the rate decreases because the concentration of the reacting species reduces.

• The rate constant \( k \) is critical because it encapsulates factors affecting the rate, like temperature and the presence of a catalyst.
• If \( k \) is high, the reaction proceeds rather quickly, allowing the formation of products in less time.

Understanding the reaction rate is essential in predicting how long a process will take and optimizing conditions for desirable outcomes.

chemical kinetics

Chemical kinetics is the field of chemistry that focuses on the speed of chemical reactions and the factors that affect this speed. It's a vital area of study that helps chemists understand and control reactions better.
Several factors could affect reaction rates, such as temperature, concentration, and the presence of a catalyst. For a first-order reaction, chemical kinetics provides insights into how these factors efficiently drive reactions.

• As a reaction occurs, monitoring the concentrations of reactants and products can indicate how far the reaction has proceeded.
• With kinetic studies, scientists can establish mechanisms or sequences of steps a reaction follows.

This understanding allows for greater control over industrial processes, environmental management, and even biochemical pathways in living organisms. Mastery of chemical kinetics is essential to effectively implementing reactions for desired results and limiting unwanted by-products.