Problem 50 - 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: Understanding the half-life concept

To begin, understand that a half-life is the time required for the concentration of a reactant to decrease by half. For a first-order reaction, the number of half-lives passed directly influences the remaining fraction of the reactant.

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

After one half-life, half of the reactant remains. Mathematically, this is expressed as: \( \rac{1}{2} \) of the original concentration.

Step 3: Calculating the fraction after multiple half-lives

The fraction of reactant remaining after \( n \) half-lives is given by the formula \( \rac{1}{2^n} \). This is a direct consequence of repeatedly halving the reactant for each half-life.

Step 4: Calculating the fraction after five half-lives

For five half-lives (\( n = 5 \)), substitute \( n \) in the formula: \( \rac{1}{2^5} \). Calculating this gives \( \rac{1}{32} \), or \( 0.03125 \).

Step 5: Conclusion

Thus, after five half-lives, \( \rac{1}{32} \) of the original reactant remains, meaning approximately 3.125% of the reactant is left.

Key Concepts

Half-Life in First-Order Reactions

The concept of half-life is pivotal in understanding how reactants decrease over time in chemical reactions, especially those which are first-order. A half-life is defined as the amount of time it takes for the concentration of a reactant to decrease to half of its initial value. Imagine it like a timer for your reactant quantity.

• In a first-order reaction, each half-life represents the same amount of time for the concentration to halve.
• For each successive half-life, the remaining concentration is half of what it was at the end of the previous half-life.

This is consistent regardless of how much reactant you start with. Therefore, you can calculate how many half-lives have passed and deduce how much of your reactant remains.If we consider five half-lives, as in the original exercise, using the formula \( \frac{1}{2^n} \) where \( n \) is the number of half-lives, approximately 3.125% of the reactant would remain.

Reactant Concentration Changes Over Time

In chemical reactions, understanding the change in reactant concentration over time helps predict the progression and completion of a reaction. Reactant concentration refers to the amount of reactant present in a unit volume of solution at any time point.

• For first-order reactions, the rate of decrease in concentration depends on the current concentration: as the reactant concentration decreases, the reaction rate slows.
• The relationship in a first-order reaction is given by the formula: \( [A]_t = [A]_0 e^{-kt} \), where \([A]_t\) is the concentration at time \( t \), \([A]_0\) is the initial concentration, \( k \) is the rate constant, and \( e \) is the base of the natural logarithm.

This formula emphasizes how reactant concentration diminishes exponentially in first-order reactions, mirroring the half-life concept.

Chemical Kinetics and Reaction Mechanism

Chemical kinetics is the branch of chemistry that studies the speed of chemical reactions and the factors affecting it. It answers questions such as how fast a reaction occurs and how different conditions influence this rate.

• The rate of a reaction can be influenced by the concentration of reactants remaining, temperature, and catalysts present.
• In a first-order reaction, the rate law is often simplified to \( rate = k[A] \), meaning the rate is directly proportional to the concentration of a single reactant \( [A] \).

This simplicity allows for straightforward calculations and predictions about how a reaction progresses over time.Studying kinetics allows chemists to better understand the reaction mechanisms, or the step-by-step sequence of elementary reactions by which a chemical change occurs, thus elucidating the pathway from reactants to products.

Understanding Reaction Rate

The reaction rate is a measure of how quickly reactants are being converted into products in a chemical reaction. In terms of first-order reactions, the reaction rate is directly proportional to the concentration of the reactant.

• Mathematically, the reaction rate can be represented by the expression \( rate = -\frac{d[A]}{dt} \), where \([A]\) is the concentration of the reactant and \( t \) is time.
• Features of the reaction rate include a constant rate constant \( k \) that is intrinsic to each reaction and remains unchanged as the reaction proceeds.

In practice, monitoring the reaction rate over time provides critical insights into the reaction dynamics and helps in designing processes for better production yields or determining optimum conditions for specific reactions.Overall, reaction rates are a fundamental aspect of chemical kinetics, providing valuable information for both theoretical understanding and practical applications in chemistry.