Given the following diagrams at t = 0 min and t = 30 min

After four half-life periods for a first-order reaction, what fraction of reactant remains? [Section 14.4]

Verified step by step guidance

1. Identify that the problem involves a first-order reaction and the concept of half-life.
2. Recall that for a first-order reaction, the half-life (t_1/2) is the time required for the concentration of the reactant to decrease by half.
3. Understand that after one half-life, the concentration of the reactant is reduced to 1/2 of its initial value.
4. After two half-lives, the concentration is reduced to (1/2)^2 = 1/4 of its initial value.
5. After four half-lives, the concentration is reduced to (1/2)^4 = 1/16 of its initial value.

Key Concepts

First-Order Reaction

A first-order reaction is a type of chemical reaction where the rate is directly proportional to the concentration of one reactant. This means that as the concentration of the reactant decreases, the rate of the reaction also decreases. The mathematical representation of a first-order reaction is given by the equation: rate = k[A], where k is the rate constant and [A] is the concentration of the reactant.

Half-Life

The half-life of a reaction is the time required for the concentration of a reactant to decrease to half of its initial value. For first-order reactions, the half-life is constant and does not depend on the initial concentration. This property allows for easy calculations of remaining reactant concentration over multiple half-lives, making it a crucial concept in kinetics.

Fraction of Reactant Remaining

The fraction of reactant remaining after a certain number of half-lives can be calculated using the formula: (1/2)^n, where n is the number of half-lives that have passed. For example, after four half-lives, the fraction remaining would be (1/2)^4 = 1/16. This concept is essential for understanding how reactant concentrations change over time in a first-order reaction.