Question

If \(t_{1 / 2}\) of a second-order reaction is \(1.0 \mathrm{~h}\). After what time, the amount will be \(25 \%\) of the initial amount? (a) \(1.5 \mathrm{~h}\) (b) \(2 \mathrm{~h}\) (c) \(2.5 \mathrm{~h}\) (d) \(3 \mathrm{~h}\)

Short answer

2.0 hours

Step-by-step solution

Understand the Second-Order Reaction Half-Life Relationship

For a second-order reaction, the half-life, denoted as \(t_{1/2}\), is inversely proportional to the initial concentration. The formula to calculate the half-life for a second-order reaction is given by \(t_{1/2} = \frac{1}{k[A]_0}\), where \(k\) is the rate constant and \([A]_0\) is the initial concentration.

Relate the Amount Remaining to the Half-Life

To determine the time when the amount is 25% of the initial amount, we can use the fact that after one half-life, the amount reduces to 50% of the original. Since we want to find when it's 25%, we need to go through another half-life period.

Calculate the Total Time

At 25%, two half-lives have passed. Therefore, the total time is \(t = 2 \times t_{1 / 2} = 2 \times 1.0 \text{ h} = 2.0 \text{ h}\).

Key concepts

Understanding Chemical Kinetics

Chemical kinetics is a branch of chemistry that studies the speed or rate at which a chemical reaction occurs, and the factors that affect this rate. In essence, kinetics is about understanding how quickly reactants are converted into products during a reaction. There are several factors that can influence reaction rates, including temperature, the presence of catalysts, reactant concentrations, and the physical state of the reactants.

When studying kinetics, it's important to know that reactions can occur at different rates. Some happen almost instantaneously, while others may take minutes, hours, or even longer. Kinetic studies are not just academic; they have practical implications in industries such as pharmaceuticals, where the rate at which a drug reacts in the body can determine its effectiveness and safety.

To observe these rates, chemists use concentration-time graphs, rate laws, and mathematical relationships, such as the one for calculating the half-life of a reaction, to model the kinetics of the chemical processes. The rate of a reaction is often expressed as the change in concentration of reactants (or products) per unit time and is a crucial part of understanding chemical behavior in schooling and industrial applications alike.

Reaction Rate Constant

The reaction rate constant, denoted as \(k\), is a proportionality constant that is part of the rate law equation and gives a measure of the speed of a chemical reaction. For a given chemical reaction at a certain temperature, it is a fixed value, and it provides insight into the kinetics of the reaction by quantifying the rate at the molecular level.

For a general reaction in which reactant A converts to product B, the rate can be expressed as \( rate = k[A]^n \), where \([A]\) is the concentration of A, and \(n\) is the order of the reaction with respect to A. The order of a reaction indicates the power to which the concentration of a reactant is raised in the rate law. It can be zero, first, second, or even fractional, representing different dependencies on the concentration.

A second-order reaction is one where the rate is proportional to the square of the concentration of a single reactant, or to the product of the concentrations of two reactants. In this case, the rate law would look like \( rate = k[A]^2\) or \( rate = k[A][B]\) if there are two reactants involved. The rate constant \(k\) can be determined experimentally and is essential for predicting how fast a reaction will proceed under different conditions.

Half-Life Calculation in Second-Order Reactions

Half-life, symbolized as \(t_{1/2}\), is the time required for the concentration of a reactant to decrease to half of its initial value. For second-order reactions, this concept is particularly important because the half-life is not constant; it varies with the concentration of the reactants. To calculate the half-life for a second-order reaction, the formula \(t_{1/2} = \frac{1}{k[A]_0}\) is used, where \(k\) is the rate constant and \([A]_0\) is the initial concentration of the reactant.

In the context of the exercise, once you know the initial half-life, you can infer subsequent half-lives by remembering that the concentration halves with each passed half-life period. Therefore, to reach 25% concentration, you will pass through two half-lives. So if the initial half-life is 1.0 hour, you simply multiply this value by 2 to find the time it takes for the reactant to reach a quarter of its initial concentration. Hence, the correct solution is 2 hours.

This sort of calculation is significant in many fields including pharmacology, where half-life determines how frequently a medication should be administered, and in environmental science, predicting how long a pollutant will remain at a harmful concentration. By mastering half-life calculations, students gain a more comprehensive understanding of the temporal aspects of chemical reactions.