Problem 101 - Chemical Principles

Question

The half-life of a substance taking part in a third-order reaction \(\mathrm{A} \rightarrow\) products is inversely proportional to the square of the initial concentration of A. How can this half-life be used to predict the time needed for the concentration to fall to (a) onehalf; (b) one-fourth; (c) one- sixteenth of its initial value?

Step-by-Step Solution

Answer

The time required for the concentration to fall to (a) one-half is t_1/2; to (b) one-fourth is 2 * t_1/2; to (c) one-sixteenth is 4 * t_1/2.

Step 1: Understanding the Third-Order Reaction Half-Life

The half-life of a third-order reaction, denoted as t_1/2, is inversely proportional to the square of the initial concentration of the reactant A, denoted as [A]_0. This relationship can be expressed as t_1/2 ∝ 1 / [A]_0^2. To predict the time needed for the concentration to fall to a certain fraction of its initial value, we can use this relation.

Step 2: Time for Concentration to Fall to One-Half

To find the time for the concentration to fall to one-half of its initial value, we use the definition of half-life. Since the half-life is the time required for the concentration to fall to one-half, the time needed for condition (a) is simply t_1/2.

Step 3: Time for Concentration to Fall to One-Fourth

For the concentration to fall to one-fourth ([A] = 1/4[A]_0), the reaction must go through two half-lives. Therefore, the time needed for condition (b) is two times the half-life: t = 2 * t_1/2.

Step 4: Time for Concentration to Fall to One-Sixteenth

For the concentration to fall to one-sixteenth ([A] = 1/16[A]_0), the reaction must go through four half-lives (since 1/16 is 1/2 raised to the fourth power). Thus, the time needed for condition (c) is four times the half-life: t = 4 * t_1/2.

Key Concepts

Reaction Kinetics

Reaction kinetics is the study of rates at which chemical processes occur and the factors that influence these rates. The core of this study lies in understanding how the concentration of reactants, the presence of catalysts, temperature, and other conditions affect the speed of a reaction.

In a third-order reaction, the rate is proportional to the cube of the reactant's concentration, which means that if you were to triple the concentration of the reactant, the rate of reaction would increase twenty-seven-fold. It's interesting to see how sensitive such a reaction is to changes in concentration, and this sensitivity is reflected in the calculation of half-lives and reaction rates.

Half-life

Half-life, in the context of reaction kinetics, is a term used to describe the time required for the concentration of a reactant to decrease to half of its original amount. It's a critical concept when evaluating the speed and progression of a chemical reaction.

As described in the exercise, in a third-order reaction, the half-life is inversely proportional to the square of the initial concentration, suggesting that a higher starting concentration would result in a quicker decrease to half its value. This principle makes it possible to forecast how long it will take for the concentration to reach not just one-half, but also one-fourth or one-sixteenth of its initial value by understanding that each subsequent half-life is the same length of time, assuming the reaction continues at the same rate.

Chemical Reaction Rate

The chemical reaction rate is an expression of the speed at which a chemical reaction occurs. It's often quantified by the change in concentration of a reactant or product per unit time. In our specific case of a third-order reaction, the reaction rate increases dramatically with increased concentration — showcasing how every participant in the chemical process has a significant role in determining the overall pace of the reaction.

By understanding that the half-life for a third-order reaction is inversely proportional to the square of the concentration, it becomes simpler to predict the duration required for the reaction to reach certain milestones. For example, once you've determined the half-life based on the initial concentration, you can easily determine the time needed for the concentration to fall to one-half, one-fourth, or one-sixteenth by multiplying the half-life by 1, 2, or 4, respectively.