Which statement below regarding the half-life of a second-order reaction is true?
A) Each half-life is half as long as the preceding one.
B) Each half-life is twice as long as the preceding one.
C) Each half-life is four times as long as the preceding one.
D) The length of the half-life remains unchanged throughout the course of the reaction.
The correct answer and explanation is :
The correct answer is B) Each half-life is twice as long as the preceding one.
Explanation:
In the context of chemical reactions, the half-life refers to the time required for the concentration of a reactant to decrease by half during the course of the reaction. The half-life of a reaction depends on its order, which is a measure of how the reaction rate is influenced by the concentration of reactants.
For second-order reactions, the rate of the reaction is proportional to the square of the concentration of the reactant. This is mathematically expressed as:
$$
\text{Rate} = k[A]^2
$$
Where:
- $\text{Rate}$ is the rate of reaction,
- $k$ is the rate constant, and
- $[A]$ is the concentration of the reactant.
For a second-order reaction, the expression for half-life ($t_{1/2}$) is given by:
$$
t_{1/2} = \frac{1}{k[A]_0}
$$
Where:
- $t_{1/2}$ is the half-life,
- $k$ is the rate constant, and
- $[A]_0$ is the initial concentration of the reactant.
This equation shows that for a second-order reaction, the half-life depends on the initial concentration of the reactant. As the concentration decreases over time, the half-life becomes longer. In other words, each successive half-life is longer than the preceding one, and it increases in a manner that is directly proportional to the inverse of the concentration. As the concentration of the reactant decreases, the time required for the next half-life increases.
Thus, for a second-order reaction, the half-life is not constant as it is in a zero-order reaction, nor does it decrease as in a first-order reaction. Instead, it increases over time. In simpler terms, each half-life is twice as long as the previous one because the reaction is slower as the concentration of reactants decreases.
To summarize, the half-life of a second-order reaction increases as the reaction progresses, and it is proportional to the initial concentration, making option B the correct choice.