Question

Which statement below regarding the half-life of a second order reaction is true? a. The length of the half life remains unchanged throughout the course of the reaction. b. Each half life is four times as long as the preceding one. c. Each half life is half as long as the preceding one. d. Each half life is twice as long as the preceding one.

Short answer

d. Each half life is twice as long as the preceding one.

Step-by-step solution

Understand Half-life for Second Order Reactions

For a second-order reaction, the half-life, denoted as \( t_{1/2} \), depends on the initial concentration. It is given by the formula: \[ t_{1/2} = \frac{1}{k[A]_0} \] where \( k \) is the rate constant and \( [A]_0 \) is the initial concentration.

Evaluate the Formula in Terms of Reaction Progress

In a second-order reaction, as the reaction progresses, the concentration \( [A] \) decreases. Since \( t_{1/2} \) is inversely proportional to \( [A]_0 \), the half-life increases as \( [A]_0 \) decreases. This means each successive half-life will be longer than the previous.

Identify the Correct Option

Given that each successive half-life is longer than the preceding one, we should look for the option that reflects this pattern. Option (d) states "Each half life is twice as long as the preceding one," which correctly aligns with the behavior of second-order reactions.

Key concepts

Second-order Reaction

A second-order reaction is unique in the way it proceeds. Unlike first-order reactions, where the rate depends solely on the concentration of one reactant, second-order reactions depend on the concentrations of either two reactants or the square of a single reactant. This means that if we have a reaction like \[ A + B \rightarrow C \], its rate law could be written as \[ ext{Rate} = k[A][B] \]if two different reactants are involved. Alternatively, if only one reactant is involved in a manner that its concentration is squared, the rate law becomes \[ ext{Rate} = k[A]^2 \]. The nature of these dependencies means that the reaction's progress can show more complexity, especially as the concentration of reactants changes over time. Understanding how the concentration affects reaction behavior is crucial in predicting how quickly a reaction will reach completion.

Reaction Kinetics

Reaction kinetics is the study of the rates at which chemical processes occur. It is a fundamental topic in chemistry as it helps us understand how different factors influence reaction speeds. In second-order reactions, the kinetics are more complex compared to first-order ones due to their dependency on concentration in a squared manner. Key factors affecting second-order kinetics include:

• Concentration of reactants: As the concentration decreases, the rate of reaction also decreases.
• Temperature: Usually, an increase in temperature results in a faster reaction rate.
• Presence of catalysts: Catalysts can accelerate the rate without being consumed.

Reaction kinetics allows chemists to optimize conditions for desired reaction speeds and to better design chemical processes.

Rate Constant

The rate constant, denoted as \( k \), is a crucial part of the rate law equation and serves as a proportionality factor in determining reaction rates. For second-order reactions, the units of \( k \) are typically \[ M^{-1} \, s^{-1} \] (molarity inverse per second), reflecting the dependence on concentration.The value of \( k \) is temperature-dependent, meaning it can change if the reaction temperature varies. However, it remains constant for a given reaction at a specific temperature. By knowing the rate constant, chemists can predict how fast a reaction will proceed under given conditions, making it an integral tool in chemical kinetics.

Initial Concentration

Initial concentration, denoted as \( [A]_0 \), refers to the concentration of a reactant at the start of the reaction. In second-order reactions, this initial concentration plays a significant role in determining the half-life of the reaction, which is different from zero and first-order reactions.The half-life \( t_{1/2} \) of a second-order reaction is given by the formula:\[ t_{1/2} = \frac{1}{k[A]_0} \].This equation shows that as the initial concentration \( [A]_0 \) decreases, the half-life increases. Thus, the reaction takes longer to reach half of its starting concentration with each successive half-life. Understanding the initial concentration's impact is essential for predicting how a reaction will change over time and can assist in planning experimental conditions.