Question

Consider two reactions. Reaction (1) has a constant half-life, whereas reaction (2) has a half-life that gets longer as the reaction proceeds. What can you conclude about the rate laws of these reactions from these observations?

Short answer

From the information about the half-lives of reactions, we can infer that reaction (1) is a first-order reaction because it has a constant half-life. Thus, the rate law is formulated as \(rate = k_1[A_1]\), where \(k_1\) is the rate constant and \([A_1]\) is the concentration of the reactant. Conversely, reaction (2) is not a first-order reaction since its half-life increases as the reaction proceeds. The rate law form of reaction (2) could be either zeroth-order, second-order, or higher-order, not in the form \(rate = k_2[A_2]\). However, without further experimental data, we can't arrive at a precise conclusion about its order.

Step-by-step solution

Understanding half-lives and rate laws

Half-life is the time it takes for the concentration of a reactant to decrease by half. The rate law is an equation that describes the relationship between the reaction rate and the concentrations of the reactants. The order of the reaction is determined by the exponents of the concentration terms in the rate law equation. For first-order reactions, the rate law equation is: \(rate = k[A]\), where k is the rate constant, and [A] is the concentration of the reactant A. For first-order reactions, the half-life is constant and independent of the initial reactant concentration. In other words, it takes the same time to go from 100% to 50% of the reactant as it does to go from 50% to 25%. For other reaction orders, the half-life depends on the initial concentration of the reactants.

Identifying the rate law for reaction (1)

Since reaction (1) has a constant half-life, it is indicative of a first-order reaction. Therefore, the rate law for reaction (1) can be represented as: \(rate = k_1[A_1]\), where k_1 is the rate constant for reaction (1) and [A_1] is the concentration of reactant A for reaction (1).

Identifying the rate law for reaction (2)

As the half-life for reaction (2) gets longer as the reaction proceeds, it suggests that the reaction is not first-order; otherwise, the half-life would be constant. Therefore, we can conclude that the rate law for reaction (2) is not of the form \(rate = k_2[A_2]\), where k_2 is the rate constant for reaction (2) and [A_2] is the concentration of reactant A for reaction (2). This reaction could be a second-order or a higher-order reaction, depending on the experimental data. In general, a zeroth-order reaction and a second-order reaction have a half-life that depends on the initial concentration of the reactants, but given the information we have, we cannot say for sure which order it is. We can only conclude that reaction (2) is not a first-order reaction.

Conclusion

From the given information about the half-lives of the reactions, we can conclude that: 1. Reaction (1) is a first-order reaction with a rate law of the form \(rate = k_1[A_1]\). 2. Reaction (2) is not a first-order reaction, and its rate law must be of a different form, which could be either zeroth-order, second-order, or a higher-order. The exact form depends on the experimental data.

Key concepts

Half-life and Reaction Order

The concept of half-life is essential in understanding reaction kinetics. It is defined as the time it takes for half of a reactant to be consumed in a reaction. This value can provide insights into the reaction order. The order of a reaction refers to the power to which the concentration of a reactant is raised in the rate law expression. Different reaction orders will impact the behavior of half-life during a reaction.

• If a reaction's half-life is independent of the initial concentration of reactants, it suggests the reaction is first-order.
• If the half-life depends on initial concentration, the reaction could be zeroth, second, or a higher order.

By analyzing how half-life changes, chemists can deduce valuable information about the kinetics and mechanisms of the reaction. This connection between half-life and reaction order is used to characterize reactions in various scientific contexts.

First-Order Reaction

In a first-order reaction, the rate of reaction is directly proportional to the concentration of one reactant. The general rate law equation is represented as:\(rate = k[A]\)Here, \(k\) is the rate constant, and \([A]\) is the concentration of the reactant. A key feature of first-order reactions is that they have a constant half-life. This means that the time required for half of the reactant to be consumed remains the same throughout the reaction.

Examples include radioactive decay and certain chemical reactions where the reaction rate is dependent solely on one molecule. Because of this, first-order reactions are straightforward to model and predict. The constant half-life also allows for simple calculations of remaining reactant over time, making it a vital concept in both theoretical and applied chemistry.

Second-Order Reaction

Second-order reactions are characterized by their dependence on the concentrations of two reactants, or the square of one reactant. The general rate law is given by:\[rate = k[A]^2 \text{ or } rate = k[A][B]\]In these reactions, the half-life is not constant but increases as the reaction proceeds. This dependence on concentration makes second-order reactions more complex compared to first-order reactions.

• For a second-order reaction involving a single reactant, the half-life is given by \(t_{1/2} = \frac{1}{k[A]_0}\), indicating it increases as the concentration decreases.
• Second-order reactions often occur in processes such as dimerization, where two monomer molecules react together.

This variability in half-life can help scientists determine that a reaction follows second-order kinetics, enabling them to refine predictions and understand reaction dynamics better.