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

In conclusion, based on the given information about half-lives, we can deduce that reaction (1) has a first-order rate law: Rate = k[A] For reaction (2), the rate law is given by: Rate = k[A]^n where n ≠ 1, indicating that reaction (2) is not a first-order reaction and could be zero-order, second-order, or any other order other than first-order.

Step-by-step solution

Recall the relationship between half-life and reaction order

A reaction's half-life is the time required for the concentration of a reactant to decrease to half of its initial concentration. The half-life depends on the reaction order, which determines the dependence of the reaction rate on the concentrations of reactants. For instance, for a zero-order reaction, the rate law is given by: Rate = k[A]^0 = k In this case, the half-life is directly proportional to the initial concentration of the reactant and inversely proportional to the rate constant k. For a first-order reaction, the rate law is given by: Rate = k[A]^1 = k[A] In this case, the half-life is independent of the initial concentration of the reactant and only depends on the rate constant k. For a second-order reaction, the rate law is given by: Rate = k[A]^2 In this case, the half-life is inversely proportional to the initial concentration of the reactant and directly proportional to the rate constant k.

Deduce the rate law for reaction (1)

For reaction (1), it is given that the half-life is constant. This information implies that the half-life does not depend on the concentration of the reactants. As recalled in step 1, the half-life of a first-order reaction is independent of the initial concentration of the reactant and only depends on the rate constant k. Thus, we can conclude that reaction (1) follows a first-order rate law: Rate = k[A]

Deduce the rate law for reaction (2)

For reaction (2), the half-life gets longer as the reaction proceeds, which means the half-life is not constant and depends on the concentration of the reactants. As we have deduced in step 1, the half-life for a zero-order reaction is directly proportional to the initial concentration of the reactant, and for a second-order reaction, the half-life is inversely proportional to the initial concentration of the reactant. Since we cannot deduce the exact order of the reaction from the given information, the rate law for reaction (2) will be given as: Rate = k[A]^n where n ≠ 1 since we concluded that reaction (1) is of first-order. Reaction (2) could be zero-order, second-order, or any other order other than first-order. In conclusion, reaction (1) has a first-order rate law and reaction (2) has a rate law of order n ≠ 1, which means it's not a first-order reaction.

Key concepts

Half-life of a Reaction

The concept of the half-life of a reaction is pivotal in understanding chemical kinetics. Half-life, often denoted as $t_{1/2}$, refers to the duration required for a reactant to reach half of its original concentration in a chemical reaction. This parameter is a clear indicator of how quickly a reaction occurs.
For different orders of reactions, the half-life behaves distinctly. In a first-order reaction, the half-life remains constant throughout the process, regardless of the concentration of the reactants. This characteristic of constancy makes it unique and incredibly useful, especially because it allows for the straightforward determination of the rate constant $k$ simply by measuring the half-life. As such, substances that decay radioactively often showcase this type of first-order behavior.
On the other hand, other reaction orders do not have this simplicity. For instance, with zero-order reactions, the half-life decreases as the concentration diminishes, and with second-order reactions, it increases as the concentration drops. Assessing a reaction's half-life can help us deduce the reaction's order and better understand the dynamics at play.

Reaction Order

Reaction order is a fundamental concept in the field of chemical kinetics, giving insight into how the rate of a chemical reaction is affected by the concentration of reactants. Mathematically, the rate of a reaction can often be expressed by the general rate law: $\text{Rate}=k[A{]}^n$, where $k$ is the rate constant, $[A]$ is the reactant concentration, and $n$ is the reaction order.
Reaction order can be integer (e.g., zero, first, second) or even fractional, and it indicates the power to which the concentration of a reactant is raised in the rate law. Zero-order reactions rate is independent of reactant concentration, first-order reactions rate is directly proportional to reactant concentration, while second-order reactions rate varies with the square of the concentration. Knowledge of the reaction order is critical as it not only affects calculations of reaction rates but also influences the strategies chosen for controlling and optimizing chemical processes.
The exercise in question illustrates a scenario where we can infer the reaction order from the behavior of the reaction's half-life – a constant half-life suggests a first-order reaction, while a half-life that changes as the reaction progresses suggests a reaction of a different order.

Rate Constant

The rate constant, denoted as $k$, is a crucial quantity in the rate law expression that provides the speed at which a reaction proceeds. It is unique for every chemical reaction and varies with temperature, typically increasing with rising temperature in accordance with Arrhenius' equation.
For a given reaction at a specific temperature, the rate constant connects the rate of the reaction to the concentration(s) of the reactant(s) as formulated in the rate law. It allows us to calculate the rate of a reaction if we know the concentration of reactants and the order of reaction. In a practical context, if we measure the half-life of a first-order reaction, we can determine the rate constant without knowing the concentration.
The role of the rate constant in determining the overall reaction rate cannot be understated. It effectively acts as a metric of a reaction's propensity to occur, with larger values of $k$ corresponding to faster reactions. The interplay between the rate constant, the concentration of reactants, and the reaction order defines the pace of a chemical process and ultimately, the half-life associated with the reaction.