Question

Consider two reactions. Reaction (1) has a constant halflife, 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 half-life behavior of the two reactions, we can deduce that Reaction 1 follows a first-order rate law with a constant half-life (\(t_{1/2} = \frac{0.693}{k}\)), while Reaction 2 follows a second-order rate law with a half-life that increases as the reaction proceeds (\(t_{1/2} \propto \frac{1}{[A]}\)).

Step-by-step solution

Recall relationship between half-life and reaction order

For different reaction orders, the relationship between the half-life (t_1/2) and reactant concentration ([A]) differs. Here's a summary of the relationships: 1. Zero-order reaction: t_1/2 = (Initial concentration) / (2 * Rate constant), \(t_{1/2} \propto \frac{1}{k}\) 2. First-order reaction: t_1/2 = 0.693 / Rate constant, \(t_{1/2} = \frac{0.693}{k}\) (constant) 3. Second-order reaction: t_1/2 = 1 / (Rate constant * Initial concentration), \(t_{1/2} \propto \frac{1}{[A]}\)

Determine Reaction 1 rate law

Because the half-life of Reaction 1 is constant, this means that the half-life does not depend on the concentration of the reactants. Based on the relationship between half-life and reaction order, this behavior is characteristic of a first-order reaction. Thus, we can conclude that Reaction 1 follows a first-order rate law.

Determine Reaction 2 rate law

For Reaction 2, the half-life increases as the reaction proceeds. As the reaction proceeds, the concentration of the reactants decreases. Therefore, in this case, the half-life is inversely proportional to the concentration of the reactants. This behavior is characteristic of a second-order reaction. Thus, we can conclude that Reaction 2 follows a second-order rate law.

Conclusion

By analyzing the behavior of half-lives in the given reactions, we can conclude that Reaction 1 follows a first-order rate law, while Reaction 2 follows a second-order rate law. This is based on the relationship between half-life and reaction order for different reaction orders.

Key concepts

half-life and reaction order

Understanding the relationship between half-life and reaction order is crucial in determining the rate laws governing chemical reactions. The half-life of a reaction, represented by \( t_{1/2} \), is the time required for the concentration of a reactant to decrease to half of its initial value. This concept is directly linked to the order of the reaction, which characterizes how the rate of a reaction depends on the concentration of reactants.
- In a **zero-order reaction**, the half-life decreases as the reaction proceeds because it is dependent on the initial concentration of the reactants.
- A **first-order reaction** features a half-life that remains constant throughout the reaction progress, since it is independent of the initial concentration.
- For a **second-order reaction**, the half-life increases as the reaction progresses, as it varies inversely with the reactant concentration.
By evaluating how the half-life behaves as the reaction progresses, scientists can deduce the order of the reaction and thus understand the underlying kinetics.

zero-order reaction

In a zero-order reaction, the rate of reaction is independent of the concentration of the reactant. This means the reaction rate remains constant over time. The equation for the half-life of a zero-order reaction is \( t_{1/2} = \frac{[A]_0}{2k} \), where \( [A]_0 \) is the initial concentration and \( k \) is the rate constant.
- As the concentration decreases, the half-life of the zero-order reaction decreases accordingly.
- This results in a linear decrease in reactant concentration over time.
Zero-order reactions are less common, and typically occur in cases where a substance is initially present in abundance, causing the reaction rate to be determined primarily by other factors such as catalyst availability or surface area of the reactant.

first-order reaction

First-order reactions have a distinct feature: their half-life remains constant irrespective of the concentration of the reactant. This means that no matter how much reactant is present, the time it takes for the concentration to halve remains the same.
The relationship is given by the formula \( t_{1/2} = \frac{0.693}{k} \), where \( k \) is the rate constant. Because the half-life is constant, - First-order reactions are common in natural decay processes, like radioactive decay where the decay rate is independent of the amount of substance present.
- They are also prevalent in many biological systems, such as enzyme reactions where the enzyme is in excess compared to the substrate.
This constant nature makes it easier to predict how long it will take for a certain amount of reactant to be consumed in a first-order reaction.

second-order reaction

For a second-order reaction, the rate of the reaction is proportional to the square of the concentration of one reactant or to the product of the concentrations of two reactants. This relationship makes the behavior of half-lives in these reactions unique. The half-life for second-order reactions is given by \( t_{1/2} = \frac{1}{k[A]_0} \), where \( k \) is the rate constant and \( [A]_0 \) is the initial concentration.
In this scenario:- The half-life increases as the concentration of the reactant decreases, meaning the process slows down over time.
- This characteristic is often observed in reactions where collisions between two reactant molecules are required, thus making kinetic dependencies more complex.Second-order reactions are typical in systems where the rate of formation of product is dependent on the collision frequency of reactant molecules, which decreases as fewer reactant molecules are available.