Question

The reaction \(\mathrm{A} \longrightarrow \mathrm{B}\) shown here follows first- order kinetics. Initially different amounts of A molecules are placed in three containers of equal volume at the same temperature. (a) What are the relative rates of the reaction in these three containers? (b) How would the relative rates be affected if the volume of each container were doubled? (c) What are the relative half-lives of the reactions in (i) to (iii)?

Short answer

(a) Rates are proportional to initial concentrations: \([A]_1 : [A]_2 : [A]_3\). (b) Doubling volume doesn't change relative rates: \([A]_1 : [A]_2 : [A]_3\). (c) Half-lives are equal in all cases.

Step-by-step solution

Describe First-Order Kinetics

In a first-order reaction, the rate of reaction depends linearly on the concentration of one reactant. The rate law for a first-order reaction is expressed as \( \text{rate} = k[A] \), where \( k \) is the rate constant and \([A]\) is the concentration of the reactant.

Determine Relative Rates

Given that the reaction is first-order, the rate is directly proportional to the concentration of \( A \). If you have different initial concentrations in the three containers, for example, \( [A]_1 \), \( [A]_2 \), and \( [A]_3 \), the relative rates are in proportion to these concentrations. Therefore, the relative rates are \( [A]_1 : [A]_2 : [A]_3 \).

Analyze Effect of Doubling Volume

Doubling the volume of the container will halve the initial concentration of \( A \) in each container, assuming the amount of \( A \) remains constant. For example, if the initial concentration was \( [A]_1 \), it becomes \( \frac{[A]_1}{2} \). Since rate is proportional to concentration, the new rates will also be halved. Thus, the overall relative rates remain \( [A]_1 : [A]_2 : [A]_3 \).

Determine Half-Life in First-Order Reactions

For first-order reactions, the half-life \( t_{1/2} \) is given by the formula \( t_{1/2} = \frac{0.693}{k} \) and is independent of the initial concentration. Thus, the relative half-lives of cases (i) to (iii) are equal, regardless of the initial concentrations.

Key concepts

Reaction Rate

When we talk about the reaction rate, we are discussing how fast a reactant turns into a product. In the context of a first-order reaction, like \( \mathrm{A} \rightarrow \mathrm{B} \), the rate is directly proportional to the concentration of the reactant \( [A] \). Simply put, if you have more of \( A \), the reaction happens faster.
- **Proportional Relationship**: The higher the concentration of \( A \), the faster the reaction rate. This tells us that the reaction rate is reliant specifically on how much \( A \) we start with.- **Rate Constant \( k \)**: This is an inherent number that stays the same for a given reaction at a specific temperature, acting as a multiplier for the reaction rate.The rate law equation for a first-order reaction is given by \( \text{rate} = k[A] \), clearly showing the direct relationship between reaction rate and concentration.

Rate Law

The rate law is an expression that relates the reaction rate to the concentration of the reactants. For first-order reactions, this is quite straightforward. The rate law is given by \( \text{rate} = k[A] \), where \( k \) is the rate constant and \( [A] \) is the concentration of reactant \( A \).
- **Linear Dependence**: The rate depends linearly on the concentration of \( A \). As the concentration changes, the rate changes in direct proportion to \( [A] \).- **Unique to Conditions**: The rate constant \( k \) does not change if the concentration changes, but it does depend on temperature. This means that different reactions or different temperatures will have different \( k \) values.This makes the rate law a handy tool for predicting how changes in concentration will affect the speed of a reaction.

Half-life

The half-life of a reaction is the time it takes for half of the reactant to be used up. In first-order reactions, the half-life is remarkably straightforward because it's independent of the initial concentration. This means it remains constant throughout the process.
For a first-order reaction, the half-life \( t_{1/2} \) is calculated using the formula \( t_{1/2} = \frac{0.693}{k} \). Here, \( k \) is the rate constant, and \( 0.693 \) is the natural logarithm of 2.
- **Independent of Concentration**: Unlike other reaction orders, the half-life for first-order reactions doesn't change even as the concentration of reactant declines.- **Predictable Reaction Time**: The constancy of the half-life makes it easier to predict how long a reaction will take to significantly decrease the concentration of \( A \).This uniformity across different concentrations is what makes first-order reactions easier to work with in various chemical scenarios.

Concentration

Concentration, in the context of reactions, refers to how much of a substance is present in a certain volume. It is measured as the number of moles of a substance per liter, denoted as \([A]\) for reactant \( A \). Variations in concentration affect the reaction rate directly, especially in first-order reactions.
- **Initial Concentration Matters**: At the start, the concentration of \( A \) will determine how fast the reaction kicks off.- **Changing volumes**: If the reaction container's volume is doubled, the concentration of \( A \) is halved, given a constant amount of reactant. This directly impacts the rate since the rate is contingent on the concentration as per the rate law.Understanding the impact of concentration shifts is crucial for predicting how a reaction proceeds over time, especially in scenarios where volume changes occur for the reacting solutions.