Problem 15 - General Chemistry: Principles and Modern Applications

Question

One of the following statements is true and the other is false regarding the first-order reaction 2A \(\longrightarrow \mathrm{B}+\mathrm{C}\). Identify the true statement and the false one, and explain your reasoning. (a) The rate of the reaction decreases as more and more of \(\mathrm{B}\) and \(\mathrm{C}\) form. (b) The time required for one-half of substance \(A\) to react is directly proportional to the quantity of A present initially.

Step-by-Step Solution

Answer

Both statements (a) and (b) are false. The rate of a first-order reaction does not change with the formation of products, and the half-life of a first-order reaction is constant and not dependent on the initial concentration of the reactant.

Step 1: Check Statement (a)

The rate of a first-order reaction is dependent only on the concentration of one reactant, in this case, A. It does not change with the formation of products B and C. Therefore, as long as there is reactant A remaining, the rate of the reaction remains constant, assuming the temperature remains constant. Hence, statement (a) that suggests the rate of reaction decreases as more B and C form is false.

Step 2: Check Statement (b)

The time required for half of the reactant A to react (half-life) in a first order reaction is a constant that is independent of the initial concentration of A. Therefore, regardless of how much A is present at the start, the half-life remains the same. Hence, statement (b), stating that the time required for one-half of A to react is directly proportional to the quantity of A present initially, is also false.

Key Concepts

Reaction rate

In first-order reactions, the reaction rate is primarily influenced by the concentration of one reactant. In our example, the reaction is given by the equation: \(2A \rightarrow B + C\). Here, the reaction rate depends on the concentration of reactant \(A\) alone. It is essential to understand that for first-order reactions, the reaction rate is not impacted by the concentration of the products \(B\) and \(C\). This remains true as long as temperature is constant.

• The rate equation can be expressed as: \( \text{Rate} = k[A] \), where \(k\) is the rate constant.

This means that the rate will decrease as the concentration of \(A\) decreases over time, but it does not decrease due to the formation of B and C. Therefore, the belief that the rate diminishes as more products form is a misunderstanding. It is crucial to focus purely on how much reactant \(A\) is left.

Half-life

The concept of half-life in first-order reactions refers to the time it takes for half of the reactant to be converted into product. Importantly, for first-order reactions, the half-life is constant and does not depend on the initial concentration of the reactant.

• The formula for the half-life of a first-order reaction is: \( t_{1/2} = \frac{0.693}{k} \), where \(k\) is the rate constant.

This means that no matter how much of \(A\) you start with, the time for half of it to convert to products remains consistent. This property can be especially helpful in predicting how reactions will progress over time without detailed initial measurements. It simplifies practical scenarios, such as calculating how much reactant remains after a series of half-lives.

Concentration dependence

Concentration dependence in first-order reactions relates to how the reaction rate depends on the concentration of the reactant. In our specific reaction example, the rate is dependent solely on the concentration of \(A\), as represented by the relation: \( \text{Rate} = k[A] \). For students, it is important to grasp that the rate constant \(k\) is a catalyst here, as it remains unaffected by how much of products \(B\) and \(C\) exists at any point. A decrease in \(A\)'s concentration directly leads to a decreased rate, but this is unconnected to the concentration of the products formed.

• This concentration dependence showcases how the reactant's quantity alone dictates the rate, ensuring a predictable rate decline as \(A\) diminishes.

Understanding this concept aids in comprehending rate laws, making it easier to predict and manipulate reaction kinetics in practical applications, such as chemical manufacturing or quality control in a laboratory setting.