Question

Which of the following statement(s) is(are) true? a. The half-life for a zero-order reaction increases as the reaction proceeds. b. A catalyst does not change the value of the rate constant. c. The half-life for a reaction, \(\mathrm{aA} \longrightarrow\) products, that is first order in A increases with increasing \([\mathrm{A}]_{0}\). d. The half-life for a second-order reaction increases as the reaction proceeds.

Short answer

Only statement d is true: "The half-life for a second-order reaction increases as the reaction proceeds."

Step-by-step solution

Determine the truth of statement a.

Zero-order reactions have a constant rate that does not depend on the concentration of the reactant. The half-life of a zero-order reaction can be defined as follows: \[t_{1/2} = \frac{[\mathrm{A}]_0}{2k}\] where \([\mathrm{A}]_0\) is the initial concentration of the reactant A and k is the zero-order rate constant. Since the reaction rate is constant, the half-life will not change as the reaction proceeds. Therefore, statement (a) is **false**.

Determine the truth of statement b.

A catalyst is a substance that increases the reaction rate by altering the reaction mechanism or by providing an alternative pathway with a lower activation energy. This change in reaction mechanism affects the rate constant of the reaction. A catalyst does change the value of the rate constant. Therefore, statement (b) is **false**.

Determine the truth of statement c.

For a first-order reaction, the half-life is independent of the initial concentration of the reactant. The half-life of a first-order reaction is: \[t_{1/2} = \frac{\ln{2}}{k}\] As we can see in the equation, the half-life does not depend on the initial concentration \([\mathrm{A}]_0\). Therefore, the half-life does not increase with increasing \([\mathrm{A}]_0\). Statement (c) is **false**.

Determine the truth of statement d.

For a second-order reaction, the half-life is dependent on the concentration of the reactant. The half-life of a second-order reaction is: \[t_{1/2} = \frac{1}{k[\mathrm{A}]_0}\] As the reaction proceeds, the concentration of the reactant A decreases. Since the equation for half-life has \([\mathrm{A}]_0\) in the denominator, as the concentration decreases, the half-life increases. Therefore, statement (d) is **true**. So, the correct answer is: Only statement d is true: "The half-life for a second-order reaction increases as the reaction proceeds."

Key concepts

Half-Life

In reaction kinetics, the concept of half-life is crucial. Half-life refers to the time it takes for half of the reactant concentration to be consumed in a reaction. It's a simple way to understand how fast a reaction progresses.
Half-life calculations are popular for comparing different types of reactions. Reactions can have constant, increasing, or decreasing half-lives depending on their order:

• For zero-order reactions, the half-life decreases as the reaction proceeds because it depends directly on the initial concentration.
• First-order reactions have a constant half-life, independent of the initial concentration.
• In second-order reactions, the half-life increases as the concentration of the reactant decreases.

This makes the concept of half-life a powerful tool for predicting how long a reaction will take or how it will change over time.

Zero-Order Reaction

Zero-order reactions are unique because their rate is independent of the concentration of reactants. This means that the reaction rate remains constant, irrespective of how much reactant is present.
The mathematical representation of half-life in zero-order reactions is given by the formula:\[t_{1/2} = \frac{[\mathrm{A}]_0}{2k}\]Here, \([\mathrm{A}]_0\) is the initial concentration, and \(k\) is the rate constant. The half-life decreases as the concentration decreases. It highlights how zero-order reactions can quickly lose half of their reactants before the constant rate catches up with the decreasing concentration.

First-Order Reaction

First-order reactions are common in chemical kinetics, where the rate depends linearly on the concentration of one reactant. These reactions are interesting because their half-life is constant, not affected by the concentration of reactants.
The half-life equation for a first-order reaction is shown as:\[t_{1/2} = \frac{\ln{2}}{k}\]Notice how the initial concentration \([\mathrm{A}]_0\) does not appear in the equation. This means no matter how much reactant is present at the start, the half-life remains the same. These reactions are predictable and stable over time, making them easy to study and model.

Second-Order Reaction

Second-order reactions have rates that depend on either the concentration of two reactants or the square of the concentration of a single reactant. This relationship affects the half-life calculation:
The equation for the half-life of a second-order reaction is given by:\[t_{1/2} = \frac{1}{k[\mathrm{A}]_0}\]This formula shows that as the reactant's concentration decreases, the half-life increases. After half of the initial reactants have reacted, the reaction slows down significantly. This makes understanding the dynamics of second-order reactions important to anticipate how long the reaction will take to proceed to completion.