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 \(\Delta \mathrm{E}\) . c. The half-life for a reaction, 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

The true statements are b and d. Statement b is true because a catalyst does not change the overall energy change of the reaction, only the activation energy. Statement d is true because the half-life for a second-order reaction increases as the reaction proceeds, as it is inversely proportional to the initial concentration of A. Statements a and c are false.

Step-by-step solution

Statement a: The half-life for a zero-order reaction increases as the reaction proceeds.

For a zero-order reaction, the half-life is described by the equation: \[t_{1/2} = \frac{[\mathrm{A}]_{0}}{2k}\] Here, \(t_{1/2}\) is the half-life, \([\mathrm{A}]_{0}\) is the initial concentration of A, and k is the rate constant. Since the half-life depends only on the initial concentration of A and the rate constant, it does not change as the reaction proceeds. This statement is false.

Statement b: A catalyst does not change the value of \(\Delta \mathrm{E}\).

A catalyst speeds up the rate of the reaction by lowering the activation energy. However, it does not change the overall energy change of the reaction. In other words, the difference between the potential energy of the reactants and the products remains the same. Therefore, this statement is true.

Statement c: The half-life for a reaction, aA \(\longrightarrow\) products, that is first order in A increases with increasing \([\mathrm{A}]_{0}\).

For a first-order reaction, the half-life is defined as: \[t_{1/2} = \frac{0.693}{k}\]. Here, \(t_{1/2}\) is the half-life, and k is the rate constant. In this case, the half-life does not depend on the initial concentration of A, so it remains constant even as \([\mathrm{A}]_{0}\) increases. This statement is false.

Statement d: The half-life for a second-order reaction increases as the reaction proceeds.

For a second-order reaction, the half-life is given by: \[t_{1/2} = \frac{1}{k[\mathrm{A}]_{0}}\] In this case, the half-life is inversely proportional to the initial concentration of A, so as the reaction proceeds and \([\mathrm{A}]_{0}\) decreases, the half-life increases. Thus, this statement is true. In summary, statements b and d are true, while statements a and c are false.

Key concepts

Reaction Order

The concept of reaction order is fundamental in chemical kinetics and refers to the power to which the concentration of a reactant is raised in the rate equation. The reaction order provides insight into how the concentration of reactants affects the speed of the reaction. Generally, it is determined by the sum of the exponents of the concentration terms in the rate law.

Different reactions can have different orders:

• Zero-order: Rate is independent of the concentration of the reactant.
• First-order: Rate depends linearly on the concentration of one reactant.
• Second-order: Rate depends either on the square of one reactant's concentration or the product of two reactants' concentrations.

Determining the reaction order is crucial for understanding and predicting how a reaction proceeds over time.

Half-Life

Half-life, often written as \( t_{1/2} \), is the time required for the concentration of a reactant to decrease to half of its initial value. It's an important metric in kinetics as it provides a clear indicator of how fast a reaction progresses. The half-life formula varies depending on the order of the reaction.

For each reaction order:

• Zero-order: \( t_{1/2} = \frac{[A]_0}{2k} \), where it actually decreases with passing time.
• First-order: \( t_{1/2} = \frac{0.693}{k} \), remaining constant regardless of concentration.
• Second-order: \( t_{1/2} = \frac{1}{k[A]_0} \), meaning it increases as the concentration decreases.

Understanding half-life is essential for applications in pharmacokinetics and radioactive decay.

Catalyst Effect

A catalyst plays a significant role in chemical reactions by altering the rate without being consumed. The primary function of a catalyst is to lower the activation energy of the reaction, thereby increasing the rate at which the reaction occurs. However, a catalyst does not affect the overall energy change \((\Delta E)\) of the reaction.

Catalysts work by:

• Providing an alternative reaction pathway with a lower activation barrier.
• Stabilizing transition states to reduce the energy required for reactions.

They are crucial in industrial processes and biological systems, ensuring efficient conversion and synthesis.

Zero-Order Reaction

In zero-order reactions, the rate of reaction is constant and does not depend on the concentration of the reactant. This means that the rate law can be written simply as \( ext{rate} = k \). For such reactions, the half-life decreases over time because the reactant is consumed linearly.

Characteristics of zero-order reactions include:

• Rate remains constant as the reaction progresses.
• Graph of concentration vs. time is a straight line.
• Common when a reactant is in excess or a surface-catalyzed reaction.

Understanding zero-order kinetics helps predict the behavior of reactions under specific conditions.

First-Order Reaction

For first-order reactions, the rate of the reaction is directly proportional to the concentration of one reactant. The rate law for a first-order reaction is \( ext{rate} = k[A] \), where the concentration \([A]\) of the reactant declines exponentially over time.

Key properties of first-order reactions include:

• Constant half-life: \( t_{1/2} = \frac{0.693}{k} \).
• Graph of ln([A]) vs. time yields a straight line.
• Commonly seen in radioactive decay and many types of chemical decompositions.

Understanding first-order kinetics is crucial for modeling decay processes and reaction mechanisms.

Second-Order Reaction

In second-order reactions, the rate is proportional to the product of two reactant concentrations or the square of a single reactant's concentration. The rate law can be written as \( ext{rate} = k[A]^2 \) or \( ext{rate} = k[A][B] \), depending on whether one or two reactant species are involved.

Notable features include:

• Half-life increases with decreasing concentration: \( t_{1/2} = \frac{1}{k[A]_0} \).
• Graph of \( 1/[A] \) vs. time is linear.
• Common in bimolecular reactions where two entities collide.

Understanding second-order kinetics helps with insights into reaction dynamics and mechanisms.