Problem 40 - Chemistry: The Central Science in SI Units, Global Edition

Question

(a) Develop an equation for the half-life of a zero-order reaction. (b) Does the half-life of a zero-order reaction increase, decrease, or remain the same as the reaction proceeds?

Step-by-Step Solution

Answer

(a) The equation for the half-life of a zero-order reaction is \(t_{1/2} = \frac{[A]_0}{2k}\). (b) The half-life of a zero-order reaction decreases as the reaction proceeds.

Step 1: Write the rate law for a zero-order reaction

A zero-order reaction is one in which the rate of the reaction is independent of the concentration of the reactant. The rate law for a zero-order reaction can be written as: Rate = k where Rate is the rate of reaction, and k is the rate constant. Since Rate is also equal to the change in concentration (∆[A]) over change in time (∆t): k = -∆[A]/∆t Now, let's integrate this equation to find the relationship between the concentration of the reactant and time.

Step 2: Integrate the rate law equation

Integrating the rate law equation, we get: \(\int_{[A]_0}^{[A]} -d[A] = \int_0^t kdt\) Solving this integral: \(-([A]-[A]_0) = kt\) Rearranging the equation for [A]: [A] = [A]_0 - kt Now we can use this equation to find the half-life of the reaction.

Step 3: Calculate the half-life equation

The half-life (t1/2) is the time it takes for the concentration of the reactant to reach half of its initial value. In our equation for [A], let [A] = [A]_0/2, and the time as t = t1/2. Substituting these values and solving for t1/2: [A]_0/2 = [A]_0 - kt1/2 Now, let's isolate t1/2 to obtain the equation for half-life: kt1/2 = [A]_0/2 t1/2 = [A]_0/(2k) This is the equation for the half-life of a zero-order reaction.

Step 4: End Task (a)

We have found the equation for the half-life of a zero-order reaction: t1/2 = [A]_0/(2k) Now, let's move on to part (b) of the exercise.

Step 4: Determine the relationship between half-life and reaction progress

We want to find out if the half-life of a zero-order reaction increases, decreases, or remains the same as the reaction proceeds. For this, let's consider the half-life equation we derived in step 3: t1/2 = [A]_0/(2k) Notice that t1/2 depends on the initial concentration [A]_0. As the reaction proceeds, the concentration of the reactant decreases. Therefore, as [A]_0 decreases, t1/2 will also decrease. #End Task (b)# The half-life of a zero-order reaction decreases as the reaction proceeds.

Key Concepts

Rate Law for Zero-Order Reactions

In a zero-order reaction, something interesting occurs: the rate of the reaction does not depend on the concentration of the reactant. This means that whether you have a lot or a little of the reactant, the rate stays the same. This is described by the rate law equation. For zero-order reactions, the rate law is expressed as:
\[\text{Rate} = k\]
where \( k \) is the rate constant, a fixed value at a given temperature. Since the rate does not change with concentration, it simplifies the study of kinetics considerably. Another way to express the rate is by noting the change in concentration over time:
\[k = -\frac{\Delta [A]}{\Delta t}\]
This allows us to investigate how much the concentration of a reactant changes over a period of time. As we move forward, this understanding will help us see how the reaction progresses.

Understanding Half-Life in Zero-Order Reactions

The half-life, \( t_{1/2} \), of a reaction is the time it takes for the concentration of the reactant to reduce by half. For a zero-order reaction, the half-life equation is:
\[t_{1/2} = \frac{[A]_0}{2k}\]
This equation shows that the initial concentration \( [A]_0 \) directly affects half-life. Thus, as the concentration decreases during the reaction, the half-life also decreases. Unlike other reactions where half-life may remain constant, in zero-order reactions, it becomes shorter as the reaction proceeds. This happens because there is less reactant available over time, affecting how quickly half the amount is used up.

Reaction Progress in Zero-Order Reactions

The progress of a zero-order reaction can be mapped out using the change in concentration over time. The integrated rate law for this is:
\[[A] = [A]_0 - kt\]
Here, \( [A]_0 \) is the initial concentration, and \( [A] \) is the concentration at time \( t \). This linear equation signifies a steady decrease in concentration as the reaction progresses. Since the rate is constant, every unit of time sees the same amount of reactant being consumed. This straight-line behavior is characteristic of zero-order reactions, making them unique and simpler to predict across a given time frame. Understanding this pattern helps in modeling how substances react and transform efficiently.