Question

When chemists are performing kinetics experiments, the general rule of thumb is to allow the reaction to proceed for 4 half-lives. (a) Explain how you would be able to tell that the reaction has proceeded for 4 half-lives. (b) Let us suppose a reaction \(\mathrm{A} \rightarrow \mathrm{B}\) takes 6 days to proceed for 4 half-lives and is first order in A. However, when your lab partner performs this reaction for the first time, he does not realize how long it takes, and he stops taking kinetic data, monitoring the loss of A, after only 2 hours. Your lab partner concludes the reaction is zero order in A based on the data. Sketch a graph of [A] versus time to convince your lab partner the two of you need to be in the lab for a few days to obtain the proper rate law for the reaction.

Short answer

In short, to determine if a reaction has proceeded for 4 half-lives, we can monitor the concentration of the reactant and check when it reaches 1/16th of its initial concentration. The graph of concentration A versus time for a first-order reaction will show that stopping the experiment after only 2 hours results in insufficient data for determining the rate law, and thus, the experiment should be conducted for a few days to get proper results.

Step-by-step solution

Part (a): Identifying 4 half-lives

A half-life is the amount of time required for the concentration of a reactant to reduce to half of its initial value. To determine if a reaction has proceeded for 4 half-lives, one can monitor the concentration of the reactant. After each half-life, the concentration of the reactant will be halved. So after 4 half-lives, the concentration of the reactant should be reduced to: \(\frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}\times \frac{1}{2} = \frac{1}{16}\) Thus, the concentration of the reactant A would be 1/16th of its initial concentration after 4 half-lives have passed. By monitoring the concentration of reactant A and checking when it reaches this 1/16th value, it is possible to tell that the reaction has proceeded for 4 half-lives.

Part (b): Sketching the graph

To convince the lab partner that more time is needed to gather enough data, we will sketch a graph of concentration of A versus time, considering the first-order reaction kinetics. The equation for concentration A over time in a first-order reaction can be represented as: \[[A] = [A]_0 e^{-kt}\] Here, [A] is the concentration of reactant A, [A]_0 is the initial concentration, k is the rate constant, and t is the time. Given that the reaction takes 6 days to proceed for 4 half-lives, it means that one half-life takes 6/4 = 1.5 days. To compare it with the 2-hour experiment conducted by the lab partner, we must convert the time units. So, 1.5 days = 36 hours. Using the half-life formula for a first-order reaction, we have: \[t_{1/2} = \frac{0.693}{k}\] Now we solve for the rate constant k: \[k = \frac{0.693}{36\,\mathrm{hours}} = 0.01925\,\mathrm{h^{-1}}\] Now, we can plot the graph of [A] versus time using the above equation for different time points. It will show that after 2 hours (the time the lab partner stopped the experiment), the concentration of A has changed very little, meaning that stopping the experiment so early would not provide enough data to determine the rate law for the reaction properly. A complete analysis of the reaction can only be achieved if the experiment is allowed to run for a few days, as initially planned.

Key concepts

Half-life

Understanding the concept of half-life is essential in the study of reaction kinetics. In the context of chemical reactions, half-life is defined as the time it takes for the concentration of a reactant to decrease to half its initial value. For example, if you start with 100 grams of a substance, after one half-life, you would have 50 grams remaining, then 25 grams after two half-lives, and so on.

This concept is vital because it helps chemists understand how long a reaction takes to reach a certain point and can be used to determine the rate at which a reactant is consumed. It's important to note that the half-life of a reaction is constant for first-order reactions, meaning that it does not depend on the starting concentration of the reactant. Therefore, if the half-life of a reactant is one hour, it will take one hour for the concentration to halve, regardless of whether the starting amount was 1 gram or 1000 grams.

Reaction Order

The reaction order is a key term in the realm of kinetics that indicates the degree to which the concentration of a reactant affects the rate of the reaction. Essentially, it tells us how the reaction rate will change when the concentration of one or more reactants changes.

A reaction can be zero order, first order, or second order with respect to a particular reactant. A zero-order reaction means that the reaction rate is independent of the reactant's concentration, while a first-order reaction implies that the rate is directly proportional to the concentration. Second-order reactions, meanwhile, have rates that change with the square of the concentration of the reactant.

For our lab scenario, when the partner concluded the reaction to be zero order, it likely meant that there was little to no change in concentration observed during the incomplete 2-hour observation period, leading to the incorrect conclusion that the rate was independent of concentration. However, with first-order reactions, time is needed to observe the exponential decay in the concentration, which is best seen over longer time frames, such as multiple half-lives.

Rate Law

The rate law of a reaction reveals the direct mathematical relationship between the concentration of reactants and the reaction rate. It is typically expressed in the form of an equation:
\[\text{Rate} = k[A]^{m}[B]^{n}\]
where \([A]\) and \([B]\) represent the concentrations of the reactants, \(m\) and \(n\) are the reaction orders with respect to each reactant, and \(k\) is the rate constant, a proportionality factor unique to each reaction.

In determining the rate law, it's crucial to conduct the reaction over a sufficiently long period that can capture the full kinetics behavior of the reacting species. Failing to do so, as happened with the lab partner's 2-hour experiment, results in insufficient data for accurate determination of the rate law. The reaction time must be adequate to witness the exponential change in concentration dictated by first-order behavior if that is the true nature of the reaction.

First-order Reaction Kinetics

When it comes to first-order reaction kinetics, the rate of the reaction is directly proportional to the concentration of a single reactant. Mathematically, this is described as:
\[\text{ln} [A] = -kt + \text{ln} [A]_0\]
or
\[[A] = [A]_0 e^{-kt}\]
where \([A]\) is the concentration at time \(t\), \([A]_0\) is the initial concentration, \(k\) is the rate constant, and \(t\) is the time elapsed. This relationship demonstrates that, as time progresses, the concentration of the reactant decreases exponentially.

In the exercise provided, understanding this relationship would suggest that more time is needed beyond the two hours the lab partner used to monitor the loss of reactant A. By extending the time to a few days as per the original half-life data, the exponential decay will become evident, allowing for correct identification of the reaction as first-order. This understanding also emphasizes the importance of patience and precision in collecting kinetic data over appropriate time scales to avoid incorrect interpretations of a reaction’s kinetics.