Question

The activity of the antibiotic penicillin slowly decomposes when stored in a buffer at \(\mathrm{pH} 7.0,298 \mathrm{~K}\). The time dependence of the penicillin antibiotic activity is given in the table below. $$ \begin{array}{lc} \text { Time (weeks) } & \text { Penicillin Activity (arbitrary units) } \\ \hline 0 & 10,100 \\ 1.00 & 8,180 \\ 2.00 & 6,900 \\ 3.00 & 5,380 \\ 5.00 & 3,870 \\ 8.00 & 2,000 \\ 10.00 & 1.330 \\ 12.00 & 898 \\ 15.00 & 403 \\ 20.00 & 167 \\ \hline \end{array} $$What is the rate law for this reaction, that is, what is the order of the reaction with respect to the penicillin concentration? Calculate the rate constant from the data if possible. (Data adapted from Ref. I.)

Short answer

The reaction is first-order with a rate constant of 0.2 weeks^{-1}.

Step-by-step solution

Understanding the Problem

We have been given data showing the decomposition of penicillin over time at a specific pH and temperature. Our goal is to determine the rate law, which involves identifying the order of the reaction and calculating the rate constant.

Review the Given Data

The data table provides penicillin activity at various weeks. The activity decreases as time increases, indicating a reaction that changes over time. Analyze this to determine the reaction order.

Determine Reaction Order via Plotting

To find the reaction order, we need to test different plots. For zero-order reactions, plot concentration vs. time. For first-order reactions, plot ln(concentration) vs. time. For second-order reactions, plot 1/concentration vs. time. We will plot each to see which yields a straight line.

Plot ln(Activity) vs. Time for First-Order Reaction

Using the data, calculate ln(activity) for each time point and plot it against time. Using Excel or graphing software, a linear plot suggests a first-order reaction. The slope of this plot gives the rate constant.

Analyze the Plots for Best Fit

After plotting, we find that the ln(activity) vs. time plot is linear. This indicates that the reaction is first-order. The slope of this line will provide the rate constant.

Calculate the Rate Constant

From the linear plot ln(activity) vs. time, determine the slope, which is the negative of the rate constant for a first-order reaction. If the slope is found to be -0.2 weeks^{-1}, then the rate constant \(k = 0.2\) weeks^{-1}.

Key concepts

Rate Law

The rate law defines how the speed of a chemical reaction depends on the concentration of its reactants. In the exercise provided, the goal was to find the rate law for the decomposition of penicillin. The rate law expression often takes the form:

\[\text{Rate} = k [A]^n\]where:

• \( k \) is the rate constant.
• \([A]\) is the concentration of the reactant.
• \( n \) is the reaction order.

The exercise aimed to deduce these variables through the analysis of provided data. By plotting various potential transformations of the data, one determines the relationship best describing how the concentration affects the reaction speed.

Reaction Order

The reaction order is an important aspect of the rate law and indicates how the concentration of a reactant affects the rate of reaction. In the given case of penicillin decomposition:

• A zero-order reaction shows that the reaction rate is independent of the concentration of the reactant.
• A first-order reaction shows that the reaction rate is directly proportional to the concentration of the reactant.
• A second-order reaction indicates an exponential dependency; for example, doubling the concentration quadruples the rate.

Through plotting ln(activity) vs. time, it was found that the reaction is first-order, meaning the change in activity (or concentration) is logarithmically dependent on time.

Rate Constant

The rate constant \( k \) is a crucial component of the rate law. It mathematically defines the rate of reaction independent of reactant concentrations. For first-order reactions, it can be determined from the slope of the linear plot of ln(concentration) vs. time:

\[ \text{Rate constant} \ (k) = -\text{slope of the plot}\]In the penicillin decomposition case, once the reaction was identified as first-order, the rate constant was calculated as follows. If, for example, the slope of the ln(activity) vs. time plot was found to be \(-0.2\) weeks\(^{-1}\), then \(k\) is 0.2 weeks\(^{-1}\). This constant is crucial for predicting how fast the penicillin will decompose over time.

First-Order Reaction

A first-order reaction is characterized by the rate of the reaction being proportional to the concentration of one reactant. This is clearly exemplified in the penicillin study, where plotting ln(activity) vs. time yielded a straight line. This linear relationship:

\[\text{ln}[A]_t = -kt + \text{ln}[A]_0\]

• \( [A]_t \) is the concentration at time \( t \).
• \( [A]_0 \) is the initial concentration.

This equation illustrates that as time (\( t \)) progresses, the logarithm of the concentration decreases linearly, defining a typical first-order kinetic behavior. Such knowledge helps in predicting the decay and managing the storage of sensitive compounds like penicillin.

Antibiotic Decomposition

Antibiotic decomposition is a key concern in medicine, especially with sensitive drugs like penicillin. Decomposition refers to the reduction in activity or potency of the antibiotic over time, often sped up by factors such as temperature or incorrect storage conditions. The exercise focused on determining the rate at which penicillin decomposes at a specific pH level and temperature.

• Understanding the decomposition kinetics helps in predicting the shelf life of the drug.
• Ensures proper dosing of effective antibiotics in clinical settings.
• Gives insights into manufacturing and distribution conditions to maintain drug efficacy.

This knowledge is especially critical for life-saving drugs that must retain their potency to prevent and treat infections efficiently.