91Ó°ÊÓ

In a chemical reaction, the rate law is a mathematical equation that describes the relationship between the rate of a reaction and the concentration of its reactants. For a first-order reaction, the rate law is significant because it reveals how the concentration of the reactant decreases over time. The typical rate law for a first-order reaction is given by:\[\ln \left(\frac{[A]_t}{[A]_0}\right) = -kt\]where:

• \([A]_t\) is the concentration of the reactant at time \(t\).
• \([A]_0\) is the initial concentration.
• \(k\) is the rate constant.

This equation helps us understand how quickly a reaction is progressing. In the given exercise, this equation was the foundation to determine how much of the compound remains after a certain time. Simply put, it is the mathematical sentence that gives insights into the change in concentration over time.

The rate constant \(k\) is a crucial part of the rate law. It is a proportionality factor that gives us an idea of the speed of the reaction. For first-order reactions, the units of \(k\) are typically \ \( \text{s}^{-1} \ \). Understanding \(k\) is key to predicting how the concentration changes over time.To find \(k\) in the exercise, use the rearranged rate law formula:\[k = -\frac{\ln \left(\frac{[A]_t}{[A]_0}\right)}{t}\]Inserting the given concentrations, \([A]_0 = 0.0700\, M\) and \([A]_{40} = 0.0569\, M\), for \(t = 40\) seconds, we calculated \(k\) as approximately 0.004965 \(\text{s}^{-1}\). This value of \(k\) tells us how quickly the concentration is decreasing per second. A larger \(k\) would indicate a faster reaction, whereas a smaller \(k\) would correspond to a slower one.

Concentration is the amount of a substance in a given volume, often measured in molarity (M). In chemical kinetics, monitoring how the concentration of a reactant changes over time is essential for understanding the reaction's progress. For a first-order reaction, the concentration decreases exponentially. Using the rate law and knowing the \(k\), you can calculate the concentration at any time \(t\). In the exercise, after determining the rate constant \(k\), we found the concentration after 91 seconds using the formula:\[[A]_{91} = [A]_0 \times e^{-kt}\]Given \([A]_0 = 0.0700\, M\) and \(k = 0.004965\, \text{s}^{-1}\), we calculated \([A]_{91} = 0.0439\, M\). This shows how concentration is predicted to diminish with time, illustrating exponential decay typical of first-order reactions. Understanding these changes helps in controlling and predicting the behavior of chemical reactions.