91Ó°ÊÓ

The rate constant, often denoted as \( k \), is a crucial element in the study of reaction rates and chemical kinetics. It is a measure of how quickly a reaction proceeds. For a first-order reaction, like the conversion of cyclopropane to propene, the rate constant has the units of \( \text{s}^{-1} \). This specificity is key because it indicates that the rate of the reaction is dependent on time. A larger rate constant means a faster reaction. In our exercise, the rate constant is given as \( 5.5 \times 10^{-4} \text{ s}^{-1} \), helping us to quantify the speed of this particular reaction at \( 500^{\circ} \text{C} \). Understanding the rate constant allows us to predict how much reactant will be converted into product over a given time period.

Half-life, denoted as \( t_{1/2} \), is the time required for the concentration of a reactant to reduce to half its initial value. For first-order reactions, the half-life is independent of the initial concentration. Instead, it is determined using the formula:

• \[ t_{1/2} = \frac{0.693}{k} \]

In our case, substituting \( k = 5.5 \times 10^{-4} \text{ s}^{-1} \) into the equation, we find that the half-life is approximately 1260 seconds, or about 21 minutes. This calculation is vital for determining how quickly a reactant is consumed in a chemical reaction, which can be especially useful in industrial applications where reaction timing is critical.

The integrated rate law for first-order reactions provides a relationship between the concentrations of a reactant at different times. The formula is:

• \[ \ln \left( \frac{[A]_t}{[A]_0} \right) = -kt \]

where \([A]_t\) is the concentration at time \( t \), and \([A]_0\) is the initial concentration. For the cyclopropane example, this equation allowed us to calculate the remaining concentration after a specified time. By inputting the initial concentration and the rate constant into the formula, and solving for \([A]_t\), we determined the concentration of cyclopropane remaining after 2 hours. This tool is incredibly useful as it helps in making predictions about concentration changes over time during a reaction.

Chemical kinetics is the study of the speed at which chemical reactions occur and the factors affecting these rates. It encompasses the analysis of rate laws, rate constants, and the mechanisms of reactions. First-order reactions like the conversion of cyclopropane to propene are straightforward and follow a simple relationship between the concentration of reactants and time. By understanding kinetics, chemists can influence reaction rates, which is vital in fields such as pharmaceuticals, engineering, and environmental science. The concepts and calculations demonstrated in our exercise highlight how kinetic principles are used to manage and optimize chemical processes.