Cyclobutane decomposes to ethylene according to the equation $$ \mathrm{C}_{4} \mathrm{H}_{8}(g) \longrightarrow 2 \mathrm{C}_{2} \mathrm{H}_{4}(g) $$ Determine the order of the reaction and the rate constant based on the following pressures, which were recorded when the reaction was carried out at \(430^{\circ} \mathrm{C}\) in a constant-volume vessel. $$ \begin{array}{rc} \hline \text { Time (s) } & P_{\mathrm{C}_{4} \mathrm{H}_{8}}(\mathrm{mmHg}) \\\ \hline 0 & 400 \\ 2,000 & 316 \\ 4,000 & 248 \\ 6,000 & 196 \\ 8,000 & 155 \\ 10,000 & 122 \\ \hline \end{array} $$

Chemical kinetics delves into the rate at which chemical reactions occur and the factors affecting these rates. Kinetics isn't just about the speed of a reaction, but also the detailed path taken during the transition from reactants to products—known as the reaction mechanism. Understanding kinetics is vital in various fields, including developing new chemicals, pharmaceutics, and environmental science.

In chemical kinetics, we investigate aspects like the concentrations of reactants and products over time. This allows us to establish the reaction order, which dictates how the reaction rate is affected by the concentrations of reactants. Different reactions have distinct orders, and identifying them is key to understanding how the reaction will proceed under varying conditions and how to control or optimize it.

A first order reaction is a chemical reaction where the rate is directly proportional to the concentration of a single reactant. This implies that as the concentration of the reactant is halved, so is the rate of the reaction. A common way to identify a first order reaction is by analyzing concentration or pressure data over time and looking for a linear relationship when plotting the logarithm of concentration or pressure against time.

In mathematical terms, the rate law for a first order reaction is given by the expression: \(\text{rate} = k[\text{Substance}]\), where \(k\) is the rate constant and \( [\text{Substance}]\) is the concentration of the reactant. The integration of this rate law yields an equation linking concentration or pressure to time, which can be used for extracting the rate constant from experimental data.

The rate constant, often denoted by \(k\), is a crucial quantity in chemical kinetics that provides the relationship between the concentration of reactants and the rate of the reaction. The value of \(k\) is determined by the specifics of the reaction under consideration, including the reaction's temperature, and it characterizes the speed at which a reaction progresses.

For a first order reaction, you can calculate the rate constant from experimental data by using the formula \(2.303 \log(\frac{P_{\text{initial}}}{P_{\text{final}}}) = kt\), as seen in the given exercise. Finding the rate constant allows chemists to predict the reaction speed under different conditions and to control the synthesis of products accordingly.

A pressure vs time graph is a useful tool in understanding the kinetics of reactions involving gases, as pressure can be related to concentration via the Ideal Gas Law. By plotting the pressure of a gaseous reactant over time, we can infer information about the reaction's order and calculate the rate constant.

In a first order reaction, plotting the natural logarithm (logarithm to the base \(e\)) of pressure or the common logarithm (logarithm to the base 10) multiplied by 2.303 against time yields a straight line, from which the rate constant can be determined. The slope of this line corresponds to the rate constant for the reaction. It's important to note that only for zero order reactions will a plot of pressure itself versus time be linear. For first and second order reactions, transformations of the pressure data, like those described above, are necessary to obtain a straight-line graph, indicative of a simple relation between variables.