The following gas-phase reaction was studied at \(290^{\circ} \mathrm{C}\) by observing the change in pressure as a function of time in a constant-volume vessel: $$ \mathrm{ClCO}_{2} \mathrm{CCl}_{3}(g) \longrightarrow 2 \mathrm{COCl}_{2}(g) $$ Determine the order of the reaction and the rate constant based on the following data: $$ \begin{array}{rr} \hline \text { Time (s) } & \boldsymbol{P} \text { (mmHg) } \\ \hline 0 & 15.76 \\ 181 & 18.88 \\ 513 & 22.79 \\ 1164 & 27.08 \\ \hline \end{array} $$ where \(P\) is the total pressure.

The rate constant is a critical element in the study of chemical kinetics as it quantifies the speed of a chemical reaction. In a mathematical sense, the rate constant, denoted as \( k \), appears in the rate law equation that describes how the rate of a reaction is related to the concentration of reactants. For a reaction of order \( n \), the rate law can be written as \( rate = k[A]^n \), where \( [A] \) represents the concentration of reactant \( A \).

Essentially, the rate constant provides a measure of how rapidly a reaction proceeds when the reactants are at certain concentrations. It is specific to each chemical reaction and can be influenced by factors such as temperature, pressure, and the presence of a catalyst. Temperature, in particular, has a profound effect on the rate constant where typically, as the temperature increases, the rate constant also increases, leading to a faster reaction rate.

To determine the rate constant, experiments are carried out under controlled conditions, and data such as pressure or concentration changes over time are collected, much like the exercise provided. The calculation from the exercise involves plotting a graph of the logarithm of pressure changes versus time, and from the slope and intercept of the resulting line, the rate constant can be derived. This method demonstrates the direct connection between observational data and the numerical value of \( k \), making understanding this concept essential for interpreting experimental results.

The reaction order is a term used in chemical kinetics to indicate the power to which the concentration of a reactant is raised in the rate law equation. It reflects the dependency of the rate of reaction on the concentration of that reactant. The overall order of the reaction is the sum of the orders with respect to each reactant involved.

In the given exercise, determining the reaction order is critical for understanding the relationship between the change in pressure and the rate at which the reaction proceeds. A reaction can be zero-order, first-order, second-order, or a mixed-order depending on how the rate responds to concentration changes. For example, a zero-order reaction means the rate is constant and does not depend on the concentration of the reactants. On the other hand, a first-order reaction implies that the rate is directly proportional to the concentration of a single reactant.

As demonstrated in the exercise, graphing techniques are often employed to determine the reaction order by plotting data in specific ways, such as logarithmic plots, and observing the resulting trends. The slope of the log-log plot gives us the reaction order in this case. It is important for students to grasp that the exactness of this method hinges on the quality and range of the experimental data, and the correct identification of the reaction order is foundational in deducing the complete rate law of a reaction.

A gas-phase reaction is a chemical process that occurs between substances in the gaseous state. These reactions are particularly interesting because the behavior of gases, and hence the course and speed of gas-phase reactions, can be predicted and described by laws such as the Ideal Gas Law. In these types of reactions, pressure and volume can serve as key indicators of the reaction’s progress since they are related to the number of gaseous molecules present.

In the context of the provided exercise, observing the change in pressure in a constant-volume vessel allows us to track the progression of the gas-phase reaction. The nature of gases means that the increase in pressure corresponds to the production of gas molecules as the reaction proceeds. Because gas molecules move freely and can collide with one another, their concentrations significantly impact the rate of reaction, akin to reactions in other phases.

Understanding the kinetics of gas-phase reactions is vital for many practical applications, like in combustion engines or atmospheric chemistry where the gas-phase reactions take place under varying pressure and temperature conditions. Students should appreciate how these variables interplay with the inherent properties of gases and how such reactions are studied using a combination of theoretical models and empirical data, as seen in this problem where pressure change is correlated with reaction rate to deduce both the order of the reaction and the rate constant.