As described in Exercise \(14.43,\) the decomposition of sulfuryl chloride \(\left(\mathrm{SO}_{2} \mathrm{Cl}_{2}\right)\) is a first-order process. The rate constant for the decomposition at \(660 \mathrm{~K}\) is \(4.5 \times 10^{-2} \mathrm{~s}^{-1}\). (a) If we begin with an initial \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) pressure of 450 torr, what is the pressure of this substance after \(60 \mathrm{~s} ?\) (b) At what time will the pressure of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) decline to one-tenth its initial value?

In the realm of chemical kinetics, the rate constant is a crucial value that determines the speed at which a reaction progresses. In the context of a first-order reaction, this constant directly correlates with the reaction's half-life; it remains constant irrespective of the reactant concentrations.

Take for instance the decomposition of sulfuryl chloride, which adheres to first-order kinetics according to our textbook exercise. The rate constant here, denoted as \(k\), not only helps in calculating how fast the substance breaks down but also in predicting concentrations, or in our case, pressures at any given time.

When measuring a first-order reaction, the formula \(P(t) = P_0 \times e^{-kt}\) is essential. In this expression, \(P_0\) signifies the initial pressure, \(P(t)\) is the pressure at time \(t\), and \(e\) stands for the mathematical constant that is the base of natural logarithms. It's imperative to understand that the rate constant, \(k\), is unique for each reaction and is influenced by various factors such as temperature and the presence of catalysts, making it a pivotal factor in our reaction's investigation.

The decay of sulfuryl chloride (\(SO_2Cl_2\)) serves as a good illustration of first-order kinetics. This reaction involves the breakdown of sulfuryl chloride into sulfur dioxide (\(SO_2\)) and chlorine gas (\(Cl_2\)), without the need for its concentration to dictate the rate at which this happens.

Under the set experimental conditions, which include a specific temperature of 660 K, the decomposition rate is determined solely by the rate constant. This type of reaction is characterized by its independence from reactant concentration, making it simpler to analyze compared to other reaction orders.

In practical terms, if you started with a certain amount of sulfuryl chloride, you could predict how much of it would remain after a period of time, which is precisely what the textbook problem requires us to do. This predictability is what makes studying chemical kinetics so valuable for scientists and engineers planning experiments or designing chemical processes.

Changes in pressure over time can provide insightful information about the progress of a chemical reaction, especially when dealing with gases. In our exercise, we use the reduction of pressure as an indirect method of measuring the decomposition of sulfuryl chloride.

By applying the first-order kinetic equation, we can determine the pressure of \(SO_2Cl_2\) at any moment throughout the reaction. This approach is particularly beneficial as it allows us to anticipate how pressure decreases as the reaction advances and how this decrease is exponentially related to time due to the nature of first-order kinetics.

It's crucial for students to grasp that in the case of a first-order reaction, the pressure does not drop linearly but exponentially—a fact often corroborated by a negative exponential graph. This ties back to the constant proportionality which is the rate constant \(k\). The larger its value, the sharper the decline in pressure over time, illustrating the swift progression of the reaction.