Phosgene \(\left(\mathrm{Cl}_{2} \mathrm{CO}\right)\), a poison gas used in World War I, is formed by the reaction of \(\mathrm{Cl}_{2}\) and \(\mathrm{CO}\). The proposed mechanism for the reaction is: \(\mathrm{Cl}_{2} \rightleftharpoons 2 \mathrm{Cl} \quad\) (fast, equilibrium) \(\mathrm{Cl}+\mathrm{CO} \rightleftharpoons \mathrm{ClCO} \quad\) (fast, equilibrium) \(\mathrm{ClCO}+\mathrm{Cl}_{2} \longrightarrow \mathrm{Cl}_{2} \mathrm{CO}+\mathrm{Cl} \quad(\) slow \()\) What rate law is consistent with this mechanism?

Understanding the rate law is crucial for determining how the concentration of reactants affects the speed of a chemical reaction. In the context of the exercise, the rate law for the formation of phosgene gas is derived from the rate-determining step, which is the slowest step in the reaction mechanism and thus dictates the overall reaction rate. The rate law describes the relationship between the rate of the reaction and the concentration of reactants.

The rate law derived from the exercise: \[\text{Rate} = k[\mathrm{ClCO}][\mathrm{Cl}_2]\] includes the concentration of the intermediate \(\mathrm{ClCO}\) and the reactant \(\mathrm{Cl}_2\), raised to the power of their stoichiometric coefficients. However, since \(\mathrm{ClCO}\) is an intermediate and not a reactant present initially, we use equilibrium constants to express its concentration in terms of the initial reactants \(\mathrm{Cl}_2\) and \(\mathrm{CO}\). This provides a practical rate law that can be used to predict the rate of the reaction under different conditions.

In many chemical reactions, there are species formed during the reaction which do not appear in the overall balanced equation, these are known as reaction intermediates. The exercise given elucidates the importance of reaction intermediates like \(\mathrm{ClCO}\).

Understanding the role of intermediates helps elucidate a reaction's pathway. Intermediates are typically formed in one step of a reaction mechanism and consumed in subsequent steps. They are crucial for writing correct rate laws, as they impact the relationship between reactant concentrations and the reaction rate. However, since they are transient species, their concentrations are often not directly measurable. Instead, their concentrations are expressed using equilibrium constants from fast, preceding steps in the reaction mechanism, as demonstrated in the solution to the exercise:

\[K_2 = \frac{[\mathrm{ClCO}]}{[\mathrm{Cl}][\mathrm{CO}]}\]
\[K_1 = \frac{[\mathrm{Cl}]^2}{[\mathrm{Cl}_2]}\]
These equilibrium constants allow for the elimination of the intermediate in the final rate law expression.

The concept of chemical equilibrium is used to understand the point at which the forward and reverse rates of a reversible reaction are equal, resulting in no net change in the concentration of reactants and products over time. It is a dynamic state where the concentrations of reactants and products remain constant.

In the exercise about phosgene formation, two of the steps are fast and are assumed to be at equilibrium. This assumption allows the use of the equilibrium constants \(K_1\) and \(K_2\) to describe the concentrations of the intermediates in terms of the concentrations of stable reactants and products:

\[K_1 = \frac{[\mathrm{Cl}]^2}{[\mathrm{Cl}_2]}\]
\[K_2 = \frac{[\mathrm{ClCO}]}{[\mathrm{Cl}][\mathrm{CO}]}\]
These constants facilitate the creation of a rate law that reflects the realities of the reaction's progress under equilibrium conditions. The final expression showcases the power of equilibrium in connecting the reaction steps and simplifying the representation of the reaction rate to a form that can be studied and manipulated in experiments and industry.