Question

The reaction \(2 \mathrm{NO}(g)+\mathrm{Cl}_{2}(g) \longrightarrow 2 \mathrm{NOCl}(g)\) was performed and the following data obtained under conditions of constant \(\left[\mathrm{Cl}_{2}\right]\) : (a) Is the following mechanism consistent with the data? $$ \begin{aligned} \mathrm{NO}(g)+\mathrm{Cl}_{2}(g) & \longrightarrow \mathrm{NOCl}_{2}(g) \text { (fast) } \\ \mathrm{NOCl}_{2}(g)+\mathrm{NO}(g) & \longrightarrow 2 \mathrm{NOCl}(g) \text { (slow) } \end{aligned} $$ (b) Does the linear plot guarantee that the overall rate law is second order?

Short answer

Yes, the proposed mechanism is consistent with the given data, and the linear plot guarantees that the overall rate law is second order with respect to \(\mathrm{NO}\), as we derived the rate equation \(Rate = k'[\mathrm{NO}]^2[\mathrm{Cl}_{2}]\), where \(k' = k \cdot \frac{k_1}{k_{-1}}\).

Step-by-step solution

Analyze the Reaction Mechanism and Derive a Rate Equation

We are given the following reaction mechanism: \( \text{Step 1 (fast): }\mathrm{NO}(g) + \mathrm{Cl}_{2}(g) \Longrightarrow \mathrm{NOCl}_{2}(g) \\ \text{Step 2 (slow): }\mathrm{NOCl}_{2}(g) + \mathrm{NO}(g) \longrightarrow 2 \mathrm{NOCl}(g) \) Since step 2 is the slow step in this mechanism, it is the rate-determining step. Therefore, the rate equation can be written as follows: \(Rate = k[\mathrm{NOCl}_{2}][\mathrm{NO}]\) However, in order to compare it to the given data, we need to express the concentration of \(\mathrm{NOCl}_{2}\) in terms of the concentration of the reactants, \(\mathrm{NO}\) and \(\mathrm{Cl}_{2}\). From the fast step, we create a pseudo-equilibrium, which means that the rate constant for the forward reaction is equal to the rate constant for the reverse reaction: \(k_1[\mathrm{NO}][\mathrm{Cl}_{2}] = k_{-1}[\mathrm{NOCl}_{2}]\) Now, we can solve for [\(\mathrm{NOCl}_{2}\)]: \([\mathrm{NO_{Cl_2}}] = \frac{k_1}{k_{-1}}[\mathrm{NO}][\mathrm{Cl}_2]\)

Compare the Derived Rate Equation with the Given Data

Now, we need to substitute the expression of \([\mathrm{NO}_{\mathrm{Cl}2}]\) into the rate equation: \(Rate = k \cdot \frac{k_1}{k_{-1}}[\mathrm{NO}][\mathrm{Cl}_2][\mathrm{NO}] = k'[\mathrm{NO}]^2[\mathrm{Cl}_{2}]\) where \(k' = k \cdot \frac{k_1}{k_{-1}}\). We can now compare this rate equation with the given data. According to the problem, the concentration of \(\mathrm{Cl}_{2}\) is constant. It means that the given data should be consistent with the rate equation as long as \([\mathrm{NO}]^2\) plays a significant role on the rate.

Determine If the Linear Plot Indicates the Overall Rate Law is Second Order

If we observe a linear plot with the overall rate on the y-axis and \([\mathrm{NO}]^2\) on the x-axis, then the overall rate law is determined by \([\mathrm{NO}]^2\). Since the rate equation we derived also depends on \([\mathrm{NO}]^2\), the linear plot guarantees that the overall rate law is second order with respect to \(\mathrm{NO}\), and the proposed mechanism is consistent with the data. In conclusion, based on the step-by-step analysis, the proposed mechanism is consistent with the given data, and the linear plot guarantees that the overall rate law is second order with respect to \(\mathrm{NO}\).

Key concepts

Rate-Determining Step

The concept of a rate-determining step is crucial in the study of reaction mechanisms in chemical kinetics. Imagine the reaction sequence as a chain of events where each step is like a hurdle in a race. The rate-determining step is analogous to the highest hurdle that takes the most time to cross, hence dictating the pace for all runners—or in our case, the entire reaction.

This step is the slowest among all the steps in a reaction mechanism and thus controls the overall rate of the reaction. To make it more relatable, think of it as the bottleneck in a funnel where the rate of flow through the funnel is controlled by the narrowest part, not by how full the funnel is. In the given exercise, the second step is slower and is therefore the rate-determining step. The rate law derived from this step (Rate = k[NOCl2][NO]) reflects the concentration of the reactants influencing this slowest phase, much like the width of the bottleneck controls the flow rate.

Understandably, by targeting the rate-determining step in chemical processes, chemists can alter the speed of the overall reaction, which is immensely valuable in industrial applications where time is money.

Chemical Kinetics

Diving into the river of chemical kinetics, we navigate the currents of reaction rates and the factors that affect them. At its heart, chemical kinetics is the study of the speed at which chemical reactions occur and the path or steps through which these reactions proceed. It's not just about the 'what' of the end products, but the 'how' and 'how fast' of the journey from reactants to products.

In our exercise, chemical kinetics helps us to establish a relationship between the reaction mechanism and the rate at which the products are formed. By monitoring how the concentration of NO affects the rate when the amount of Cl2 remains constant, students can gain insights into how the interactions between molecules dictate the frequency and energy with which they collide, leading to successful reactions. Chemical kinetics is the scientific compass guiding chemists to predict and control reactions, ensuring that each molecular dance leads to the desired outcome in the most efficient way possible.

Order of Reaction

The order of a reaction is a key piece of the puzzle in the vast field of chemical kinetics. It refers to the exponents of the concentration terms in the rate law equation—the mathematical formula that describes the rate of a chemical reaction. This is not something you guess; it is determined experimentally and sheds light on the intricate relationship between the concentration of reactants and the reaction rate.

In our textbook example, the rate law is found to depend on the square of the concentration of NO, thus indicating a second-order reaction with respect to NO. Don't think of these 'orders' as ranks in a hierarchy but more like ingredients in a recipe—the quantity of each ingredient (reactant) affects the final product (rate of reaction). Knowing the order of reaction is not just about predicting the rate; it's a guide to understanding how changes in concentration can pace up or slow down the chemical reaction, akin to how altering the heat can affect the cook time of your dish. With this knowledge, a chemist can mix reactants with precision to control the tempo of product formation, much like a chef masters the flame to perfect a gourmet meal.