Question

In the discussion of the Lindemann mechanism, it was assumed that the rate of activation by collision with another reactant molecule \(A\) was the same as collision with a nonreactant molecule M such as a buffer gas. What if the rates of activation for these two processes are different? In this case, the mechanism becomes \\[ \begin{array}{l} \mathrm{A}+\mathrm{M} \quad \underbrace{k_{1}}{=} \mathrm{A}^{*}+\mathrm{M} \\\ \mathrm{A}+\mathrm{A} \stackrel{k_{-1}}{=} \mathrm{k}_{-2} \\ \mathrm{A}^{*} \stackrel{\mathrm{k}_{3}}{\longrightarrow} \mathrm{P} \end{array} \\] a. Demonstrate that the rate law expression for this mechanism is \\[ R=\frac{k_{3}\left(k_{1}[\mathrm{A}][\mathrm{M}]+k_{2}[\mathrm{A}]^{2}\right)}{k_{-1}[\mathrm{M}]+k_{-2}[\mathrm{A}]+k_{-3}} \\] b. Does this rate law reduce to the expected form when \([\mathrm{M}]=0 ?\)

Short answer

In this modified Lindemann mechanism, the rate law expression is given by \(R = k_3 \left(\frac{k_1[A][M]}{k_2[A] + k_3}\right)\). When the concentration of the nonreactant molecule M is 0, the rate law reduces to R = 0, indicating that the reaction won't proceed without M.

Step-by-step solution

Write the Rate Equations for Each Step

Write the rate equation for each step of the mechanism. They are given by 1. \(A + M \xrightarrow{k_1} A^* + M\): Rate = \(r_1 = k_1[A][M]\) 2. \(A + A^* \xrightarrow{k_2} 2A\): Rate = \(r_2 = k_2[A][A^*]\) 3. \(A^* \xrightarrow{k_3} P\): Rate = \(r_3 = k_3[A^*]\)

Use the Steady State Approximation for A*

Since \(A^*\) is an intermediate, we can use the steady-state approximation which implies that the net rate of its formation and consumption is zero. That is: \(0 =\frac{d[A^*]}{dt} = r_1 - r_2 - r_3\)

Solve for the A* Concentration

Rearrange the steady-state equation to solve for the concentration of \(A^*\), we have: \([A^*] = \frac{k_1[A][M]}{k_2[A] + k_3}\)

Find the Rate Law for Product Formation

Now, we want to find the rate of product formation, R, which is given by \(r_3 = k_3[A^*]\), so plug in the expression obtained in step 3 for \([A^*]\), we get \(R = k_3 \left(\frac{k_1[A][M]}{k_2[A] + k_3}\right)\)

Check if the Rate Law Reduces to the Expected Form when [M]=0

When \([M] = 0\), the rate law equation becomes, \(R = k_3 \left(\frac{k_1[A][0]}{k_2[A] + k_3}\right) = 0\) This means that when concentration of M is zero, the rate is zero. This makes sense considering that the first step in the mechanism involves a collision between the reactant A and the nonreactant molecule M. Therefore, if there is no M available, the process won't proceed, and the reaction rate will be zero.

Key concepts

Steady-State Approximation

The steady-state approximation is an important method used in chemical kinetics to simplify complex reaction mechanisms involving intermediates. It is based on the assumption that the concentration of an intermediate species in a multi-step reaction remains constant over the time course of the reaction. This occurs when the rate of formation of the intermediate equals the rate of its consumption.
Therefore, the derivative of the concentration of the intermediate with respect to time is set to zero. For example, in the Lindemann mechanism, the intermediate \(A^*\) is assumed to be in a steady state. This approach allows chemists to solve for the concentration of intermediates and thereafter obtain an expression for the overall reaction rate law.
When applying this approximation, it simplifies the mathematical analysis of the mechanism by reducing the number of differential equations needed to describe the system. This makes it possible to focus on the observable reactants and products, offering a more practical perspective for experimental and theoretical studies.

Reaction Rate Law

The reaction rate law expresses the rate of a chemical reaction in terms of the concentration of reactants. It's an equation that allows chemists to calculate the speed at which products are formed from reactants under certain conditions.
In the context of the Lindemann mechanism, the rate law is derived using the steady-state concentration of the intermediate \(A^*\). The final rate law relates the observable rate of product formation, denoted \(R\), to the concentrations of the reactants and the rate constants. The rate law captures the essence of the reaction's dependency on the concentration of species involved. Each term in the rate law has a corresponding rate constant that reflects the speed of the respective step of the mechanism.
Understanding the rate law for a given reaction is crucial because it provides insight into the factors that influence the speed of the reaction, allows for the optimization of reaction conditions, and offers a deeper understanding of the reaction mechanism.

Chemical Kinetics

Chemical kinetics is the branch of chemistry that deals with the rates of chemical reactions and the mechanisms by which they proceed. It analyzes how different experimental conditions, such as temperature, pressure, and concentration of reactants, affect the speed of chemical reactions.
Kinetics is also interested in the reaction pathways and the formation and consumption of intermediates, which are essential for understanding the steps involved in the reaction. These insights into the reaction mechanisms can help predict the kinetics of similar reactions and design chemical processes that are more effective and economically viable. In chemical education, kinetics is a foundational concept, providing a quantitative understanding of how reactions occur and with what velocity they transform reactants into products.

Intermediates in Chemical Reactions

Intermediates are species that are formed during a chemical reaction but are neither reactants nor final products. They are typically short-lived and cannot usually be isolated. In a reaction mechanism, intermediates play a crucial role as they appear in the elementary steps that compose the overall reaction. These steps can be unimolecular, bimolecular, or even termolecular reaction events, involving the transient intermediate molecules.
In the modified Lindemann mechanism given in the exercise, the species \(A^*\) is an intermediate that forms during the reaction between \(A\) and a molecule \(M\), which may be a buffer gas. Understanding the behavior of intermediates like \(A^*\) is essential in order to accurately describe the overall kinetics and mechanism of the reaction. Chemical kinetics considers the lifespan, concentration, and reactivity of intermediates to unravel the complexity of chemical reactions and predict product formation rates.