Question

Consider the gas-phase isomerization of cyclopropane: Are the following data of the observed rate constant as a function of pressure consistent with the Lindemann mechanism? $$\begin{array}{cccc} \boldsymbol{P}(\text { Torr }) & \boldsymbol{k}\left(\mathbf{1 0}^{4} \mathbf{s}^{-1}\right) & \boldsymbol{P}(\text { Torr }) & \boldsymbol{k}\left(\mathbf{1 0}^{4} \mathbf{s}^{-1}\right) \\ \hline 84.1 & 2.98 & 1.36 & 1.30 \\ 34.0 & 2.82 & 0.569 & 0.857 \\ 11.0 & 2.23 & 0.170 & 0.486 \\ 6.07 & 2.00 & 0.120 & 0.392 \\ 2.89 & 1.54 & 0.067 & 0.303 \end{array}$$

Short answer

To determine if the given data is consistent with the Lindemann mechanism, first normalize the rate constants and pressures by dividing each value by the maximum value in its respective column. Next, plot the normalized rate constant (k) versus the normalized pressure (P) and examine the relationship between the two variables. If the data fits the linear relationship form \(k_{normalized} = (\frac{k_1 k_2 [A]}{k_1 [M] + k_2})\), then the Lindemann mechanism is consistent with the given data for the gas-phase isomerization of cyclopropane.

Step-by-step solution

Normalize the rate constants and pressures

To analyze the pressure dependence of the rate constants, it is helpful to normalize the rate constants (k) and pressures (P) in the tabular data to their maximum values by dividing each value by the maximum value in its respective column. This way, the relationship between the rate constant and pressure becomes easier to examine.

Determine the relationship

Now, plot the normalized rate constant (k) versus the normalized pressure (P) and look for any trend or correlation between the two variables. The graph should show either a linear, exponential, or inverse relationship between the rate constant and pressure. If the relationship fits the linear relationship form \(k_{normalized} = (\frac{k_1 k_2 [A]}{k_1 [M] + k_2})\), then the data are consistent with the Lindemann mechanism.

Confirm the mechanism

Having obtained the relationship in Step 2, the final step is to determine whether the observed relationship is consistent with the Lindemann mechanism. If the data fits the linear relationship form \(k_{normalized} = (\frac{k_1 k_2 [A]}{k_1 [M] + k_2})\), then we can conclude that the Lindemann mechanism is consistent with the given data for the gas-phase isomerization of cyclopropane. In conclusion, by normalizing the data, plotting the normalized rate constants versus the normalized pressures, and checking whether it follows the equation \(k_{obs} = \frac{k_1 k_2 [A]}{k_1 [M] + k_2}\), we can determine whether the given data is consistent with the Lindemann mechanism for the gas-phase isomerization of cyclopropane.

Key concepts

Gas-phase Isomerization

Gas-phase isomerization involves the transformation of one molecular structure into another in the gas phase. This process occurs without a change in molecular formula but involves the rearrangement of atoms within a molecule. Cyclopropane, a molecule with three carbon atoms forming a ring, can undergo isomerization. During this transformation, it changes to propene, a more stable form. Such reactions are important in both chemical synthesis and industry.
Isomerization reactions may seem simple, but they involve various kinetic factors that influence the rate and path of the reaction. Understanding these factors helps in controlling the reactions for desired outcomes. In cyclopropane's case, its ring strain makes it a good candidate for gas-phase isomerization, often analyzed under different conditions like pressure, to understand reaction mechanisms such as the Lindemann mechanism.

Rate Constant Pressure Dependence

The rate constant pressure dependence is a key feature of certain reaction mechanisms, including the Lindemann mechanism. As pressure influences molecular collisions, it can significantly affect reaction rates in gas-phase reactions, like the isomerization of cyclopropane. When the pressure changes, the frequency of molecular encounters alters, impacting how likely they are to undergo transformation.
In order to analyze this effect comprehensively, data are typically presented in a tabular format showing rates at different pressures. Normalizing these values helps in identifying patterns and relationships, like in the provided exercise. This approach makes it easier to spot trends and determine if the Lindemann mechanism is followed. Such understanding is crucial in both theoretical chemistry and practical applications where controlling reaction rates is necessary.

Reaction Kinetics

Reaction kinetics is the study of the speed at which chemical reactions occur and the factors that affect these speeds. It encompasses the analysis of rate laws, reaction orders, and the effects of variables like concentration and pressure. For gas-phase reactions, mechanisms like the Lindemann mechanism provide frameworks for understanding these dependencies.
In the case of cyclopropane isomerization, analyzing the rate constant at various pressures helps elucidate the kinetics involved. By plotting normalized rate constants against normalized pressures, researchers can observe the reaction's adherence to theoretical models.

• This practice allows chemists to predict reaction behavior under different conditions.
• It supports advancements in reaction control and optimization.

Understanding kinetics is vital for fields ranging from industrial chemistry to environmental science.

Cyclopropane Isomerization

Cyclopropane isomerization is a classic example of a gas-phase isomerization reaction where a strained ring compound converts to a more stable, open-chain form. This reaction is significant because it illustrates how energy stored in molecular structures can be manipulated.
The isomerization of cyclopropane to propene is often used to study reaction mechanisms like the Lindemann mechanism due to its simplicity and well-documented behavior. Scientists use such examples to develop and validate kinetic models.

• By studying how cyclopropane rearranges, insights into the energy landscape of molecules can be gained.
• It helps in refining techniques for energy management in chemical processes.

Therefore, cyclopropane isomerization serves as a fundamental study object in organic chemistry and reaction dynamics.