Question

In the unimolecular isomerization of cyclobutane to butylene, the following values for \(k_{u n i}\) as a function of pressure were measured at \(350 \mathrm{K}\) \\[ \begin{array}{lcccc} \boldsymbol{P}_{\mathbf{0}}(\mathbf{T o r r}) & 110 & 210 & 390 & 760 \\ \boldsymbol{k}_{\boldsymbol{u n i}}\left(\mathbf{s}^{-1}\right) & 9.58 & 10.3 & 10.8 & 11.1 \end{array} \\] Assuming that the Lindemann mechanism accurately describes this reaction, determine \(k_{1}\) and the ratio \(k_{-1} / k_{2}\)

Short answer

Based on the provided data and the Lindemann mechanism, we obtain that the rate constant for the formation of the activated complex, \(k_1\), is approximately \(11\ \text{s}^{-1}\), and the ratio between the rate constants for the dissociation of the activated complex and its isomerization to the products, \(\frac{k_{-1}}{k_2}\), is approximately \(7.25 \times 10^{-3}\ \text{mol/L}\).

Step-by-step solution

Understanding the Lindemann Mechanism

The Lindemann mechanism is a model used to describe the kinetics of unimolecular reactions. It postulates that molecules must first undergo a collision to form an activated complex before they can go on to react, which can then proceed in two different pathways. In our case, for the isomerization of cyclobutane to butylene, the steps in the Lindemann mechanism can be represented as: 1. \(A + M \overset{k_1} \underset{k_{-1}}\rightleftharpoons A^* + M\) (formation and dissociation of activated complex) 2. \(A^* \overset{k_{2}}\rightarrow products\) (reaction of activated complex to form products) Here, \(A\) represents cyclobutane, \(M\) is a generic molecule (often called a "bath gas"), and \(A^*\) is the activated complex. \(k_1, k_{-1}\), and \(k_2\) are the rate constants for each step. Also, notice that step 1 is reversible, with a rate constant for both the forward and the backward reaction.

Calculating the Uncatalyzed First-Order Rate Constant

For a unimolecular reaction, the first-order rate constant is given by: \(k_{uni} = \frac{k_1k_2[A^*]}{k_{-1}[M] + k_2}\) As given, \(k_{uni}\) is measured as a function of pressure. The pressure dependency can be expressed as: \([M] = \frac{P_0}{RT}\) where \(P_0\) is the partial pressure of the bath gas in Torr, \(R\) is the ideal gas constant in suitable units (here, \(62.36\ \frac{L\ \text{Torr}}{\text{mol}\ K}\)), and \(T\) is the temperature in Kelvin. We need to plot a graph to determine the value of \(k_1\) and \(k_{-1}/k_2\). We can do this by realizing that for high pressure (large \([M]\)), the reaction behaves as a first-order process, and the expression for \(k_{uni}\) simplifies to: \(k_{uni}^{\text{high pressure}} = k_1\) Similarly, at low pressure, the expression for \(k_{uni}\) approaches: \(k_{uni}^{\text{low pressure}} = \frac{k_1k_2[P_0]}{k_{-1}RT}\)

Plotting the Graph

We now need to plot a graph of \(k_{uni}\) as a function of \(P_0\), using the provided data. The data can be plotted as: $$ \begin{array}{|c|c|} \hline P_0\ (\text{Torr}) & k_{uni}\ (\text{s}^{-1}) \\ \hline 110 & 9.58 \\ \hline 210 & 10.3 \\ \hline 390 & 10.8 \\ \hline 760 & 11.1 \\ \hline \end{array} $$ Plotting the data, we observe a linear relationship between \(k_{uni}\) and \(P_0\), with the graph approaching an asymptote at high pressure. The point where this asymptote is reached can be extrapolated and used to deduce the value of \(k_1\), which, as seen earlier, is equal to the \(k_{uni}\) at high pressure.

Determining \(k_1\) and \(k_{-1}/k_2\)

From the plot, we can extrapolate the value of \(k_1\), which corresponds to the asymptote: \(k_1 \approx 11 \text{ s}^{-1}\) Now we can determine the ratio \(k_{-1}/k_2\). We can use the low-pressure expression for \(k_{uni}\) (with an arbitrary choice of one data point, e.g., \(P_0 = 110\ \text{Torr}\) and \(k_{uni} = 9.58\ \text{s}^{-1}\)): \(k_{uni}^{\text{low pressure}} \approx \frac{k_1k_2[P_0]}{k_{-1}RT}\) An approximation for the ratio is: \(\frac{k_{-1}}{k_2} \approx \frac{k_1[P_0]}{k_{uni}^{\text{low pressure}}RT} = \frac{(11 \text{ s}^{-1})(110 \text{ Torr})}{(9.58 \text{s}^{-1})(62.36 \text{ L Torr/mol K})(350 \text{ K})} \approx 7.25 \times 10^{-3} \text{ mol/L}\) So, the values we obtained are: \(k_1 \approx 11 \text{ s}^{-1}\) \(\frac{k_{-1}}{k_2} \approx 7.25 \times 10^{-3} \text{ mol/L}\)

Key concepts

Unimolecular Reactions

Unimolecular reactions are a fascinating category of chemical reactions involving the transformation of a single molecule. These reactions are distinct because they don't require a direct collision between two reactant molecules to proceed. Instead, the reactant molecule undergoes a kinetic process that leads to the formation of products, often following the mechanisms described by Lindemann.
In the Lindemann mechanism, an unimolecular reaction progresses in two major steps. First, the reactant molecule collides with another molecule to form an intermediate complex. This step is crucial and involves the energy transfer needed to activate the molecule. The activated complex then has a chance to move forward and transform into products.
Unimolecular reactions are exemplified in the isomerization of cyclobutane to butylene, which follows the Lindemann mechanism. Understanding these processes is important in chemical kinetics, especially when analyzing reactions that appear to be simple from the balanced chemical equation but hide intricate mechanistic details.

Rate Constant

The rate constant is a fundamental aspect of chemical kinetics, providing insights into the speed of a reaction under certain conditions without altering the reaction's stoichiometry.
In the Lindemann mechanism for unimolecular reactions, we encounter several rate constants: \[k_1, k_{-1},\] and \[k_2.\] Each represents a step within the reaction mechanism.

• \(k_1\) is the rate constant for the forward reaction where the reactant forms an activated complex.
• \(k_{-1}\) is the constant for the reverse reaction, where the activated complex reverts back to the reactant.
• \(k_2\) is the rate at which the activated complex progresses to form the products.

The observable rate constant \(k_{uni}\) relates to these constants when examining how they translate to a first-order reaction at varying pressures. It's important for students to realize how intrinsic parameters like \(k_1, k_{-1},\) and \(k_2\) interplay with conditions like pressure to affect observed reaction kinetics.

Pressure Dependence

The rate at which unimolecular reactions occur can depend significantly on pressure, primarily because the concentration of a "bath gas" influences the formation of the activated complex.
In the Lindemann mechanism, pressure affects the stabilization of the activated complex. At high pressure, there are more collisions, increasing the chance of stabilizing the complex. Thus, the reaction mimics a first-order one, with \(k_{uni}\) reaching a maximum value related to \(k_1\).

• High pressures indicate a linear relationship between \(k_{uni}\) and the concentration of the bath gas.
• At low pressures, the rate of formation of the activated complex is still significant, but the activated complex doesn't efficiently proceed to product formation.

Students should observe that the apparent rate constant \(k_{uni}\) varies with pressure, a hallmark of unimolecular reactions governed by the Lindemann mechanism.

Activated Complex

The activated complex is a critical concept in understanding the kinetics of unimolecular reactions, especially within the framework of the Lindemann mechanism.
The notion of an activated complex reflects a transient species that forms during the reaction's conversion from reactants to products. It's the high-energy state achieved when sufficient energy has been accumulated in the molecule due to collisions.
In the isomerization of cyclobutane to butylene, the activated complex symbolizes the energy threshold the molecule crosses before rearranging into the product. Crucially, it is the formation and stability of this activated complex that determine the overall rate of the reaction. While the activated complex only exists fleetingly, its formation is an essential step in understanding how energy is transferred into the system and how reactions are able to overcome the activation energy barriers effectively.