Problem 55 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

In \(80 \%\) ethanol at \(55^{\circ} \mathrm{C}\), isopropyl bromide reacts with hydroxide ion according to the following kinetics: $$ \begin{array}{l} -\frac{\mathrm{d}[\mathrm{RX}]}{\mathrm{d} t}=\left(4.8 \times 10^{-5} \mathrm{M}^{-1} \mathrm{~s}^{-1}\right) \\ {[\mathrm{RX}]\left[\mathrm{OH}^{-}\right]+2.4 \times 10^{-6} \mathrm{~s}^{-1}[\mathrm{RX}]} \end{array} $$ What percentage of isopropyl bromide reacts by the \(S_{\mathrm{N}_{2}}\) mechanism when \(\left[\mathrm{OH}^{-}\right]=0.01 \mathrm{M} ?\) (a) \(16.67 \%\) (b) \(83.33 \%\) (c) \(66.67 \%\) (d) \(33.33 \%\)

Step-by-Step Solution

Answer

16.67% of isopropyl bromide reacts via the SN2 mechanism.

Step 1 - Write the rate equation for the reaction

According to the given kinetics, the rate of the reaction can be described as: \(-\frac{d[\mathrm{RX}]}{dt} = (4.8 \times 10^{-5} \mathrm{M}^{-1} \mathrm{s}^{-1})[\mathrm{RX}][\mathrm{OH}^{-}] + (2.4 \times 10^{-6} \mathrm{s}^{-1})[\mathrm{RX}].\) This equation has two terms representing the two different mechanisms, where the first term indicates the rate of the SN2 mechanism and the second term indicates the rate of the SN1 mechanism.

Step 2 - Calculate the rate of reaction via the SN2 mechanism

Insert the given concentration of hydroxide ion into the first term of the reaction rate to find the rate of the SN2 mechanism: \(Rate_{\text{SN2}} = (4.8 \times 10^{-5} \mathrm{M}^{-1} \mathrm{s}^{-1}) [\mathrm{RX}] [0.01 \mathrm{M}].\)

Step 3 - Calculate the rate of reaction via the SN1 mechanism

Similarly, use the second term to determine the rate of the SN1 mechanism: \(Rate_{\text{SN1}} = (2.4 \times 10^{-6} \mathrm{s}^{-1}) [\mathrm{RX}].\)

Step 4 - Determine the percentage that reacts via the SN2 mechanism

The percentage of the reaction that occurs via the SN2 mechanism is the rate of the SN2 mechanism divided by the total rate (sum of rates via both SN2 and SN1 mechanisms) times 100: \[\text{Percentage}_{\text{SN2}} = \frac{Rate_{\text{SN2}}}{Rate_{\text{SN2}} + Rate_{\text{SN1}}} \times 100\% .\]

Step 5 - Insert values into the percentage formula

Substitute the calculated rates into the formula to get: \(\text{Percentage}_{\text{SN2}} = \frac{(4.8 \times 10^{-5} \mathrm{M}^{-1} \mathrm{s}^{-1}) [\mathrm{RX}] [0.01 \mathrm{M}]}{(4.8 \times 10^{-5} \mathrm{M}^{-1} \mathrm{s}^{-1}) [\mathrm{RX}] [0.01 \mathrm{M}] + (2.4 \times 10^{-6} \mathrm{s}^{-1}) [\mathrm{RX}]} \times 100\%.\) Notice that \([\mathrm{RX}]\) cancels out because it appears in both the numerator and the denominator.

Step 6 - Simplify the equation and solve for the percentage

After simplification, we get: \(\text{Percentage}_{\text{SN2}} = \frac{4.8 \times 10^{-7}}{4.8 \times 10^{-7} + 2.4 \times 10^{-6}} \times 100\% = \frac{4.8}{4.8 + 24} \times 100\% = \frac{4.8}{28.8} \times 100\% = \frac{1}{6} \times 100\% = 16.67\%.\) The correct answer is (a) 16.67%.

Key Concepts

Physical Chemistry

Physical chemistry involves the study of how matter behaves on a molecular and atomic level and how chemical reactions occur. It combines principles of physics and chemistry to understand the physical properties of molecules, the forces that act upon them, and the energy changes associated with chemical reactions.

Within the realm of physical chemistry, reaction mechanisms are a critical area of study. A reaction mechanism details the step-by-step sequence of elementary reactions by which overall chemical change occurs. For instance, the SN2 mechanism—short for 'nucleophilic substitution, second order'—is a common topic in physical chemistry. This mechanism involves a nucleophile attacking an electrophilic carbon and displacing a leaving group all in one step, leading to an 'inversion of configuration' at the carbon center. Understanding the kinetics of such reactions also fits within the purview of physical chemistry since it involves the rate at which the reaction occurs under various conditions, such as different temperatures or solvent environments.

Reaction Kinetics

Reaction kinetics, a subdivision of physical chemistry, is concerned with understanding the rates of chemical processes. It involves the study of how different variables such as concentration, temperature, and catalysts affect the speed of chemical reactions. The rate equation expresses this speed as a function of the concentration of reactants, and the rate constant provides the necessary quantitative measure.

For example, analyzing the rate equation provided in the SN2 reaction of isopropyl bromide with hydroxide ion in an ethanol solvent, we can discern insights about the reaction's dynamics. The equation includes terms for both SN2 and SN1 (nucleophilic substitution, first order) mechanisms, indicating that both are possible pathways for the reaction. Comparing the magnitude of rate constants can reveal the predominant mechanism under given conditions, which is essential for predicting product distribution and understanding how the reaction progresses.

Competitive Examinations Chemistry

Chemistry is a fundamental subject in various competitive examinations, including college admission tests, medical entrance exams, and engineering qualifiers. These exams evaluate a student's understanding of key concepts across organic, inorganic, and physical chemistry, including the SN2 mechanism and concepts of reaction kinetics. It's crucial for examinees to not only memorize facts but also develop a deep understanding of the processes and apply critical thinking to solve problems.

To efficiently tackle chemistry questions in competitive exams, students should focus on conceptual clarity, practice problem-solving skills, and familiarize themselves with the format of the questions. For instance, understanding the rate equation and being able to calculate the percentage of isopropyl bromide that reacts via the SN2 mechanism requires a grasp of reaction kinetics and the ability to perform calculations under time constraints—skills that are honed through rigorous practice and study.