Question

The reaction $$ \left(\mathrm{CH}_{3}\right)_{3} \mathrm{CBr}+\mathrm{OH}^{-} \longrightarrow\left(\mathrm{CH}_{3}\right)_{3} \mathrm{COH}+\mathrm{Br}^{-} $$ in a certain solvent is first order with respect to \(\left(\mathrm{CH}_{3}\right)_{3} \mathrm{CBr}\) and zero order with respect to \(\mathrm{OH}^{-} .\) In several experiments, the rate constant \(k\) was determined at different temperatures. A plot of \(\ln (k)\) versus 1\(/ T\) was constructed resulting in a straight line with a slope value of \(-1.10 \times 10^{4} \mathrm{K}\) and \(y\) -intercept of 33.5 . Assume \(k\) has units of \(\mathrm{s}^{-1}\) a. Determine the activation energy for this reaction. b. Determine the value of the frequency factor \(A\) . c. Calculate the value of \(k\) at \(25^{\circ} \mathrm{C}\) .

Short answer

a. The activation energy for this reaction is \(92154\,\text{J/mol}\). b. The frequency factor A is \(3.49 \times 10^{14}\,\text{s}^{-1}\). c. The rate constant k at 25°C is \(1.64\times 10^{-2}\,\text{s}^{-1}\).

Step-by-step solution

Use the Arrhenius equation

The Arrhenius equation is given by: \(k = Ae^{-E_{a}/RT}\) where: - k is the rate constant - A is the frequency factor - \(E_{a}\) is the activation energy (in J/mol) - R is the gas constant (8.314 J/mol·K) - T is the temperature (in K) Taking the natural logarithm of both sides, we get: \[\ln(k) = \ln(A) - \frac{E_{a}}{R} \cdot \frac{1}{T}\]

Find the activation energy (Ea)

Since we're given the slope of the graph as \(-1.10 \times 10^{4}\) K, we can directly relate it to the slope of the equation above: Slope = \(-\frac{E_{a}}{R}\) We need to find the activation energy, so we can rearrange this to: \(E_{a} = -R \cdot (-1.10 \times 10^{4})\) Substitute the value of R and calculate Ea: \(E_{a} = (8.314\,\text{J/mol·K}) \times (1.10 \times 10^{4}\,\text{K})\) \(E_{a} = 92154\,\text{J/mol}\) So, the activation energy of this reaction is \(92154\,\text{J/mol}\).

Find the frequency factor A

The y-intercept of the graph is given as 33.5. This corresponds to the term \(\ln(A)\) in the equation. We can find A using the following equation: \(A = e^{\ln(A)}\) Plug in the given y-intercept value: \(A = e^{33.5}\) Calculate the value of A: \(A \approx 3.49 \times 10^{14}\,\text{s}^{-1}\) The frequency factor A is \(3.49 \times 10^{14}\,\text{s}^{-1}\).

Calculate the rate constant k at 25ºC

Now, we will use the Arrhenius equation to find the rate constant at 25ºC. First, convert the temperature to Kelvin: \(T = 25 + 273.15 = 298.15\,\text{K}\) Now, substitute the values of A, Ea, R, and T into the equation: \(k = (3.49 \times 10^{14}\,\text{s}^{-1}) \cdot e^{(-92154\,\text{J/mol})/(8.314\,\text{J/mol·K} \cdot 298.15\,\text{K})}\) Calculate the value of k: \(k \approx 1.64\times 10^{-2}\,\text{s}^{-1}\) The rate constant k at 25°C is \(1.64\times 10^{-2}\,\text{s}^{-1}\). In conclusion: a. The activation energy for this reaction is \(92154\,\text{J/mol}\). b. The frequency factor A is \(3.49 \times 10^{14}\,\text{s}^{-1}\). c. The rate constant k at 25°C is \(1.64\times 10^{-2}\,\text{s}^{-1}\).

Key concepts

Understanding Activation Energy

Activation energy (E_a) is a key concept in the field of reaction kinetics, representing the minimum energy required for a chemical reaction to proceed. Imagine activation energy as the initial push needed to get a stationary object moving. Without this energy, the reactants will not transform into products.In the context of the Arrhenius equation, activation energy is what determines how temperature influences the rate at which a reaction occurs. A higher activation energy means that it takes a greater temperature increase to speed up the reaction. In mathematical terms, it is given by the slope of a plot of \(\ln(k)\) versus \(1/T\). The formula is: \[E_a = -R \cdot \text{slope}\]Where:

• R is the gas constant (8.314 J/mol·K)
• Slope is the \(1/T\) slope of the line in the Arrhenius plot

Understanding activation energy enables us to predict how the speed of a reaction changes with temperature, thus crucial for controlling and optimizing chemical processes.

Reaction Kinetics Simplified

Reaction kinetics is the branch of chemistry that examines the rate at which chemical reactions occur and the factors affecting these rates. It helps us understand why some reactions happen quickly while others take time. This knowledge allows chemists to manipulate conditions to either speed up or slow down reactions as needed. There are several factors that influence reaction kinetics, such as:

• Concentration of reactants: Generally, higher concentration increases the reaction rate.
• Temperature: Raising the temperature usually speeds up reactions by providing more energy to reactant molecules.
• Presence of a catalyst: Catalysts lower the activation energy needed, hence quickening the rate without being consumed themselves.

Additionally, reaction order is a fundamental concept in kinetics. In the given problem, we had a first-order reaction with respect to \(\text{(CH}_3\text{)}_3\text{CBr}\) and zero order with respect to \(\text{OH}^-\). This means the reaction rate depends only on the concentration of \(\text{(CH}_3\text{)}_3\text{CBr}\), not on \(\text{OH}^-\).
Comprehending reaction kinetics is essential for anyone involved in chemistry, from industrial processes to academic research, as controlling how fast a reaction goes can greatly impact outcomes.

The Role of Frequency Factor

The frequency factor (A), also known as the pre-exponential factor, is part of the Arrhenius equation. It accounts for the frequency of collisions between reactant particles and the probability that these collisions will lead to successful reactions. In simpler terms, it is a measure of how often molecules collide in the correct orientation. A higher frequency factor generally indicates a higher likelihood of a reaction occurring upon collisions.The Arrhenius equation is given by:\[k = Ae^{-E_{a}/RT}\]Where:

• A is the frequency factor.
• E_a is the activation energy.
• R is the gas constant.
• T is the temperature in Kelvin.
• k is the rate constant.

The frequency factor can be determined from the \(y\)-intercept in an Arrhenius plot. When you have \(\ln(k)\) vs. \(1/T\) plotted, the intercept \(C\) gives \(\ln(A)\), from which you can retrieve A by calculating \(e^C\).
Getting to grips with the frequency factor allows one to better understand how molecular behavior at a microscopic level influences reaction rates in real-world conditions.