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 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

The activation energy for this reaction is -91,654 J/mol. The frequency factor A is approximately \(3.10 \times 10^{14}\). The rate constant k at 25°C is approximately \(1.37 \times 10^{-10} s^{-1}\).

Step-by-step solution

Write the Arrhenius equation and identify the unknowns

The Arrhenius equation is given as follows: \[k = Ae^{\frac{-E_a}{RT}}\] where: - k is the rate constant, - A is the frequency factor or pre-exponential factor, - \(E_a\) is the activation energy, - R is the gas constant (8.314 J/mol·K), and - T is the temperature in Kelvin. Taking the natural logarithm of both sides, we get: \[\ln(k) = \ln(A) - \frac{E_a}{RT}\] This is the equation of a straight line with \(y = \ln(k)\), slope = \(- \frac{E_a}{R}\), and intercept = \(\ln(A)\). We have been given the slope, the y-intercept, and we need to find the values of \(E_a\), A, and the rate constant k at 25°C.

Determine the activation energy

We are given the slope of the plot, and we know the relationship between the slope and the activation energy: \[- \frac{E_a}{R} = -1.10 \times 10^4 K\] Now, we can solve for \(E_a\): \[E_a = R \times (-1.10 \times 10^4 K)\] Substitute the value of R (8.314 J/mol·K) into the equation: \[E_a = 8.314 J/(mol \cdot K) \times (-1.10 \times 10^4 K)\] \[E_a = -91, 654 J/mol\] The activation energy for this reaction is -91,654 J/mol.

Determine the frequency factor A

Next, we will find the value of the frequency factor A using the y-intercept, which is equal to \(\ln(A)\): \[\ln(A) = 33.5\] Now, we can solve for A by taking the exponential of both sides: \[A = e^{33.5}\] \[A \approx 3.10 \times 10^{14}\] The frequency factor A for this reaction is approximately \(3.10 \times 10^{14}\).

Calculate the rate constant k at 25°C

Finally, we'll find the rate constant k at 25°C (298.15 K) using the Arrhenius equation: \[k = Ae^{\frac{-E_a}{RT}}\] Substitute the values of A, \(E_a\), R, and T (298.15 K) into the equation: \[k = (3.10 \times 10^{14} s^{-1})e^{\frac{(-91, 654 J/mol)}{(8.314 J/mol \cdot K)(298.15 K)}}\] \[k \approx 1.37 \times 10^{-10} s^{-1}\] The rate constant k at 25°C is approximately \(1.37 \times 10^{-10} s^{-1}\).

Key concepts

Understanding Activation Energy

Activation energy (\(E_a\)) is a fundamental concept in chemical kinetics, an area of chemistry focused on the rates of reactions. It represents the minimum amount of energy that reacting molecules must possess for a chemical reaction to occur. Imagine activation energy as a barrier that reactants need to overcome to transform into products. The higher the barrier, the fewer the molecules that have enough energy to react at a given temperature.

In the context of the Arrhenius equation, understanding activation energy is crucial. The equation tells us that the rate constant (\(k\)) for a reaction increases as the activation energy decreases. This makes sense because a lower energy barrier means more molecules can collide with sufficient energy to react, leading to a faster reaction rate. In our example, a negative value for the activation energy suggests a calculation error, as energy cannot be negative. In reality, activation energy is always a positive value.

For example, a reaction between substrate ((CH_3)_3CBr) and hydroxide ions (OH^-) requires a certain amount of activation energy to proceed. The amount of energy can be determined experimentally by measuring the rate constant at various temperatures and then using the Arrhenius equation to calculate the activation energy. The negative sign of activation energy in the given slope should be interpreted as the rate constant decreasing with an increase in temperature, which is contrary to expected behavior and indicates an error that needs to be corrected before proceeding.

The Significance of the Rate Constant

The rate constant (\(k\)) is quite literally the constant of the rate, as it appears in the rate laws that describe the speed at which reactants are converted into products. This value is a measure of how quickly a reaction occurs and is specific for each chemical reaction at a given temperature. It includes the effects of temperature, activation energy, and the frequency factor and is influenced by the presence of catalysts.

When we analyze a reaction such as the given conversion of (CH_3)_3CBr to (CH_3)_3COH, knowing the rate constant is crucial. It allows chemists to predict how fast the reaction will proceed under certain conditions. In the example provided, the rate constant at 25°C is found using the Arrhenius equation and the calculated activation energy and frequency factor. It's crucial to remember that the rate constant doesn't stay constant if environmental conditions, such as temperature, change. The relationship between temperature and the rate constant is precisely what the Arrhenius equation helps to clarify, linking the kinetic molecular theory with observable reaction rates.

Frequency Factor: A Closer Look

The frequency factor (A), also known as the pre-exponential factor, plays a pivotal role in the Arrhenius equation. It relates to the frequency of collisions and the orientation of reactant molecules. A higher frequency factor indicates that molecules collide more often and with the proper orientation necessary for a reaction to take place. It's a reflection of both the number of collisions and the fraction of collisions that are effective in leading to a reaction.

The frequency factor is influenced by many variables including the physical states of the reactants, the complexity of the reaction, and molecular sizes. For example, in gas-phase reactions where molecules move more freely, the frequency of effective collisions may be higher than in reactions involving solids or in solution. In our specific reaction, the value of the frequency factor tells us how often effective collisions occur between (CH_3)_3CBr and OH^- ions in the solvent.

It is important to note that while the frequency factor can give an indication of the likelihood of effective collisions, it does not guarantee that a reaction will occur. Activation energy must also be considered, as even frequent collisions will not result in a reaction if the energy threshold is not met.