Problem 88 - Chemistry The Science in Context

Question

Values of the rate constant for the decomposition of \(\mathrm{N}_{2} \mathrm{O}_{5}\) gas at four temperatures are as follows: $$\begin{array}{cc} T(\mathrm{K}) & k\left(\mathrm{s}^{-1}\right) \\ \hline 658 & 2.14 \times 10^{5} \\ \hline 673 & 3.23 \times 10^{5} \\ \hline 688 & 4.81 \times 10^{5} \\ \hline 703 & 7.03 \times 10^{5} \\ \hline \end{array}$$ a. Determine the activation energy of the decomposition reaction. b. Calculate the value of the rate constant at \(300 \mathrm{K}.\)

Step-by-Step Solution

Answer

Question: Determine the activation energy (Ea) of the decomposition reaction of N2O5 gas, and calculate the value of the rate constant (k) at 300K.

Step 1: Calculate the natural logarithm of rate constants and reciprocal of temperatures

For each of the given temperatures and rate constants, calculate the natural logarithm of the rate constants (ln(\(k\))) and the reciprocal of the temperatures (\(\frac{1}{T}\)).

Step 2: Plot ln(\(k\)) vs \(\frac{1}{T}\)

Using the values calculated in Step 1, plot ln(\(k\)) on the y-axis and \(\frac{1}{T}\) on the x-axis. The graph should be a straight line.

Step 3: Calculate the slope of the graph

Based on the linear graph from the previous step, calculate the slope, which represents \(\frac{-E_{a}}{R}\).

Step 4: Calculate the activation energy (\(E_{a}\))

To determine the activation energy \(E_{a}\), use the slope value obtained in the previous step and the gas constant, \(R=8.314\: J\:mol^{-1}K^{-1}\): $$E_{a}=-slope \times R$$

Step 5: Use one of the given temperatures and rate constants to determine the pre-exponential factor (\(A\))

Now that we have the activation energy, we can choose one of the given temperature-rate constant pairs to determine the pre-exponential factor \(A\) using the Arrhenius equation: $$A=\frac{k}{\operatorname{e}^{-\frac{E_{a}}{RT}}}$$

Step 6: Calculate the rate constant at \(300\mathrm{K}\)

Using the activation energy \(E_{a}\) and the pre-exponential factor \(A\) calculated in the previous steps, we can now apply the Arrhenius equation to calculate the rate constant (\(k\)) at \(300\mathrm{K}\): $$k = A\operatorname{e}^{-\frac{E_{a}}{R(300)}}$$

Key Concepts

Activation Energy

Understanding activation energy (often denoted as E_a) is crucial for deciphering how chemical reactions occur. It's defined as the minimum amount of energy that reacting particles must have for a reaction to take place. Essentially, it's like a barrier that reactants must overcome to transform into products.

In the exercise, determining the activation energy involves using experimental data from the rate constant at different temperatures. By plotting ln(k) against 1/T, the slope of the resulting line is related to the activation energy, through the equation \(slope = -\frac{E_{a}}{R}\), where R is the gas constant. This relationship is part of the Arrhenius equation, and knowing the activation energy helps chemists understand how temperature affects the reaction rate and the likelihood of a reaction occurring.

Rate Constant

The rate constant, represented by the symbol k, is a crucial parameter in the Arrhenius equation and chemical kinetics as a whole. It quantifies the speed of a chemical reaction; a larger rate constant means a faster reaction. The value of k changes with temperature, as seen in the provided data.

Using the Arrhenius equation, one can calculate the rate constant at different temperatures, once the activation energy and the pre-exponential factor—symbolized as A—are known. The equation is given as \( k = Ae^{-\frac{E_{a}}{RT}}\), where T is the temperature. By experimentally determining k at several temperatures, we can use this information to predict the rate constant at other temperatures, which is what was done in the exercise's final step.

Chemical Kinetics

The study of chemical kinetics delves into the speed or rate at which reactions occur and the various factors that influence these rates, such as concentration, catalysts, and temperature. Through kinetics, we can understand reaction mechanisms — the step-by-step pathway from reactants to products.

In the context of the exercise, the rate at which the decomposition of \(\mathrm{N}_{2}\mathrm{O}_{5}\) gas occurs is the focus of our analysis. By calculating the temperature-specific rate constants, we can begin to draw connections between the controlled experiment variables and the behavior of the reaction over time. Understanding kinetics is essential for controlling reactions in industrial processes and for developing new technologies.

Temperature Dependence of Reaction Rates

The temperature dependence of reaction rates is a reflection of how a slight change in temperature can significantly affect how quickly a reaction will proceed. According to the Arrhenius equation, even a small temperature increase can lead to a dramatic rise in the rate constant, k. This is because higher temperatures usually provide more energy to the reacting molecules, increasing the chances of successful collisions leading to a reaction.

In the textbook example, this principle is applied to calculate the unknown rate constant of a reaction at 300 K, using the known values at higher temperatures. Temperature's impact on reaction rates is key to various applications, from industrial synthesis to food preservation.