Problem 87 - Chemistry: An Atoms-Focused Approach

Question

The rate constant for the reaction of ozone with oxygen atoms was determined at four temperatures. Calculate the activation energy and frequency factor \(A\) for the reaction $$\mathrm{O}(g)+\mathrm{O}_{3}(g) \rightarrow 2 \mathrm{O}_{2}(g)$$ given the following data: $$\begin{array}{cc}T(\mathrm{K}) & k\left[\mathrm{cm}^{3} /(\text { molecule } \cdot \mathrm{s})\right] \\\250 & 2.64 \times 10^{-4} \\\\\hline 275 & 5.58 \times 10^{-4} \\\\\hline 300 & 1.04 \times 10^{-3} \\\\\hline 325 & 1.77 \times 10^{-3} \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

Question: Determine the activation energy and frequency factor A for the given reaction of ozone with oxygen atoms using the provided data points and the Arrhenius equation. Solution: 1. Write down the Arrhenius equation. 2. Take the natural logarithm of the Arrhenius equation. 3. Rearrange the equation and note down data points. 4. Plot the data points, and calculate the slope and intercept. 5. Calculate the activation energy (Ea) and frequency factor (A) using the obtained slope and intercept.

Step 1: Write down the Arrhenius equation

The Arrhenius equation is given by: $$k = A\mathrm{e}^{-Ea/RT}$$ where k is the rate constant, A is the frequency factor, Ea is the activation energy, R is the gas constant, and T is the temperature in Kelvin.

Step 2: Take the natural logarithm of the Arrhenius equation

Taking the natural logarithm of both sides of the Arrhenius equation will get: $$\ln{k} = \ln{A} - \frac{Ea}{RT}$$

Step 3: Rearrange the equation and note down data points

In order to calculate the activation energy and frequency factor A, we need to create a linear equation that can be used to determine the slope and intercept of the provided data: $$\frac{Ea}{R} = -slope$$ $$\ln{A} = intercept$$ Now, we will take each pair of temperature (T) and rate constant (k) data points and write down their corresponding values for \(\ln{k}\) and \(1/T\). $$\begin{array}{ccc}T(\mathrm{K}) & 1/T\left(\mathrm{K}^{-1}\right) & \ln{k} \\\\\hline 250 & \frac{1}{250} & \ln{(2.64 \times 10^{-4})} \\\\\hline 275 & \frac{1}{275} & \ln{(5.58 \times 10^{-4})} \\\\\hline 300 & \frac{1}{300} & \ln{(1.04 \times 10^{-3})} \\\\\hline 325 & \frac{1}{325} & \ln{(1.77 \times 10^{-3})} \\\\\hline\end{array}$$

Step 4: Plot the data points and calculate the slope and intercept

Using the data points obtained in Step 3, plot the graph of \(\ln{k}\) vs \(1/T\). A linear graph should be observed. Using the graph, calculate the slope and intercept.

Step 5: Calculate activation energy and frequency factor

From the values of slope and intercept obtained in the previous step, we can calculate the activation energy (Ea) and frequency factor (A): $$Ea = -slope \cdot R$$ $$A = e^{intercept}$$ Remember that the value of the gas constant R should be in the units of \(\mathrm{cal} \cdot \mathrm{K}^{-1} \cdot \mathrm{mol}^{-1}\). Replace the values of slope, intercept, and R, and solve for Ea and A.

Key Concepts

Arrhenius Equation

The Arrhenius Equation is a fundamental formula that describes how the rate constant ( k ) of a chemical reaction is dependent on temperature. This equation is crucial for predicting how a reaction will proceed under different conditions. It is given by \( k = A \mathrm{e}^{-Ea/RT} \), where each symbol represents a specific aspect of the reaction:

• k: the rate constant, measuring the reaction rate.
• A: the frequency factor, relating to the fraction of molecules that have the right orientation to react.
• Ea: the activation energy, or the energy barrier that must be overcome for a reaction to occur.
• R: the gas constant, a constant that relates energy scales in the equation.
• T: the temperature, which must be in Kelvin for the equation to be valid.

To solve problems involving the Arrhenius equation, you often need to transform it using logarithms, allowing for linearization which is critical for graphical methods of analysis.

Rate Constant

The rate constant, denoted as \( k \), is a central concept in the realm of reaction kinetics. It quantifies the speed of a chemical reaction at a given temperature and is essential for calculating reaction rates. The magnitude of \( k \) depends on both the inherent characteristics of the reaction and the temperature:

• Inherent Reaction Characteristics: Reactions have intrinsic features that can make them inherently fast or slow. This is reflected in different rate constants.
• Temperature: As temperature increases, normally, the rate constant also increases, leading to a quicker reaction.

Thus, the rate constant helps predict how quickly reactants are converted to products. Understanding \( k \) is important for controlling chemical processes like those in industrial manufacturing or biological systems.

Temperature Dependence

Temperature is a key player in determining the rate of chemical reactions. According to the Arrhenius equation, the rate constant \( k \) is dependent on temperature, implying that even slight changes in temperature can significantly affect the reaction rate. There are a few reasons why temperature influences reaction rates:

• Increased Molecular Activity: Higher temperatures generally increase the kinetic energy of molecules, leading to more frequent collisions.
• Overcoming Activation Energy: With increased kinetic energy, more molecules have enough energy to surpass the activation energy barrier.

When plotted, the relationship between the natural logarithm of the rate constant (\( \ln{k} \)) and the inverse of the temperature (\( 1/T \)) produces a straight line, showcasing the linearity suggested by the Arrhenius plot. This plot can help determine the activation energy when the slope is evaluated.

Frequency Factor

The frequency factor, denoted as \( A \), is a component of the Arrhenius equation that indicates how often molecules collide with the correct orientation to lead to a reaction. It affects the rate constant but is independent of the temperature. Here is why it matters:

• Frequency of Collisions: \( A \) accounts for the number of reactant collisions per unit time. This assumes that favorable conditions for reactions (proper orientation and sufficient energy) occur often.
• Reaction Pathway Efficiency: A larger value of \( A \) implies more effective collisions, potentially leading to faster reactions if the activation energy is not prohibitive.

The frequency factor is essential for estimating the reaction rate's intrinsic likelihood and combining it with the effect of the activation energy and temperature. Understanding \( A \) aids in predicting how changes in conditions can influence the rate of reaction.