Problem 19 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

Rate constants at a series of temperatures were obtained for the decomposition of azomethane \\[ \mathrm{CH}_{3} \mathrm{N}_{2} \mathrm{CH}_{3} \rightarrow 2 \mathrm{CH}_{3}^{*}+\mathrm{N}_{2} \\] $$\begin{array}{llllll} \hline T / \mathrm{K} & 523 & 541 & 560 & 576 & 593 \\ k / 10^{-6} \mathrm{s}^{-1} & 1.8 & 15 & 60 & 160 & 950 \end{array}$$ Use the data in the table to find the activation energy, \(E_{a}\), for the reaction. (Section \(9.7)\)

Step-by-Step Solution

Answer

The activation energy \( E_a \) can be calculated using the Arrhenius equation in linear form with slope from the plot of \( \ln k \) vs. \( 1/T \).

Step 1: Understanding the Relationship

The rate constant \( k \) is related to temperature \( T \) through the Arrhenius equation: \[ k = A e^{-E_{a} / RT} \] where \( A \) is the 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.

Step 2: Arrhenius Equation in Logarithmic Form

To find \( E_{a} \), we convert the Arrhenius equation to its linear form by taking the natural logarithm of both sides: \[ \ln k = \ln A - \frac{E_{a}}{R} \cdot \frac{1}{T} \] This equation takes the form of \( y = mx + c \), where \( y = \ln k \), \( x = \frac{1}{T} \), \( m = -\frac{E_{a}}{R} \), and \( c = \ln A \).

Step 3: Transform Data Points

Calculate \( \ln k \) and \( \frac{1}{T} \) for each temperature:- At \( T = 523 \) K, \( k = 1.8 \times 10^{-6} \): - \( \ln k \approx -13.12 \) - \( \frac{1}{523} \approx 0.001912 \)- At \( T = 541 \) K, \( k = 15 \times 10^{-6} \): - \( \ln k \approx -11.80 \) - \( \frac{1}{541} \approx 0.001849 \)- Continue similarly for the rest.

Step 4: Perform Linear Regression

Plot \( \ln k \) versus \( \frac{1}{T} \). Using a linear regression technique, determine the slope \( m \) of the line that fits the plotted points. This can be calculated or estimated using software or graphing tools.

Step 5: Calculate Activation Energy

Since the slope \( m = -\frac{E_{a}}{R} \), we solve for the activation energy \( E_{a} \):\[ E_{a} = -m \cdot R \] Use the calculated slope from Step 4 and \( R = 8.314 \) J/mol·K to find \( E_{a} \).

Step 6: Interpret and Verify

Ensure that the calculated \( E_a \) is reasonable by checking it against typical values for such reactions and verifying through literature if necessary. Precision in the slope calculation is crucial for accuracy.

Key Concepts

Arrhenius Equation

The Arrhenius Equation is a fundamental principle in chemistry that expresses how the rate constant \( k \) of a reaction depends on temperature \( T \). It is given by the formula:

• \( k = A e^{-E_{a} / RT} \)

Here, \( A \) represents the pre-exponential factor, sometimes known as the frequency factor, which gives an idea of the number of times reactants approach the activation barrier per unit time. \( E_{a} \) is the activation energy needed for the reaction to occur, \( R \) is the universal gas constant (8.314 J/mol·K), and \( T \) is the temperature in Kelvin.
This equation essentially tells us that as temperature increases, so does the rate constant \( k \), indicating that reactions typically proceed faster at higher temperatures. This is because higher temperatures provide the energy needed to overcome the activation energy barrier.

Rate Constants

Rate constants are crucial for understanding chemical kinetics, as they quantify the speed of the reaction under specific conditions. For the decomposition of azomethane, the rate constants vary significantly as the temperature changes:

• At 523 K, \( k = 1.8 \times 10^{-6} \, \mathrm{s}^{-1} \)
• At 593 K, \( k = 950 \times 10^{-6} \, \mathrm{s}^{-1} \)

As seen from these values, an increase in temperature causes the rate constant to increase, demonstrating the temperature dependence of reaction rates as described by the Arrhenius Equation. This clearly highlights the sensitivity of reaction rates to temperature changes and forms the basis for using these constants to determine other kinetic parameters such as activation energy.

Linear Regression

Linear regression is a statistical method used to determine a straight line that best fits a set of data points. In the context of chemical kinetics, we use linear regression to analyze the relationship given by the linearized form of the Arrhenius Equation:

• \( \ln k = \ln A - \frac{E_{a}}{R} \times \frac{1}{T} \)

By plotting \( \ln k \) versus \( \frac{1}{T} \), we obtain a straight line whose slope \( m \) is equal to \(-\frac{E_{a}}{R} \). The process involves computing \( \ln k \) and \( \frac{1}{T} \) for each data point, plotting these values, and using a linear regression tool to calculate the best-fitting line. This technique allows us to extract the activation energy from the slope, an essential part of understanding reaction kinetics.

Temperature Dependence

Temperature dependence is a critical aspect of reaction kinetics, determining how the rate constants of a chemical reaction change with temperature. It is highlighted through the Arrhenius Equation and makes it possible to explore how reaction rates increase with temperature:

• Each 10-degree rise in temperature approximately doubles the reaction rate for many reactions.
• The steep rise in the rate constant \( k \) with temperature, as observed for the decomposition of azomethane, illustrates this relationship.

Being able to plot \( \ln k \) against \( \frac{1}{T} \) helps us understand this dependency, as the slope of this line directly links to the activation energy. Understanding temperature dependence allows chemists to predict reaction behavior under various thermal conditions, which is critical in fields such as materials science and pharmaceuticals.