Problem 67 - Chemistry and Chemical Reactivity

Question

Data for the reaction $$\begin{aligned}\left[\mathrm{Mn}(\mathrm{CO})_{5}\left(\mathrm{CH}_{3}\mathrm{CN}\right)\right]^{+}+\mathrm{NC}_{5} \mathrm{H}_{5} & \\\\\longrightarrow &\left[\mathrm{Mn}(\mathrm{CO})_{5}\left(\mathrm{NC}_{5}\mathrm{H}_{5}\right)\right]^{+}+\mathrm{CH}_{3} \mathrm{CN}\end{aligned}$$ are given in the table. Calculate \(E_{\mathrm{a}}\) from a plot of \(\ln k\) versus \(1 / T.\) $$\begin{array}{ll}T(\mathrm{K}) & k\left(\min ^{-1}\right) \\\\\hline 298 & 0.0409 \\\308 & 0.0818 \\\318 & 0.157 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

The activation energy \( E_a \) is approximately 50.76 kJ/mol.

Step 1: Calculate ln(k) and 1/T

First, we need to calculate the natural logarithm of the rate constant \( k \) and the reciprocal of temperature \( T \) for each given temperature. For \( T = 298 \) K, \( k = 0.0409 \) min\(^{-1}\): \( \ln k = \ln(0.0409) \approx -3.198 \) and \( 1/T \approx 0.00336 \) K\(^{-1}\).For \( T = 308 \) K, \( k = 0.0818 \) min\(^{-1}\): \( \ln k = \ln(0.0818) \approx -2.508 \) and \( 1/T \approx 0.00325 \) K\(^{-1}\).For \( T = 318 \) K, \( k = 0.157 \) min\(^{-1}\): \( \ln k = \ln(0.157) \approx -1.855 \) and \( 1/T \approx 0.00314 \) K\(^{-1}\).

Step 2: Plot ln(k) versus 1/T

Create a plot with \( \ln k \) on the y-axis and \( 1/T \) on the x-axis. Plot the points calculated in Step 1: - Point 1: (0.00336, -3.198)- Point 2: (0.00325, -2.508)- Point 3: (0.00314, -1.855)This plot should resemble a linear relationship, which follows the Arrhenius equation.

Step 3: Determine the slope of the line

Since the Arrhenius plot is linear, use two points to calculate the slope, \( m \), of the line. The slope \( m \) is given by the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Using the points (0.00336, -3.198) and (0.00314, -1.855):\[ m = \frac{-1.855 - (-3.198)}{0.00314 - 0.00336} = \frac{1.343}{-0.00022} \approx -6104.55 \]

Step 4: Calculate Activation Energy, E_a

The slope of the Arrhenius plot is related to the activation energy \( E_a \) by the equation: \[ m = -\frac{E_a}{R} \]where \( R \) is the universal gas constant \( 8.314 \text{ J/(mol K)} \). Thus, \( E_a = -m \times R \). Substitute \( m = -6104.55 \), we get: \[ E_a = 6104.55 \times 8.314 \approx 50761 \text{ J/mol} \approx 50.76 \text{ kJ/mol} \]

Key Concepts

Arrhenius Equation

The Arrhenius Equation is a fundamental mathematical representation used to describe the temperature dependence of reaction rates. It is expressed as:\[k = A \cdot e^{-\frac{E_a}{RT}}\]where:

• \(k\) is the rate constant
• \(A\) is the pre-exponential factor, often considered a measure of the frequency of successful collisions
• \(E_a\) is the activation energy
• \(R\) is the universal gas constant \(8.314\, \text{J/(mol K)}\)
• \(T\) is the temperature in Kelvin

This equation shows that the rate constant increases exponentially as the temperature rises or as the activation energy decreases. In chemical kinetics, the Arrhenius Equation is instrumental in predicting how changes in temperature affect reaction rates.
It also helps calculate the activation energy, which is the minimum energy required for a reaction to occur. You can derive valuable insights into the reaction dynamics by studying the Arrhenius Equation.

Reaction Rate

The reaction rate is a measure of how fast a chemical reaction occurs. It is often represented by the rate constant \(k\), which depends on factors such as temperature, activation energy, and the nature of the reactants.
Reaction rates can vary widely:

• Some reactions are near-instantaneous, like the mixing of acids and bases.
• Others, such as rusting of iron, may take years to observe significant progress.

The study of reaction rates, known as kinetics, seeks to understand and quantify these speed differences. Factors that affect reaction rates include:

• Concentration of reactants: Higher concentration generally leads to faster reactions.
• Temperature: Increasing the temperature typically speeds up the reaction.
• Catalysts: Substances that increase reaction speed without being consumed.
• Surface area of reactants: Finely divided materials react faster than bulk materials.

Understanding reaction rates not only allows chemists to control industrial processes but also provides insight into natural phenomena and biological systems.

Temperature Dependence

The rate of a chemical reaction is heavily dependent on temperature. This is because temperature influences kinetic energy:

• At higher temperatures, molecules move faster.
• Faster molecules lead to more frequent and energetic collisions.
• More collisions increase the likelihood of surpassing the activation energy barrier.

The Arrhenius Equation illustrates this temperature dependence, defining the relationship as an exponential increase in the reaction rate with rising temperature.
It highlights the sensitivity of reaction kinetics to temperature changes, often described as a doubling of reaction rate with each 10°C increase.
In practical terms, understanding temperature's effect on reaction rates is crucial in fields ranging from industrial manufacturing to biochemical processes in living organisms.

Arrhenius Plot

An Arrhenius Plot is a graphical method to analyze reaction rates in terms of temperature. By plotting \( \ln k \) (natural logarithm of the rate constant) against \( 1/T \) (inverse of temperature in Kelvin), you can determine the activation energy of a reaction.
The plot typically results in a straight line due to the linearization of the Arrhenius Equation:

• \( \ln k = \ln A - \frac{E_a}{R} \cdot \frac{1}{T} \)
• The slope of the line is \(-\frac{E_a}{R}\).
• The y-intercept is \( \ln A \).

Using these plots, you can easily extract both the activation energy and the pre-exponential factor. The linear relationship aids in visualizing the temperature dependence of a reaction's rate constant.
This method simplifies the determination of \(E_a\) and provides a clear interpretation of kinetic data, making Arrhenius plots a valuable tool in chemical kinetics.