Problem 38 - Chemistry and Chemical Reactivity

Question

If the rate constant for a reaction triples when the temperature rises from \(3.00 \times 10^{2} \mathrm{K}\) to \(3.10 \times 10^{2} \mathrm{K},\) what is the activation energy of the reaction?

Step-by-Step Solution

Answer

The activation energy is approximately 84.9 kJ/mol.

Step 1: Understand the Arrhenius Equation

The Arrhenius equation describes the temperature dependence of the rate constant and is given by \( k = Ae^{-\frac{E_a}{RT}} \), where \( k \) is the rate constant, \( A \) is the pre-exponential factor, \( E_a \) is the activation energy, \( R \) is the gas constant, and \( T \) is the temperature.

Step 2: Setting Up the Given Information

We are given that the rate constant triples when the temperature increases from \(300\, \text{K} \) to \(310\, \text{K}\). Hence, \( k_2 = 3k_1 \), where \( k_1 \) is the rate constant at \( T_1 = 300 \text{ K} \) and \( k_2 \) is the rate constant at \( T_2 = 310 \text{ K} \).

Step 3: Use the Arrhenius Equation Ratio

Using the Arrhenius equation for both temperatures, we can write \( \frac{k_2}{k_1} = \frac{Ae^{-\frac{E_a}{RT_2}}}{Ae^{-\frac{E_a}{RT_1}}} = e^{\frac{E_a}{R}(\frac{1}{T_1} - \frac{1}{T_2})} \). Since \( \frac{k_2}{k_1} = 3 \), it becomes \( 3 = e^{\frac{E_a}{R}(\frac{1}{300} - \frac{1}{310})} \).

Step 4: Solve for Activation Energy \( E_a \)

To find \( E_a \), take the natural logarithm of both sides: \( \ln(3) = \frac{E_a}{R} \left( \frac{1}{300} - \frac{1}{310} \right)\). Simplify this to solve for \( E_a \): \[ E_a = R \cdot \ln(3) \cdot \left( \frac{1}{300} - \frac{1}{310} \right)^{-1} \].

Step 5: Calculate the Activation Energy

Substitute in the values: \( R = 8.314 \text{ J/mol} \cdot \text{K} \) and calculate the difference in reciprocal temperatures: \( \frac{1}{300} - \frac{1}{310} = \frac{310 - 300}{300 \times 310} = \frac{10}{93000} \approx 0.0001075 \).Therefore, \( E_a = 8.314 \cdot \ln(3) / 0.0001075 \approx 8.314 \cdot 1.0986 / 0.0001075 \approx 84857 \text{ J/mol} \) or \( 84.9 \text{ kJ/mol} \).

Key Concepts

Arrhenius equation

The Arrhenius equation is a fundamental concept in chemistry that explains how reaction rates depend on temperature. This equation is expressed as \( k = Ae^{-\frac{E_a}{RT}} \). Here:

• \( k \) is the rate constant, indicating how fast a reaction proceeds.
• \( A \) is the pre-exponential factor, representing the frequency of collisions.
• \( E_a \) is the activation energy, the minimum energy required for a reaction to occur.
• \( R \) is the gas constant, valued at approximately 8.314 J/mol·K.
• \( T \) is the absolute temperature in Kelvin.

The equation shows that as the temperature increases, the rate constant usually increases, leading to faster reaction rates.
The exponential function \( e^{-\frac{E_a}{RT}} \) highlights how an increase in the temperature reduces the fraction \( \frac{E_a}{RT} \), thereby increasing \( k \), assuming all other factors remain constant.

Rate constant

The rate constant \( k \) is a crucial parameter in kinetics, providing specific information about the speed at which a chemical reaction occurs. It is a proportional constant in the rate law expression of a reaction. For example, in the rate equation \( ext{Rate} = k[A][B] \) for a bimolecular reaction, \( k \) determines the rate based on concentrations \( [A] \) and \( [B] \).

The value of the rate constant is influenced by several factors:

• Temperature: As indicated by the Arrhenius equation, the rate constant typically increases with temperature.
• Activation energy: A lower activation energy results in a higher rate constant, meaning the reaction can proceed more easily.
• Reaction mechanism: Different mechanisms can have different rate constants.

Understanding these factors helps in controlling reaction speeds in various applications, such as industrial synthesis or biochemical reactions.

Temperature dependence

Temperature plays a crucial role in determining chemical reaction rates. According to the Arrhenius equation, the rate constant \( k \) is directly correlated with temperature. This is often referred to as the "temperature dependence" of reaction rates.

When temperature increases, more molecules have the requisite energy (equal to or greater than the activation energy \( E_a \)) to react. This results in:

• Increased frequency of effective collisions.
• Higher probability of overcoming the activation energy barrier.
• An increase in the rate constant \( k \), speeding up the reaction.

This dependency highlights the importance of temperature control in chemical processes, ensuring optimal reaction rates for efficiency and safety. Even a small change in temperature can significantly affect the pace of a reaction.

Natural logarithm

The natural logarithm function (denoted \( \ln \)) is essential in the context of the Arrhenius equation, especially when deriving activation energies. This logarithm is based on the constant \( e \approx 2.718 \,\), which is fundamental in exponential growth and decay problems.

In practical applications, the natural logarithm helps in linearizing the Arrhenius equation. By taking the \( \ln \) of both sides of \( k = Ae^{-\frac{E_a}{RT}} \), the equation becomes:\[ \ln k = \ln A - \frac{E_a}{R} \cdot \frac{1}{T} \]This linear form allows comparison and graphing of experimental data. If we plot \( \ln k \) against \( \frac{1}{T} \), the slope of the resulting straight line gives \(-\frac{E_a}{R}\).

Utilizing the natural logarithm simplifies calculations, making it easier to determine the activation energy \( E_a \) from empirical data.