Understanding the Activation Energy of a Reaction: Analyzing Rate Constants at Different Temperatures

The Arrhenius equation is a cornerstone in physical chemistry, providing a way to understand the relationship between a reaction's rate constant and temperature. In this article, we will explore how the activation energy of a reaction can be calculated when given the rate constants at two different temperatures. Specifically, we will use a practical example to illustrate the concept and solve for the activation energy.

Introduction to the Arrhenius Equation

The Arrhenius equation is given by the formula:

k A e-Eact/RT

In this formula, k is the rate constant, A is the pre-exponential factor, Eact is the activation energy, R is the gas constant (8.314 J/mol·K), and T is the temperature in Kelvin. Using this equation, we can analyze the influence of temperature on the rate constants of chemical reactions.

Revisiting the Given Problem: Understanding the Relationship at Different Temperatures

The provided problem states that the reaction rate constants at 298 K and 373 K are 0.050 s-1 and 20 s-1 respectively. To find the activation energy, we will make use of the equation:

lnk2/k1 Eact/RT1 - Eact/RT2

This equation allows us to calculate the activation energy without needing the pre-exponential factor, A.

Step-by-Step Solution: Calculating the Activation Energy

Let's break down the problem step-by-step:

Assign the given values to the variables:

k1 0.050 s-1 k2 20 s-1 T1 298 K T2 373 K

Substitute the values into the equation:

ln(20 / 0.050) Eact / (8.314 × 298) - Eact / (8.314 × 373)

This simplifies to:

ln(400) Eact / 2477.572 - Eact / 3101.262

Calculate the natural logarithm of the rate constant ratio:

ln(400) 5.991

Combine the terms involving Eact to simplify the equation:

5.991 (Eact × 3101.262 - Eact × 2477.572) / (2477.572 × 3101.262)

5.991 (Eact × 623.69) / (7682500.324)

Isolate Eact to find its value:

Eact (5.991 × 7682500.324) / 623.69

Eact ≈ 70425.22 J/mol

Conclusion: Interpretation of Activation Energy

The calculated activation energy, Eact, is approximately 70425.22 J/mol (or 70.425 kJ/mol). This value indicates the amount of energy required for the reactants to transition to the products, effectively determining the reaction's speed and the factors influencing it at different temperatures.

Summary and Further Reading

This article has explored the Arrhenius equation and its application in calculating the activation energy of a reaction. By using the rate constants at two different temperatures, we were able to solve for the activation energy with precision. Understanding this concept is crucial in various fields, including chemical engineering, biochemistry, and environmental science. For further reading on this topic, consider exploring:

Chemguide: Arrhenius Equation Journal article: Applications of the Arrhenius Equation CHEMstation: Activation Energy

Understanding the relationship between activation energy and reaction rates can provide valuable insights into the nature of chemical reactions, their mechanisms, and how they can be optimized under different conditions.