What is the Activation Energy of a Reaction?

In this article, we will delve into the fascinating world of chemical kinetics by calculating the activation energy of a reaction where the rate doubles upon a 10°C increase in temperature. This calculation utilizes the Arrhenius equation and the principles of the temperature dependence of reaction rates.

Introduction to Activation Energy and the Arrhenius Equation

The rate at which a chemical reaction proceeds can be significantly influenced by the temperature of the reaction mixture. This is often quantified by the activation energy (Ea), which represents the minimum energy required for reactants to successfully transform into products. The Arrhenius equation, a cornerstone in chemical kinetics, provides a mathematical relationship between the temperature of a reaction and its rate constant (k).

The Arrhenius equation is expressed as:

[ k A e^{- frac{E_a}{RT}} ]

Here:

( k ) is the rate constant.

A is the pre-exponential factor, which is temperature-independent and relates to the frequency of effective collisions between reactant molecules.

Ea is the activation energy, the energy barrier that must be overcome for the reaction to proceed.

R is the universal gas constant, with a value of 8.314 J K?1 mol?1

T is the temperature in Kelvin.

Application of the Arrhenius Equation

Given a scenario where the rate of a reaction doubles for every 10°C increase in temperature, we can use the Arrhenius equation to determine the activation energy. This is particularly useful in understanding the thermodynamics and kinetics of various biochemical and industrial processes.

Let's denote:

T1 as 37°C, which is 310.15 K in Kelvin.

T2 as 47°C, which is 320.15 K in Kelvin.

Since the rate doubles, we have:

[ frac{k_2}{k_1} 2 ]

Using the Arrhenius equation for both temperatures:

[ frac{k_2}{k_1} frac{A e^{- frac{E_a}{RT_2}}}{A e^{- frac{E_a}{RT_1}}} e^{- frac{E_a}{RT_2} frac{E_a}{RT_1}} ]

This simplifies to:

[ frac{k_2}{k_1} e^{E_a left(frac{1}{RT_1} - frac{1}{RT_2}right)} ]

Substituting ( frac{k_2}{k_1} 2 ):

[ 2 e^{E_a left(frac{1}{RT_1} - frac{1}{RT_2}right)} ]

Calculation Steps

Taking the natural logarithm of both sides:

[ ln(2) E_a left(frac{1}{RT_1} - frac{1}{RT_2}right) ]

We then calculate the difference in the fractions:

[ frac{1}{T_1} - frac{1}{T_2} frac{T_2 - T_1}{T_1 T_2} frac{320.15 - 310.15}{310.15 times 320.15} frac{10}{310.15 times 320.15} ]

Calculating (310.15 times 320.15):

(310.15 times 320.15 approx 99248.0225)

So:

[ frac{1}{310.15} - frac{1}{320.15} approx frac{10}{99248.0225} approx 1.007 times 10^{-5} text{ K}^{-1} ]

Substituting back into the equation:

[ ln(2) E_a cdot frac{10}{R times 99248.0225} ]

Using ( R 8.314 text{ J K}^{-1} text{mol}^{-1} ):

[ E_a frac{ln(2) cdot R cdot 99248.0225}{10} ]

Calculating ( ln(2) approx 0.693 ):

[ E_a approx frac{0.693 cdot 8.314 cdot 99248.0225}{10} ]

Calculating the final value:

[ E_a approx frac{0.693 cdot 8.314 cdot 99248.0225}{10} approx frac{0.693 cdot 825.76}{10} approx 57.26 text{ kJ mol}^{-1} ]

Therefore, the activation energy ( E_a ) is approximately:

[ boxed{57.26 text{ kJ mol}^{-1}} ]

Conclusion

This detailed calculation demonstrates the importance of the Arrhenius equation in understanding and predicting the behavior of reactions under different temperature conditions. The activation energy is a critical parameter that influences the rate of a reaction and is fundamental in the design and optimization of industrial processes, as well as in the study of biological and chemical systems.