5.2.5 Finding Activation Energy

Arrhenius Equation and Activation Energy

The Arrhenius equation links the rate constant $k$ to temperature $T$ and provides a way to determine the activation energy $E_a$, which is the minimum energy required for a reaction to occur:

$$k = A e^{-\frac{E_a}{RT}}$$

where:

• $A$ is the Arrhenius constant (or pre-exponential factor), representing the frequency of collisions with proper orientation.
• $E_a$ is the activation energy (J mol$⁻¹$), the minimum energy required for the reaction to occur.
• $R$ is the gas constant (8.31 J K$⁻¹$ mol$⁻¹$).
• $T$ is the temperature in Kelvin.

Rearranging the Arrhenius Equation for Graphical Analysis

To find $E_a$ experimentally, we rearrange the Arrhenius equation into a form that allows us to plot a straight-line graph:

$$\ln k = -\frac{E_a}{R} \cdot \frac{1}{T} + \ln A$$

This equation is in the form $y = mx + c$

Where:

• $y = \ln k$
• $x = \frac{1}{T}$
• Slope $m = -\frac{E_a}{R}$
• y-intercept $c = \ln A$

Plotting to Find Activation Energy

1. Measure $k$ at Different Temperatures: Obtain rate constants k by conducting experiments at different temperatures and calculate $\ln k$ for each value.
2. Plot $\ln k$ Against $\frac{1}{T}$: Plotting $\ln k$ (y-axis) versus $\frac{1}{T}$ (x-axis) should produce a straight line.
3. Calculate the Gradient: The gradient of this line is $-\frac{E_a}{R}$. Use this relationship to calculate $E_a$ by rearranging:

$$E_a = -\text{gradient} \times R$$

Example Calculation

Suppose the gradient of the $\ln k$ versus $\frac{1}{T}$ graph is -5000. Then:

$$E_a = -(-5000) \times 8.31 = 41550 \, \text{J mol}^{-1} = 41.55 \, \text{kJ mol}^{-1}$$

Significance of Activation Energy

Activation energy reveals the energy barrier for a reaction:

• Higher $E_a$ means a greater energy requirement for reactants to convert to products, leading to a slower reaction.
• Lower $E_a$ suggests a faster reaction as less energy is needed to reach the transition state. By using the Arrhenius equation and plotting $\ln k$ against $\frac{1}{T}$, you can determine $E_a$ experimentally and gain insights into reaction kinetics and temperature dependence.