Calculate Activation Energy from Arrhenius Rate Constant Equation

The given rate constant equation for the decomposition of a hydrocarbon is: $k = (5.5 \times 10^{11}\text{ s}^{-1}) e^{-\frac{28000\text{ K}}{T}}$

According to the Arrhenius equation, the temperature dependence of the rate constant is represented as: $k = A e^{-\frac{E_a}{RT}}$

where:

• $A$ is the pre-exponential factor (frequency factor),
• $E_a$ is the activation energy of the reaction,
• $R$ is the universal gas constant ($8.3\text{ J K}^{-1}\text{mol}^{-1}$),
• $T$ is the absolute temperature in Kelvin ($\text{K}$).

By comparing the exponential terms of both equations, we have: $-\frac{E_a}{RT} = -\frac{28000\text{ K}}{T}$

Canceling $-T$ from both sides: $\frac{E_a}{R} = 28000\text{ K}$

Rearranging to solve for $E_a$: $E_a = 28000\text{ K} \times R$

Substitute the value of $R = 8.3\text{ J K}^{-1}\text{mol}^{-1}$: $E_a = 28000 \times 8.3\text{ J mol}^{-1}$ $E_a = 232400\text{ J mol}^{-1}$

To convert the activation energy into $\text{kJ mol}^{-1}$: $E_a = \frac{232400}{1000}\text{ kJ mol}^{-1} = 232.4\text{ kJ mol}^{-1}$

Rounding off to the nearest integer gives: $E_a \approx 232\text{ kJ mol}^{-1}$