Question

The rate of a chemical reaction doubles when the temperature is raised from 298 K to 308 K. Calculate the activation energy (\(E_a\)) for this reaction assuming it does not change with temperature. (Given: \( R = 8.314 \, \text{J mol}^{-1}\text{K}^{-1} \), \( \log 2 = 0.30 \))

Hint:

If rate doubles with 10 K rise → \(E_a\) usually ~50–60 kJ/mol. Use Arrhenius logarithmic form for temperature change problems.

Answer 1

Step 1: Use the Arrhenius Equation.
The Arrhenius equation relates the rate constant (\(k\)) of a reaction to the temperature and activation energy (\(E_a\)):
\[ k = A e^{-\frac{E_a}{RT}} \] where:
- \( k \) = rate constant,
- \( A \) = pre-exponential factor,
- \( E_a \) = activation energy,
- \( R \) = universal gas constant (8.314 J/mol·K),
- \( T \) = temperature in Kelvin.

Step 2: Use the Integrated Form of the Arrhenius Equation.
We are given that the rate of the reaction doubles when the temperature increases from 298 K to 308 K. To calculate the activation energy, we can use the following form of the Arrhenius equation:
\[ \frac{k_2}{k_1} = e^{\frac{E_a}{R} \left( \frac{1}{T_1} - \frac{1}{T_2} \right)} \] where: - \( k_2 \) and \( k_1 \) are the rate constants at temperatures \( T_2 \) and \( T_1 \), respectively,
- \( T_1 = 298 \, \text{K} \) and \( T_2 = 308 \, \text{K} \).
We are also given that \( k_2 = 2k_1 \), so:
\[ 2 = e^{\frac{E_a}{8.314} \left( \frac{1}{298} - \frac{1}{308} \right)} \]

Step 3: Solve for Activation Energy.
Taking the natural logarithm of both sides:
\[ \log 2 = \frac{E_a}{8.314} \left( \frac{1}{298} - \frac{1}{308} \right) \] Substitute \( \log 2 = 0.30 \) and calculate the difference in reciprocals of temperatures:
\[ \frac{1}{298} - \frac{1}{308} = \frac{308 - 298}{298 \times 308} = \frac{10}{91704} \approx 1.091 \times 10^{-4} \] Now substitute these values into the equation:
\[ 0.30 = \frac{E_a}{8.314} \times 1.091 \times 10^{-4} \] Solving for \( E_a \):
\[ E_a = \frac{0.30 \times 8.314}{1.091 \times 10^{-4}} \approx 53 \, \text{kJ/mol} \]

Conclusion.
The activation energy (\(E_a\)) for this reaction is approximately 53 kJ/mol.