Question

The velocity constants of a reaction are 0.02 s\(^{-1}\) and 0.07 s\(^{-1}\) at 500 K and 700 K, respectively. Calculate the value of \( E_a \).

Hint:

The activation energy \( E_a \) can be estimated from temperature dependence using the Arrhenius equation.

Answer 1

Step 1: Using Arrhenius Equation The Arrhenius equation: \[ \ln k_2 - \ln k_1 = \frac{E_a}{R} \left( \frac{1}{T_1} - \frac{1}{T_2} \right) \] Given: \[ k_1 = 0.02, \quad k_2 = 0.07, \quad T_1 = 500K, \quad T_2 = 700K, \quad R = 8.314 \text{ J mol}^{-1} K^{-1} \] Step 2: Substituting Values \[ \ln \left( \frac{0.07}{0.02} \right) = \frac{E_a}{8.314} \left( \frac{1}{500} - \frac{1}{700} \right) \] \[ \ln (3.5) = \frac{E_a}{8.314} \times 0.000571 \] \[ 1.25 = \frac{E_a \times 0.000571}{8.314} \] \[ E_a = \frac{1.25 \times 8.314}{0.000571} = 18.2 \text{ kJ mol}^{-1} \]