Question

A first order reaction follows the equation \(k = (4 \times 10^{10}\ \text{s}^{-1}) e^{-2000/T}\). The value of \(E_a\) for the reaction is about (\(R = 8.314\ \text{J K}^{-1}\text{ mol}^{-1}\))

• A. $17.4\ \text{kJ mol}^{-1}$
• B. $18.5\ \text{kJ mol}^{-1}$
• C. $17.5\ \text{kJ mol}^{-1}$
• D. $16.6\ \text{kJ mol}^{-1}$
• E. $17.8\ \text{kJ mol}^{-1}$

Hint:

From Arrhenius form: - Compare exponent directly - $E_a = (\text{given constant}) \times R$

Answer 1

The Correct Option is D

Concept: Arrhenius equation: \[ k = A e^{-E_a/RT} \] Comparing with given equation: \[ k = 4 \times 10^{10} e^{-2000/T} \]

Step 1: Compare exponents.
\[ \frac{E_a}{RT} = \frac{2000}{T} \]

Step 2: Cancel $T$.
\[ \frac{E_a}{R} = 2000 \]

Step 3: Calculate $E_a$.
\[ E_a = 2000 \times R = 2000 \times 8.314 \] \[ E_a = 16628\ \text{J mol}^{-1} \approx 16.6\ \text{kJ mol}^{-1} \]