The rate of a reaction quadruples when temperature changes from $27^{\circ} ext{C}$ to $57^{\circ} ext{C}$.

Question

The rate of a reaction quadruples when temperature changes from $27^{\circ}	ext{C}$ to $57^{\circ}	ext{C}$. Calculate the energy of activation. Given $R = 8.314 	ext{ J K}^{–1} 	ext{ mol}^{–1}$, $\log 4 = 0.6021$

Accepted Answer

According to the Arrhenius equation,
$\log \frac{k_2}{k_1} = \frac{E_a}{2.303 R} \left[ \frac{T_2 - T_1}{T_1 T_2} ight]$
Given:
$T_1 = 27^{\circ}	ext{C} = 27 + 273 = 300 	ext{ K}$
$T_2 = 57^{\circ}	ext{C} = 57 + 273 = 330 	ext{ K}$
$\frac{k_2}{k_1} = 4$ (since the rate quadruples)
$\log 4 = \frac{E_a}{2.303 	imes 8.314} \left[ \frac{330 - 300}{300 	imes 330} ight]$
$0.6021 = \frac{E_a}{19.147} \left[ \frac{30}{99000} ight]$
$0.6021 = \frac{E_a}{19.147} 	imes \frac{1}{3300}$
$E_a = 0.6021 	imes 19.147 	imes 3300$
$E_a = 38044.1 	ext{ J mol}^{-1} \approx 38.04 	ext{ kJ mol}^{-1}$.

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