The rate constants of a reaction are 1 × 10−3 s−1 and 2 × 10−3 s−1 at 27 °C and 37 °C respectively. What is the activation energy of the reaction?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 53.6 kJ mol−1
The Arrhenius equation written for two temperatures relates the two rate constants to the activation energy Ea:
\( \displaystyle \log\frac{k_2}{k_1}=\frac{E_a}{2.303R}\left[\frac{T_2-T_1}{T_1T_2}\right] \)
Data:
\( \displaystyle T_1=27+273=300\,\mathrm{K},\quad T_2=37+273=310\,\mathrm{K} \)
\( \displaystyle \frac{k_2}{k_1}=\frac{2\times10^{-3}\,\mathrm{s^{-1}}}{1\times10^{-3}\,\mathrm{s^{-1}}}=2,\quad R=8.314\,\mathrm{J\,K^{-1}\,mol^{-1}} \)
Rearranging for Ea and substituting:
\( \displaystyle E_a=2.303R\left[\frac{T_1T_2}{T_2-T_1}\right]\log\frac{k_2}{k_1} \)
\( \displaystyle E_a=2.303\times8.314\times\frac{300\times310}{310-300}\times\log2 \)
\( \displaystyle E_a=19.147\times9300\times0.3010 \)
\( \displaystyle E_a=53600\,\mathrm{J\,mol^{-1}}=53.6\,\mathrm{kJ\,mol^{-1}} \)