Question

The rate constant for a first order reaction is 0.58 s^-1 at 300 K and 0.026 s^-1 at 290 K. What is the energy of activation? (R = 8.314 J K^-1 mol^-1)

Options

• 124.48 kJ

• 224.55 kJ

• 348.18 kJ

• 513.21 kJ

Answer 1

224.55 kJ

Explanation:

\[\log_{10}\frac{\mathrm{k}_{2}}{\mathrm{k}_{1}}=\frac{\mathrm{E}_{\mathrm{a}}}{2.303\mathrm{R}}\left(\frac{\mathrm{T}_{2}-\mathrm{T}_{1}}{\mathrm{T}_{1}\mathrm{T}_{2}}\right)\]

Substituting the given values

\[\begin{aligned} \log_{10}\frac{0.58}{0.026}= & \frac{\mathrm{E}_{\mathrm{a}}}{2.303\times8.314\mathrm{J}\mathrm{K}^{-1}\mathrm{mol}^{-1}} \\ & \times\left[\frac{300\mathrm{K}-290\mathrm{K}}{290\mathrm{K}\times300\mathrm{K}}\right] \end{aligned}\]

\[\log_{10}22.307=\frac{\mathrm{E_a}}{2.303\times8.314\mathrm{Jmol^{-1}}}\times\frac{10}{290\times300}\]

\[1.3484=\frac{\mathrm{E_a}}{19.15\mathrm{J~mol}^{-1}}\times1.15\times10^{-4}\]

\[\mathrm{E_a}=\frac{1.3484\times19.15}{1.15\times10^{-4}}\mathrm{Jmol}^{-1}\]

\[=224538\mathrm{Jmol}^{-1}\approx224.55\mathrm{kJmol}^{-1}\]