The rate constant of a reaction


The rate constant of a reaction increases by five times on increase in temperature from $27^{\circ} \mathrm{C}$ to $52^{\circ} \mathrm{C}$. The value of activation energy in $\mathrm{kJ} \mathrm{mol}^{-1}$ is_________ (Rounded-off to the nearest integer) $\left[\mathrm{R}=8.314 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right]$


$\mathrm{T}_{1}=300 \mathrm{~K}, \mathrm{~T}_{2}=325 \mathrm{~K}, \mathrm{~K}_{2}=5 \mathrm{~K}$

$\ln \frac{\mathrm{K}_{2}}{\mathrm{~K}_{1}}=\frac{\mathrm{E} \mathrm{a}}{\mathrm{R}}\left[\frac{1}{\mathrm{~T}_{1}}-\frac{1}{\mathrm{~T}_{2}}\right]$

or, $\ln 5=\frac{\mathrm{Ea}}{8.314}\left[\frac{1}{300}-\frac{1}{325}\right]$

or, $\mathrm{Ea}=0.7 \times 2.303 \times 8.314 \times 12 \times 325$

$=52271 \mathrm{~J}=52.271 \mathrm{~kJ}$

Nearest integer answer will be $52 \mathrm{~kJ}$

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