Question

The half-life of a first order reaction is $900$ min at $820K$. Estimate its half-life at $720$K if the energy of activation of the reaction is $250kJmo{l}^{-1}$.

Answer 1

From Arrhenius equation

$lo{g}_{10}\frac{t_{1/2}}{t_{1/2}^{\prime }}=lo{g}_{10}\frac{k^{\prime }}{k}=\frac{E_a}{2.303R}\left[\frac{T^{\prime }-T}{T{T}^{\prime }}\right]$

$lo{g}_{10}\frac{900min}{t_{1/2}^{\prime }}=lo{g}_{10}\frac{k^{\prime }}{k}=\frac{250kJ/mol\times 1000J/kJ}{2.303\times 8.314J/mol/K}\times \left[\frac{720K-820K}{820K\times 720K}\right]$

$lo{g}_{10}\frac{900min}{t_{1/2}^{\prime }}=13057\times [-1.694\times {10}^{-4}]$

$lo{g}_{10}\frac{900min}{t_{1/2}^{\prime }}=-2.212$

$\frac{900min}{t_{1/2}^{\prime }}=0.006145$

$t_{1/2}^{\prime }=146473min$

The half life at 720 K is 146473 min.