Question

At ${380}^{\circ }C$, the half-life period for the first order decomposition of $H_{2}O$ is 360 min. The energy of activation of the reaction is 200 kJ $mo{l}^{-1}$. Calculate the time required for 75% decomposition at ${450}^{\circ }C$:

• A. $t=20.4minutes$
• B. $t=10.2minutes$
• C. $t=40.8minutes$
• D. None of these

Answer 1

Answer: (A)

$t=20.4minutes$

The rate constant at ${380}^{0}C$ is
$k=\frac{0.693}{t_{1/2}}=\frac{0.693}{360min}=1.925\times {10}^{-3}/min$
$log\left(\frac{k_2}{k_1}\right)=\frac{E}{2.303}(\frac{1}{T_1}-\frac{1}{T_2})$
$log\left(\frac{k_2}{1.925\times {10}^{-3}/min}\right)=\frac{2000\times {10}^{3}J/mol}{2.303\times 8.314J/K/mol}(\frac{1}{653K}-\frac{1}{723K})$
$k_2=6.81\times {10}^{-2}/min$
$t=-\frac{2.303}{k}log\frac{\left[A\right]}{[A{]}_0}=-\frac{2.303}{6.81\times {10}^{-2}}log\left(\frac{25}{100}\right)=20.4min$