Question

In the Arrhenius equation for a certain reaction, the value of $A$ and $E$ (energy of activation) are $4\times {10}^{-13}se{c}^{-1}$ and $98.6kJ$ $mo{l}^{-1}$ respectively. If the reaction is of the first order, the temperature at which its half-life period will be $10$ minutes in $K$ will be $x$, then find the value of $x/6$ to the nearest integer.

Answer 1

Given: $A=4\times {10}^{13}se{c}^{-1}$

$E_a=98.6\times {10}^{3}Jmo{l}^{-1},t_{1/2}=10\times 60sec$

$\because K=A{e}^{-E_a/RT}$

$\therefore K=4\times {10}^{-13}\times e^{-\left[\right(98.6\times {10}^3\left)/\right(8.314\times T\left)\right]}$

$\because$ For 1st order reaction $K=\left(0.693/t_{1/2}\right)=\left(0.693/600\right)se{c}^{-1}$

$\therefore \left(0.693/600\right)=4\times {10}^{-13}\times e^{-\left[\right(98.6\times {10}^3\left)/\right(8.314\times T\left)\right]}$

Take log both side-

$log\left(0.693/600\right)=log(4\times {10}^{-13})-\frac{(98.6\times {10}^3)}{(8.314\times 2.303T)}$

$\therefore T=-415.3K$
$x/6=70$

Answer is 70.