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Q.
In a first order reaction, a substance is reduced to $\frac{1}{3^{rd}}$ in $100$ seconds. In how much time will it be reduced to $\frac{1}{9^{th}}$ ?
NTA AbhyasNTA Abhyas 2020
Solution:
According to first order kinetic equation,
$K=\frac{2 . 303}{t}\left(log\right)_{10}\left(\frac{a}{a - x}\right)$
where $a=$ initial concentration
$x=$ change in the concent-
ration after time $'t'$
$k=\frac{2 . 303}{100}log_{10}\left[\frac{a}{\frac{a}{3}}\right]$
$k=\frac{2 . 303}{100}\left(log\right)_{10}3...\left(1\right)$
$\frac{1}{9}$ part will be left at time t
$K=\frac{2 . 303}{t}\left(log\right)_{\left(\right. 10 \left.\right)}\left[\frac{a}{\frac{a}{9}}\right]$
$K=\frac{2 . 303}{t}log_{10}9$
$K=\frac{2 . 303}{100}log3^{2}$
$=\frac{2 . 303}{100}\cdot 2log3$
From equations $\left(\right.i\left.\right)$ and $\left(\right.ii\left.\right)$
$\frac{2 . 303}{100}log_{10}3=\frac{2 . 303}{t}log_{10}3^{2}$
$\frac{1}{100}=\frac{2}{t};t=200sec$