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  4. In a first order reaction, time required for completion of 99.9 % is x times of half-life (f1/2) of the reaction, x is

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Q. In a first order reaction, time required for completion of $99.9 \%$ is $x$ times of half-life $ (f_{1/2}) $ of the reaction, $x$ is

Chemical Kinetics

Solution:

When reaction is completed $99.9 %, $
$\left[R\right]_{n} =\left[R\right]_{0} - 0.999\left[R\right]_{0} $
$k=\frac{2.303}{t} log \frac{\left[R\right]_{0}}{\left[R\right]}=\frac{2.303}{t } log\frac{ \left[R\right]_{0}}{\left[R\right]_{0}-0.999\left[R\right]_{0}} $
$=\frac{2.303}{t} log 10^{3} \Rightarrow t = 6.909 k $
For half-life of the reaction,$ t_{12} =0.693/k$
$ \frac{t}{t 12} log 10^{3}\Rightarrow t =6.909/k $