What is half life of first order reaction if time required to dec…

Question

What is half life of first order reaction if time required to decrease concentration of reactants from $0.8\ M$ to $0.2\ M$ is $12$ hours?

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Answer

Given: $[ A ]_0=0.8 M ,[ A ]_{ t }=0.2 M , t =12$ hours
To find: Half life of reaction $\left( t _{1 / 2}\right)$
Formulae:
1. $k =\frac{2.303}{ t } \log _{10} \frac{[ A ]_0}{[ A ]_{ t }}$
2. $t _{\frac{1}{2}}=\frac{0.693}{ k }$
Calculation: Substituting given value in
$ k =\frac{2.303}{ t } \log _{10} \frac{[ A ]_0}{[ A ]_{ t }}$
$k =\frac{2.303}{12 hr } \log _{10} \frac{0.8}{0.2}$
$=\frac{2.303}{12 hr } \log _{10}(4)$
$=\frac{2.303}{12 hr } \times 0.6020$
$=\text { Antilog }_{10}\left(\log _{10} 2.303+\log _{10} 0.6020-\log _{10} 12\right)$
$=\text { Antilog }_{10}(0.3623+\overline{1} .7796-1.0792)$
$=\text { Antilog }_{10}(\overline{1} .0627)$
$=0.1115 hr ^{-1}$
$t _{\frac{1}{2}}=\frac{0.693}{ k }=\frac{0.693}{0.1155 hr ^{-1}}=6 hr $

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