Question

The half life a first order reaction is $10$ seconds. Calculate the time for completion of $99.9\%$ of the reaction.

Answer 1

$k=\frac{2.303}{t}log\frac{a}{a-x}$

Where $k$- rate of reaction

$t$- time taken

$a$- initial concentration

$x$- reactants those converted into products

We know that

$t_{\frac{1}{2}}=halflife=10s$

$k=\frac{0.693}{t_{50\%}}$

$k=\frac{0.693}{10}=0.0693$ --------------(1)

For $99.9$ reaction to be completed we have,

$a=100,x=99.9$

$\therefore 0.0693=\frac{2.303}{t_{99.9\%}}.log\frac{100}{100-99.9}$

$0.0693=2.303/t_{99.9\%}log\left[1000\right]$

$t_{99.9\%}=\frac{2.303\times 3}{0.0693}$ -------- $log\left[1000\right]=log\left[{10}^3\right]=3log10=3$

$t_{99.9\%}=99.6969s\approx 100s$