Question
The time taken for $$90\% $$ of a first order reaction to complete is approximately
A.
1.1 times that of the half-life
B.
2.2 times that of the half-life
C.
3.3 times that of the half-life
D.
4.4 times that of the half-life
Answer :
3.3 times that of the half-life
Solution :
$$\eqalign{
& {t_{90\% }} = \frac{{2.303}}{k}\log \frac{1}{{1 - 0.9}} \cr
& \,\,\,\,\,\,\,\,\,\,\, = \frac{{2.303}}{k}\log 10 = \frac{{2.303}}{k} \cr
& {\text{As}}\,\,{t_{\frac{1}{2}}} = \frac{{0.693}}{k} \cr} $$
$$\,\,\,\,\,\frac{{{t_{90\% }}}}{{{t_{\frac{1}{2}}}}} = \frac{{2.303}}{k} \times \frac{k}{{0.693}}$$ $$ = 3.3 \Rightarrow {t_{90\% }} = 3.3\,{t_{\frac{1}{2}}}$$