# Question 406c5

##### 1 Answer
Mar 26, 2014

The half-life of the isotope is 2.5 min.

The number of half-lives n = t/t_½#

For each half-life, you divide the total amount of the isotope by 2.

Amount remaining = original amount × $\frac{1}{2} ^ n$ or

$A = {A}_{0} / {2}^{n}$

${A}_{0} / A = {2}^{n}$

$\ln \left({A}_{0} / A\right) = n \ln 2$

$n = \frac{\ln \left({A}_{0} / A\right)}{\ln} 2$

$n = \frac{\ln \left(\left(100 \text{ mg")/(1.50" mg}\right)\right)}{\ln} 2 = \ln \frac{66.7}{\ln} 2 = \frac{4.20}{0.693}$ = 6.06

So 15 min = 6.06 half-lives

1 half-life = $\frac{15 \min}{6.06}$ = 2.5 min