# The half-life of iodine-131 is 7.2 days. How long will it take for a sample of this substance to decay to 30% of its original amount?

Dec 11, 2017

To decay 30% of original amount it will take $12.51$ days.

#### Explanation:

Half life of iodine -131 is $t = 7.2$ days

We know $p \left(t\right) = p \left(0\right) \cdot {e}^{k t} \mathmr{and} {e}^{k t} = \frac{p \left(t\right)}{p \left(0\right)} = \frac{1}{2} = 0.5$

Taking natural log on both sides we get,

$k t = \ln \left(0.5\right) \mathmr{and} 7.2 k = \ln \left(0.5\right) \mathmr{and} k = \ln \frac{0.5}{7.2} \approx - 0.09627$

When $\frac{p \left({t}_{0.3}\right)}{p \left(0\right)} = 0.3 \therefore {e}^{k {t}_{0.3}} = \frac{p \left({t}_{0.3}\right)}{p \left(0\right)} = 0.3$ or

$k \cdot {t}_{0.3} = \ln \left(0.3\right) \therefore 0.09627 \cdot {t}_{0.3} = \ln \left(0.3\right)$ or

${t}_{0.3} = \ln \frac{0.3}{-} 0.09627 \approx 12.51 \left(2 \mathrm{dp}\right)$ days

To decay 30% of original amount it will take $12.51$days. [Ans]