# An isotope has a half life of 90 days. How many days will it take for a 5 mg sample to decay to 1 mg?

Jun 8, 2017

The number of days is $= 209 \mathrm{da} y s$

#### Explanation:

The radioactive decay is represented by the equation

$M \left(t\right) = M \left(0\right) {e}^{- \lambda t}$

The radioactive constant is

$\lambda = \ln \frac{2}{t} _ \left(\frac{1}{2}\right)$

Here, the half life is

${t}_{\frac{1}{2}} = 90 d$

$\lambda = \ln \frac{2}{90} = 7.7 \cdot {10}^{-} 3$

The initial mass is $M \left(0\right) = 5 m g$

The final mass is $m \left(t\right) = 1 m g$

$1 = 5 {e}^{- {7.710}^{- 3} t}$

${e}^{- {7.710}^{-} 3} t = \frac{1}{5} = 0.2$

$- {7.710}^{-} 3 t = \ln 0.2 = - 1.609$

$t = \frac{1.609}{7.710} ^ - 3$

$= 209 d$