# If a scientist purifies 10 gram of radium-226, how many years must pass before only 0.50 gram of the original radium-226 sample remains unchanged?

Dec 26, 2017

The number of years is $= 6911 \text{years}$

#### Explanation:

The half life of $\text{Radium} - 226$ is ${t}_{1 / 2} = 1599 \text{years}$

The radioactive constant is $\lambda = \ln \frac{2}{t} _ \left(1 / 2\right)$

The equation for radioactive decay is

$\frac{N \left(t\right)}{{N}_{0}} = {e}^{- \lambda t}$

Therefore,

${e}^{- \lambda t} = \frac{N \left(t\right)}{N} _ 0 = \frac{0.5}{10} = 0.05$.

Taking natural logs of both sides,

$- \lambda t = \ln \left(0.05\right)$

$t = - \frac{\ln 0.05}{\lambda}$

$= - \ln \frac{0.05}{\ln 2 / {t}_{1 / 2}}$

$= - \ln \frac{0.05}{\ln} 2 \cdot {t}_{1 / 2}$

$= - \ln \frac{0.05}{\ln} 2 \cdot \text{1599 years}$

$=$ $\text{ 6911years}$