Radium has a half-life of 1,660 years.The number of years until only 1/16th of a sample remains is 6641 years.
Answer: 6641 years
Explanation:
Given that the half-life of Radium is 1660 years. The number of years to be calculated when the sample of radium could just be remained of 1/16th of its part, which can be calculated by the following means,
Half life, [tex]T_{1 / 2}=1660 \text { years }[/tex]
[tex]\frac{N}{N_{0}}=\frac{1}{16}[/tex]
t is the taken time, then,
[tex]\frac{N}{N_{0}}=e^{-\lambda t}[/tex]
[tex]\frac{N}{N_{0}}=e^{\lambda t}[/tex]
Taking ln on both sides, we get,
[tex]\lambda t=\ln \left(\frac{N}{N_{0}}\right)[/tex]
[tex]t=\frac{1}{\lambda} \ln \left(\frac{N_{0}}{N}\right)[/tex]
[tex]=\frac{T_{1 / 2}}{0.693}\left(\ln \left(\frac{N_{0}}{N}\right)\right)[/tex]
By substituting the given values,
[tex]t = \frac{1660}{0.693}(\ln 16)[/tex]
Therefore, t = 6641 years
