Question

A certain radioactive material has a one-fourth of its life in 4000 years. Find the time required for a given amount to become one-tenth of its original mass.

Answer 1

Let the equation be $m=m_0{e}^{-\alpha t}$

A certain radioactive material has a one-fourth of its life in 4000 years.

So $\frac{m_0}{4}=m_0{e}^{-4000\alpha }$

$\frac{1}{4}=e^{-4000\alpha }$

Applying log both sides

$\log \left(\frac{1}{4}\right)=\log \left(e^{-4000\alpha }\right)$

$-2\log 2=-4000\alpha$

$\alpha =\frac{\log 2}{2000}$ (1)

Now time required for a given amount to be one-tenth of original mass can be found by:

$\frac{m_0}{10}=m_0{e}^{-\alpha t}$

$\frac{1}{10}=e^{-\alpha t}$

Applying log both sides

$\log \left(\frac{1}{10}\right)=\log \left(e^{-\alpha t}\right)$

$-\log \left(10\right)=-\alpha t$

$\log \left(10\right)=\alpha t$

Substituting (1)

$\log \left(10\right)=\frac{\log \left(2\right)}{2000}t$

$t\approx 6646.6$ years