Question

The radioactive element americium-241 has a half-life of 432 years. How many years will it take a 10 gram mass of americium-241 to decay to 2.7 grams.

Answer 1

Answer: 816 years

This problem can be solved using the Radioactive Half Life Formula:

$A=A_{o}.2^{\frac{-t}{h}}$   (1)

Where:

$A=2.7g$ is the final amount of the material

$A_{o}=10g$ is the initial amount of the material

$t$ is the time elapsed  (the quantity we are asked to find)

$h=432y$ is the half life of americium-241

Knowing this, let's find $t$ from (1):

$2.7g=(10g).2^{\frac{-t}{432y}}$  

$\frac{2.7g}{10g}=2^{\frac{-t}{432y}}$  

$0.27g=2^{\frac{-t}{432y}}$  

Applying natural logarithm in both sides:

$ln(0.27g)=ln(2^{\frac{-t}{432y}})$  

$-1.309=-\frac{t}{432y}ln(2)$  

$-t=\frac{(-1.309)(432y)}{0.693}$  

Finally:

$t=816y$