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^{(-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^{(-t)/(432y)}`

`(2.7g)/(10g)=2^{(-t)/(432y)}`

`0.27g=2^{(-t)/(432y)}`

Applying natural logarithm in both sides:

`ln(0.27g)=ln(2^{(-t)/(432y)})`

`-1.309=-(t)/(432y)ln(2)`

`-t=((-1.309)(432y))/(0.693)`

Finally:

`t=816y`