Question

The element AoPSium has a half-life of 120 years. After a certain sample of AoPSium was stored for 600 years, only 7 grams was left. How many grams were in the original sample?

Answer 1

Initial amount of the element taken was 224 grams.

Explanation:

Half life of the element AoPSium is 120 years.

We know the radioactive decay is represented by the formula

`A_(t)=A_(0)(e)^(-kt)`

where A(t) = Element remaining after time t

A(0) = Initial amount

t = time in years

k = decay constant

`(A_(0) )/(2)=A_(0)(e)^(-k* 120)`

`(1)/(2)=e^(-120k)`

Now we take natural log on both the sides of the equation

`ln((1)/(2))=ln[(e)^(-120k)]`

-0.69314 = - 120k

k =
`(0.69314)/(120)`

k = 0.005776

Now we have to calculate the amount taken of the element after 600 years when amount remaining was 7 grams.

`A_(t)=A_(0)(e)^(-kt)`

7 =
`A_(0)(e)^(-(0.0057762* 600))`

7 =
`A_(0)(e)^(-3.4657)`

`A_(0)=7* e^((3.4657))`

`A_(0)=31.99885* 7`

`A_(0)=224` grams

Therefore, Initial amount of the element taken was 224 grams.