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

Answer:

Initial amount of the element taken was 224 grams.

Step-by-step 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

$\frac{A_{0} }{2}=A_{0}(e)^{-k\times 120}$

$\frac{1}{2}=e^{-120k}$

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

$ln(\frac{1}{2})=ln[(e)^{-120k}]$

-0.69314 = - 120k

k = $\frac{0.69314}{120}$

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 = $A_{0}(e)^{-(0.0057762\times 600)}$

7 = $A_{0}(e)^{-3.4657}$

$A_{0}=7\times e^{(3.4657)}$

$A_{0}=31.99885\times 7$

$A_{0}=224$ grams

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