Question

Will medal/fan! :)
Atoms of element A decay to atoms of element B with a half-life of 20,000 years. If there are 10,000 atoms of A to begin with (and 0 atoms of B), how long will it take for there to be 2,500 atoms of A?
A. 20,000 years
B. 40,000 years
C. 60,000 years
D. 100,000 years

Answer 1

\[A = A_{0}(1/2)^(\frac{ t }{ h}\]

Where A = the amount of sample left over
A0 is how many samples we have
h = half-life
And t = time.

we are looking for time right?

so we can actually solve this for time.

now here is the long way

Answer 2

1. \[A = A_{0}*(\frac{ 1 }{ 2 })^(\frac{ t }{ h })\]

Step #1 since we are solving for t, we need to do our best to isolate t. We can do this first by dividing both sides by A0

2. \[\frac{ A}{ A_{0} } = (\frac{ 1 }{ 2 })^\frac{ t }{ h}\]

Then we can take the log of both sides. or natural log.

Answer 3

3. \[Ln(\frac{ A }{ A_{0} }) = \frac{ t }{ h }*Ln(0.5)\]

4. rearrange

\[h*Ln(\frac{ A }{ A_{0} }) = t*Ln(0.5)\]

\[\frac{ h*Ln(\frac{ A }{ A_{0} } )}{ Ln(0.5) } =t \]

Then plug in your numbers.

\[\frac{ 20,000*\ln(\frac{ 2,500 }{ 10,000 }) }{ Ln(0.5) } = 4.0x10^{4}, years \]

That's the really long math way

This way is much easier

you know you start out with 10,000 right atoms right?
and you know that the half life is 20,000 years we can just simply re-write all this.

10,000

after 1 half life we have

5,000 atoms, 20,000 years 1 half life
2,500 atoms, 40,000 years 2 half lives

Answer 4

Oh, jeez! Thank you very much, I couldn't figure it out for the life of me! Now I at least have a bit of guidance for my other problems ;) Thank you again!

Answer 5

@lameobraino there are two ways to do this: the math way and the easy way whatever method works best for you.