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Exponential Decay

  • #1

Homework Statement



A radioactive substance has a half-life of 20 years. If 8 mg of the substance remains after
100 years, find how much of the substance was initially present.


Homework Equations



A=A0ekt

The Attempt at a Solution



I set the equation up so that 8=A0e100k and figured I could solve from there taking the ln but that gives me ln8/A0=100k... I still have two variables so I can't solve. How do I fix this? Can I plug in 20 for k?
 

Answers and Replies

  • #2
Char. Limit
Gold Member
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14
Just a thought: an easier way to do this might be to recognize that the object has gone through half-life five times, namely that the object has been halved five times. So if you multiply by 2^5, you'll get the answer you're looking for.
 
  • #3
Yes, I do realize that, but my prof will give 0 marks for an answer that doesn't use the equation.
 
  • #4
Char. Limit
Gold Member
1,204
14
I assume that k here is the decay constant. Just remember that one of the definitions for k is...

[tex]k=\frac{log(2)}{t_{\frac{1}{2}}}[/tex]

Since you know the half-life, finding the decay constant should be easy.
 
  • #5
Thanks!
 
  • #6
Char. Limit
Gold Member
1,204
14
No problem. Have a great day!
 
  • #7
HallsofIvy
Science Advisor
Homework Helper
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955
What Char Limit said is excellent: 100= 5(20) so the substance must have been halved 5 times. Work backwards- what is the opposite of "halving"?

As for your method, you have more variables than equations because you did not use all of the information- in particular you made no use of the fact that the half-life is 20 years.

If you start with any amount C, after 20 years, you will have C/2 left:
[tex]Ce^{20k}= C/2[/tex]
so
[tex]e^{20k}= \frac{1}{2}[/tex]
You could solve that for t and then use
[tex]Ce^{100k}= 8[/tex]


But using [itex]Ce^{kt}[/itex] at all is the "hard way". All "exponentials", to any base, are interchangeable.
[tex]Ca^{\alpha t}= Ce^{ln(a^{\alpha t}}= Ce^{(\alpha ln(a))}[/tex]
which is the same as [itex]Ce^{kt}[/itex] with [itex]k= \alpha ln(a)[/itex].

Knowing that the decay is exponential and that the half life is 20 years tells you that you can use
[tex]C\left(\frac{1}{2}\right)^{t/20}[/tex]
where dividing t by 20 tells you how many "20 year periods" there are in t years.

But, again, the way Char Limit suggested is simplest and best.
 
  • #8
HallsofIvy
Science Advisor
Homework Helper
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Yes, I do realize that, but my prof will give 0 marks for an answer that doesn't use the equation.
If you explain clearly what you are doing I cannot imagine any teacher giving "0 marks" for an easier solution.
 
  • #9
My prof is a psycho. He's given me a 9% on an assignment before because I didn't use the methods he liked, even if I got the correct answers.
 

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