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Thread: Finding half life given decay rate

  1. #1
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    Finding half life given decay rate

    Hey guys, I'm working on a problem right now and I'm having some trouble.
    The decay of a certain chemical is 9.3% per year. Using the exponential decay model P(t) = P0-kt d where k is the decay rate, and P0 is the original amount of chemical find the half-life.

    Thanks!
    Last edited by jkort13; Apr 26th 2012 at 12:31 AM. Reason: found the answer, feel free to remove
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  2. #2
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    Re: Finding half life given decay rate

    Just out of interest did you get the answer 7.1 years?
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  3. #3
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    Re: Finding half life given decay rate

    Yes that is the correct answer.
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  4. #4
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    Re: Finding half life given decay rate

    I'm a bit surprized- that formula doesn't make sense. If $\displaystyle P(t)= P_0^{kt d}$, then the "initial value" is $\displaystyle P(0)= 1$, not $\displaystyle P_0$. Did you mean $\displaystyle P(t)= P_0e^{kt d}$? And if $\displaystyle P_0$ is the initial value, t is the time, and k is the decay rate, what is "d"?

    More common is the formula $\displaystyle P(t)= P_0e^{kt}$ for decay. In one year, we will have $\displaystyle P(1)= P_0e^k$. If the chemical decays 9.3% per year, P(1) should equal $\displaystyle (1- .093)P_0= .907P_0$ so $\displaystyle P_0e^k= .907P_0$. That is, k must satisfy $\displaystyle e^k= .907$ so k= ln(.907)= -.0976.

    Once you know that, the "half life" is the time until $\displaystyle P_0$ becomes $\displaystyle P_0/2$. Solve $\displaystyle P_0e^{-.0976t}= P_0/2$ which is the same as $\displaystyle e^{-.0976t}= 1/2$. Solve that for t.
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