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Thread: Half-life

  1. #1
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    Question Half-life

    Radium-221 has a half-life of 30 seconds. How long will it take for 85% of a sample to decay?


    Ok so I really don't know what to do here. I know I can find r because its $\displaystyle r=ln(2)/30$ but what about the 85%? Any help is appreciated.
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by n8thatsme View Post
    Radium-221 has a half-life of 30 seconds. How long will it take for 85% of a sample to decay?


    Ok so I really don't know what to do here. I know I can find r because its $\displaystyle r=ln(2)/30$ but what about the 85%? Any help is appreciated.
    Let $\displaystyle N_0$ be the initial amount of Radium-221, and let $\displaystyle N$ be the current amount of Radium.

    Since 85% decayed, that means that the current amount is 15% of the original amount.

    Thus, this implies that $\displaystyle N=.15N_0\implies \frac{N}{N_0}=.15$

    Since the decay is modeled by $\displaystyle N=N_0e^{-kt}$, we see now that $\displaystyle .15 = e^{-kt}$

    Now, k can be found using the half-life formula: $\displaystyle \lambda=\frac{\ln2}{k}\implies k=\frac{\ln 2}{\lambda}$, where $\displaystyle \lambda$ is the half-life.

    So our equation now has the form $\displaystyle 0.15=e^{-\frac{\ln2}{\lambda}t}$

    Can you take it from here?

    --Chris
    Last edited by Chris L T521; Oct 3rd 2008 at 11:20 PM.
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