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Thread: Radio active decay

  1. #1
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    Radio active decay

    Radioactive substance has a half life of 18 years. Decay rate remains constant how long will it before 95 percent of the sample has decayed?

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  2. #2
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    Hello, av8or91!

    A radioactive substance has a half life of 18 years. Decay rate remains constant.
    How long will it be before 95 percent of the sample has decayed?
    The half-life function is: .$\displaystyle A \;=\;A_o\,e^{-kt}$
    . . where $\displaystyle A_o$ is the intial amount and .$\displaystyle k \:=\:\frac{\ln(2)}{\text{half-life}} $

    Our function is: .$\displaystyle A \;=\;A_o\,e^{-\frac{\ln 2}{18}t} $


    If 95% of the sample has decayed, then: .$\displaystyle A \,=\,0.05A_o$

    We have: .$\displaystyle A_oe^{-\frac{\ln2}{18}t} \:=\:0.05A_o \quad\Rightarrow\quad e^{-\frac{\ln2}{18}t} \:=\:0.05 $

    Take logs: .$\displaystyle \ln\left(e^{-\frac{\ln2}{18}t}\right) \:=\:\ln(0.05) \quad\Rightarrow\quad -\tfrac{\ln2}{18}t\ln(e) \:=\:\ln(0.05)$

    . . $\displaystyle t \:=\:\frac{18\ln(0.06)}{-\ln2} \;=\; 77.79470571 \;\approx\;77.8\text{ years}$

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