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Math Help - finding the half life of a radioactive element

  1. #1
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    Exclamation finding the half life of a radioactive element

    Hi,


    Find the half-life of a radioactive element that decays according to the rule:
    dA
    dt = −0.012 A
    where A is the amount in kg present after t years.


    I know the equation is A=A0e^(-0.012t), but how do you find the half life?
    please help, thanks.
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  2. #2
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    Quote Originally Posted by [nuthing] View Post
    Hi,


    Find the half-life of a radioactive element that decays according to the rule:
    dA
    dt = −0.012 A
    where A is the amount in kg present after t years.

    I know the equation is A=A0e^(-0.012t), but how do you find the half life?
    please help, thanks.
    Substitute A = A0/2, simplify and then solve for t.
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  3. #3
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    That's probably the simplest way to do it. You can also argue that if "T" is the half life, then in time "t" there are t/T "half lives" so the quantity is multiplied by 1/2 t/T times: A= A_02^{t/T}. Since A= A_0e^{-0.012t} we have A_0 2^{t/T}= A_0e^{-0.012t} which immediately gives
    2^{t/T}= \left(2^{1/T}\right)t= e^{-0.012t}= \left(e^{-0.012}\right)^t
    so that 2^{1/T}= e^{-0.012} and take the logarithm of both sides as The Second Solution suggested.

    (Where, oh, where, has my LaTeX gone?)
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  4. #4
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    Alternatively half life is related to the decay constant (k = 0.012):

    t_{1/2} = ln(2)/|k|
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