# Thread: finding the half life of a radioactive element

1. ## 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.

2. Originally Posted by [nuthing]
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.

3. 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?)

4. Alternatively half life is related to the decay constant (k = 0.012):

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