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TFM
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Homework Statement
Unobtainium is a radioactive (and fictional) element, with its rate of decay being proportional to the amount of Unobtainium, [tex] u' = −k u [/tex]. The half-life of Unobtainium, in other words the time it takes for half the initial amount to decay, is [tex] 2.3 × 10^9 [/tex]yr.
(a) Find the value of k.
(b) How many decay events will there be per second in a 1 kg block of Unobtainium, given
that it has an atomic mass of 415?
Homework Equations
N/A
The Attempt at a Solution
I think I am doing the right thing, but I can't seem to work out k. I have a feeling I need limits to me integrals. so far I have:
[tex] \frac{du}{dt} = -ku [/tex]
[tex] \frac{1}{u} dt = -k dt [/tex]
[tex] ln u = -kt + C [/tex]
Where C is a constant of integration
taking exponentials of both sides gives:
[tex] u = e^{-kt} + C [/tex]
If we had a [tex] u_{0} [/tex] somewhere, we know the half life, and thus could say that [tex]u = \frac{1}{2} u_o [/tex], stick some values for u and u0 in, and thus could probably work out k.
Any ideas where I am going wrong?
TFM