How to determine half life $T_{1/2}$ in terms of the decay constant $k$ and then solve for $P(t)$ in terms of $T_{1/2}$?

Question

I am trying to figure out a radioactive decay problem, and I am a little stuck. I have the decay equation $$\frac{\mathrm{d}P}{\mathrm{d}t} = -kP$$ And I am trying to determine the half life $T_{1/2}$ in terms of the decay constant $k$, and then solve for $P(t)$ in terms of $T_{1/2}$

However I dont really know how to go about this...

Any help would be much appreciated Cheers

Answer 1

$$\frac{\mathrm{d}P}{\mathrm{d}t} = -kP \implies$$

$$P(t)=p(0)e^{-kt}$$ and the half life is found to be $$T_{1/2} = \ln 2/k$$

Thus $$ k= \ln 2/T_{1/2}$$ $$ P(t)=p(0)e^{-kt} =p(0)2^{-\frac {t}{T_{1/2}}}$$