1. ## half-life

3) Californium-252 has a half-life of 2.645 years. If the equation describing the decay of the readioactive nuclei is given by $\displaystyle y=y_0e^{kt}$, find:
a) the value of k
b) how long it will take for 95% of the radioactive nuclei to disintegrate.

Please offer a step-by-step solution; I don't know how to do half-lifes really, but need to for my exam.

2. Originally Posted by jangalinn
3) Californium-252 has a half-life of 2.645 years. If the equation describing the decay of the readioactive nuclei is given by $\displaystyle y=y_0e^{kt}$, find:
a) the value of k
b) how long it will take for 95% of the radioactive nuclei to disintegrate.

Please offer a step-by-step solution; I don't know how to do half-lifes really, but need to for my exam.
This is the second problem in which you say you have an exam coming up but have no idea how to do a problem. That's a very bad situation!

Half life MEANS "time until only half is left"- if you start with $\displaystyle y_0$, after one "half life" you will have $\displaystyle y_0/2$ left.

Put that into your given equation: $\displaystyle y_0/2= y_0e^{kt}$ when t is the half life- which you are told is 2.045 years: $\displaystyle y_0/2= y_0e^{2.045k}$. The "$\displaystyle y_0$"s cancel leaving $\displaystyle 1/2= 0.5= e^{2.045k}$. Solve for k by taking the logarithm of both sides.

Once you know k, use $\displaystyle 0.95y_0$ instead of $\displaystyle y_0/2$ and the value of k you just found: $\displaystyle 0.95y_0= y_0e^{kt}$ and solve for t.

3. Originally Posted by HallsofIvy
This is the second problem in which you say you have an exam coming up but have no idea how to do a problem. That's a very bad situation!

Half life MEANS "time until only half is left"- if you start with $\displaystyle y_0$, after one "half life" you will have $\displaystyle y_0/2$ left.

Once you know k, use $\displaystyle 0.95y_0$ instead of $\displaystyle y_0/2$ and the value of k you just found: $\displaystyle 0.95y_0= y_0e^{kt}$ and solve for t.
The general solution for finding half-life for a radioactive decay is $\displaystyle t_{1/2} = \frac{ln2}{2}$. It is derived from finding k in terms of t