half-life

• Jun 15th 2009, 02:12 AM
jangalinn
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.
• Jun 15th 2009, 03:33 AM
HallsofIvy
Quote:

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.
• Jun 15th 2009, 03:53 AM
e^(i*pi)
Quote:

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