The probability of an atom disintegrating in any time interval is indenpendent of its past history and is the same for all atoms of the same kind.

(dN/N)/dt = -p where p is the probability that decay will occurs.
so dN/N = -p dt
then we will have

N = N_o e^-pt = N_o e^-t/T where p=1/T

N_ois the number of atoms at t=0, at t=T (called mean lifetime) N(T)=N_o e^-1
We also define half-life T_1/2 such that N(T_1/2)=N_o/2
so T_1/2 = log(2.) T = 0.693 T

A given radioactive decay can be a single step in a long sequence of transformation from one unstable nuclide to another.
There are three naturally occuring radioactive series beginning with ²³⁸U, ²³⁵U, and ²³²Th , all ending in different isotopes of lead.

For example: nuclide A will decay and become nuclide B, and nuclide B will decay and become nuclide C, ...

This applet will show how the population of each nuclide changed during radioactive decay series:

 

The applet shown radioactive decay from nuclide A(black) -> B(red) -> C(green) -> D(yellow).
Each circle represent a nuclide.

For example:

For nuclide A decay into nuclide B : black circle will change into red circle.
For nuclide A decay into nuclide B : red circle will change into green circle.
...

If you just want to see decay from A to B, and B is stable nuclide -> click the checkbox at the top-right.

Your suggestions are highly appreciated! Please click hwang@phy03.phy.ntnu.edu.tw

Author：Fu-Kwun Hwang, Dept. of physics, National Taiwan Normal University