In calculus terms, we write: d N(t)/dt = -K * N(t) or d N(t)/N(t) = -K dt The minus sign means that each decay decreases the total number of atoms.Integrating both sides, we get: ln N(t) = -Kt + C C is the constant of integration that we can often ignore, but not here.When t = 0, ln N(0) = C Taking exponentials of both sides, we get N(t) = N(0)exp(-Kt) If t = one half life, then N(t)/N(0) = 1/2 = exp(-Kt), and: ln(1/2) = -ln2 = -Kt, so t = ln2 / K So what we do in practice is determine the decay constant and calculate half life from it.
In other words there was originally 4 parts per million Parentium-123 and 0 parts per million Daughterium-123.
Since there is now only 1/4 of the original amount of Parentium-123, we know that two half-lives of Parentium-123 have elapsed.
We could be sure that a mineral containing parentium originally had no daughterium.
If the mineral contained 1 part per million Parentium-123 and 3 parts per million Daughterium-123, we could be sure all the Daughterium-123 was originally Parentium-123.
What radioactive materials actually do is decay according to a law: Decays/Time = K * Number of atoms K is a constant called the decay constant.Let t stand for time and N(t) stand for the number of atoms at time t .