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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.
Furthermore, Parentium and Daughterium are so different in chemical properties that they don't otherwise occur together.
If there were such a pair of isotopes, radiometric dating would be very simple.
Imagine we have an undiscovered element, Parentium, that has a radioactive isotope, Parentium-123, which decays to stable Daughterium-123.
It is one of the simplest examples of a differential equation.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.
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.