# With these two elements how can the isotope half-life be determined using first order kinetics?

With these two elements how can the isotope half-life be determined using first order kinetics?

nobelium-259 half-life recorded @ 58 minutes
fermium-253 half-life recorded @ 3 days

Although I do not know, there is probably an average starting concentration that can probably be determined by the name of the element. Maybe in the comments someone can hook me up with some estimate averages so the expansion of the math could be given in the answer.

• What is first order kinetics? Whatever it is, to correctly predict half-lifes full quantum mechanics is needed, as decay is an inherently quantum phenomenon. – ACuriousMind Jun 27 '14 at 19:04
• I want to understand the rate of decay. From a formula based calculation instead of a "visual measurement". – Decrypted Jun 27 '14 at 19:49
• @ACuriousMind Are quantum phenomenons formula based? Not that I'm limited to a formula based result. – Decrypted Jun 27 '14 at 20:08
• The two isotopes you picked decay either by alpha emission or electron capture. Are you asking if there's a theoretical means of predicting the decay rate for one of those modes? – paisanco Jun 28 '14 at 0:01
• @Onlyheisgood. this is the publication of how the 58 minutes was determined: sciencedirect.com/science/article/pii/0375947473905204 – DavePhD Jun 30 '14 at 15:33

First order reaction kinetics only states that the system obeys the decay equation $$\frac{d[X]}{dt}=-k[X],\quad k>0.$$ The solution is usually written $$[X(t)]=[X_0]e^{-kt},$$ where $[X_0]$ is the starting concentration (or simply number of particles), but in terms of the half-life, $k=(\ln 2)/\tau_{1/2}$, so $$[X(t)]=[X_0]2^{-t/\tau_{1/2}}.$$ So using kinetics, you'd need to know the concentrations/numbers of particles at two different times (or at least their ratio) to measure the half-life. Otherwise, as stated in the comments, you need the full quantum mechanical theory of nuclear kinetics, which is generally not referred to as first order kinetics.