Radiometric dating allows us to measure the age of a rock or a fossil by measuring the object's proportions of atoms and isotopes. Considering an object with both atoms of type A and B, my measuring the ratio of these atoms we can find the age. A radioactive isotope's nucleus undergoes spontaneous radioactive decay such as a proton turns into a neutron (This is known as beta plus decay, where the proton decays into a neutron, positron and an electron neutrino. The positron is the antiparticle of an electron and has positive like the proton). During the decay, A is the parent isotope and B is the daughter isotope, i.e., the decay products. The number of atoms of isotope A reduces in half in a certain amount of time known as the half-life. The following formula for the age of the object is given here:
T = τ ln ((A +B)/A)/ln (2),
Where τ is the half-life and ln (2) comes from the definition. The value of ln (2) is 0.693, a number worth noting. Radiometric dating is understood as we understand today the underlying physics of nuclear processes. For our purposes we don't need to know these mechanisms; to understand them, one needs quantum mechanics and some basic quantum field theory.
To derive this we use the equation Division gives
Taking the natural log i.e., ln = loge we get
A = (A+B) 2-t/τ.
A/ (A+B) = 2-t/τ.
-t/τ ln (2) = ln (A/ (A+B)),
Where we used the fact that ln (xy) = yln(x). Multiplying -τ on both sides gives the required result. One should note that -ln(x/y) = ln(y/x).
4th EditionJames S. Walker
3rd EditionDavid J. Griffiths
6th EditionDouglas C Giancoli