"Thus mixing lines can be mistaken for isochrons and visa [sic] versa."

W. M. White, "Isotopic Geochemistry Of Subduction Zone Magmas". (N.B.: this quote is not related to the topic of this page.)

Radiogenic evolution and binary mixing

The similarity between binary mixing lines and the loci traced out by radiogenic species on various isotope ratio plots is not coincidental. The chemical evolution of radiogenic species can be understood as a binary mixture of a radiogenically primitive componentWhat is a "radiogenically primitive component"? This question suggests the idea of model age. and a radiogenically evolved (stable) component.

Suppose a chemical species P with mass \(P\) is transformed into a distinct chemical species D with mass \(D\) at a rate \(dP/dt\) given by some differential equation. The fundamental mass balance equation describing this transformation is \(-(P-P_0)=D-D_0\): the mass of species P lost is equal to the mass of species D gained. This equation implies that a point with starting composition \((P_0, D_0)\) on a plot of \(D\) vs. \(P\) will evolve along a line with a slope of -1 and reach a final composition \((0, P_0 + D_0)\). (Of course, this line is not an isochron.)

We can interpret this line as a mixing line. (Take \(P\) and \(D\) to be concentrations.) Any composition on this line is a binary mixture of a primitive component \((P_0, D_0)\) and an evolved component \((0, P_0 + D_0)\); that is, every point \((P,D)\) on this line is given by a vector-valued, single-variable function \((P(f),D(f))\) where \(f\) is taken to be the mixing fraction of the evolved component. Considering the first coordinate function \(P(f)\) of the composition vector gives \(f=(P-P_0)/P_0\). We can now find \(f\) as a function of time \(t\). Assuming that P is undergoing radioactive decay to D, \(dP/dt = -\lambda P\). It can be seen that \(f=1-e^{-\lambda t}\), so \(f=0\) corresponds to \(t=0\) and \(f=1\) corresponds to \(t=\infty\). Notably, \(f=1/2\) when \(t=t_{1/2}\).

This approach is especially revealing for more complicated radiometric evolution diagrams, in particular the various isotope ratio diagrams used to understand the U-Th-Pb system.