# Carbon dating mathematical modelling

We end up with a solution known as the "Law of Radioactive Decay", which mathematically is merely the same solution that we saw in the case of light attenuation.We get an expression for the number of atoms remaining, N, as a proportion of the number of atoms N, where the quantity l, known as the "radioactive decay constant", depends on the particular radioactive substance.To learn more or modify/prevent the use of cookies, see our Cookie Policy and Privacy Policy.Modern imaging techniques can provide sequences of images giving signals proportional to the concentrations of tracers (by emission tomography), of X-ray-absorbing contrast materials (fast CT or perhaps NMR contrast), or of native chemical substances (NMR) in tissue regions at identifiable locations in 3D space.Note that that the domain of F is the interval from zero to 1, which corresponds to the interval of time from zero to infinity.Plotting t against F with a value of l=1 gives the graph on the right. The equivalent thickness for the medium in radiation attenuation is known as "half-value thickness".The steps are the same as in the case of photon survival.

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When translated into a self-consistent set of differential equations, the model becomes a mathematical model, a quantitative version of the hypothesis. However, the next step is to reduce the mathematical model to a computable form; anatomically and physiologically realistic models account of the spatial gradients in concentrations within blood-tissue exchange units, while compartmental models simplify the equations by using the average concentrations.

The former are known as distributed models and the latter as lumped compartmental or mixing chamber models.

This question can be answered using a little bit of calculus. Once we have an expression for t, a "definite integral" will give us the mean value of t (this is how "mean value" is defined).

From the equation above, taking logarithms of both sides we see that lt = -ln(N/N.   The raw 'uncalibrated' radiocarbon ages are given in radiocarbon years before the present day (BP) but these do not equate directly to actual calendar years due to variations in atmospheric carbon over time.

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