# Carbon 14 dating calculus problems

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The amount of carbon-14 decreases relative to the amount of normal carbon.Radiocarbon dating seizes on that fraction, which decreases over time, to estimate age. The problem is that the fraction can decrease not only as carbon-14 decays but also as normal carbon increases.It describes how fossil fuel emissions will make radiocarbon dating, used to identify archaeological finds, poached ivory or even human corpses, less reliable.As scrolls, plant-based paints or cotton shirts age over thousands of years, the radioactive carbon-14 that naturally appears in organic objects gradually decays.To do this we use the following equation for finding E(x) of a probability density function: and if we substitute in our equation we get: Now, we can integrate this by parts: So the expected (mean) life of an atom is given by 1/λ.In the case of Carbon, with a decay constant λ ≈0.000121 we have an expected life of a Carbon-14 atom as: E(t) = 1 /0.000121 E(t) = 8264 years.: is the initial quantity of the element λ: is the radioactive decay constant t: is time N(t): is the quantity of the element remaining after time t.So, for Carbon-14 which has a half life of 5730 years (this means that after 5730 years exactly half of the initial amount of Carbon-14 atoms will have decayed) we can calculate the decay constant λ. We can then manipulate this into the form of a probability density function – by finding the constant a which makes the area underneath the curve equal to 1. Therefore the following integral: will give the fraction of atoms which will have decayed between times t1 and t2.

The half-life of a radioactive isotope expresses the time needed for a sample of that isotope to reach half of its original mass.After 5730 years, N(5730) will be exactly half of N and if we take the natural log of both sides and rearrange we get: λ = ln(1/2) / -5730 λ ≈0.000121 We can now use this to solve problems involving Carbon-14 (which is used in Carbon-dating techniques to find out how old things are). You find an old parchment and after measuring the Carbon-14 content you find that it is just 30% of what a new piece of paper would contain. We could use this integral to work out the half life of Carbon-14 as follows: Which if we solve gives us t = 5728.5 which is what we’d expect (given our earlier rounding of the decay constant).We can also now work out the expected (mean) time that an atom will exist before it decays.By 2100, a dead plant could be almost identical to the Dead Sea scrolls, which are more than 2,000 years old.These well-known “aging” properties of atmospheric carbon were pinpointed for different emissions scenarios in a paper published in the yesterday.