I was in History class the other day and I asked how people are able to get “accurate” dates for how old certain objects are. He said something along the lines of there are certain elements in materials and they decay overtime, half life this so we know roughly how long ago it was made.
I’m a mathematics major and am proficient in physics. I understand the concept of exponential decay and half-life’s. My questions is how do we know how much of a material we are measuring there initially was? To me, without knowing that, we could say that something was made whenever we want it to be made. Clearly I’m missing something but I can’t quite figure it out.
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> My questions is how do we know how much of a material we are measuring there initially was?
Geochronology has developed methods where the initial amounts can either remain as an unknown, or become part of the solution when determining the age.
The former relies upon using systems where the initial decay product in the mineral being measured is zero or close to zero. This is achieved by selecting a mineral which does not include the daughter product in its crystalline structure, eg. using the U-Pb system to date zircons, which routinely include trace amounts of U in them but reject Pb as it is not compatible with the crystalline structure of Zircon.
The idea is that any Pb is thus the product of U decay. Nature is messier than that, and contamination with Pb (or loss of Pb somewhere along the way) *is* possible, but this can be accounted for using the Concordia method, which relies on the fact that there are two different decay chains between U238 and U235 to different isotopes of Pb. The rate that these progress at dictates a set curve when the two parent-daughter ratios are plotted against each other. If the ratios measured are concordant with the curve the it’s a valid age. If not then the direction of discordance indicates Pb contamination or Pb loss, [something like this.](https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcRpl_6EPostwtL-9mxH_bxUMddDvXFQiKD2yw&usqp=CAU) The two for one decay system gives a kind of internal check of itself and the U-Pb Concordia method is pretty much the gold standard for geochronology (of large ages anyway) in part for this reason.
The other key technique — where the initial isotope amounts are solved for as well as the age — is the isochron method. This relies on measuring isotope ratios in several different minerals and plotting the results on a graph using a reference isotope in the ratios so you have D:R against P:R (where D = daughter, R = reference, and P = parent). The straight line plotted will intersect the y-axis at the initial value of D:R, something like [this](https://upload.wikimedia.org/wikipedia/commons/4/4e/Isochron.jpg). The age can be determined from the slope of the isochron, which is shown as equal to the exponential factor in that diagram. So the whole thing essentially gets reduced to a *y = mx + c* type problem; though it requires some careful field sampling to put into geological context, some very high precision measurements usually using a mass spec, then application of a least squares regression method to measured values in order to get a meaningful final result.
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