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This is because there is carbon dioxide (CO exchange, and so the ratio of C-14 to the far more common carbon isotope, C-12, will begin to decrease as the C-14 atoms decay, yielding nitrogen (N-14) with the emission of an electron (or "beta particle") plus an anti-neutrino.
The ratio of C-14 to C-12 in the atmosphere's carbon dioxide molecules is about 1.3×10, and this value is assumed constant for the main part of archaeological history since the formation of the earth's atmosphere.
Exactly the same treatment can be applied to radioactive decay.
However, now the "thin slice" is an interval of time, and the dependent variable is the number of radioactive atoms present, N(t). If we have a sample of atoms, and we consider a time interval short enough that the population of atoms hasn't changed significantly through decay, then the proportion of atoms decaying in our short time interval will be proportional to the length of the interval.
From the equation above, taking logarithms of both sides we see that lt = -ln(N/N.
Let's look further at the technique behind the work that led to Libby being awarded a Nobel prize in 1960.
Carbon 14 (C-14) is a radioactive element that is found naturally, and a living organism will absorb C-14 and maintain a certain level of it in the body.
Here isotopes with longer half lives are used, which enables dating of geological formations and rocks. For example, in lava form, molten lead and Uranium-238 (standard isotope) are constantly mixed in a certain ratio of their natural abundance.
Once solidified, the lead is "locked" in place and since the uranium decays to lead, the lead-to-uranium ratio increases with time.The steps are the same as in the case of photon survival.