How to calculate age using carbon dating
And then either later in this video or in future videos we'll talk about how it's actually used to date things, how we use it actually figure out that that bone is 12,000 years old, or that person died 18,000 years ago, whatever it might be. So let me just draw the surface of the Earth like that. So then you have the Earth's atmosphere right over here. And 78%, the most abundant element in our atmosphere is nitrogen. And we don't write anything, because it has no protons down here. And what's interesting here is once you die, you're not going to get any new carbon-14. You can't just say all the carbon-14's on the left are going to decay and all the carbon-14's on the right aren't going to decay in that 5,730 years.It's just a little section of the surface of the Earth. And that carbon-14 that you did have at you're death is going to decay via beta decay-- and we learned about this-- back into nitrogen-14. So it'll decay back into nitrogen-14, and in beta decay you emit an electron and an electron anti-neutrino. But essentially what you have happening here is you have one of the neutrons is turning into a proton and emitting this stuff in the process. So I just said while you're living you have kind of straight-up carbon-14. What it's essentially saying is any given carbon-14 atom has a 50% chance of decaying into nitrogen-14 in 5,730 years.The resulting radiocarbon combines with atmospheric oxygen to form radioactive carbon dioxide, which is incorporated into plants by photosynthesis; animals then acquire in a sample from a dead plant or animal such as a piece of wood or a fragment of bone provides information that can be used to calculate when the animal or plant died.
A fossil found in an archaeological dig was found to contain 20% of the original amount of 14C. I do not get the $-0.693$ value, but perhaps my answer will help anyway.The ratio of the amount of 14C to the amount of 12C is essentially constant (approximately 1/10,000).When an organism dies, the amount of 12C present remains unchanged, but the 14C decays at a rate proportional to the amount present with a half-life of approximately 5700 years.$$ So either the answer is that ridiculously big number (9.17e7) or 30,476 years, being calculated with the equation I provided and the first equation in your answer, respectively. The method was developed by Willard Libby in the late 1940s and soon became a standard tool for archaeologists.
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Libby received the Nobel Prize in Chemistry for his work in 1960.