Carbon dating calculus
The ratio of Carbon-14 remaining indicates the times since the death of a living substance.Carbon-14 only works for things between 3 and 40 thousand years old. Carbon dating is based on an isotope of carbon, carbon 14, that's unstable. We breathe in carbon dioxide, we eat carbon, we take in carbon and so our bodies continually renewing our supply of carbon 14.And if you type that in your calculator you'll find that this specimen is 700, oh sorry, 7860 years dead. So that's the way that we can do these calculations. Let's do it a different, let's do a different one.It's always the same thing and if you're having trouble in going from this step to this step, make sure you know how to do that. We take the natural log of both sides and then we solve for t. Let's say that a specimen has been dead for 10,000 years and I want to know its carbon 14 ratio.And that's useful, but what if I care about how much carbon I have after 1/2 a year, or after 1/2 a half life, or after three billion years, or after 10 minutes? A general function, as a function of time, that tells me the number, or the amount, of my decaying substance I have.So that's what we're going to do in this video.When an organism dies it ceases to replenish carbon in its tissues and the decay of carbon 14 to nitrogen 14 changes the ratio of carbon 12 to carbon 14.
At time is equal to two half-lives, we'd have 25% of our substance, and so on and so forth.It decays with a half life of 5700 years into nitrogen 14 and electron and an electron antineutreno. So for that reason, every living thing that is interacting with its environment is expected to have this natural abundance of carbon 14. But when something dies, now it's not interacting with the environment anymore. We know that the amount at time t is equal to the initial amount times one half to the time over the half life, alright?So this is just an ordinary beta decay process and this carbon fourteen's half life is way way way too short for any carbon to just kind of exist naturally in the atmosphere, you'd think, not quite right. So that mean that 1.3 times 10 to the -12 carbon 14 atoms, exist for each and every carbon 12 atom in nature. So you'd think that if you got this 1.3 times 10 to the -12 carbon 14 atoms for each carbon 12 atom at some time, well then 5700 years later, half of the carbon 14 will have decayed. But in fact what happens is, cosmic rays from the sun interact with the upper atmosphere and they actually create carbon 14, at this rate so that in equilibrium, 1.3 times 10 to the -12 carbon 14 atoms will exist for every carbon 12 atom. It's no longer replenishing its carbon 14 supply. This is our standard radioactive decay formula, always works.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.This change in the amount of 14C relative to the amount of 12C makes it possible to estimate the time at which the organism lived.