A useful application of half-lives is radioactive dating.This has to do with figuring out the age of ancient things.(Whatever you're being treated for is the greater danger.) The half-life is just long enough for the doctors to have time to take their pictures.

Knowing about half-lives is important because it enables you to determine when a sample of radioactive material is safe to handle.

The method of carbon dating makes use of the fact that all living organisms contain two isotopes of carbon, carbon-12, denoted 12C (a stable isotope), and carbon-14, denoted 14C (a radioactive isotope).

And carbon-14 is constantly doing this decay thing. So over the course of 5,730 years, roughly half of them will have decayed. Well, if you know that all living things have a certain proportion of carbon-14 in their tissue, as kind of part of what makes them up, and then if you were to find some bone-- let's just say find some bone right here that you dig it up on some type of archaeology dig.

But what's interesting is as soon as you die and you're not ingesting anymore plants, or breathing from the atmosphere if you are a plant, or fixing from the atmosphere. Once a plant dies, it's no longer taking in carbon dioxide from the atmosphere and turning it into new tissue. And this carbon-14 does this decay at a specific rate. And you say, hey, that bone has one half the carbon-14 of all the living things that you see right now.

If we assume Carbon-14 decays continuously, then $$C(t) = C_0e^,$$ where $C_0$ is the initial size of the sample. Since it takes 5,700 years for a sample to decay to half its size, we know $$\frac C_0 = C_0e^,$$ which means $$\frac = e^,$$ so the value of $C_0$ is irrelevant.

Now, take the logarithm of both sides to get $$-0.693 = -5700k,$$ from which we can derive  k \approx 1.22 \cdot 10^.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.And then you can use that rate to actually determine how long ago that thing must've died. It would be a pretty reasonable estimate to say, well, that thing must be 5,730 years old.If you're seeing this message, it means we're having trouble loading external resources on our website. So carbon by definition has six protons, but the typical isotope, the most common isotope of carbon is carbon-12. And then that carbon dioxide gets absorbed into the rest of the atmosphere, into our oceans. When people talk about carbon fixation, they're really talking about using mainly light energy from the sun to take gaseous carbon and turn it into actual kind of organic tissue.

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