# Dating equations

### BioMath: Carbon Dating Although we now recognize lots of problems with that calculation, the age of 25 my was accepted by most physicists, but considered too short. In this section we will explore the use of carbon dating to determine the age of decay to calculate the amount of carbon at any given time using the equation. Radiocarbon dating can be used on samples of bone, cloth, wood and plant fibers. The half-life We can use a formula for carbon 14 dating to find the answer.

## Half-life and carbon dating

I could call this N0. This is going to be equal to-- and I won't do any of the math-- so we have 1 milligram we have left is equal to 1 milligram-- which is what we found-- plus 0. And then, all of that times e to the negative kt. And what you see here is, when we want to solve for t-- assuming we know k, and we do know k now-- that really, the absolute amount doesn't matter. What actually matters is the ratio. Because if we're solving for t, you want to divide both sides of this equation by this quantity right over here. So you get this side-- the left-hand side-- divide both sides. You get 1 milligram over this quantity-- I'll write it in blue-- over this quantity is going to be 1 plus-- I'm just going to assume, actually, that the units here are milligrams.

So you get 1 over this quantity, which is 1 plus 0. That is equal to e to the negative kt. And then, if you want to solve for t, you want to take the natural log of both sides. This is equal right over here. You want to take the natural log of both sides. So you get the natural log of 1 over 1 plus 0. And then, to solve for t, you divide both sides by negative k. So I'll write it over here.

And you can see, this a little bit cumbersome mathematically, but we're getting to the answer. So we got the natural log of 1 over 1 plus 0. Well, what is negative k? We're just dividing both sides of this equation by negative k. Negative k is the negative of this over the negative natural log of 2 over 1.

And now, we can get our calculator out and just solve for what this time is. And it's going to be in years because that's how we figured out this constant. So let's get my handy TI First, I'll do this part. So this is 1 divided by 1 plus 0. So that's this part right over here. That gives us that number. And then, we want to take the natural log of that. So let's take the natural log of our previous answer.

So it's the natural log of 0. It gives us negative 0. So that's this numerator over here. And we're going to divide that.

So this number is our numerator right over here. We're going to divide that by the negative-- I'll use parentheses carefully-- the negative natural log of that's that there-- divided by 1. So it's negative natural log of 2 divided by 1. This can be corrected for. Most minerals will lose Ar on heating above oC - thus metamorphism can cause a loss of Ar or a partial loss of Ar which will reset the atomic clock.

If only partial loss of Ar occurs then the age determined will be in between the age of crystallization and the age of metamorphism. If complete loss of Ar occurs during metamorphism, then the date is that of the metamorphic event. The problem is that there is no way of knowing whether or not partial or complete loss of Ar has occurred. Thus the ratio of 14C to 14N in the Earth's atmosphere is constant.

Living organisms continually exchange Carbon and Nitrogen with the atmosphere by breathing, feeding, and photosynthesis. When an organism dies, the 14C decays back to 14N, with a half-life of 5, years. Measuring the amount of 14C in this dead material thus enables the determination of the time elapsed since the organism died. Radiocarbon dates are obtained from such things as bones, teeth, charcoal, fossilized wood, and shells. Because of the short half-life of 14C, it is only used to date materials younger than about 70, years.

Other Uses of Isotopes Radioactivity is an important heat source in the Earth. Elements like K, U, Th, and Rb occur in quantities large enough to release a substantial amount of heat through radioactive decay. Thus radioactive isotopes have potential as fuel for such processes as mountain building, convection in the mantle to drive plate tectonics, and convection in the core to produce the Earth's magnetic Field. Carbon is a key element in biologically important molecules.

During the lifetime of an organism, carbon is brought into the cell from the environment in the form of either carbon dioxide or carbon-based food molecules such as glucose; then used to build biologically important molecules such as sugars, proteins, fats, and nucleic acids. These molecules are subsequently incorporated into the cells and tissues that make up living things. Therefore, organisms from a single-celled bacteria to the largest of the dinosaurs leave behind carbon-based remains.

Carbon dating is based upon the decay of 14C, a radioactive isotope of carbon with a relatively long half-life years. While 12C is the most abundant carbon isotope, there is a close to constant ratio of 12C to 14C in the environment, and hence in the molecules, cells, and tissues of living organisms. This constant ratio is maintained until the death of an organism, when 14C stops being replenished. At this point, the overall amount of 14C in the organism begins to decay exponentially.