### How to figure out radiometric dating

Some examples of isotope systems used to date geologic materials. To see how we actually use this information to date rocks, consider the following: To account for this, we first note that there is an isotope of Sr, 86 Sr, that is: If we divide equation 4 through by the amount of 86 Sr, then we get: Note also that equation 5 has the form of a linear equation, i. How can we use this? In nature, however, each mineral in the rock is likely to have a different amount of 87 Rb. Thus, once the rock has cooled to the point where diffusion of elements does not occur, the 87 Rb in each mineral will decay to 87 Sr, and each mineral will have a different 87 Rb and 87 Sr after passage of time.

The Concordia curve can be calculated by defining the following: The discordia is often interpreted by extrapolating both ends to intersect the Concordia. Pb leakage is the most likely cause of discordant dates, since Pb will be occupying a site in the crystal that has suffered radiation damage as a result of U decay. U would have been stable in the crystallographic site, but the site is now occupied by by Pb. An event like metamorphism could heat the crystal to the point where Pb will become mobile. Another possible scenario involves U leakage, again possibly as a result of a metamorphic event.

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U leakage would cause discordant points to plot above the cocordia. The Age of the Earth A minimum age of the Earth can be obtained from the oldest known rocks on the Earth. So far, the oldest rock found is a tonalitic Gneiss metamorphic rock rock from the Northwest Territories, Canada, with an age of 3.

This gives us only a minimum age of the Earth. Is it likely that we will find a rock formed on the Earth that will give us the true age of the Earth?

From the Pb-Pb isochron equation 11 we can make some arguments about meteorites. First, it appears that meteorites have come from somewhere in the solar system, and thus may have been formed at the same time the solar system and thus the Earth formed. If all of the meteorites formed at the same time and have been closed to U and Pb since their formation, then we can use the Pb-Pb isochron to date all meteorites.

First, however, we need to know the initial ratios of the Pb isotopes. We recognize two major types of meteorites: Fe- meteorites and stony or chondritic meteorites The Fe meteorites contain the mineral troilite FeS that has no U.

Since the mineral troilite contains no U, all of the Pb present in the troilite is the Pb originally present, and none of it has been produced by U decay. We can then determine the Pb ratios in other meteorites and see if they fall on a Pb-Pb isochron that passes through the initial ratios determined from troilite in Fe-meteorites. The slope of this isochron, known as the Geochron, gives an age of 4. K-Ar Dating 40 K is the radioactive isotope of K, and makes up 0.

Thus the ratio of 14 C to 14 N 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 14 C decays back to 14 N, with a half-life of 5, years. Measuring the amount of 14 C in this dead material thus enables the determination of the time elapsed since the organism died.

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• Radiocarbon dates are obtained from such things as bones, teeth, charcoal, fossilized wood, and shells. Because of the short half-life of 14 C, 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.

Initial isotopic ratios are useful as geochemical tracers.

## 5.7: Calculating Half-Life

Such tracers can be used to determine the origin of magmas and the chemical evolution of the Earth. Short-lived isotopes Isotopes made during nucleosynthesis that have nearly completely decayed away can give information on the time elapsed between nucleosynthesis and Earth Formation. Lastly, accuracy of C dating has been affected by atmosphere nuclear weapons testing. Fission bombs ignite to produce more C artificially. Samples tested during and after this period must be checked against another method of dating isotopic or tree rings.

To calculate the age of a substance using isotopic dating, use the equation below:. How long will it take for Ra has a half-life of years. Radioactive dating can also use other radioactive nuclides with longer half-lives to date older events. For example, uranium which decays in a series of steps into lead can be used for establishing the age of rocks and the approximate age of the oldest rocks on earth. Since U has a half-life of 4. In a sample of rock that does not contain appreciable amounts of Pb, the most abundant isotope of lead, we can assume that lead was not present when the rock was formed.

Therefore, by measuring and analyzing the ratio of U Pb, we can determine the age of the rock. This assumes that all of the lead present came from the decay of uranium If there is additional lead present, which is indicated by the presence of other lead isotopes in the sample, it is necessary to make an adjustment. Potassium-argon dating uses a similar method. K decays by positron emission and electron capture to form Ar with a half-life of 1.

If a rock sample is crushed and the amount of Ar gas that escapes is measured, determination of the Ar K ratio yields the age of the rock. Other methods, such as rubidium-strontium dating Rb decays into Sr with a half-life of As of , the oldest known rocks on earth are the Jack Hills zircons from Australia, found by uranium-lead dating to be almost 4.

An ingenious application of half-life studies established a new science of determining ages of materials by half-life calculations. After one half-life, a 1. Present day estimates for the age of the Earth's crust from this method is at 4 billion years. Isotopes with shorter half-lives are used to date more recent samples. Chemists and geologists use tritium dating to determine the age of water ocean and fresh. In addition, tritium dating can be useful in determining the age of wines and brandies. I'm just going to make up these numbers. And usually, these aren't measured directly, and you really care about the relative amounts.

But let's say you were able to figure out the potassium is 1 milligram. And let's say that the argon-- actually, I'm going to say the potassium found, and let's say the argon found-- let's say it is 0. So how can we use this information-- in what we just figured out here, which is derived from the half-life-- to figure out how old this sample right over here?

How do we figure out how old this sample is right over there? Well, what we need to figure out-- we know that n, the amount we were left with, is this thing right over here. So we know that we're left with 1 milligram. And that's going to be equal to some initial amount-- when we use both of this information to figure that initial amount out-- times e to the negative kt.

And we know what k is. And we'll figure it out later. So k is this thing right over here. So we need to figure out what our initial amount is. We know what k is, and then we can solve for t. How old is this sample? We saw that in the last video. So if you want to think about the total number of potassiums that have decayed since this was kind of stuck in the lava.

And we learned that anything that was there before, any argon that was there before would have been able to get out of the liquid lava before it froze or before it hardened. So maybe I could say k initial-- the potassium initial-- is going to be equal to the amount of potassium 40 we have today-- 1 milligram-- plus the amount of potassium we needed to get this amount of argon We have this amount of argon 0.

The rest of it turned into calcium And this isn't the exact number, but it'll get the general idea. And so our initial-- which is really this thing right over here. 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.