And so we could make this as over 1. Let me write it over here in a different color. The negative natural log-- well, I could just write it this way. If I have a natural log of b-- we know from our logarithm properties, this is the same thing as the natural log of b to the a power. And so this is the same thing.

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Anything to the negative power is just its multiplicative inverse. So this is just the natural log of 2. So negative natural log of 1 half is just the natural log of 2 over here. So we were able to figure out our k. It's essentially the natural log of 2 over the half-life of the substance. So we could actually generalize this if we were talking about some other radioactive substance.

## Radioactive Dating

And now let's think about a situation-- now that we've figured out a k-- let's think about a situation where we find in some sample-- so let's say the potassium that we find is 1 milligram. 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.

## K-Ar dating calculation

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.

## Carbon 14 dating 1

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. For geologic dating, where the time span is on the order of the age of the earth and the methods use the clocks in the rocks , there are two main uncertainties in the dating process:.

Starting with the simplest case where there are no daughter atoms present and no mass is lost from the sample, the age can be determined by measuring the relative amounts of the isotopes. This can be done by chemical means, but for precise determinations, mass spectrometry can be used. From the radioactive decay equations, an expression for elapsed time can be developed. Using the common nuclear practice of calling the isotopes "parent" and "daughter", we use P and D to indicate the associated numbers of atoms.

The requirement of keeping the same number of nuclei gives. Now suppose that there was an original amount of the daughter element present at the formation time of the sample being studied. This adds an additional unknown in the process, and requires an additional piece of data to permit a solution for elapsed time. The requirement on the populations is now. Fortunately for radioactive dating processes, additional information is available in the form of other isotopes of the elements involved in the radioactive process.

Any argon present in a mineral containing potassium must have been formed as the result of radioactive decay. F, the fraction of K40 remaining, is equal to the amount of potassium in the sample, divided by the sum of potassium in the sample plus the calculated amount of potassium required to produce the amount of argon found.

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The age can then be calculated from equation 1. In spite of the fact that it is a gas, the argon is trapped in the mineral and can't escape. Creationists claim that argon escape renders age determinations invalid. However, any escaping argon gas would lead to a determined age younger, not older, than actual. The creationist "argon escape" theory does not support their young earth model. The argon age determination of the mineral can be confirmed by measuring the loss of potassium. In old rocks, there will be less potassium present than was required to form the mineral, because some of it has been transmuted to argon.

The decrease in the amount of potassium required to form the original mineral has consistently confirmed the age as determined by the amount of argon formed. See Carbon 14 Dating in this web site. The nuclide rubidium decays, with a half life of Strontium is a stable element; it does not undergo further radioactive decay. Do not confuse with the highly radioactive isotope, strontium Strontium occurs naturally as a mixture of several nuclides, including the stable isotope strontium If three different strontium-containing minerals form at the same time in the same magma, each strontium containing mineral will have the same ratios of the different strontium nuclides, since all strontium nuclides behave the same chemically.

Note that this does not mean that the ratios are the same everywhere on earth. It merely means that the ratios are the same in the particular magma from which the test sample was later taken. As strontium forms, its ratio to strontium will increase. Strontium is a stable element that does not undergo radioactive change. In addition, it is not formed as the result of a radioactive decay process.

The amount of strontium in a given mineral sample will not change. Therefore the relative amounts of rubidium and strontium can be determined by expressing their ratios to strontium It turns out to be a straight line with a slope of The corresponding half lives for each plotted point are marked on the line and identified.

It can be readily seen from the plots that when this procedure is followed with different amounts of Rb87 in different minerals , if the plotted half life points are connected, a straight line going through the origin is produced.

### Radioactive Dating

These lines are called "isochrons". The steeper the slope of the isochron, the more half lives it represents. When the fraction of rubidium is plotted against the fraction of strontium for a number of different minerals from the same magma an isochron is obtained. If the points lie on a straight line, this indicates that the data is consistent and probably accurate.

An example of this can be found in Strahler, Fig If the strontium isotope was not present in the mineral at the time it was formed from the molten magma, then the geometry of the plotted isochron lines requires that they all intersect the origin, as shown in figure However, if strontium 87 was present in the mineral when it was first formed from molten magma, that amount will be shown by an intercept of the isochron lines on the y-axis, as shown in Fig