Atomic number, atomic mass, and isotopes. Video transcript In the last video, we give a bit of an overview of potassium-argon dating. In this video, I want to go through a concrete example. And it'll get a little bit mathy, usually involving a little bit of algebra or a little bit of exponential decay, but to really show you how you can actually figure out the age of some volcanic rock using this technique, using a little bit of mathematics.

So we know that anything that is experiencing radioactive decay, it's experiencing exponential decay.

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And we know that there's a generalized way to describe that. And we go into more depth and kind of prove it in other Khan Academy videos. But we know that the amount as a function of time-- so if we say N is the amount of a radioactive sample we have at some time-- we know that's equal to the initial amount we have. We'll call that N sub 0, times e to the negative kt-- where this constant is particular to that thing's half-life. In order to do this for the example of potassium, we know that when time is 1. So let's write it that way. So let's say we start with N0, whatever that might be.

It might be 1 gram, kilogram, 5 grams-- whatever it might be-- whatever we start with, we take e to the negative k times 1. That's the half-life of potassium We know, after that long, that half of the sample will be left. Whatever we started with, we're going to have half left after 1. Divide both sides by N0.

## Carbon 14 Dating - Math Central

And then to solve for k, we can take the natural log of both sides. The natural log is just saying-- to what power do I have to raise e to get e to the negative k times 1. So the natural log of this-- the power they'd have to raise e to to get to e to the negative k times 1. Or I could write it as negative 1.

That's the same thing as 1. We have our negative sign, and we have our k. And then, to solve for k, we can divide both sides by negative 1. And so we get k. And I'll just flip the sides here. And what we can do is we can multiply the negative times the top. Or you could view it as multiplying the numerator and the denominator by a negative so that a negative shows up at the top. 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. 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.

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.

## Radiometric Dating Isochron exercise

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.

### Radiometric dating

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.

C The percent completion after 5 half-lives will be as follows:.

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Ethyl chloride decomposes to ethylene and HCl in a first-order reaction that has a rate constant of 1. Radioactivity, or radioactive decay, is the emission of a particle or a photon that results from the spontaneous decomposition of the unstable nucleus of an atom. The rate of radioactive decay is an intrinsic property of each radioactive isotope that is independent of the chemical and physical form of the radioactive isotope. The rate is also independent of temperature.

In this section, we will describe radioactive decay rates and how half-lives can be used to monitor radioactive decay processes. In any sample of a given radioactive substance, the number of atoms of the radioactive isotope must decrease with time as their nuclei decay to nuclei of a more stable isotope. Activity is usually measured in disintegrations per second dps or disintegrations per minute dpm.

The activity of a sample is directly proportional to the number of atoms of the radioactive isotope in the sample:. Here, the symbol k is the radioactive decay constant, which has units of inverse time e. Because radioactive decay is a first-order process, the time required for half of the nuclei in any sample of a radioactive isotope to decay is a constant, called the half-life of the isotope.

The half-life tells us how radioactive an isotope is the number of decays per unit time ; thus it is the most commonly cited property of any radioisotope. For a given number of atoms, isotopes with shorter half-lives decay more rapidly, undergoing a greater number of radioactive decays per unit time than do isotopes with longer half-lives. The half-lives of several isotopes are listed in Table In our earlier discussion, we used the half-life of a first-order reaction to calculate how long the reaction had been occurring.

Because nuclear decay reactions follow first-order kinetics and have a rate constant that is independent of temperature and the chemical or physical environment, we can perform similar calculations using the half-lives of isotopes to estimate the ages of geological and archaeological artifacts. The techniques that have been developed for this application are known as radioisotope dating techniques. The most common method for measuring the age of ancient objects is carbon dating. As a result, the CO 2 that plants use as a carbon source for synthesizing organic compounds always includes a certain proportion of 14 CO 2 molecules as well as nonradioactive 12 CO 2 and 13 CO 2.

Any animal that eats a plant ingests a mixture of organic compounds that contains approximately the same proportions of carbon isotopes as those in the atmosphere. Comparing the disintegrations per minute per gram of carbon from an archaeological sample with those from a recently living sample enables scientists to estimate the age of the artifact, as illustrated in Example A plot of the specific activity of 14 C versus age for a number of archaeological samples shows an inverse linear relationship between 14 C content a log scale and age a linear scale.