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.

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.

### Learning Goals

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. The above equation makes use of information on the composition of parent and daughter isotopes at the time the material being tested cooled below its closure temperature. This is well-established for most isotopic systems. Plotting an isochron is used to solve the age equation graphically and calculate the age of the sample and the original composition. Radiometric dating has been carried out since when it was invented by Ernest Rutherford as a method by which one might determine the age of the Earth.

In the century since then the techniques have been greatly improved and expanded. The mass spectrometer was invented in the s and began to be used in radiometric dating in the s. It operates by generating a beam of ionized atoms from the sample under test. The ions then travel through a magnetic field, which diverts them into different sampling sensors, known as " Faraday cups ", depending on their mass and level of ionization.

On impact in the cups, the ions set up a very weak current that can be measured to determine the rate of impacts and the relative concentrations of different atoms in the beams. Uranium—lead radiometric dating involves using uranium or uranium to date a substance's absolute age. This scheme has been refined to the point that the error margin in dates of rocks can be as low as less than two million years in two-and-a-half billion years. Uranium—lead dating is often performed on the mineral zircon ZrSiO 4 , though it can be used on other materials, such as baddeleyite , as well as monazite see: Zircon has a very high closure temperature, is resistant to mechanical weathering and is very chemically inert.

Zircon also forms multiple crystal layers during metamorphic events, which each may record an isotopic age of the event. One of its great advantages is that any sample provides two clocks, one based on uranium's decay to lead with a half-life of about million years, and one based on uranium's decay to lead with a half-life of about 4.

This can be seen in the concordia diagram, where the samples plot along an errorchron straight line which intersects the concordia curve at the age of the sample. This involves the alpha decay of Sm to Nd with a half-life of 1. Accuracy levels of within twenty million years in ages of two-and-a-half billion years are achievable.

## Radiometric Dating

This involves electron capture or positron decay of potassium to argon Potassium has a half-life of 1. This is based on the beta decay of rubidium to strontium , with a half-life of 50 billion years. This scheme is used to date old igneous and metamorphic rocks , and has also been used to date lunar samples.

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Closure temperatures are so high that they are not a concern. Rubidium-strontium dating is not as precise as the uranium-lead method, with errors of 30 to 50 million years for a 3-billion-year-old sample. A relatively short-range dating technique is based on the decay of uranium into thorium, a substance with a half-life of about 80, years. It is accompanied by a sister process, in which uranium decays into protactinium, which has a half-life of 32, years.

While uranium is water-soluble, thorium and protactinium are not, and so they are selectively precipitated into ocean-floor sediments , from which their ratios are measured. The scheme has a range of several hundred thousand years. A related method is ionium—thorium dating , which measures the ratio of ionium thorium to thorium in ocean sediment.

Radiocarbon dating is also simply called Carbon dating.

### Radiometric dating

Carbon is a radioactive isotope of carbon, with a half-life of 5, years, [25] [26] which is very short compared with the above isotopes and decays into nitrogen. Carbon, though, is continuously created through collisions of neutrons generated by cosmic rays with nitrogen in the upper atmosphere and thus remains at a near-constant level on Earth. The carbon ends up as a trace component in atmospheric carbon dioxide CO 2.

A carbon-based life form acquires carbon during its lifetime.

## Radiometric Dating

Plants acquire it through photosynthesis , and animals acquire it from consumption of plants and other animals. When an organism dies, it ceases to take in new carbon, and the existing isotope decays with a characteristic half-life years. The proportion of carbon left when the remains of the organism are examined provides an indication of the time elapsed since its death. This makes carbon an ideal dating method to date the age of bones or the remains of an organism.

The carbon dating limit lies around 58, to 62, years. The rate of creation of carbon appears to be roughly constant, as cross-checks of carbon dating with other dating methods show it gives consistent results. However, local eruptions of volcanoes or other events that give off large amounts of carbon dioxide can reduce local concentrations of carbon and give inaccurate dates. The releases of carbon dioxide into the biosphere as a consequence of industrialization have also depressed the proportion of carbon by a few percent; conversely, the amount of carbon was increased by above-ground nuclear bomb tests that were conducted into the early s.

Also, an increase in the solar wind or the Earth's magnetic field above the current value would depress the amount of carbon created in the atmosphere. This involves inspection of a polished slice of a material to determine the density of "track" markings left in it by the spontaneous fission of uranium impurities.

The uranium content of the sample has to be known, but that can be determined by placing a plastic film over the polished slice of the material, and bombarding it with slow neutrons. This causes induced fission of U, as opposed to the spontaneous fission of U.

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The fission tracks produced by this process are recorded in the plastic film. The uranium content of the material can then be calculated from the number of tracks and the neutron flux. This scheme has application over a wide range of geologic dates. For dates up to a few million years micas , tektites glass fragments from volcanic eruptions , and meteorites are best used. Older materials can be dated using zircon , apatite , titanite , epidote and garnet which have a variable amount of uranium content. The technique has potential applications for detailing the thermal history of a deposit.

The residence time of 36 Cl in the atmosphere is about 1 week. Thus, as an event marker of s water in soil and ground water, 36 Cl is also useful for dating waters less than 50 years before the present. Luminescence dating methods are not radiometric dating methods in that they do not rely on abundances of isotopes to calculate age.

Instead, they are a consequence of background radiation on certain minerals. Over time, ionizing radiation is absorbed by mineral grains in sediments and archaeological materials such as quartz and potassium feldspar. The radiation causes charge to remain within the grains in structurally unstable "electron traps".

Exposure to sunlight or heat releases these charges, effectively "bleaching" the sample and resetting the clock to zero. The trapped charge accumulates over time at a rate determined by the amount of background radiation at the location where the sample was buried. Stimulating these mineral grains using either light optically stimulated luminescence or infrared stimulated luminescence dating or heat thermoluminescence dating causes a luminescence signal to be emitted as the stored unstable electron energy is released, the intensity of which varies depending on the amount of radiation absorbed during burial and specific properties of the mineral.

These methods can be used to date the age of a sediment layer, as layers deposited on top would prevent the grains from being "bleached" and reset by sunlight. Pottery shards can be dated to the last time they experienced significant heat, generally when they were fired in a kiln. Absolute radiometric dating requires a measurable fraction of parent nucleus to remain in the sample rock. For rocks dating back to the beginning of the solar system, this requires extremely long-lived parent isotopes, making measurement of such rocks' exact ages imprecise.

To be able to distinguish the relative ages of rocks from such old material, and to get a better time resolution than that available from long-lived isotopes, short-lived isotopes that are no longer present in the rock can be used. At the beginning of the solar system, there were several relatively short-lived radionuclides like 26 Al, 60 Fe, 53 Mn, and I present within the solar nebula. These radionuclides—possibly produced by the explosion of a supernova—are extinct today, but their decay products can be detected in very old material, such as that which constitutes meteorites.

By measuring the decay products of extinct radionuclides with a mass spectrometer and using isochronplots, it is possible to determine relative ages of different events in the early history of the solar system. Dating methods based on extinct radionuclides can also be calibrated with the U-Pb method to give absolute ages. Thus both the approximate age and a high time resolution can be obtained. Generally a shorter half-life leads to a higher time resolution at the expense of timescale. The iodine-xenon chronometer [32] is an isochron technique. Samples are exposed to neutrons in a nuclear reactor.

This converts the only stable isotope of iodine I into Xe via neutron capture followed by beta decay of I.

After irradiation, samples are heated in a series of steps and the xenon isotopic signature of the gas evolved in each step is analysed. Samples of a meteorite called Shallowater are usually included in the irradiation to monitor the conversion efficiency from I to Xe.

This in turn corresponds to a difference in age of closure in the early solar system. Another example of short-lived extinct radionuclide dating is the 26 Al — 26 Mg chronometer, which can be used to estimate the relative ages of chondrules. The 26 Al — 26 Mg chronometer gives an estimate of the time period for formation of primitive meteorites of only a few million years 1. From Wikipedia, the free encyclopedia.

Earth sciences portal Geophysics portal Physics portal. The disintegration products of uranium". American Journal of Science.