Radiometric Age Dating - Carbon-14 Age Determination

Radiometric Age Dating, Carbon-14, C-14

  • This method is a surprisingly accurate way to tell SOME very long time intervals!
  • The most famous of these methods uses Carbon-14, and a complete discussion is presented here in order to fully understand WHY it is pretty accurate and why, when careful science is done, the results are very reliable.
  • The discussion also explains that amazingly small numbers of atoms of Carbon-14 are involved in such analysis, and therefore why there are limits on the time intervals for really accurate results.

Science uses radioactive isotopes to determine the age of many objects from the ancient past. One of the most well-known is the Carbon-14 dating method. These methods are generally very accurate and valuable, but a couple of considerations must be made regarding certain of these methods.

Radioactive dating techniques rely on a statistical assumption, but a really solid one. Any one of the many trillions of unstable atoms in a piece of radioactive material could individually decay or fission (break apart) at any moment. It is impossible to predict exactly when any specific atom will decay, but there is actually an overall pattern that exists. With so many separate random events, their timings of the decay events fit into a statistical 'bell-shaped curve'. With such an enormous number of separate events, the bell-shaped curve is extremely accurate and consistent. The very middle of that curve is its highest point, and it also represents the length of time where half the atoms decay before that moment and half decay after.

This statistical situation is the definition of a "half-life" of any radioactive material. Because of the enormous number of separate events involved, the half-life can be determined extremely accurately in a laboratory for each isotope. Virtually nothing can influence the rate of such radioactive decay, so it can represent a very accurate timing method for physical processes.

Technical Stuff

There are many physical phenomena that occur at a rate that depends on how much of the material exists at any time. A population of deer has new babies approximately in proportion to the number of father and mother deer that are around! The number of bacteria that divide and multiply is clearly dependent on the number of bacteria that exist at any moment. The rate of radioactive decay is dependent on the number of remaining undecayed atoms still remaining at any moment. These are generally referred to as "exponential growth" or "exponential decay". We are about to show why that is, due to mathematics!

We might say that the number of remaining undecayed atoms at any instant is f(t), meaning some function of time. This does not say WHAT that function is, just that the number remaining is somehow dependent on time. However, we DO specify that f(t) must be greater to or equal to zero, for a real value of however many atoms are actually left at any moment. We can similarly say that f'(t) could represent the RATE of the atoms decaying, the apostrophe symbol indicating the mathematical DIFFERENTIAL of the quantity f(t). Differential simply refers to the quantity f(t) changing, and that the DIFFERENCE over some period of time is the RATE of the decays occurring. In a second, or a millionth of a second however many atoms decayed is therefore the differential during that interval of time.

To say that the rate of radioactive decay is (linearly) dependent on the quantity of the undecayed atoms left could therefore be written:

f'(t) = k * f(t)

where k is some constant number.

This is called a differential equation for radioactive decay.

We can try to solve this equation, which means to Integrate that equation, which essentially means to add up the effects of the decays for every fraction of a second to see what the total result would be.

To make solving it easier, we first divide both sides by f(t) to get:

(f'(t) / f(t)) = k

SOLVING this means "mathematically Integrating" this equation. This really means adding up the effect of all those seconds or millionths of seconds mentioned above. For a simple example, which is involved below, the Integral of "t" (the time) is just the total of adding up all those individual millionths if a second, which then totals the whole time interval being considered, in other words, t. When you actually calculate an Integral, you specify a starting value and an ending value, in this case the time of starting and the time of ending. Therefore, the result of that Integral of the time is simply the whole interval of time specified.

We are now going to Integrate the equation above, over a time period that we specify as starting at some chosen moment, where there is some specific number of atoms that have not yet decayed, and specify as an ending time EITHER some specific time later OR that there is some fraction of the atoms that have not yet decayed.

There is an Integral symbol that indicates this process, resembling a giant S.

So Integrating that equation is written:

Integral Notation-a

This is actually written a little differently, to indicate the limits of the Integration, in this case the starting and ending times. We are consider the starting instant to be time 0, and the ending time to be t.

Integral Notation-b

We then make a "change of variable" in defining u = f(t), which makes du = f'(t) * dt. We now have:

Integral Notation-c

This is a standard equation for an Integral, and there are tables of Integral solutions that show that the Integral of the 1/u on the left side is the natural log of u, or ln(u). We therefore now have as a solution to our problem:

Integral Notation-d

This is ln(f(t)) - ln(f(0)) = k * t


ln (f(t)/f(0)) = k * t

This is the same as:

(f(t)/f(0)) = ekt

We therefore have:

f(t) = f(0) * ekt

This is establishing that the remaining quantity of atoms remaining at any time t is proportional to the EXPONENTIAL of the time. This is why it is called an exponential decay.

For radioactive decay, it is usually advantageous to write this equation in a different form, with a different base than e on the right side, specifically TWO. Also, since this is a DECAY, the number f(t) will always be less than f(0), so the exponent will always be negative, and it is usually written:

f(t) = f(0) * 2-ct

Using this form, if we consider the situation where f(t) is exactly 1/2 of f(0), we have:

0.5 = 2-ct

The product c * t therefore would have to be exactly 1. The time interval represented by 1/c is therefore called the half-life. (The half-life of radium is around 1600 years, so our c would then be 1/1600. If we do this equation for a date 1600 years after a start, our exponent would be 1600/1600 or the 1 that we just described. After one half-life, the remaining undecayed atoms have been reduced to half, after a second half-life, to one-fourth, and for any other time interval, similar.

This is the rigid mathematical proof which establishes the relationship between the half-life interval and any interval of time; if you know one you can calculate the other. For radioactive age dating, the ending proportion of the decayed and undecayed atoms can be counted/measured, and this equation therefore calculates the number of half-lives since the process began. By knowing the length of the half-life interval for any specific radioactive isotope, we can know actual numbers of years.

Notice that we did not need to ASSUME anything regarding an equation, and we derived this final equation purely by mathematics.

In the source radioactive material, each atom is called a "parent" atom. At the moment of radioactive decay, radiation is emitted and the parent atom becomes converted to a "daughter" atom. The daughter atom is always either a different isotope of that element or (much more often) it has changed into an entirely different element. In either case, it is usually possible to determine the presence of the daughter atoms.

Early in the radioactive decay process (a young sample), an analysis of a sample would show very few daughter atoms among a vast majority of parent atoms. After one half-life of time has passed, such an analysis would show about an equal number of parent and daughter atoms, indicating that half of the parent atoms had already decayed into daughter atoms and half of them had not yet decayed, to do so in the future.

For example, Radium-226 has a half-life of 1600 years. When it decays, this atom splits into Radon-222 and Helium-4. As it happens, Radon-222 does not naturally occur by itself. Therefore, if a sample of a material was found that showed about the same number of Radium-226 and Radon-222 atoms in it, it is reasonable to conclude that 1600 years ago there was all Radium-226 and no Radon-222 present. The Helium-4 is a gas that would probably have seeped out of the rock during the 1600 years. Also, Helium-4 is normal Helium and, if any was found, it would not be certain whether the Helium found was natural or from the decay process.

If the sample was found instead to show three times as many Radon-222 atoms as Radium-226 atoms, that would imply that 3200 years ago, it was all Radium-226. After 1600 years, it would have been half Radium-226 and half Radon-222. The half that was Radium would have continued to decay, so that half (of the original) would have become (after another 1600 years) one-fourth (of the original) Radium-226 and one-fourth Radon-222. Adding that up, we get one-fourth Radium-226 and three-fourths Radon 222 being present after two half-lives or 3200 years.

This is the basis of radiometric age dating. It is usable for samples of material from about one-tenth of a half-life to about ten half-lives. For example, an object that initially included Radium-226 16,000 years ago, would have gone through ten half-lives by now. The proportion of Radium-226 to Radon-222 would be one part in 210 or 1/1024. The reason for this limit of ten half-lives is just because of practical measurement matters. The procedures of accurately measuring numbers of atoms are less accurate when such large differences exist between numbers of parent and daughter atoms. This is why radiometric dating is not accurate beyond about ten half-lives.

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The other important factor is that at least one of the daughter atoms is somehow unusual, not naturally occurring. If a decay process only resulted in normal Nitrogen, for example, it would be impossible to tell the resultant atoms from the many naturally occurring Nitrogen atoms, so the count-comparison could not be done. If it can be absolutely known that the resultant atoms were permanently fixed in place, then even common atoms could be accurately counted.

There is a visual aid that can help in understanding part of this concept, and it is fun, too! And possibly thirst-quenching!

Open a new bottle or can of any carbonated soda and pour it in a clear glass, with no ice. Now sit and watch the surface of the liquid. Shortly after you have poured it, have a friend time out ten seconds and you try to count how many bubbles break the surface in that time. It will be a lot! Three minutes later, there might be half as many bubbles breaking the surface in a ten-second interval. That would be similar to being after one half-life. Without shaking or disturbing the glass of soda, you know that fewer and fewer bubbles will break the surface as time goes on. As some of the carbonation escapes, there is less carbon dioxide still dissolved in the liquid, so less is still available to make new bubbles. Have you ever seen a glass of carbonated soda after a full day. We call it "flat" because it has so few bubbles that still break the surface, because nearly all of the carbon dioxide has escaped from the liquid. But even after a full day, if you watch carefully, every once in a while, a bubble comes up! When you had just opened it, you had thousands of bubbles to count, and it was pretty easy to tell when there were half as many. But now, when you only see one bubble every other minute, how accurately will you be able to tell when it slows to half of that? That is sort of the reason why there is a practical limitation of around ten half-lives for accurate results in radioactive dating procedures.

Fortunately for science, there are many radioactive isotopes, and they have a wide range of half-lives. For example, Iodine-131 has a half-life of 8.1 days, and Uranium-238 has a half-life of 4.5 billion years.

Carbon-14 or C-14

For historical dating of artifacts of ancient human civilizations, one especially useful radio-isotope is Carbon-14. Its half-life is very accurately known to be 5730 years. Normal carbon is Carbon-12, which is not radioactive. The specific value in Carbon-14 as a dating tool lies in the fact that both isotopes of carbon exist in our atmosphere and both can become carbon dioxide when combined with oxygen. If you're curious as to how there could be "natural"Carbon-14, then you're thinking! Any Carbon-14 that existed when the earth formed, billions of years ago, would obviously have decayed LONG ago! The Carbon-14 in the atmosphere is ONLY there because it is continuously being formed high in the atmosphere, as a result of very high energy cosmic rays hitting and damaging atoms (of the standard Nitrogen which is the bulk of our atmosphere) very high in the atmosphere. It is generally thought by scientists that the number of cosmic rays hitting our atmosphere has been pretty constant, at least for such a brief period as 50,000 years, so the amount of new Carbon-14 being constantly created is also pretty constant. The total amount (percentage) of Carbon-14 that exists is therefore due to the balance between the continuous production and its natural continuous radioactive decay. The actual amount of Carbon-14 happens to be a VERY small fraction, compared to the amount of Carbon-12. For every 1,000,000,000,000 normal Carbon atoms, there is between one and two Carbon-14 atoms (actually around 1.47)

Interestingly, we know the total mass of the Earth's atmosphere to be around 5.136 * 1018 kg. Carbon dioxide is around 380 parts per million of it, by volume, and after we also account for the density of carbon dioxide gas, we find that there is currently 3.0 * 1015 kg of carbon dioxide in the atmosphere. The actual total amount of carbon in the entire atmosphere (in those carbon dioxide molecules) is therefore around 8.1 * 1014 kg. The proportion above tells us that the TOTAL amount of C-14 in the entire Earth's atmosphere RIGHT NOW is around 1190 kg! A grand total of only around 2600 pounds of Carbon-14 in the entire Earth's atmosphere! A REALLY small amount!

Let's say that a tree lived 5730 years ago. In the process of living, that tree would take in carbon dioxide from the atmosphere and release oxygen, with BOTH KINDS of the carbon atoms becoming parts of the structure of the tree. As long as the tree lived, this process would continue, so the proportion of Carbon-14 to carbon-12 in the wood of the tree would be constant (at the expected proportions). However, after the tree died, it no longer collected any carbon atoms from the carbon dioxide of the atmosphere. Its carbon content would be fixed at what it was when it died. However, AFTER it died, the Carbon-14 atoms in it would do their radioactive decay thing. The daughter atoms from Carbon-14 decay are Nitrogen-14, which is common nitrogen in the atmosphere. Therefore, it is not possible to generally measure the parent-daughter proportion to determine how many half-lives have passed. Instead, the proportion of Carbon-14 to Carbon-12 is measured. If that proportion is half of what is the known natural proportion, the tree or other organic material is considered to have died one half-life ago. In the example of our tree, we would find just that, that the proportion of Carbon-14 to Carbon-12 was half of what would normally be expected. The conclusion is that the other half of the original Carbon-14 atoms had already decayed into Nitrogen-14, which probably seeped out of the object to the atmosphere.

Now, with C-14 only being about one one-trillionth of the carbon in an object, it might seem that there would only be a few such atoms in a sample. Nope! A one-pound piece of wood contains about half a pound of carbon. Using Avogadro's number, we can calculate that there are around 1.1 * 1025 atoms of carbon in it. one one-trillionth of that is 1.1 * 1013 atoms of Carbon-14. That is 11,000,000,000,000 atoms of Carbon-14 in that single chunk of wood! It turns out that around 3,800 of those C-14 atoms would decay every minute, plenty to be measurable!

It is fortunate that humans have long cut down trees to use for wood throughout human activities! Wood tools, decorations, structures, weapons, or anything else which contains carbon, can be sampled for C-14 dating, as long as contamination does not occur. (A sample would not be taken from any surface but a piece would be cut with a sterilized blade, to get to an inner piece of the wood that was less likely to be contaminated.

C-14 dating would work for animals, too, but it is relatively rare that any portions of an actual animal remain after even a few hundred years. If any bones of the skeleton remain, or any hair, or fingernails, C-14 might be usable. When frozen carcasses are found, there are many possible sources, so several different C-14 analyses can be done to compare the results. Keep in mind that C-14 only works for plants or animals that once absorbed or used Carbon from the atmosphere, usually as Carbon Dioxide), so it is useless for rocks, metals, glass, etc.

Note also that the C-14 method determines the date the plant or animal DIED (and stopped absorbing carbon dioxide from the atmosphere) and not a birth date. Current methods are not precise enough to be that accurate anyway. This method was only invented around 60 years ago, and improvements in precision of the results keep occurring. But because of the extremely tiny fraction of carbon that is C-14, the accuracy will probably always be limited to the nearest decade or two.

Most radiometric dating methods do not have the problem of decaying into an extremely common isotope like Nitrogen-14. They decay into easily recognized and unique daughter isotopes, so the parent-daughter proportion can be directly determined. The only remaining possible source of error for those radio-isotopes is in whether any daughter atoms had escaped from the sample, which would affect the proportion we would measure, generally making the object actually older than measured.

Carbon-14 dating, because of this inability to compare parent-daughter numbers, has the possibility of a couple of errors in accuracy. In order for this Carbon-14 method to be used, it must be assumed that the natural proportion of Carbon-14 to Carbon-12 was the same when the tissue was living as it is now. There is no easy way to confirm that that is true for the ancient past. If, for example, 5730 years ago, the Carbon-14 to Carbon-12 proportion was HALF of today's value, a modern sample of such an organic tissue would show one-fourth of the present expected amount, implying an age of two half-lives or 11,460 years.

Related to this problem is the fact that any original Carbon-14 from the Earth's formation is long gone, and the only Carbon-14 that exists (or that existed when an ancient plant or animal lived) was created in the Earth's upper atmosphere by cosmic rays converting atoms of the atmosphere's normal Nitrogen-14 into radioactive Carbon-14. This process happens continuously (and randomly). The question is whether its rate of creation of Carbon-14 is constant or not. Usage of Carbon-14 for age dating organic materials is based on assuming that the rate at which cosmic rays enter our atmosphere is relatively constant, and that the amount of Nitrogen-14 in our upper atmosphere has been relatively constant, and that no external effect, higher in the atmosphere, could affect the constancy of the rate of this creation process. Again, there is no easy way to confirm that the rate of creation of Carbon-14 or even the rate of entry of cosmic rays into our upper atmosphere, are constant.

Fortunately, Carbon-14 dating has such a short half-life (5730 years) that major changes in the structure of our atmosphere seem unlikely during the short time periods involved. Even though it might be true that Carbon-14 creation was considerably different millions or billions of years ago, it is probably safe to assume that it has been reasonably constant for the past 30,000 years. A similar relative safety can be therefore assumed in believing that the relative proportion of Carbon-14 to Carbon-12 has not significantly changed in such a short geologic period of time. Some researchers feel that possibly a 5% variation (fluctuation) might occur in the proportion of C-14 in the atmosphere. There is really not any reliable way to confirm even this, but it seems at outer limit to the size of errors. It seems likely that if a sample is of a large enough size and is uncontaminated, better than 1% accuracy is usually possible. For our example tree, we could probably be really sure that it died within about 57 years of the 5730 years ago that our analysis would indicate.

For these reasons, even though Carbon-14 dating is actually subject to some possible errors, the reality is that it is very likely to be virtually as accurate as possible based on the method used for determining the relative proportions of Carbon-14 and Carbon-12. The dates established by Carbon-14 dating for artifacts from known civilizations (Greek, Roman, Egyptian, etc), where archaeological dates have been accurately established (often from date inscriptions on the objects!) C-14 testing has generally given results within ten or twenty years of what archaeologists had previously determined.

As long as a researcher makes sure of a reliable sample, and it is uncontaminated, pretty accurate C-14 dating is reliable. A NEW sample that contains exactly one gram (1/28 ounce, a very small amount) of carbon, is expected to produce around 14 decay events each minute of the C-14 in it. Instead of trying to confirm that there is actually 0.000000000001 gram of C-14 in that tiny sample with a scale, researchers have found that counting the electrons that get emitted during the (beta) decay is far more accurate. Those electrons ALWAYS carry away (have kinetic energy of) 156,480 electron-volts of energy. There are several experimental procedures that can identify such electrons. If a sample is known to have exactly one gram of carbon in it (pretty easy to determine), and we get the 14 decays counted each minute (or actually 840 in an hour, a more accurate way to count), then we would know that the object was essentially "new", such as wood that was very recently cut down. If we only got 420 counts in that hour, like from our example tree, we would know that one half life had passed for it and it was 5730 years since it died.

Considering our hypothetical one-hour sample counting further: If we had gotten a count of 840 in an hour, it would mean that 0 years had passed since the tree died. If we had gotten 420 it would have meant 5730 years. But how about if we missed just a single count during that hour, where we counted 419 instead? That would give a result of about 20 years more, or 5750 years. This is basically why such results (for samples of that age) are often given as "within 20 years".

Now, let's consider that same hour-long measurement for a sample that is ten times as old, around 57,000 years old. Well, a total of ONE count during the entire hour would represent 55,570 years while TWO counts would represent 49,850 years. In this case, if we missed just a single count, we could be 6,000 years off! THIS is why C-14 is not considered reliable for artifacts of ten or more half-lives old.

We could now consider an object around 30,000 years old. Well, 22 counts in our hour-long measurement would represent 30,060 years while 23 counts would represent 29,690 years. This is a 370 year difference, a little more than 1% of the age, and we could probably be fairly confident that the artifact was very close to 30,000 years old. Carbon-14 is therefore considered to be of usable accuracy for artifacts up to around 30,000 years old, but no older.

OK. When we consider artifacts from the time of Jesus, two thousand years ago, accuracy gets much better! A count of 659 represents 2002 years ago, while a count of 660 represents 1990 years ago. This suggests that we can get C-14 dates that are within around ten years of the actual dates, for that era!

For a sample from around 1300 AD (700 years ago) such as the Shroud of Turin, we start getting nearer the other end of the limitation of the precision accuracy of C-14 dating. A count of 772 would represent an age of 697 years, while a count of 771 represents 707 years. This is only a ten year range, which seems excellent. But keep in mind that our expectation for a NEW sample would be 840. If there were slight variations in the proportion of carbon in the sample, or if there was a slight contamination, the whole scale could be shifted. Maybe the contaminated sample, when new, would have given 843 counts instead of 840. That effect would shift the time scale by around 30 years. For the more ancient artifacts, such a shift of 30 years would not be too significant, but for objects from recent history, we expect better precision than that!

In principle, C-14 dating could be used to determine the date that the wood of your house was cut down. It actually has been used to try to confirm the construction date of some houses that may have been important in Revolutionary times (around 1776) and has shown that a number of such houses were actually built of trees cut down in the 1810s and 1820s! Yes, for your house, a count of 839 would represent 10 years old, but that depends on the precise chemical composition of the wood and the precise accuracy of the 840 we are basing everything on! Again, an experimental error of 30 years either way is realistic. So, to pay for a procedure to analyze the wood of your house (from the 1950s), and only get a result of 30 years would seem a waste of money! This is why C-14 is rarely used for relatively recent artifacts.

By the way, this subject offers an example of just how tiny atoms are! Considering that there are only around 450 kg of C-14 in the entire atmosphere at any time, and the whole atmosphere weighs around 5,100,000,000,000,000,000 kg, one might wonder how much C-14 we might actually encounter! Well, I did! If you breath in 0.5 liter of air in a breath, an average-sized breath, calculations show that there are around 2,000,000 C-14 atoms in that breath! So, yes, we encounter plenty of C-14, and in a whole life of breathing, we get the chance to absorb a lot of C-14 into the structures of our bodies! So plants and trees and all the animals, including us, that eats them, certainly have the expected proportion of c-14/C-12 in us. There really is no way that any of us could avoid it! As long as we keep eating plants (and animals that ate plants), and as long as those plants and trees are alive, this is certainly true. It is only after the tree, plant or animal dies that the proportion of C-14 begins to drop, because the C-14 atoms that are decaying are no longer being replaced by new ones.

There seem to be some people who want to disagree with ages determined by C-14 dating, and they simply dismiss the C-14 dating method or radioactive dating in general as worthless. Often, such people have no knowledge about how the process is done or why it works. This presentation is intended to show how and why these are generally reliable methods. The primary reason they could give results that were seriously wrong has to do with the tester being extremely careless and using a sample that is fouled with contamination. If a researcher is very careful, that chance can be nearly completely eliminated and extremely accurate and useful dates can be obtained.

This presentation was first placed on the Internet in December 1998.

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C Johnson, Theoretical Physicist, Physics Degree from Univ of Chicago