Science uses radioactive isotopes to determine the age of many objects from the ancient past. One of the most wellknown is the Carbon14 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 'bellshaped curve'. With such an enormous number of separate events, the bellshaped 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.


Technical StuffThere 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:
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.
We then make a "change of variable" in defining u = f(t), which makes du = f'(t) * dt. We now have:
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:
This is ln(f(t))  ln(f(0)) = k * t Or ln (f(t)/f(0)) = k * t This is the same as: (f(t)/f(0)) = e^{kt} We therefore have: f(t) = f(0) * e^{kt} 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 halflife. (The halflife 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 halflife, the remaining undecayed atoms have been reduced to half, after a second halflife, to onefourth, and for any other time interval, similar. This is the rigid mathematical proof which establishes the relationship between the halflife 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 halflives since the process began. By knowing the length of the halflife 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 halflife 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, Radium226 has a halflife of 1600 years. When it decays, this atom splits into Radon222 and Helium4. As it happens, Radon222 does not naturally occur by itself. Therefore, if a sample of a material was found that showed about the same number of Radium226 and Radon222 atoms in it, it is reasonable to conclude that 1600 years ago there was all Radium226 and no Radon222 present. The Helium4 is a gas that would probably have seeped out of the rock during the 1600 years. Also, Helium4 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 Radon222 atoms as Radium226 atoms, that would imply that 3200 years ago, it was all Radium226. After 1600 years, it would have been half Radium226 and half Radon222. The half that was Radium would have continued to decay, so that half (of the original) would have become (after another 1600 years) onefourth (of the original) Radium226 and onefourth Radon222. Adding that up, we get onefourth Radium226 and threefourths Radon 222 being present after two halflives or 3200 years.
This is the basis of radiometric age dating. It is usable for samples of material from about onetenth of a halflife to about ten halflives. For example, an object that initially included Radium226 16,000 years ago, would have gone through ten halflives by now. The proportion of Radium226 to Radon222 would be one part in 2^{10} or 1/1024. The reason for this limit of ten halflives 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 halflives.
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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 tensecond interval. That would be similar to being after one halflife. 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 halflives for accurate results in radioactive dating procedures.
Interestingly, we know the total mass of the Earth's atmosphere to be around 5.136 * 10^{18} 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 * 10^{15} 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 * 10^{14} kg. The proportion above tells us that the TOTAL amount of C14 in the entire Earth's atmosphere RIGHT NOW is around 1190 kg! A grand total of only around 2600 pounds of Carbon14 in the entire Earth's atmosphere! A REALLY small amount!
Now, with C14 only being about one onetrillionth of the carbon in an object, it might seem that there would only be a few such atoms in a sample. Nope! A onepound piece of wood contains about half a pound of carbon. Using Avogadro's number, we can calculate that there are around 1.1 * 10^{25} atoms of carbon in it. one onetrillionth of that is 1.1 * 10^{13} atoms of Carbon14. That is 11,000,000,000,000 atoms of Carbon14 in that single chunk of wood! It turns out that around 3,800 of those C14 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 C14 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.
C14 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, C14 might be usable. When frozen carcasses are found, there are many possible sources, so several different C14 analyses can be done to compare the results. Keep in mind that C14 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 C14 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 C14, the accuracy will probably always be limited to the nearest decade or two.
Carbon14 dating, because of this inability to compare parentdaughter numbers, has the possibility of a couple of errors in accuracy. In order for this Carbon14 method to be used, it must be assumed that the natural proportion of Carbon14 to Carbon12 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 Carbon14 to Carbon12 proportion was HALF of today's value, a modern sample of such an organic tissue would show onefourth of the present expected amount, implying an age of two halflives or 11,460 years.
Related to this problem is the fact that any original Carbon14 from the Earth's formation is long gone, and the only Carbon14 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 Nitrogen14 into radioactive Carbon14. This process happens continuously (and randomly). The question is whether its rate of creation of Carbon14 is constant or not. Usage of Carbon14 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 Nitrogen14 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 Carbon14 or even the rate of entry of cosmic rays into our upper atmosphere, are constant.
Fortunately, Carbon14 dating has such a short halflife (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 Carbon14 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 Carbon14 to Carbon12 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 C14 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 Carbon14 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 Carbon14 and Carbon12. The dates established by Carbon14 dating for artifacts from known civilizations (Greek, Roman, Egyptian, etc), where archaeological dates have been accurately established (often from date inscriptions on the objects!) C14 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 C14 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 C14 in it. Instead of trying to confirm that there is actually 0.000000000001 gram of C14 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 electronvolts 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 onehour 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 hourlong 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 C14 is not considered reliable for artifacts of ten or more halflives old.
We could now consider an object around 30,000 years old. Well, 22 counts in our hourlong 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. Carbon14 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 C14 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 C14 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, C14 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 C14 is rarely used for relatively recent artifacts.
Conservation of Angular Momentum  An Exception or Violation (Sept 2006)
Galaxy Spiral Arms Stability and Dynamics A purely Newtonian gravitational explanation (Nov 1997, Aug 1998)
Twins Paradox of Relativity Is Absolutely Wrong (research 19972004, published Aug 2004)
Perturbation Theory. Gravitational Theory and Resonance (Aug 2001, Dec 2001)
Origin of the Earth. Planetary Gravitational Resonances (Dec 2001)
Rotation of the Sun (Jan 2000)
Origin of the Universe. Cosmogony  Cosmology (more logical than the Big Bang) (devised 1960, internet 1998)
Time Passes Faster Here on Earth than on the Moon (but only a fraction of a second per year!) (Jan 2009)
Globular Clusters. All Globulars Must Regularly Pass Through the cluttered Galaxy Plane, which would be very disruptive to their pristine form. (Nov 1997, Aug 1998)
Existence of Photons. A Hubble Experiment to Confirm the Existence of Individual Photons (experimental proof of quanta) (Feb 2000)
Origin of the Moon  A New Theory (June 2000)
Planetary Rotation of Jupiter, Saturn, and the Earth (Jupiter has a lot of gaseous turbulence which should have slowed down its rapid rotation over billions of years) (March 1998)
Cepheid Variable Stars. Velocity Graph Analysis (Feb 2003)
Compton Effect of Astrophysics. A Possible New Compton Effect (Mar 2003)
Olbers Paradox Regarding Neutrinos (Oct 2004)
Kepler and Newton. Calculations (2006)
Pulsars. Pulsars May Be Quite Different than we have Assumed (June 2008)
Sun and Stars  How the Sun Works  Nuclear Fusion in Creating Light and Heat (Aug 2006)
Stars  How They Work  Nuclear Fusion. Lives of Stars and You (Aug 2004)
Sundial Time Correction  Equation of Time. Sundial to ClockTime Correction Factor (Jan 2009)
General Relativity  A Moon Experiment to Confirm It. Confirming General Relativity with a simple experiment. (Jan 2009)
General Relativity and Time Dilation. Does Time Dilation Result? (Jan 2009)
Geysers on Io. Source of Driving Energy (June 1998)
Mass Extinction, a New Explanation. A New Explanation for Apparent Periodicity of Mass Extinctions (May 1998, August 2001)
Precession of Gyroscopes and of the Earth. Gyroscope Precession and Precession of the Earth's Equinoxes (Apr 1998)
Ocean Tides  The Physics and Logic. Mathematical Explanation of Tides (Jan 2002)
Earth's Spinning  Perfect Energy Source (1990, Dec. 2009)
Earth's Magnetic Field  Source and Logic. Complex nature of the magnetic field and its source (March 1996)
Earth Spinning Energy  Perfect Energy Source From the Earth's Spinning (1990, Nov. 2002)
Nuclear or Atomic Physics Related Subjects:
Nuclear Physics  Statistical Analysis of Isotope Masses Nuclear Structure. (research 19962003, published Nov 2003)
Quantum Defect is NOT a Mathematical Defect It Can Be Calculated The Quantum Defect is a Physical Quantity and not a Fudge Factor(July 2007)
Atomic Physics  NIST Atomic Ionization Data Patterns Surprising Patterns in the NIST Data Regarding Atomic Ionization (June 2007)
Nuclear Physics  Logical Inconsistencies (August 2007)
Neutrinos  Where Did they all Come From? (August 2004)
Neutrinos  Olbers Paradox Means Neutrinos from Everywhere (Oct 2004)
Quantum Nuclear Physics. A Possible Alternative (Aug 2001, Dec 2001, Jan 2004)
Quantum Physics  Quantum Dynamics. A Potential Improvement (2006)
Quantum Physics is Compatible with the Standard Model (2002, Sept 2006, Oct 2010)
Quantum Dynamics (March 2008)
Ionization Potential  NIST Data Patterns. Surprising patterns among different elements (March 2003)
Mass Defect Chart. (calculation, formula) (research 19962003, published Nov 2003)
Assorted other Physics Subjects:
Precession of Gyroscopes and of the Earth. Gyroscope Precession and Precession of the Earth's Equinoxes (Apr 1998)
Earth's Magnetic Field  Source and Logic. Complex nature of the magnetic field and its source (March 1996)
Earth Spinning Energy  Perfect Energy Source (1990, Nov. 2002)
Earth Energy Flow Rates due to Precessional Effects (63,000 MegaWatts) (Sept 2006)
Accurate Mass of the Earth. Gravitational Constant  An Important Gravitation Experiment. (Feb 2004)
Tornadoes  The Physics of How They Operate, including How they Form. Solar Energy, an Immense Source of Energy, Far Greater than all Fossil Fuels (Feb 2000, Feb 2006, May 2009)
Radiometric Age Dating  Carbon14 Age Determination. Carbon14, C14 (Dec 1998)
Mass Extinction, an Old Explanation. An Old Explanation for Apparent Periodicity of Mass Extinctions (Aug 2003)
Hurricanes, the Physics and Analysis A Credible Approach to Hurricane Reduction (Feb 2001)
Sundial Time Correction  Equation of Time. Sundial to ClockTime Correction Factor (Jan 2009)
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C Johnson, Theoretical Physicist, Physics Degree from Univ of Chicago