Carbon dating and the Math

One would have to be a hermit not to have heard about carbon dating.  This is the dating, for instance, of a piece of wood in an old building or a piece of charcoal in an archaeological dig.

At a first approximation, the physics is pretty straight forward.  An atom consists of a nucleus with electrons whizzing around the nucleus.  Which element the atom is depends on the number of electrons and the number of electrons, in turn, depends on the number of protons in the nucleus.  In a normal, unionized atom, the number of electrons and protons are equal and the atom is neutrally charged.

The glue that holds these positively charged protons together in the nucleus (remember like charges repel each other) are the neutrons.  Don't ask me how they do this.  The explanation is way above my pay grade.  Very roughly speaking, there are the same number of neutrons as protons but this can vary.  Carbon, for instance, can exist in a state with 6protons and 6 neutrons for an atomic mass number of 12. It can also exist in a form with 6 protons and 8 neutrons for a mass number of 14.

These two types are called isotopes of Carbon.  There is a third one but it is not needed for this explanation.

Some isotopes are stable, some are not (why is also above my pay grade).  In the case of Carbon, 12 is stable, 14 is not. 

Carbon 14 disintegrates into Nitrogen 14 with the ejection of an electron from one of it's neutrons.  The neutron becomes a proton so the atom is now a new element with 7 protons and 7 neutrons, hence ¹⁴N.

No one knows when any individual Carbon 14 atom is going to disintegrate.  There is a very small probability at any one moment but when you have a lot of ¹⁴C, you can predict how many atoms will change to ¹⁴N in any given time period.  This results in something interesting which has been observed experimentally.  If you know how much of the radioactive element you have, you will observe that half of it will break down in a given time, referred to as it's half life.  The half life of various radioactive isotopes varies from tiny fractions of a second to many millions of years.

In the case of ¹⁴C, it's half life is 5730 years give or take 40 years.

In 5730 years you will have half left, in another 5730 years, a quarter of the original amount, in one more half life, one eighth of the original amount and so forth.


So now we need the math for this.  We will work out what I call the straight forward formula and then we can change it around (solve for other parts) so that each component of the formula becomes the subject.

First a note on mathematical notation.

What is meant when you see a symbol.

xA means multiply the A by x.  If A is 2 and x is 3 then xA is 6

A^x means multiply A by itself x times.  If A is 2 and x is 3 then A^x is 8.  In words, A is raised to the xth power.

However, in the symbols A_x,  x is not an operator.  ie, it doesn't say to do anything.  It is a label.  It means the xth A.  For instance you could have A₁, A₂, A₃ etc.  This is the first, second and third A.  Or Ao and At which for our purposes will mean A at time zero and A at a specified future time.

There is a special one in Chemistry.  I'll use Carbon since this is what we are talking about.  For instance ¹⁴C.  This means the carbon atom with 14 nucleotides.   ie, The sum of neutrons and protons adds up to `14.  There also exist ¹²C and ¹³C.  Of course both have 6 protons or it wouldn't be Carbon.  The number of neutrons varies.

And one more in Math.  If the subscript is after the word log such as log₅ then it means log to the base 5.  If only log is used, it is understood it is to the base 10.   That is to say, log = log₁₀ and if ln is used it is to the base 'e'.  Don't worry about it, we don't need 'e'.  I only mention it because it is on your little hand held computer and you might wonder.

Lets go back to the basics.  Every half life period, (h) the amount is halved. In the case of Carbon, the half life is 5730 years but half lives for other isotopes varies hugely.   Lets call the amount we start with as A_o (A at time zero) and the amount we are left with as A_t (A at some time t in the future).  The amount we will have left after one half life is:

1.   A₁ = A_o(1/2)¹

After two half lives
2.    A₂ = Ao(1/2)²

After three half lives
3.    A₃ = Ao(1/2)³
Remember 1/2 times 1/2 is 1/4.   Multiply once more by 1/2 and you have 1/8.  When you see a times sign between fractions, replace it in your mind with "of".  then 1/2 x 1/2 becomes one half of one half.

The 1,2 and 3 are the number of half lives that have gone by.

4.  So A_n - A_o(1/2)ⁿ  or in words, to find the amount of a substance after n half lives have gone by, multiply A_o, the initial amount, times 1/2 raised to the nth power.

Note that in the notation A_x,  x means the amount at time x expressed in half lives.

Also note that even if the n is not a whole number and therefore would take a wee bit of higher math (knowing logarithms), to solve, your computer does this with no problem.  Your high school computer can solve, for instance, 6^3.22 without raising a sweat.

Suppose we start with one gram of a radioactive substance and one half life has gone by.  We simply multiply 1gram times 1/2

Suppose 4 half lives have gone back.  We multiply the one gram times (1/2)⁴.  that is to say by 1/2 times 1/2 times 1/2 times 1/2 which equals 1/16th times the original amount.

Now suppose we know what the half life (h) of a particular isotope is.  Say it is 10 years, for simplicity.  Say 30 years have gone by.  Obviously 3 half lives have past.  In other words n, the number of half lives equals the time elapsed (t) divided by the Half life (h).  In this case n = 30/10 = 3.

5.   n=t/h.

And, as I said, it doesn't have to be a whole number.  If the half life is 10 years and 75 years have gone by then n = 75/10 = 7.5.  With simple math we would have a problem raising a number to a fractional exponent but your computer has no such problem so don't sweat it.

You can see where this is leading.  Since n=t/h, we can substitute t/h into the formula where we see n.

The radioactive decay formula then becomes

6.  A_t = A_o(1/2)^t/h
or in words, to find the amount of radioactive material remaining after time t, multiply A_o, the initial amount, times one half raised to the power of t/h.


Good heavens!  I forgot to tell you where the radioactive Carbon comes from.  If it's half life is only 5730 years, in about 50,000 years there will be so little of it that carbon dating is out of the question and the world has been here for over 4b years.  Clearly, ¹⁴C must be being created somewhere.  the 'Where',, is in the upper atmosphere.  As cosmic rays hit the upper atmosphere, they are so energetic that they cause some nuclear reactions and one of these is to change some¹⁴N into ¹⁴C.  It is a very small amount but enough to be detected in living material with modern methods so we have a clock we can use.  When an organism dies it stops taking up carbon and the clock starts to tick.  If we  analyze it sometime in the future, we can know when it died (up to about 50,000 years).

Now we can do what a mathematician calls solving for A_o or for t or for h.  In other words we re-arrange the formula so that each of these terms in turn become the subject of the formula (ie. is by itself on the left and everything else is  on the right). I'll tell you what each variation of the formula is good for as we rearrange them.

The basic principle of solving for a factor (one of the letters) in a formula is that we can do anything we want to one side as long as we do the same to the other side.  After all if I have a formula that 7 = 3+4, if I multiply both sides by, say, 5, the formula is still correct.  Of course we don't just do random things to both sides of the formula. The trick is to do something that gets us closer to the solution we are looking for.

One other thing.  At one point in the procedure I am going to have to take a log of both sides.  Even if you don't understand logarithms, this should pose no emotional problem since I am doing the same to both sides.  Then, however, you are going to have to take my word for a 'log identity'.  If you are into logarithms, you will understand why the identity holds but if not, don't sweat it.  It is true.  This identity is:

log_ab^c = clog_ab.  Incidentally, the inverse of the left side of this formula is a^c =b.  That may give you a clue why the identity works.

In words:   log to the base 'a' of 'b' raised to the 'c'th power equals c times the log to the base a of b.

So let's start.  I want to end up with a formula for each of the terms, in turn, on the left side of the equation.

The original equation is

A_t = A_o(1/2)^t/h

Let's divide each side by (1/2)^t/h.  Note that this cancels out the (1/2)^t/h on the right side and leaves it on the left in the denominator*.  It is more conventional to have the subject of the formula on the left so we will exchange them.  After all if 7 = 3+4 then 3+4 = 7.  Our formula then becomes

* The bottom part of a fraction.

A_o = A_t divided by (1/2)^t/h. Don't know how to get my computer to write this so I will leave you to write it down on a piece of paper.

Use
So what is this formula good for.  It was noted early on in the use of carbon dating that there were some discrepancies.  With artifacts for which the exact date was known, the Carbon date did not agree.  The hypothesis was that the rate of ¹⁴C production in the upper atmosphere might not have been constant over the years.  So cores were drilled into very old trees, the rings were separated and carbon dated.  The above formula was used to work out the concentration  of carbon 14 which had been present for each year  that a ring was laid down.  And indeed it was found that the true curve diverged by a small but significant amount over time from the theoretical curve.  When the true curve was used, the dates all fell into place.
 

Now let's work on t and h.  The first thing I will do is to divide both sides by A_o.  This cancels A_o on the right side and leaves us with

A_t/A_o = (1/2)^t/h

Now I'll take the log of both sides

log (A_t/A_o) = log[(1/2)^t/h]

Remember our identity.  I can take t/h to the front of the right side so

log(A_t/A_o) = t/h(log1/2)

Now it is simple.  I simply divide both sides by log1/2 and we have t/h by themselves on the right side.  You take it from here.  Isolate t and h.  If you do it right you will find that

t = [hlog(A_t/A_o]/[log(1/2)]

and

h = [tlog1/2}/[log(A_t/A_o}

Use
How about the formula for t.  This is pretty obvious.  Now that we have the needed correction of the production of 14C over the past , we can date any object that was once alive up to about 50,000 years.  This is carbon dating.

Use
How about h.  We can't actually wait around for 5730 years to see when we have half of a quantity of radioactive carbon left.  We can, thought, observe the rate of disintegration on a shorter time span.  Using the h formula we can work out the half life of each radioactive isotope and some of them are multi millions of years.

It is never that easy

There are always complications.  Charcoal, for instance, if it is in ordinary soils or even in a cave can be colonized by micro-organisms.  If in active soil, the micro-organisms will have a modern carbon signature.  One has to first clean the charcoal of the modern material in order to get the correct date for the charcoal

Add to that, that we have been spewing carbon into the atmosphere from fossil fuel.  This is old carbon and hence contains no Carbon 14.  On the other side we have had nuclear tests in the air.  They have added Carbon 14 to the air.  For future anthropologists, they will have to take this into account.

Other types of radioactive dating have their own special requirements.  For instance when a rock melt cools, crystals form and just as a solution of salt and sugar, as it crystallizes, will  produce crystals of pure salt and pure sugar, the  crystals in a melt are of one type of molecule.  If one of these is a radioactive species and it's end product is known you can measure the concentraton of both and calculate when the rock  was melted. 

The Anthropocene

This is a book review of William F Ruddiman's book, Plows, Plagues and Petroleum.  It's premise is that the Anthropocene* didn't start some 200 years ago with the beginning of the industrial revolution and hence the burning of fossil fuels but actually started 6000 to 8000 years ago.

* The age in which humans have started to have a significant effect on the climate

In the popular literature you will often find comments such as 'we live in a very unusual period.  Our climate, compared with previous times, has been remarkably stable for thousands of years'    That is not to say completely stable.  We have had the so called little ice age for instance and the medieval warm period but compared to the climate as read in ice cores, this has been a period of great stability.

Prof. Ruddiman basis much of this contention on information from ice cores.  In Antarctica, cores have been drilled which reach ice which was deposited around 800,000 years ago.  Over this period the alteration between glacial periods and interglacial periods* has had a cycle of about 100,000 years.  Here is a most amazing graphic of the past cycles.

* Note that I say glacial and interglacial period, not ice age.  Strictly speaking, despite popular usage, an ice age is the approximately 3m year period we are in with approximately 50 or so glacials and interglacials.  If we want to use the term ice age, for instance, for the time between the previous interglacial (the Eemian) and the present interglacial (the Holocene) then we need another name for the approx. 3m year period of alternating cold and warm periods that we are in the middle of right now.

What has caused these warm and cold periods has been pretty well established as the Milankovitch cycles.  There are three of these which have different periodicities.  There is the tilt of the earth  which varies between 21.2 and 24.5 degrees from the plane of it's orbit.  it is called Obliquity for some reason.  It's period is about  41,000  years.  There is the eccentricity of the orbit which varies from round to elliptical and back with a period of 100,000 years* and there is the orientation of this ellipticity in space which will result in the earth being closest to the sun in summer or closest in winter.  This has a period of 23,000 years and is called axial precession

*  It is a little more complicated than this.  For instance Eccentricity has a number of components.  It is not a simple sin wave but that will do for now.

Adding these three cycles together you get a variability in the strength of the sun on the surface of the earth and most important, in the mid to high latitude area of the Northern Hemisphere (where most of the land is).  To go into a glacial (glacial period), the insolation (Amount of radiation reaching the earth's surface) must be low in the Northern Hemisphere summer.  This allows snow to remain over the summer and to be increased during the next winter. Then the more land that is covered continually with snow, the more solar radiation is reflected back into space and we have a feedback which accelerates the process.  I won't go into how glacials end but you can go here and here for some ideas on how this occurs.

Over many many glacial-interglacial periods it has been observed that Carbon dioxide rises as the ice melts (some controversy on why) and a little before maximum melt, Carbon dioxide begins to fall.  Following this, with the odd up-tick CO2 falls continually.  At a certain level of Carbon dioxide, combined with the right part of the Milankovitch cycle, snow begins to accumulate, bringing on the start of the next glacial.

Since the Milankovitch cycle is the sum of three cycles, each with a different period, each glacial-interglacial cycle is somewhat different.  Looking at these cycles, the two which are most like the present one that we are in are the 4th and the 9th back from our present one.

In both these cycles (and in other less similar cycles) Carbon dioxide began to fall and just continued to do so, starting a little before maximum melt and falling to about 185ppm.

Our recent (Holocene) interglacial started some 20,000 years ago by definition since that was when the ice sheet was at it's greatest extent but melting really got under way about 11,500 years ago.  And as with all other cycles, Carbon dioxide began to rise.

Then, as usual, just before maximum melt, Carbon dioxide began to fall.

If it had continued, then at a certain point, snow would have begun to accumulate again.  Apparently the 'epicenter' of ice accumulation is on the high lands of Baffin Island and somewhat later in Labrador.  It didn't happen.  Around 6000 to 8000 years ago, the concentration of Carbon dioxide began to climb in complete contrast to other cycles.  It wasn't enough to fully counteract the downswing in the  Milankovitch cycle  but greatly slowed down the cooling.

It had almost reached the level for snow accumulation when there were two catastrophic events in human History.  One was the Black Death which scythed down huge numbers of people* in Asia, the Middle East and Europe.

*  It is often noted that this was the beginning of the rise of the rights of the serfs since they were in such short supply that they could demand better conditions in exchange for their labor.

The second was the invasion of South America by the Spanish.  The Spanish brought with them a plethora of deadly diseases for which the local population had no resistance.  Disease spread through south, central and North America and decreased the population*, by some estimates, by 90%.  In both plagues forests grew up on deserted farm lands and drew down Carbon dioxide below the level needed for the beginning of snow accumulation.

*Contrary to popular opinion, archeology has now confirmed that North America was populated by a large number of people, many of them living in what we would characterize as  advanced civilizations.

There is some very interesting evidence that glaciation  started.  Around the high lands of Baffin island there is a 'halo' of dead lichen with young new lichen beginning to grow here and there.  What happened?

Apparently, snow began to accumulate and last through the summer and occupy more and more area and of course smothered the lichen.  Then  along came the industrial revolution and the snow retreated again leaving this halo of dead lichen.  We were that close to beginning, once more, to slide into a glacial.

So what did man do to slow the advent of a new glacial for long enough for the Industrial Revolution to take over and really up the concentration of this green house gas.

First there was the burning down of forests to simply roast and catch animals. Areas burnt off, and especially if burnt off regularly, became grass lands which attract grazing animals and in which it is much easier to hunt.   In Australia, this probably started around 50,000 years ago when man first reached that continent.  Then as agriculture started, forests were cleared to plant crops.  An early technique was to simply ring bark a tree and then plant a fire at the base once it had died and dried out.  As the bronze age and then the iron age took hold, we could simply fell the trees.

Very soon after that, the plow was invented.  We have seen the tremendous damage the plow can do in modern times with the destruction of the soils of the great plains in America.  These were reservoirs of huge amounts of carbon which the plow released into the atmosphere.  If you travel through the Middle East you see clearly all the exposed rock.  The soils there have not only released their carbon but have been washed into the sea.  Farming with the plow is mainly responsible.

In the Far East the cultivation of rice in ponds was developed.  Anaerobic ponds give out large amounts of Methane which is a very powerful green house gas.  It oxidizes to the less potent Carbon dioxide and so stays around in a less toxic form.  This development reversed the methane trend.  Of course to build the extensive rice ponds, often terraced up the sides of mountains, you first have to eliminate the forests.

Composting barns

I've just read an article on composting barns  in our local farming magazine.  We are re-inventing the wheel but that is OK.  I saw this system in 1989 in South Africa and they had been using it for some time.  So what are they.  First a little background science.

You can classify the break down of organic material into two main types.  Both result in simpler substances which are available for the growth of plants.  The two types are anaerobic and anaerobic.  The results are different.  With anaerobic break down, the processes are less energetic and two significant by-products are ammonia, NH₄, and Hydrogen sulphide, H₂S, (which in the air oxidizes to Sulphur dioxide and water.  SO₂  H₂0). Both Ammonia and Hydrogen sulfide are gases and go off into the air.  In doing so, they  take with them the valuable nutrients Nitrogen (N₂) and Sulfur (S). 

 

Of course the other result of the anaerobic digestion of organic material is a nasty smell.

The third produce is methane which can be used in an internal (or external) combustion engine to run an electric generator with the waste heat going to heat water, used in, for instance, a the milking shed.

Aerobic processes are far more vigorous since the strong oxidizer Oxygen (O₂) is present and aerobic breakdown only gives off Carbon dioxide (CO₂) and water. (and is virtually odor free). In aerobic break down a whole ecology of microfauna build the available nutrients into their body mass. Aerobic processes can use cellulose and lignin as a source of Carbon and energy*.  In anaerobic processes, both are refractory. As long as the source of organic carbon lasts, the aerobic waste products of each trophic level are built back into body mass by the primary producers*.  Finally, in this system, as organic carbon runs out, nutrients are released in a form that plants can use.  The ecology runs down and the final product left is Humus which has some interesting benefits for the soil.

Note that the release of nutrients from an aerobic compost is in a slow release controlled fashion which the plants have time to take up and utilize.  With the use of soluble nitrate chemical fertilizer, the plants get a heavy initial jolt of Nitrogen and rain or irrigation can wash nitrogen down into the ground water.

*  In a photosynthesis system, the primary producers are plants.  In the sea, they are single cell algae and sea weed and pytoplankton for the most part.  In a compost pile they are micro-organisms and if the source of carbon and energy is wood (cellulose) then the micro-organisms which produce cellulase, the enzyme that can cut off the sugar molecules from the cellulose, are the primary producers.


In a composting barn, you provide a source of carbon in the form of saw dust and/or wood shavings and/or straw. You could also use pelleted paper or any other source of cellulose.  Cellulose is an interesting substance.  It is a poly-sacaride.  In other words a chain of sugar molecules joined together in an insoluble form.  No multi-celled animal can digest this material.  Some bacteria, on the contrary, produce cellulase*.  While algae are the primary producers in the sea, cellulase producing micro-organisms are the base of the food chain  in a cellulose rich compost.

*  Enzymes are named for the substance that they can catalyze the use of.  Hence the enzyme that helps metabolize sucrose would be called sucrase while the enzyme that metabolizes cellulose is cellulase.

Of course the cellulose is not enough for these micro-organisms.  It is just a source of carbon and energy.  Micro-organisms need the other nutrients such as nitrates, phosphates, sulphates and all the other 'ates' to build their bodies.  They scavenge these from the environment and they themselves become food for a whole range of grazers who build these substances into their bodies.

As a rough rule of thumb, each level in the trophic chain can incorporate about a tenth of the material from the level below it.  A ton of phytoplankton can make a tenth of a ton of Krill which can make a hundredth of a ton of whale.  The remaining 90% at each transfer goes back into the soup to be used again by the primary producers.

As long as there is a source of energy, such as sunshine in the case of phytoplankton or cellulose in the case of a compost pile, all these nutrients are re-incorporated into biomass.  When the energy source runs out, there is a net release of nutrients as the various micro-organisms feed on each other but now, without enough energy and Carbon to power  the uptake of the released nutrients*. Compost is a slow release fertilizer.

* This is why it is so bad to mix saw dust into your soil.  All the free nutrients will be scavenged until the saw dust is used up.  Then nutrients will be released and the plants can start to grow again. Saw dust, by contrast, can be used as a mulch because it is only incorporated at the boarder between the soil and the sawdust.

So how about composting barns.  In these barns there are a number of requirements.  First, you need a thick layer of cellulose as bedding.  The urine and dung of the animals living-in, or visiting the barn  is absorbed by the saw dust, wood shaving or straw.  The farm we visited in South Africa used the coarse saw dust from a saw mill.  But that is not sufficient.  The bedding must be kept aerobic.  In Africa, where I first saw this method, they were growing chickens.  This is possibly easier than growing cows because the urine of birds is almost solid.  Cows, by contrast, produce copious amounts of liquid urine.  Labor in South Africa at the time was not expensive and the saw dust bedding of the chickens was stirred each day by hand.

In the case of a cow shed, one would have to have a mechanical method of stirring the bedding.  Cows go for milking and in some systems, go to graze during the day. giving a perfect time to aerate the bedding.

Note that the metabolism of all these wee beasties in the compost give off heat just as you and I do when we metabolize.  The bedding is warm and it has been reported that given a choice, cows will bed down in these barns in preference to staying outside or going into stalls with straw on the floor.

As you can imagine, ventilation is of the greatest importance as well.  No poisonous gases such as Ammonia or Hydrogen sulfide are given off but Carbon dioxide is produced.  A sloping roof with vents at the top of the slope and good access for air from the sides is vital.  The heat from the bedding and the not inconsiderable heat from the cows will create a natural convective circulation if the barn is suitably designed.  It is also useful to place the watering troughs outside the shed wall so that the cows can access it but so it does not drip into the bedding. Moisture is needed for the activity of the compost bed but too much makes it very difficult to maintain aerobic conditions.

 Also useful would be to have drop down curtains, especially on the side where the heavy weather comes from so that rain can be excluded from reaching the bedding.

In really cold climate, one could employ a really large heat exchange ventilation systems which uses a counter flow system to pass outgoing air past incoming air to keep the heat while exchanging the air.  Such systems are used on as smaller scale in air-tight houses today.

When we talked to the farmer in South Africa who was using this system for Chickens, he mentioned as an aside how disease-free his chickens were under this system.  Apparently any pathogens that fall into the bedding are on a 'hiding to nothing'.  The environment is inimical to their survival and they are destroyed by the rich fauna of composters.  Another article I read on cow sheds using this system emphasized the same phenomenon.  A reduction in pathogens can only increase production and reduce the cost of antibiotics.

 

The South African chicken farm was completely odorless.

To recap, what are the benefits of this system.

* Animal welfare.  The very fact that cows vote with their feet and choose to bed down on the compost in preference to staying out in the cold or going to a straw lined byre shows how beneficial such a system is.  It is highly likely that in such a system, the amount of milk per unit feed would increase as the cows are using less energy to keep warm and are less stressed.

* Nutrient retention.  All the nutrients from the waste products of the cows are held in the compost to be later used to enrich the soil of the farm.  Nitrogen and Sulfur do not go off as gases to be lost to the farm as is the case with anaerobic systems..

* The bedding when used on the fields is a slow release fertilizer, thus not polluting nearby water ways as is the case with chemical fertilizer.

* Odor control.  The smell of a well aerated compost is faint and pleasant in great contrast to an anaerobic compost.  The neighbors are not annoyed.

* Disease control.  There are strong indications that diseases are reduced with this system.  It is likely, for instance,  (though not yet reported on) that mastitis would be reduced when the cows bed down on a compost bedding.