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Wednesday, January 17, 2018

Conservation agriculture

There is a 'new' sort of agriculture practiced by a handful of farmers in diverse locations around the world.  Conservation Agriculture is not a descriptor although the words describe to some extent what it is.  It is, rather, a name given to a suit of farming methods which taken together are called Conservation Agriculture.

This so called Conservation Agriculture involves a) not ploughing the soil, (and hence, direct drilling) b) rotating crops in a random fashion, with longer periods between growing the same crop, c) leaving all the unused parts (stems, leaves and, of course roots) of the past crop on the land as a mulch and d) the planting of cover crops between commercial crops.  It is not absolutely against using chemical fertilizers but results in great reduction or even  elimination of the use of such chemicals.  In addition it may involve grazing down the standing crop residue and/or cover crops, and thus converting them into dung and urine. If grazing is used, it is very intense, very infrequently.  It may also involve,  the incorporation of char into the soil.  To find more detail on the methods go to this site or to get a historical perspective on the fate of societies that didn't preserve their soils, to this site.  What I would like to explore in this blog is the logic behind the methods.

Let's take a corn plant as an example.  The seed grows into a plant and the plant uses water, carbon dioxide and various minerals to build it's roots, stems, leaves, and seeds (the corn we eat).  The energy to transform these simple, low energy substances into complex, high energy compounds comes from the sun and this captured energy is now in the form of chemical energy.  The resulting chemicals (largely cellulose along with many other  compounds in lesser amounts) can be burnt as a fuel but can also be 'burnt' by soil organisms just as we 'burn' the corn in our bodies for energy.  The soil organisms  incorporate some of this stover into the substance of their bodies, especially proteins and vitamins just as we do with the corn seeds.

Saprophytes (funguses) are specialist in using dead plant material for their sustenance.  Think of the fungus growing on dead wood in moist conditions.  But these are the fruiting bodies of the fungus.  Most of the fungus consists of thin filaments (mycelia) that extend through the media and collect nutrients.  Of particular importance in-so-far as we are talking about soil health for crop production is that the funguses not only use dead organic material for energy but can also mobilise minerals in the soil that are in insoluble, mineral form and make them available to plants.  Many of the funguses grow their mycellia inside or around root hairs and exchange the nutrients they have mobilized  for energy rich compounds that the plant provides. Anything to encourage the growth of these funguses and to avoid disrupting the mycellia that extend throughout the soil is good for the crop.  Therefore we put lots of organic material on the soil where the fungus can access it and we do not plough

 The obvious question is why don't we mix this material into the soil.  Firstly, this would involve ploughing and hence the disruption of the mycellia of the funguses but there is another reason.  If there is a large amount of reduced carbon (cellulose and other compounds) in the soil, the micro-organisms that produce cellulase* and hence can access this source of carbon and energy, will scavenge all the available soluable nutrients from the soil to build their bodies.  The funguses are not the only organisms that can utilise cellulose.  Many single cell soil organisms have the same ability.  If a considerable amount of cellulose is incorporated into the soil, there will be nothing available for the growth of the crop you have planted.  Put the organic material on the surface and it is gradually incorporated into the soil and nutrients are still available for the crop.  But the surface layer of mulch has other benefits.

* The enzyme that can break down cellulose.

The surface mulch shades the soil and keeps it from heating up so much.  The soil looses less water by evaporation, leaving more for the crop.  The mulch softens the blow of the rain and slows the flow across the ground and hence avoids sealing the surface of the soil and increases infiltration.  Again more water for the crop.

A word here about trophic levels.  As a first approximation, only ten percent of the material consumed is fixed into the next trophic level.  10 tons of algae will make one ton of krill and one ton of krill will make a tenth of a ton of whale.  Sounds good since the 90% excreted is mineralized. Some of it is in a form that can be taken up by plants, but here is the rub.  If there is lots of cellulose around, the micro-organisms which can break down cellulose will use the cellulose as energy and scavenge all the mineralised material, which has been released by other organisms, leaving none for the plants.  Of course as the quantity of remaining cellulose decreases, more and more of the mineralised nutrients will be available for the plants.

So, the next thing is why do we plant a cover crop when the main crop has been harvested.  First we capture more sun energy in the form of the chemical energy of the cover crop and hence produce more organic carbon for the soil organisms.  Secondly we scavenge any left over soluble nutrients from the soil and turn them into a slow release fertilizer (the bodies of the plants).  As this organic material breaks down it releases its nutrients into the soil over time.

If we include a deep and a shallow rooter, we scavenge nutrients throughout the depth of the soil as well as spreading roots through the soil which will not only disintegrate over time but will provide passages for water and air to penetrate the soil.

If we include a legume that is either inoculated with the appropriate rhyzobium bacteria or finds the correct bacteria in the ground, atmospheric nitrogen will be taken from the air and turned into a nitrogen compound that can be used by the next cash crop.  Since most of the nitrogen compounds produced will be incorporated into the leaves, stems and seeds of the legume, it is important that this material be left in the field to enrich the soil.

If we include a root crop such as a radish or turnip, as they later disintegrate, in addition to releasing their nutrients, they create tunnels for water to infiltrate.  They also often are deep rooted which will help to scavenge nutrients from lower levels.

It is important to cut down the cover crop or trample or roll it into the surface of the soil before it sets ripe seeds.  You don't want the plants of the cover crop to themselves become weeds.

If you decide to graze the cover crop, it is grazed very heavily for only a day or two.  This tramples some of the crop into the surface of the soil, ensures that all plants are utilized and not only the favorites, including weeds that you have not planted, and turns the cover crop into urine and manure.  This short sharp grazing leaves lots of time for the soil organisms to  sort out any surface damage and to incorporate the animal excretement into organic material.

When the cash crop is then planted by direct drilling, it has all the best of the soil structure and soil organisms to support it.

Sunday, December 3, 2017

Mitigating Dairy Farm Harm to the Environment

We are having a debate in New Zealand for and against irrigation.  It really boils down to a debate on our national dairy herd.  With irrigation, you can put cows on land that otherwise would not support them.  Our dairy herd can then increases and with it the pollution of our environment.

True, there are some concerns about irrigation itself.  For instance, the need to dam a stream in some cases to provide the water or the misuse of irrigation water.  Actually, using more water than is needed is a thing of the past for any farmer worth his air conditioned tractor.  Sensors tell the farmer just how much water he should apply.  The primary concern about irrigation is that it allows expansion of our dairy herd with the possibility of increased pollution.

To come out for or against irrigation  may be good for radio sound bites but as with most cases in the affairs of man, the devil is in the details.Clearly we need irrigation for our farmers to fill in the gaps left by nature. Even in the best areas, there are periods without rain.  A farmer needs reliable inputs to be able to run his business.

Equally clearly, if we can not find ways of farming that preserve our environment then the crude sledge hammer method of reducing herds and restricting where they can graze must be taken.  The question is, can we have dairy herds and not pollute.  The answer may be yes for some areas and no for others and will depend, to quite a large extent, on the details of how we farm.

The core of the problem is to be able to apply the waste output of the cows back on to the  land a) in a way, b) in a concentration and c) at the right time such that it constitutes a valuable fertilizer and not an environmental pollutant. If this can be done, dairying is no longer a source of pollution.

Throughout history, societies that trashed their soils, declined and disappeared.  One factor in trashing soils is not returning nutrients to the soil that are extracted. so far  as is possible, nutrients must be returned in an organic form that benefits the soil organisms.  Quite clearly, the urine, manure and spilt milk from a dairy herd constitutes a valuable resource for the enhancement of the soil.

That is not to say that chemical fertilizer should not be used but as you will see, much less of them can be used if farming methods are tweaked.

If farming remains a process of plow, add chemical nutrients, sow the seeds and irrigate then our soils will degrade, pollution will be rampant and we will go the way of many previous societies that mined their soils  instead of farming them.

It takes a lot more 'smarts' to farm in a way that improves the soil, reduces  inputs, increased water infiltration, improves the bottom line and leaves you with a much better farm to pass on to your children or to sell than when you started.

Let's look at some of the tools we have available.

Riparian Zones
Fencing off streams and encouraging the growth of trees, shrubs and grasses between the fence and the stream is a great help.  Not only does it stop the cows from entering the stream and urinating and defecating into it but the roots of the vegetation of the riparian zone take nutrients from the water table which is slowly flowing toward the stream.  However, it has been reported that 70% of the nutrients entering the streams comes from the very small feeder streams and ditches.  It is simply not possible to fence off every little feeder stream. We need some other measures in the pasture.

Composting Barns
Composting barns use deep layers of wood shavings or coarse saw dust as bedding and the cows are allowd in to bed down at night.  They also have free access to the barn to escape inclement weather.  The bedding is stirred mechanically every day, keeping it aerobic.  It has been found that cows prefer such an environment to bed down in even choosing it ahead of a straw-lined byre.  The composting process produces heat which reduces the feed need of the cows and a rich compost eats up pathogens.  The compost captures all the nutrients from the waste of the cows including N and S which in an anaerobic system tend to go off as the gases NH3 and H2S.

The bedding can be applied to the fields at the correct concentration and correct time which most benefits the soil and the pasture and hence causes no pollution.  Some research needs to be done on what portion of the effluent of a cow is released while in such a barn compared to what proportion is released out on the pasture*.  Do they mainly urinate and defecate at night or in the day, while they are grazing or when they are chewing their cud.  this would give an indication of how much of the nutrient stream could be captured by a composting barn.

*Great job for some long suffering masters student

Bio-Gas Generators.
Finally, a farm in Southern New Zealand is using the waste produced in the milking shed* to generate bio-gas.  The biogas is use  to produce electricity. The waste heat from the motor which drives the generator is used to heat the water used in the milking shed.  This combination, utilizing the waste heat from the motor that powers the generator, makes for a very efficient system, energy wise.  The effluent from the biogas generator contains almost all the nutrients in the waste stream since mainly C and H have been taken off as biogas (and some of the S).   As with compost-bedding it can be applied to the fields when and in what concentrations most benefits the pasture and hence least pollutes the environment.

*More work for that long suffering student.

Managing the Pasture
We have now removed a portion of the waste stream with a)Riparian zones, b) compositing barns and c)biogas generators.  Let's see what we can do out on the pasture.  There is a fantastic book by David R Montgomery called Growing A Revolution; Bringing back our soils.  In it he describes visiting farmers all over the world who have independently come up with a way of farming.  The methods they use would be familiar to any farmer before the advent of cheap chemical fertilizers but each method is updated in light of modern knowledge. Farming this way results in an improved bottom line, slashed pollution to the environment, reduced farming costs, increased infiltration of rain, continually improving soils  and as a bonus sequesters significant amounts of carbon in the soils.  It also, due to the greatly increased organic content of the soil, results in the capture of much of the Nitrogen when a cow urinates. The urine is soaked up by the organic material giving the soil organisms time to scavenge the nitrogen.

Before we go off half cocked and reduce one of our most valuable industries, we must pay attention to the details.  Farming can not be allowed to degrade our environment but there are farming methods which address this problem.  The devil is in the detail.

Tuesday, November 28, 2017

The ice pump

It has been a bit of a mystery why the floating ice around Antarctica has been increasing in area over the last few decades despite global warming.  After quite a bit of research and some reference to some well known physics, there is a pretty plausible theory/story to explain this.  It is called the ice pump.  First we need a fact or two before we tie it all together.

1.  Sea level is rising but only some of this rise is  due to the melting of land ice.  The remainder is due to the expansion of the water of the oceans as it heats up.  The heat is being gradually stirred into deeper and deeper water.  The salty deep 'circumpolar water' around the Antarctic is  a case in point.

2.  H2O expands when it freezes, contracts when it melts.  It makes intuitive sense that as you apply pressure to ice, it will melt at a temperature below zero degrees centigrade.  Indeed this is observed experimentally.  If you have ever skated you have used this phenomenon.  the blades of an ice skate are very narrow and apply high pressure to the ice which melts under the blade and allows the skates to slide over the ice.

 Image result for table melting point of ice under pressure
 100MPa equals about 9950m so one interval across the horizontal axis is about 2480m.  At this depth the melting point of ice is depressed about 2.4 degrees C.  As you can see from the following illustration, the depth of the bottom below sea level in West Antarctica is well below 2000m

3.  A few glaciers on East Antarctica and most on West Antarctica are on a retrograde slope.  The ice is so heavy that it has depressed the land and the land bottom below the ice gets deeper and deeper as you go inland. In East Antarctica some outflowing glaciers have carved deep channels well below sea level.   Most of West Antarctica land is way below sea level.

So let's put all this together.

The deep circumpolar water over-tops the sill at the outlet of some of the glaciers.  It is salty which keeps it below the surface, fresher water despite the fact that it is a little warmer.

Being heavier, it flows down the sloping sea bottom under the floating ice until it comes to the grounding line.  There it comes into contacts with ice.  Not only is it salty and warm but ice melts at below zero under pressure so this salty bottom water melts the ice at the grounding line making the grounding line retreat landward.

The glacier is moving seaward under the pressure of ice from the interior but grounding lines have been observed to be retreating so clearly the melting is  faster than the flow of ice seaward.

As the grounding line retreats it is at greater and greater depth and hence at a higher pressure where ice melts at lower and lower temperatures.  The melting becomes greater for a given quantity and temperature of circumpolar deep water flowing down the slope.

When you mix the water from the melting ice with this  salty deep polar water, the mix is fresher and hence lighter than the deep water.  It flows up the slope of the ice ceiling in a sort of up side down river and flows out on to the surface of the ocean.  The deep water is often described as seeping under the ice or some such gentle term.  We can see that as the light super cooled water flows out on to the surface of the ocean, deep water is being sucked in under the ice.   The more water flowing out on the surface the greater the 'suck'.

As the lighter water flows upward into a zone of reduced pressure, it is below the freezing point of ice at that depth.  It begins to freeze and for some reason freezes in thin sheets called platelets which form a sort of mushy layer below the sea ice ceiling.    This is the ice pump.  It is in effect taking ice from the grounding line and depositing it in shallower water under the ice ceiling.  The deeper the grounding line, the more effective the pump.

The sea ice around the Antarctic continent disappears every year or two so this ice from the grounding line is lost to the continent.  ie contributes to sea level rise.

The water which flows out on to the surface of the ocean, either at the edge of the ice shelf or into a lead is still super cooled and freezes readily, especially as it comes into contact with Arctic air which is well below freezing.  Here is one small part of  the explanation of the increasing ice around Antarctica.  Any leads which open up due to wind and currents, fill rapidly with ice  and hence can not close up again if the wind changes.

As the ice is eroded from underneath the glacier, the floating part of the glacier deflates and increases the slope of ice from the interior, seaward.  The glacier speeds up, pushing more ice seaward.  This is another part of the expansion of the floating ice.

The increased flow of ice seaward should push the grounding line seaward but apparently, at present,  melting trumps glacier flow.  In addition as the glacier deflates it floats up off the ground.  This also contributes to moving the grounding line landward.

There are a couple of further wrinkles to this story.

The rising water flowing up the ice ceiling apparently, in at least some locations, carves out up side down valleys in the ice and the light water collects in these and flows seaward.  This will, of course, reduce the surface area where this light up-flowing water is in contact with the surrounding water.  It is not quite a pipe but will reduce mixing compared to a sheet flow.

In addition, these valleys have reduced buoyancy compared to the surrounding ice so will weaken the ice shelf, contributing to it's break up.  If, for instance, you had one valley running along the middle of an ice shelf, the surrounding ice would have a force on it trying to make the ice tip toward the valley from both sides.

Another factor in the expansion of the surface area of floating ice is that the air flowing off the Antarctic continent is apparently getting stronger and this will tend to push ice outward (North).  As mentioned, leads opened up will rapidly freeze, stopping the ice from moving back south.

The winds flowing clockwise (looking down on the continent) around Antarctica are apparently also increasing in velocity.  They push on the ice.  Anything moving in the southern hemisphere and especially if it is near the pole, is veered to the left by Coriolis.  To the left is away from the continent.  Again we have ice moving North and leads freezing over, stopping the ice from returning south.

The bottom line of all this is that for a while, we would expect the floating ice to increase in area around the Antarctic due, ultimately, to the warming of the deep salty circumpolar water.  At the same time, we should expect to see coastal glacier deflating and the floating ice shelves breaking up.  Already two of the Larson Ice shelves along the Arctic peninsula have disintegrated.  They are the Northern most Antarctic ice shelves.  The third Larson Ice Shelf may be on its way and the rest should follow in time.  This will remove the plug and allow inland glaciers to flow more quickly and we will see if this movement can reverse the retreat of the grounding line.  This is unlikely as the glacier deflate and float upward.

What is interesting is that we have probably passed a tipping point in the break down of glaciers which are grounded way below sea level.  When the salty deep circumpolar water contacts ice at relatively shallow depths, it will erode the ice but the flow of ice seaward may be able to balance the melting.  However, when this circumpolar water is contacting ice at greater depth, its erosion ability is greatly increased due to the suppression of the melting temperature of the ice at greater depth and hence pressure.  The removed ice is transfered to the underside of the ice shelf at shallower depth and this ice is lost each summer as it floats off into the ocean.  Even if the deep circumpolar water cooled to its previous temperature, the depth effect has so increased the ability of this water to melt ice that the process would likely continue.  Since there is no prospect that such a cooling will occur, it is doubly likely that the ice sheets which are grounded well below sea level will collapse.

The disintegration of the Antarctic ice which is grounded below sea level is now probably inevitable, even if we were to stop all green house gases tomorrow.

I wouldn't be buying any coastal property

Friday, November 10, 2017


This is a book review of David R Montgomery's book Dirt which he wrote before Growing a Revolution.  In Growing a Revolution he describes how a few farmers scattered far and wide across the planet have worked out a better way of farming which restores the soil with all the benefits this brings.  In this book Dirt, he describes how civilization after civilization,with very rare exceptions, have destroyed themselves by trashing their soils.

Clearly there are other factors involved in the demise of civilizations but at the core, if you can't feed your population, you are on a slipery slide.

A common sequence Prof Montgomery describes is a move into a new valley and a build-up of farming.  With a reliable source of food, human populations  increase at a truly astounding rate.  In the words of one of my favorite authors, Richard Dawkins,  "If ever there is an increase in food production, population will rise until the previous state of misery is re-established." It is not inevitable but very very common. 

In a few countries the population increase and with it the destruction of ever more sensitive soils has been reversed and would you believe it, we are fighting it tooth and nail. (see above link).

As the bottom land is completely occupied, the new generation of farmers move up slope and farm ever steeper land.  When the plow is used, the die is cast.  Plowing moves soil down hill and the removal of ground cover greatly accelerates natural erosion by rain and wind which, moves the soil even faster down hill.  Soils either accumulate on the valley bottoms and/or are washed into the stream or river to be exported to the sea.

  For instance, early in American (European) farming, they plowed straight up and down the slopes, would you believe???  Contour plowing was a "great innovation" and even this "innovation" only slowed down the destruction.

 The Americans eventually reached the great central Loes plains, leaving destruction behind them and proceeded to destroy these soils as well.

On a visit to Virginia I saw many stone gates leading into a young forest with no drive way visible.  When I asked the locals about this curious occurrence, they told me that these were abandoned tobacco and cotton farms.  The farmers had moved west when the soil ran out.  In fact, it was common for a farm to last only for a decade or two when the farmer had to move west.  This, more than anything might explain the constant western movement of the Americans into lands owned by the first people.

On a recent visit to Otterton, in Devon to see the return of the beavers we were told that Otterton was once a sea port.  Soil erosion had filled the estuary and Otterton is now land locked.  We found out later that this is a very common situation around the UK.

Just last month, we took a trip to Bulls in North Island (New Zealand)  There we saw plowed fields all over the place and the streams ran brown with silt.  Our streams here in Canterbury are the same when there is anything above a very gentle rain.

the present zeitgeist is climate change and we are finally waking up to its dangers.  The more serious crisis may just possibly be the destruction of our soils. This is exacerbated by our short term rush to the maximum short term profit rather than a greater long term profit.

Friday, October 13, 2017

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

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

However, in the symbols Ax,  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 Ao (A at time zero) and the amount we are left with as At (A at some time t in the future).  The amount we will have left after one half life is:

1.   A1 = Ao(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 An - Ao(1/2)n  or in words, to find the amount of a substance after n half lives have gone by, multiply Ao, the initial amount, times 1/2 raised to the nth power.

Note that in the notation Ax,  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, 63.22 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.  At = Ao(1/2)t/h
or in words, to find the amount of radioactive material remaining after time t, multiply Ao, 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 Ao 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:

logabc = clogab.  Incidentally, the inverse of the left side of this formula is ac =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

At = Ao(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.

Ao = At 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.

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 Ao.  This cancels Ao on the right side and leaves us with

At/Ao = (1/2)t/h

Now I'll take the log of both sides

log (At/Ao) = log[(1/2)t/h]

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

log(At/Ao) = 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(At/Ao]/[log(1/2)]


h = [tlog1/2}/[log(At/Ao}

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.

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.