Skylights

 Skylights really annoy me.  Not the having of them but the way they are integrated into a house.  It's an engineering abortion.  We are living in a house in which I built the sky lights. We are downsizing to a house that is being built for us.  And we have 4 sky lights, bought a great expense with silly engineering.

The bought sky lights stick up above the corrugated iron cladding so any rain that runs down the roof above the sky light wants to flow into the roof cavity.  To avoid this you need some sophisticated flashing and lots of sealant.  There is a far simpler and vastly cheaper option.

The thickness from the cladding of a roof to the ceiling is typically about 6 inche-sometimes more.  This is the width of the perlons (the horizontal boards which are laid on edge across the rafters).  There are lots of other configurations but generally there is lots of space between the ceiling and the cladding.  So what do we do.  

First build what is essentially a picture frame with dimensions to take a glass sandwich plus 5mm or so in both directions. This is built into the roof at the level of the ceiling.

The glass sandwich you can get made up by any glazier.  It consists of a lower sheet made of that safety glass which itself is a sandwich of plastic between two sheets of glass.  If you ever have some sort of a disaster, this glass will ensure that no shards fall and hurt someone.  

The upper layer I would recommend be some sort of diffusing glass.  Basically the sort of thing you would have in a bathroom window.  You don't want a sharp edged beam of light with deep shadows around it but rather light that is dispersed over the whole room. The two sheets of glass are separated by an aluminum  strip of around 5mm thickness.  If you really want to get sophisticated, you can get the hollow between the layers of glass, filled with Argon.

If you decide to seal this glass into the frame, make sure to use a thick layer of rubber or silicon sealer so the glass can expand and  contract as it heats and cools without cracking the glass.

So how about the rest of the sky light.  Under a corrugated iron roof, you usually put a layer of tar paper or similar membrane to take any condensation drips from the iron, down toward the eves.  At the location where you have the sky light, you use a layer of transparent polycarbonate instead.  Then we only have to clad the roof.  From the ridge down to the sky light (hang tough, I'll start from the eves).  From the eves up to the bottom of the sky light is regular corrugated iron.  This is overlapped by a sheet of corrugate transparent poly-carbonate over the glass, up to the top of the sky light and beyond a bit.  From the top of the sky light to the ridge is another piece of corrugated iron.  

No way is this going to leak, it is cheap and simplicity itself to construct.

How to destroy China

 It will take a little time but China should be gone as a major state in 50 to 100 years.  And it is simple.  Do you realize that, while Greece and Rome and many other empires rose and fell, China remained as an organized, prosperous country through it all.  Greece and Rome destroyed their soils and fell.  Rome ended up being dependent on countries in North Africa for her food.  It was so extreme that it was a capital offense to interfere in any way with the transport of food to Rome from Africa.    What we have to do to destroy China as a world power is to get them  to destroy their soils.  And it would be easy to do.  Simply mock them about their use of human manure on their fields.  Convince them that it is backward and yucky and the modern way is to use chemical fertilizers and to use flush toilets.  Much more aesthetic and look at the great harvests America gets. (steer the conversation away from the destruction of American soils by using chemical fertilizers)..

The reason that China has survived and prospered for some 5000 years is that they have always cycled every bit of organic material back into their soils.  Stover (the part of the plant that we don't utilize) animal manure, human manure, waste wood - in fact anything that will compost is put back into their soils.  And this has allowed them to maintain prosperity over such a long time.  Sure they had floods and droughts and this caused severe social dislocation but over all, they bounced back based on a working agriculture.

And if we get them to adopt flush toilets, it is a double (or triple) whammy.  If they can be convinced to do only primary treatment or even secondary treatment and to pour the used water into their rivers, they will kill off the life in the rivers.  In addition this added use of water for flushing, may tip them over the edge in some of the drier parts of China.  Then when the river pours out into the ocean, it will likely result in an anaerobic area which trashes the fish and other marine life in the area.  

Then, of course, they will have to mine more phosporous, fix more nitrogen and obtain all the other elements that makes for good crops.  All this will put more strain on their economy.  Oh and convince them that the use of the plow is the formula for better crops.  If they start using the plows extensively, for a short period, their crops will improve* while at the same time, destroying their soils.  By the time they have woken up to the harm the plow does, it will be too late.  With the very best of management it takes about three years to restore a soil to a passable level of production.

* Using the plow is like taking the capital from your bank account rater than just using the interest.  Over time you bank account falls to zero.  

And don't let them know about the sewage plants that turn sewage into valuable fertilizer.

Batteries for Static Applications

 To my way of thinking, we should try very hard to discontinue the use of batteries based on Li chemistry for static applications such as home batteries and grid storage.  One reason for this is that doing so would reduce the demand for Li and reduce its price.  This would bring down the price of Li batteries for mobile applications, reduce the price of EVs (Electric Vehicles) and hence widen the market for these cars and increase how fast we shift away from fossil fuels. 

 

Better still, some of the alternate chemistries produce batteries that are superior to Li batteries for static applications.  The reason we stick with Li for mobile applications is weight.  I'm not aware of any other battery that holds more power per kg(Kilogram) of battery than Li batteries.   Weight is of very little significance for batteries which are used for static applications.  So what other possibilities are there and how are they superior to Li batteries.

 

Liquid metal batteries

Conceptually one of my favorite types is the liquid metal battery.  These were developed by professor Sadoway and his team at MIT.  His favorite comment when he is talking about his battery is that if you want a battery that is dirt cheap, make it from dirt.  OK, it is not exactly dirt that is used but fairly close.  His batteries are based on Ca(Calcium), Sb(Antimony) and a calcium chloride salt.  They operate at about 500 degrees C(centigrade).  And, many experiments are ongoing to use different chemistries that need lower temperatures.

These batteries are kept molten by the electric current flowing through them during charging and discharging so there is some waste of power.  The effectiveness of the insulation is of primary importance.  When shipped the chemicals are simply dumped into the battery and the battery sealed.  When they arrive at their destination, the chemicals are heated up until they melt, separate into their layers and the battery is ready to go.  This sort of battery works well under heavy use but if left unused for some time, the components will eventually solidify and have to be melted again to regain function.  If left fully charged, when the components solidify, the battery is fully charged and ready to go when melted.  So what are the characteristics of these batteries.

They can be cycled between 0 and 100% charge with no damage.  This has a lot to do with their liquid nature.  Because all the components are liquid, they can't form spicules which in other batteries, grow and short out the plates.

They last far longer than Li batteries with no fade.  In fact I am not sure if they have even found a limit to the number of times they can be cycled from 0 to 100% and back again, before they show some fade.

They are completely safe to ship and so can be shipped by air, sea or land.  Ingredients are solid, mixed and you could short out the terminals with no effect.

They are made from elements which are readily available and cheap from multiple sources.  Antimony is mainly from the sulfide mineral known as stibnite.  It is not particularly expensive, ranging from US$2 to US$6 per pound at various times in the past.  Calcium metal is obtained from the electrolysis of a calcium salt such a Calcium cloride.

Liquid metal batteries operate under a wider range of external temperatures than Li batteries

ZnBr Batteries

There are two types of ZnBr batteries.  On is the flow/plating batteries.  It is essentially a Zn electroplating unit.   Zinc Bromide is dissolved in an aqueous solution and there are plastic horizontal 'shelves' in the battery dosed with C to make them conductive.  Under charging, the Zinc comes out of solution and electro-plates the plastic shelves.  The bromine which is heavier than the solution accumulates at the bottom of the battery.  Under discharge, the Zn and the Br recombine and zinc bromide dissolves into the solution.  It is not only possible but is recommended that the battery be discharged completely from time to time as this eliminates any build up of dendrites (spicules) that could short out the battery.  Thus it is useful to have your battery pack operate in multiple stacks.  One stack is completely discharged into the other pack on an established schedule.

The other type of ZnBr battery is the gel battery.  Instead of a solution, the electrolyte is in the form of a gel.  From the outside, these batteries look very much like a Lead-Acid battery and in fact the manufacturing process is very similar.  Apparently a Lead-Acid battery manufacturing plant can be converted to making ZnBr gel batteries with only modest modifications.  Gel batteries also need to be completely discharged from time to time.  To me they sound like a real winner since you simply stack them in series and parallel to get the voltage and capacity that you need. And there are no moving parts.  It remains to be seen what their cost per kWh stored will be and what their working life will be.

For their advantages, look back at the advantages of liquid metal batteries.  They are the same.

Redox Batteries

These batteries are based on elements with two or more valence states.  For instance Fe (iron) compounds exists in the Ferric (+3) state and the Ferrous (+2) state.  You have two tanks, one with an iron salt in the +3 state and one in the +2 state.  And you have a reactor through which these solutions flow separated by a membrane.  What is notable about such batteries is that if there is a small amount of leakage through the membrane, it is of little consequence.  It is still an Iron salt.  If you were using two different elements, you could quickly contaminate your solution.  If you want more capacity you simply need bigger tanks.  If you want more power you need more reactors.  I don't know how they get around the solubility problem.  The ferric state is far less soluble than the ferrous state*.

The other element used so far is V(Vanadium) which has multiple valence states.

*By the way, this is why there are huge deposits of iron in some parts of the world.  In the early years of the world there was no oxygen.  The oceans were anaerobic.  Iron was dissolved in the oceans in the soluble ferrous state.  Then stromatolites  developed.  These are sort of like reef building algae that take in Carbon dioxide and put out oxygen.  Apparently the oceans were full of ferrous iron.  As oxygen was produced, the ferrous iron used up the oxygen, converting it to ferric iron which precipitated.  Western Australia is a region where you can see this and in the shallow water there are still some living stromatolites.

Iron oxygen Batteries

These are also being worked on and apparently a very large one is about to be built.  As they discharge, iron is combined with oxygen producing rust.  When charging the rust is converted back into Iron and Oxygen.  An interesting aside here is that this is a way to produce iron from Iron ore which is essentially rust.  This would be a carbon free method of refining iron ore.  I have read speculation that you could put the oxides of other metals into the retort and produce alloys of iron such as stainless steel.

I'll add more batteries as they are shown and described.  So far it is hard to get reliable data on the longevity and other characteristics of these alternate batteries.

Inheritance (death) taxes

I tried to come up with a single word or phrase that describes Inheritance (death) taxes.  Illegal doesn't work because what is illegal is whatever the government says it is.  Immoral isn't quite right because it tends to refer to  human relationships.... but it is closer.  The best I could come up with is 'it just ain't right'

 

Part of it has to do with one's world view.  People who are for inheritance tax tend to look at a family as a mom and dad and their children.  I look at a family and I see an extended family including the deceased great grand parents, the living grand parents, the current generation, their children and as time goes on, their children's children.  All these element of a family are giving support to each other both up and down the generations in many ways both pecuniary and non-pecuniary.   

Lets take an example.  A man working at a crap job has scraped and saved and can finally buy that farm he has dreamed about.  Every cent he has earned, he has paid taxes on.  Every bite of food he has put into the mouths of his family likewise.  He has paid taxes on the fuel to get back and forth to his job and on the car he bought.  If he has saved his money in a bank, the bank has paid taxes which reduced the return on his loan* to the bank.  But finally he can buy the farm with the residual that has been left to him.

*That's right.  When you deposit your money in the bank, you are loaning it to the bank.

He needs a place to live (more taxes on every bit of material to build the house) and with luck he is able to build his own house.  Alternately, he gets a contractor to build his house.  The contractor pays tax on everything he does and has to cover this by charging our farmer more for his construction work.  He will spend more to have the municipality inspect every stage of construction.  Then he needs machinery, seed, chemicals, fencing etc etc.  The list is endless and you guessed it.  Taxes on everything he buys.  And his wife is probably working outside the farm to bring in needed funds to run the farm until it can stand on its own.  More taxes.  And every bite he buys outside to feed his family has been taxed

In the fullness of time the farm is successful but Dad is getting a bit long in the tooth by this time.  He still does a lot around the farm but as his strength wanes, his sons and daughters take over more and more of the functions of the farm.

One of his sons realizes that the model his father is using is destroying the soil with the plow and the chemicals dad uses and he begins to convert the farm to what is know as regenerative farming.  This could show a constant bottom line but is more likely to have anything up to three years with the bottom line suffering.  From then on he has greatly reduced input costs, has improved the bottom line and has made his farm much more resilient to vulgarities in the weather and crop prices.  He has stopped polluting nearby streams and rivers, is helping to mitigate floods and is sequestering carbon.  None of this is recognized by his government despite the fact that it decreases their expenses (cleaning water of pollution, compensating flood victums and paying for their carbon output).

But it is actually a bit more complicated than this.  Let's look at the customer who buys a loaf of bread made from the wheat our farmer grew in his field.  The farmer sells his wheat to a mill.  They pay a whole range of taxes and have to recover this when they sell the flour to the bakery.  The bakery pays a whole range of taxes which they have to recover in the price of a loaf of bread.  Each stage in an economic chain of this sort pays taxes and here is the rub.  Some of the taxes are on the product they sell so they are paying taxes on the taxes from the former level.  How does this concern the farmer.  I puts considerable downward pressure on what he can charge for his wheat. Compound taxes make it so.

 

Now in a well run country, all these taxes are necessary.   The infrastructure that only the government can implement is needed for all the enterprises of the country.

Then Dad dies.

The one son or daughter that wants to remain on the farm has to buy out his siblings or split the revenue of the farm with them so he is faced with added expense.  In a country that has death duties, the farmer then has to find  as much as 40% of the value of the farm that he and his father have carved out of a virgin piece of land, greatly increasing its value, to pay the government.  Not only has the family paid taxes on every single financial interaction  throughout the life of the farm since his dad bought it but the value of the farm is mainly due to the blood, sweat and tears of his dad and his siblings.  They have created this value and now they have to pay taxes.  The more succesful they were, the greater their taxes.

 

Along comes the government evaluator.  The farm is worth far more than the piece of land that Dad bought and the value is mainly due to the efforts of the family.  The family now has to pay taxes on this added value that they created.

Like I said, it just ain't right.  He will probably have to sell the farm - often at a loss, just to pay the death duties and this at a very stressful time when he has lost his dad.  It just ain't right.

Blow out holes in the Arctic

 Over the past few years, blow out holes have appeared in the Arctic.  They are often many tens of meters diameter and many tens of meters deep.  They are close to round in shape with more or less vertical sides.

Typical blow out hole.  The people give scale. 

 

 

To my way of thinking, many of the commentators have got their wires crossed regarding how these features are formed.  Let me run this by you. 

 

1/  Much of the Arctic has permafrost, some of it multi-kilometers deep but much of it much less thick.  By definition, permafrost is soil that is frozen over the summer and on into the next winter.  Some of it has been frozen for thousands of years.  Now with the warming of the climate, it is melting, especially at its more southerly extent.    Leave that aside for a moment.

 

 

 

 

2/  When you mix methane with water, nothing happens.  A very small amount of the methane dissolves in the water and the rest bubbles out.  However cool the water to a few degrees above freezing and pressurize it to a pressure equivalent to about 250m deep in the sea or lake, and the methane combines with the water and forms an ice.  It is called a methane ice, methane hydrate or Methane Clathrate*.  A kg of water which is fully saturated with methane (forming an ice) can hold about 160liters of methane (measured at Standard Temperature and Pressure (STP)

*You can think of a Clathrate as a solid solution.

 

 

In fact under greater and greater pressure the methane clathrate can form at a higher and higher temperature. Note that to a first approximation, one bar is equal to one atmosphere.  As you dive down in the ocean, again to a first approximation, for every 10 meters you descend, the pressure increases by one atmosphere (or one Bar).  If you look at the above graph, you can see that a clathrate will form at about 18⁰C at about 50 atmospheres(490m).  This graph doesn't go far enough to the left but at a degree or two above freezing, a clathrate will form at about 25 atmospheres (Bar) or at about 250m*.  Leave that aside for a moment.

*Methane ice has been dredged  from the bottom of the ocean.  A piece left on the deck slowly melts into a puddle of water.  You can light it on fire and it burns as it gives up its methane.

A piece of methane Clathrate burning.

3/  When organic material is buried and especially if it finds itself at considerable depth where the heat of the earth is high*, the material breaks down into a range of hydrocarbons.  It is basically a pyrolysis process.  Depending on the starting material you can produce shale, coal or liquid hydrocarbons (oil) but in all cases, methane is one of the products.  Being a gas the methane will seep upwards until it either meets an impermeable layer or it vents into the atmosphere.  The impermeable layer can be a geological layer such as a layer of clay, for instance, or can be permafrost.  

*The temperature in the earth is created by the decay of radioactive atoms and this heat slowly conducts and convects upwards to escape into the atmosphere.  It is a minuscule amount compared to the heat we receive from the sun but since the earth is a great insulator, it accumulates and melts the center of the earth.  As a rough rule of thumb, temperature increases about 25⁰C per km depth.  

4/  If the methane vents into the atmosphere, it comes into contact with OH (hydroxide) radicals and is slowly oxidized into Carbon dioxide and is incorporated into the biosphere.  The half life of methane in the atmosphere is around 7 years.  If it comes into contact with an impermeable layer, it accumulates.  If there is moisture where it accumulates, if the pressure is sufficient, it forms a clathrate.  A clathrate can form at much less than 250m depth if the impermeable layer confines the gas, like a pressure cooker lid.  So let's put all this together.

 

Say you have a really solid, really cold impermeable layer of permafrost of 100m depth and somewhere below the permafrost there are layers of coal, oil or shale (or even a buried swamp or peat bog).  They are venting methane but there is no geological impermeable layer between the hydrocarbon layers and the bottom of the permafrost.  The bottom of the permafrost is at zero degrees C.  If it was colder the permafrost would thicken (and there is moisture there).  The permafrost acts like the lid of a pressure cooker and the methane begins to form a clathrate with the available moisture.  

 

Now we humans come along and start to warm the atmosphere.  The permafrost begins to warm and weaken and the clathrate warms as well.  Gas comes out of the 'solid solution'.  Pressure builds up.  As mentioned a kg of clathrate can hold as much as 160liters of methane.  If the pressure becomes high enough it blows a hole through the permafrost.

 

There is a corollary to this.  Where we have continental glaciers, we should expect that there will be places where the glacier is underlying with coal, oil or shale.  In such locations there should be a layer of methane clathrate at the bottom of the ordinary ice.  The ice of a continental glacier can be a few km deep so the pressure is great and there are lakes under the ice of Antarctica. They remain liquid because of the heat seeping up from the earth and the insullating ability of the overlying ice.  As the ice thins, the pressure decreases and so you would expect huge outputs of methane as the glaciers of Greenland and Antarctica retreat.

 

Incidentally, I have heard some commentators say that the ignition of the methane is what blows out these holes.  Completely impossible.  Methane only burns or explodes when mixed with air or oxygen at certain ratios.  For methane this is between 5 and 60% methane.  More than 60% methane and the mixture will not burn.  Below the permafrost there is no oxygen, it is pure methane (probably with some Carbon dioxide).  Once it enters the atmosphere, it could ignite but would need a source of ignition to burn or explode*.  

*Owners of older style ICE (internal combustion engine) cars will be familiar with this.  If you flood the engine the car will not start.  You depress the accelerator, hold it there and try again.  What you are doing is flushing out the excess fuel vapor.  At too high a concentration it will not ignite.   

The radioactive formula

 The math for calculating how old something is from a study of it's radioactive elements is often presented in a very obscure fashion when it is really quite simple.

A formula is a mathematical expression or 'sentence' that describes some phenomenon in nature (or not in nature).  There is a straight forward expression of any formula - the one that can be understood intuitively.  Then you can rearrange the formula (solve for) to make any of the other factors in the equation the subject.  These 'derived' formula are often less easy to understand intuitively. 

First, for any youngsters that haven't yet learned algebraic notation, let's divert for a moment.  Skip this if you already know algebra.

 

 Notation in Algebra

First why do we bother to make formulas with letters.  Why not just put in the numbers.  The reason is so that this gives us a general formula which can be used for the same phenomenon but for different numerical examples.  Also with a different letter representing each different factor, it is much easier to make any one of the factors the subject of the formula.

ab

First what does it mean when two letters are presented beside and touching each other.  For instance ab.  

This means you are to multiply the value of 'a' by the value of 'b'.  With letters, we don't separate them by a times sign such as axb because this can be confused for telling you to multiply a times x times b.  On the other hand if you are using numbers you can include the times sign so 7 x 9.  Alternatively, a dot is often used in algebra in the middle of the line like this 7^.9.  This notation says to multiply 7 by 9 while 7.9 with the dot at the bottom of the line means seven decimal (or point) 9 - nearly 8.

 

3a

How about 3a.  

This instructs you to multiply 'a' by 3 so it is the same as a+a+a.

 

a³

And then we have a³.  

This tells you to multiply 'a' by 'a' by 'a' which is also called 'a' to the third power 

a₃

We also have a_{3 also known as asub3}.  This isn't an operator.  It doesn't tell you to do anything.  Series are a powerful tool in math and this notation could mean the third 'a' in  a series.  You could also have a₀.  This could mean a at time 0.  We will use this just now.  Whatever the notation,  generally we described what each of the factors in a formula means at the beginning of the calculation unless it is such a well known formula that an explanation is not needed

The Physical Situation

Math is used to describe something in nature.  I fine it quite gob smacking that so many things in nature can be described by quite simple math.  In our case we are looking at the breakdown of a radioactive isotope.  Perhaps we should describe what an isotope is for the young ones.  There are about 92 separate natural occurring elements on the periodic table and a whole bunch more that are artificially created.  Their chemistry (how they react with other elements) and hence their identity is determined by the electrons whizzing about the nucleus and especially the outer shell of electrons.  In the nucleus are positive particles called protons and the number of electrons in an uncharged atom* exactly match the number of protons.  But there are also uncharged neutrons in the nucleus.  They have almost the same mass as a proton.  The sum of the protons and neutrons gives you the Atomic number of the element.  The Electrons are very light compared to protons and neutrons.  The number of  neutrons approximately equal the number of protons but can vary quite a bit.  The different forms of atoms of a given element, due to the different number of neutrons, are called isotopes of that element.  All the atoms of a specific element contain the same number of protons but the neutrons can vary.  Some of these isotopes are radioactive.  All this means is that they break down spontaneously into simpler, lighter elements. They aren't stable.   At break down, they give off alpha, beta or gamma radiation.  I'll tell you about them in an appendix.

* Perhaps you have heard of ions and might confuse an ion with an isotope.  An ion is an atom which is temporarily missing or has an excess of one or more electron.  Ions have a very strong tendency to gain or loose electrons to bring them back into equilibrium.  Actually ions have a part to play in the dating of radioactive isotopes as we shall see.

Here is where we get to the critical observation about these isotopes which allows us to date them.  Early researchers noted that if you observed a given radioactive element for a while, at some time in the future, half of it will be gone - changed into something else.  Nothing surprising here.  Then if you continued to observe this isotope, for the that same time period, half of what was left will be gone.  Starting with some initial amount, each time, that particular period of time (which is unique for each different isotope) elapses, you would have half, then a quarter, then an eighth, then a sixteenth and so forth until there is too little to observe.  This period has, not surprisingly, being called its half life.  Half lives vary from milliseconds to millions of years for different isotopes.

The Formula

So we can start to build a formula.  We will put each formula into words as well.

Suppose you know how much of a radioactive isotope you have today and want to know how much you will have after one 'half life' has gone by. 

A_1/2 = 1/2A₀  In words, the amount after one half life is equal to one half times the amount at time 0. This is what we observed in the physical world. So far pretty simple - no? Remember for this problem we define A_{1/2 to mean The Amount after one half life.  We could put 't' instead of 1/2 meaning the Amount after time t has gone by after the initial condition.}

Now suppose that two half lives have gone by.  We have to multiply again by 1/2 so we have A_t = (1/2)(1/2)A₀  Or we could write (1/2)²A₀.  ie. after two half lives one half squared times the Amount at time 0 which, of course, is one quarter as much as we started with.

We could go on like this or we could put n (number of half lives) in the formula so it becomes A_t = (1/2)ⁿA_{0.  In words, if you want to know how much of a radioactive element remains after a given time, raise 1/2 to the power of the number of half lives that have past and multiply this number by  the original amount.} 

Now we will have a short break and I'll ask you a question.  Suppose we have some radioactive isotope that has a half life of 5 (h = 5)years and 15(t = 15) years have gone by.  How many half lives have gone by and what fraction of the original isotope is left.

If you have understood the concept, you calculated 3(n = 3) half lives  and there is one eighth left of the original amount.   Putting this into letters, n = t/h or the number of half lives that have passed equals the time elapsed divided by the half life.  Since n= t/h we can put t/h where n appeared in our formula.

We now have A_t = (1/2)^(t/h)A₀  and that is the whole radioactive formula.  One of the main uses of this formula is to find t, the time that has elapsed since, say, Carbon 14 was incorporated into a plant or animal or perhaps the time since a rock melted and solidified and reset the clock.  Below a quick description of what is meant and then we will learn how to rearrange the formula to make t or h or A₀ the subject of the formula instead of At

Note that your little hand held calculator isn't bothered at all if 't' is not an even number of half lives.  Not an easy thing to do by hand without another branch of math called logarithms but the calculator takes it in its stride.

The physical situation

There are two main fields where radioactivity is used for dating.  First Carbon fourteen.

Carbon 14

The usual, (common) non-radioactive form of Carbon is Carbon twelve.  This means that the sum of the number of protons plus the number of neutrons in the nucleus is 12. In this case the number of neutrons and protons are equal.  That is to say, 6 of each.   This sort of Carbon is stable, it is not radioactive.  However, when high energy particles know as cosmic rays hit the atmosphere from outer space and high energy particles from the sun do likewise, some of the Nitrogen 14 is changed into carbon 14.  and this diffuses into the atmosphere.  It becomes part of the biosphere and any plant or animal takes up some of this radioactive Carbon along with the non-radioactive type.  The quantities are very small but the methods of detecting the relative proportions of radioactive and non radioactive carbon in a plant or animal are very very accurate.  I will describe them later.  

 

What this results in, is that any living animal or plant is more or less in equilibrium with its environment and to a close approximation all have the same proportion of radioactive carbon to non radioactive carbon in their bodies. 

 

However, when an organism dies, it is no longer taking up either form of carbon and the proportion of  Carbon 14 begins to decrease as it changes back into N14.  The half life of C14 is 5730 years and at a pinch, the amount can be measured with modern techniques to about 10 half lives.  Of course the accuracy decreases, the older the sample but this takes us back to about 50,000 years.  To put this into perspective, we can date materials back to half way into the most recent glacial period but not to the most recent interglacial, the Eemian which occurred about 125,000 years ago. 

 

Rocks 

Rocks can also be dated using radioactive isotopes.  Here you can do a wee experiment in your kitchen to illustrate the process.  Get some Copper sulfate which is a blue crystal.  Dissolve as much as you can in some water.  Now add sugar and dissolve as much of this as you can in the same water.  Pour off the clear liquid (colored blue) into a clean glass and suspend a piece of string that you have dipped in powdered crystals of both solutes.  Watch what happens.  The Copper Sulfate and the sugar will crystalize separately as the water evaporates so you will have sugar and copper sulfate crystals back again.  The sugar crystalizes with the sugar and the Copper sulfate with the Copper sulfate.  The same thing happens with rocks.  If you melt them and let them  cool and solidify, the separate mineral crystalize  separately.  Think of a granite rock.  melting re-sets the clock.  If there is a Uranium containing mineral in the melt, it will crystalize out separately.  Then it will continue to break down and the final result of a series of radioactive decays is Pb (Lead).  By measuring the relative proportions of U and Pb, you can date when the rock cooled from a molten state.  However we need to re-arrange the formula to make 't' the subject of the formula.  Let's do it.

 

Solving for 't'

We start with the basic formula  A_t = (1/2)^(t/h)A₀

      and we want to get 't' by itself and all the rest on the other side of the equation.  This is called 'solving for t'.  So let's divide both sides by A0.  Remember, we can do anything we want to an equation as long as we do exactly the same to both sides.  Of course, the trick is to choose the right thing to do.

               A_t/A0 = (1/2)^(t/h)

^{This cancels out Ao on the right side and leaves it in the denominator (bottom) of the left side.}  

^{Now we need a wee log identity.  I will give you a hint at the bottom of this blog of how logs work but for the moment, take my word for it that:}

^{log_abc = clog_ab}         Or in words, Log to the base 'a' of 'b' raised to the 'c'th power equals c times log to the base a of b.  ie You can move the power to the front.  I'll explain more about this in an appendix.  Just remember that we can do anything to one side of an equation if we do the same to the other side.  So the formula becomes.

log At/A0 = log (1/2)(t/h)   and this becomes

log At/A0 = (t/h)log(1/2)

 

Oh, I nearly forgot to mention something.  When you use the log function without any explanation, 'log' it always means to the base 10.  If you want it to a different base, you must note it and ln is the natural log to base 2.718. Both Log and ln are on your computer.

Now all we have to do is to divide both sides by log(1/2) to move log(1/2) to the other side and  multiply both sides by h to move h to the left side.  't' remains in glorious isolation on the right side.  It is conventional to put the subject of the formula on the left side so we can reverse them.  After all if a = b then b = a

Our formula for t then becomes

t = (logA_t/A₀₎

       (log1/2)

The Analysis System

  I suppose one could extract the lead from a rock and the Uranium and weigh how much of each there was.  These figures could be inserted into the formula to age a rock.  But that wouldn't work with Carbon.  You are comparing the amounts of Carbon fourteen with the amount of Carbon twelve and both have the same chemistry.  There is a better and very sensitive method that works much like the first television sets.  

In the early TV sets, the screen was one side of a large tube which tapered to the back behind the screen.  At the back of the tube was a filament which, when heated, gave off electrons.  These were accelerated in an electrical field and then passed through two sets of magnets oriented at right angles to each other.  A charged particle moving through a magnetic field is bent.  One set of magnets bent the beam of electrons back and forth horizontally and the other set up and down.  The magnets were varied in such a way that the beam of electrons sped back and forth over the inside of the screen, causing the layer of phosphorescent material on the inside of the screen to light up.  The electrical field was varied to give a stronger field which would provide a brighter point or less strong field which would give a darker point.  The 'refresh' rate was so fast that you saw moving picture.  

A machine to measure, for instance, the relative amounts of Carbon 14 and Carbon 12 works pretty well the same way.  The sample of carbon to be measured is heated so hot that charged atoms of carbon are produced.  This can be done with a laser.  The charged atoms are accelerated through an electrical field and then between electric magnets, one North and the other South.  This bends the beam of Carbon atoms.  The neat part is that the heavier atoms are bent less than the lighter atoms.  Where the beams hit the side of the container are detectors and the electronics connected to them can measure individual atoms.  The strength of the magnets can be varied to bend the beam more or less to ensure each beam hits a detector.  Clearly, this is a very accurate method of measuring very tiny amounts of Carbon 14 and Carbon 12.  The number of hits on each detector can be inserted into the formula.

Logarithms

I promised I would try to explain logarithms.  First a bit of notation.  If you see a notation log1000 = 3, it is understood that this is the log to the base 10.  If you use another base, you have to state it.  There is another notation ln.  It means the log to the base 'e'.  e is the base of the natural logarithms, whatever that means.  I don't understand it.  e is 2.71828.  It is known as Euler's number.  If anyone out there has a good explanation for the natural logarithms, please put it in the comments.  Anyway the two following notations are equivalent.

10⁵ = 100000

log 100000 = 5

You can see that the log of a number is the exponent that you have to raise 10 to in order to get the number.  Note again that it is understood that if you use the term log, it is understood that it is to the base 10.  You could also write it this way.

log₁₀100000 = 5

Late Capitalism

 All sorts of articles declare the demise of capitalism and call the state we are in at present 'late capitalism' as if this is it's final gasp.  This could well be so but to my way of thinking it has little to do with capitalism itself.  The problem is not capitalism, communism or any other 'ism'.  The problem is the lack/demise of democracy and the uneven application of 'the rule of law'.  Whatever system you introduce, there will be an attempt by some members of your community to subvert the system to their own ends and accumulate all the 'filthy lucre' and power in their own hands.  In the extreme, the psychopaths rise to the top by trampling all in their way to get there and work hard to subvert the rule of law by the use of the wealth they have accumulated.  Look at who heads major businesses.

Look at our election systems as an example.  We allow businesses and rich individuals to finance our elections and then are amazed that the resulting government does the will of these contributors.  Stopping this nonsense would result in a better class of politicians, ones that are not primarily motivated by greed and make all the much needed reforms  that much easier.  Lets start with Communism.

  Communes could well have been a good solution to their economic woes in China and Russia but two sociopaths took over - Mao in China and Stalin in Russia.  Both led to the deaths of millions of their own citizens and both suppressed the rule of law and any influence of their people in the running of their government.

Look, also, at the various religious communes that have sprung up over the years.  They typically have a charismatic leader who seems to think that every attractive woman in the group should be his private hand maiden and the women, sensing the alpha male, go along with it.  Don't get me wrong.  There is nothing (or not much) wrong with a commune based on free love.  Hypocrisy is the problem.  The glorious leader preaches celibacy and abstinence for this flock while going full-licentiousness for himself and perhaps a couple of lieutenants.  If the whole group agreed to 'willing partner' it would greatly reduce the tension.

One last example is the Kibbutz in Israel.   This was true commune-ism.  The difference was that it was fiercely democratic with decisions taken by all the members.  Leaders of the various branches of the Kibbutz (fish ponds, dairy, factory, kitchen, horticulture, field crops) and of the whole kibbutz were changed around, typically, every three years as was the representative from the kibbutz to the central organization of that particular movement.  It worked an absolute treat but I'm sad to say the system has collapsed.  Reasons had to do with fiduciary responsibility and the desire of the  women to have a larger part in the lives of their offsprings.  Such a shame.  The system worked very well. 

And then we have America.  It is highly questionable if she ever actually had a democracy but now it is going South at a rapid rate.  We mock Russia with her oligarchy.  If you think that is bad look at the way America is going.  It is nothing that couldn't be cured by a strict following of 'one man one vote', a strict following of application of the rule of law evenly across all levels of society and the cessation of the system in which you allow corporations and rich individuals to finance elections, make politicians rich while they are in office and give them sweet jobs when they retire.  Such simple solutions but Republicans would completely loose control of their sphincters if anyone proposed such a solution.  Capitalism is not the problem but the corruption of one man one vote and the application of the rule of law.