## Saturday, November 5, 2016

### Some basic science

If you want to  to make sense of the weather, the climate and climate change, at the very least you need some basic science.  As our education curriculum gets more and more crowded by IT and other subjects, many of these concepts are not being taught and they are interesting.  Way back, when I was in High School, the basics were taught and became part of our core knowledge.  (Sorry to sound like an old fart but I guess I am)  So hear goes.  No particular order.  I will type them as I think of them.

Sensible Heat (as in 'to sense something' - not that the heat is somehow morally superior to some other unsensible type of heat)

This is the heat that it takes to raise the temperature of something or the heat that must be removed to cool something.  The small calorie (with a small 'c') was defined as the amount of heat needed to raise one gram of water by one degree centigrade.   There is also a large Calorie (with a capital 'C') which is the amount of heat to raise a kilogram of water by one degree.  Clearly since there are 1000 grams in a kilogram, there are 1000small calories in one large Calorie. Incidentally, if you need to work in SI units, One calorie (small 'c') is equal to 4.1813 joules*.

*A Joule is one watt second.  In other words a watt (of electricity) acting for one second.  If a one watt heater was immersed in a gram of water and operated for 4.183 seconds, it would heat the water (assuming no heat loss) by one degree C (or K if you like).

Latent Heat
I defined sensible heat first in order to make it easier to define Latent heat.
Latent heat  is the amount of heat needed to melt or evaporate a substance or conversely, the heat that must be removed to condense or freeze something.
This is a core concept when talking about weather and for a whole range of other subjects.  I'll use the old calorie units here because it makes the explanation easier.

When you melt ice, it takes a lot of heat. Specifically 80cal (334j) per gram.  This is the same amount of heat that would be needed to raise a gram of water from 0 degrees   to 80 degrees C.  Conversely, when water freezes, each gram gives out 80 calories.  Some people ask; "so if heat is given out, doesn't that warm the water".  No, but to a good first approximation, it keeps the water at zero degrees until all the water is frozen.  Similarly, when you boil water, it stays at 100 degrees while the water is being boiled off.

https://en.wikipedia.org/wiki/Latent_heat

The energy needed to change  liquid water to water vapor is even greater.  To evaporate one gram of water takes 532 cal (2264j) or enough heat to raise a gram of water from 0 degrees C to boiling 5 times over and then a bit.  Of course, when water vapor condenses into a liquid this same 532cal is given out.

Since we have these two, we can add them and find the heat of sublimation.  This is the heat needed to change solid ice to a vapor directly without going through the liquid phase.  This is 532+80 =  612cal (2598j) per gram of ice.

As an interesting corollary* of this: if moist air blows across ice and the cooling of the air by the ice causes the water vapor to condense out as water, every kg of water condensed out of the air will melt 532/80 = 6.65 litres of water from the ice.  Think of a moist foen wind blowing across Greenland for instance.

*A corollary is something that results directly from some previous fact(s).  Usually used in Math.

If you look at the table in the link above, you will note that two substances, Ammonia and Water have much greater phase change energies (latent heat) than other substances in the table.  The reason is of interest.

You have probably heard of the experiment of Earnest Rutherford who worked out that atoms are not like plumb puddings but more like solar systems.  Most of the mass is concentrated in the center and most of the rest of the atom is empty space.  The solar model was an improvement as far as it went but later it was worked out that while the so called 's' sub orbits were pretty well circular, the 'p' sub-orbitals were dumb bell shaped.     In water, the hydrogen attaches to the ends of two of these dumb bell sub orbits making the water molecule an angular shape.  Move your thumb and index finger as far apart as you can and look at them.  The Oxygen atom is located where your thumb and fore finger join and the hydrogen atoms are on the ends of the finger and thumb.

The electrons spend much of their time around the oxygen atom leaving a naked proton at the other "end" of the molecule.  Not only is the water molecule charged (positive at the Hydrogen side and negative at the Oxygen side) but there are no electron orbits closer to the hydrogen nucleus to keep other molecules away.  A negative charge can get much closer to the Hydrogen end than with other atoms and hence the bond is stronger*.  This is the famous Hydrogen bond.

*In general, a force field decreases with the square of the distance from the object creating that field.

This is the explanation why it takes so much energy to melt or evaporate H2O.  The water molecules, because they have a positive and a negative side, cling together.  This is also why their melting and vaporization temperatures are so high compared to other molecules of a similar molecular weight.

Incidentally, if you want to visualize ammonia, spread your first two fingers and your thumb far apart.  The Nitrogen atom is where they all meet with the three Hydrogen atoms on the ends of your fingers and thumb.  Just like water, ammonia has a positive and a negative end (side) and the positive ends are naked hydrogen atoms (protons) and hence form the famous hydrogen bond.

You might wonder how a molecule of water can leave the surface of the water and go into the air. After all, it is at the temperature of the water and the molecules cling together.  It is a little like a rocket leaving the earth.  It has to have enough energy to break free of gravity.  The water molecule has to have enough energy to break free of the electrostatic attraction to other water molecules. The answer is that temperature is a measure of the average energy of the molecules As they bounce off each other, as long as energy is not being added or removed from the body of water, the average energy stays the same.  However this is only an average.  As they randomly bounce off each other, there will be some that have more velocity and some less.  Some molecules on the surface will have enough energy to rocket into the air.

Incidentally, this is the explanation why evaporating water cools the surface it is on.  The energetic (hot) molecules leave, leaving behind the less energetic (cooler) molecules.  They absorb heat from the surface they are on and in turn some of the energetic ones rocket into the air.

Some genius worked out that a given volume of any gas at the same temperature and pressure contains the same number of molecules. (it wasn't Avogadro)  Oxygen exists in the air as a molecule of two oxygens joined together as does hydrogen and nitrogen.  This is rather convenient since if you know the molecular weight of any gaseous molecule, you know it's relative specific gravity (how heavy it is compared to other gases).

Hence, Oxygen has an atomic weight of 16 (rounded up), it's molecule is 32.  Nitrogen is 14 so its molecular weight is 28 and water is 16 plus 2 equals 18.  Air (very approximately) has a molecular weight of 30 (between Oxygen and Nitrogen*).  Water vapor therefore has a density of 18/30 = 3/5th or 60% of air.  Counter to what might think, a mix of air and water vapor (humid air) is  lighter than dry air.

* I haven't taken into account that 4/5th of air is N and 1/5 is oxygen or the actual mollecular weight of the gasses.  This would alter the calculation slightly.  The above is what one calls a first approximation.

Note here that when you dissolve sugar or salt into water, the solute (solid) to some extent fits between the water molecules so the volume of the resulting solution is less than the volume of the original water plus the volume of the original solute.  With gas this is not so.  If you add a gas to an existing gas, it increases the volume by exactly the amount that you added.

Temperature
Ignoring the Fahrenheit system that only a few primitive societies still use, the centigrade system is as follows. (conceptually - there are a few minor whichevers and we will have a look at them later).

Using a thermometer, you mix ice and water and note where the liquid in the thermometer settles down.  You make a mark and call this 0 degrees Centigrade.  You then boil the water and once more make a mark where the liquid comes to in the shaft of the thermometer.  You call this 100 degrees Centigrade.  You divide the difference into 100 gradations.

Absolute Temperature
Using the same gradations you go downward until you can't remove any more heat from whatever substance you are examining.  You find that as low as you can go is about 273 degrees below the freezing point of ice.  This is called absolute zero and is a point at which no more heat is held in the substance. In Kelvin,(the absolute system)  the freezing point of water then becomes 273K and the boiling point of water 373K

A way of getting a first estimate of absolute temperature is to cool a gas and note its volume as you hold the pressure constant.  Draw a graph.  Where the volume goes to zero is a good first approximation of absolute zero.

How about the whichevers.  First then a word on isotopes.

Isotopes
The nucleus of atoms contains positively charged protons and no charge neutrons.  The neutrons, somehow hold the protons together from flying apart. Don't ask me how.  That is above my pay grade. The protons are all positively charged and you might remember from science that same charges repel each other.  In any atom, there are approximately the same number of neutrons and protons.  This is not quite correct and becomes a little less so for the heavier atoms but it is a fair first approximation.  A given element, let's say Carbon, always has six protons.  This is why it is carbon or to be more accurate, it has the same number of electrons as protons and 6 electrons results in the physical and chemical properties that we know as carbon.

However, within limits, it can have various numbers of neutrons.  Carbon can have 6, 7 or 8 neutrons and hence Carbon 12, Carbon 13 and Carbon 14.  Carbon 12 and 13 are stable but Carbon 14 is not.  If you have an atom of Carbon 14 it will at some point fly apart.  You can't know when this will happen for any given atom but it has been observed that with large quantities, you know how much of the carbon 14 will break down in any period.  It turns out that half will break down in 5730 years and half of the remaining half in another 5730 years and half of the remaining quarter in another 5730 years.  This makes it very useful for dating organic material and I will explain this later and give you the math needed to date objects.  It is not difficult.

You might wonder why there is not an equal amount of Carbon 12 as 13 if both are stable.  I don't know.  Carbon 12 is far more prevalent.  Perhaps for some reason more Carbon 12 is produced in super novas that carbon 13.  If someone knows, put a note on the bottom of this  blog.

Most elements have a number of isotopes and some are stable and some are not.  The unstable ones have half lives which vary in the different elements from miliseconds to millions and millions of years.  It is not always the heavier that is the most unstable.  For instance Uranium 235 is less stable than U238. Apparently certain configurations are in a sweet spot.  The search carries on to find trans-uranic elements which have sweet spots.

So back to Temperature
Water can be made from any of the isotopes of Hydrogen and Oxygen. Hydrogen has three isotopes, namely ordinary hydrogen with one proton in the nucleus, Deuterium with one proton and one neutron in the nucleus and, you guessed it, Tritium with one proton and two neutrons.

Oxygen has three isotopes, O16, 17 and 18.  Since she has 8 protons, these isotopes have 8, 9 and 10 neutrons in the nucleus.  In this case, O16 is the stable one. (often although not always, the lighter one is the stable one.) So what does this mean.

If the lightest Hydrogens reacted with the lightest Oxygen, you would have a molecule with 18 nucleons.  This is as light as water can get.  If the heaviest of both linked up you could have a water molecule with 24 nucleons.  Since virtually all the weight of an atom is due to the nucleons, you can have a wide range of weights.  Say for a first approximation that all the atoms in the water you are using to establish your thermometer have the same energy.  To have the same energy, the lighter molecules are moving faster. (energy is equal to half the mass times the velocity squared) so they will more likely have escape velocity if they find themselves at the surface of the water.  So what happens.

The light molecules fly off preferentially leaving the heavier molecules and as you continue to boil the water, the temperature rises.  The reverse is also true. The heavier molecules of water condense more easily (at a higher temperature) than the lighter ones.

This, of course, makes the calibration of a thermometer a tad difficult as the boiling point of water (if you want to be picky and scientists are very picky) keeps changing as you boil it.  Of added difficulty, some sources of water have slightly different proportions of isotopes than others.

Boyls Laws
These are pretty simple and also come from observation.   Simply stated they are as follows.

If you increase the pressure on a volume of gas, the volume decreases.  If you double the pressure you half the volume.  Nature could have given us some other relationship between pressure and volume.  Isn't it nice that it is such a simple relationship.

If you heat up a gas, its volume increases.  Here though, we are talking about absolute temperature.  If you double the absolute temperature, you double the volume. (if the pressure is kept constant). Again, nice that nature provides such a simple relationship.

For instance if you were to raise the temperature of a gas from the freezing point of water to the boiling point of water you would increase its volume by 373/273 or by about 1.366 times.  (that is why I told you about absolute temperature first).

Coriolis
The earth is about 25000 miles around its equator.  It rotates on its axis once a day.   Hence if you are standing on the equator you are traveling at about 1000miles per hour eastward.  If you were standing on one of the poles, you would rotate once per day but are moving at 0 miles per hour (we are in an earth reference frame).  Of course the earth is moving through space and you with it but that is neither here or there for this example).  If you fire a cannon ball northward from the equator in the northern hemisphere, In addition to its northward velocity, it is traveling sideways toward the East at 1000mph.  It will still be traveling toward the East as it flies through the air but the ground over which it flies is traveling slower and slower, the further north you go.  Looked at from above, the object veers to the right in relationship to the earth below.  As you can see, the effect is greater, the further north you go.  If you go from the equator to a degree north of the equator, the sideways velocity hardly changes.  If you go from one degree south of the North Pole to the North Pole, the change in velocity is large.  The same occurs when you move something southward in the Northern Hemisphere.  It veers to the right.  This effect has some profound implications on our weather.

As I mentioned, radioactive isotopes break down into simpler atoms.  Certain proportions of protons and neutrons in the nucleus are not stable.  It was observed fairly early on that if you measure the time it takes for half of the radioactive isotope to break down, then half of the remaining will break down in the same time, half of what remains in the same time and so forth.  This is called the half life.  I'll use Carbon as the example since it is valuable for the dating of organic material. Carbon 14 has a half life of 5730 years.  You might ask, if it has a half life so short in comparison with the age of the earth, how come there is any of it around now.  Also, you would have to know how much was in a living organism when it died in order to measure how much there is now and use these two figures to date it.  The answer is rather neat.

Carbon 14 is continually being produced in the upper atmosphere by an atomic reaction.  When high energy cosmic rays hit Nitrogen 14, some of it is converted into Carbon 14.  To a first approximation (more of this later) its rate of production has been constant over time.  Living organisms incorporate carbon into their bodies throughout their lives but when they die, no more is taken in.  The Carbon clock starts and if you can measure the proportion of Carbon 14 in relation to "ordinary" carbon, you can tell the age of the artifact.  Now for the math. Co is the amount of carbon in the artifact (piece of wood) at time zero.  That is to say, when the wood died and stopped taking up carbon.  Ct is the amount of carbon at time 't'.  That is to say when you took the artifact and decided to measure it.  1/2 is just what it says.  One half.  and 'n' is the number of half lives that have gone by so we have the formula.

Ct=C0 X (1/2)n. Also written in algebra without the times sign as Ct=Co(1/2)n

Lets look at this.  Suppose you start with 8 grams of a radioactive element and one half life has gone by.  You raise 1/2 to the first power which leaves it as 1/2 and multiply by 8.  Answer 4 grams.  Suppose two half lives have gone by.  You raise 1/2 to the second power (multiply 1/2 by 1/2) and you get one quarter.  Multiply this times 8 grams and you have two grams.  Let's do one more.  Three half lives have gone by.  You raise 1/2 to the third power (multiply 1/2 times 1/2 times 1/2) to get 1/8.  Multiply 1/8 times 8 and you get one gram.  So this formula works.  Now let's make it a little more sophisticated.  We will define 't' as the time that has gone by and 'h' as the half life of the isotope we are working on.

Clearly the number of half lives that have gone by equals t/h. Say the half life of an isotope is 10 years and 30 years have gone by.  Clearly three half lives have gone by.  In other words n = 30/10.   We can now substitute this into our first formula.  Where we had n we will put t/h so Ct = C0 x 1/2(t/h).

Now we have the formula in what I call the forward or straight forward form. In other words in the form that is easily understood, we can now "solve for" any of the terms. In other words make t or n or Co the subject of the formula. If you remember your algebra, since the right side is equal to the left side, as long as I do exactly the same to both sides, the formula will still be valid. I could multiply both sides by some number, square or take the square root of both sides and so forth.   The trick, of course,  is to choose the correct thing to do to both sides to get the term I want as the subject of the formula.  Why bother. Well, sometimes I might want to work out the age of the artifact 't' or the half life of an isotope 'h' so it is useful to change the formula around to make the desired factor the subject of the formula.  Of course to work out 't' or 'h' I would have to know the value of the other terms in the equation.

Let's solve for 't', the time that has gone by.  Then we will have a formula we can use for dating an artifact.

I divide both sides by Co resulting in

Ct/Co = 1/2(t/h)

Now you will have to take my word that the following identity is correct.  To explain logs at this point would take a tad too long.

logABC (log to the base A of B to the Cth power) = C x logAB (C times log to the base A of B).  In other words, you can put the exponent before the log and the value remains the same.   So first we will take the log of both sides.  You don't have to understand logarithms but only the principle that if we do the same to both sides of an equation, the formula is still correct.

Log10(Ct/Co) = log101/2(t/h)

Using the conversion (moving the exponent, (t/h) in front of the term on the right side, I get:

Log10 (Ct/Co) = (t/h) log10(1/2)

Now to get t by itself, I simply divide both sides by (log10((1/2)) and multiply both sides by h to get: t=hlog10(Ct/Co) / log101/2

I said that this was a first approximation.  It was seen that the value obtained for objects of known age differed slightly from the theoretical value.  For instance, in the high mountains of America is a species of tree known as the Bristle Cone Pine. As with many trees it has growth rings and live trees have been found that are 5000 years old.  In addition, in the area, there is dead wood which with Dendrochronology can take the age back another 5000 years.  Carefully shaving off individual growth rings and carbon dating them showed a small variance from the theoretical value.

The best explanation for this is that the rate of C14 production has not been exactly the same over time.  Cosmic rays come from violent events in the universe and have varied over time.  What is good, though, is with the application of this correction, the age of artifacts of known age slotted into place.