A Medley of Potpourri

Chapter Five -- All Molecules Great and Small

2012-01-11T12:37:00.001-05:00

AllMolecules Great and Small

Iused the word bond(s) quite frequently in the last chapter, and by now youshould have some grasp of what they are: when two atoms approach closely enough, the most stable state of theiroutermost, or valence, electrons usuallycenters between the atoms. Whenever twoor more atoms can bond this way, the resulting structure is called a molecule;i.e., two hydrogen atoms and an oxygen atom will bond together to make thestable water molecule, of H₂O. We were also introduced to the idea that different atoms can formdifferent numbers (and types) of bonds: one for hydrogen, four for carbon, three for nitrogen, two for oxygen,and so forth. (The four bonds carbon canform will prove most convenient in coming chapters.)

Thereare numerous ways of portraying bonds. In first year college chemistry courses the student is usuallyintroduced to the Lewis dot structure. Whereas before we represented bonds as barsbetween atoms, e.g., H-H,a better representation is to show the electrons themselves and where they arelocated in space: H˸H. Here it is clear that the two hydrogen atomsshare their electrons in a region focused between the two nuclei. You can see the stability this creates, andso why the bond is stable.

You’realready suspicious though, I’ll bet. TheLewis dot structure still commits a cardinal sin when portraying electrons; forthey are not hard little dots or spheres but spread out regions of uncertainty. Remember how in the quantum world, we cannotsay that objects cannot have exact locations, speeds, and directions; that ifwe do devise an experiment to measure one quantity exactly, we lose allinformation about the others. Probability distributions rule the day. Thus, an improvement on the dot structure representation would looksomething like this:

Figure XIV.

Here,the black spheres represent the hydrogen nuclei (usually just a proton), andthe smeared out red region the electrons in the bonded state. Note that we don’t even distinguish betweenone electron and the other, only that their combined region of highest density(probability distribution, that is) is directly between the nuclei. This is about the best representation of howelectrons behave in atoms and molecules. To answer your next question, the dotted line surrounding the moleculesencompasses ninety percent of the electrons’ distribution – in fact, this goesoff to infinity.

* * *

What’scoming is, I think, rather grimmer than the material covered so far, but is soessential to describing atoms and molecules and bonds that I see no way ofavoiding it. Some advanced equationswill be presented, but don’t worry, because there’s no need to understand themor know how to solve them. I displaythem for you so that you will see there is a mathematical underpinning to allthis talk of electron states and bonds. Truth be told of course, there is a mathematical underpinning to allphysics, so this should not come as a surprise.

Butfirst, a switch in terminology. Insteadof states we will use the word orbital(both atomic and molecular) to describe an electron’s characteristics. Orbital is, in my opinion, an unfortunatechoice of word, as it implies electrons orbiting the nucleus, a fallacy I havestruggled to help you overcome. But ithas been used so long now that we are apparently stuck with it; just rememberthat it is a synonym for state, not orbit, and you should be fine. With this in mind, let’s plunge into thefray:

Figure XV.

Thisis the Schrödinger equation for the hydrogen atom. You may recognize or know some of the symbolsit contains, but the one I want you to concentrate on is y,which is the equation’s solution. Interestingly, it is a wave function, i.e., a function that describeswave motion, such as that of a piano string vibrating back and forth. The general form of this function, thesolution to the equation, is:

Figure XVI.

Again,don’t worry about most of the details; deriving this solution requires someheavy duty math. There are certainvariables you should pay heed to, however. These are n,ℓ, and m,known respectively as the principle quantum number, the angular momentumquantum number, and the magnetic moment quantum number. The combined values of these numbersdetermines the exact form of y. There are relationships among them as well: while ncan have any integer value from one on up (1, 2, 3, 4, 5, … ¥, ℓcan range only from 0 to n – 1, and mfrom – ℓ to + ℓ. So for example, if n = 1 (the lowestenergy states of the atom), then l and m can only be 0; while for n = 2 ℓcan have the values 0 and 1, while mbe one of three values, -1, 0, and 1.

The orbitals I mentioned earlier aredetermined by their quantum numbers. Thenumber specify their shapes, the probability distributions of the electronsoccupying them (a maximum of two, known as the Pauli Principle), and theenergies of those electrons:

Figure VII. (repeated)

Theelements in the periodic table are built up by adding additional protons (andneutrons) to the nucleus, and then filling the atomic orbitals from inwards tooutwards with the required electrons to maintain the atoms electricalneutrality (atoms that are not neutral are called ions, but they are stillatoms). For the working chemist theelectrons are described by orbital designations; say, for selenium, or element34, the designation would be 1s²2s²2p⁶3s²3p⁶4s²3d¹⁰4p⁴,where 1, 2, 3, and 4 represent the shell or primary quantum number, s, p, and dthe ℓnumbers in the shell, and the exponent the number of electrons occupying eachorbital group in a shell (thus, there are 3p orbitals, 5 d orbitals, 1s orbital, and so on). I won’t describe this scenario in any greaterdetail because: a) it still befuddles mesome after all these years and I doubt I could describe it clearly and simply;and b) our main thrust will be bonding/molecular orbitals, and only a basicgrasp of the atomic kind is needed to do so.

* * *

Thereis one more revelation about atoms that must be made before turning tomolecules. In truth, all the atoms thathave been built up in the periodic table should be analyzed by constructing andsolving a Schrödinger-type equation for each one, yet we have only done so forhydrogen and assumed the results applied to all atoms. Amazingly, and fortunately, they almostdo! In fact, we could set up an equationfor each elements; could set up, but never solve by any known mathematicaltechnique known to Homo sapiens. The reason? In hydrogen we are dealing with only two interacting bodies, but forevery other atom and molecule, we are working with more than two. The so-called three-body equation is intractable to all math and physics; notjust in the quantum world but everywhere, at all times, in the solar system andthe universe as a whole – though whether it explains why there are only twosexes I have no idea: nature seems tohave no difficulty with it, but then nature is just doing what it doesaccording to its laws; it (She?) isn’t trying to calculate anything in advance.

Returningto the equation of the hydrogen atom we are lucky indeed that its solutiongives us a sound working model for the other atoms as well; only minormodification are needed to obtain excellent results for them. However, and this is important to emphasize,any two or more solutions (functions) to the equation that have approximatelyequal energy and obey the mathematical requirement for “orthogonally” can becombined to make new functions/solutions. For example, on carbon the single 2sand three 2porbitals can be combined to create four new “sp³“orbitals, which stick out at tetrahedral angles from the carbon nucleus:

Figure XVII.

Isuspect I have presented you with a puzzle. Why not use these orbitals to hold carbon’s four valence (outer)electrons, instead of the original one from solving the Schrödingerequation? The answer is simply that theoriginal orbitals are at lowest energy when considering lone carbon atoms; onlywhen chemically bonded to other atoms – including other carbon atoms – is thisconfiguration preferable, for reasons we shall see soon enough.

Thisis not the only possible combination. Another “hybrid” (as they are know) orbital is the sp²orbital, in which one of the porbitals – technically the p_z)is left out:

Figure XVIII.

Noticethat the omission one a porbital has changed the geometry of the atom from tetrahedral to a flattriangle; the new hybrid orbitals comprising the triangle is called, reasonablyenough, sp². Other hybrids or combinations produce a widevariety of configurations, especially when they involve d or higherorbitals.

* * *

Imade the blithe statement that all of this has to do with bonding with otheratoms to produce molecules, and it is time to explain why.

Atomstakes up considerable space in molecules, and therefor can not simply assumeany arrangement. A good example of thisis the compound methane, or CH₄. We should like to spread the hydrogen atomsand their electrons as far apart as necessary, in order to reduce interelectronrepulsions. As a first guess we couldtry a flat structure:

│

H ¾ C ¾ H

│

Figure XIX.

Thatthis is inadequate should not be difficult to see. We have here a two dimensional representationof methane; and although it is often convenient to draw it out this way, thelack of a third dimension means that the hydrogens are not spread as far apartas possible.

Goback to the tetrahedral arrangement of electrons around carbon, and you can seethat the carbon’s electrons and the hydrogen’s electrons finally put as muchspace between each other as possible, to produce the configuration with lowestenergy:

Figure XX.

Comparingthis with Figure XVII.We see that the tetrahedral arrangements of hydrogens about the central carbonexactly matches its 3p³bond arrangement. This is why these arethe preferred orbital pattern in the molecule. The only thing left to do in our preliminary exploration is to describejust what a bond is. I’ve used anequivalent phrase molecular orbital, but with out explanation what that means.

Thefact that I’ve used bond and molecular orbital interchangeability should besuggestive however, implying that bonds are orbitals too, and this is quiteright. Just as atoms can combine theirbase orbitals into hybrid orbitals, so can orbitals on different atoms also merge into molecular orbitals, given the rightconditions of fairly equal energy and orthogonality. Remember the depiction of the hydrogenmolecule earlier in the chapter:

Figure XIV.repeated

Ofcourse, we really need to solve the Schrodinger equation for the system, but wecan’t so we must find another way. Again, that is the linear combination of orbitals technique. The 1s orbitals on each hydrogen atom addtogether to produce the molecular orbital or bond, within which the twoelectrons have even lower energy than single atoms. Bear in mind, though, that the number ofhybrid or molecular orbitals must equal the number of base orbitals we beganwith. Thus there are four sp³orbitals about carbon (and nitrogen and oxygen, and many of atoms). So combining the 1sorbitals on the two hydrogens must yield two new orbitals. And indeed they do; in addition to thebonding orbital, there is also an “anti-bonding” orbital, an orbital which hasthe effect of repelling the hydrogens if there are electrons in them as much asthe occupied bonding orbital which attracts the hydrogen:

Figure XXI.

Noticethat the main electron density is on the outside of the atoms instead ofbetween them in Figure XIV. Some things ought to bestarting to gel by now. Normally,hydrogen exists as a diatomic molecule, H₂. This is because the two electrons from theatoms can fill the bonding orbital, but the anti-bonding are empty. On the other hand, the next element in theperiodic table, helium or He, exists a monoatomic atoms. He₂, if it existed, would have two more electrons than H₂,and those two electrons would fill the anti-bonding orbital, pushing theheliums apart and breaking the bond. Hence,only He atoms are found in nature.

* * *

Thereis one more step until we can start letting the horses out of the barn. We’ve spoken so far of base atomic orbitals(1s,2p,3d,etc.), hybrid atomic orbitals (sp³andsp²),and molecular orbitals resulting from combining base atomic orbitals. Look back on the picture of the methanemolecule (Figure XX). Can you see how the sp³orbitals on carbon are combining with the 1son the four surrounding hydrogens? Remember, any proper linear combination of orbitals (strictly speaking,their underlying wave functions) yields new orbitals. So hybrid orbitals can combine with atomic,other molecular, or hybrid orbitals. Thepossibilities are mind-boggling, perhaps even infinite, especially as you addmore and more atoms to the potential molecules. Many millions of molecules are already known to science. Obviously, I can only cover a very tinyfraction of them, but fortunately that will do to get the idea in your mindclearly.

Let’slook at methane again, and try an experiment in our minds. What if we were to take a methane and pop offone of the hydrogens? Why then, we wouldhave something the looks like this:

│

H ¾ C ·

│

Figure XXII.

Forgivethe flat structure, which we know not to be true, but there is method inmadness here. The first thing the hitsyour eye is that the right hydrogen has been replaced by a dot, whichrepresents an unbonded electron. Inremoving the hydrogen with its lone electron, we leave carbon’s contribution tothat bond hanging out there in space, so to speak. More importantly, it is now available forbonding with another atom or molecule with a lone electron. For example, themolecule here is actually called a methyl radical, radical meaning containingunbonded electrons. We can certainlycombine it with another methyl radical:

H H

│ │

H ¾ C ¾ C ¾ H

│ │

H H

Figure XXIII.

Using the flat structure for methane makes it mucheasier to draw this new molecule, called ethane. The bonds about each carbon are stilltetrahedral, however.

Incidentally, bonds of this nature, with head-onoverlap of the constituent orbitals are known as sigma (σ)bonds. The way this chapter hasprogressed, you can be excused for thinking it is the only kind of bond. There are several more, and right now I’dlike to concentrate on a specific one: the pi (π) bind. In keeping with methane and ethane, I’ll showyou the simplest case, that of ethylene:

Figure XXIII.

Before discussing the “double” bond between thecarbon atoms in ethylene, I want to draw your attention to the flat, trigonalgeometry about them. We have alreadyseen this, in Figure XVIII,the sp²hybrid orbitals which creates this geometry. Four of these orbitals combine with the 1sorbitals on the hydrogens to create those bonds, while the two carbon pointing sp²overlap to create a sigma bond, just as with H₂. But this leaves something unaccountedfor. Both carbon’s p_zorbitalsare now free to make other combinations, and each contains a single electron itcan use for bonding. In fact, the moststable bond it can form is by a sidewaysoverlap of thep_z orbitals. Such resulting bonds are called pi bonds, andthey look like this:

Figure XXIV.

Imaginethe two orange balloons on the left as the carbons’ p_zorbitals;then the yellow region on the right is the region of maximum overlap orcombination. This orbital (althoughthere are two lobes, it is one orbital, above and beneath the sigma, represented by the straight line) is thebonding; the psalso combine to make an anti-bonding orbital, which on ethylene is empty,stabilizing the pi structure.

* * *

Thischapter could cover more, much, much more; and we will pursue more materialalong these lines. Especially, ourexploration will heavily revolve around hydrogen and carbon, as well as oxygenand nitrogen; for these are the main atoms which make up the molecular natureof life. The basics have been laid downhere.

Chapter Four: Quantum Cats

2011-12-22T08:23:00.000-05:00

The last chapter, on relativity, was awarm up to the problem of thinking about weird things. Remember, things only get weird when theyfall outside, usually well outside, our common range of experiences. We all have this biologically built-inunderstanding of the world, of physics more precisely, which we’ve acquiredfrom our genetic ancestry and life’s experiences. We rarely give it much thought, unlessmagicians play havoc with it to fool us, or physics teachers demand we absorball the symbolic definitions and formalities and math we need if were going to– say, if we’re going to figure out how to send men and instruments to worldsmillions of miles away. Fortunately, fewof us are called on for such flights of fancy.

Special relativity is weird because itinvolves travelling at speeds no one has ever come close to. General relativity is weird because it triesto understand the universe as a whole, or at least things much more massive thanourselves; ambitions few of us have ever gotten the itch for.

It’s time now for a differentdirection. From now on, we will befocussing on the realm of the vastly small and, often, extremelyshort-lived. To give you an idea of justhow small, how many molecules of water are in a glassful? If the volume is about a cup, or, more usefully,let’s use the metric system and call it 100 milliliters or 0.1 liter, then thatnumber is about 33,000,000,000,000,000,000,000,000 or 3.3×10²⁴molecules. To capture a feel for justhow large a number than is (i.e., how small water molecules are), all of thewaters of this planet Earth would take somewhere between 10¹⁹ and 10²⁰glasses to hold it. That number is atleast ten thousand times smaller thanthe number of water molecules in an ordinary drinking glass.

Myopening salvo is, objects that small definitely fall deep into our notions ofweird. You really can’t imagine it all,you can only do calculations with the numbers. I’m tempted to try to give you a feeling for it; say, by going down tothe beach and building a sand castle, and all that. But sand grains themselves are so large thatthey fall into the range of common-sense physics, and are barely one iota alongour quest for the ultra-small. If sandgrains behaved as atoms and molecules do, this would be a very, very differentworld indeed.

* * *

If you were to go down to the beachanyway, and try to follow the whims and whereabouts and whichways of a sandgrain buffeted about by the sea breezes, you would probably feel like a catchasing the spot of a laser pointer about the house (they go nuts chasing it). Nevertheless, come, let me take you – a childversion of you works best because of its innocence – down to the sea shore tochase a grain of sand. You will no doubtquickly report to me that it is impossible, you will never be able to do it. In response, I say, “I know what you say sounds right, but itcan’t be. Think about it. At every moment in time the sand grainhas: a) an exact mass; b) an exactspeed; c) an exact direction it’s travelling; d) an exact momentum; e) an exactkinetic energy; f) an exact temperature; g) an exact albedo (how much of thesun’s light it reflects); and – I’m sure I could come up with othermeasurements, but you get the idea. Youjust need to apply yourself more diligently to follow these quantities.”

A true child would not comprehend thisquandary, but just find something else to play with. But the adult you gets the point: whilein practice we simply cannot move quickly enough to follow a sand grain’s variousdances, in theory it could bedone. Could be. Are you certain? What if I were to start shrinking that grain,ever smaller and smaller, until it was so small it fell beyond the range of themost powerful light microscopes to see it, down to something of the size of awater molecule? What would happen then?

We suddenly feel as though we’re in themiddle of Paul McCartney’s song “Lucy in the Sky With Diamonds” or perhaps“Penny Lane”; a surreal, dreamlike state where all the ordinary rules of realityhave been suspended, and we have to figure out the new ones.

And in truth, that’s not justmetaphor. How reality looks and actslike at this scale is truly, and astonishingly, very, very little like sand grainsand castles and rail cars. And it is, atleast at first, just a little unsettling. For it means, for example, we have to throw away things like ourprevious, common-sense idea of exact measurements. Oh, you can still make them, but at aprice. That price is complete loss ofall information about something else equally measurable.

As a concrete example of this (alwaysturns abstract concepts into concrete visions, when possible, remember), try tomeasure the position where our sand grain scaled down to water molecule can befound at any given moment. Yes, you canstill set up a measuring system that gives you a 100% accurate value for thatposition. But only for that moment whenyou make the measurement. And where hasthe grain gotten too after that? Strangely, oddly, bizarrely, utterly incomprehensibly we have no idea! In order to measure its position exactly, wehave completely scrambled all information about its speed and direction fromthat position. It, literally, and I meanliterally in the literal sense here as in quite absolutely literally, could nowbe anywhere in the known universe!

You see, we must hand over ourmagnifying glasses and microscopes, and radar guns, and all other mechanismsfor measuring the quantities of things, over to the police of this Brave(perhaps with a few shots of gin) New World, and accept new tools for spying onthe world about us. These tools onlyallow us to make probabilistic approximations on things. By accepting limits on how well we canmeasure, we can measure other things too, albeit within their limits as well.

IfI sound heavy-handed in repeating again that we are not dealing with metaphoror analogy, but with nature as it actually is, please be patient. We are so accustomed to our “normal” rules ofreality that different ones do sound like artistic inventions or flights of theimagination. I merely want to emphasizeover and over that what we are faced with is as real as the computer I’m typingthese words on. But it is only in theland of the ultra-miniature do our new rules come into play, that is, in a waythat we cannot fail to notice even if we try.

* * *

I just used the word probabilistic, Isuspect not for the first time. Intalking about probability we all, I think, have a good grasp of what thisconcept means. We don’t know – sportsanalogies always work best – who is going to win the World Series next year butwe can analyze the different teams and the caliber of their players andcompute, even if roughly, probabilities – likelihoods, odds, where to place ourmoney, and so on. In a similar fashionwe cannot predict the height of a human being chosen at random, but we can useprobability to come up with a good ballpark (pardon the pun) figure. In fact, the particular method of probabilityused here is simply called the Gaussian Distribution function, illlustrated below:

Figure V.

Don’t concern yourself with the symbols, like μ (mu) andσ (sigma), although we will return to them later. Note simply that the distribution is highestin the middle of the curve, and drops off on either side, coming close to butnever reaching zero.

A measurement of a large number of human heights, plottedon a graph, would look much like this. There would be a high central region where most of the heights foundthemselves, and then declining tails of progressively shorter and taller – andfewer – people. A person chosen atrandom has a 68.2% chance of falling within one standard deviation, or 1σ, ofthe highest, most central height, called μ or the norm or average; a 95.4% ofcoming within 2σ of the norm; and so forth.

And that’s it. Allwe can say. We simply cannot speak withcertainty the height of an individual chosen randomly. There is a big difference, however, betweenthis problem, which is part of ordinary reality, and the “same” problem in the quantumworld. In the world we’re used to we canstill measure the selected individual and reveal himself to be an individual(e.g., David Strumfels) with a height of six feet two inches. But in the quantum world, no. First, there are no individuals; and if wetry to make measurements which with to create individuals, then other importantinformation about them is irretrievably lost. In the quantum world, probability rules with an iron fist.

We can place this in more prosaic contexts however. According to the laws of the quantum world,we cannot say exactly where a water molecule is in a glass; but we cancalculate a probability distribution, similar to the Gaussian but not eactlythe same. There is, in fact, a famous calculation chemistryand physics students perform that is quite similar: a lone electron in a (one-dimensional) box. You cannot say precisely where the electronis, you can only construct an equation showing the probabilities of being inthe box somewhere at any given moment.

Given that I’ve raised the subject of electrons …

* * *

Atoms. We’ve allheard about them. Unless you truly sleptthrough all your schooling you heard that: a) everything is made from them (not true); b) they’re so tiny nobodycan even see them (also not true); and c) they’re composed of a tiny, densenucleus made from protons and neutrons, with even tinier electrons spinningaround it (partially true, but critically untrue for where this discussion isheaded).

It is time to continue on our journey. We were at the level of water molecules, andas as we work our way down past that, past individual atoms even, down finally toelectrons, things change little. Mainly,those probability curves just keep getting bigger. They get so big, in fact, that by the timewe’ve reached the level of electrons the curves are complete masters of theirbehaviors.

Our interest in electrons is simple enough; they are whatmake chemistry possible, and control most of what it can do. Not everything – remember the discussion onorganic chemistry and geometry before – but without electrons there is no suchthing as chemistry at all.

Some further evidence of just how strange the quantumworld is worth the effort here. In thequantum world probability is truly just that; by that I mean that if, say, weknew all the forces and effects working on a coin toss we could predict, withabsolute certainty, whether a given toss would be heads or tails. And it is always one or the other, nothing …well, nothing in-between, curious as that may sound. In quantum probabilities, no such luckhowever. All we can do is makeobservations, in which we will see this or that or something else, an infinityof possibilities with each one having its given probability.

This might sound purely academic, but study the picturebelow:

Figure VI.

The gist of this, Erwin Schrödinger’s famous quantum catexperiment, is a sealable container (usually a box), within which resides: a) a live cat; b) a vial of deadly poison; c)a hammer and a lever which can drop the hammer, breaking the vial and killingthe cat; and d) a single radioactive atom with a 50% probability of decayingwithin an hour and which, if it does decay will trip the lever, again killingthe cat. Unlike the picture however, thebox is not open but completely sealed so that no information on the cat’s state(alive or dead) can get out.

You see, the quantum world is even stranger than I’vedescribed. I’ve spoken of makingobservations and getting results according to probabilities, as though the results were already therewaiting for us to unveil them. It’sworse than that. The results don’t even exist until we make the observation! Quantum physicists speak of things like “collapsingthe state function” among other arcane words and phrases, but what they mean ismaking an observation. The act ofobservation reduces the infinitely many probabilities into a single, actualresult, and that result doesn’t exist until the observation is made.

Sounds like magic, I know, or some kind oftomfoolery. But it is the simple truth. A truth that has been demonstrated withabsolute certainty by thousands of experiments, making measurements to manynaughts beyond the decimal point.

* * *

Let’s return to our concrete example, the quantumcat. Seal the box, wait an hour –remember, I said that there is a 50% probability of the atom within decayingwithin that time span – then open it again. What will you see?

This part of the answer is simple. Either the atom has decayed, the levertripped, the vial containing the point released, and the cat dead; or the atomdidn’t decay and the cat is very much alive.

The hard part of the answer is, what’s the state of thecat just before opening the box? Yourfirst reply I’ll bet is the same as before, though you’re probably suspiciousby this point. You smell a rat followingthe cat and you think I’m trying to pull a fast one on you.

If you feel that way, that’s actually good; it meansyou’re following the discussion. (See, Itold you you could do it!) Remember Ijust said that, at the quantum level, the result of an observation doesn’texist until it is made. The cat isn’tdead or alive prior to opening the box – unless of course, you decide that thecat is an observer who saw the vial being smashed or not, which, to be honest,is a valid argument here. So: what does exist?

Recall that I said that the result of the coin tossexperiment is never somethin “in-between?” I meant that it’s in a head state or tail state even before we lift ourcovering hand and make the observation. Again, no such luck with the quantum cat. The convenient way of treating it, that iswhat scientists do, is to say that it is in a “superposition” of bothstates. It’s both alive and dead; untilyou open the box that is, and collapse the state fuction into one of the othercomponent states.

I advise not over-thinking about this. Scientists and philosophers and other punditsto this day argue about what quantum physics implies about reality and ourrelation to it. Even the cat experimentis taken seriously by some, while a joke by others. My personal take about the experiment, by theway, is that it is not physically possible; you can’t construct a containerwhich allows no information to leak out about the cat being dead or alive,meaning that it is always possible to make a measurement about the cat’s statewithout opening the box; whether or not the measurement is actually made or not(what exactly is a measurement or who or what makes it, anyway?) isirrelevant. But their are problems withthis stance too. So please – don’t loseany sleep over it.

The pragmatic truth about quantum physics is thatscientists use it because it works. Andthere are very real versions of the cat experiment that can and have been done,experiments which have shown that the superposition of states with the resultnot existing until observation is made is correct. As for the philosophical implications, most scientistsare too practical-minded to worry about them. Too much.

* * *

It is now time to return to electrons, and how theybehave in atoms and molecules. Remember,this is the fundamental requirement just to have chemistry, so we need areasonable feel for this before we can continue. Fortunately, a reasonable feel is all weneed. To get that, all is needed is to takewhat we learned about cats in the quantum world and apply them to this newarena.

Like cats in a box, electrons in atoms and molecules existin quantum states; these states describe such (indirectly) observablequantities as their energies, magnetism, where they’re likely to be found, andso on. Also, like the cat’s fate, the states are quite distinct from eachother, and the electron can only move from one to the other, never being foundin-between.

If you have been wondering why the world of theinfinitesimally small is called quantum, the answer lies here. In the quantum world things generally existin discrete states and not somewhere along a continuum. To give another example, imagine getting inyour car, pulling out along along a highway, and accelerating to 50 mph. It seems to you that the acceleration iscontinuous, with no breaks anywhere; there are no discrete sepeed states, andyou don’t make jumps from one state to another. But if quantum rules applied strongly at “ordinary” levels of reality,you might find that you could only do the speeds 0, 10, 20, 30, 40, 50 mphexactly, jumping from 0 – 10 – 20 instantly. You could not do 36.568 mph, for that “state” doesn’t exist. Of course, it’s possible that the actualstates don’t comprise a continuum anyway, they are just too close together tomeasure. In fact, this is the actualcase, and quantum physics applies at the medium, large, and enormous scales ofreality too. It’s just that as themasses involve increases, the states grow ever and ever close together. Even at the level of sand grains they are fartoo close together to be observed.

So: electrons inatoms and molecules do not orbit around nuclei, like planets around the sun,but exist in appropriate (usually lowest energy) quantum states. All the images of atoms being like miniaturesolar systems are innacurate in this respect. So why do we use them? Simply,the actual picture of an atom is too difficult to illustrate at our realitylevel, and the solar system works best here. It is accurate in a number of ways that justify its use. Just bear in mind that, when it comes toelectron behavior, and it’s relationship to chemistry, it really doesn’t workat all that way.

Before I go on, let me remind you about thermodynamicsand entropy again. This is a verycomplex subject, but all we need to know is that electrons, like everythingelse, “perfer” to be in their lowest energy states. In a multi-atom molecular, electrons areattracted to more atomic nuclei (made of protons, and particles of oppositecharge attract one another), and this alone would be large molecules the moststable states of matter. Careful. The larger an atom/molecule the moreelectrons there are too, and these tend to repel each other, thereby raisingthe energy. Exactly what types andnumber atoms and molecules which from atoms is largely a balancing act betweenthese two forces.

The simplest, and most abundant atom in the universe ishydrogen, consisting of a one proton nucleus and one electron distributed aboutit. This is not the most stable atom;it’s abundance is purely the result of how the universe has evolved. Next to hydrogen, helium, with a two-protonnuclear and two electrons about it, is most common; it is more stable thathydrogen, but is still far from maximum stability and again it’s abundance isexplain by cosmic evolution.

As for molecules, hydrogen usually exists in diatomicmolcules, in this case H₂. This is because the two electrons are more strongly drawn toward the twonuclei than they repel each other. Helium exists as monoatomic He atoms, however, because the four electronsexert a more profound effect than the two, albeit di-proton nuclei, cancounter.

As we climb the periodictable of elements (just a diagrammic listing of all elements, to be readleft to right and top to bottom) you see hydrogen and helium in the top row:

Figure VII.

The elements Li through Uuo don’t follow an easily madeout trend when it comes to stability and molecule making, but there are generaltrends that can be worked out and understood, usually with a lot of hairy mathand computer time.

As this section is in fact about molecules and not loneatoms, what we want is a feeling for how the constituent atoms form them, withthe particular shapes they have. A goodplace to start is to return to good old water, a simple molecule that canillustrate many of the principles.

Water is made from two hydrogen and one oxygenatoms: the famous H₂O. But why that particular formula, and why dowater molecules take on a bent shape:

Figure VIII.

No need to go into the details here. Basically, the available basic shapes of theoxygen atom make up a tetrahedron (remember, I said you need to recall somebasic geometry):

Figure IX.

The ogygennucleus sits at the center of the tetrahedron, and the electrons are “at”(actually around) the vertices (the connecting lines are the chemical bonds). If you imagine sticking the hydrogen atoms(whose electrons make up spherical states) at two of the vertices, the waterwater molecule shape should pop out at you. Each electron on the hydrogen atoms pair up with a “lone” electron onogygens atoms pair up; this is another quantum oddity in which a state can holdonly up to two electrons; any more and they spill over into other states,usually larger, more energetic ones. This limitation, by the way, is why atoms and molecules can build up.

A more complex example of a molecule is glucose (bloodsugar):

Figure X.

Here, the black spheres are carbon atoms, the red oxygen,and the white hydrogen. Ignore the HC=Opart at the upper right end (the = is called a double bond) and for the moment,concetrate your attention on the carbon atoms. Notice how the bonds sticking out of them make for tetrahedronstoo? Carbon can bond with as many asfour other atoms so it is very clear here. But the H-O-C parts are also, like water, bent, indicating the sametetrahedral shape as in water. Butoxygen can bond with only a maximum of two other atoms, so the other points ofthe tetrahedra are empty.

I am not implying that all bonds have tetrahedralshapes. Far from it. It is merely one of a myriad shapes which isfairly common, especially among atoms in the top two rows of the periodictable. The HC=O group, for example, isflat and triangular. Another shape isrepresented below

Figure XI.

This molecule (which has a negative three charge) iscomrpised of two shapes. The 6 CºN portions are linear (the º is called a triplebond), while the central FeC₆ part is an octahedron, the carbon atthe center:

Figure XII.

As said, there are numerous other shapes as well. And shapes can be bonded to shapes, creatinglarge, complex molecules, such as proteins and DNA and many other moleculesfound in our bodies. Still, othermolecules possess even simpler shapes, such as O=O (O₂), H-H (H₂), NºN (N₂), O=C=O (CO₂), and soon. The possibilities are, for apractical purposes, infinite, and many millions of molecules are now knownabout.

Many of these molecules are called “organic” and it istime to know why.

Chemistry at Venus' Cloud Tops

2011-12-17T16:53:00.003-05:00

Thereis a curiosity I keep wondering about concerning the inner solar system’splanet’s atmospheres. I have often readhow Earth’s powerful magnetic field helps to deflect the solar wind andmaintain our dense atmosphere. Mercury,the moon, and Mars have only weak magnetic fields and it is asserted that theirlittle to no atmospheres are the result of the wind ionizing the gasses andthen literally stripping them away.

Butthere was always one obvious exception to this generality: Venus. Venus too has no magnetic field but easily possesses the densestatmosphere of the inner worlds, much denser than Earth’s. Part of this is probably due to the carbondioxide composition of Venus’ atmosphere; CO₂ is a large, heavymolecule that is no doubt easier to retain than Earth-like gasses of oxygen,nitrogen, argon, and water vapor. Furthermore, the active (though periodic) heavy volcanism on Venus meansthat this gas is being regularly generated and introduced into the planet’satmosphere (the early loss of heavy volcanism is probably why the same hasn’toccurred on Mars), perhaps as fast or even faster than the solar wind can stripit away.

ButI wonder if there is another reason. Venus is topped by a thick layer ofsulfuric acid clouds, which largely (I think) overlays the carbon dioxidenearer the surface. This layer could actas a shield against incoming electrons and protons from the sun. Sulfuric acid is a very heavy molecule whichwould probably fracture into smaller radicals and ions when struck by suchparticles. Possible reactions could be:

H₂SO₄+ e^- ® OH^- + HSO₃·

H₂SO₄+ p⁺ ® H₂O₂ + SO₂²⁺

IfOH^- ions are formed they are probably to weak to be helped by Venus’gravity, but they might recombine with other radicals/ions first. The other species will probably also hangaround long enough to recreate sulfuric acid, while scattering the electronsand protons at lower energies. Bear inmind, this is probably a very small sampling of the kind of chemistry thatoccurs at Venus’ cloud levels. If it is happeningthen Venus protects its atmosphere with chemistry instead of magnetism. Similar chemistry could be going on at Saturn’smoon Titan’s cloud tops, although the solar wind is much weaker there.

A Favorite Way of Picturing Deep Time

2011-12-17T10:48:00.001-05:00

Portraying geology and geology in terms of "deep time" (essentially goin back to the beginning of Earth and the solar system) isn't easy, but I think this picture does a reasonably good job of it.

Spice, Salt, and Civilizations?

2011-12-17T08:08:00.001-05:00

Some years ago a writer I much admire named Jared Diamond wrote Guns, Germs, and Steel, the main premise of which (though it isn't in the title, oddly) is that Eurasian civilizations took off sooner and faster than African or American due to the large number of domesticatable plants and animals living there (e.g., cattle, swine, chickens, wheat and other grains, etc.), which is why Eurasia came to dominate the rest of the world so badly once we developed ocean going vessels.

If we define civilization as the transition from nomadic to settled cultures, there may be another factor, this one concerning food. The reason uncivilized cultures are nomadic is that food is a constant requirement but, with obvious exceptions like tropical areas, is not plentiful anywhere all year round. Their is a time of plenty, and a time of -- well, what? Without means to preserve (many) foods, it's either go hungry or move to new places of plenty. Hense nomadism.

We take freezing and refrigeration (in technologically modern cultures) as the obvious way to preserve meat, fish, and other perishable food items, but that's only been quite recent (and still is no where near universal). Prior to cold, the main preservatives used by humans were salt and spices -- even if you went to market everyday such places often had to be kept going by sellers and producers using salt & spices to keep their foods reasonably fresh for their buyers.

I don't know what the archeological evidence here is (which means I'll have to check it), but may I suggest that at least part of what triggered settled behaviors in Homo sapiens was the discovery of substances which can preserve perishable foods for fairly long periods? At least this makes sense to me. Nomadism is a brutal way of life, happily abandoned the moment people can do so. On the go most of the time, carrying all you own with you, over mountain passes and deserts and other nasty places? In truth, it doesn't even sound romantic.

When you first settle, you're still uncivilized in all other ways. A stone-age culture that doesn't move about. Although there are other obvious reasons, perhaps one of the causes of tropical people living "primitive" lives long after the rest of us have abandoned them, is the year-round presence of food. They're settled because they don't have to move. And even though many of them do use preservation methods, they're not an absolute essential; you literally can get fresh food every day.

Well. I suppose I'm not the first person to think of this. But I haven't seen a book entitled Spice, Salt, and Civilizations, anywhere, by anyone. So proably not too much has been done with this idea amongst the lay public, and maybe not even among scientists. If so, I'd love to hear about it.

The implications of this picture are either highly disturbing or highly exciting, or both. But they can't be neither.

2011-12-16T05:15:00.002-05:00

Some Belated Tricks with Java and Excel VBA

2011-12-16T04:44:00.000-05:00

Several years ago, whenI was working for Dr. Jean-Claude Bradley at Drexel University, I posted on anumber of programming projects on the UsefulChem blog site (http://usefulchem.blogspot.com/). I recall a flurry of interest in how I gotExcel VBA programs to call Java programs and how Java could make Excelspreadsheets. It’s probably a lot easiernow, but here were my tricks below. First part is how to get VBA to run any software and wait for itscompletion:

________________________________________

‘Declarations

PrivateType PROCESS_INFORMATION

hProcess As Long

hThread As Long

dwProcessId As Long

dwThread As Long

EndType

PrivateType STARTUPINFO

cb As Long

lpReserved As String

lpDesktop As String

dwX As Long

dwY As Long

dwXSize As Long

dwYSize As Long

dwXCountChars As Long

dwYCountChars As Long

dwFillAttribute As Long

dwFlags As Long

wShowWindow As Integer

cbReserved2 As Long

lpReserved2 As Long

hStdInput As Long

hStdOutput As Long

hStdError As Long

EndType

PrivateDeclare Function CreateProcess Lib "Kernel32" Alias"CreateProcessA" _

(ByVallpApplicationName As String, ByVal lpCommandLine As String, _

ByVallpProcessAttributes As Long, ByVal lpThreadAttributes As Long, _

ByValbInheritHandles As Long, ByVal dwCreationFlags As Long, _

lpEnvironmentAs Any, ByVal lpCurrentDirectory As String, _

lpStartupInfoAs STARTUPINFO, lpProcessInformation As PROCESS_INFORMATION) As Long

PrivateDeclare Function CloseHandle Lib "Kernel32" (ByVal hObject As Long)As Long

PrivateDeclare Function WaitForSingleObject Lib "Kernel32" _

(ByValhHandle As Long, ByVal dwMilliseconds As Long) As Long

‘The actual function.

PrivateFunction RunWaitApp(lpCommandLine As String, wShowWindow As Integer, bWait AsBoolean, _

Optional lWaitTimeAs Long) As Boolean

Dim sinfo As STARTUPINFO

Dim pinfo As PROCESS_INFORMATION

Dim res As Long

Dim lWait As Long

If bWait Then

lWait = lWaitTime

Else

Shell lpCommandLine, wShowWindow

RunWaitApp = True

Exit Function

End If

sinfo.cb = Len(sinfo)

sinfo.wShowWindow = wShowWindow

res = CreateProcess(vbNullString, lpCommandLine,0, 0, True, &H20, ByVal 0&, vbNullString, sinfo, pinfo)

If res <> 0 Then

res = WaitForSingleObject(pinfo.hProcess,lWait)

If res <> &H102& Then

Exit Do

End If

DoEvents

Loop While True

CloseHandle pinfo.hProcess

RunWaitApp = True

Else

RunWaitApp = False

End If

EndFunction

_______________________________________

A bit windy, I know,but you need write the module only once, then copy, paste, and call it whereverneeded.

The code to createExcel spreadsheets from within Java then required a special library import, ofwhich I could only find one at the time but am sure now there are many others. The full code is:

________________________________________

importjava.io.*;

importjava.util.*;
importorg.apache.poi.hssf.usermodel.*; // The special import

publicclass CreateXLS
{
public static void main(String[] args)
{

    if (args.length < 1)
    {
      System.out.println("Usage: java CreateXLS ");
      System.exit(0);
    }

try

    {
      String saveDirectory = args[0];
      String feedFile = args[1];
      System.out.print("Creating XLS filesfor items from " + feedFile + " ... ");
      File dd = new File(saveDirectory);
      if (!dd.isDirectory()) dd.mkdir();
      String separator =System.getProperty("file.separator");
      String newLine =System.getProperty("line.separator");
      String tab = "\t";
      FileWriter fw;
      FileOutputStream os;
      HSSFWorkbook wb;
      HSSFSheet sheet;
      HSSFRow row;
      Feed feed = Feed.loadFeed(feedFile);
      ArrayList itemList = feed.getItemList();
      Item item;
      Molecule molecule;
      Field suppliers;
      ArrayList suppliersList;

      for (int i = 0;i <itemList.size();i++)
      {
        item = (Item) itemList.get(i);
        molecule = item.getMolecule();
        fw = new FileWriter(saveDirectory +separator + item.toString() + ".txt");
        os = new FileOutputStream(saveDirectory+ separator + item.toString() + ".xls");
        wb = new HSSFWorkbook();
        sheet = wb.createSheet();
        row = sheet.createRow((short) 0);
        row.createCell((short) 0).setCellValue("UCNumber:");
        row.createCell((short)1).setCellValue(item.toString());
        fw.write("UC Number:" + tab +item + newLine);
        row = sheet.createRow((short) 1);
        row.createCell((short)0).setCellValue("SMILES");
        row.createCell((short)1).setCellValue(molecule.getSMILES().getFieldContents());
        fw.write("SMILES:" + tab +molecule.getSMILES().getFieldContents() + newLine);
        row = sheet.createRow((short) 2);
        row.createCell((short)0).setCellValue("InChI:");
        row.createCell((short)1).setCellValue(molecule.getInChI().getFieldContents());
        fw.write("InChI:" + tab +molecule.getInChI().getFieldContents() + newLine);
        row = sheet.createRow((short) 3);
        row.createCell((short) 0).setCellValue("ImageURL:");
        row.createCell((short)1).setCellValue(molecule.getImageURL().getFieldContents());
        fw.write("Image URL:" + tab +molecule.getImageURL().getFieldContents() + newLine);
        row = sheet.createRow((short) 4);
        row.createCell((short)0).setCellValue("Substructure Search:");
        row.createCell((short)1).setCellValue(molecule.getSubStructureSearch().getFieldContents());
        fw.write("SubstructureSearch:" + tab + molecule.getSubStructureSearch().getFieldContents() +newLine);
        row = sheet.createRow((short) 5);
        row.createCell((short)0).setCellValue("Item Page:");
        row.createCell((short) 1).setCellValue(molecule.getItemPage().getFieldContents());
        fw.write("Item Page:" + tab +molecule.getItemPage().getFieldContents() + newLine);
        row = sheet.createRow((short) 6);
        row.createCell((short)0).setCellValue("Canonical MW:");
        row.createCell((short)1).setCellValue(molecule.getCanonicalMW().getFieldContents());
        fw.write("Canonical MW:" +tab + molecule.getCanonicalMW().getFieldContents() + newLine);
        row = sheet.createRow((short) 7);
        row.createCell((short)0).setCellValue("Natural MW:");
        row.createCell((short)1).setCellValue(molecule.getNaturalMW().getFieldContents());
        fw.write("Natural MW:" + tab+ molecule.getNaturalMW().getFieldContents() + newLine);
        row = sheet.createRow((short) 8);
        row.createCell((short)0).setCellValue("Suppliers:");
        fw.write("Suppliers:");

suppliers = molecule.getSuppliers();

suppliersList =suppliers.getFieldContentsList();

if (suppliersList != null)

        {

          for (int j = 0;j <suppliersList.size();j++)
          {
            row.createCell((short)1).setCellValue((String) suppliersList.get(j));
            row = sheet.createRow((short) (j +9));
            fw.write(tab + (String)suppliersList.get(j) + newLine);
            row.createCell((short)0).setCellValue("");
          }

}

wb.write(os);

        os.close();
        fw.flush();
        fw.close();
      }

System.out.println("[ok]");

    }
    catch (Exception e)
    {
      System.out.println("[failed]: " + e.getMessage());
    }

}

________________________________________

This is all five yearsafter the fact, and much has changed since then, but hopefully someone willfind it useful.

Chapter Three -- The Universe as we Don’t See It

2011-12-15T10:49:00.000-05:00

I want to take you on ajourney. For me it started quite young,but when at age twelve or thirteen my parents gave me a six-inch Newtonianreflector, it began in earnest. I amtold that my largesse was the result of my sister getting (for a while) a pony,and their was fear I would be envious, but the two events were never connectedin my mind and I never remember a trace of envy or resentment toward mysister. I was just so darn happy to havethe telescope. .

Even in the ratherlight-polluted suburbia USA we lived in the telescope revealed a marvel ofheavenly capital the naked eye never suspected. We all know there are craters on the moon, but with but 96Xmagnification I could see them, bright and clear. And the Galilean satellites of Jupiter. The rings of Saturn. And stars beyond stars, nebulae, galaxies –by any professional standards it was just a child’s toy, but what it broughtinto my backyard most of the greatest philosophers of history could not havedreamt about.

It took me … outthere. Away from this secluded andnarrow viewpoint of tiny spot on planet Earth where I stood, toward placeshundreds or thousands of years of light speed travel that would have beenneeded to actually be there in the flesh (not that I would have survived long,but I never thought about that). It wasa Asperger’s child’s vision of paradise, to this day probably the best thing myparents ever did for me. I was not tobecome an actual astronomer, but this gift opened that door to me better thananything else.

I’d like to repeat afigure from chapter one:

Figure I. (repeated)

Getting away from ourordinary, Earth-bound existence and planting ourselves somewhere in space –here, a position millions of miles beyond the sun-Earth system – we see alreadyhow much our perspective on things have changes. For one thing, the cause of the seasons,which had baffled us before, becomes obvious. We also see Earth (and the sun) as spherical objects in space, insteadof as flat, infinite surfaces which or may not have boundaries. If the picture were to be fully fleshed out,we would see other planets too (specifically, the inner world of Mercury andVenus, and the fourth planet Mars; whether Jupiter and/or Saturn would showfrom here is not as clear).

Whether you realize itor not, I have done something profound to your senses; more precisely, to yourbrain’s interpretation of reality. Butall I have really done is change your point of view, as my telescope changedmine. I call it profound, however,because the brilliant insights of Albert Einstein, in the beginnings of thetwentieth century, demonstrated that it must be so.

* * *

If you have ever takenhigh school physics, or a general college physics course, you may havediscovered that it is, in a very real sense, boring. It’s boring because you’re not learninganything you hadn’t already intuitively learned by about age two or so. If you don’t believe that, then watch nexttime a magic show is performed before a group of toddlers. They are just as dumb-founded and thrilledseeing the laws of ordinary physics seemingly violated as you or I. And even a small baby can tell when somethingis amiss; if you convince it something is in a certain place and then revealthat it isn’t, their eyes will open wide with surprise, and they may evenbecome distressed.

“But that’s just commonsense,” you might be tempted toprotest. Yet what is common sense? If it were as easy and as obvious as itseems, artificial intelligence would be a snap to accomplish, and would have beenyears ago.

The fact is, our brainsevolved to perceive and “understand” reality in ways necessary for ourstone-age ancestors and further back, and thus it is unsurprising that weshould possess, even at a very young age, the common-sense concepts wecollectively call reality. Recall what Isaid about magicians and how they do what they do; they use those “wired-in”,common-sense, laws of physics and manipulate our senses and points of view tocause us to see impossible things. Likebending a spoon with your fingers, if I may cite a rather common trick by“psychics” – magicians who pawn themselves off as special people with specialpowers.

What you learn in basicphysics course are the details, the precise definitions, and the math behindthe ordinary. And, despite what I said,it isn’t boring at all; I strongly recommend taking such a course (my motherdid, and got a B, which somehow didn’t convey to her that she could graspscientific thinking), perhaps even before reading what’s coming up.

* * *

The lesson of the lastsection, I hope, is that although we possess common-sense intuition about“ordinary” reality – the reality all of us spend all our lives in, a realitywithin a narrow range of space and time – at the same time evolution could nothave bestowed us no gifts about reality outside those strict ranges, becauseour ancestors never encountered them. And indeed, it hasn’t. But untilyou understand that, it is only natural that you should think the Laws ofPhysics, as we somewhat pompously and arrogantly call them, will applyeverywhere, all the time, across all scales of time and place.

That’s why I startedout talking about my childhood telescope, and showing Figure I again. Even this is not too far a deviation from ourhum-drum down on this planet’s lives, but there are some noteworthydifferences. The biggest one may be thatthe light from the sun or that reflected from Earth will take several minutesto reach our new vantagepoint. Wecertainly aren’t used to significant (or any) delays between the time somethinghappens and when we observe it, for light travels – well, it travels fasterthan anything known in the universe, a full 186,282 miles per second. That being the case, when a bank of lights ata stadium are turned on, the stadium is full alit “at once” , though itactually takes around a fraction of a millisecond or one thousandth of a secondfir this miracle to happen. Since ourbrains can’t measure time intervals that short (we’re a tad slow, to tell thetruth), this is instantaneously as far as we are concerned. Indeed, by all common-sense measurements thespeed of light is for all practical purposes infinite.

Yet it is in fact not infinite, as our hovering overFigure I. shows. That light –electromagnetic radiation I should call it, including radio, microwaves,infrared (heat) rays, ultraviolet (black) light, x-rays, and gamma rays –journeys at a specifically defined speed came out of work on electricity andmagnetism in the 1800’s, already dents our common sense view of things. But what comes next tramples it into unrecognizableshards.

I have to backtrack someto explain why. I’ll ask you to closeyour eyes (but don’t stop reading!) and imagine the following: someone else and I are on a rail car,travelling at fifty mph down the track. We both have baseball mitts (not actually necessary), and one baseball,and we are playing a game of catch between us. Some kindly passenger (you, as it turns out) on the train is timing howfast we throw, and reports us both hurling the ball at 50 miles per hour, or 73feet per second. You can easilycalculate this because the rail car is 73 feet long and it takes exactly onesecond from throw to catch.

Open your eyesagain. That was probably easy to picturebefore your mind’s eye, I’m certain; we do things like this all the time, ifnever exactly this.

Okay, close ‘em again,and this time picture yourself on a train platform at a station, watching thetrain whizz by at fifty mph (it’s an express, and doesn’t stop there). You can easily look through the car’s windowsand watch the game of throw and catch.

Question: what do you see now?

You are probablyalready uncertain as to whether you will see the same thing, but if you haven’tquite figured out what you do see, I’ll hand you the answer and then explainit. Using the same clock, you now seethe thrower at the rear of the car throwing, like a top flight major leaguepitcher, the ball at 73 + 73 = 146 feet in the one second that ticks off yourclock, or one hundred mph; while the thrower at the front can accomplish a mere73 – 73 = 0 feet in that second, or zero mph.

The explanation is thatspeeds add. The train is travellingfifty mph forward, and this speed must be added to the rear of car throwerspeed and subtracted from the front of car’s thrower’s velocity. If all the window shades are drawn, however,the passengers have no way of knowing their speed with respect to the station, because we can’t sense constant speed,only acceleration (speeding up,slowing down, or changing direction). Everything here, keep in your mind, is at constant speeds anddirections. And I’ll wager it doesn’tgall you too much. You’ve actuallywitnessed it first hand many times in your life; you know you can’t really sayan object is travelling at such and such a speed without specifying thereference point that speed is being measured from. Ever sat in a motionless train and, whilewatching another train moving slowly by you, actually sense yourself moving inthe opposite direction? This iswhy. Like magic. it confuses your brainagain.

Let’s get back tolight. Nineteenth century physicistsshowed that visible light was in fact an electromagnetic wave, or possibly somekind of particle like a baseball only infinitely tinier, and the speed of thatwave was a well-measured 186,282 mps. The natural question now? With respect to what point of reference? The answer is either: either none, because all the laws of physicsup to that point demonstrated that there are no privileged or special points ofreference in the universe, that they are all equal; or, despite those laws(and, to their defense, a law of physics is so only because we humans to our meagerabilities have been able to estimate it well enough to call it a law, so we canat least bend if not outright break it when needed) their was something calledthe universal aether, a fluid havingno density, no color, no resistance to movement through it (viscosity), indeed no observable properties at all, whichpermeated all space and actually was stationary in some absolute sense.

A note of personalpreference before I go on here. Lots ofbooks on relativity have lots of pictures of trains and trolleys and clocks andother pertinent things – only natural, as this is what got Einstein thinkingabout the issues raised here – but I won’t, because if I haven’t presented myconcepts in a simple, straightforward enough manner, then I’ve already failedin the main aim of this book. I assureyou, it has nothing to do with my incompetence in drawing pictures (OK, well it does some).

I’ve given you the two possibilitiesand, if I have been successful, you are probably up a creek paddleless tryingto choose. How can there be no point ofreference for light’s speed; or, alternatively, there is a point, the aether,that has no physical properties that can be observed and measured? You should be reeling a bit by this point,because by the late 19’th century almost all scientists were reeling trying to answerthis seemingly impossible conundrum. Soyou are in good company.

I’ve suggested that thereare only two solutions to this problem, but, and in fair warning this is wherethings get strange (but still logical!), a third solution does present itself. Oh, how I wish I could say I thought of itmyself. No, it took the genius of anEinstein to see what was so unobvious to the rest us. The third solution is that light is in fact it’s own reference point,and all other speeds must pay homage to it. It is light, electromagnetic radiation, to which we must bow to andfollow it’s rules, however absurd they may seem to us.

Again, an aside beforeI move on. The speed of light is soenormous that, compared to it, the differences between the ordinary speeds weencounter (and the 50,000+ mph of space probes and objects, as fast as theyseem to us, are still way below light’s 670,000,000 mph, by a factor of tenthousand and more) are so insignificant to render the light speed problem moot. This is why we never notice it in our lives,or ever imagine it in our minds. It iscertainly not part of our intuitive understanding of physics for it never hadto be in our evolutionary history.

* * *

I apologize here, for Imust lay down some equations for your edification. They are not really complicated, not unlessyou’re going to go into full-fledged physics mode, which there is no need todo. The first equation involves aquantity that you’ve probably heard of, momentum. It is simply the mass of an object multipliedby its velocity (speed + direction, remember). Prior to Einstein, and as still presented in all general physics coursestoday it is:

momentum( or p) = mass(or m) × velocity(or v)

p = mv

Equation I.

This is the equationNewton derived for what, essentially, he called inertia, and so derived the nowfamous Law of Conservation of Linear Momentum (quite similar to the one forAngular Momentum we’ve already encountered). Newton derived it assuming that all velocities were relative to somepoint of reference, even light, although he had little idea what light was orhow/how fast it travelled. Science wasn’tdeveloped well enough in his day, through no fault of his.

In Einstein’s newformulation, the equation of momentum must be modified from p = mvto:

p = gm₀v

Equation II.

Wheregor gamma (mathematics is all symbolized, even X + Y = Z, so don’t let thisthrow you), stands for:

Equation III.

The naught (0) on mindicates rest mass, which we’ll come to in a tick, and c means the speedof light.

Again, don’t let any ofthis get too heavy on you; you should have had all the symbolisms in highschool, or can reference them easily. Anyway,this factoring in of the square root (what √^¯means) of 1– (v/c)² to the momentum equation, whichremember, applies to all objects, takes into account c (speed of light)being this supreme reference point in the universe we must all be subservientto. Note something critical here: if v << c,as is the case with all velocities we normally encounter, then (v/c)²goes to essentially zero, and gjust as essentially goes to 1, meaning that gamma is just 1/1 or 1 and dropsout of the equation, leaving us with our Newtonian original, except for thatlingering naught on m which we’ve let to explain beyondcalling it the rest mass. Is there such a thing as a different not at restmass?

Yes there is, and alittle examination of Equation III. should show why. Imagine we make vvery close to c,or even make it equal to c. Then (v/c)²becomes just 1, and as 1 - 1 = 0 the bottom half of g becomes infinite,meaning that p becomes infinite too!

Even as v getscloser and closer to c,p grows rapidly. It is as though all the energy we are throwing into our object to makeit go faster and faster end up only increasing its mass as the speed of lightis approached. If we could get to c the mass wouldbe infinite in fact.

The only possiblephysical interpretation of this is that energy and mass are somehow equivalent,and dumping more of the former onto an object means it also has more of thelatter. The math for that is just hairyenough to excuse us from examining it but the bottom line is the equation weall know and love: E = m₀c². The naught after mis, as said, the object’s rest mass; multiplying it by the speed of lightsquared gives its equivalent in energy, and since c²is a very large number you will see (as scientists in the 1930’s were beginningto see, the results being Hiroshima and Nagasaki), if you can make the conversationyou will release a very large amount of energy indeed.

Other, equally strangephenomena crop up when we move close to the light speed. Because all observers must find the samevalue for the this speed, regardless of their reference points, time and spacebecome, well, malleable; it can be different with different observers.

Let’s go back to ourbaseball throwers in the rail car. Thebig difference, instead of throwing baseballs back and forth and each other,each one now has a laser pointer. As soon as the light from one pointer reachesthe corresponding “catcher”, he in turn flashes his collaborator with his ownpointer.

This experiment, asdescribed, may sound absurd. The laserlight travels so fast that it takes only 7.422×10^-8 seconds, or 74billionths of a second, for the light to exit the pointer and reach the otherplayer, still fifty feet away. Very wellthen; let’s makes this easier to visualize by making the rail car 186,282 mileslong, so that it now takes our full second for the traverse time. Never mind that this makes the experimentphysically impossible (the rail car would stretch around our world over seventimes!). Experiments can, in many casesare, done in our minds; as long as we get the math right and imagine thingscorrectly, this is a perfectly valid approach to the subject. (Such experiments are called gedanken, a German word meaning literally”done in thought”).

Do you have the picturein your head, though admittedly it is a bit tougher this time? Good. Next: imagine yourself, as before, the measurer in the rail car, timinghow long it takes the laser light to get from pointer to receiver, andback. Naturally you find this number tobe 186,282 mps, just the speed of light. No surprises there.

Now place yourself onthe station again. You time the laserpulses again, just as you timed the baseball throw before from the platform. Now, one, crucial, last part of theexperiment to stitch into your mind’s eyes: the rail car is also travelling 186,282 mps pass the station, in thesame way as it was travelling 50 mph before. So: what do you expect to see?

With the baseballs, weobtained the answer that the speeds were either 50 + 50 = 100 mph for the backof car thrower, or 50 – 50 = 0 mph for the front of car, for now our referencepoint is the platform at the train station. Speeds, velocities more precisely, are additive because they depend onthe point of reference you are measuring them from.

Amazingly,the result is completely different for the laser pointers and the cartravelling at the speed of light! Recallmy statement that the speed of light is its own reference. This means that its speed is always the same,regardless of any other point of reference. The platform observer, pocket watch in hand, finds that the laserpointers still fire their light beams at each other at c, just as thecar observer does. Indeed, everyobserver, wherever he is in space and time, obtains the same values. There is no adding of speeds, at least notthe way it was with baseballs.

If you are trying tomake sense of this, and failing, I can tell you why. You are making the common-sense assumptionthat space and time are the same for all viewers. But that is incorrect. It feelsright only because all your experience comes from speeds much below c, and evolutionby natural selection has been geared to that. But our, rather simple, gedanken experiment has shown that both time and space are malleable,and depend on the observer’s state of motion.

Space and time are notflat and absolute. Special relativityshows this beyond a doubt. But if theyare not flat and absolute, then – what are they? It was to take Einstein ten more years towork that out, and so much of modern cosmology depends on what he discovered. It is time to turn to there, the nextstate: the general theory of relativity, published in 1916.

* * *

Einstein was a persongifted with deep imagination and insight into nature. He knew that if he tried to work out hisideas about light and space and time not only with static motion but withaccelerated motion as well, he would probably never have succeeded ateither. So he started with static,meaning straight line, constant speed, motion, and in 1905 arrived at hisspecial theory. It’s special because itspecifically excludes all motions which involve changes in speed or direction(collectively known as accelerations), and concentrates purely on staticmotion. It was a maneuver which rewardedhim with pure gold. It showed that massand energy were equivalent, that the speed of light was the ultimate speed inthe universe, and perhaps most importantly, that time and space were not flat,abstract constructs, the same for everyone everywhere, but ebbed and floweddepending on different circumstances. Anenormous triumph, which earned him little notice at the time (he was stillworking at the same Swiss patent office well after publication), and whichdecidedly did not impress those who bequeathed Nobel Awards (though he did earnone some years later for a different line of work, which we’ll discuss in thenext chapter).

Getting back to thetheme of this section, special relativity was just that; a special case ofrelativity at work. Einstein’s main goalwas a general theory of relativity,one which included accelerated motion as well as static. It was to take him another ten years to grindout the final, so-called fieldequations for relativity, equations I will not present here, fully confessing Idon’t know enough math to do so.

We don’t need tounderstand that math (few do) to get a decent sense of what general relativityis about. One of the things Einsteinnoticed when dealing with accelerated motion was the results were virtuallyindistinguishable from being in a gravitational field. Here on Earth, as we would on all mass bodiesin the universe, we feel ourselves being pulled downward by some unknown,almost magical force, which we call gravity. And indeed, if the ground beneath our feet were to give way, we wouldfall, ever faster and faster, into the resulting abyss, until we were smashedto our deaths by whatever we finally landed on (if the opening goes deepenough, however, we will be incinerated by the heat of Earth’s depths andcrushed by the overwhelming pressures first – this all assumes the fall is farenough, of course).

An aside to describeaccelerated motion. Our falling wouldoccur under a force of one g(the gravitational force at Earth’s surface), meaning that we would plunge everfaster downward, at a rate of 32 feet per second per second, or 32fps². All this means is that, after the firstsecond, we are travelling at 32 fps, after the second second 64 fps, after thethird 96fps, and so on. (All this ignores air friction, which counteracts theacceleration and limits us to 200-300 fps final speed, depending on ourorientation.) Acceleration means achange in velocity, faster or slower (or in a different direction) over time.

All sorts of phenomenacall cause accelerated motion. Take, asI believe Einstein did, an elevator. Ifyou were sealed inside a windowless elevator somewhere out in deep interstellarspace, you would be weightless and float freely all about the car. Now imagine that some unknown being ofprodigious powers attaches a rope to the top of the elevator car, and startspulling it upward with a force of oneg(32fps², remember). I thinkyou intuitively sense that you would be slammed onto the floor of the car, andyou would be right – though be careful about intuitions! I ask you a question: now that you are standing on a floor with a32fps² acceleration force trying to pull you downward, what is thedifference between this situation and being in an elevator on Earth, at astopped floor level? Remember there areno windows, nothing to tell you what is going on outside the elevator exceptthis force pulling you down.

If I tell you basicallynothing, you are probably not surprised. After all, we have all been in elevators, and felt the lurch in ourstomachs, and the temporary lightness or heaviness in our bodies as we moved upor down in the car (try this at the Empire State Building or another monsterskyscraper with express elevators if youreally want to feel this effect). Thesensation of falling if an elevator plunges down rapidly can be quiteunnerving, rather like being on a roller coaster. I still have dreams about it, once in awhile.

So our super being acceleratingus at 32fps² “upwards” (there is no such thing as direction in deepspace, remember) feels exactly the same as being on an elevator on Earth,stopped at a specific floor. The beingcould even lurch us “upward” at more or less than one gif it really wanted to mimic the effects of the ground-based car. The point being, you simply could not tellthe difference.

Here’s where anEinstein mind works on a different plane than our own. He made the deduction that, the scenariodescribed above being true, there was nodifference, according to the laws of physics, between them. They are in essence one and the samephenomena. Like me, I suspect you’llhave to put this book down at this point and chew on this idea. If there is no difference in the perceptionof two or more experiences, than physics tells us they are intimately linked bysome thread; essentially, they are the same thing. This most certainly does violate ourintuitions, which accounts for the warning I gave before.

* * *

But what is the threadconnecting the two? Einstein finallyrealized that space and time, and their malleability, had to be brought intothe situation to make that thread. Butanother, hopefully short, aside on another needed prerequisite is needed here toshow how. You have probably heard of theconcept called entropy, though youmight (and should be) puzzled as to its exact meaning. We won’t get into that here, except to saythat one meaning is that physical systems are inexorably (actually, probabilistically)drawn to their lowest energy states. Anexample is the falling we’ve been talking about; think about the energy,derived from its load of jet fuel, it takes to keep an airliner miles above theground, or the far greater amount of rocket fuel it takes to put an astronauton the moon or a robot about the planet Saturn. Falling things lose energy, which, according to the laws of entropy, isexactly what they should and must do, unless something stops them, like thefloor of an elevator car.

The elevator, whetheron Earth or in deep space, is providing energy to keep its occupants out of astate of free fall, i.e., what they’dbe doing without the forces acting on them, floating about in a zero gravity (g)field. In both cases we are dealing withacceleration, even for an elevator stationary on Earth’s surface. What could account for such a curiousconundrum?

Space and time. Spacetime. Special relativity had indicated that the two were indivisiblyconnected, and the fertile mind of the Swiss patent clerk finally become modestphysics professor worked instinctively along these lines. If spacetime were a flexible, or curvable, concept that changed as anobserver’s reference point changed, then maybe it was more that that: if was a realthing, and not only real but highly variable depending on perhaps manyconditions. One of those conditions, herealized, was the presence of mass. Herealized that mass objects somehow distorted spacetime, in a way that causedother objects to be attracted toward them. Using this insight, he finally solved the problem of the cause ofgravity, which Newton knew better than to destroy his reputation on solving twohundred years earlier.

To explain howspacetime operates under general relativity I could reproduce a fairly standardpicture, chances are you’ve seen, which appears to illustrate it:

FigureIV.

Thegrid represents the “curvature” of spacetime about the massive body of theplanet Earth. Although it gives a goodfeeling for what is going on – you can easily picture objects approaching Earthbeing drawn in by the curvature of spacetime (the white grid), but the pictureworks only because Earth is at the bottom of a deep well and we intuitivelyknow that objects near a well will be drawn into it. What I am objecting to in this picture asthat it assumes what it is supposed to be explaining. It’s not a bad start but it must not be anend to the explanation. The reader wantsto know what is really happening.

Ifinstead of falling downwards the gridlines (representing spacetime, remember)get closer together as they approach Earth, then I believe we have a betterpicture of why objects fall toward other objects. Objects cause spacetime to compress, ratherthan to indent; a very small compression for objects the size of Earth, largerfor the size of the sun, and quite large for nasty things like neutron stars andblack holes, which compress them into singularities,a place which we won’t cover here.

Recallentropy. Objects “prefer” to be at themost compressed regions of spacetime because they can lose energy that way andobtain entropy by this particular meaning. That’s why they’re attracted toward each other, and will either fallinto each other or orbit each other, the latter until loss of energy results inan eventual (trillions of years and more for the Earth-sun system, so don’tworry) collision.

If spacetime can curvein one way, it can – at least in theory – curve many ways – and many of thesolutions to Einstein’s field equations describe the entire universe as being “closed”(spherically even) or “open”, or “saddle-backed” or plain old flat. New solutions crop up once every few years ordecades, describing how the universe must be or how it will evolve in time, orwhat phenomena (like black holes) it might reveal. There is also a “standard” solution, whichitself gets itself upgraded once in a while, as the result of new astronomicalobservations (e.g., that the universe is not only expanding but expanding moreand more so, due to something called “dark energy”) and other considerations. .

* * *

Iam going to stop here, because if you have been following me reasonably well,then I’ve succeeded in what I set out to do in this chapter. I wanted to describe for you a field ofphysics which, although it violates all our common-sense notions about howthings behave and why, was within your grasp to at least get a good feelfor. Don’t worry about all the sundrydetails I’ve left out (though if this has left you hungering for a deeperunderstanding of relativity, all power to you!); I just wanted to give you ataste for how strange and wonderful reality can be when we drag ourselves awayfrom our evolutionary-derived intuitions. Again, if I’ve succeeded, than we are ready to talk about what I’ve beensetting forth to cover, some chemistry and biology that should be within yourunderstanding as well. This is where westart on a very new road from the physics of Einstein, to the physics (andchemistry) of the quantum, a field many great minds have contributed to and towhich, I think, no one can be designated the creator of. On to quantum cats.

Reaction of bleach With Detergent -- Part 2

2011-12-14T07:44:00.000-05:00

I did this again this morning and, although their was still foaming, I felt no heat in the reacting vessel. Perhaps I was wrong in inferring a chemical reaction last time -- I'm sure if you poured ordinary water into bleach you'd still get some foaming.

Lesson is, science is built on repeatable observations, and I couldn't repeat my own. Who knows why the mixture grew warm last time; maybe it was nothing but overstimulated imagination.

I guess I'll have to try it one last time, just to make sure.

Wondering About Ourselves: An Excerpt from WONDERING ABOUT

2011-12-13T18:42:00.000-05:00

Toour astonishment, our common sense view of things biological begins to disintegratethe moment we apply curiosity and imagination to it, to dissect it and lookinto it at the finest levels science allows us to probe. In doing so, try as we might, we neverencounter this special essence or quality which seems so obvious at firstsight. Instead, what we do find, when webreak out our detectors and other scientific instruments, is that living thingsare composed of atoms and moledcules like everything else, albeit not in thesame elemental proportions, yet acting according to the same laws of physicsand chemistry as everything else. Themechanical, Newtonian universe of objects and forces, modified by quantumeffects on the smallest scales, appear all that is needed to explain why butterfliesfly, or mate, or find food, or stare at us with the seeming same curiosity thatwe feel gazing upon it. All our initialimpressions, and all the stories that have been told and retold aside, there appearsno miraculous special something that we can affix to or inject matter with tomake it come alive; no energy fields, no forces, no protoplasm, no elixir ofthe living, nothing we can pump into Dr. Frankenstein’s reassembled parts of corpseswhich will make it groan and open its eyes and have thoughts and feelings andbreak its bonds to move in accordance with them. There is nothing like that whatsoever. No, whatever it is that characterizes lifelies elsewhere.

Butthe impression of such a force is so strong, so deep, so instinctual that, tryas we might, we cannot simply abandon it without at least wondering why it isthere, where it comes from, and what it tells us. Somethingis there, of that there can be no question.

Intentionality. Complexity. Design. Try to put aside your ordinaryimpressions and perceptions of things, and seed your mind, germinate in yourmind, take root and push out of the soil and put forth leaves and vines in yourmind, the theme that to satisfy our curiosity we must look at the world from adifferent perspective, the one that imagination unlocks. Very often, we find that when we lookclosely, what we thought we were seeing fades away, yet is replaced bysomething just as amazing – no, more so.

Letus start with the simplest of things that could be called living. Consider the virus. Here is something both considerably smallerand simpler than the smallest, simplest bacterium, all biologists wouldagree. But on the most microscopic of scales,that of individual atoms and molecules, even the simplest virus turns out to bea machine of remarkable complexity. Atthe very least it has to be able to recognize a host cell it can parasitize,whether it is a cell in your body or a bacterium (in which case it is called abactaeriaphage), somehow figure out the molecular locks and other gizmos whichcells use to protect themselves from invasion, penetrate the defenses, thenusurp the molecular machinery the cell uses to replicate itself, perverting thecell into a factory for producing many more copies of the virus, copies whichthen have to figure out how to break out of the cell in order to repeat thecycle on other cells or bacteria, all the while avoiding or distracting themany other layers of defenses cells and bodies use to protect themselves fromsuch invasions.

Biologistsstill debate whether viruses can be legitimately counted among the variouskingdoms and domains of life, but there is no doubt that their hosts, whetherbacteria or other single celled organisms or multicellular organisms, can beclassified in the great Tree of Life, from which all other living things, bethey plants, animals, fungi, or you, diverge from. And what dominates this tree, right down to themost primitive beginnings we have yet been able to detect, is a level ofcomplexity that we simply do not encounter among the great many more thingsthan don’t belong on this tree, from rocks to stars to solar systems togalaxies.

Soafter all this, have we cornered our quarry? We started with the at first sight idea that life possessed some specialquality or substance or essence, then realized that we could not find thatessence however hard we looked. But whatwe did find was that living things, even the simplest of them, showed a levelof complex organization well beyond the most complex of non-living things.

Lifeis special. I don’t want to lose sight of that. We are fully justified in our grand divisionof matter into the non-living – things we explain only by the laws of physicsand chemistry at a simple level – and the living, all the things we must alsoapply whatever biology has to teach us. What I have been trying to show is that, whatever that specialness is,it isn’t as obvious as it appears upon first sight. It is more subtle, involving a number ofcharacters and qualities, one of which is complexity and another the appearanceof design or purpose.

* * *

Again,I say that life truly is special. It isearly May, and I have just come home from a walk through Pennypack Park, one ofthe many lovely natural places which skirt the city where I live, Philadelphia,one of several cities along the easternedge of North America. I would love oneday to walk on the moon or on the red soil of the planet Mars, but what I havejust experienced would be utterly lacking in those dead, albeit fascinatingplaces. In the spring in this part ofthe world, as in many other parts of our planet, every sense is roused to lifeby the call of the wild. Not only areyou surrounded by the verdant green of new buds and flowers and grasses, but alsoby a cacophony of whistles, chirps, tweets, and other rhythmic sounds which remindsyou that you that new life is all about, some of it still rustling itself tofull wakefulness after winter but much of it already in the air and alit on themany twigs and branches. And even withoutvision and sound, you can still smell the musty beginnings of stirrings things,the scents of enticing blossoms and irritating pollens, and you can still feelthe grass between your toes and the softness of young leaves on your skin as youbrush by the undergrowth.

HereI have spoken of complexity and the appearance of purpose and meaning, andperhaps that is exactly what our scientific mission into the heart and soul ofbiology requires, but this is one place where, I have to submit, we will neverreally capture the essence of what we are studying. Life is something that has to be experienced,and only living things themselves have the capacity, as far as we know, toexperience anything. So, in a sense, ourquest to satisfy our curiosity begins with the admission that, at least for theworld of the living, we never can completely satisfy it.

Am I going to give up, then? No, because, as I have maintained up to thispoint, curiosity combined with imagination and the scientific method can undoany knot, unlock any riddle, however baffling and impervious it may seem. I have even suggested a starting place even,this idea of complexity combined with apparent purposefulness, an idea I hopeto build upon and demonstrate just how powerful it is. I think we can agree that it is a goodstarting place. Biological things, eventhe simplest of them, are highly complex, we now see, and there does seem to besomething to this notion of being imbued with purpose, however that comesabout. If we can make some progress onthis front, then perhaps in the end we will satisfy our intellects after all,as impossible as that seems looking at things from their beginnings.

Quadratic Fitting of Data

2011-12-13T06:43:00.001-05:00

I recently had to tear my hair out finding an algorithm for the coefficients of a second order fit for a set of data points ( first order fits are easy). I found this below, in FORTRAN, and translated it into VB:

Sub QuadFit(N As Double, P As Double, i As Integer, X() As Double, _
             Y() As Double, Q As Double, N As Double, R As Double, _
              S As Double, T as Double, U As Double, V As Double, _
              W as Double, as As Double, b As Double, c As Double)

    P = 0

    For i = 1 To N
      P = P + X(i)
    Next i

    Q = 0

    For i = 1 To N
      Q = Q + X(i) ^ 2
    Next i

    R = 0

    For i = 1 To N
      R = R + X(i) ^ 3
    Next i

    S = 0

    For i = 1 To N
      S = S + X(i) ^ 4
    Next i

    T = 0

    For i = 1 To N
      T = T + Y(i)
    Next i

    U = 0

    For i = 1 To N
      U = U + X(i) * Y(i)
    Next i

    V = 0

    For i = 1 To N
      V = V + X(i) ^ 2 * Y(i)
    Next i

    W = N * Q * S + 2 * P * Q * R - Q ^ 3 - P ^ 2 * S - N * R ^ 2
    a = (N * Q * V + P * R * T + P * Q * U - Q ^ 2 * T - P ^ 2 * V - N * R * U) / W
    b = (N * S * U + P * Q * V + Q * R * T - Q ^ 2 * U - P * S * T - N * R * V) / W
    c = (Q * S * T + Q * R * U + P * R * V - Q ^ 2 * V - P * S * U - R ^ 2 * T) / W

End Sub

I hope you can use it.

2011-12-12T09:33:00.000-05:00

I love this illusion. Squares A and B are "clearly" of different shades of grey, eh? So why do when I connect them with a line with A's shade (or B's shade), a line which is in fact of one shade because I selected it that way (with Paintbrush), the lines lighten or darken depending on which square that it is in.

Of course you're brain is fooling you. Lesson? Take caution listening to your brain!

How We Know What We Know -- Chapter Two

2011-12-11T09:51:00.001-05:00

When I was in my early twenties I was in love. It was unrequited, but we still got alongwell and to this day I still say that she is one of the finest persons I haveever met. I won’t say it wasn’t painful– as an aspie I probably just didn’t have the knowledge and maturity to win herheart (or maybe it just wasn’t “meant to be”, whatever that means) – but I havealways been glad that I knew her. Shewas a terrific friend and companion.

One day, we were discussing ancient pyramids, of boththe old and new worlds. You might have noticed that among the very largestructures built by these ancient cultures (Maya, Mesopotamia, and Egyptmainly) were a variety of pyramids, from step to flat-faced, with the smallerstep ones coming first because they are easier to build, and often evolving, asin Egypt about 2500 BC, toward the huge, flat-faced structures (e.g., Giza). That this came about by improvements in theneeded engineering skills is fairly certain (I doubt they needed ancientastronauts, though of course it is possible), and probably also by largerpopulations, a decreasing portion of which didn’t have to grow food and so wereavailable as necessary labor. Betterpolitical and cultural organization no doubt played a role too.

In chatting on the subject, she made a claim which Iimmediately found hard to swallow. I canaccept the Egyptians and Mesopotamians influencing each other; the areas arenearby in the Middle East, while excursions (and even conquests) between thetwo are common in history. It’s quitereasonable to imagine (though I am not certain) the two cultures shared andcontributed to each others’ pyramid construction techniques andstrategies. I am ignorant of whetherthis really happened, but it is plausible and easy to believe.

But Egypt/Mesopotamia influencing the Maya? She was quite sure of this; but it was onlybecause she couldn’t imagine two separate cultures building such commonstructures, especially such massive ones, without their being a physical connection. To be fair, the idea sounds superficiallyreasonable and even compelling, this idea of Egyptian boats making the trans-Atlanticvoyage to the Yucatan peninsula and instructing the Mayans on the time-honoredart of pyramid building.

What a minute, though. Ancient Egyptian boats making trans-Atlantic voyages? In fact, this is a real problem. As the Europeans were to find out in the15’th and 16’th centuries AD (and the Chinese about the same time), there arehuge differences between large ocean-traversing vehicles and those who stick torivers, bays, and small seas and lakes. You need deep, complex keels in the ocean variety to handle the higherand more violent waters and storms, deep harbor ports to handle such vehicles, whichare larger, sturdier boats (made with hard wood at least, which Egypt hadlittle of) with more men and much more supplies (to handle journeys of monthsinstead of days or weeks at most), and so on. Now, I have never heard of any discoveries of these things being made inEgypt through the many centuries she was a great power in the western world;and we certainly would have found them if they’d existed, for there is no lackof archeological exploration there. Whatwe do know is that Egyptian boats were mostly made from papyrus and other reeds,hardly up to ocean travelling needs. Indeed,these ancient, (mostly Mediterranean and Black) sea travelling boats stuckclose to shorelines for safety, something you couldn’t do in a large ocean.

This would seem to make it virtually impossible for anyancient Egyptians/Mesopotamians to reach the Yucatan Peninsula in CentralAmerica. Even if one did, by accidentsay (this is possible, with incredible luck), why would they carry pyramidbuilders and technology with them? Theywould have had no ideas what to expect, besides, perhaps, an end of the worldto fall off (the ancient Egyptians didn’t know Earth was spherical, a fact thatwas discovered by the Greeks many centuries later).

I think all this alone destroys this hypothesis, thoughit is not always easy to make such statements with certainty. For there is still the fact of similarpyramids in old and new world cultures, something that still needsexplaining. To be complete, the one factthat fits poorest for my friend’s idea (perhaps even worse than the ship dilemma)is that the new world pyramids were built many centuries and even millenniaafter the old world ones. If an Egyptianboat were to somehow cross the Atlantic at its pyramid building times it wouldnot have encountered a culture that could imitate much of the Egyptian/Mesopotamiantechnology/political/cultural levels even to save its life. Yet by the time the Mayans (and some otherCentral American cultures) were ready for it, the old world was far beyondpyramids, having acquired the ability to build more complex and usefulstructures (oh, say, like the Valley of the Kings, the Greek Parthenon andRoman aqueducts, maybe even medieval castles).

* * *

Think about it. You are a well organized, powerful, and highly command-centered Neolithicstone-age culture, with a good supply of available manual labor (including, nodoubt and, alas, slaves) and rock. Time,as in decades, you have in abundance too, or so you hope. As the leader of this culture you want toconstruct huge monuments to your greatness, both to intimidate the masses andyour neighbors, and to make you remembered for “all of time.” What would you construct?

Your engineering skills are still pretty primitive forsuch tasks, so you need the easiest to build, strongest, and most sturdieststructure you can manage. Is it hard tosee that this would be a pyramid, starting off with small, steps ones andbuilding them larger, with smaller steps, as your engineering and architecturalskills were acquired over decades and centuries? A pyramid is in fact very strong, with a stable,a broad, flat bottom combined with tapering construction above it. I’m sure it requires the least engineeringand architectural mastery, as you are just basically carving out (shaped) stonesfrom a quarry, dragging them to the pyramid, and using scaffolding or leveringto get them on top of the existing stones. You may or may not have wheels (as in logs?) and animal power to helpyou, but that just increases the time it takes. Enough people, time, and sophisticated enough stone carving tools, andit can be done in a lifetime or less, maybe a decade or less.

Apparently, my friend didn’t think of all theseobjections to her “hypothesis” (better just called a belief). She’d stumbled across one fact, the similaritiesbetween old and new world pyramids, and that was good enough for her; there was little or no further researching,or thinking, or skepticism. I have anunpleasant feeling that that is the way many if not most people think,especially B people (As can’t do this). They find one or two facts (or factoids even) which suggest an excitingidea, or one that fits a pre-existing idea, and if they look or think furtherit is only to confirm the idea, which becomes a simple article of faith fromthereon.

I have used the word hypothesis occasionally here, asthough it is interchangeable with belief or idea, or even speculation, but tothe scientific mind the words hardly approach each other in theirmeanings. I haven’t used the word theoryyet, which I will now, for again in many minds sets up an equivalency:

Belief/Idea/Speculation= Hypothesis = Theory = Truth

It’s clear to me that my friend, though quiteintelligent, thought largely along these lines, while it’s a pretty standardphilosophical approach for most of Earth’s population. Unfortunately, it is wrong, dead wrong, a mistake no scientists worth rock salt wouldever make. I also think it is why B-typepeople are much more prevalent than they ought to be. I also connect it with the authoritarianthinking, mentioned in the last chapter, which can bury human curiosity under athink, wet, cold, woolen blanket; for it is seriously, and even dangerously,fallacy supporting. My friend wasintelligent, but she didn’t know how to think or question things. Shame, though I still respect her.

* * *

Belief/Idea/Speculation (BIS) = Hypothesis = Theory =Truth.

Is this truly the way type Bs (not all, to be fair)think? Type As, definitely not: they could not perform their jobs, or carryon with their enthusiams, if they did. Butis it as common as I have implied? Andif it is, what is really wrong with it? We are pretty much all after the truth, after all, and this could be aformula for it, one I simply don’t appreciate for prejudices of my own.

Actually I don’t think it is all that too common as pessimismwould suggest, at least not in so pure a form. But people do routinely make confusions here. This is important: a big part of science is giving words andconcepts precise, accurate meanings, ones that can then be used in almostmathematical formulations. And so, if weare to use the words/concepts here in like fashion, we must do the same. Then, perhaps, we can answer the question Iraised at the beginning of the section.

BIS’s are what most of our minds are filled with mostof the time, even, I strongly expect, most scientists. E.g., we Believe in an Idea called God, ormaybe various gods; or if we don’t, we still Speculate about whether oursentience is a soul, and whether it survives death, by becoming, say, part ofsome BIS called “cosmic consciousness”. Or, to be less esoteric, we have plenty of BIS’s about the people in ourlives, about politics, economics, religion, and the many, many other things we “thinkabout without thinking about.”

I am not criticizing here. The human mind probably has to work this way,if for no other reason that if we were as meticulous about science as we areabout everything else, it would be difficult to get anything done! Remember, too, our brains have been largelywired by genes we’ve inherit from our stone age, uncivilized ancestors. Making “snap decisions” or acting on gutfeeling, without too much asking and exploring, was, for most of our evolution,the better way to save your life and pass on your genes. But the result is, we’re stuck with them, at least for the time being.

I think my friend’s idea about Egypt helping with newworld pyramid building is a textbook example of a BIS. It is so easy to bring this Idea down, bybeing skeptical and thinking aboutit, that she must have never done those things. No doubt she just liked the idea so much, and, having one fact tosupport it, simply assumed that meant it was true. Man BIS’s are based on the one fact fallacy.

* * *

Let’s focus our microscopes on the other three words ofthe equation: Hypothesis, Theory, and Fact.

First thing that needs to be said is that, despite allthe = signs, from a scientific view they are not equals at all, but distinctlydifferent entities. At the same time I’lladd up front that in fact they are also not really so distinctly different, butoverlap to considerable degrees.

Let’s start with the word hypothesis, and as usual, anexample of it. I think mycounter-arguments to my friend’s idea constitute a valid hypothesis. It is not theory, and certainly not fact, butsimple hypothesis. First of all, afterall I not only attacked the idea (with gusto, of course; all ideas should beattacked with gusto), but presented counter-ideas of my own; for example whypyramid building is natural for a well organized, stone age culture at an earlyage, and why.

But I did not present any supporting evidence for that,other than the “it should be obvious and here’s why” implication. Given that, you might dispute my claim tohypothesis status! But I did give, Ibelieve, some pretty sound logic for it; more important than that, logic that canbe explored and tested to see whether it holds up to test.

“Whether it holds up to test” is a great deal of whattrue hypotheses, the ones in type A minds, concern themselves with. For an hypothesis is a concept that proposessomething, or explains some phenomenon, andwhich fits all known facts, contradicts none, and can be further tested (thatis, can it make predictions). I believemy friend’s pseudo-hypothesis has actually failed this concept (in her defense,though, she isn’t here to counter her critic, which really isn’t fair), whilemine passes muster – probably; I am not an expert in the subjects and therecould be facts difficult to fit into it – if only by the skin of its teeth. And, to reemphasize, it is nowhere near tobeing a theory, or a fact itself.

I am taking a conservative approach here, as should allscientists. At heart, we’re curmudgeons whohale from Missouri and often don’t believe things even when we see them with ourown eyes (not a good reason to believe just about anything, by the way). Propose something to a scientist and the bestyou’re likely to get is, “That’s interesting” along with appropriate bodylanguage, or something like that. Believe it or not, it’s a compliment.

Such are the basics behind hypotheses. So, next time some fascinating soundingthought comes to you, wait until you’ve checked it against all the facts andlogic you can find, and think of some ways it could be further tested, beforeyou announce it to the world. Not thatthe thought is automatically useless if you don’t; but then, you’ve just beenlucky. My friend was not lucky.

* * *

Theory and fact are more difficult to pin down, becausethey really have multiple, sometimes interlocking, meanings. In common parlance, and often in science too,theory just means an explanation for something, even if not a necessarilyproven true one (though it must have good evidence for it); in the former, commonparlance, case, but decidedly not for scientists, it is not even a necessarily clear,well-supported explanation. So if, forexample, I propose an explanation for how stars form (already been done!), andit passes the hypothesis examinations, people will call it a theory. But they might not call it a fact because itstill hasn’t passed enough testing.

Charles Darwin’s idea of evolution by natural selectionwas initially an idea, then a hypothesis, and is now, as it is usually called,a theory. It’s an explanation, true; butit is also, because it has passed so many tests and has so much evidence on itsside, a fully-fledged fact as well. Einstein’s theories of special and general relativity also get similar,justified, status. As does the atomictheory of matter. They’re explanations and they are facts. Nobody seriously disputes this.

At the same time, as a theory is in another sense also justan hypothesis that has stood up to further testing and observation, such that itcan be a claim to fact that may or may not (though most facts do) explain otherfacts, or support other theories. I’llput Alfred Wegener’s theory of continental drift in this arena. The theory says that the different continentsmove around on the ocean beds, occasionally joining each other and then breakingup, as shown below:

Figure II.

In fact, for much of its existence this theory wasn’teven taken seriously even as an hypothesis by most of the scientificcommunity. This was partly Wegener’sfault, for he proposed causes for continental drift that were clearly absurd –I emphasize however that this really should not be regarded as evidence againstan hypothesis – and mostly (I believe) that community’s fault for notsupporting an out of league player (Wegener was a meteorologist by training,not a geologist).

Currently, the theory now is not only clearly true, butis a theory in both senses: continentaldrift is a fact (with clear, proven causes), and it is a theory that explainsmany other phenomena about Earth, ones that had puzzled scientists for a longtime. We now call it rightfully the theory of plate tectonics, after the truecauses of drift.

* * *

Fact. Now, don’tgo thinking that fact means “naked observation by the senses” or anything likethat. I already alluded to this, butthis is a good time to go further. Ifobservation really is equal to fact, then the (fact? – maybe you’re lying, orpsychotic) that you just saw someone walk through a wall of solid concretewithout smashing it apart in someway a fact, or merely an observation – that isto say, a visual illusion? I’m sure you’llconclude the latter, even if you have no idea how the illusion was pulled offor how convincing it is.

This may put us in a pickle. Facts aren’t observations, but don’t theyhave to be, somehow, supported by observations? But how do we know whether we’re being fooled or not by these otherobservations?

One of the problems of science is that it really can’t makeindisputable proclamations about the universe. This makes science vulnerable to “straw men” arguments, often easy to demolish,but unfortunately inevitable if we want to keep it pure. Yet we can still make real progress here. For example, sticking with ourconcrete-traversing man scenario, what would happen if we were to view it fromall viewpoints, even those slowed or speeded up in time? Why, somewhere the illusion would certainlybe revealed, for a lot of magic is based on the magician having his/heraudience in a chosen viewpoint. Thebrain insists on interpreting sensoryinput in certain ways, another evolutionary trap which actually is reasonablebut sometimes leads us to error.

This suggests a good way of determining fact (if notwith infinite certainty). We make ourobservations from as many viewpoints as we can, and compare the results. If they agree, especially repeatedly, weaccept them as true; otherwise, they are spurious observations, fascinatingpossibly but of little scientific value. Of course, this is not always easy to do! Do two astronomers, gazing at the same phenomenona billion light-years in space, really constitute two viewpoints? In some ways yes, in others certainlynot. But it is the best we can do inthis case.

* * *

One conclusion of this chapter is that the dividingline between hypothesis, theory, and fact is not always clear, in fact it canbe quite broad and grey, the subject of innumerable, passionate, debates. But, I maintain, the line between the firstpart of the equation, the BIS, and the others is night and day. And, I emphasize further, this is the linethat is so precise in type A’s minds, but can get so muddled in type B’s. I think this is the main cause of why B’s(say they) don’t get science and math, beyond any natural talents in eitherareas.

So remember: youcan have all the ideas you want, but if you want them widely accepted as true,you must eschew the BIS approach and embrace the scientific one. And good luck to you, for it can be and oftenis a hard trek.

Kepler-22B; A Warning or Two

2011-12-10T11:01:00.001-05:00

The recent discovery ofplanet Kepler-22B (orbiting the sunlike star Kepler-22) has set off a flurry ofarticles and other printed/spoken material speculating that we have at lastfound another Earth-like planet in our immediate stellar neighborhood (the starKepler-22 is about 600 light-years from our sun, making it quite close insofaras intra-galactic distances are concerned; the Milky way is approximately100,000 light-years across).

In science however,excitement must be tempered by sober examination of evidence, and there are somegood reasons why we should not get too excited by Kepler-22B just yet. In the first place, we don’t have a good estimateof its mass yet, and probably won’t for a few more months. This is the most critical consideration as towhether the new planet is a (relatively small) gas-giant world, like Neptune orUranus, only about half their diameters, or is truly an Earth-like planet, onewith a rocky core probably covered with deep oceans.

If the first scenariois true is found to be the truth, this doesn’t automatically rule out life onthe new world. It might still possessliquid water, here as cloud layers, and life could possibly begin in dropletsor drops of water seeded with ammonia, methane, hydrogen cyanide, and carbondioxide, a lá the Stanley Milgrim experiments of the 1950s. A distinct planetary surface is not actuallyneeded for life, or so current thinking runs. However, its seems doubtful that such life would have evolved far beyondthe single cell, or prokaryotic, stage. Definitelyworth knowing however, if it turns out true.

The other,mass-determined, probability is that of a “super Earth”, a planet like our own,only considerably larger, and one probably covered by ocean-girdling waters anda thick, greenhouse atmosphere. Again,primitive life is a good candidate for the place, and here even complex,multi-cellular organisms may have gotten a toe-hold. They could be swimmers and flyers, thoughalmost certainly little in the way of land dwellers, for there would be littleof any land to dwell on. Still, polaricecaps might provide some of this. Alot depends on the depth of the greenhouse effect, driven largely by watervapor, carbon dioxide, and methane. Allthree gasses should be copiously produced by volcanism, so we shall see.Volcanism in turn is driven by a hot liquid core containing sufficient amounts of radioactiveatoms, atoms like uranium-235/-238, thorium-232, potassium-40 and strontium-87. Earth has significant amounts of them (creatingalso our strong magnetic field which protects us from the solar wind) becausethe creation of our solar system was probably initiated by a supernova type-IIexplosion, seeding us with heavier elements, but it is not clear whetherKepler-22 was born under similar circumstances (it is not all that unlikelyhowever, so we can reasonably speculate it). If not however, Kepler-22B might be frozen over, with little internalheat or heavy elements, leading to few prospects for life.

All this is speculationright now, but it may be of the purely academic kind, for other conditions areneeded for life. The biggest problem isthe apparent lack of large gas giant worlds, situated further out than Kepler-22B. They may still exist, in slightly differentorbital places than 22B, such that we don’t see their occultations from Earth;doppler “wobbles” in the star’s spectrum might yet root them out.

If they are not found,however, this is troubling for life’s prospects on 22B. Jupiter and Saturn stand as staunch shieldsagainst a large number of asteroid and comet impacts to our planet, impactsthat nevertheless occur to a disturbing degree and which could wipe out alllife here if large enough ones occurred with sufficient frequency. But we have a couple of heavy duty barbouncers that either suck up those impacts themselves, or hurl the offendingrock/ice worldlets out of the solar system, or park them in the asteroid belt.

If Kepler-22B doesn’thave its own bouncers, then it is probably being regularly pounded by asteroidsand comets, so much so that life can’t get started there. Now perhaps its not that bad a problem outthere because the Kepler-22 system was not the result of a supernova explosion;but then there might not be enough heavy elements to make a hot, molten core,with its attending strong magnetic field and copious atmospheric components.

Then there’s the otherproblem, which I’m not certain is truly severe or not. Earth has an axial tilt of 23° , which isalmost perfect for our seasons and the life adapted to it. The tilt does not vary greatly, and supposedly we haveour large moon to largely thank for that. The reasonable length of our day is also due largely to the moon. Frankly, I don’t know how large of a problemthis really is; with the exception of Uranus (with a 98° axial tilt, rotatingvirtually on its side), all the planets rotate on roughly vertical axes to thesolar system’s orbital plane, and only Mercury and Venus have unusual days, inthe first case one locked in a 2:3 resonance with its solar orbit, and in thesecond, Venus’, case a slow retrograde axial orbit (opposite to its movementabout the sun).

If these parameters areimportant (as suggested by the “Rare Earth” hypothesis) then 22B could bein big trouble, though this is not certain. But all of these considerations, taken together, should keep ourenthusiasm in check as we explore Kepler-22B further.

Another Mixing Snafu

2011-12-07T13:41:00.001-05:00

I just gotnailed on another reaction of household products, this time regular bleach withtoilet bleach. Remember that regularbleach is just sodium hypochlorite dissolved in water:

NaOCl (H2O) ® Na⁺+ OCl^-

OCl^- + H2O ↔ HOCl + OH^-

H⁺ + Cl^- + OCl^-®OH^- + Cl₂

H⁺+ Cl^- + HOCl ® H₂O + Cl₂

In both of thebottom reactions gaseous chlorine is generated in serious quantities. This is because often, not always but in thiscase, which I would have known had I bothered to read the labels instead ofjust assuming, toilet bleach is fairly concentrated hydrochloric acid. Needless to say it starting foaming at once,and when I came had just enough to flush before being driven out, coughing andtearing.

Lesson: read labelson chemicals before using!

Making Hydrochloric Acid from Household Ingredients

2011-12-07T09:17:00.001-05:00

MakingHydrochloric Acid from Household Ingredients

Iused to do this when I was young. I’muncertain now: could it not beconsidered a terroristic threat? Thetimes, they are a changing! Anyway, intothe science.

Hydrochloric(HCl) acid is simply a solution of the gas HCl (hydrogen chloride) inwater. The basic acid-forming reactionis:

HCl + H₂O ®Cl^- + H₃O⁺

H₃O⁺↔ H⁺+ H₂O

Themain acidic species can be considered either H₃O⁺ or H⁺, although the latter isusually used as it is clearer and more consistent. In almost (but not all) all water basedacids, this is the actual acidic species, whatever the starting acid (nitric,sulfuric, acetic, etc.) is.

Thereare a number of industrial and lab process to make HCl acid, usually from otherstrong mineral acids. Another way,however, is to generate HCl gas directly and dissolve it in water (it is highlysoluble, almost as much as ammonia). Thehousehold method uses this approach. Questions: how do you make HClgas, and how to you get it into the water?

Warning, Warning! HCl gas isvery irritating and corrosive, so you have to set up some kind of protectionfor your lungs and throat and eyes before generating it!

Ata young age, I loved to tinker with chemicals (perhaps not a good idea when Ilook back on it, but I was usually reasonably careful), both those I found inthe house and those I got in chemistry sets. And I loved to read chemistry books and ponder what might happen if youmixed such and such with so and so and heated them or dissolved them inwater. Amazingly, I still have all mybody parts and they all work well, which might be something of a wonder.

Inthis case I noticed something. It seemedas though if you mixed ordinary table salt and baking soda and heated themstrongly, you might get the following reaction:

NaCl+ NaHCO₃ ® HCl + Na₂CO₃

Inwhich the two reactants swapped the hydrogen and chlorine, Further, since HCl was a gas, it would escapethe reaction mixture (the upward arrow) and constantly drive the reaction tothe right.

Ofcourse I had to try it. Now, if I’d hada balance, I’d weigh out 5.85 grams of salt and 8.8 grams of baking soda. This is one tength of a mole of each product,thus an equal number of molecules of each, perfect for the 1:1 reaction. It would have yielded 3.85 grams of gaseousHCl ( and 9.4 grams of Na₂CO₃) . The two weight combinations on either side ofthe arrow equal, as they should. I didnot have a balance however, and so used a teaspoon or tablespoon of bothreactions – good enough.

Nowhere comes the part where you shouldn’t have done what I did. I would mix both reactants in an Kimex glass laboratorygrade Ehrlenmeyer flask (the triangular shaped one), place the flask on one ofour electric stoves, and (at least have the sense to) gradually heat the flaskuntil the stove temperature was at or near high. I know that gaseous HCl was irritating andcorrosive, so I would carefully smellfor any gasses coming through the top of the flask. Sure enough, I found myself tearing andcoughing pretty soon, and I knew my hypothesis was a triumph. The question now was, how to deliver the gasinto (preferably cold) water?

You’vealready noticed that household ingredients aren’t quite enough, you also needsome laboratory equipment, mostly glassware. I had such from my chemistry sets: Ehrlenmeyer flasks, beakers, corks/rubber stoppers that fitted the flaskand had a hole large enough for the glass tubing, the tubing, and an alcoholburner I could use to bend the tubing from the top of the Ehrlenmeyer over tothe beaker (more than a ninety degree angle) – not as easy as it might soundfor glass work requires some practice and experience. (You can no doubt still get these things,though I don’t know if you’ll attract unwanted attention doing so).

Let’sassume you have a desktop balance for weighing chemicals, though don’t ask mehow much they cost; anyway, you don’t need a highly priced one. Now, if you weigh amount of reactants in theflask as described above, you should generate 0.1 mole (8.8 grams) of HCl gaswhen you heat it strongly. After pouringthe reactants into the flask, next, assemble the apparatus. The stopper shouldfit tightly inside the Ehrlenmeyer, the bent glass tube pass through thestopper (not too far, though, just enough to pick up any gasses and deliverthem!), and the other end of the tube should reach the bottom of the beaker,which should hold about 100 milliliters (~ 1/10 of a quart, or half a pint orso – use a graduated cylinder if youcan) of cold water. Now strongly heatthe mixture in the flask. What you’llobserve is curious. First, a stream ofbubbles will emerge from the beaker end of the tube, rising and escaping intothe air. Don’t be alarmed; this is just theheated air being forced from the flask through the tube. What happens next is the main show. The bubbles stop, and the gas level in thebeaker stays pretty much flush with the water. What is happening here is that HCl gas is now being generated rapidly and,being highly soluble in water, immediately dissolves when it hits it, leavingno more bubbles. Your HCl acid isstarting to form!

Youshould keep this reaction/process going until you observe the following. As the reactants are consumed, the HCl isproduced in smaller and smaller quantities; and, again because it is so solublein water, begins to suck liquid up the tube from the beaker. At this point you should stop the reaction(turn off the heat and move the flask off the stove, remove the flask + stopper+ tube from the beaker, etc.). You DONOT want water pouring back through the tube into the Ehrlenmeyer under strongheat – I never tried this, but I assume the water will flash into steam, atleast cracking if not exploding the Ehrlenmeyer, thereby releasing a lot ofacid and HCl gas into the atmosphere, any probably other nasties I haven’tthought about. All in all, don’t letthis happen!

Letit all cool down for a while, before disassembling everything and thoroughlywashing out everything but the beaker and its contents (use lots of water, onyour hands too). Now, if the reactionhas gone to completion (though remember, some HCl is lost), I figure theconcentration to be 0.1 mole HCl gas dissolving into 0.1 liter water, givingaround a 1.0 molar (M) solution. This isa fairly potent concentration (if you get it on yourself, wash thoroughly withwater). It’s more than enough to dissolvealuminum and tin foil, magnesiumzinc, probably lead and iron and some othermetals, giving off streams of bubbles of hydrogen gas as it does so (this isalso potentially hazardous, and hydrogen gas is highly flammable). Remember mixing vinegar (a dilute solution of acetic acid, CH₃CH₂COOH)with baking soda and watching it fizz up?

CH₃CH₂COOH+ NaHCO₃ ® Na⁺ + CH₃CH₂COO^-+ H₂O + CO₂

TheCO₂, or carbon dioxide, is the gas that fizzes up, just as from acan of beer or soda. If you substitutethe weak and highly diluted acid vinegar with fairly concentrated hydrochloric acid,the reaction ought to be considerably stronger:

HCl+ NaHCO₃ ® Na⁺ + Cl^- + H₂O+ CO₂

Notthat I remember trying this. Oh, onemore thing; I’m pretty sure that you can make the acid highly concentrated(though I don’t recommend this, however, as it is VERYHAZARDOUS at very high concentrations), simply by upping the amount ofreactants. Multiply the reactants byfive or ten (you may have to run the reaction several times, or find a largeenough Ehrlenmeyer flask), and you should get five-ten molar acid. Again, something you really shouldn’t playaround with, unless you know how to do so safely).

On Curiosity (From WONDERING ABOUT)

2011-12-07T07:15:00.001-05:00

The humility Ihave described here is not the humility we see (not always in sincere form) invarious Eastern religious leaders and the like, although it is related. I am speaking of intellectual humility: the ability to accept that anything one hascome to believe, whether it be from schooling or a church, from books, parentsor other authorities, or even as the product of one’s own observations andthoughts, could genuinely be mistaken; mistaken no matter how much observationand thought or the weight of authority or time lend to it. Or how many people hold the belief, for howmany centuries. It is the recognition ofhuman limitations and fallibility, even among the most brilliant, well-educatedminds. My personal favorite example ofthis is Einstein adding the so-called Cosmological Constant to his equationsfor General Relativity to prevent, for what were mainly esthetic reasons ofhis, an expanding (or contracting) universe, something which his raw equationsimplied. When Edwin Hubble was withinbarely a decade to demonstrate by his observations of the red shifts of distantgalaxies that the universe is in fact expanding, Einstein pronounced this ad-hoc addition of the CosmologicalConstant the greatest blunder of his career. What makes this example my favorite is how a more recent discovery incosmology, that the universe is not only expanding but that, contrary to allexpectations the expansion rate is accelerating (the mutual gravitational pullof the galaxies ought to be slowing it down, yet it is speeding up), hasresurrected Einstein’s self-disavowed constant, albeit in somewhat differentform. Einstein’s confession of hisgreatest blunder may thus prove itself an even greater error, an irony I haveto expect he would have enjoyed.

Another, importantaspect to humility is the overwhelming feeling, shared by most of us I suspect,at looking upon a universe not only greater than our ability to fullyunderstand, but, as the biologist J.B.S. Haldane observed (though he used theword queerer rather than greater), greater than we can understand. One of themost wondrous and compelling things about science, which is such a large partof the reasons I have spent a lifetime immersed in it, is how strange andwonderful it can make the most “ordinary” of things, simply by the act ofexplaining them.

Walt Whitman, "Leaves of Grass"

2011-12-07T07:04:00.001-05:00

WHEN I heard the learn'd astronomer,
When the proofs, the figures, were ranged in columns beforeme,
When I was shown the charts and diagrams, to add, divide,and measure them,
When I sitting heard the astronomer where he lectured withmuch applause in the lecture-room,
How soon unaccountable I became tired and sick,
Till rising and gliding out I wander'd off by myself,
In the mystical moist night-air, and from time to time,
Look'd up in perfectsilence at the stars.

Reaction of Bleach with Detergent?

2011-12-06T13:43:00.001-05:00

Here’ssome more interesting household chemistry, one that I just discovered. I wrote about the dangers of mixing bleachand ammmonia before, but naïvely thought that bleach and pure detergent (likedishwashing liquid) was perfectly safe. In fact, it’s not particularly hazardous, but there is some chemistrygoing on and some precautions one should take.

It’sobvious that chemistry is transpiring because when the mixture is made, thereis some some significant foaming and a mild temperature rise that results. Again, neither is dramatic, and both ceasesoon, giving you a stable liquid (which is great to use on sinks, pots andpans, and on other kitchen or bathroom surfaces).

Ofcourse, I’ve been trying to squirrel out why all this happens. What follows doesn’t come from any probingresearch but are just my own ideas and chemical knowledge. The first thing that strikes me as that whenyou mix bleach and detergent, you’re setting up reaction between bleach, which is a solution of sodiumhypochlorite, and detergent (usually), a sulfonic acid:

NaOCl + H₂O® Na⁺ +OCl^- + ( H₂O) ;

OCl^-+ H₂O ® HOCl + H⁺+ Cl^-;

R-(S(=O)₂)-OH + OCl^- and/ or HOCl ® ?;

Thereare other possible reactions going on too; this is not as straightforward as Ithought it was going to be! There’s anotherclue to what’s going on, which is that I found the evolved gas odorless and colorless. Now if it had been something like chlorine orsulfur dioxide, there was enough that I should have picked up the pungency or even color ofthese two compounds. That leaves us withhydrogen and oxygen, which are both colorless and odorless. Now, I’making oxygen the more likely of thetwo because I espy a straightforward way of evolving it (and one consistentwith all observations of the reaction), while a I can’t see too many ways itcould be hydrogen.

Takethe S(=O)₂)-OHpart of the sulfonic acid. I can easilyimagine it reacting with HOCl:

S(=O)₂)-OH+ HOCl ↔ S(=O)₂)-Cl+ HOOH

Thereaction yields the chlorinated version of the sulfonic acid (detergent), theproperties of which should not be too strongly altered; and hydrogen peroxide,which in the vigor of an exothermic chemical reaction can break down into waterand ogygen, foaming it up and realeasingheat. That’s right, I’m suggestingthe gas is oxygen (this is easy to test, by the way; just stick a smolderingmatch stick end into it and see if it flares up brightly).

Also,note the ↔ symbol I use for the reaction direction, instead of the singleheaded arrow. I’m suggesting that thisis a reversible reaction; it can goeither way, as long as some other process doesn't contiuously consume one or more of thereactants/products. Many chemicalreactions proceed this way. In thereaction above, as long as the HOOH is breaking down into H₂O and O₂then it must keep proeceeding to the right, because both the oxygen escapes themixture. Two things to bear in mindhere, however; first, the reaction is obviously not very strong (or it wouldget hot and foam up dramatically, perhaps even explode; concentrated HOOH ismost unpleasant stuff), and second, the moment the bottle of bleach + detergentis closed tightly the back pressue of oxygen building up in the bottle essentially brings thebreakdown of HOOH to at least a near dead stop, and then the entire reactioncan go merrily back in forth in equibrium mode. That’s why it quickly cools and stops foaming.

Sowhat I am proposing that you end up with an equibrium mixture of bleach,detergent, chloronated detergent, and HOOH (hydrogen peroxide). If so, that makes it an especially effectingcleaning/bleaching mixture, as all three components will contribute their share.

Atleast, this is the best I can make of it on short notice. I’d be fascinated by alternative hypothesis.

From Quantum Cats to Cats' Paws: Chapter One

2011-12-06T09:43:00.001-05:00

^{Chapter One: Those Who Get Science and Those Who Don’t}

Sincemy childhood, when my love of science and nature was first sparked, I foundmyself surrounded by an idea about humanity that at the time I could neitherunderstand nor accept. This idea was, atheart, that there were basically two kinds of people: Those who got math and science, and those whodid not. It was such a strange idea tome, one which I saw duplicated nowhere else.

Actually,there was one similar idea, this one about sports. I noticed here too the idea ofthere being those who were good at sports, those who weren’t, and littlein-between. I thought this very odd too;both ideas actually, as the whole point of school – or so I presumed – was toteach kids skills and knowledge which they otherwise would lack. This point apparently didn’t apply toscience/math and sports. I proved that it did apply for sports in the fourth orfifth grade however, when, by a combination of studying batting mechanics,concentration, keeping my eye on the ball all the way, and controlled aggressionand plain old confidence, I could hit a softball just as well as anyone, if notalways as far (I am not very much athletically gifted).

Imust have determined, sometime around then, that if the can/cannot dichotomyaround sports was untrue, then it was probably equally untrue aboutscience/math: people weren’t dividedinto can do / can’t do camps, at least not in any straightforward way. All I needed to do was to find the right wayof showing/teaching/presenting science and math so that anyone could “get it”,at least to a reasonable degree. I justhad no idea what that right way was. Infact, I had no idea what the problem was for a very large portion of my life. One of the points of this book is my strugglewith and hopes for solving it.

Don’tget me wrong. Different people havedifferent talents and abilities in their lives, and scientists plusmathematicians (which I will from now on collectively call scientists)obviously do grasp ideas in science and math better than those not so gifted. I just don’t understand why there should besuch a vast gap between the two. Indeed,I don’t believe in any such gap at all, only a continuum of talents. Most of life is this way, so why not here too?

* * *

Thisis not an autobiography. Still, I shouldstop here and tell you some things about myself. One of them is about a condition which isfairly well understood today; one for which children get routine diagnoses, andall kinds of special help and training is available for those diagnosed. You’ve probably heard of it yourself. It’s called Asperger Syndrome, named afterthe Austrian pediatrician Hans Asperger who, during WWII, first described theconditions and its symptoms. But it tooksome fifty years for the condition to be widely accepted (for those of you inthe medical field, this is not a goodnumber) and for children to begin getting diagnosed with it and receivingvarious but consistent treatments.

Unfortunatelyfor me, this was all before my time, so I was left to my own devices inhandling the “strangeness” about me that I didn’t understand and thought I waslargely imagining.

Theinteresting thing about Asperger Syndrome (which I will call AS from now on) isthat it is placed in the “autism spectrum” of pervasive developmental disorders. Now, most of us have heard of classicalautism, in which a child’s intellectual and social development are locked intoan almost infantile state. However thereare also versions of autism which don’t impair intellectual development, which,paradoxically perhaps, may lead tosuperior intellectual development. Thewriter and animal specialist Dr. Temple Grandin is a wonderful example of thelatter, and you have to read some of her written works to get a feel for whatit is like to be brilliant yet autistic.

YetGrandin is not an “aspie” (a favorite term by those who have AS) but a (very)high functioning autistic, and there are important differences. Both can have high intelligence and mentaltalents, this is true. The maindifference to me seems to be that, although both are still socially deficient (I’mtempted to say “retarded”, just likethose few people who really cannot, intellectually, grasp science and math atalmost any level are deemed “retarded” in unofficial circles), aspies still atheart crave social acceptance and love, while high functioning autistics (HFAs?)like Grandin seem perfectly comfortable without them.

Thereare other differences too. My position is that I think that aspies are worseoff than HFAs because they so desperately crave what they cannot figure out howto get: friends, acceptance, normalcy,popularity, and so forth. These frustrationsof course only get worse as one proceeds through adolescence andadulthood, Because aspies are oftenhighly intelligent, they can learn to “fake their way” through the adult world,with more or less success. But theanxiety and frustration and despair at feeling so deeply disconnected canultimately prove to be too much. Thiswas the case for me, but clearly not for all aspies.

* * *

Whydo I raise this subject? It is not, Iassure you, to gain cheap sympathy from readers. A treatise on what Asperger’s has driven meinto and through could be an entire book, perhaps one worth writing. But here I am concentrating on what, at leastI believe, is one particular consequence of it.

Evenas a young child I was often absorbed in my own world (a common theme amongaspies), and because I had some intellectual precociousness, I developed a verystrong sense of curiosity about myself and the world around me. I also developed some ways of satisfying thatcuriosity. Thus, for example, I learnedhow to read and write at an unusually early age (this was, however, in partbecause I was fascinated by the sounds of different letters and words – probablyalso due to Asperger’s, who can get fixated/obsessed on things and ideas and,to the annoyance and worse to others, people). I had also, at least by age five, fully developed the scientificapproach of not simply believing things because authorities (parents, teachers,etc.) told me them, but of trying to figure out how to test those claimsmyself. I won’t repeat in detail myfavorite example of the color of the disk of the sun (it really doesn’t lookyellow, as we are all taught) and my struggles with my kindergarten teacher todraw it as I had learned to observe it.

(Incidentally,why the sun seems yellowish-white (if you don’t stare strongly at it, which youshouldn’t do if you don’t know how to while protecting your eyes) is a true andfascinating scientific tale, one I won’t tell here except to hint that it’s thesame reason why the sky is blue.)

Nevermind these early clashes with teachers on such things. I was to have more, in which I was sometimesright and sometimes wrong, but in all cases was fortunate to have teachers whoaccepted or at least tolerated a child who thought for himself (thank my luckystars for all of you). The point I’mtrying to drive home is how an insatiable curiosity in me was forged by acombination of my Asperger’s and my intellectual/cultural/family environment. How much better it might have been has weknown about Asperger’s at the time! Instead I was regarded a somewhat precocious child combined with a somewhatrebellious nature. Since I was never aserious behavior problem I never came to the attention of school psychologists(I think). I liked my teachers too, andnever wanted to disrespect them or show them up – no, I was decidedly a goodboy. But all this stuff was festeringinside me nonetheless, and it finally came out in high school and beyond. Again, however, never mind that; I’ve onlysought to explain the origins of my unerring need to know and understand, whichI (luckily!) have within me to this day.

* * *

Curiosityis an essential ingredient in science, and in the minds of those who work inthe field, either professionally or as amateurs. I’m also certain that practically everyonehas it, at least to some degree; but I’m also just as certain that for mostpeople it has been blunted and buried and snuffed down to a slow simmer becausethe adult world in general doesn’t encourage it. I’m sorry to have to say this, that even inthis, possibly the freest of societies/cultures in history, people are still oftenhamstrung by the need to accept authority and its proclamations about thenature of things; and that those who do so are rewarded while those who fail toconform are sufficiently driven to near extinction to drive the point home. Ironically, I believe there is some truth tothis even in the scientific establishments themselves (though nowhere near asmuch), as controversial such a claim might be. It may, indeed, be necessary to have it to some degree, forsocial/cultural adherence and order. Well, I’d better drop the issue now.

* * *

Bearin mind, this is just my two cents, and not the nexus of the discussion. I meant to concentrate on curiosity as essentialif we are to be among those people who “get” science (and math). But is curiosity enough? What other powers of the brain needemployment here?

Thisis not so obvious, and I had to think about it for a long time before I came upwith a sensible sounding idea. Of course,not all scientists think exactly the same way (thank God!), but there does seemto be a basic pattern, a fundamentalmode, in their thinking, just as there are fundamental nodes in the plucking ofa musical instrument’s strings. This, Ithink, is their ability to take abstract ideas and place them in their minds asconcrete pictures and/or processes.

Togive a personal example of what I mean, I did very well in my undergraduatecourses in organic chemistry (not without some serious studying, mind you),while many other students struggled terribly. Now, organic chemistry is a subject concerning large (carbon-based,which we’ll get to later) molecules, often with complex shapes. I didn’t find it particularly difficult topicture these molecules in my mind, even without the help of molecular modelingkits. It seemed to me that all I had todo was to combine this ease of picturing with certain things you learn ingeneral chemistry (like electronegativities, and the different kinds and strengthsof bonds, also things we’ll get to). Youcould almost figure out anything from just these two sources and lick theorganic chemistry bear without working up too much of a sweat.

Otherstudents, however, wrestled mightily with the bear. Sometimes I would try to help them, butneither of us could figure out what I was doing right and they were doingwrong.

Thenone day I was happened to be reading a book on how the mind worked and cameacross a fascinating puzzle. The author presented a picture of twoblock-composed objects (that is, objects made of, say, wooden blocks gluedtogether). I wish I could remember orfind the objects, so you could do this test for yourself. Having sketched (or photographed) the objects– this is all in two dimensions, bear in mind – the author made the boldassertion that the human mind could not imagine them in 3D space being arrangedin such a way in which an extension of one could fill a gap or hole of theother.

Inearly fell over, for I realized at once that I could easily picture this situation. It was as easy as sitting down! Then I remembered taking geometry in thetenth grade (with dear Dr. Israel Nolan, wherever you are), and having to dopractically no studying or homework because the problems looked so easy to me Icould work out the geometric principles on the tests and get an A for thecourse. The two abilities, the one inorganic chemistry and the other in geometry, I realized were really two aspectsof the same gift!

Giftis perhaps a poor choice of words, for it implies that you either have it oryou don’t. The truth, I believe, is thateveryone has it, just not to the same degree. With me it is obvious. And,naturally, there are many who are far superior to me in it – I suspect thatEinstein could actually picture the shaping and warping of space-time even inhis equations for it, something we more ordinary mortals struggle mightily with(I think I’ve got it down a little bit, but … help!).

* * *

Let’sget to the bottom line. Once again thisisn’t actually about me, or what modest talents I seem to possess. It’s about the issue raised by the title ofthe chapter: “Those Who Get Science andThose Who Don’t”, and why. Myconclusion, or I should call it hypothesis (an hypothesis is an “educated guess”about the nature of things, drawn from existing observations; to become a theory it must pass more stringent testsand many more observations, after which it may even achieve fact status), is that the main reasonsare: those in column B simply don’t, forwhatever reason (lack of Asperger’s?) have the probing and insatiable curiosityto the degree those in column A do; or/and that the A types are better (thoughnot infinitely so) at turning abstract ideas into reasonable concrete images in their mind. I say reasonable because there are no doubtother criteria, such as logical thinking, involved – you can, after all,imagine all sorts of absurd, illogical things, something we all do frequentlyand sometimes deliberately.

Onething I hope the reader is taking away from this chapter is that few peoplereally fall into either A or B perfectly, that this is a fallacy foisted onto us by psycho-sociological forces Idon’t claim to understand. I also hopethat, if you have always thought of yourself as the classic B type (most peopledo), don’t despair; you almost certainly have some A coursing through your veins,and you can understand science to a degree beyond what you believe. With hope comes invitation, and I amwelcoming you pseudo-B’s to come exploring the possibilities with me (alongwith all you A’s, of course).

* * *

Again,there must still be something missing, however, to this hypothesis I’ve laidout about why people are of type A or B when it comes to science. I think everyone knows what I mean: we all know people who are clearlyintelligent but shake their heads in fogged embarrassment (to be bitingly truthful,not all of them appear embarrassed, but even smug and proud!) at theirineptitude in matters of the scientific intellect.

I’lltake my own mother as a personal example of that, partly because she hasrecently passed (and will be sorely missed by all her children andgrandchildren) and is much on my mind still, and also because she was adecidedly intelligent and educated person, one of the most I’ve ever known (sothis is out of respect, mom). But shewas a textbook example of what the physicist and novelist C.P. Snow lamented asthe breakdown of intellectualism, even society as a whole, into twofactions: literary/artisticintellectuals, and scientific intellectuals. This division is clearly quite real and has been become quite rancorousover the last several centuries up until today. It’s almost impossible not to see it, especially in the halls ofacademia. In mom’s defense, she admiredmany scientists and their accomplishments, and well understood that the highand healthy standard of living she and her family enjoyed was because ofscientific work. I’m certain she also understoodthe scientific principle and could apply it effectively.

Ihave at my fingertips a good example of what I mean. Some years ago, as we were preparing to leaveher house, the question of what caused the Earth’s seasons came up. I immediately jumped into my professorialrobes (always keep ‘em around, just in case) to explain the seasons, but wasfirmly stopped before I could even begin: “I don’t understand scientific reasons; I don’t have a mind for thosethings,” she insisted, or something likethat, to my utter astonishment, giving me no chance to protest that even achild could understand the science underlying Earth’s seasons. Worse, shedidn’t even want to try! I rememberbeing crushed.

Whatreally has me puzzled here is that those in the literary/artistic camp are notdevoid of this ability to picture abstractions I mentioned earlier. For, after all, this is just the action of imagination,and who can imagine better than artists and writers? There’s something to the reasonable aspect ofimagination that sometimes comes into effect here. I also sense that literary/artisticintellectuals regard scientists as dangerous and even naïve (which of coursethey are sometimes, as all of us are).

Isit envy? Scientists’ equations andproclamations are difficult to understand, yet they wield considerable power andinfluence in society, power and influence the competition, well, just can’tstomach? I’m tempted, but must rejectthis hypothesis, as writers/artists can be equally dense and incomprehensible,and they too have their influence in the halls of government and academia. Besides, as I said, my mother had little butadmiration for scientists, even if she didn’t think she could follow theirexplanations and equations to save her life.

* * *

PersonallyI see nothing natural or inevitable about this division of society into two,almost warring, camps. And indeed, manyscientists do appreciate literature and the arts, and vice-versa. I suspect this is a temporary division,brought on by nuclear and other weapons technology, and other abuses of some scientists who see satisfaction ofcuriosity as an end justifying any means. If I am right about this it heartens me, because I like many havewitnessed the many recent attempts of the scientific community to ethicallypolice itself, and the strides of many in the literary/arts camp to gainscientific education so that they can have a say in ethical scientific philosophytoo.

* * *

PerhapsI should have emphasized earlier this idea of the perversion of curiosity asmeans, however immoral in specific cases,to its own end of self-gratification. Ithink you’ll agree it is not only important, but will only become more so asscience progresses. As noted, this hasbeen going on for some time now: MaryShelley’s famous book Frankenstein, written two hundred years ago, is probablythe most influential tome along these lines.

Idon’t want to elaborate on this, however, because again I return to one of thefundamental aims of this book. Let meask you: do you see yourself as a type Aor a type B; and when I say type B I include our artistic/literary brethren aspart of this group. Perhaps you arestraddling on the seemingly wide saddle between the two, one foot in onestirrup and the other in its counterpart. Perhaps you aren’t certain whether you even care; though, if you’re inthis camp, you’ve probably stopped reading by now, so we can safely eliminate youfrom the discussion.

* * *

Moreto the point, how should I proceed? Ithink perhaps here that an explanation of the seasons should make a good a startingpoint as any, given that it really isn’t a difficult scientific problem andthat someone so dear to me proved a classical type B in refusing to listen tothe solution. You can judge as well atthis point: do I make a clear, coherenttheory of the seasons such that a school child could understand it, or do Ileave you still scratching your head?

Letus begin. I know that somewhere in yourprimary education you learned that Earth moves by a double motion: it revolvesaround the sun, in a time period known as a year,and it also rotates on its own axis (an imaginary line connecting the north andsouth geographical poles), in a time period we call a day. Abstract knowledge isnot enough here, remember; an act of reasonable, logical imagination isneeded. Thus, I’ve provided a picture ofthis double motion, as seen from some vantagepoint way out in space:

Herethe sun is at the center of the picture, and Earth is the blue spheresrevolving about her (of course, there’s only one Earth; the six in the picturesimply show it at different points in its orbit). Although it doesn’t demonstrate Earth’srotation about its own axis, it does show the axis, as the faint blue lines drawnthrough the various Earths, tilting from bottom left up toward the right. Earth spins on that axis, which, you’llnotice, doesn’t change direction as the planet orbits the sun – hint, this isthe key to the explanation. Oh: Theterms periapsis and apoapsis refer to the points in Earth’s orbit when it isclosest and when it is furthest from the sun; for the orbit is not a perfectcircle, but is actually, as Johannes Kepler realized in the 16’th century, anellipse in which the sun is at one focus (of two foci) of the ellipse.

[Ishould not assume anything. You can makean ellipse yourself by using the following directions: place a piece of paper on a table; stick twopins some distance apart from each other (not too far from each other or toofar from the center of the paper, or the ellipse will fall off the paper andthe experiment won’t work); take a piece of string with the two ends tiedtogether (making it a loop) and place the loop around the two pins and a pencilthat is in contact with the paper (it should yield a triangular shape for thestring); with the pencil inside the fully stretched out string (remember, thestring is stretched about only the pencil and the two pins), draw the naturallyclosed shape on the paper, keeping it taut as you draw all the way around. This shape is an ellipse and the two pins thefoci of the ellipse. It looks, as youcan see in the Earth/sun picture, like a squashed circle; and that’s a good waydescribe it.]

Allplanetary orbits, including Earth’s, are ellipses with the sun at one focus;that, recall, is one of (the three of) Kepler’s Laws of planetary motion. Now, you might think at once that thisexplains the seasons; for when Earth is closest to the sun it will be summer,and when it furthest, winter will grip the planet.

Youmight be tempted towards such an hypothesis. Instead, what should immediately smack you on the head is somethingyou’ve known for a very long time: Earth’sseasons aren’t neatly divided into summer and winter. When it is summer in the northern hemisphereit is winter in the southern, and vice-versa. Indeed, if you look at the picture of Earth’s orbit, you’ll see that thenorthern winter solstice (the firstday of winter) occurs just two weeks before periapsis, or the closest approachto the sun. The same is true of the northernsummer solstice and apoapsis.

Sothis hypothesis won’t wash (though I have seen it seriously proposed). What hypothesis will pass muster, as in beingconsistent with observable facts? Ihinted before that it lay with Earth’s axis, and indeed in here the solutionpresents itself. Go back and look at theEarth/sun picture yet again, especially at the different Earths’ axes, andobserve something I haven’t pointed yet. Do you see it? Don’t worry if youdon’t because the significance isn’t all that obvious until you think about it.

Ifyou haven’t spotted it yet, here it is. The axes are not straight up and down with respect to Earth’s orbitalplane (the imaginary flat and infinite surface the orbit naturally fits inside),but are tilted with respect to that plane; and furthermore, as said, theyretain the same tilt all the way through Earth’s orbit – um, at the risk ofgetting ahead of myself, this is an example of Conservation of AngularMomentum, one of the great conservation laws of physics (and is truly whatmakes the world go round!).

Theaxis tilt in this case is about 23° (degrees), in the system where 90° or aright angle is just two connecting lines perpendicular to each other (_└)– you should have picked this up from high school geometry, but may haveforgotten so I repeat it here.

Lookat the periapsis Earth point, near the upper right corner of the diagram. The whole explanation for the seasons can bemade here, because what I am about to say will apply to all the Earth points. This is almost the particular point (it isreally at the winter solstice, the first day of winter, December 21) where inthe northern hemisphere the axis tilts furthest away from the sun, and,conversely, in the southern hemisphere tilts furthest towards the sun. This is thekey to the seasons.

Lookat this particular point I’ve chosen carefully. Northern Earth appears to be, in fact is, leaning away from the sun, whilethe southern part of the planet is leaning toward it. For observers standing on both hemispheres,the northern one not only sees fewer sunlit hours during a winter’s day (whichis why we say winter has shorter hours, though this is not really true; alldays are 24 hours long, nighttime and daytime parts combined), but also theangle of the sun is lower in the sky, spreading the portion of sunlight perground area (space on Earth’s surface) thin. For the southern observer, it is high summer for the precise samereasons: more sunlit hours during theday, and a high sun in the sky, concentrating its rays on a minimum area. Can you see this? I hope that, maybe with some effort, you can.

Nowonder it is cold in winter and warm in summer, and that the two hemisphereshave opposing seasons! The smalldifference in overall sunlight received from Earth’s orbit being an ellipse (itis, in fact, only very slightly elliptical, nowhere near as much as implied inthe picture) makes only a small perturbation (change) to the effects of Earth’stilted axis.

Inthe interests of not over-simplifying things and so “dumbing science down” (allattempts will be made to avoid doing so in this book, or at least keep it on theshortest leash possible), I need to be a little more forthcoming – although youcan skip the following if you feel the need to do so, a need which I’ll justhope you will resist. I have pretty muchstated flat out that Earth’s orbital axis is about 23° and always points in thesame direction, and that is due to the so-called immutable Law of Conservationof Angular Momentum. In fact, looked atfrom year to year, or even century to century, this is basically true, which iswhy my explanation of the seasons can stay afloat. But, as always, the real picture is morecomplicated than this. There are othermotions of the Earth. First, the angleis not always 23° but “wobbles” about somewhat over hundreds of thousands ofyears. Also, the direction of the tiltalso moves, in a circular fashion, over a period of about 25,000 years. This means that 12-13,000 years from now thenorthern and southern seasons will have switched and we’ll be celebrating thewinter holidays (Christmas, Chanukah, Kwanza, Festivus, etc.) during thebeginning of summer, just as those in Australia and South America do today. Easter too will come in the fall, notspring. (This all assumes any of ushumans will still be here to celebrate them, by no means an automatically trueproposition.) These relatively minormovements can also be perturbed into larger ones, at least in theory, by thegravitational influences of the other planets, or by passing stars or otherlarge objects as they come and go near the solar system over periods lasting millionsof years. If you are interested by theway, our large moon largely, and fortunately I hasten to add, shields us from mostof the more extreme perturbations; I say fortunately, because it is unclearwhether life, or at least complex life, could exist if Earth were that unstablein its motions. Interesting are thewhims and wills of the universe!

* * *

Thisconcludes the explanation of the seasons. How do you feel? Did you followit, perhaps with effort and several rereads, or did you get it instantly, orare you still in a thick fog? If thelast possibility I’d like to know (unless you were simply bored because youhave no scientific curiosity, in which case why are you still reading?). Are you feeling at least a little more A’ish,or does B still have you in its vise? Ifthe latter, I invite you to reread the section (perhaps yet again), payingcloser to the parts where you started to get confused. I think most of you will make progress,though you shouldn’t have to reread it too many times to do so.

Inany case, we have encountered our first scientific explanation of a naturalfact, and will do so many more times in this book. Most of these facts will be more challengingto explain than the seasons, so I shall have to work harder to satisfy youwithout snowing you over or treating you like the fool I’m convinced you’re not.

I’veentitled this book From Quantum Cats to Cats Paws, meaning I intend tocover a hopefully comfortable handful of interesting and important concepts inphysics, chemistry, and biology. Incidentally, don’t fret if you don’t understand or have even heard thephrase “quantum cats”, for anything in the field of quantum mechanics needs alot of careful work to explain to anyone at any level. It is something even great physicists don’tclaim to fully understand.

Furthermore,we won’t be starting there. I’ll bestarting with something that, strange enough to say, I believe is easier tounderstand, at least at the basic level we’re aiming at here, than most peoplethink: Einstein’s theory of specialrelativity (it’s the general theorythat has even the brightest pulling our hair out, though even here somefundamentals can be outlined and given a reasonably good feel for). But first, I want to say more abouthypotheses and theories and facts, a subject I only touched lightly upon here. I also want to touch on other areas, like howart/literature and science/math are alike and how they are different. I’ll do this periodically as we go along.

Herewe go!