Translate

Google+ Badge

Follow by Email

Search This Blog

Friday, December 21, 2012

Chapter One of The Third Row

A Panoply of Elements

On the Nature of Substances

Look around you. I do not know about your environment, but I can describe mine in considerable detail; actually, in more detail than you would probably be willing to slog your way even if I were to write it down. A computer, a lamp, a desk, television … the objects in my environment are (at the moment) pretty mundane, or so they seem at first sight. I’ll bet that your environment is much the same way. Instead of focusing on the objects in our environments however, consider instead the substances they are composed of. These substances too are quite likely fairly common, and chances are there is a great overlap between your environment and mine. Wood, glass, living flesh, plastic, metal, paint, cardboard … or, if you are outside, plant and animal life, clouds, sunlight (or starlight or moonlight), dirt, rock, air, water … the list would appear to go on and on, with no end in sight.

 Or would it? This is an interesting point to ponder. There could be an infinite number of substances that things are composed of; or there could be a limited number, perhaps even a rather small number, of basic substances that combine in innumerable, different ways to make up the objects in our lives and in our universe.

The latter option – a limited number of basic substances of which everything is composed – seems preferable, if only because it makes figuring out the world around us a much simpler task. And indeed, there appears to be good evidence that this is so. Take the substance water, for example: we find it in all kinds of things, from milk to soda pop to our own bodies, to the great oceans of our home planet and even elsewhere in the universe. Reflecting on it, water is found in a great variety of things. Perhaps this means that water is one of these basic, or fundamental, substances, that we are trying to classify.

 Taking stock of things, we notice that water is not the only possibly basic substance. What about air? Although it is invisible, we are constantly aware of the existence of air merely by the act of breathing it in and out, or feeling a breeze on our face, or by watching it make the leaves of a tree rustle and sway as we walk through a park on a spring day. Noticing all this, we might want to classify air as one of our fundamental substances too, just like water.

 What about the earth beneath our feet? If we dig our fingers into the soil and pull some of it up, we see that earth is a substance as well. A fundamental substance? Well, we do find that it is almost always there, wherever we go, although it is not always of the same quality. Sometimes our digging will pull up sand, or rock, or clay, materials of different color and hardness and other attributes. Yet all of these things may simply be variations on the main theme, that of earth. So we will, at least for the time being, call earth a fundamental substance, adding it to water and air.

 Clearly, there are many directions we can take in all this classifying of substances. What about fire? This is a very interesting substance, one reason being because it can turn one substance into another. For example, it can boil the substance water, converting into the substance air, or what we call steam. It can also, when quite hot, be used extract metals like copper and tin and iron from certain rocks, which is how we have most of these materials. Quite an amazing substance, isn’t this fire? Perhaps we should list it among our fundamental substances too.

 Let us stop here and recapitulate our findings. We have selected four substances, water, air, earth, and fire, and labeled them as fundamental substances. Before we proceed, I would like to introduce another term for fundamental substances. The term I am proposing is elements. An element is a fundamental substance in the sense that it cannot be broken down into, or reduced to, other elements. Each element stands on its own, composed of nothing but itself. If this is true, then all substances and objects that we perceive in the world are a combination, in one form or another, of these four elements we have identified.

 Using this kind of analysis, we seem to have made some progress in understanding the world around us. We have reduced all things into a combination of four elements. If indeed, this is how the world works, we are very fortunate to have stumbled upon its basic constitution. Using the right combination of the four elements, perhaps tempered in the right way by the element fire, we should be able to create any object or substance we desire, from gold and diamonds, to modern computers and all the other electronics which have made the Information Age possible. Amazing!

The question is, are our elements truly elements by our definition – fundamental substances which themselves cannot be broken down or reduced to any other elements? If not, then our quest is not finished. Furthermore, how can we make the determination whether they are or aren’t, and if not, what are these elements that we seek?

 For an example, let us take our earth element, weigh it carefully in some kind of container such as a flask, and mix it with our water element, also carefully weighed in another flask, and stir the resulting mixture very thoroughly so that they are as completely blended together as possible. We then take this mixture, which we all recognize from our childhoods as plain old mud, and pass it through a filter, collecting the resulting filtrate, which is the liquid that passes through the filter, in yet a third flask; preferably we use a scientific filter designed for such purposes but a simple coffee filter should be very effective as well. If the filter is good enough, meaning if the holes are small enough only to pass the water plus anything dissolved (this is a suggestive concept in and of itself) in the water and not the entire mixture, something very interesting will happen. We will notice that the residue that remains behind in the filter after all the water has passed through it probably looks essentially the same as the earth we originally placed in its flask, with the exception that this residue is wet, or muddy, looking; while the watery filtrate, at the bottom of the flask we collected it in also still resembles ordinary water, though it too maybe somewhat colored, probably a color much like our muddied earth.

Now here is the interesting part. If we take the water filtrate out of its flask and set it in the sun, or heat it over a kitchen stove – it’s amazing how much science you can do in a kitchen – unlike plain ordinary water once all the water has been evaporated there will be a remaining dry sediment left behind. Or at least I will bet there will be. This sediment might be white, or one of a number of different colors, or even the same brown or other hue like the earth it was extracted from.

 Wait a minute, you say. Extracted? What exactly does that mean? How do I know that? Thinking about this, it would seem to be that, at the very least, we have separated the earth into at least two simpler substances: the filtrate, which passed through the filter, and whatever remains in the filter. But how can that be if earth itself truly is an element? By our definition of the word element, it can’t be.

 There is something else highly suggestive about this experiment, which is the concept of filtration. The whole idea of a filter is that it presents a solid barrier with very small holes, or perhaps passages is the better term, in it which allow particles smaller than the passage to go through, while blocking all larger particles. What’s suggestive is that the earth + water mixture, or mud, is composed of small particles of varying size, such that they can be separated by filtration. This whole idea, the particulate concept of matter, is of course not at all surprising to us because this is the twenty-first century and we all know about atoms, but what I want to emphasize is that the idea of atoms is not as obvious as it might seem. It only seems obvious to us because we have gone to school where we were taught the atomic theory of matter; but if we hadn’t been so taught, or indoctrinated is perhaps the better term, then like most people throughout history and even today we wouldn’t know about atoms at all and probably wouldn’t stumble upon this explanation of filtration and how it works. I won’t mention more about this idea here, the particulate nature of matter, because I am going to return to it in force fairly soon; but I hope you can see how it relates to the idea of elements and how they answer the riddle of matter. Our experiment with filtering earth + water mixtures gives us a small window of insight into this powerful idea.

 There is a great deal more to our filtration experiment and how it might be interpreted. For example, in addition to the process of separation, maybe what I am looking at is a result of a reaction between the two original elements, earth and water, when I mixed them together. The filtrate, as well the residue in the filter, may very well be the result of such a reaction. How can I determine the difference among the various possibilities?

 One way of going about this would be to weigh the original earth, the dried residue in the filter, and the dried filtrate in its flask, and add the various weights together. When we do this, and assuming that we have been very accurate and precise in our weighings, we discover that, as if magic ran the universe instead of blind physical laws – no, actually the reverse – the combined weights exactly equal the weight of the original earth we started out with. This is very revealing, for if there had been a reaction with the water, that would have increased the weight by the amount of water consumed, or perhaps decreased by some fraction perhaps. But this has not happened. To clinch the issue, if instead of allowing it to go free we have instead been diligently collecting all the evaporated water during our experiment and weigh it with the remaining liquid form, again we are chagrined – well, perhaps not too chagrined by now – to find that it too matches the mass of the original water.

 Even so, having done all these additional measurements turn out we can be certain that we have taken one of our original elements, earth, and broken it down it into at least two new substances, one that passed through the filter and one that does not. That being so, we can hardly call earth an element any longer! And yet, contemplate this fact: should this really surprise us? After all, we never did have good reason for saying earth was an element in the first place. We just assumed it because earth is so ubiquitous that it seemed reasonable to call it an element; we followed common sense and our intuitions instead of investigating nature closely and clearly and methodically, as science teaches us we must do. So perhaps, in retrospect, we shouldn’t be surprised at all.

 The next question is, how about the water? Unfortunately, this turns out to be a little trickier. It was certainly not separated into different substances by the filter, so at first sight we might be justified in calling it one of the elements we are searching for. In fact, water passes a lot of tests to determine elementhood, and so it is easy to conclude that it is an element. But there is a very well known experiment that will show elsewise: the electrolysis of water. This experiment is not as easy to set up as the filtration experiment, and we require some special materials and equipment. But it is still not that complicated. What is needed is two glass test tubes, filled with water and connected near their tops – their open ends – by a glass tube or corridor. At the very top of each tube is a water tight cork or rubber stopper, through which has been inserted an electrode of platinum or other suitably chemically inert, electrically conducting, metal (even graphite, a form of carbon that conducts electricity, can be used). The connected tubes are filled with water – this of course is done before the electrode containing stoppers have been inserted. It is critical, for reasons I won’t go into right now but are also suggestive, that the water be slightly salty, or into which some other substance which helps its electrical conductivity has been dissolved. The entire apparatus is then turned upside down. The wires now coming down from the electrodes / bottoms of the stoppers are then connected to a source of direct current electricity, usually a battery or a set of batteries wired in series, one that can provide sufficient current and voltage. The final apparatus looks like this:
 
 
 
 Here the “tops” (remember are actually the bottoms) of the test tubes also have tubular holes in them, and collection balloons have been placed, tightly, around the holes. When the wires from the electrodes are connected to the anode and cathode of the battery, something very interesting starts to happen at the surfaces of the electrodes, also shown in the drawing. Bubbles of gas start to form around them, bubbles which, when they have grown large enough to break their adherence to the electrode, rise up and collect in the balloons. This process continues as long as the water level is high enough to reach the electrodes and the connecting tube. At this point the gasses stop forming.

 At the end of the experiment, we weigh the collected gasses (again, not an easy thing to do), and the remaining water, and again we find that the summed weights equal the original weight of water placed in the apparatus. This is because we have taken our “element” water and broken it down into two new substances, which I will now admit are the gasses hydrogen and oxygen. Oh, and incidentally, if you mix the hydrogen and oxygen and burn them together, the product is … one guess … that’s right, water. Voila! Water is no more an element than earth.

 Air suffers the same fate. If we take a weighed volume of air and burn a weighed quantity of something – anything flammable, say paper – in it, we find that after the burning that the air has gained some weight while the burned material has not only changed appearance but is also now lighter, by exactly the same weight; that is, the air plus paper is the same weight after the burning as before. Something in the paper has been transferred to the air somehow.

 Not only thus, but if you liquefy air, again not an easy process, you find you can distill – that is, boil out fractions at different temperatures – from it a number of separate liquefied gasses, mostly nitrogen and oxygen, with a little argon and other gasses.

 So air is not an element either. As is neither water, or earth. As for fire, how can something that appears and then vanishes, into thin air one might say, be an element? It isn’t even clear that fire can be called a substance; or if so, it is certainly a very mysterious one.

 After all this discussion, we seem to have come full circle with the most fundamental question: just what, precisely, is an element, and how do we determine it?

 One part of our definition is that it is a substance that cannot be broken down into other substances by ordinary physical or chemical means. If you take a chunk of gold, for example, no matter how you heat it, combine it with other materials, chop it up, or otherwise afflict it, you cannot reduce it to anything simpler. You can make more complicated substances from it, like the various alloys and compounds of gold, but not something simpler. All of this is not obvious, of course. It requires a great deal of careful experimentation to show that it is true. But chemists have been working with gold long enough that they can call it an element with great confidence. Of course, that’s the way science usually works; a lot of time and people and material, and many, many experiments done over many years by people just to come to a firm conclusion. And even then we are not absolutely certain beyond any doubt, just adequately sure beyond any reasonable ones.

 I said that an element cannot be reduced to simpler substances by ordinary physical or chemical means. By that I meant we could heat it, freeze it, mix it with other substances (and then heat or freeze it) – all the things chemists do in their laboratories – and though you might yield materials with interesting properties, the gold or other elements it contains can still be extracted; the processes we put it through have not transformed it. Using other physical and/or chemical means we can restore the same gold, in its original condition.

 This leads me to another interesting subject, not just about gold but any physical substance: we can take a piece of it, divide that into two pieces, divide each of those pieces so that we have four, and divide again and again and again in this manner, each division yielding progressively smaller pieces of the substance. Actually, we are aware of course that we can only take this process so far; eventually we will reach a point where we cannot find a knife, or whatever we’re cutting the substance with, small enough to continue. But assuming you could, just how far can we go with this division and sub-division process? Could we go on forever? What exactly would happen?

 It is possible with modern scientific instruments to divide a piece of gold into many very tiny pieces. And lo and behold, each piece is still gold. But “very tiny” is a relative term. At some point, if we are somehow able to sub-divide it enough times, we may yet find gold to be composed of simpler things. Fortunately, there are ways of probing well beyond our method of divisions. But a brief discussion on the subject of radioactivity is necessary first.


Radioactivity and the Discovery of Sub-Atomic Particles

 By the end of the nineteenth century / beginning of the twentieth, a number of scientists had discovered an interesting property of certain kinds of substances. They appeared to be unstable at some very fundamental level, decomposing into other substances and emitting a variety of “rays” or radioactive emissions while doing so. The Curies, Marie and Pierre, are the most historically famous contributors in this field of work, although others were involved as well. Altogether, three main kinds of rays were initially discovered and labeled, using the first three letters in the Greek alphabet: alpha rays, beta rays, and gamma rays. Other kinds of rays were to be discovered later, but this is where the story begins. It was also not until later that the nature of these rays were determined: it turned out that alpha rays were actually particles, now known to be composed of two protons and two other particles, the latter of which we today we call neutrons; beta rays were also particles, in fact what we now know as electrons, albeit moving at high velocities from the radioactive atomic nuclei emitting them; and gamma rays were electromagnetic radiation, like light but of very high energy, even higher than X-rays, which can easily penetrate flesh and show the bones in our bodies.

 One of the interesting things about these rays, or particles, or both, are their penetrating powers. Alpha rays, although being the most massive of the three, have the least penetrating ability; a simple sheet of paper can stop (most of) them cold in their tracks. Beta rays / particles, are more penetrating and can get past your skin and well into the underlying flesh. Gamma rays are, as just noted, the most penetrating of all, even more so than X-rays. Alas, these penetrating power of the radioactive emissions and what they do to living tissues make them extremely hazardous to living organisms such as ourselves, a fact which tragically was not really recognized for several decades after their discoveries, resulting in many unnecessary terrible diseases and deaths due to the handling radioactive substances (Marie Curie, for one example, died of leukemia).

 Back to the beginning of the twentieth century. In 1911 the physicist Earnest Rutherford and his scientific team performed a remarkable experiment using alpha particles on a very thin sheet of gold, which can be beaten very thin. The sheet was so thin that they expected the alpha particles to pass through it with very few if any deflections, in much the same manner as a hard thrown baseball will go through tissue paper with virtually no resistance. I say “expected” with some reservation; if they were certain that this would happen they of course would never have bothered to do the experiment. In science you always start with some doubt or incomplete knowledge, and hope to be surprised, at least once in a while.

 Rutherford and his team were very much surprised by the results of their experiment. To their utter incredulity, although most of the particles did, as predicted, pass through the gold sheet without hindrance, a very small number of them were instead deflected; and not just deflected but by very large angles at that. It was as though, to paraphrase Rutherford’s description of the phenomenon at the time, a cannon ball had bounced off something in the gold sheet, to come straight back at the experimenter and strike him on the nose!

 Such behavior was very hard to explain unless one assumed that almost all of the mass of the gold was concentrated in a very large number of very tiny regions, regions spread throughout the sheet like raisons in a pudding. But if this were true then gold clearly is not infinitely divisible into ever smaller and smaller pieces. There is a smallest piece which may or may not be subdivisible into other things.

 My guess is that none of this really surprises you, because you live in the year 2010 and almost everyone has heard about atoms by now. What those few alpha particles were bouncing off of were the tiny but quite massive nuclei of the gold atoms, while the rest of them blasted through the extremely light electrons circling, or doing whatever electrons do, around the nuclei. In fact, Rutherford’s experiment is usually considered the proof of the basic structure of atoms. At the time it was groundbreaking work however, because only recently had the truth of the existence of atoms been established beyond a reasonable doubt by men like Einstein and J. J. Thompson (who discovered the electron), although John Dalton, a century earlier, is usually given credit for the modern version of the atom.

 All this returns us to answer the question of whether gold should be regarded as an element, in the modern chemical sense. As the protons and neutrons of the gold nucleus cannot be subdivided by an ordinary physical or chemical means, and gold is composed solely of gold atoms, the answer is a clear yes; gold is an element. But what I want to emphasize is that this answer is not at all obvious; it took many people many years and enormous amounts of work to establish this, what seems to us today so straightforward elementary school a fact that we take it for granted.

 At this point there are many more, although not necessarily easy, experiments we can do on, say, the hydrogen and oxygen generated by breaking down water, which shows these two gasses to be elements as well. We could also work on our various pieces of earth and show that they too are composed of simpler elements, such as silicon, aluminum, oxygen, iron, magnesium, and others. The nitrogen and oxygen and argon in air are also elements (though other minor gasses in it, such as water vapor and carbon dioxide, or CO2, are not). As for fire, it too is a mixture of elements, or compounds of elements, all undergoing a number of chemical reactions with each other and with oxygen in the air at high temperature, reactions which gives fire its various colors.


The Modern Conception of Elements

 I hope you are asking the next logical question in this lecture. Gold is an element; nitrogen and oxygen and hydrogen are elements; silicon and aluminum are elements; and so on. In fact, there are some ninety elements in nature, and about two dozen manmade ones as of this writing. The question is, what makes them all different from each other? And, more to the point of this book, are their differences and similarities organized in any way?

 To answer these questions, we must look at the nuclei of the atoms which compose each element, remembering that in doing so we are jumping over the enormous amount of scientific work that had to be done to establish, not only the very existence of atoms, but also the fact that they have nuclei. In doing so, we find that each element is characterized, no, defined, by the specific number of protons – relatively massive, positively charged particles – in the nucleus. Hydrogen has one proton, helium two, oxygen eight, iron twenty-six, and so on. This is called the element’s atomic number. To maintain electrical neutrality, an equal number of electrons surround the nucleus: one electron for hydrogen, twenty-six for iron, ninety-two for uranium, the most massive naturally occurring element, and also so on. As mentioned, a second kind of particle also resides in the nucleus, approximately the same mass as the proton but electrically neutral: the neutron, discovered by James Chadwick in 1932 and earning him the 1935 Nobel prize in physics. The total number of protons and neutrons in the nucleus is what is called the atomic mass of the element.

 I mentioned John Dalton a few moments ago as the author of the modern concept of the atom in the early 1800s. Yet what we see is that, despite Dalton’s elegant reasoning for his atomic theory, it took an entire century for scientists and philosophers to fully accept atoms as real things, not merely some bookkeeper’s way of keeping track of quantities in chemical reactions. Part of the reason for this lack of acceptance is that the scientific instrumentation capable of probing matter at the atomic level didn’t exist in Dalton’s time. Another part is that the concept of atoms didn’t fit neatly with either the edifice of Newtonian physics or the laws of thermodynamics as they unfolded in the 1700s and 1800s.

 Yet if atoms and the atomic theory of elements (an element is characterized by one and one only type of atom) had to wait until the twentieth century to be fully accepted, the modern concept of the chemical element was the offspring of work done in the late 1700s / early 1800s, by men like Lavoisier and LaPlace and Scheele and Priestly, among others. Hundreds of years of (failed) experiments in alchemy plus the Enlightenment and Scientific Revolution had driven home the idea that there were certain substances which simply could not be broken down into simpler ones, or turned into other ones, by any known chemical or physical processes. Thus, the main dream of alchemy – to turn “base” metals into gold – was finally seen as a delusion, even if the greatest minds of the day still did not know why. Yet some substances that had been thought of as elements, water being the prime example used in this chapter, were shown to be chemically reducible to simpler substances, in this case hydrogen and oxygen, which in turn proved to be elemental in character. The element fire, as already mentioned, which had been thought of as the release of a mysterious substance called phlogiston, was shown in fact to be the chemical breakdown or reassembly of a variety of substances, followed by their reaction with atmospheric oxygen in the vapor phase. And so on, with most of the original substances believed to be elements.

 Throughout the nineteenth century, as scientific instruments and theory became better and better honed, many new elements came to be added to the list, while some substances, like carbon and sulfur and iron and copper, which had been known since antiquity, also found their way in. The net result of all this innovation and exploration was that by the latter half of the nineteenth century a veritable zoo of elements had been identified and characterized. So large was this zoo, in fact, that scientists began to wonder if there were an underlying order to them, some schemata which naturally organized them according to their properties, both chemical and physical.


The Periodic Table
  Enter the brilliant Russian chemist Dmitri Ivanovich Mendeleev. Although others before him had noticed periodical trends in the elements, and even attempted to create tables of them, in which each column represented a series of similar elements, it wasn’t until 1869 that Mendeleev, via his own independent work, presented a table both complete and sophisticated enough that it was accepted by the scientific community. What was probably the most powerful feature of Mendeleev’s table, and what set it apart from others, was that it provided a means of testing it. It did this by predicting the existence of new, hitherto undiscovered elements to fill gaps in it. Specifically, he predicted the existence of what he called ekaaluminium and ekasilicon, amongst several others, and the properties these elements would have. When the elements gallium (Ga) and germanium (Ge) were found in 1875 and 1886, with properties that almost perfectly matched those predicted for ekaaluminum and ekasilicon, Mendeleev’s periodic table and his fame were secured. There were still more gaps to be filled, but over the next half century or so scientists teased them out from minerals in Earth’s crust (or in the case of helium, discovered it via spectroscopic lines in the sun’s atmosphere), to the point where today the aptly named periodic table of elements is now complete:

As noted, there are ninety naturally occurring elements, the rest having been man-made through nuclear transmutation of existing elements. Some terminology is in order. The table is called periodic because each row is a period, one that begins at an “alkali” metal (Li, Na, K, etc.) and ends at a “noble” gas (He, Ne, Ar, Kr, etc.). Incidentally, hydrogen (H), while sitting atop the alkali metals, doesn’t fit neatly anywhere, for reasons we shall come to. Complementary to this designation, each column is dubbed a group. In modern terminology there are eighteen groups, numbered in order from left to right; thus, the greatest length a period can be is eighteen members.

 So: we have made a little headway into understanding the elements, and their relationships to each other. Just a little, however; I still need to explain what these groups and periods actually mean, in both the physical and chemical senses. What exactly was Mendeleev’s brilliance, that has made him one of the most important scientists in history?

 Go back and study the modern periodic table as just presented. In particular, single out groups 1 and 2 (known as the alkali metals and alkaline earths) as well as groups 17 and 18 (the halogens and the noble gasses). Remember to exclude hydrogen, as it doesn’t neatly fit into any group. If you specifically examine group 1, the alkali metals, the similarity in their properties as you go up and down the group is remarkable: not only are they all highly metallic, they are also soft and malleable (becoming more so as you go down the group), react strongly with oxygen (O2) and water (H2 O) to form highly basic oxides and hydroxides in which the ratio of metal to oxide (O2-) and hydroxide (OH-) is exactly the same, react with other elements and compounds in very similar ways as well, and so on. The same can be said for the other groups I mentioned, the alkali earths, the halogens, and the noble gasses; as you go up and down the group/column, the physical and chemical properties bear a strong resemblance.

 These resemblances are the rational basis – no, the heart and soul – of the periodic table’s structure. Of equal if not greater importance is the way that the groups repeat themselves to form the rows, or periods; notice that although the groups are numbered 1 to 18, only the fourth through sixth periods actually have eighteen members (period seven would, and will, have them once we synthesize all of its elements; they are too radioactive to exist in nature). Period one has only two members, hydrogen and helium, while periods two and three have eight. If you look at periods six and seven, you will notice a break after the first two groups, filled in by the detached “sub”-periods beneath them known as the lanthanides and actinides, each of which having fourteen members. Believe it or not, if the number of elements were extended far enough, by artificial transmutations as they don’t exist in nature, the number and types of these sub-periods would continue to grow (as would their lengths – the next one would hold eighteen members). Indeed, theoretically there is no end to the table and how far it can be built; it goes on indefinitely. We should thank nature that there are only ninety naturally occurring and (as of this writing) around twenty man-made elements!

 It should go without saying that there is a good reason, founded in chemistry and physics, why the periodic table is built up this way, that it is not merely the way it is in order to baffle and befuddle poor students of chemistry. There is, and we shall get to it, but first we should note some other interesting aspects about the table. The one that should be staring you in the face is that, to demonstrate the family resemblances in groups/columns, I specifically singled out only the two left-most and two right-most ones. You might wonder why I was so persnickety about my choices, and you would be right to do so.

 The reason is that only in groups 1, 2, 17, and 18 do the resemblances of group members remain strong as you go all the way up and down the group. For the middle groups, 3 through 16, the top two series (He and Li through Ne) show distinct differences from their heavier brethren beneath. Specifically, boron, carbon, nitrogen, and oxygen, or B, C, N, and O, appear quite set apart in their properties than the elements beneath them, Al, Si, P, and S, or aluminum, silicon, phosphorus, and sulfur. As one example, carbon dioxide (CO2), which makes soda water fizzy and is a waste material we dispose of every time we exhale (as well as the main culprit behind global warming), is a colorless, essentially odorless gas at ordinary temperatures and pressures, while silicon dioxide (SiO2) is a hard, crystalline, more or less transparent solid under the same conditions. Likewise, water (H2O) is an almost colorless (it is actually slightly blue, as the color of the oceans attest), odorless, and tasteless liquid with a number of important and remarkable properties – life on this planet would not exist without copious amounts of it, in both liquid and gaseous form – while its sulfur analog, hydrogen sulfide (H2S), is a foul-smelling, highly toxic gas, as are H2Se and H2Te.

 Why the first two periods should display such differences from the periods beneath them is another topic we shall come to soon enough. First, however, let’s return to atoms.


The Idea of the Atom

 When Mendeleev created his first periodic table in 1869, atoms were not widely believed to exist, at least not as real physical entities, that is. Moreover, scientists had yet to discover the components of atoms, of which we are so familiar today: protons, neutrons, and electrons. Given this ignorance, based on what feature, or features, of the elements did Mendeleev and others base their tables from?

 If I were to answer that the feature were their atomic masses, your first response should be to object that that number is also derived from an atomic view of nature: it is, just as I said earlier, simply the combined mass of the protons and neutrons and binding energy (this is what holds them together) that characterize each element, averaged out over the percentages of each the element’s isotopes (different isotopes of an element have different numbers of neutrons in their nuclei).

 However, even though I told you this, it is not exactly true. There is another definition of atomic mass, one that doesn’t require any mention of sub-atomic particles. This definition is that it is the mass, in grams, of an Avagadro’s number of atoms – or elemental particles, if we do not know about atoms – of the element in question. Avagadro’s number is slap in the face enormous, being approximately 6.022 × 1023, although nobody knows its exact value. Note that it is just a number, or constant; one can have an Avagadro’s number of anything, from atoms to sand grains to basketballs to Ford model T’s to galaxies – anything you like. To give you a rough idea of just how large a number it is, if we are talking about sand grains, then by my estimate it is on the order of a hundred billion to a trillion beaches worth of sand or so – far, far more than all the grains on all the beaches and deserts on our Earth. Yet, large as it is, it is a very convenient number for dealing with things as small as atoms; an Avagadro’s number of atoms of any element is a quite manageable quantity of it, weighing from grams to hundreds of grams, depending on the element we are dealing with.

 One thing, however, that is a problem with talking about an Avagadro’s number of something is that it is a long, fumbling mouthful of syllables which would leave us needing a glass of water everytime we invoked it. Fortunately, chemists have come up with a short hand way of saying it, which is the word mole. A mole of something is simply an Avagadro’s number of the something, and again we can talk about a mole of atoms or sand grains or anything else. Whatever it is, I’m sure you’ll agree is a lot easier on the tongue. More to the point, using this much easier word the definition of atomic mass of an element is simply the mass, in grams, of a mole’s amount of it. In the case of the element carbon this is 12.011 grams/mole, and of the element gold is 196.97 grams/mole.

 Mendeleev did not know about the reality of atoms or anything about their sub-atomic components, and so his initial periodic table could only use atomic mass as a guide to where to place the various elements – a fact that made his construction of a workable table that much more difficult and his success in doing so that much more remarkable. Today we not only know about the reality of atoms but also all of their constituents, down to electrons, protons, and neutrons, the latter two of which can be furthered sub-divided into various quarks, as well as the various force particles which hold them together. This is important, because the true, modern, correct version of the table uses atomic numbers, the number of protons in the atomic nucleus (and the number of electrons swirling about that nucleus if it is an electrically neutral atom). None of this should be surprising, by the way; for as I keep emphasizing and re-emphasizing, science rarely if ever proceeds from zero knowledge to 100% understanding in one all-encompassing leap but largely from simpler, cruder models of reality to gradually more sophisticated, complete ones. The fact that we can make progress this way is one of the most fascinating features of science, not to mention one of the most curious features of reality; there is no reason, a priori, that we know of why this should be so. Why shouldn’t it be that to understand anything, you must understand everything first? Why should we be so fortunate that this is so? Feel free to speculate on that little philosophical conundrum.

 But first finish reading this book. As I noted, the modern periodic table is divided into columns or groups of similar elements, each repeating themselves in ever increasing sizes of periods; excepting that, as I have said, the first two periods are really not all that similar to those beneath them. The first period has only two members, hydrogen and helium, the second and third periods have eight members, the fourth and fifth eighteen members, the sixth and seventh thirty-two members (if you add in the lanthanides and actinides, that is), and so on.

 I can’t resist talking about this in more detail, as it fascinated me as a child who didn’t understand the reasons why nature is organized this way. It turns out that it takes a while to tease out the pattern to the increases, but it works out to be: (first period) = two protons/electrons; (second / third) = eight; (fourth / fifth) = eighteen; (six / seventh) = thirty-two. Putting it in tabular form, these increases go as the following:

2 = 2
2 + 6 = 8
2 + 6 + 10 = 18
2 + 6 + 10 + 14 = 32
2 + 6 + 10 + 14 + 18 = 50

 The pattern is, I hope, clear: each new row in this table adds an additional column, which is equal to the previous row’s last column entry plus four. A rather strange pattern, one must admit; but we must also be grateful for it, for all patterns in nature are the evidence of underlying structures or principles, and so are keys to understanding those structures/principles. The patterns in the periodic table are no different in this regard, as we shall come to see.

 The title of this book, The Third Row, refers to the third period in the periodic table, which has a total of eight elements. As I have already mentioned, without explanation, that the first two periods possess substantially different properties from those beneath them, not to mention the first significantly different from the second, this period is the first in which the strong similarities up and down the group become more apparent for all of the groups. For example, once again, hydrogen sulfide (H2S), hydrogen selenide (H2Se), and hydrogen telluride (H2Te), are much more alike to each other than they are to hydrogen oxide, or water (H2O).

 I think a natural question which arises here is: just why are there so many elements – or, to be more precise, atomic nuclei – in nature, and how did they come to exist?

 Where do the Elements come from? Why are There so Many of Them?

 To answer this question, we must segue from chemistry to, first, nuclear physics, and then to astrophysics and cosmology. The first segue, nuclear physics, is necessary because the elements, or again more specifically their atomic nuclei, are created by the joining together, or fusing, of smaller nuclei. To use the most common example of this, four hydrogen nuclei or protons (1H, where the 1 superscript indicates the total number of protons and neutrons in the nucleus, one proton in the case of hydrogen) are fused together, in one of a number of pathways, to make a helium four nucleus, or 4He, containing two protons and two neutrons. The overall reaction can be written, with some simplification, as:

 1H + 1H + 1H + 1H = 4He + 2e+ + 2νe

 The last two particles in this reaction, e+ and νe, are called the positron, or anti-electron, and the so-called electron anti-neutrino (neutrinos are particles with very small mass and no charge, which travel very close to the speed of light). Their emission is needed to turn two of the 1H nuclei, which are protons, into the two neutrons in the 4He nucleus. Another example of a fusion reaction is the so-called “triple alpha” process, in which three 4He nuclei, which are also the alpha particles mentioned earlier, are fused together to make one 12C nucleus:

 4He + 4He + 4He = 12C

 These and many other fusion reactions are employed by nature to build up the complement of chemical elements she has so generously provided to us. However, even the simplest of these reactions, hydrogen to helium, can happen only under very specific, and hence uncommon, conditions. To answer why this is so, stand back and take a better look at what we are doing. Remember how in school you learned that unlike electric charges attract each other while like charges repel? Well, atomic nuclei are composed of protons and neutrons, and while the neutrons are electrically neutral the protons carry a very strong positive electric charge and so should, and in fact do, repel each other, even in atomic nuclei. So what then even holds them together in the nucleus, let alone allowing them to fuse together into even larger nuclei? Why don’t atomic nuclei go around exploding like miniature firecrackers as a result of these mutual like charges in their nuclei, leaving us with an atom free universe?

 This turns out to be a very good question, and one again that took many years to answer, by scientists incessantly scratching their heads and trying innumerable experiments. You might attempt, as a first approach to solving this conundrum, to speculate that there might be other forces in nature which provide us with the solution to it. What about gravity, for example? We know a good deal about gravity; for example, that it causes all mass objects, regardless of their electric charge or any other factor, to be attracted to each other, via the relationship:

 F = G(m1m2)/(r*r)

In this equation, F is the gravitational force, m1m2 the product of the objects’ masses, r2 the square of the distance between the objects (and so the factor which shows how quickly the force between the objects diminishes with distance), and G the proportionality constant in the equation, being equal to 6.673×10−11 N m2 kg−2 if you are interested. Gravity would, indeed, seem to be a good candidate for holding atomic nuclei together; after all, it is what holds our Earth, not to mention the sun and all the other planets and most of their moons, together, keeps us secure on the surface of our planet instead of being hurled out into space from the centrifugal force its spin generates, keeps the moon revolving about Earth, and Earth and all the other planets in our solar system in their orbits about the sun. In fact, we are much more aware of gravity than of the electric force, and so can be excused for thinking it to be the stronger of the two, and by a considerable ratio.

 Not only could we be excused for thinking this way, we would have to be excused, because reality is in the opposite direction, and by a very large factor at that. In truth, the electromagnetic force of attraction or repulsion is approximately one thousand trillion trillion trillion (1039) stronger than gravity! The equation of this force is:

F = ke(q1q2)/(r*r)

 where now, instead of m1m2 we have q1q2, the product of the electric charges on the objects (whether attractive or repulsive), and ke as the proportionality constant. As with gravity, we also see that the force diminishes as the square of the distance between the objects.

 The fact that the electromagnetic force can be either attractive or repulsive, whereas gravity is always attractive, is the cause of our error in thinking gravity the stronger of the two. When matter is accumulated on the scales we are accustomed to, and larger, there are almost always as many negative as positive charges, and the net effect of this equality is to mutually cancel these charges out so that at most only a very, very small excess exists in either direction, if indeed there is any excess at all. As for gravity, however, this force is always cumulative, so that massive objects can build up an appreciable attractive charge, the larger the accumulation resulting in the greater the charge. Build up enough mass in a small enough volume in fact, and you will have yourself a something called a black hole, which is an object whose gravity is so intense that not even light can escape its clutches.

 So much for gravity, then; it has no chance of solving our dilemma. What then does hold the protons together in the nucleus and, more to our point, allows them to be fused together into ever increasingly larger nuclei? The answer, as Hamlet says to Horatio, involves realizing that “There are more things in heaven and earth than are dreamt of in your philosophy.” Just because gravity and electricity (more correctly, electromagnetism) are the only fundamental forces in the universe that we are directly aware of, thanks to their squared distance attenuation, so there are other forces we are rarely cognizant of solely because they diminish over much shorter distances. Physicists call these forces nuclear forces precisely because they drop to virtually zero over distances of even a small amount greater than atomic nuclei. There are two such forces, named, perhaps unimaginatively, the strong nuclear force and the weak nuclear force. The weak nuclear force comes into play in certain kinds of nuclear decay and will not be discussed further here. The strong nuclear force is what catches our interest because it is both attractive only (but only to protons and neutrons and other particles collectively known as hadrons) and is some one hundred times as strong as the electromagnetic force. Again, the reason we almost never directly encounter it is its extremely short range, approximately that of several protons and/or neutrons or the atomic nucleus at most; beyond that distance, it rapidly diminishes to essentially nothing.


How Does Nature Build Elements Beyond Hydrogen?

 Let us return to the simplest fusion reaction, that of hydrogen to helium:

1H + 1H + 1H + 1H = 4He + 2e+ + 2νe

 We can now see that what holds the two protons and two neutrons in the resulting helium nucleus must be the strong nuclear force. This is how nature creates, not just in helium but in all of the elements larger than hydrogen, by fusing together smaller nuclei. This present us with a problem, however. The four hydrogen nuclei, or protons, start out at a distance from one another much larger than the strong force’s range, while at the same time they are close enough to feel the electromagnetic force keeping them apart, a force which still immensely powerful. Somehow, some way, we must push the protons closer and closer together until they start feeling the strong force more strongly than their mutual electric repulsion and so stick together to form a two plus particle nucleus. (What happens after this in the creation of helium and other small nuclei, if you are interested, is that the weak nuclear force causes one or more protons to decay into neutrons, in the process emitting positrons and anti-neutrinos as we saw in the 41H → 4He reaction.)

 There are only two ways of forcing the protons close enough to overcome their repulsion, and that is either by pressing them together under extremely high density and/or raising by their temperature very high, generally in the millions of degrees, so that they will be moving fast enough to overcome their repulsion and fuse. In practice, one usually has to do both. In nature, there are only two places/times that these conditions exist: one is in the first few minutes of the Big Bang, the primordial beginning to our universe, while the other is in the extremely hot, dense cores of stars like our sun, both active today and in the past. The reason these conditions existed during the Big Bang is that it was the way our universe began, as either a singularity (single point in space-time) or a volume very close to it, so that the density and temperature must have passed through a time near its beginning when such fantastic conditions existed. The reason they exist now in the cores of stars is due to the massive gravitational compression and heating existing there; the so-called “proton – proton nucleosynthesis” fusion reaction to helium is in fact the primary source of most stars’ prodigious energy outputs, including our own sun. However, although there have been many quadrillions of stars in our universe carrying out this reaction since stars first started to form some thirteen billion years ago, most of the helium in the cosmos today is in fact the result of Big Bang nucleosynthesis – this is actually one of the facts that have been used to prove the Big Bang theory correct. The Big Bang is also responsible for most of the trace amounts of lithium, beryllium, and, I believe, boron, atomic numbers three through five, in the present cosmos.


Creation of Elements Beyond Helium

What about the other elements, including the ones in the third period we will be discussing? I’ve already shown one fusion reaction, the triple-alpha process, that yields carbon. This reaction requires much higher densities/temperatures than the proton-proton reaction however, and it should be by now pretty obvious why. Helium nuclei contain twice the number of protons as hydrogen, and so the electromagnetic repulsion between them is proportionately higher, at least twice as high – while the fact that the strong force is also one hundred times as strong does not help us here because of its very short range. Another reason is that now we are trying to fuse three nuclei into one, meaning you have to start by fusing two and hoping this nuclei will last long enough to be struck by the third 4He. This intermediate nucleus 8Be, however, is extraordinarily unstable, fissioning or breaking back down into two 4He in a fraction of a trillionth of a second.

 The triple-alpha process couldn’t happen during Big Bang nucleosynthesis because by the time enough helium had been created to do so, the temperature and density of the expanding universe had dropped to below what is needed for this reaction. The only place it can still happen, and still does happen, like the proton-proton process, is in the cores of stars; not just any stars, however, but only those significantly more massive than our sun. The reason for this is straightforward: hydrogen fusion in stars creates a helium “ash” which, as it is both heaver than hydrogen and has no energy source itself, collects in the center of the star. This core of helium grows throughout the star’s lifetime, in doing so raising the temperature of the core by its unchecked gravitational compression. As the core’s temperature thereby rises, the hydrogen fusion surrounding it becomes more intense; this leads to more helium accumulation, leading to still higher temperatures from gravitational compression in their cores, higher rates of proton-proton fusion, and so on, in a positive feedback mechanism that causes even sunlike stars like the sun to grow steadily hotter and brighter throughout this part of their evolution. The end result of this positive feedback loop is the creation of a “red giant” phase, in which stars like our sun become hundreds of times brighter than during their “Main Sequence” phase, their outer regions expanding to some hundred times their current diameters or more, while their surface temperatures cause a drop in color from yellow/white to red as these expanded atmospheres cools.

 For a sun-like star, that is pretty much it (and it’s about five billion years in our future for our sun, so don’t worry about it). The greatly increased radiation pressure from the red giant’s core eventually blows away most of its atmosphere and other outer regions into interstellar space. In doing so, the remnant central region, exhausted now of hydrogen fuel to fuse, shrinks until it is approximately the size of Earth or smaller. It’s surface is still white hot from the core, hence the name “white dwarf”, but it gradually cools over billions of years back down to red and then infrared invisibility. Finally it is cold as space itself.

 This is the fate of most stars, but not, as I have said, those significantly more massive than the sun. In massive stars the hydrogen fusion is much more profligate, as it must be to generate the enormous radiation pressure needed to hold the star up against gravitational collapse. This means the core temperature is much higher than our sun’s, and by the star’s red giant phase will be in the hundreds of millions to billions of degrees (instead of a “modest” fifteen million degrees C in the sun). At these temperatures the triple alpha process can and does occur. This “helium flash” in the core will consume most of its helium (and quite quickly), converting it not only into carbon but also turning some of that carbon further into oxygen, neon, and magnesium as additional 4He nuclei are fused in. Other elements are also created by the fusion of protons, neutrons, and other small nuclei.

 All these processes, in the most massive of stars, can continue to build heavier nuclei all the way up to iron and nickel. However, to create nuclei larger than iron and nickel requires an input of energy rather than its release, as the most stable nuclei (those having the smallest binding energies) end with these metals; all the heavier elements, up to uranium and beyond, are created mainly by neutron capture, a process that absorbs energy rather than releasing it (thereby hastening the end of the star’s life). Using this process, however, even the most massive stars can create nuclei only up to a certain number of protons, to between roughly ninety and one hundred. The reason for this is that most of these larger nuclei (some of the thorium and uranium nuclei are exceptions) are intensely radioactive and decay to smaller ones in short periods of times.

 All this of course only says how the elements are created; it does not tell us how they then found their way into other stars and their planets, including Earth, the compositions of which they primarily constitute. What completes the tale is also told by the most massive stars; for not only do they build these heavy elements, they then blast them into interstellar space via the supernova explosions which ends their brief lives, leaving them as either neutron stars or black holes. Newer generation stars and their planetary entourages then sweep these elements up during their formation. This also explains why the lighter elements, essentially the first two rows of the periodic table and to a lesser extent the third, make up the bulk of the matter in our universe, for, as we have seen, as you progress to heavier elements they become increasingly challenging to create.


Summary

So. We have explained what the chemical elements are, as well as how they are organized, how they were created, and why there are as many of them as there are. Excuse me, I should say how their atomic nuclei were created; talking about the elements themselves means talking about their constituent atoms, which includes their electrons, how the electrons are organized about the nucleus, and how they behave. This finally is the subject of chemistry, and we will make our first inroads into it in the next chapter.

 

Wednesday, January 11, 2012

Chapter Five -- All Molecules Great and Small

All Molecules Great and Small
I used the word bond(s) quite frequently in the last chapter, and by now you should have some grasp of what they are:  when two atoms approach closely enough, the most stable state of their outermost, or valence, electrons usually centers between the atoms.  Whenever two or 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 the stable water molecule, of H2O.  We were also introduced to the idea that different atoms can form different numbers (and types) of bonds:  one for hydrogen, four for carbon, three for nitrogen, two for oxygen, and so forth.  (The four bonds carbon can form will prove most convenient in coming chapters.)
There are numerous ways of portraying bonds.  In first year college chemistry courses the student is usually introduced to the Lewis dot structure.  Whereas before we represented bonds as bars between atoms, e.g., H-H, a better representation is to show the electrons themselves and where they are located in space:  H˸H.  Here it is clear that the two hydrogen atoms share their electrons in a region focused between the two nuclei.  You can see the stability this creates, and so why the bond is stable.
You’re already suspicious though, I’ll bet.  The Lewis dot structure still commits a cardinal sin when portraying electrons; for they are not hard little dots or spheres but spread out regions of uncertainty.  Remember how in the quantum world, we cannot say that objects cannot have exact locations, speeds, and directions; that if we do devise an experiment to measure one quantity exactly, we lose all information about the others.  Probability distributions rule the day.  Thus, an improvement on the dot structure representation would look something like this:
Figure XIV.
Here, the black spheres represent the hydrogen nuclei (usually just a proton), and the smeared out red region the electrons in the bonded state.  Note that we don’t even distinguish between one 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 how electrons behave in atoms and molecules.  To answer your next question, the dotted line surrounding the molecules encompasses ninety percent of the electrons’ distribution – in fact, this goes off to infinity.
*          *          *
What’s coming is, I think, rather grimmer than the material covered so far, but is so essential to describing atoms and molecules and bonds that I see no way of avoiding it.  Some advanced equations will be presented, but don’t worry, because there’s no need to understand them or know how to solve them.  I display them for you so that you will see there is a mathematical underpinning to all this talk of electron states and bonds.  Truth be told of course, there is a mathematical underpinning to all physics, so this should not come as a surprise.
But first, a switch in terminology.  Instead of states we will use the word orbital (both atomic and molecular) to describe an electron’s characteristics.  Orbital is, in my opinion, an unfortunate choice of word, as it implies electrons orbiting the nucleus, a fallacy I have struggled to help you overcome.  But it has been used so long now that we are apparently stuck with it; just remember that it is a synonym for state, not orbit, and you should be fine.  With this in mind, let’s plunge into the fray:

Figure XV.
This is the Schrödinger equation for the hydrogen atom.  You may recognize or know some of the symbols it 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 describes wave motion, such as that of a piano string vibrating back and forth.  The general form of this function, the solution to the equation, is:
Figure XVI.
Again, don’t worry about most of the details; deriving this solution requires some heavy duty math.  There are certain variables you should pay heed to, however.  These are n, , and m, known respectively as the principle quantum number, the angular momentum quantum number, and the magnetic moment quantum number.  The combined values of these numbers determines the exact form of y.  There are relationships among them as well:  while n can have any integer value from one on up (1, 2, 3, 4, 5, … ¥, can range only from 0 to n – 1, and m from – to +.  So for example, if n = 1 (the lowest energy states of the atom), then l and m can only be 0; while for n = 2 can have the values 0 and 1, while m be one of three values, -1, 0, and 1.

The orbitals I mentioned earlier are determined by their quantum numbers.  The number specify their shapes, the probability distributions of the electrons occupying them (a maximum of two, known as the Pauli Principle), and the energies of those electrons:
Figure VII. (repeated)
The elements in the periodic table are built up by adding additional protons (and neutrons) to the nucleus, and then filling the atomic orbitals from inwards to outwards with the required electrons to maintain the atoms electrical neutrality (atoms that are not neutral are called ions, but they are still atoms).  For the working chemist the electrons are described by orbital designations; say, for selenium, or element 34, the designation would be 1s22s22p63s23p64s23d104p4, where 1, 2, 3, and 4 represent the shell or primary quantum number, s, p, and d the numbers in the shell, and the exponent the number of electrons occupying each orbital group in a shell (thus, there are 3 p orbitals, 5 d orbitals,  1 s orbital, and so on).  I won’t describe this scenario in any greater detail because:  a) it still befuddles me some 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 basic grasp of the atomic kind is needed to do so.
*          *          *
There is one more revelation about atoms that must be made before turning to molecules.  In truth, all the atoms that have been built up in the periodic table should be analyzed by constructing and solving a Schrödinger-type equation for each one, yet we have only done so for hydrogen and assumed the results applied to all atoms.  Amazingly, and fortunately, they almost do!  In fact, we could set up an equation for each elements; could set up, but never solve by any known mathematical technique known to Homo sapiens.  The reason?  In hydrogen we are dealing with only two interacting bodies, but for every other atom and molecule, we are working with more than two.  The so-called three-body equation is intractable to all math and physics; not just in the quantum world but everywhere, at all times, in the solar system and the universe as a whole – though whether it explains why there are only two sexes I have no idea:  nature seems to have no difficulty with it, but then nature is just doing what it does according to its laws; it (She?) isn’t trying to calculate anything in advance.
Returning to the equation of the hydrogen atom we are lucky indeed that its solution gives us a sound working model for the other atoms as well; only minor modification 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 approximately equal energy and obey the mathematical requirement for “orthogonally” can be combined to make new functions/solutions.  For example, on carbon the single 2s and three 2p orbitals can be combined to create four new “sp3“ orbitals, which stick out at tetrahedral angles from the carbon nucleus:
Figure XVII.

I suspect 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 Schrödinger equation?  The answer is simply that the original orbitals are at lowest energy when considering lone carbon atoms; only when chemically bonded to other atoms – including other carbon atoms – is this configuration preferable, for reasons we shall see soon enough.
This is not the only possible combination.  Another “hybrid” (as they are know) orbital is the sp2 orbital, in which one of the p orbitals – technically the pz) is left out:
Figure XVIII.
Notice that the omission one a p orbital has changed the geometry of the atom from tetrahedral to a flat triangle; the new hybrid orbitals comprising the triangle is called, reasonably enough, sp2.  Other hybrids or combinations produce a wide variety of configurations, especially when they involve d or higher orbitals.
*          *          *
I made the blithe statement that all of this has to do with bonding with other atoms to produce molecules, and it is time to explain why.
Atoms takes up considerable space in molecules, and therefor can not simply assume any arrangement.  A good example of this is the compound methane, or CH4.  We should like to spread the hydrogen atoms and their electrons as far apart as necessary, in order to reduce interelectron repulsions.  As a first guess we could try a flat structure:
H
H ¾ C ¾ H
H
Figure XIX.

That this is inadequate should not be difficult to see.  We have here a two dimensional representation of methane; and although it is often convenient to draw it out this way, the lack of a third dimension means that the hydrogens are not spread as far apart as possible.
Go back to the tetrahedral arrangement of electrons around carbon, and you can see that the carbon’s electrons and the hydrogen’s electrons finally put as much space between each other as possible, to produce the configuration with lowest energy:
Figure XX.
Comparing this with Figure XVII. We see that the tetrahedral arrangements of hydrogens about the central carbon exactly matches its 3p3 bond arrangement.  This is why these are the preferred orbital pattern in the molecule.  The only thing left to do in our preliminary exploration is to describe just what a bond is.  I’ve used an equivalent phrase molecular orbital, but with out explanation what that means.
The fact that I’ve used bond and molecular orbital interchangeability should be suggestive however, implying that bonds are orbitals too, and this is quite right.  Just as atoms can combine their base orbitals into hybrid orbitals, so can orbitals on different atoms also merge into molecular orbitals, given the right conditions of fairly equal energy and orthogonality.  Remember the depiction of the hydrogen molecule earlier in the chapter:
Figure XIV. repeated
Of course, we really need to solve the Schrodinger equation for the system, but we can’t so we must find another way.  Again, that is the linear combination of orbitals technique.  The 1s orbitals on each hydrogen atom add together to produce the molecular orbital or bond, within which the two electrons have even lower energy than single atoms.  Bear in mind, though, that the number of hybrid or molecular orbitals must equal the number of base orbitals we began with.  Thus there are four sp3 orbitals about carbon (and nitrogen and oxygen, and many of atoms).  So combining the 1s orbitals on the two hydrogens must yield two new orbitals.  And indeed they do; in addition to the bonding orbital, there is also an “anti-bonding” orbital, an orbital which has the effect of repelling the hydrogens if there are electrons in them as much as the occupied bonding orbital which attracts the hydrogen:
Figure XXI.
Notice that the main electron density is on the outside of the atoms instead of between them in Figure XIV.  Some things ought to be starting to gel by now.  Normally, hydrogen exists as a diatomic molecule, H2.  This is because the two electrons from the atoms can fill the bonding orbital, but the anti-bonding are empty.  On the other hand, the next element in the periodic 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 the heliums apart and breaking the bond.  Hence, only He atoms are found in nature.
*          *          *
There is 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 (sp3 and sp2), and molecular orbitals resulting from combining base atomic orbitals.  Look back on the picture of the methane molecule (Figure XX).  Can you see how the sp3 orbitals on carbon are combining with the 1s on 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.  The possibilities are mind-boggling, perhaps even infinite, especially as you add more and more atoms to the potential molecules.  Many millions of molecules are already known to science.  Obviously, I can only cover a very tiny fraction of them, but fortunately that will do to get the idea in your mind clearly.
Let’s look at methane again, and try an experiment in our minds.  What if we were to take a methane and pop off one of the hydrogens?  Why then, we would have something the looks like this:
H
                H ¾ C ·
H
Figure XXII.

Forgive the flat structure, which we know not to be true, but there is method in madness here.  The first thing the hits your eye is that the right hydrogen has been replaced by a dot, which represents an unbonded electron.  In removing the hydrogen with its lone electron, we leave carbon’s contribution to that bond hanging out there in space, so to speak.  More importantly, it is now available for bonding with another atom or molecule with a lone electron. For example, the molecule here is actually called a methyl radical, radical meaning containing unbonded electrons.  We can certainly combine 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 much easier to draw this new molecule, called ethane.  The bonds about each carbon are still tetrahedral, however.
Incidentally, bonds of this nature, with head-on overlap of the constituent orbitals are known as sigma (σ) bonds.  The way this chapter has progressed, you can be excused for thinking it is the only kind of bond.  There are several more, and right now I’d like to concentrate on a specific one:  the pi (π) bind.  In keeping with methane and ethane, I’ll show you the simplest case, that of ethylene:
Figure XXIII.
Before discussing the “double” bond between the carbon atoms in ethylene, I want to draw your attention to the flat, trigonal geometry about them.  We have already seen this, in Figure XVIII, the sp2 hybrid orbitals which creates this geometry.  Four of these orbitals combine with the 1s orbitals on the hydrogens to create those bonds, while the two carbon pointing sp2 overlap to create a sigma bond, just as with H2.  But this leaves something unaccounted for.  Both carbon’s pz orbitals are now free to make other combinations, and each contains a single electron it can use for bonding.  In fact, the most stable bond it can form is by a sideways overlap of the pz orbitals.  Such resulting bonds are called pi bonds, and they look like this:
Figure XXIV.
Imagine the two orange balloons on the left as the carbons’ pz orbitals; then the yellow region on the right is the region of maximum overlap or combination.  This orbital (although there are two lobes, it is one orbital, above and beneath the sigma, represented by the straight line) is the bonding; the ps also combine to make an anti-bonding orbital, which on ethylene is empty, stabilizing the pi structure.
*          *          *
This chapter could cover more, much, much more; and we will pursue more material along these lines.  Especially, our exploration will heavily revolve around hydrogen and carbon, as well as oxygen and nitrogen; for these are the main atoms which make up the molecular nature of life.  The basics have been laid down here.