c22 - 223 Chapter 22 MANY-ELECTRON ATOMS ©2004 2008 2011...

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Unformatted text preview: 223 Chapter 22 MANY-ELECTRON ATOMS ©2004, 2008, 2011 Mark E. Nob/e To this point, we've identified the many types of orbitals which are available for electrons in atoms, according to the laws of Nature and as revealed by quantum mechanics. Given this list of orbitals which are available, which orbitals are actually used by the electrons in a given atom? In the case ofan atom or monatomic ion with only one electron, we have already seen that 15 is best (lowest energy), so the one electron will be in the 15 orbital when in the ground state. This was based on the relative energies of the various orbitals and those energies were based on the principal quantum number, n. This much even goes back to Bohr's energies, just as we showed in Chapter 20. Although Bohr had n = 1 as best, he didn't do the 5 part. It wouldn't have mattered anyway: since energy is determined only by n, all subshells of the same shell have the same energy. Thus, 25 has the same energy as 2p, and both of these are higher energy than 15. 3s, 3p and 3d all have the same energy and all of these are higher energy than 25 and 2p. Et cetera. Et cetera. Et cetera. As we go along here, remember that higher energy is less favorable and vice versa. Keep in mind the ladder notion. Nature wants you on the bottom rung, lowest energy. A higher energy rung is less favorable. With atoms and with ions which have two or more electrons, the energy will depend on factors other than just n. By the way, anything with two or more electrons is called many—electron or multi—electron. These many-electron cases are where we are heading now. This will get more complicated, so keep the eyeballs open. There will be a pattern to this. First, there are new points to introduce. 22.1 Spin The Schrodinger equation led to three quantum numbers and each possible combination of these three quantum numbers identifies a specific orbital in an atom. It turns out that Nature wasn't finished with just that much. Nature also gave each individual electron its own unique property with a separate quantization, and this means that there is another quantum number to cover. This fourth quantum number is not related to the Schrodinger equation and it is totally independent of orbitals; in fact, the electron has this property even when it's not in an atom. This property is called "spin"; its quantum number is called the "spin" or "spin magnetic" quantum number and it is designated ms. Electron "spin" is a magnetic property. Each and every electron has magnetism, in addition to charge. The quantization of electron magnetism gives only two possible numbers: +1/2 or —1/2. The reason for the one-half part is not important to us right now; the really important part is that there are only two possibilities for this quantum number and that they are opposites (positive or negative). Do not confuse the "spin magnetic" quantum number, ms, with the "magnetic" quantum number, m,. The reason m, is called magnetic is not related to why m5 is called magnetic. Besides, m, applies to an orbital and mS applies to an electron. Why is the electron's magnetic property called a spin? This is another historical thing: the term simply stuck. In earlier days, the physical interpretation of ms was related to the particle picture: the electron was taken to be spinning on its own axis, just like a globe representing Earth. But the reality is that electrons don't spin like a globe. Although the term "spin" is still used, you need to remember that it is a magnetic property, not a globe property. While we're at it, don't confuse spin with charge. Many quantum particles have this spin property, and that is totally separate from charge. For example, neutrons and even photons have spin even though they have no charge at all. SPIN IS MAGNETIC. The + or — value for mS for the electron corresponds to a magnetic property, much like north and south correspond to the opposite poles ofa permanent magnet. In fact, the two different electron spins interact differently with the north and south poles of a magnetic field. There are several important consequences of electron spin, as we shall see. Here's one you're familiar with: permanent magnets, such as the normal, everyday bar magnet or horseshoe magnet. What makes these magnetic? The magnetism arises from the electron spins in the atoms of the magnet. This notion of opposite spin values is extremely important. If you have two electrons and one has m5 of +1/2 and the other has m5 of —1/2, then their magnetisms cancel each other. In fact, this is why most things are not magnetic and are not attracted to a magnetic field. Half of their electrons have spin of +1/2 and half have spin of —1/2. The spins cancel, and there's no magnetism. OK, now we have four quantum numbers. We have three quantum numbers for an orbital and one quantum number for each electron. We can now specify each and every electron in an atom by the orbital it is in (n, l, m,) and by its spin (m5). 224 Chapter 22: Many-Electron Atoms The next point which I need to introduce is spin exclusion. Two electrons of the same spin exclude each other from close proximity; in other words, they avoid each other. You might wonder, "So what?", since electrons are negative charges and negative charges avoid each other anyway. That is correct, but that is charge (electrostatic) repulsion and that is different from spin exclusion. Charge repulsion exists between all electrons because all electrons have a negative charge. Spin exclusion (avoidance) exists between electrons ofthe same spin but not between electrons ofdifferent spin. These are totally separate effects. When operating, spin exclusion is much more important than charge repulsion for keeping electrons away from each other. Spin exclusion is a bit strange and it does not have an easy explanation. It is a law of the quantum realm so we must deal with it even if we don't understand it. We need it right now because it leads to a very important principle, called the exclusion principle: Nature excludes two electrons with the same spin from being in the same orbital. This has two really big consequences. First, if there are two electrons in the same orbital, then they must have opposite spins. Second, an orbital can hold up to two electrons but no more; any more than two would require two electrons of the same spin and that is not allowed. Let me restate these points for emphasis. NATURE DICTATES THAT A SINGLE ORBITAL CAN HOLD AT MOST TWO ELECTRONS AND, IF TWO ARE PRESENT, THEN THEY MUST HAVE OPPOSITE SPINS. When two electrons have opposite spin, we say that they are "paired". This term is very important. We can say that the electrons are paired or we can say that the spins are paired or we can say that the electrons are "spin-paired". Some compounds have all of their electrons paired, and some compounds don't. In the latter cases, at least one electron is "not paired" or "unpaired". Here's the thing to remember about this "pair" business: pairing refers to two opposites. It's just like a pair of shoes or a pair of gloves. Two opposites. We are now ready to tackle the problem of the many-electron atom. 22.2 Splitting subshells Well, the easy part is over. Now things get a bit complicated. The easy part was the one—electron atom or ion, such as H or He". The energy of the orbitals was directly calculated from n, and all subshells within one shell had the same E. All of this arises from the attraction of the negative electron for the positive nucleus. I'll start referring to this as electron-nuclear attraction. When only one electron is present, that's all we need to worry about. This is not true in the multi—electron case. As soon as you bring a second electron into the picture, you get into new complications. There are now charge repulsions between the different electrons; collectively, these are called electron-electron repulsion. There is also spin exclusion between electrons of the same spin. These factors now influence the total energy picture in a way which cannot be easily calculated. Let's illustrate this effect with the simplest case. Start with one electron in is (for example, as in H). Now, put a second electron into the same orbital (for example, to get H'). The two electrons must be spin-paired in order to satisfy the exclusion principle. That much is OK in this case. Charge repulsion, however, cannot be so readily dismissed. The two electrons are separately attracted to the nucleus but they also are repulsive to each other. Electron—electron repulsion always detracts from total energy. Pay particular attention here: I said total energy. Total energy is the grand sum of all electron- nuclear attractions plus all electron-electron repulsions. So if electrons repel each other, isn't the second electron better off in 25 instead of cramming into the ls? This would eliminate some of the repulsion from being in the same orbital. No, not necessarily. This simple question underscores one of the most important matters for dealing with the problem of many—electron atoms. You see, we are again dealing with the question of balance. I've been telling you off and on since Chapter 1 about the importance of balance in so many different aspects of chemistry. We are now dealing with another case where balance is absolutely, critically important. All multi-electron cases are the result of the balance between electron—nuclear attractions and electron—electron repulsions. This is not entirely straightforward, and there are numerous variations which can arise due to the subtleties involved. Nevertheless, a majority pattern does emerge, and we shall adopt this as our general result. Let's first consider the two sides to the story separately. We already covered the part fairly well about electron—nuclear attraction in the last two Chapters. The strength of this attraction is dominated by the size of the orbitals and the charge of the nucleus. Chapter 22: Many-Electron Atoms 225 Larger orbitals from higher n shells will have less attraction and are less favorable. More protons in the nucleus give greater attraction due to higher positive charge. The part about electron—electron repulsion is more tedious to explain. There are several aspects here to consider. Let's say you're an electron in some orbital in some multi-electron atom. The question is: how repulsive are you to other electrons in the atom? Well, you're repulsive to another electron in your orbital, if there is one. You're repulsive (to a different extent) to any electrons in other orbitals of your same subshell. You're repulsive (to another different extent) to any electrons in other subshells of your shell. You're repulsive (to another different extent) to electrons in other shells. You have to consider all of those electrons in all of those different orbitals, all in proximity but still separate and still sensing one other. It gets complicated. Let's try a different picture. You have an atom with one electron in 15. Now add a second electron, which is where we left off above. The second electron will also end up in 15 despite the fact that it detracts to be in the same orbital with another electron. The reason it does this, instead of going to 25 (or to any other orbital), is a trade—off: it still comes out ahead overall by pairing in ls because 25 is larger around the nucleus and therefore less favorable. The trade—off is the balance. Now add a third electron. Where will it end up? Exclusion limits all orbitals to two electrons, so ls is booked up. Sorry, no vacancy. Let's go to the next larger group, n = 2. Will the third electron go into 25 or 2p? In a one—electron atom, 25 and 2p have the same energy, so they are equally favorable. Now, however, we have other electrons in the atom and now we have more problems. It's still TOTAL ENERGY which must be considered. The determining factor here is electron-electron repulsion between the two electrons in 15 and the third electron we are dealing with. This repulsion differs depending on whether that third electron is in 25 or in 2p, since these orbitals differ by their shapes with their different regions of different concentration of the orbital field. Because of differing repulsions, SUBSHELLS WITHIN A SHELL ARE NO LONGER EQUAL in terms of the total energy picture. The answer here is that the third electron goes to 25. By the way, a fourth electron will also go to 25. It's still better to pair up in 25 than go to 2p because the total energy is better. On the other hand, a fifth electron has no choice. 25 is booked up, and 2p is the next available. Let me summarize this so far. For a multi—electron atom, the general trend remains that energy is less favorable with highern shell. Now, in addition, we have a new consideration: within a given shell, the subshells are no longer equal towards the total energy outcome. This gives an energy spread, and the different subshells will differ in total energy for electrons. This relationship goes by the 1 values as follows. l = O l = 1 l = 2 l = 3 5 p d f etc..... most H H H less preferred preferred Keep in mind that this is for subshells WITHIN THE SAME SHELL. Don't go applying this to 2p versus 35. That won't work. This relationship only works in the same shell. The inequality of the subshells is what makes the multi-electron cases more complicated. This becomes even worse when the subshells spread out so far that they overlap the next shell. Although this is complicated, we do need to cover this; we need to see what orbitals are actually in use for a given atom. The general notion for putting all this together is that the electrons will be in the orbitals which give the best (lowest) total energy for the atom as a whole. We've already considered the orbitals for the first five electrons that would be present in an atom: two go into 15, two go into 25 and the fifth goes into one of the 2p orbitals. Which 2p? It doesn't matter: 2px or 2py or 2pz. At this stage of the game, all three are equal in energy, all three are equally favorable, and all three are equally available, so it doesn't matter which and we don't need to specify which. But I'm getting ahead; let me back—up a moment and introduce a few more items. As we add more electrons to the picture, these additional electrons do not benefit from the full nuclear charge because they are repelled to some extent by the electrons already present. This gives us two new terms. We say the additional electrons are "screened" or "shielded" from the full nuclear charge by the other electrons. The nouns for the process are "screening" or "shielding". Since an electron may not benefit from the full nuclear charge due to other electrons present, we can say that the electron senses a reduced nuclear charge; this is called the "effective nuclear charge". Effective nuclear charge 226 Chapter 22: Many-Electron Atoms (symbolized Zeff) is always less than true nuclear charge, Z, for any multi—electron atom or ion. How much less? It depends, and there are ways of determining this but we won't go into that. We'll stick with the general idea. For the general idea, there are two effects of significance to note as more and more electrons are added to the picture. > Screening effects are quite substantial between the added electrons and the electrons already residing in smaller n—shells. These cases weaken the nuclear attraction (decrease Zeff) the most for additional electrons. > Screening effects are weak between the additional electrons and other electrons already in the same n—shell. These cases have little effect on nuclear attraction (little effect on Zeff). These effects play important roles in setting up the total energy picture. The sequence ls then 25 then 2p is based on total energy and it represents the sequence by which electrons will be found in the orbitals in an atom or monatomic ion. We now extend this sequence further in order to accommodate any realistic number of electrons. As I had mentioned before, variations are possible for different circumstances due to the subtleties involved but there is a general pattern which emerges for the majority of cases. I shall be calling this pattern a "fill sequence", since it refers to the sequence by which the electrons will fill the orbitals in an atom in its ground state. Keep in mind that all of this is the overall result of the total energy balance of all electron—nuclear attractions and electron— electron repulsions. Let's just work with fill sequence for now, without worrying about how many electrons are involved. We‘ll come back to number of electrons in the next Section. So far, we have the fill sequence ls-Zs—Zp, as derived by comparing total energies. After 2p fills up, the n = 2 shell is done and we then turn to n = 3. In n = 3, there are three subshells, 3s, 3p and 3d, in that order of preference. This brings our fill sequence to ls—Zs—Zp—fi—fig—fl and that ends n =3. We turn to n = 4, which has 4s, 4p, 4d and 4f. We now encounter our first complication. When we place this into the general sequence, 45 comes out better than 3d. The reason for this is that the prior subshell spread, 3s—3p—3d, went so far as to overlap 3d into the n = 4 range of total energy. This is our first example of an overlap between different shells. There can be exceptions to this; although most cases will indeed prefer 45 over 3d, some will prefer it the other way. We will follow the majority, and so we place 45 ahead of 3d in our fill sequence. 15—25—2p—3s—3p—4s—3d The rest of n = 4 adds on to the end. ls—25-2p-3s-3p-4s-3d—42—fl-4f Next: with n = 5, we have the subshell spread 55, 5p, 5d, 5f and By. We have another overlap problem: just as 45 came out ahead of 3d, so also does 55 come before 4d, in the majority of cases. So, we squeeze 55in ahead of 4d in our fill sequence. ls-25-2p—3s—3p—4s—3d—4p—g—4d—4f We also have another overlap problem, but now of a different sort. As you can see so far, d's get spread out pretty far but f's get spread out even further. 4f gets kicked out so far that it is actually out past where 5p will be. Overall, in terms of total energy, 5p comes in ahead of 4f. 15-25-2p-3s-3p—4s—3d—4p—Ss—4d—§Q—4f The other 5's add on to the end so far. 15-25-2p-3s-3p-4s-3d—4p-55-4d—5p—4f—ij—fl—fig Now we introduce n = 6. 65 comes in ahead of 5d, just like the above cases. What's even stranger is that 65 comes in ahead of 4f. This just shows you how much these subshells are spreading out and overlapping. It also shows you how messy this can get. 15—25—2p—3s—3p—4s—3d—4p—Ss-4d- 5p-§_s_-4f—5d-5f-Sg Chapter 22: Many-Electron Atoms 227 It turns out that 65 and 4f and 5d are close. All three constitute an overlap region which will have frequent exceptions. Again, we're sticking with the majority of cases right now. After 65, we turn to 6p, which gets inserted before 5f. ls-Zs—Zp—3s—3p-4s—3d—4p—55—4d—5p—6s—4f—5d—§Q—5f-Sg I could continue this story on and on, but we are approaching the end of all known monatomic cases. At this point, I shall go to the final result. 15-25-2p—3s-3p-4s-3d-4p—55—4d- 5p-6S-4f—5d—6p-Z§-5f—_6_g-ZQ Notice how I brought in some from n = 7. Also notice that 59 got dropped off, since it got spread out well past the others. Now, don't get me wrong: there is still a Sg, but no known elements have enough electrons for one to be in Sg in the ground state. You can do an excited state with an electron in 59, but we're doing ground states here. This fill sequence business is for ground states. Alright, let's recap. This crazy-looking string of subshells is the fill sequence for the ground states of most multi-electron atoms. The sequence represents the balance of opposing forces related to electron-nuclearattraction and electron—electron repulsion. Exceptions occur often, ...
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