The Periodic Table Of The Chemical Elements

Mendeleev's famous table (not to be confused with the periodic table of the internet) suffers from the minor inconvenience of having to insert the lanthanides and actinides as rows outside the table. It would be nice to have a way of in-lining them without breaking up the structure of the table. Before I describe ways of doing that, however, I'd better make some preliminary remarks.

The question of whether H appears as a group I element (above Li, as witnessed in the H⁺ ion of inorganic chemistry) or a group VII element (above F, as witnessed routinely in organic chemistry) is open to debate; equally, it is worth asking whether the chemical differences between ²H (a.k.a. D) and ¹H suffice to warrant mentioning both, possibly with one in group I and the other in group VII. I've opted for putting H into group VII, below, since the resulting table is then somewhat neater.

In the following variations on Mendeleev's table, I've italicised the symbols of elements that don't arise in nature. The data used below is somewhat out of date – published in 1972 – but is provided mainly for illustrative purposes in any case; my aim is to indicate how to re-draw the table, not to take sides in arguments about what to call the elements; and to show how the re-drawn table lends itself to carrying more information, though the information used to illustrate that may be imperfect.

Column Shuffling

One simple way to make it convenient to in-line the actinides and lanthanides is to move the leftmost two columns of Mendeleev's table, groups I and II, over to the right of the inert gases (and up a row, of course):

The alternate position for H (discussed above) is where I've put p (proton – the ¹H nucleus), alongside n (neutron – with atomic number zero and the same electron configuration as the H⁺ ion, just as He matches the Li⁺ ion; and note that the neutron is radioactive, with half-life 613.9 seconds, β-decaying to a proton) as nominal zeroth noble gas. (Are group VIII's members called noble gases, or Nobel gases ?)

One obvious complication is that Lu and Lr, rather than La and Ac, now appear in the column below Sc and Y; it would be interesting to know which pair is the closer fit for being chemically similar (which is what columns are meant to encode) to Sc and Y. Since I don't know the relevant chemistry, the best I can do is examine the pattern of filling of the assorted relevant electron shells:

• Sc has 1 in 3d and 2 in 4s; the subsequent elements to Zn fill up the 3d shell, mostly one at a time but with occasional borrows of 1 from 4s, causing a 2 step and then a no change in 3d as 4s claims the next.
• Y has 1 in 4d and 2 in 5s; the subsequent elements to Cd fill up the 4d shell with rather more extensive borrowing from 5s (which Pd even has empty), making the filling of 4d somewhat jumpy.
• La and Lu both have 1 in 5d and 2 in 6s; the elements between fill up the 4f shell, mostly borrowing the 1 from 5d (without any borrowing from 6s); 4f is empty up to La and full from Lu onwards (and is never borrowed from). The subsequent elements to Hg fill up 5d, with a minor borrow from 6s at Pt and Au.
• Ac and Lr both have 1 in 6d and 2 in 7s; the elements between mainly fill up the 5f shell, without any borrowing from 7s, but start with Th which fills 6d (to 2), then Pa with 2 in 5f, 1 in 6d; subsequently, the 1 in 6d is borrowed for some of the time as 5f fills.

all of which rather suggests, at least to me, that the extent of chemical similarity is unlikely to be a drastically compelling argument for choosing between Lu and Lr or La and Ac as successors to Sc and Y. Of course, you'd have to consult an actual chemist to settle this matter properly.

Further cycling

If La and Ac really are the better candidates, that means we should cycle the table (at least) one more step round to the left; we can then put some or all of the lanthanides and actinides with La and Ac, the rest positioned as above. Since pulling Sc, Y, La and Ac round to the right breaks up the tidy right edge we got above, I'll (arbitrarily, for illustration) shunt H from group VII to group I in the illustration; and to fit with the new shape of the right margin, I'll split the lanthanides and actinides after one column – but note that this cut is arbitrary.

Of course, as for Sc and Y, so for the elements following them: if Ti and Zr resemble Ce and Th better than they resemble Hf and Rf, we can likewise cycle them round; if V and Nb resemble Pr and Pa better than Ta, they too can go round; and so on. Ultimately the correct way to determine where to break must be the chemical properties of the elements involved; which are determined by the the outer electron orbital shells – if electrons can easilly be added to an outer orbital, or dislodged from it, then the element can readilly form ions, whose charge is determined by the number of electrons added or dislodged. Adding electrons to a nearly full orbital generally takes little energy (the nuclear charge is already holding several electrons in that orbital; and the other occupants have little impact on the ease with which more may be held in it) and produces a stable ion when the orbital is full; removing electrons from an orbital with few occupants is also generally easy (the nuclear charge is just barely able to hold the electrons in that orbital to begin with).

It is thus interesting to look at the filling up of electron orbitals; in the following, elements are coloured according to the type of orbital they add electrons to – S, P, D or F. Let's start with a nice pretty variant (each rightwards growth is by two; leftwards expansions are by successive multiples of four; this structure is suggested by the orbital-filling pattern described below) that could, in principle (but only if the chemistry genuinely supports it), be appropriate, and see how the colours line up (orbitals from Rf onwards are purely conjectured):

We can predict, from what we already know, that element 118 will be in group VIII; this last table's pattern would predict eight more elements to the right of 118 followed by sixteen on the next row to the left of Np and 26 from below Np through to group VIII again at element 118+8+16+26 = 168. One should not, however, place any faith in a prediction based only on unexplained pretty patterns (a.k.a. numerology), as this is; particularly given how easily we can improve the column-alignment of the orbital pattern. (However, below, I'll give a better argument for this pattern of lengths of periods.)

Of course, nothing stops us cycling the whole lanthanide/actinide block:

(which improves the orbital matching down columns) and even the whole next block:

which has the intriguing consequence of lining up the first three artificial elements as a column; however, its relevance to chemistry should be judged on whether the elements it places in columns together are chemically similar; and, it rather messes up the orbital alignment in columns, so it doesn't look so good on that score. It certainly isn't so good at lining up orbitals as the previous; which is close to as good as we can hope for, though the slant-cycled and original simple variants above are roughly as good as it on that score. Here's another that's about as good as I can manage for orbital column alignment:

The important thing is that the structure [grammar] of the table allows us arbitrary amounts of cycling in the manner illustrated here (movement between the left end of one row and the right end of the row above), while the meaning [semantics] of the table says we should do such cycling in so far as it brings chemically similar elements into the same column as one another.

Equivalent cycling can be applied to the following, if appropriate, but I'll assume Lu and Lr are as like Y and Sc as La and Ac are, for the present; i.e. that my first version of the table, the simply cycled one, is apt.

Transpose

The table above comes out rather wide: one way to fix this is to transpose it, making each period a column and each group a row. Of course, the result comes out tall just as the above comes out rather wide; but the accompanying narrowness lends itself to including rather more information about each element – with the result that my first illustration is actually wider than what we've seen before:

Well, I guess listing all the isotopes is a bit extreme, but hey. (For reference: bold means more than 50% of the given element is the indicated isotope, in natural samples; italic means radioactive; radioactives are only listed here if either they're naturally occurring or the element is unnatural; in the latter case, the element itself is italic, as noted above – and I've taken the opportunity to throw in what little is known about some of the more recent discoveries. ⁵⁶Fe gets special treatment because it has lower mass per nucleon than any other nuclide.) Maybe average atomic mass would be better (albeit somewhat bogus for the unnatural elements – I'll just mention a known atomic mass for those):

Quite what happens after element 120 is yet to be discovered, though doubtless one could do the theoretical (or computational) analysis to discover what electron orbitals arise, which would answer the question. The pattern of what we have, however, would seem to suggest that 121 would begin a column even taller than the La and Ac ones. (The pretty pattern played with earlier would point to a height of 50 for this column.) I've heard some speculation to the effect that there may be an island of stable nuclei around atomic number 120 to 140; and even that strange nuclei (i.e. ones in which some of the down quarks – of which each proton has one, each neutron two – are replaced by strange quarks, which have the same charge (and isospin) but are heavier) may be stable at atomic numbers of roughly this order. All of that would happen in the next column.

The lengths of the cycles

Mendeleev's table is described as periodic because it has patterns that repeat; however, every second repeat the length of the cycle increases. This is caused by a new family of electron orbitals coming into play. The states an electron can take in an atom (of given atomic number, Z) are characterised by three quantum numbers (aside from their own intrinsic spin): a positive integer n, a natural number b < n and an integer ranging from −b to b. The over-all length scale of the state is proportional to n/Z and its total angular momentum is proportional to b.(1+b).

For given n and b there are 2.b+1 states, all with energies very close together, each with a degeneracy of two due to the electron's intrinsic spin; these are collectively known as an orbital. For given n, summing 2.(2.b+1) over all the allowed values of b, we get 2.n.n states collectively known as a shell. Many characteristics of an orbital depend only on b, regardless of n; notably, the number of states; but also whether an electron in a state in that orbital spends its time mostly near to (low b) or far from (high b) the nucleus. This last causes low-b orbitals to tend to have lower energy, within any given shell, than high-b orbitals.

The states of an electron and a nucleus follow the above pattern and have energies which depend only on n, the shell number. However, once we've filled the lowest shell, with n = 1, and maybe some other orbitals, the electrostatic potential caused by the nucleus is partially masked by the electrons in that shell. Outside the volume where these inner electrons mostly are, another electron sees an apparent nucleus with atomic number less than the true nucleus by the number of electrons already present. Inside that inner volume, however, the number of masking electrons drops off the closer we get to the actual nucleus, so that any further electrons we add will see the nucleus more or less masked according as they fall into orbitals concentrated near to or far from the nucleus. Thus orbitals with higher n are at higher energy than those with lower n, for given b; and, for given n, orbitals with higher b have higher energy than those with lower b.

I can thus specify a mapping ({shells}:Ψ|{naturals}) with each shell being itself a mapping ({orbitals}:Ψ(n)|n), interpreting each natural as the set of smaller naturals (so 0 = {}, 1 = {0}, …, 1+n = {0,…,n}). This gives us a nominal empty 0 shell, which we can ignore (because it's empty) or interpret as the collection of electron orbitals occupied in the neutron, interpreted as element zero. Each orbital Ψ(n,b) has degeneracy 2.(1+2.b); this is how mahy electrons it can hold. The Ψ(n,0) are commonly known as S shells, Ψ(n,1) as P shells, Ψ(n,2) as F shells.

As noted above, the filling of orbitals and shells determines the properties of elements. However, in larger atoms (where the outer orbitals are further from the nucleus), physical and chemical behaviour may be less tightly controlled by the filling up of orbitals. Indeed, the source I've used for orbital filling data (Nuffield Advanced Science Book of Data, ISBN 0 582 82672 1, pp 50–51, table EIA) notes that there is some uncertainty about some of the configurations, especially Pt and Np and Beyond ₉₄Pu, the assignments are conjectural. This source gives its data in a table which has columns (none of which have any entries; nor has 6f) for 5g, 6g and 6h. Working outwards from the empty Ψ(0) shell, orthodox notation's 1s is Ψ(1,0), 2s is Ψ(2,0), 2p is Ψ(2,1), 3s is Ψ(3,0), 3p is Ψ(3,1), 3d is Ψ(3,2) and so on, adding 4f as Ψ(4,3), 5g as Ψ(5,4), 6h as Ψ(6,5), albeit these last two, along with 6f and 6g, never get any electrons in them. The filling of orbitals follows this pattern:

• a P orbital, Ψ(N,1) has higher energy than Ψ(N,0) and each Ψ(N−i,1+i) with i > 0 (and N > 1+2.i); once these are all full, Ψ(N,1) fills, ending with a group VIII element; once full it – along with all lower energy orbitals – remains full;
• the next S orbital out, Ψ(N+1,0) fills;
• each Ψ(N−i,2+i) orbital begins filling, in order of increasing i, while N > 2.(i+1), starting with i=0 at the D oribital Ψ(N,2); each starts with one or two electrons before the next takes over;
• the last of these fills, tending to borrow an electron or two off the ones that started before it; as each fills, the last one to start before it fills up in its turn;
• as the D orbital Ψ(N,2) fills, it borrows from Ψ(N+1,0); when this D orbital is full, the S orbital Ψ(N+1,0) re-fills;
• all Ψ(N+1,0) and Ψ(N−i,2+i) orbitals are now full; the next P orbital Ψ(N+1,1) is now ready to fill and we begin the cycle again.

We already knew that Ψ(N,j+1) and Ψ(N+1,j) have higher energy than Ψ(N,j) for all N, j. What the above tells us is that the orbitals Ψ(N+1,0) and Ψ(N−i,2+i), with i natural and N > 2.(i+1), all have roughly equal energies, distincly greater than the energy of Ψ(N,1) but less than that of Ψ(N+1,1) – the i=−1 member of the given family of orbitals. We only witness this pattern for i = 0 and 1, so it might plausibly break down for larger i; we can't know until we see some element broaching 5g, which can't possibly happen before element 112 (and the above pattern tells us to not expect until after 120) so that we can compare its energy with 6f, 7d and 8s or 8p. However, if we assume that the above pattern holds true, we can predict the lengths of future periods of the table; it gives us a basis for understanding the previously noted pattern of two cycles of equal length followed by a jump by four more than the previous jump.

Indeed, each cycle starts after Ψ(N,1)'s filling (at a group VIII element) and ends when Ψ(N+1,1) is full. In between, we have filled Ψ(N+1,0) and all Ψ(N+1−i,1+i) with i natural and N+1−i > 1+i, i.e. N > 2.i. This gives us orbitals with b running from 0 to the largest i with 2.i in N; thus cycles with odd N = 2.n−1 and even N = 2.n both get each b in n, so have the same patterns of orbitals. Summing 2.b+1 over b in n yields n.n, so each cycle of the table should be twice a perfect square; sure enough, we have cycles of lengths 2, 8, 18 and 32, twice the squares of 1, 2, 3 and 4. We can thus expect the cycle following ₈₆Rn to take 32 elements, ending with a group VIII element at Z = 118 (followed by members of group I and II before we expect to embark on 7d, 6f and 5g). After that we can expect two cycles of length 50 yielding group VIII elements at Z = 168 and 218; then two of length 72 ending at 290 and 362; and so on. Each perfect square differs from the previous by an odd number, the next such after the previous such difference, and so each growth in period, being twice such a difference, is indeed four more than the previous one.

Further thoughts

Different isotopes of an element will have very slightly different spectra, due to the difference in inertia of their nuclei. Consequently, elements with only one stable mass number have narrower spectral lines; this makes them more precise standards. Thus, indeed, Caesium's spectrum is used to define the second, whence the metre is now defined; and these two imply the kg/A/A (kilogramme per square amp) and other related units, such as the Ohm. The single-isotope elements are (from one of the tables above): Beryllium, Fluorine, Sodium, Aluminium, Phosphorus, Scandium, Manganese, Cobalt, Arsenic, Yttrium, Niobium, Rhodium, Iodine, Caesium, Holmium and Gold – plus Praseodymium, Terbium, Thulium, Bismuth, Polonium, Francium, Thorium and Protactinium, except that these are radioactive, which may undermine the benefits of only having one naturally-occuring isotope.

One can extend Planck's system of units to include charge and current. One of the candidate units for charge is √(h/Z₀), which comes out at 8.28 times the proton's charge; just slightly more than the charge on the Oxygen nucleus. Using Dirac's constant in place of Planck's in that gives a charge of 3.3 protons, between those of the Lithium and Beryllium nuclei. Using 4.π.ε₀, c and h implies a charge of 29.34 times that on the proton, between Copper and Zinc. In contrast, the Planck mass is about 33e18 atomic mass units; the mass of one mole of hydrogen is less than nineteen thousand Planck masses. The heaviest quark, Truth, has mass around 43 AMU, which falls between those of Ca and Sc; the proton-analogue obtained from two Truth quarks and a Beauty quark (presumably) has mass around 90 AMU, close to those of Y and Zr.

Written by Eddy.