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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, (in most cases) below, since the resulting table is then somewhat neater.

In the following variations on Mendeleev's table, I've italicized the symbols of elements that don't arise in nature, or that only arise in trace quantities as radioactive isotopes. The data used below are somewhat out of date – published in 1972 – but are provided mainly for illustrative purposes in any case (although I haphazardly supplement them from modern sources); my aim is to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):

(Hover entries to get clues as to why they have the names and symbols they have. For more on the linguistic complications behind those names, see here. The repeatedly-mentioned Swedish village Ytterby is near Stockholm; several elements were first discovered in ores from a mine there.)

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.

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 (which is relatively orthodox) in this 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. While I'm here, let's throw in the old alchemical symbols for the few actual elements that got such symbols (some of which exist in several variants, even within Unicode; some of which are also used to denote other things; some of which might be missing from the fonts your browser supports; hover for the name and modern symbol) and the traditional group numbers (using modern numerals: hover for the Latin numerals more conventionally used – some of which were long enough they messed with column spacing – and their names), albeit there's several other ways the groups (i.e. columns) can be labelled:

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. Those are determined by the the outer electron orbital shells – if electrons can easily be added to an outer orbital, or dislodged from it, then the element can readily 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 – compared to the preceding element. 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 already know enough to expect that Og, element 118, is in group VIII (it's been created, but not in the sort of quantity that would let folk determine its chemistry; and it's not stable). 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. When looking at similarity, it's also instructive to consider various classifications of the elements.

Let's see what happens when we distinguish the non-metals (in the chemist's sense, not the astrophysicists' one, which considers everything but Hydrogen and Helium to be metals) and metals whose ions prefer an Oxygen donor (class A) from the other metals (that either prefer Nitrogen or Sulphur donors, or vary between these and Oxygen; this is one of the competing specifications for the confused term heavy metal), with a fuzzy boundary (including the semiconductors) between these last and the non-metals (aside from some simple cycling, this is my closest match to Brian Clegg's Metals and non-metals version in his Royal Institution lecture from 2021 December 23rd):

Notice that, compared to the orbital-aligned version preceding it, the only difference here (aside from the different classification, hence colouring scheme) is in the placement of Aluminium. The important thing is the combination of the sequence of the elements (by how many protons the nucleus contains, hence how many electrons are engaged in mediating the atom's interaction with the rest of the universe) with the recurrence of chemical properties, as patterns of filling of electron orbitals later in the sequence reiterate patterns seen earlier. That can be represented in divrse ways. The diagrams above all, fundamentally, hang on the sequence being mapped to reading text from left to right and wrapping to a next line that's below a previous line; below I'll look at columns read downwards, each followed by the next column to its right. One can, just as well, make a snaking path in two (or more) dimensions that starts as a circle, spirals outwards as it goes and spits off a fresh bulge as each new set of orbitals opens up, returning to the prior pattern once it's exhausted the new set of orbitals.

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. (In fact, in Mendeleev's paper in 1869, at least one of his presentations of the periodic character of the elements' properties is indeed in columns, not hugely different from the following – aside from having gaps that showed where some elements were clearly missing, that we've since found.) 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 (naturally occuring) isotopes (and including the element names if you hover them) 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 occur in nature or the element is unnatural; in the latter case, the element itself is italic, as noted above – and I've only reported the most stable confirmed isotope. (In several cases, reports of more stable heavier isotopes were unconfirmed when I looked this up.) 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 longer 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 164 (elements with densities around 36 to 68.4 kg/litre, well above Osmium's 22.59 kg/litre, the densest we currently have); 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, with 164 lining up with carbon and silicon.

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 characterized 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, which can be understood as element zero. Each orbital Ψ(n,b) has degeneracy 2.(1+2.b); this is how many electrons it can hold. The Ψ(n,0) are commonly known as S shells, Ψ(n,1) as P shells, Ψ(n,2) as D shells, Ψ(n,3) 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 orbital 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, distinctly 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.

In an attempt to depict that filling, ignore the borrowing and re-filling that actually happens during the Ψ(N−i,2+i) phase and pretend each of those orbitals takes exactly two elements to get started, then finishes off with the remaining 8 +4.i elements adding electrons in the orbital after those with higher i have filled. This then gives us the shape I sketched in my earlier Block-Cycled Periodic Table of the Chemical Elements, so replace each block of elements in that by the parameters to Ψ that describe the orbital nominally filling in that block. Extending that to include the next cycle we get:

The next cycle would then repeat the bottom row here – 16 in Ψ(6,4) = g6, 12 in Ψ(7,3) = f7, 8 in Ψ(8,2) = d8, 6 in Ψ(9,1) = p9, then pairs in each of Ψ(10,0) = s10, Ψ(9,2) = d9, Ψ(8,3) = f8, Ψ(7,4) = g7 – followed by another pair on the right for a new Ψ(6,5) = h6 orbital. The row after that would start with the remaining 20 h6-filling elements on the left, splitting the filling of g7 between the two before h6 and the 16 after it. The borrowing I'm ignoring leads to back-filling at the third element in the Ψ(3,2) and Ψ(4,2) blocks, the fourth element in Ψ(4,3) and Ψ(5,3), the last two in each of those blocks, the last of Ψ(5,2) and the early start of Ψ(4,3) stealing the second of Ψ(5,2)'s starter pair. So, aside from Ψ(5,2)'s anomalies, the left-most block in each row – filling Ψ(i+1,i) or Ψ(i+2,i) – does some borrowing in its first quarter, that it pays back at the start of its second quarter, and borrows two more after that, which it pays back at the end of the block.

Further thoughts

Kaycie D. did a series of illustrations that represent each element as a cartoon character. Each comes with some commentary on the element. They're neat.

Isaac Arthur's video on The Phosphorus Problem includes an interesting version of the periodic table showing what processes we know of that create the various elements. The different sizes of stars and points in their life-cycle are spread in interesting ways through the diagram.

I was interested to learn that a human body's composition doesn't follow the CHONSP sequence of preponderances I dimly remember; instead, I am 65% O (Oxygen), 18.5% C (Carbon), 9.5% H (Hydrogen), 3.3% N (Nitrogen), 1.5% Ca (Calcium), 1% P (Phosphorous), 0.3% S (Sulphur), 0.2% each Cl (Chlorine) and Na (Sodium) – so I guess 0.4% table salt – and 0.1% Mg (Magnesium). Assorted traces of various other elements make up less than 0.01%, less than the rounding errors, in those with ≥ 0.1%, that leave them adding up to only 99.6% (the missing 0.4% is the sum of the rounding errors). It is notable that, in the universe at large, Phosphrous is significantly rarer, proportionately, than it is in our solar system. The talk also linked to an interactive table of the elements.

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 (kilogram 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-occurring 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 the masses 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.

There's a you-tube (or should that be U-tube ?) channel devoted to the elements, called Periodic Videos.

Written by Eddy.