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 The data used below are somewhat out of date – published in 1972 – but are provided mainly for illustrative purposes in any case (although I haphazardly supplement them from modern sources); my aim is to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):

Simply Cycled Periodic Table of the Chemical Elements

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:

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. 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).

Slant-Cycled Periodic Table of the Chemical Elements

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 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. 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):

Block-Cycled Periodic Table of the Chemical Elements
NbMoTcRuRhPdAgCd InSnSbTeIXeCsBa LaCePrNd
PmSmEuGdTbDyHoErTmYb LuHfTaWReOsIrPtAuHg TlPbBiPoAtRnFrRa AcThPaU

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:

Centre-Cycled Periodic Table of the Chemical Elements
ScTiVCrMnFeCoNiCuZn GaGeAsSeBrKrRbSr
YZrNbMoTcRuRhPdAgCd InSnSbTeIXeCsBa LaCePrNdPmSmEu GdTbDyHoErTmYb
LuHfTaWReOsIrPtAuHg TlPbBiPoAtRnFrRa AcThPaUNpPuAm CmBkCfEsFmMdNo

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

Fully Cycled Periodic Table of the Chemical Elements
GaGeAsSeBrKrRbSr YZrNbMoTcRuRhPdAgCd
InSnSbTeIXeCsBa LaCePrNdPmSmEu GdTbDyHoErTmYb LuHfTaWReOsIrPtAuHg
TlPbBiPoAtRnFrRa AcThPaUNpPuAm CmBkCfEsFmMdNo LrRfDbSgBhHsMtDsRgCn

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 (and, aside from some simple cycling, my closest match to Brian Clegg's Metals and non-metals version in his 2021 December 23rd Royal Institution lecture):

Orbital-aligned Periodic Table of the Chemical Elements
ZrNbMoTcRuRhPdAgCd InSnSbTeIXeCsBa LaCePrNdPmSmEuGd TbDyHoErTmYbLu
HfTaWReOsIrPtAuHg TlPbBiPoAtRnFrRa AcThPaUNpPuAm CmBkCfEsFmMdNoLr

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.

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.


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:

Transposed table of the chemical elements, listing isotopes
138,139 57La 225,227,228 89Ac
136,138,140,142 58Ce 230,232 90Th
141 59Pr 231 91Pa
142,143,144,145,146,148,150 60Nd 234,235,238 92U
145,147 61Pm 237,239 93Np
144,147–149,150,152,154 62Sm 238,239,242 94Pu
151,153 63Eu 243 95Am
152,154–158,160 64Gd 247 96Cm
159 65Tb 247 97Bk
156,158,160–164 66Dy 251 98Cf
165 67Ho 252 99Es
162,164,166–168,170 68Er 257100Fm
169 69Tm 258101Md
168,170–174,176 70Yb 259102No
45 21Sc 89 39Y 175,176 71Lu 255103Lr
46–48–50 22Ti 90–92,94,96 40Zr 174,176–180 72Hf 267104Rf
50,51 23V 93 41Nb 180,181 73Ta 268105Db
50,52–54 24Cr 92,94–98,100 42Mo 180,182,183,184,186 74W 269106Sg
55 25Mn 99 43Tc 185,187 75Re 270107Bh
54,56–58 26Fe 96,98–102,104 44Ru 188–190,192 76Os 269108Hs
59 27Co 103 45Rh 191,193 77Ir 278109Mt
58,60–62,64 28Ni 102,104–106,108,110 46Pd 190,192,194–196,198 78Pt 281110Ds
63,65 29Cu 107,109 47Ag 197 79Au 282111Rg
64,66–68,70 30Zn 106,108,110–114,116 48Cd 196,198–202,204 80Hg 285112Cn
10,115B 27 13Al 69,71 31Ga 113,115 49In 203,205 81Tl 286113Nh
12,13,146C 28–30 14Si 70,72–74,76 32Ge 112,114–120,122,124 50Sn 202,204,206–208 82Pb 289114Fl
14,157N 31 15P 75 33As 121,123 51Sb 209 83Bi 290115Mc
16–188O 32–34,36 16S 74,76–78,80,82 34Se 120,122–126,128,130 52Te 210 84Po 293116Lv
1,2,31H 199F 35,37 17Cl 79,81 35Br 127 53I 206,215 85At 294117Ts
3,42He 20–22 10Ne 36,38,40 18Ar 78,80,82–84,86 36Kr 124,126,128–132,134,136 54Xe 220,222 86Rn 294118Og
6,73Li 23 11Na 39,40,41 19K 82,87 37Rb 133 55Cs 223 87Fr
94Be 24–26 12Mg 40,42–44,46,48 20Ca 84,86–88 38Sr 130,132,134–138 56Ba 223,224,226,228 88Ra

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):

Transposed table of chemical elements, listing atomic masses
138.9100 57La227 89Ac
140.1200 58Ce232.038 90Th
140.9070 59Pr231 91Pa
144.2400 60Nd238.030 92U
145 61Pm237 93Np
150.3500 62Sm242 94Pu
151.9600 63Eu243 95Am
157.2500 64Gd247 96Cm
158.9240 65Tb249 97Bk
162.5000 66Dy251 98Cf
164.9300 67Ho254 99Es
167.2600 68Er253100Fm
168.9340 69Tm256101Md
173.0400 70Yb254102No
44.9560 21Sc88.9050 39Y174.9700 71Lu257103Lr
47.9000 22Ti91.2200 40Zr178.4900 72Hf261104Rf
50.9420 23V92.9060 41Nb180.9480 73Ta262105Db
51.9960 24Cr95.9400 42Mo183.8500 74W266106Sg
54.9380 25Mn99 43Tc186.2000 75Re264107Bh
55.8470 26Fe101.0700 44Ru190.2000 76Os269108Hs
58.9332 27Co102.9050 45Rh192.2000 77Ir268109Mt
58.7100 28Ni106.4000 46Pd195.0900 78Pt281110Ds
63.5400 29Cu107.8700 47Ag196.9670 79Au272111Rg
65.3700 30Zn112.4000 48Cd200.5900 80Hg285112Cn
10.81105B26.9185 13Al69.7200 31Ga114.8200 49In204.3700 81Tl286113Nh
12.01116C28.0860 14Si72.5900 32Ge118.6900 50Sn207.1900 82Pb289114Fl
14.00677N30.9738 15P74.9216 33As121.7550 51Sb208.9800 83Bi290115Mc
15.99408O32.0640 16S78.96 34Se127.600 52Te210.0000 84Po293116Lv
1.00791H18.99849F35.4530 17Cl79.9090 35Br126.9044 53I210.0000 85At294117Ts
4.00262He20.1830 10Ne39.9480 18Ar83.8000 36Kr131.3000 54Xe222.0000 86Rn294118Og
6.93903Li22.9898 11Na39.1020 19K85.4700 37Rb132.9050 55Cs223.0000 87Fr
9.01224Be24.3120 12Mg40.0800 20Ca87.6200 38Sr137.3400 56Ba226.0500 88Ra120??

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 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 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:

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.

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.

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