## The Periodic Table

I like many kinds of periodic table, but hate this one. See the problem?

Element 57 is drawn right next to element 72, replacing the element that should be there: element 71. So lutetium, element 71, is being denied its rightful place as a transition metal and is classified as a rare earth. Meanwhile lanthanum, element 57, which really is a rare earth, is drawn separately from all the rest! This is especially ironic because those rare earths are called ‘lanthanoids’ or ‘lanthanides’.

Similarly, element 89 is next to element 104, instead of the element that should be there: element 103. So lawrencium, element 103, is also being denied its rightful place as a transition metal. Meanwhile actinium, element 89, is banished from the row of ‘actinoids’, or ‘actinides’ — even though it gave them their name in the first place. How cruel!

Here Wikipedia does it right. Element 71 is a transition metal — not element 57. Similarly element 103 is a transition metal, not element 89.

This stuff is not just an arbitrary convention. Transition metals are chemically different from lanthanides and actinides. You can’t just stick them wherever you want.

In simple terms, as we move across the transition metals, they fill 1, 2, 3, … , 10 of their outermost d orbitals with electrons. Similarly as we move across the lanthanides or actinides, they fill 1, 2, 3, … 14 of their outermost f orbitals with electrons. I wrote about this here a while ago:

• John Baez, The Madelung rules, Azimuth, December 8, 2021.

There are some exceptions to the Madelung rules, but the bad periodic tables are not motivated by those exceptions. The Wikipedia periodic table accurately reflects the chemistry. The Encyclopedia Brittanica table completely ruins the story by arbitrarily sticking lanthanum and actinium in amongst the transition metals instead of the elements that should be there: lutetium and lawrencium. I see no good reason for doing this.

Here’s another common kind of periodic table that I hate. It cuts a hole into the bottom two rows of the transition metals, and moves the metals that should be there — elements 71 and 103 — into the rare earths and actinides.

This amounts to claiming that there are 15 rare earths and actinides, and just 9 transition metals in those two rows. That’s crazy: the fact that the p subshell holds 10 electrons and the d subshell holds 14 is dictated by group representation theory. Subshells hold 2, 6, 10, 14 electrons — twice odd numbers.

The periodic table is a marvelous thing: it shows how quantum
mechanics and math predict patterns in the elements. Have fun making up new designs — but if you’re going to use the old kind, use the good one!

If you don’t believe me, listen to this guy:

But unlike him, I don’t think experiments were necessary to realize that the bad periodic tables were messed up. It’s not as if they were designed based on some alternative theory about which elements are transition metals.

Interestingly, the International Union of Pure and Applied Chemistry were supposed to meet at the end of last year to settle this issue. What did they decide? If you find out, please let me know!

### 28 Responses to The Periodic Table

1. Raphael says:

Where then to put the hydrogen (H) then? Sure it has only one electron so the 1st row would be good enough, right? But then it’s shell is half filled, so why not above C, and also it misses just one electron to the noble gas configuration, so why not above F? All these choices interestingly have some chemical justification. However, I am personally quite happy with column 1, for a certain reason. There is something we call in German “Schrägbeziehung”. It’s a concept that is to my knowledge not very well established in the English speaking world. The literal translation would be “cross relation” which is not to be found in Engl. text books and extremely rarely found in the literature. You rather have the term of “ionic potential” which goes right to the origin of the phenomenon. The phenomenon is that all first elements of each column are a bit of a black sheep of the family (group). For example Li is relatively hard, it cannot be cut by a knife, B is a semi conductor or even non-metal. The observation is that these black sheep are in certain properties (mostly the properties of the elements, but also in certain aspects of coordination chemistry) more similar to their right bottom neighbors: Li to Mg, Be to Al, B to P and that is were the concept slowly breaks down. This is since it physically goes back to the ratio of charge to radius of the most stable positive ion. Which is for C not of great relevance and even has some ambiguity. You see here why you call it “ionic potential”. Now comes my point for H in the first group: I did a bit of research on the element Be and at some stage we noticed that there are certain properties in its coordination chemistry that put it into analogy with H and from a certain point of view that charge/radius ratio are quite similar for H+ and Be++ (No, the radius of H+ is NOT 0!) . For that I sometimes call it in my lectures the “versteckte Schrägbeziehung” (hidden cross relation) since no one mentioned it and it frankly is a bit of a loony idea, but I like it, since it makes some facts on chemistry stick to your mind, and lots of chemical theory is appreciated among experimental chemists mostly for that. It’s not that it ‘explains’ stuff really but it makes some important phenomena stick to your mind. In that sense many chemical concepts are more like mnemonical tools rather than really “ab-initio explanantions”. An aspect of chemistry that I guess might feel a bit alien to trained mathematicians.

• John Baez says:

Thanks for your comment! Before I delve into it more deeply, I’ll just mention that these Schrägbeziehungen are called “diagonal relations” in English and I wrote about them here:

• John Baez, Diary, December 27, 2021.

I find them fascinating!

• Raphael says:

This is good to know, thank you very much!

2. Steven says:

There are only 6 p orbitals and 10 d orbitals. Did you mean d and f orbitals?

• John Baez says:

Yes, thanks—I’ll fix that. The 10 transition metals in each row are busy filling the 10 d orbitals, while the 14 lanthanides and actinides are busy filling the 14 f orbitals.

3. Wolfgang says:

As a chemist by training I always had an odd feeling about this, because the graphical plot breaks down just for reasons of space not of science. But now I am shocked that it was that bad. Never thought about it in detail, never cared much for the lanthanoids and actinoids, never observed this discrepancy myself before. As an aside, I remember that in one of my schools we had a huge periodic table of elements in the classroom, which could be folded in such a way so that one either only saw the main group elements, or the transition metals added, and maybe the same again for the lanthanoids/actinoids, I don’t remember that (at school there are mostly neglected anyhow). It was impressive, since to fold it out during class a part of the pupils had to move their heads down first, it was really huge :). Of course, there is a rich history of attempts to capture the science of the periodic table in better graphical ways.

• John Baez says:

If you don’t mind long periodic tables you can put the lanthanides and actinides in a more logical place, but this is inconvenient.

4. Todd Trimble says:

Inconvenient certainly for Homer Simpson, who wouldn’t be able to fit it all on the palm of his hand.

• Todd Trimble says:

Hmm, that was meant to be in response to JB’s comment at 3:11am.

5. Cesar Lima says:

While I understand your rationale, I’m a bit puzzled by your comment that there the choice is not arbitrary because it involves a chemical difference. To be clear, when you mention chemical properties I think of what kind of compounds a given element forms and what properties those compounds have.

Obviously, this is difficult for Lawrencium, but what kind of chemical properties do you see Lutetium having in common with transition metals that Lanthanum doesn’t have?

For instance, I would say that the prototypical transition metal has more than one oxidation state differing by a single charge and colored solutions. It is also common for transition metals to have high melting points.

I don’t know a lot about group 3 elements, but a basic search seems to show that all group 3 elements have a single oxidation state (except for rare situations) and mostly white or pale solutions.

By the same reasoning, I strongly object to group 12 elements being called transition metals.

I agree that group theory requires your version of the periodic table, but by chemical properties alone the situation seems much more ambiguous to me.

• Raphael says:

Look for example at the ionization potentials: https://en.wikipedia.org/wiki/Ionization_energy which is (very) well defined and in the eyes of the Physicist clearly a chemicial property (while for the Chemist it’s a physical one, but that’s just kidding). You cannot obtain these from group theory, you can obtain them however from purely physical laws.

• Cesar Lima says:

I’m glad I’m not alone in being befuddled by the way chemists and physicists like to debate terminology. And Ionization energies seem to dictate Lutetium and Lawrencium in group 3. Which is the point of the video if I’m not mistaken.

But by reading the charts in the wiki link, I got the impressions that all group 3 and lanthanides seem to have roughly the same ionization energies, and that transition metals have a different pattern.

So while I agree that Lutetium and Lawrencium belong in the places John Baez has prescribed, I still think is more reasonable to call transition metals as only the elements in groups 4 to 11. What I’ve questioned is whether there is a good argument to calling Lutetium a transition metal, since that would suggest treating Scandium also as a transition metal, which I object since the element whose chemistry is most similar is Aluminium, a definitely post-transition element.

• Cesar Lima says:

Maybe I should better explain my intent. From the chemical properties point-of-view, it seems that there are several different possible periodic tables that highlight different properties. If one considers only the properties relating to chemical reactions and compound properties, it doesn’t seem clear to me to put group 3 elements inside transition metals.

Consider elements Thorium to Plutonium. All have chemistry typical of transition elements, being very similar to elements Hafnium to Rhenium, that is despite being clearly in the actinide series their chemistry does resemble transitions metals.

As so the periodic table organization seems to be more complex than what you’ll get out of single-body quantum mechanical reasoning.

• John Baez says:

Cesar Lima wrote:

By the same reasoning, I strongly object to group 12 elements being called transition metals.

I don’t want to argue about the term ‘transition metals’—though you’ll notice all 3 periodic table above call group 12 elements ‘transition metals’. What matters to me is that the ten groups 3 through 12 correspond to the ten electrons that can fit into the d subshell. Quantum mechanics and group representation theory dictate that exactly ten electrons can occupy a d subshell, as I explained here. And this physics and mathematics also explains why group 12 is dramatically different from groups 3 through 11: in group 12, the d subshell is completely full!

By coincidence, or non-coincidence, I wrote about group 12 elements in my January 18 diary entry. My focus was on mercury, but the weird properties of mercury are just an extreme version of the weird properties of the group 12 elements, which seem to arise from having their outermost electrons in a full d subshell.

I agree that group theory requires your version of the periodic table, but by chemical properties alone the situation seems much more ambiguous to me.

Since I’m a mathematician, I don’t expect that chemists will care what I think. To me, it’s wonderful how the periodic table follows an utterly simple and understandable pattern when drawn like this:

(except for putting helium above neon instead of beryllium, which we do for obvious chemical reasons), while the other two periodic tables I showed break those patterns for rather weak reasons.

My claiming to be outraged by the ‘bad’ periodic tables is mainly a way to get people interested. I don’t lie awake at night dreading IUPAC’s verdict on this case—though I’m curious to see what they’ll say.

• Cesar Lima says:

When I objected to calling group 3 and 12 elements transition metals I was trying to generate curiosity on some weird stuff in the periodic table. That you were also using moral language to garner interest completely passed me by, and now I’ve made of a fool of myself. I do apologize.

For context, which elements belong to group 3 has a big history of debate and chemists have exchanged bitter letters in public regarding it. This is probably why IUPAC was compelled to say something. In case you are interested, the book “The Periodic Table – Past, Present, and Future” by Geoff Rayner-Canham summarizes this question (as well as many other debates regarding what are the correct placements in the periodic table).

As a last note, in your diary entry, you pose the question of the similarity between zinc and magnesium. As you correctly note the full d shell makes the group 12 elements to behave similarly to main-group elements. As the book I mentioned outlines, there is a general connection between groups n and groups n+10 because of this. So not only zinc is similar to magnesium, the chemistry of chlorine has parallels with manganese and so on. The exception being group 11 that is very dissimilar from group 1.

• John Baez says:

Cesar Lima wrote:

When I objected to calling group 3 and 12 elements transition metals I was trying to generate curiosity on some weird stuff in the periodic table. That you were also using moral language to garner interest completely passed me by, and now I’ve made of a fool of myself. I do apologize.

No need to apologize! I’m fascinated by everything about the periodic table right now, so anyone who knows about these things is my friend.

For context, which elements belong to group 3 has a big history of debate and chemists have exchanged bitter letters in public regarding it.

That’s really interesting. I’m sort of blundering into this myself, noticing things like how Encyclopedia Britannica counts them as honorary rare earths.

This is probably why IUPAC was compelled to say something. In case you are interested, the book The Periodic Table – Past, Present, and Future by Geoff Rayner-Canham summarizes this question (as well as many other debates regarding what are the correct placements in the periodic table).

Thanks, that sounds really fun!

I read an article that mentioned the similarity between manganese and chlorine. The similarities listed seemed a bit too subtle for me: I can’t imagine mixing up these elements. But maybe when I learn more I’ll appreciate it.

• Cesar Lima says:

Regarding manganese and chlorine, the idea is not that cryptic. Mendeleev noted that fluor, chlorine, bromine, and iodine all formed similar compounds in the same proportions (e.g. all form acids of formula HX, X standing for one of the halogens). In the same vein the permanganate ion $MnO_4^{-}$ and perchlorate ion $ClO_4^{-}$ are strong oxidizing agents. Both elements share other compounds of similar properties. Indeed Mendeleev’s first periodic table had only eight columns, with Manganese and Bromine sharing the same box.

I suppose the strangeness comes from one element being a metal and the other a nonmetal. But as you showed the periodic table is really dominated by quantum mechanics and group theory. My favorite example is the ion $NH_4^{+}$. This ion is isoelectronic to the sodium ion, and as such, they form very similar compounds. So strong is their resemblance that even though ammonium ion is definetly not a metal it will form an amalgam with mercury very much like sodium (and other metals) do. There is a interesting video on youtube of a guy preparing this amalgam.

In case you (or anybody else) would like to check on this king of things without reading the whole book I recommended above, there is a four page paper that summarizes all of this, Rayner-Canham, G. (2000). Periodic Patterns. Journal of Chemical Education, 77(8), 1053.

• Where to put helium? You mentioned the chemical reasons, but by analogy with the yellow, blue, and green elements, one could argue that it should go where it is above.

• “ My claiming to be outraged by the ‘bad’ periodic tables is mainly a way to get people interested. I don’t lie awake at night dreading IUPAC’s verdict on this case—though I’m curious to see what they’ll say.“

It‘s only a matter of time before some TRA attacks you on twitter for abusing the term “transition”. :-)

• ”My claiming to be outraged by the ‘bad’ periodic tables is mainly a way to get people interested. I don’t lie awake at night dreading IUPAC’s verdict on this case—though I’m curious to see what they’ll say.”

That shows that you’re a mathematician. A physicist would take it more personally:

A colleague who met me strolling rather aimlessly in the beautiful streets of Copenhagen said to me in a friendly manner, “You look very unhappy,” whereupon I answered fiercely, “How can one look happy when he is thinking about the anomalous Zeeman effect?

—Wolfgang Pauli

• John Baez says:

Phillip write:

Where to put helium? You mentioned the chemical reasons, but by analogy with the yellow, blue, and green elements, one could argue that it should go where it is above.

In my work on mathematics related to the periodic table I put it in group 2 directly above beryllium, because like beryllium and the other group 2 elements its last electron completes an s shell. But of course it acts like a noble gas, not an alkali earth. So I can completely understand why chemists put it in group 18 above neon—even though the rest of the group 18 elements have their last electron filling a p subshell!

Wikipedia’s periodic table has it both ways, putting helium in group 18 but coloring it pink like the other group 2 elements:

• My gut feeling is that having an outermost shell complete is the important bit. But I’m neither a mathematician nor a chemist, but rather something of an astronomer. In astronomy, all elements apart from hydrogen and helium are known as “metals”. :-)

• John Baez says:

The subshells get filled in approximately this order, as explained in my article on the Madelung rules:

There are some exceptions but I don’t think they matter for the noble gases. The noble gases occur when the 1s, 2p, 3p, 4p, 5p, … subshells are filled. So you’ll see noble gases occur when the 1 and 2 shells are filled, but not when the 3, 4, 5, … shells are filled.

6. I may never sleep again now that I know of this travesty. The OCD (or rather CDO as I place this diagnosis appropriately in alphabetical order), is astounded such “brilliant” scientists could impose such shenanigans!

7. By the way…I only look at this table periodically and thus never noticed before.

8. frankwilhoit says:

Obviously, this question is going to fit right in next to the question of whether centuries begin with years congruent to 0 or 1 mod 100.

• John Baez says:

No, because the periodic table is making claims about about the physical and chemical nature of the elements on the chart: elements in a given column should have a similar outermost subshell.

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