Four diamonds in the sky.
Three chevrons dropping by.
Two bits, Spin and Twin.
One diamond left untwinned.
A concern expressed by many across India was that whereas classes up to 10 teach science to all students, 11 and 12 specialize to physics, chemistry, mathematics, and biology. A student taking biology but not chemistry might never encounter the periodic table in their high school education.
Finding this a bit hard to believe, I downloaded the NCERT's science texts up to class 10. In Unit 4 of class 9's science textbook, on page 43, I found a detailed table of the electron configurations for the first 18 elements of the Periodic Table.
Now I graduated from high school in 1961, taking physics and chemistry as an alternative to biology and geology. Although we covered the first 20 elements in chemistry, we did not learn as much detail about their electron configurations as today's Class 9 students do in India. So I didn't feel that the latter were missing out on all that much.
That said, 1961 was 62 years ago. Why should India be satisfied with such an old standard?
And why should India, or any other country, feel that the Periodic Table is so complicated that it cannot be taught to Class 9, or at least Class 10, as part of a general science education?
Can the Periodic Table be made more accessible?
In 1957, just when I was entering high school, Edward Mazurs, a Latvian chemist who had fled Latvia for the US via Germany during WW2, had just published Types of graphic representation of the periodic system of chemical elements. It comprehensively surveyed and classified some 700 ways of organizing Mendeleev's periodic table. Seventeen years later, the American Chemical Society re-issued Mazurs' book, more nicely bound, under the title Graphic Representations of the Periodic System During One Hundred Years.
One very recent (1956) entry in his 1957 book (on page 85 of the 1974 edition) was a design due to K.K. Sugathan and T.C.K. Menon of the chemistry department of Sree Kerala varma College in Thrissur, a city in the south-west of India. Of all 700 designs, theirs was the only one to exploit Friedrich Hund's Rule of Maximum Multiplicity.
Although Mazurs was only mildly impressed by that design, it seemed to me to have great potential as an alternative to the IUPAC's table. So I came up with a way of packaging it, and, more importantly, relating it to the IUPAC's design via a continuous transformation that illustrates the structure of the Periodic Table dynamically. This journey-is-the-destination approach makes this linking of the two designs more than just the sum of their respective parts. So much so in fact that it should make perfect sense even to a high school student in Class 10, what we called in Australia half a century ago the Intermediate Certificate.
In the process of improving the arrangement of the 1956 version of the IPT, I belatedly came to realize that more could be said about the Aufbau Principle than could be found in either textbooks or online. In the following I'll give my perspective on it, and leave it to others to take it even further than I have.
These 118 elements are laid out in seven rows numbered 1-7 and called Periods, consisting of respectively 2, 8, 8, 18, 18, 32, and 32 elements. (These are the first seven integers in Sequence A093907 of the Online ncyclopedia of Integer Sequences.) To keep the layout narrow, each of periods 6 and 7 is split up into 18 + 14 elements, with the former 18 aligned with rows 4 and 5 and the latter 14 listed below periods 6 and 7 as respectively the Lanthanides (starting with lanthanum) and the Actinides (starting with actinium).
Periods 3,5, and 7 mimic Periods 2,4, and 6 respectively. We shall call these three odd-numbered periods the Tweedledum twin of their three even-numbered predecessors, the Tweedledee twin.
The first transformation slides the three periods with T = 1, Tweedledum, up and under their respective predecessors, Tweedledee. T = 0. (So T is the parity bit of each period.)
Even though Tweedledum can now no longer be seen (with the notable exception of Period 1 because there is no Period 0 to hide it), it will mimic all subsequent transformations of Tweedledee until the time comes to bring it back out of hiding, namely at Stage (v).
The original seven visible rows have now been reduced to four, numbered H = 0 to 3; we shall call these period pairs. Any references to periods will refer to the seven original periods, three of which are currently hidden.
Each subshell has a left half and a right half. Hund's Rule fills the left half with electrons having independent spins; we can think of those electrons as each being "spin-up". It then fills the right half by pairing each new electron with an electron from the left half, thinking of these new electrons as "spin-down".
When the subshell is full, all electrons are paired, with one pair
per orbital. The Indian Periodic Table represents this pairing by
aligning the right half with the left halt.
By inspection, the visible periods 1, 2, 4, and 6 contain 1, 4, 9,
and 16 small diamonds, namely the first four squares (as integers).
Summing these would seem to show that Tweedledee contains 30 small
diamonds.
Since each small diamond represents two elements, Tweedledee accounts
for 60 elements. So does the hidden Tweedledum, which would bring
the count up to 120 elements. However Period 1 actually belongs to
Tweedledum and there is no Period 0 in Tweedledee so we are missing
two elements. Hence there are only 118 elements.
The quantum numbers n and ℓ serve to determine the
shell and subshell respectively of each new electron as
we progress through the periodic table, guided by the Aufbau
Principle described below.
The main property of Pauli's quantum numbers for any given electron in
any given element is that together they uniquely determine the position
of that electron in that element's electron configuration in its
ground state, known as the Pauli Exclusion Principle.
But this principle holds equally of the quantum coordinates defined
at Stage (iii) of the IPT transformation. The main difference is
that the quantum number n is L ∸ 1 less than the period 2H+T,
giving a slightly "denser" packing of numbers in Periods 4 through
7 where ℓ can be 2 or more.
1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s 4f 5d 6p 7s 5f 6d 7p
A superscript after each subshell gives the number of electrons in that
subshell, with a maximum of 2, 6, 10, and 14 for respectively s, p, d,
and f. The periodic table starts with the single term 1s1
for hydrogen and ends with the 19 terms
The latter sequence is then broken up into seven periods. The first
period consists of 1s2 alone, and each of the remaining
six periods ends in np6 for n from 2 to 7. This yields the
seven periods on the left of the following table.
The next nine elements of subshell d in Period 4 are titanium
through zinc, with zinc being [Ar] 4s2 3d10
on the left and [Ar] 3d10 4s2 on the right.
Period 4 is then completed with [Ar] 4s2 3d10
4p6 on the left and [Ar] 3d10 4s2
4p6 on the right.
Period 5 is the same as Period 4 except with n increased by 1.
Periods 6 and 7 include subshell f before d.
However it can also be done more directly from the IPT quantum
coordinates H and L without using T, producing ten terms instead of 19.
Start with the four lines
Why this difference? Well, the Aufbau principle governs which subshell
each newly inserted electron "officially" belongs to as we step through
the periodic table one electron (and proton) at a time. Except for
Tweedledum being hidden in the Indian case, this classification of
electrons is the same for both the IUPAC and Indian periodic tables.
The electron configuration on the other hand specifies the distance
of the new electron from the nucleus. For Periods 1 through 3, new
electrons simply go "on top" (further from the nucleus). For Periods
4 onwards however, the d and f subshells go under the s
subshell, even though they entered later. Furthermore f almost
always goes under d, with lanthanum, cesium, and gadolinium as the
three exceptions we'll describe shortly.
Unlike the Aufbau principle, which by convention has no exceptions,
electron configurations have a number of exceptions. (These exceptions
are sometimes referred to as exceptions to the Aufbau principle,
which however only makes sense when treating that principle as the
same thing as electron configurations, deprecated here.)
The most notable exceptions are with H = L = 2 involving chromium Cr,
copper Cu, molybdenum Mo, and silver Ag. (This is one exception
expanded to four via the two bits Spin and Twin.) Their expected
configuration would end with four or nine d electrons, one less than
a full half-subshell. Inspection of their spectral absorption lines
reveals that one of the two s electrons has moved to the d subshell,
thereby reducing energy by completing a half-subshell. The next
element after each of these four, respectively manganese Mn, zinc Zn,
technetium Tc, and cadmium Cd, restores normalcy by "paying back"
the "borrowed" s electron.
There are additional scattered exceptions involving the d subshell
for niobium (Nb), ruthenium (Ru), rhodium (Rh), and palladium (Pd)
in Period 5, and platinum (Pt) and gold (Au) in Period 6, all of
which send one s electron to the d subshell except for palladium
which sends two.
One might expect similar behavior with Seaborgium (Sg)
and Roentgenium (Rg) in Period 7. However these have
superheavy nuclei, leading to the following
paragraph in the Wikipedia article for seaborgium.
Very few properties of seaborgium or its compounds have been measured;
this is due to its extremely limited and expensive production and
the fact that seaborgium (and its parents) decays very quickly. A few
singular chemistry-related properties have been measured, but properties
of seaborgium metal remain unknown and only predictions are
available.
The same is true a fortiori for roentgenium Rg, #111.
Three further exceptions of a quite different kind, foreshadowed
earlier, are found in the Lanthanides (the f subshell of Period
6) with lanthanum, cesium, and gadolinium, #57, #58, and #64.
But instead of H = L = 2, these involve H = L = 3. While the Aufbau
principle suggests that the new electrons for these should be in
subshell f, one of them is in d. So (using Pauli quantum numbers),
instead of [Xe]4f1 6s2 for lanthanum, the
configuration is [Xe]5d1 6s2. For cesium
it is [Xe] 4f1 5d1 6s2. And for
gadolinium it is [Xe] 4f7 5d1 6s2.
But for the rest of the f subshell, from praseodymium to ytterbium,
no f electrons have been demoted to the d subshell.
The Indian table's recognition of Hund's rule makes this
alignment of chromium with copper and molybdenum with silver more
visible. On the other hand the IUPAC table has the advantage that isodiagonality,
relating lithium to magnesium, beryllium to aluminum, etc. is more
directly depicted as diagonally related elements. And the six metalloids boron,
silicon, germanium, arsenic, antimony, and telluriam lying diagonally
in the four groups 13 through 16 and the four periods 2 through 5
are not so clearly connected in the Indian table, at least as laid out
in two dimensions.
As noted in the beginning, the interest with the Indian table is not so
much that it is somehow better than the IUPAC table. What is arguably
more interesting are the five steps (i) through (v) connecting the two
layouts, "the journey is the destination". That said, isodiagonality
becomes more visible with the three-dimensional stacked connection
of the IPT periods described in the following section.
It is not hard to see how to connect Periods 3 and 4, and likewise 5 and
6, forming Periods 2 through 7 into a chain of six periods.
Assuming no more elements beyond oganesson, this chain can be made
a loop by inserting Period 1 into the bottom of Period 7. We then
have a loop consisting of seven "large" diamonds, of sizes 1x1 up
to 4x4. All but 1x1 are twins, accounting for the formula Whether fabricated as a chain or a loop, there are weak links
between consecutive large diamonds that call for reinforcement.
One approach would be to attach each of the seven large diamonds to
a supporting cord running the full length of the chain or loop.
As an item of jewelry, for the sake of symmetry a necklace could be
made as a loop with the three even periods on one side and their three
successors on the other. Period 1 could go in between, either at the
clasp at back, or more visibly as a pendant at the front. There are
many gemstones of the appropriate colors for the four orbital forms.
Earrings to match can be organized as a chain of periods 2,4,6
in one earring and 3,5,7 in the other, omitting Period 1. Since
earrings should not be too heavy, the color scheme should ideally be
fabricated from sufficiently light glass, ceramics, or gemstones of
the appropriate colors.
This stacking leaves the yellow s orbital forms "out on a limb".
A simple solution is to start from the loop form of the above
sequential connection, break the "large diamonds" at their weak links,
and stack them so that the s orbital forms are also aligned vertically.
The obvious complaint with that stacking is that it associates Period
1 with Period 7, and the other s orbitals with the preceding period.
Thus Li-Be becomes the 1x1 "large" diamond on its own, Na-Mg sits at the
bottom of Period 2, K-Ca at the bottom of Period 3, and so on.
We can address that complaint by sliding the column of s orbitals down
one place in order to associate them with the period they belong to.
This makes Period 1 the 1x1 diamond, Li-Be is now part of Period 2 but
at the bottom instead of the top, and so on.
The upshot is that each period has now been made an intact large
diamond by moving the s orbital to the bottom. Those s orbitals can
be viewed as forming the vertical spine of a tilted pyramid with base
Period 7. Hence when enumerating the elements in a period, we start
with the two s orbital elements at the spine and then we jump to the
top of the period at the far leaning edge to enumerate the remaining
elements, exiting the period at the bottom right p orbital.
The quantum coordinates (H, L, M, S, T) described in Stage (iii)
of Section 2 can be interpreted as five axes for five dimensions.
H moves vertically up from 0 to 3, with T = 0 then 1 as a minor
upward move. L moves horizontally outwards from the spine, starting
at 0 and stopping at H. M gives the position within chevron L,
namely from -L to L. Lastly S gives the spin, 0 for up, 1 for down,
of the most recently added electron.
This three-dimensional geometry for the IPT makes isodiagonality
more visible. In particular, lithium is now adjacent to magnesium,
beryllium is adjacent to aluminum, and so on.
This completes our description of the Indian Periodic Table. We now
move on to mnemonics for the elements in the first three periods.
Henry Higgins lives. Besides, Billy can not overlook Flossy Newman's
nature magazine, although sick people should climb after killing cats.
More recently, it occurred to me to associate Period 1 with the two
colors of the chess board. Starting with a white board representing
hydrogen, adding the 32 black squares to it represents helium.
Ignoring the pawns, there are eight pieces at each end, white
and black. These can represent the eight elements of Periods 2 and 3
respectively. The one difference from regular chess is that Chemical
Chess interchanges the knight and the castle at all four corners.
Starting with Period 2, we view the four pieces on the left as female
and the four on the right as male.
In the middle are the two royal pieces, Queen Lizzie for lithium and
King Bertie for beryllium. Moving to the right, we have Bishop Boron,
Castle Carbon, and Knight Nitrogen. To the left we have an Ordained
female Bishop for oxygen, a Female Fort or Fortress for fluorine,
and a female knight or Dame Nellie Melba for neon. Alternatively,
Officer Florence Nightingale can cater for all three.
Moving on to Period 3, in the middle are Mother Nature for sodium
and The Magistrate for magnesium. To the right, the ASP that killed
Cleopatra can serve for aluminum, silicon, and phosporus. And to the
left, we have a sultana for sulfur, a castle for chlorine,
and Arthur as the King of the knights of the round table for argon.
To summarize:
Period 1: Hydrogen, helium.
Period 2:
Period 3:
Chorus: We're all made out of chemical chess, chemical chess,
chemical chess.
Hydrogen, and helium.
Repeat Chorus
Sodium, magnesium.
Repeat chorus
For Periods 4 through 7, there are a hundred elements, far too many to
have to memorize by name. The next section addresses their structure.
Again we can ask, is there an easier proof? Well, before proving it, we
can observe a famous instance of it in the Periodic Table.
The two large diamonds in Tweedledee are of respective sizes 3x3 and
4x4, therefore consisting of 9 and 16 small diamonds respectively.
These correspond to the Pythagorean triple (3,4,5).
Can this instance of the Pythagorean Theorem be easily generalized?
Among the great many ways of proving the Pythagorean Theorem,
what is special about tangram proofs is that neither algebra nor
area distortion is involved, merely rearrangements of finitely many
polygonal areas.
A natural question then is, what is the least number of polygons
needed for a tangram proof?
Here we prove the theorem with three polygons. Two of them are copies
of the triangle itself, and the third polygon is not convex, but can
be dissected into two convex polygons. (The white line separating the
two squares can serve to separate the body into two convex polygons.)
The two copies can be viewed as the two wings of a flying proof whose
body is the third non-convex polygon.
Click here to see the proof in action for
three different values of a and b.
The key to how this proof works is the equation a + b = b + a, which
is the key equation defining Euclidean space. We create two copies of
the triangle with sides a and b by ignoring the point where the squares
a and b meet at the bottom. Instead we look at the point where b and
a would meet at the bottom if the squares were interchanged. At that
point, we make two cuts that create two triangles with sides a and b,
and swing them up like wings of a bird. Rotating them both through
exactly three quarters of a revolution, they land neatly on the body
of the "bird" to form a square of side h where h is the hypotenuse
of each wing triangle. The four sides of that square are created at
the instant the two cuts are made, each of length h, that separate the
wings from the body. The wings then fold neatly onto the body leaving
only the four cuts of length h at the boundary of the resulting square.
Stage (iv). BEND the subshells down to create chevrons.
A chevron is a V shape, inverted in this case, and segmented into small
(1x1) diamonds. As stacked, they form a diamond of side H+1, but with
the s subshell relocated to the top.
Stage (v). SHOW Tweedledum: pull it from behind Tweedledee.
This almost doubles the number of elements. But as noted above,
Period 1 belongs to Tweedledum, and has no twin in Tweedledee.
(Apparently nature abhors zero electrons.) So instead of doubling
60 elements to 120, we only have 118. We can express this number
as The Pauli quantum numbers
The Pauli
quantum numbers (n, ℓ, mℓ,
ms) correspond to the IPT quantum coordinates
(H, L, M, S, T), with the exception of the principal quantum number n.
The Aufbau principle
The Aufbau
Principle, also known as the Madelung rule, is a rule for ordering
subshells based on the first two of Pauli's quantum
numbers, n and ℓ. Each subshell is represented as nℓ but with
ℓ written as the corresponding letter. For example, when n =
4 and ℓ = 2, nℓ is written 4d. The rule is that nℓ
precedes n'ℓ' when n+ℓ < n'+ℓ', or if n+ℓ =
n'+ℓ' then when n < n'. This rule gives the following ordering of
subshells.
1s2
2s2 2p6
3s2 3p6
4s2 3d10 4p6
5s2 4d10 5p6
6s2 4f14 5d10 6p6
7s2 5f14 6d10 7p6
for Oganesson.
The Indian Aufbau principle and electron configurations
The Aufbau principal and the electron configurations use Pauli's
quantum numbers. A straightforward way of doing this with the IPT
would be to just translate from Indian quantum coordinates to Pauli
quantum numbers using H, L, and T.
0s
1s 1p
2s 2p
3s 3p.
Then expand the third and fourth lines with respectively 2d and 3f 3d
to give
0s
1s 1p
2s 2d 2p
3s 3f 3d 3p
for the Aufbau
principle and
0s
1s 1p
2d 2s 2p
3f 3d 3s 3p
for the
electron configurations. The difference is whether the expansion
with d (plus f for the fourth line) takes place after or before s.
Connecting the IPT Periods
With physical artifacts in mind such as jewelry, we give two ways
of connecting the periods as laid out in the Indian table, namely
sequential and stacked.
Sequential Connection of the IPT Periods
In our layout of the Indian Periodic Table, Periods 2 and 3 are already
connected by inserting the s-orbital diamond of Period 3 in the gap
below the p-orbital chevron of Period 2. Moreover the corresponding
connection is made between Periods 4 and 4, as well as between Periods 6
and 7.
Stacked Connection of the IPT Periods
There have been several proposals to organize the periodic table
in three dimensions. The Indian Periodic Table lends itself to an
obvious stacking that aligns each of the p, d, and f orbital forms
vertically, with Period 2 at the top and Period 7 at the bottom.
II. Chemical Chess
Like many students, when I was in high school I made up a nonsense
mnemonic for the first 20 elements of the periodic table. Here's mine.
Lizzie (lithium), Bertie (beryllium).
Bishop (boron), Castle (carbon), kNight (nitrogen)
Ordained bishop (oxygen), Fortress (fluorine), Dame Nellie (neon)
Nature (sodium), Magistrate (magnesium)
A (aluminum) S (silicon) P (phosphorus)
Sultana (sulfur), Castle (chlorine), Arthur (argon).
III. We're all made out of chemical chess
(Tune: Yellow Submarine)
We're all made out of chemical chess, chemical chess, chemical chess.
Lithium, ber-yllium.
Boron, carbon, nitrogen.
Oxygen and fluorine.
(Basso profundo) Neon, neon.
Aluminum and silicon.
Phosporus and Sulfur, too.
Chlorine, and (rising) Ah-ah-ah-ah-ar-gon!
IV. The Pythagorean theorem
The NCERT also removed the Pythagorean Theorem from classes up to X,
presumably for the same reason it removed the periodic table: it is
too hard for those younger students.
The right-hand half of the IPT
The IPT has 18 elements on the left and 100 on the right. The latter
can be related to the Pythagorean triple (6,8,10) solving the equation
a² + b² = h² where h is the length of the hypotenuse of a right
triangle with the other two sides of lengths a and b. There are 6²
= 36 elements with H = 2 (Periods 4 and 5) and 8² = 64 with H =
3 (Periods 6 and 7).
A 3-tangram proof of the Pythagorean Theorem
The standard
tangram consists of seven convex pieces. If you google for
"tangram pythagorean theorem" you will find many ways of proving the
Pythagorean Theorem using tangrams. None of these use the standard
7-piece tangram set. The main point is to exhibit how pieces that
can be assembled as two squares of sides a and b can be reassembled
as one square of side h satisfying a² + b² = h².
