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Torah and the Periodic Table
Kabbalah and Chemistry
1. Theoretical Background
One of the most well known
and ubiquitous symbols of modern science in general and chemistry in
particular is the Periodic Table of the Elements. The modern
periodic table has been almost 300 years in the making. Early efforts
to group elements produced the tables of Geoffroy (1718) and Lavoisier
(1787). The atomic theory formulated by Dalton in the early 1800s provided
chemists with a solid basis to classify elements, and the theory stimulated
vigorous experimentation that culminated in the development of the modern
form of the periodic table in 1869 (See figure 1).
1
H |
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2
He |
3
Li |
4
Be |
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5
B |
6
C |
7
N |
8
O |
9
F |
10
Ne |
11
Na |
12
Mg |
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13
Al |
14
Si |
15
P |
16
S |
17
Cl |
18
Ar |
19
K |
20
Ca |
21
Sc |
22
Ti |
23
V |
24
Cr |
25
Mn |
26
Fe |
27
Co |
28
Ni |
29
Cu |
30
Zn |
31
Ga |
32
Ge |
33
As |
34
Se |
35
Gr |
36
Kr |
37
Rb |
38
Sr |
39
Y |
40
Zr |
41
Nb |
42
Mo |
43
Tc |
44
Ru |
45
Rh |
46
Pd |
47
Ag |
48
Cd |
49
In |
50
Sn |
51
Sb |
52
Te |
53
I |
54
Xe |
55
Cs |
56
Ba |
57
La |
72
Hf |
73
Ta |
74
W |
75
Re |
76
Os |
77
Ir |
78
Pt |
79
Au |
80
Hg |
81
Tl |
82
Pb |
83
Bi |
84
Po |
85
At |
86
Rn |
87
Fr |
88
Ra |
89
Ac |
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58
Ce |
59
Pr |
60
Nd |
61
Pm |
62
Sm |
63
Eu |
64
Gd |
65
Tb |
66
Dy |
67
Ho |
68
Er |
69
Tm |
70
Yb |
71
Lu |
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90
Th |
91
Pa |
92
U |
93
Np |
94
Pu |
95
Am |
96
Cm |
97
Bk |
98
Cf |
99
Es |
100
Fm |
101
Md |
102
No |
103
Lr |
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Figure 1
From the early rudimentary groupings of chemical compounds to
our modern classification that recognizes the periodicity of atomic elements together,
chemical tables are usually based on an implicit theory of the composition
of matter . These theories
have shared a common axiom: that all of the matter in the Universe is
composed of a finite variety of basic building blocks. These building
blocks have been known from the ancient Greeks to the present as atoms.
In our modern table of the chemical elements, the different atoms are
identified and ordered by their atomic number. Atoms are defined as the
smallest unit of an element that can combine with another element. Atoms
are theorized as composing of a nucleus, made of protons and neutrons,
and electrons that move around the nucleus. The atomic number identifies
the number of protons in an element’s nucleus. Atoms can lose or gain
electrons, and the ease with which they do so is a measure of their reactivity.
In our modern periodic table of elements, elements are arranged in columns
and rows. As its name implies, the modern table is periodic in
nature, meaning that elements are placed in it based on their shared
and recurring (periodic) characteristics. Periodicity of element properties
is found to be strongest down columns of the table. Primary among these
periods is that of the 6 noble (or inert) gases which populate the far
right column of the table. The property shared by the inert gases is
a lack of reactivity ensuing from their inability to gain or lose electrons.
Another example: the first element in the table, Hydrogen (H) is a gas,
the second, Helium (He), is a noble gas, and the third, Lithium (Li),
is a soft, reactive metal. Going down the table, we find eight elements
later Fluoride (F), Neon (Ne) and Sodium (Na), a gas, a noble gas, and
a soft, reactive metal, and eight elements later, Chloride (Cl), Argon
(Ar) and Potassium (K)—again: a gas, a noble gas, and a soft, reactive
metal.
The remarkable predictability of element properties revealed by the periodic
table allowed chemists to ‘describe’ as yet unidentified elements based
on their supposed location in the table. Such was the case when in 1871
Dimitry Mendeleev, the Russian chemist who originally formulated the periodic
law, correctly described the properties of the element between Silicon
(14) and Tin (50) which he called ekasilicon. The element in question was
not identified until 1886 by a German chemist who dubbed it Germanium.
2. The Kabbalistic counterparts to the Periodic Table
It is our goal in this article to present an exact and full analogy
to the modern periodic table within Torah. The motivation for this goal
is explained in preceding chapters. To do so in a methodological manner,
we must first ascertain that the Torah does indeed include examples of
the two central concepts underlying the content and form of the Periodic
Table: (1) atoms and (2) periodicity.
The notion of the entirety of creation being constructed out of a finite
variety of basic building blocks is central to the earliest
Kabbalistic source known (and incidentally the first book of Hebrew grammar)—the Book
of Formation (Sefer Yetzirah). From there this notion assumes a
central role throughout the entire Kabbalistic and esoteric tradition
within Torah.
Specifically, the Book of Formation turns to Genesis and, following
a (spiritually) linguistic perspective, identifies 32 non-corporeal elements
or atoms. They are the 10 sefirot [which correspond to the 10 utterances (ma’amarim, מאמרים )
spoken by God when He created the world ]
and the 22 letters of the Hebrew alphabet [out of which the utterances are
constructed]. Together, these 32 atoms form the basis for language
and speech, the conduits of the creative act itself.
However, though the Book of Formation provides us with the notion
of basic building blocks of the Universe, the atoms it identifies are
ill suited for our purposes of correspondence. First, because they are
of two separate categories: one (utterances) clearly hierarchically above
the other (letters). Second, because we are searching for a one-to-one
correspondence between the atoms of the periodic table and some corresponding
unit in Torah.
However, one piece of valuable insight to be gained from the Book
of Formation is the idea that should the Torah’s equivalent of elements
or atoms be found, it would be in the first chapter of Genesis, where
the act of creation is described. What better place to search for the Torah
atoms from which Creation is constructed?
* * *
To explain the correspondence we have found, let us first mention
that of the more than 100 elements, only 92 are naturally occurring.
Atoms of elements with atomic number higher than 92 can be artificially
synthesized, however, they are generally not stable and
undergo nuclear rearrangement resulting in radioactive decay shortly
after being synthesized.
And now to our correspondence: one of the most important contributions
to Jewish scholarship in the recent past has been the work of Rabbi Zalman
Pinchas Horowitz . Rabbi
Horowitz was (to the best of our knowledge) the first to correctly count
the number of times the Tetragrammaton (YHVH) appears in the Pentateuch:
1820 times. Even more surprising and innovative was Rabbi Horowitz’s
cataloging of all the distinct words in
the Pentateuch, which he also found to be exactly 1820 in number.
This equality still warrants much research, but here we will note a fact
related to our own particular interest: of the total 1820 unique words
in the Pentateuch, the section describing creation (Genesis 1:1 to 2:3,
inclusive) contains exactly 92 distinct words. Indeed, as mentioned
already, this section of the Torah literally describes the creation of
matter in the universe—it is only fitting that it is here that we find
our sought after parallel for the 92 natural elements identified by modern
science.
Before proceeding let us copy the familiar periodic table of elements
with the 92 distinct words of Genesis placed in order:
1
H
בראשית |
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2
He
ברא |
3
Li
אלהים |
4
Be
את |
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5
B
השמים |
6
C
הארץ |
7
N
היתה |
8
O
תהו |
9
F
ובהו |
10
Ne
וחשך |
11
Na
על |
12
Mg
פני |
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13
Al
תהום |
14
Si
ורוח |
15
P
מרחפת |
16
S
המים |
17
Cl
ויאמר |
18
Ar
אור |
19
K
וירא |
20
Ca
כי |
21
Sc
טוב |
22
Ti
ויבדל |
23
V
בין |
24
Cr
ויקרא |
25
Mn
יום |
26
Fe
לילה |
27
Co
ערב |
28
Ni
בקר |
29
Cu
אחד |
30
Zn
רקיע |
31
Ga
בתוך |
32
Ge
ויעש |
33
As
אשר |
34
Se
מתחת |
35
Gr
כן |
36
Kr
שני |
37
Rb
יקוו |
38
Sr
אל |
39
Y
מקום |
40
Zr
היבשה |
41
Nb
ימים |
42
Mo
תדשא |
43
Tc
עשב |
44
Ru
מזריע |
45
Rh
עץ |
46
Pd
פרי |
47
Ag
למינו |
48
Cd
בו |
49
In
ותוצא |
50
Sn
שלישי |
51
Sb
לאותות |
52
Te
ולמועדים |
53
I
ושנים |
54
Xe
הגדולים |
55
Cs
לממשלת |
56
Ba
הקטן |
57
La
הכוכבים |
72
Hf
בהמה |
73
Ta
האדמה |
74
W
בצלמנו |
75
Re
כדמותנו |
76
Os
וירדו |
77
Ir
בדגת |
78
Pt
זכר |
79
Au
ונקבה |
80
Hg
להם |
81
Tl
וכבשה |
82
Pb
הנה |
83
Bi
לאכלה |
84
Po
ירק |
85
At
מאד |
86
Rn
הששי |
87
Fr
ויכלו |
88
Ra
צבאם |
89
Ac
השביעי |
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58
Ce
ויתן |
59
Pr
רביעי |
60
Nd
ישרצו |
61
Pm
נפש |
62
Sm
חיה |
63
Eu
ועוף |
64
Gd
התנינים |
65
Tb
כל |
66
Dy
הרמשת |
67
Ho
כנף |
68
Er
ויברך |
69
Tm
ורבו |
70
Yb
ומלאו |
71
Lu
חמישי |
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90
Th
מלאכתו |
91
Pa
וישבת |
92
U
ויקדש |
93
Np |
94
Pu |
95
Am |
96
Cm |
97
Bk |
98
Cf |
99
Es |
100
Fm |
101
Md |
102
No |
103
Lr |
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Following our methodology, we should now seek periodicity, the second
organizing principle identified above. To do so, we will first examine
and understand in-depth the periodic nature of the structure and form
of the table of elements and the model used to explain this periodicity.
We will then explore parallel spiritual models found in Kabbalah and
Chassidut. In a forthcoming article we will use our findings to examine
the periodicity inherent in our parallel Torah table of elements pictured
above.
3. On the relationship between spiritual and mundane in the Torah
Before starting our analysis, let’s take a few steps back to say a few
words about the rationale for looking to the Torah for models that can
describe (directly or indirectly) natural phenomena.
The physical world and its attributes are often spoken of as a reflection
or manifestation of the spiritual realm, and as such, by studying the
physical we may come to know more about the spiritual worlds, and ultimately
our Creator .
It is explained in Jewish tradition that there are two ways to describe
the relationship between the Torah and physical reality:
The first, more commonly held view, is that the Torah speaks of mundane
matters (e.g. laws of commerce, liability, etc.) but as it were, these
mundane matters are also to be found reflected in the higher (or inner)
spiritual dimensions of the universe. So we might say that the Torah
can be interpreted as saying something about the spiritual worlds as
well as the mundane. This interpretation can be as simple as talking
about the spirit of the law (as opposed to the letter or the law). Or,
it can form the basis of a complex and intricate (anthropomorphic) analysis
of the Divine, based on the Torah.
The second approach, advocated by assidism, holds that the Torah’s actual
subject matter are the higher (or inner) spiritual dimensions of the
universe, and it is actually they that are also reflected, or mimicked,
in the lower mundane material dimensions .
Thus we may say that the literal meaning of the Torah is spiritual, while
a non-literal, or allegorical interpretation of this meaning teaches
about the mundane physical world.
The second approach may seem troubling because the Torah does not seem
to employ ‘spiritual’ language (note the lack of mention of angels or
any other ‘heavenly’ artifacts). In fact, the opposite is more the case—the
stories related and the commandments of G-d found in it all seem to speak
directly about physical reality as it was a few thousand years ago. The
response to this point comes in the shape of the Talmudic dictum that
“Torah speaks in the language of men” .
In other words, though the subject matter of the Torah is indeed spiritual,
its language is mundane—“the language of men”—such that it employs language
that refer to objects and states of affair familiar to humans.
Armed with these two basic notions regarding the subject matter and language
of Torah, we argue that by studying the physical world using scientific
methods (which should hopefully give us a clear picture of physical phenomena)
we expect to find parallels between the Torah’s ‘physical’ terminology
and the findings of experimental science regarding those phenomena. Relating
our knowledge about such physical phenomena to the Torah’s vocabulary
(or other non-linguistic forms of communication, as will be explained)
will, in turn, lead us to a better understanding of the ‘spiritual’ issues,
which are the Torah’s ‘actual’ subject matter. Thus we come to learn
more about the spiritual realm using scientific knowledge.
The Torah contains varied types of communicable information, alluded
to by the famous acronym: PaRDeS. PaRDeS stands for the four types of
textual analysis traditionally used to explore the Torah in order to
recover its informative content. These are: pshat (literal analysis),
remez (symbolic, or numerical analysis), drash (hermeneutic analysis)
and sod (associative, or model-based analysis). In order to quickly orient
the reader we will note that drash (hermeneutic analysis) was utilized
in the study and development of Halachah (Jewish Law). Sod (associative,
model-based) analysis was most fully developed in Lurianic Kabbalah.
Our present study will make use of all four types of textual analysis.
At times, we refer to the knowledge arrived at using remez and sod analysis
as the ‘inner (or esoteric) wisdom of the Torah.’
4. Nature and the Divine
One of the most basic findings in the Torah using remez analysis (numerical,
in this case) is that the numerical value of the hebrew word for nature
(הטבע ,
hateva) = 86 – is equal to the numerical value of the name of G-d associated
with the creation of the natural world: Elokim (א־להים )
= 86. This numerical equivalency is usually understood to indicate that
there is an aspect of Divinity that is enclothed within the natural world.
As we shall see, this basic equivalency will form the backdrop for much
of our present discussion.
5. 92 naturally occurring elements
The first possibility would be to map each element to its corresponding
Hebrew root, simply based on order of appearance (see Table 1 in Appendix
A).
Further reflection though reveals an alternative. The 92 distinct roots
of the story of creation are divided such that the first 86 appear in
the verses relating the first six days of creation (Genesis 1:1 through
1:31), while the last 6 are found in the verses relating the Sabbath
(ibid 2:1 through 2:3). This motivates us to correspond the 6 noble gases
with the 6 distinct roots found in the Sabbath section in Genesis, while
the remaining 86 elements will be corresponded in order to the distinct
roots found in the 6 Days section of Genesis.
We mention this second possible mapping here because of our interest
in the inert gases, as follows.
6. Inert and non-Inert Elements
Scientifically speaking, there are many ways in which the chemical elements
can be arranged to accent different attributes of their periodicity.
Briefly, when looking at a periodic table, the elements are normally
presented with their name, atomic number, and often their valence electron
configuration. The commonly found table of elements highlights various
types of periodicity, one of the most central ones being that of the
noble or inert gases.
One of the most important and outstanding features of the 92 naturally
occurring elements is that they may be divided into two groups, based
upon their ability to form compounds: there are 6 which do not form compounds,
also known as inert (or noble) gases, while the other 86 do form compounds
with other elements.
On the periodic table in Figure 1, the inert gases form the
far right hand column. Graphically, our modern version of the table of
elements is structured such that the periodicity of the inert gases is
highlighted, though, as we shall see below, the table could be (and historically
was) arranged entirely around this periodicity.
This basic division into 6 and 86 observed in the naturally occurring
elements is to be found (again, using a non-literal analysis of the Torah
text) in the very first verse of the Torah:בראשית ברא
אלקים את השמים ואת הארץ (usually
translated as “In the beginning G-d created the heavens and the earth”).
The first chapter of the Torah recounts the act of creation of the natural
world. It is important to note that G-d here is referred to as Elokim
alone (and not by His other names), thus leading to the association of
this name with ‘nature’.
The 6 letters of the first word, Bereishit (בראשית )
can be split into 2 separate words, each with 3 letters and read as ‘barah
sheet’ (ברא שית ),
meaning “created six” (in Aramaic, the lingua franca of the
ancient near-east).
The next two words in the verse are ‘barah Elokim’. As noted
above, the numerical value of Elokim, one of the names of G-d is 86.
We have then that the first 3 words can be understood to say: ‘six were
created’ (“barah sheet”) and ‘86 were created’ (“barah Elokim”). The
sum of these two acts of creation is 86 ^ 6 = 92, the number of the naturally
occurring elements.
As mentioned above, 86, the value of Elokim, is also the numerical value
of the word hateva in Hebrew, or ‘nature’. Thus “barah Elokim” can be
read as “created nature” as well.
There is yet another appearance of the name Elokim regarding the 6 inert
gases:
The atomic numbers of the inert gases are 2, 10, 18, 36, 54, 86. The
heaviest inert gas, radon (Rn), has an atomic number of 86 = Elokim.
7. Spiritual ‘Wholeness’
We now turn to reflect on the spiritual parallel to the presence of
both inert and non-inert elements in the natural world.
The spiritual (or psychological) counterpart of physical inertness in
the elements can be found in the Torah’s description of Jacob and Lavan’s
(Jacob’s father-in-law) working relationship. Regarding the wages that
Jacob received for tending Lavan’s flocks the Torah writes (Genesis 30:42):
והיֻה העטֻפים ללבן והקשֻרים ליעקב
This is usually translated literally as:
the weaker (atufim) [flocks] were to Lavan and the stronger (k’shurim)
[flocks] were to Jacob.
The literal meaning is that the sheep were characterised as stronger
or weaker; the weaker remained the property of Lavan, the stronger were
given to Jacob as wages.
However, Rashi, the basic (literal) Medieval commentary on the
Torah interprets the meaning of atufim differently. This
Hebrew word can be analyzed to stem from the root atf(עטף )
that yields the infinitive la’atof, to wrap. It would then mean “those
that are wrapped”.
Likewise, K’shurim the word used to describe the type of flocks given
to Jacob, can be analyzed to stem from the root k.sh.r. (קשר )
and the infinitive likshor, to bind. Its meaning would then be “those
that are bound”
If these characteristics of the sheep are seen as metaphors for two different
types of personalities, then an atuf describes one who is wrapped in
wool, keeping warm all to himself, while a kashur symbolizes one who
is incomplete without forming bonds with others outside of himself. A
‘wrapped’ (atuf) individual is not in need of a mate and finds sufficient
warmth alone. Such an individual needs not receive from nor give to another.
On the other hand, a person with a ‘tied’ (kashur) personality seeks
completion in relationships with others, at times giving at times receiving.
For the sake of rigor we note that spiritually speaking, these two types
of personalities are usually associated with negative and positive qualities,
respectively. An atuf attitude (especially in the present case where
these flocks are indicated as being the property of Lavan) is considered
analogous to that found in Biblical Sodom :
“That which belongs to me is mine, that which belongs to you is yours.” While
a kashur attitude is normally associated with holiness (though at times
it can drift to an extreme form of wantonness which is of course negative).
However, in assidic writings it is explained that a truly whole individual
is one who has both qualities. To better understand why how this is so
we may take the kaballistic principle stated by Rabbi Abraham Abulafia
(1240 – c.1291), the 13th century philosopher and mystic: “being whole is
being one and a half”. Or in the famous words of Rebbe Nachman
of Bretzlov: “nothing is more whole than a broken heart”.
We coin the term whole and half (שלם וחצי ,
shalem va’chetzi) to designate this special quality of wholeness.
A truly whole (and holy) person is does not feel self sufficient,
thereby not requiring others, but rather one who is, existentially speaking,
both complete and incomplete at the same time. By virtue of their half-ness,
they need to connect or bond with others. By virtue of their whole-ness
they are able to offer support and help to others. Real wholeness (and
holiness) comes by virtue of an existential feeling of incompleteness
– of insufficiency and inadequacy to single-handedly prevail, empowered
and strengthened by a sense of whole-ness which saves one from
a sense of an inability to rise to the task at hand.
Scientifically speaking, we can immediately note the analogy between
these two basic definitions of atuf and kashur and the distinction between
inert and non-inert elements. Bond formation is possible only when an
electron orbital is half or incomplete. But when an orbital
is whole or filled, the element in question is not in need of
accepting or receiving electrons and thus does not form bonds. Yet, both
exist in nature. Nature reflects these two basic qualities.
The inert gases are also called the Noble gases. The mark of nobility
is the air of whole-ness surrounding it. The ‘nobility’ of the
elements do not react with any other elements. It is only the 86, Elokim,
elements that can do so. Nonetheless, the six noble gases form a sort
of axis around which the other 86 elements revolve.
8. Whole-ness of the Patriarchs
The attribute of bonding is found to be associated with the name Elokim
in another manner: Elokim is the name of G-d related most closely with
the Patriarch Isaac as both manifest the quality of judgment (or din
– דין ).
When departing from Lavan, his father-in-law, Jacob says:
לולי אלקי אבי אברהם ופחד יצחק היה לי, כי עתה ריקם שלחתני…
If the G-d of my father, the G-d of Abraham, the fear of Isaac, was not
with me, then you would have sent me away empty-handed…
Jacob refers to the way in which Isaac (his father) knew G-d as ‘the
fear of Isaac’ (pachad Yitzchak, פחד
יצחק ). The numerical
value of pachad, or fear, is 92, which is again Elokim (86)
plus six – the total number of naturally occurring elements.
Yet, Isaac was not always whole in the sense of being both whole and half at
the same time. The sages tell us that Isaac was actually 37 years old
at the time that Abraham (his father) was commanded to sacrifice him
to G-d (see Genesis 22), known as the ‘test of the Akeida’ – the test
of the binding of Isaac. The Zohar, the
basic book of the inner teachings of the Torah, relates that Isaac was
entirely whole, exclusively of ‘noble’, or inert, character and was therefore
not suited for marriage, not suited to bond with another. It was the
Akeida – literally, ‘the binding’ – which brought him to complete his
character with the quality of half-ness. It was only then that
he became suited for marriage, to bond with a wife. Thus pachad
Yitzchak (= 92) can be understood as the attribute of Elokim
(86) plus another 6, the addition of something to Yitzchak’s own wholeness.
By the same token the Zohar explains that Abraham was not truly whole either,
as he did not have the quality of Might or Judgment.
It was the act of the Akeida – the binding of Jacob – done out of fear
and awe of God (as the angel spoke to him following the binding: “for
now I know that you are indeed fearful of God” (Genesis 22:12) which
complemented his essence with this quality.
9. Inert Periodicity Historically
Now that we have spent some time studying the periodicity of the inert
elements, let us delve a bit into its history. Using the periodicity of
the inert elements as the basis for the table of chemical elements was
first proposed in 1895 by J. Thomsen and
was itself based on an earlier model by T. Bayley (1882). A table similar
to Thomsen’s appears in Figure 2. Note that the principal disadvantages
of this table was the large space required by the period of 32 elements
and the difficulty of tracing a sequence of closely similar elements (for
purposes of illustration the inert elements have been marked in blue, and
the non-metals, marked in green, in the contemporary table form a triangular
shape, but here do not align similarly).
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1
H |
2
He |
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3
Li |
4
Be |
5
B |
6
C |
7
N |
8
O |
9
F |
10
Ne |
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11
Na |
12
Mg |
13
Al |
14
Si |
15
P |
16
S |
17
Cl |
18
Ar |
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19
K |
20
Ca |
21
Sc |
22
Ti |
23
V |
24
Cr |
25
Mn |
26
Fe |
27
Co |
28
Ni |
29
Cu |
30
Zn |
31
Ga |
32
Ge |
33
As |
34
Se |
35
Gr |
36
Kr |
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37
Rb |
38
Sr |
39
Y |
40
Zr |
41
Nb |
42
Mo |
43
Tc |
44
Ru |
45
Rh |
46
Pd |
47
Ag |
48
Cd |
49
In |
50
Sn |
51
Sb |
52
Te |
53
I |
54
Xe |
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55
Cs |
56
Ba |
57
La |
58
Ce |
59
Pr |
60
Nd |
61
Pm |
62
Sm |
63
Eu |
64
Gd |
65
Tb |
66
Dy |
67
Ho |
68
Er |
69
Tm |
70
Yb |
71
Lu |
72
Hf |
73
Ta |
74
W |
75
Re |
76
Os |
77
Ir |
78
Pt |
79
Au |
80
Hg |
81
Tl |
82
Pb |
83
Bi |
84
Po |
85
At |
86
Rn |
87
Fr |
88
Ra |
89
Ac |
90
Th |
91
Pa |
92
U |
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Table 1
10. Orbital Filling of the 6 Inert Gases
It was only in 1922 that Niels Bohr proposed the quantum-theoretical
model that forms the basis for our current understanding of the subatomic
construct of the elements, and explains the observed periodicity of the
inert gases. According to Bohr’s model, the structure of each atom could
be singularly described using 4 quantum numbers to identify the ‘orbitals’
in which electrons organize around the atom’s nucleus. The orbitals (sometimes
called sub-shells) are grouped into shells, the shells being designated
by the letters: K, L, M, N,…, or simply 1, 2, 3, 4,….
Every orbital is classified by two quantum numbers: the primary quantum
number and the angular momentum quantum number. The angular
momentum quantum number is replaced by the letters s, p, or d. Two other
quantum numbers – the magnetic quantum number and the spin quantum
number – determine the number of electrons that can ‘fit’ in an orbital.
Looking at the periodic table using Bohr’s model, we find that the naturally
occurring elements can be described exhaustively using 7 shells and 4
orbitals, namely (designating the shells by their number, not letter):
1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p, 4d, 4f, 5s, 5p, 5d, 5f, 6s, 6p, 7s. The
number of electrons that can fit in each orbital is: 2 in s orbitals,
6 in p orbitals, 10 in d orbitals, 14 in f orbitals
To truly understand the theoretical basis for Bohr’s model is beyond
our scope. However, we would like to take a closer look at the mathematical
regularities that this model produces. So let us order the elements in
a table that will show us how their electrons ‘fill’ the various shells
and orbitals:
Shells |
1 |
1 |
2 |
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(2) |
H |
He |
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2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
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(8) |
Li |
Be |
B |
C |
N |
O |
F |
Ne |
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3 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
21 |
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30 |
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(18) |
Na |
Mg |
Al |
Si |
P |
S |
Cl |
Ar |
Sc |
— |
Zn |
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4 |
19 |
20 |
31 |
32 |
33 |
34 |
35 |
36 |
39 |
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48 |
57 |
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70 |
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(32) |
K |
Ca |
Ga |
Ge |
As |
Se |
Br |
Kr |
Y |
— |
Cd |
La |
— |
Yb |
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5 |
37 |
38 |
49 |
50 |
51 |
52 |
53 |
54 |
71 |
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80 |
89 |
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92 |
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(50) |
Rb |
Sr |
In |
Sn |
Sb |
Te |
I |
Xe |
Lu |
— |
Hg |
Ac |
— |
U |
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6 |
55 |
56 |
81 |
82 |
83 |
84 |
85 |
86 |
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(72) |
Cs |
Ba |
Tl |
Pb |
Bi |
Po |
At |
Rn |
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7 |
87 |
88 |
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(98) |
Fr |
Ra |
Orbitals |
s(2) |
p(6) |
d(10) |
f(14) |
g(18) |
h(22) |
i(26) |
Table 2
The rows designate the Shells, while the columns the orbitals in each
shell. Thus for instance, the first shell (K) can accommodate up to
2 electrons and therefore has room for 2 elements. The second shell
(L) can accommodate 8 electrons and therefore has room for 8 elements,
and so on. In the K shell, all the electrons are available in the s
orbital only. In the L shell, electrons ‘spots’ are available in both
the s and p orbitals.
Note that shells are not filled entirely before the next shell is started,
due to the fact that electrons in elements always seek the lowest possible
energy state they can reach. This is clear if we follow the elements
in this table. Up to Argon (Ar), the first shell (K) and second shell
(L) are filled completely. Then the s orbital of the third shell (M)
is filled, followed by a complete filling of its p orbital. Argon then
is the 18th element in the table with electrons completely filling the
3p [3rd shell (M), p orbital] orbital. But the next element, Potassium
(K) does not continue to fill the 3rd shell’s d orbital, but rather skips
to the 4th shell’s (N) s orbital, because electrons in that orbital actually
have a lower energy level then electrons in the 4d orbital. This is due
to the interactions between the electrons themselves, an effect known
as ‘shielding’. So Potassium’s ‘extra’ electrons do not locate in the
3d orbital but rather in the lower-energy orbital 4s. The rest of the
table follows this general trend, with electrons always vying for the
lowest energy level orbitals.
There are two interesting facts about this table that should be noted
are:
- that the number of ‘spots’ in each orbital is equal
to the differences between the total number of elements that can
populate each shell.
- all the orbitals that are used are ‘filled’ or ‘populated’
to capacity by elements except for the 5th shell’s f orbital. Though
5f has room for 14 electrons, only 4 ‘spots’ are used by the heaviest
naturally occurring elements from Actinium (89) to Uranium (92).
The first fact is the reason that we have drawn the table as 7 x 7 even
though the entire g, h and i orbitals have been left blank. Let us explicitly
write the first fact out: The total numbers of electrons in the shells
are (We designate these as set A):
A = {2, 8, 18, 32, 50, 72, 98}
The numbers of electrons in each orbital are (we will designate these
as set B):
B = {2, 6, 10, 14, 18, 22, 26}
Now note that the numbers in set B are the differences between the numbers
in set A. This can be clearly illustrated by writing the two sets, A
and B, one beneath the other, as follows:
total electrons in
shells |
2 |
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8 |
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18 |
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32 |
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50 |
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72 |
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98 |
electrons in orbitals |
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6 |
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10 |
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14 |
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18 |
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22 |
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26 |
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This is truly one of the most astonishing mathematical properties of
the periodic table.
But, arranging the elements in the above table also reveals an interesting
property of the inert gases and that is that an inert gas is formed each
time the p orbital fills. The p orbital of each and every shell has room
for 6 electrons. That is to say that each time 6 electrons fill the p
orbital, an inert gas if created (excluding Helium, which does not use
the p orbital).
Recollect that above we noted that in the story of creation, the first
word of the Torah, Bereishit, which can be understood to mean ‘created
6’, should be seen as corresponding to the formation of the 6 inert gases!
In fact, the excluded inert gas Helium, is hinted to in the word Bereishit
as the letter bet, the first letter of the word is written in the Torah
scroll as a large bet (בית רבתי ,
bet rabati) and the numerical value of the letter bet is 2.
11. Mathematical Patterns in Inert Elements
Let us now look at the numbers of the inert gases from another perspective.
If we take the atomic numbers of the inert elements and note the differences
between them we can construct the following table:
element |
atomic number |
difference in atomic number |
n, where difference = 2n2 |
He (Helium) |
2 |
2 |
1 |
Ne (Neon) |
10 |
8 |
2 |
Ar (Argon) |
18 |
8 |
2 |
Kr (Krypton) |
36 |
18 |
3 |
Xe (Xenon) |
54 |
18 |
3 |
Rn (Radon) |
86 |
32 |
4 |
Table 3
The rightmost column shows that the differences between the elements
are all values, in order, of the mathematical series f[n] = 2n2 (n starting
at 0).
These numbers are known in the inner teachings of the Torah as the double
squares (רבועים כפולים ,
ribu’im k’fulim). Their significance is related to the 32 paths of
Wisdom (ל"ב נתיבות חכמה ,
lamed beit netivot chochmah). The Book of Formation,
mentioned above, begins:
ב-לב נתיבות פליאות חכמה חקק י-ה הוי' צבאות וברא את עולמו בשלשה
ספרים בסופר וספר וסיפור
Using 32 wondrous paths of wisdom Kah Havayah [God] Lord of Hosts engraved…
and created His world, using three books: author and book and story.
It is known that the textual source in the Torah for these 32 paths
of Wisdom is to be found in the 32 times that the name Elokim is
used in the verses describing the six days of Creation. This,
again, is the Name which we have recognized as central in our discussion
of the periodic table. We note that no other Name of the Almighty appears
in the creation story, and
it appears exactly 32 times.
32 is thus the number associated
with Wisdom (chochmah). In the inner teachings of the torah we find
the number 50 associated with Understanding: 50 Gates of Understanding (חמשים
שערי בינה , chamishim
sha’arei binah). There is also another, less well known concept of 72 Bridges (ע"ב
גשרים , ayin beit gesharim).
Actually, all three of these concepts are closely related and are part
of one larger picture. This basic conceptual scheme identifies the type
of energy related to each sefirah and the type of conduit through which
it flows:
Thus the energy of Wisdom is identified as ‘mind’ that flows
through a path (נתיב ,
nativ); the energy of Understanding is identified as ‘intelligence’
that flows through a gate (שער ,
sha’ar); finally the energy of Knowledge is termed ‘psyche’
and flows through a bridge (גשר ,
gesher). This model is summarized in Table 2.
sefirah |
energy type |
conduit type |
number of conduits |
Wisdom |
mind |
Path |
32 |
Understanding |
intelligence |
Gate |
50 |
Knowledge |
psyche |
Bridge |
72 |
Table 4
Of course, 72 is also a double square (particularly, 72 = 2
. 62). We have thus, so far, found the mental significance of the double
squares for n = 4, n = 5 and n = 6. To complete our understanding
of the significance of double squares we need to complete the
series beginning with n = 1.
The basic model of the sefirot in Kabbalah indicates that above Wisdom resides
the Crown (כתר ,
keter) that is explained in the Zohar to consist of three
heads (תלת רישין שבכתר ,
tlat reishin sheba’keter). In
our present model we will map these 3 parts of the Crown to
correspond to the first 3 values of n.
Continuing our previous discussion regarding the various mental powers
we note that Wisdom marks the first conscious mental power.
Thus, the Crown – which resides, both figuratively and in our
Kabbalistic model, above the head – corresponds to the super-conscious
faculties. The three heads of the Crown, or the 3 super-conscious
mental powers are known as: Belief (אמונה ,
emunah), Pleasure (תענוג ,
ta’anug) and Will (רצון ,
ratzon).
Table 3 illustrates the double squares for values of n from
1 to 6 with their corresponding mental faculties.
sefirah |
mental faculty |
n |
f[n] = 2n2 |
Crown |
belief |
1 |
2 |
pleasure |
2 |
8 |
will |
3 |
18 |
Wisdom |
mind |
4 |
32 |
Understanding |
intelligence |
5 |
50 |
Knowledge |
psyche |
6 |
72 |
Table 5
Using the sefirot as a model for the series of double squares,
we could continue the series until n = 13. For example, corresponding
to the double square 128 (n = 8) we would have the sefirah of Might (גבורה ,
gevurah). For 338 (n = 13) we would have the sefirah of Kingdom (מלכות ,
malchut).
We have now taken a look at the series of double squares, the
differences between the atomic numbers of the inert gases. This series
is essentially the backbone of the whole periodic table of the elements.
Extrapolating from our knowledge of double squares in the periodicity
of inert elements we would expect the next inert element to be of quantum
number:
86 (Radon) ^ 32 = 118. This element has been dubbed Uuo (Ununoctium)
by the International Union of Applied Chemists (IUPAC) until its existence
is proven at which time its properties will be ascertained.
We would expect to find the next inert element at quantum number:
118 ^ 50 = 168. This element has been dubbed Uho (Unhexoctium).