The Periodic Table: What is it Showing?
Periodic Tables & The Philosophy of Science

The Periodic Table of the Chemical Elements is a cultural icon and an extraordinary object in science space. This page explores what the periodic table is in terms of basic & simple elemental substance, quantum theory and the philosophy of science.


A classic periodic table can be viewed at WebElements:

WebElements employs the most common of many possible formulations, and these can be explored using the INTERNET Database of Periodic Tables and Periodic Table Formulations, on the next page of this webbook, here.

The vast majority of Periodic Tables – and the excellent WebElements is a perfect example – are used to arrange physical, chemical, technological & historical data/information about the chemical elements in a systematic way.

Check out the various way that physical, chemical, technological & historical data/information are mapped to the Periodic Table, here.

The chemogenesis web book explores how chemical reactivity emerges from the periodic table of the elements using a root-trunk-branch chemistry-tree metaphor, with the periodic table at the base of the trunk:

The periodic schema is used in many ways and so it is interesting to ask what the periodic table actually is, and what it is showing?

It transpires that matters are a little more involved than they may at first appear...

Philosophy, Chemistry & The Periodic Table

Philosophers of chemistry consider the chemical elements in two distinct ways:

Firstly, there is the chemical element as the basic element, that is the abstract or transcendental element, the essence of the element, a bearer of properties but not having any actual properties, except for [historically, atomic mass, but now] atomic number Z. Chemical symbols, such as H and Cu, are assigned to the basic element.

Secondly, there is the element as simple substance, for example, a real piece of copper metal placed on a table has numerous, measurable, intrinsic properties such as: purity, density, conductivity, colour, melting point, molar volume, etc.

Crucially, only the basic element survives in a compound. Sodium's metallic properties and chlorine, the green gas, do not exist in the ionic salt, sodium chloride.

These matters are discussed in a paper by Eric Scerri, Some Aspects of the Metaphysics of Chemistry and the Nature of The Elements, available here. Briefly summarising these arguments:

  • There is a metaphysical view about the nature of the elements as basic substances and bearers of properties that goes back to the ancient Greeks, long before the discovery of atoms.

  • Mendeleev insisted that his periodic classification system concerned the elements as basic substances possessing only one attribute, atomic weight.

  • Paneth, one of the founders of modern radiochemistry took Mendeleev's view about the nature of basic and simple substance, but changed the basic/transcendental/abstract property of an element from atomic weight/mass to atomic number, Z.

  • Elements as basic substance represent natural kinds, a well understood philosophical position concerning the nature of classification. Elements as simple substances fail the natural kind test, due to the existence of isotopes and allotropes, etc.

Eric Scerri points out that the periodic table has, at times, been characterised as a:

  • representation
  • ordered domain
  • classification
  • system
  • model
  • law
  • theory

This author agrees with Scerri that the periodic table is an ordered domain. But it is also a schema, a 'map', that that can be used to organise information, data & knowledge concerning the chemical elements.  

Periodic Tables on Walls and In Books: What are they showing?

Most periodic tables in books, most periodic tables on classroom walls and most periodic tables on web sites use the periodic table as an organising schema to present physical data & material properties of the elements.

The boiling point of oxygen is -182.9 °C, but this is the bp of the molecular substance dioxygen, O2.

In this author's opinion, there has been a logical sleight-of-hand.

The metaphysical periodic table of abstract, basic substances is being passed off as a periodic table of the material properties of simple substances, which is not the same thing at all. This causes confusion as to what exactly it is that a particular periodic table is showing.

Morphing Multi-PTs

At least three periodic tables can be identified:

  • The periodic table of basic elements with atomic number Z.

  • The periodic table of gas phase atoms with their associated spectra.

  • Then there is the periodic table of chemicals in bottles, the actual materials under standard conditions, 25°C and 1.0 atm.

There are other periodic tables:

  • Phase changes (melting points, boiling points, etc., that show the real materials at 1.0 atm but not at a standard temperature)
  • Dates of discovery
  • NMR properties
  • etc...

The various periodic tables morph into each other to give the compound object commonly presented as The Periodic Table.

When moving across these various PTs the system complexity increase.

There is also the important notion of the elements in a compound; ie the nature of the element sodium and the nature of the element chlorine in the ionic substance sodium chloride.

Periodic Table of Basic Elements

Periodic tables generally show the chemical elements as the basic substance:

For example, oxygen is show as O and not as the common molecular form O2. Likewise, sulfur is shown as S and not S8. The usual periodic table schema simply shows the element symbols in their respective periods, groups & blocks.

This, or an equivalent formulation, and there are many – see the next page of this web book – is the periodic table as Mendeleev would have intended it: a schema showing the elements as basic substances with their positions in the schema emphasising the periodic law:

"The periodic law is the principle that certain properties of elements occur periodically when arranged by atomic number. These similarities can be reflected best by a table, so that commonalties between elements appear both in rows and in columns of the table." Wikipedia

It is commonly held [Scerri] that there is only one basic elemental property: atomic number, Z, where the element's name and symbol are assigned to Z.

But arranging the chemical elements by mass or atomic number, Z, will only give a simple list.

Placing the chemical elements into a table that explicitly displays periodicity assigns x, y (group, period) co-ordinates to the entities that make up the periodic table: the chemical elements. It follows that properties that pertain to the periodic table schema itself – block, group, period & periodicity – map to, and are properties of, the basic element. Quantum numbers, here, can be used (mapped) to the various periodic table formulations. Therefore – in this author's opinion – quantum numbers must be a basic elemental property.

Basic Element Properties:

  • Atomic number, Z, plus name & symbol (assigned to Z)
  • Quantum Numbers

    Block of periodic table: s, p, d, f
    Group number: 1-18
    Period: 1-7
    Electronic configuration

Periodic Table of Gas Phase Atoms

The chemical elements as real, simple substances can be physically normalised by studying ground-state, monoatomic gas phase atoms of the material substance.

  • For some elements this is trivial: the group 18 rare gases are already monoatomic, gas phase entities at 25°C and 1.0 atm. In other words, they are naturally in the desired state.
  • Oxygen is a molecular gas at room temperature, but it must be converted into an atomic gas.
  • Carbon boils at 4027°C and it difficult to obtain a vapour of ground-state carbon atoms, carbon gas, but it is possible.

The periodic table of ground state gas phase atoms is known, and it is the periodic table of the very simplest of simple substances:

Apart from the title, the graphic is exactly the same as the periodic table of basic substances, but this periodic table represents real chemical entities with actual, measurable, physical properties, including:

  • Average atomic mass
  • Atomic radius
  • Accurate mass & abundance of the isotopes
  • Effective nuclear charge
  • Electron affinity
  • Electron binding energies
  • Ionisation energies
  • Emission spectra. The University of Oregon Department of Physics has a dynamic periodic table, here, that shows the atomic spectra of all the elements:

Many modern technologies utilise gas phase atoms, including: the sodium vapour lamps used for street lighting and atomic clocks. Many man-made trans-uranium elements are only known as isolated gas phase atoms.

It may be thought that the common form of the periodic table on walls and in books is showing ground state gas phase atoms, but this seems unlikely as a substance like atomic carbon gas, C(g), is very uncommon.

Periodic Table Chemical of Substances Under Standard Conditions

Under standard conditions, 25°C & 1.0 atm, the chemical elements as simple substances – real chemicals in bottles – present as:

  • Gases, liquids or solids
  • Metals, metalloids or non-metals
  • Metallic, network or molecular materials

The chemical elements as material substances have many properties, including [from WebElements: Copper]:

  • Properties at standard conditions 25°C and 1.0 atm: crystal structure, molar volume, hardness, etc.
  • Phase changes under non-standard conditions: boiling point, etc.
  • History
  • Biology
  • etc.

Abundance of elements (Earth's crust)
Abundance of elements (oceans)
Abundance of elements (meteorites)
Abundance of elements (stream)
Abundance of elements (sun)
Abundance of elements (Universe)
Abundances in humans
Biological role
Boiling point
Bond enthalpy (diatomics)
Bulk modulus
Covalent radius
Critical temperature
Crystal structure
Electrical resistivity
Element bond length
Enthalpy of atomization
Enthalpy of fusion
Enthalpy of vaporization
Examples of compounds
Hardness - Brinell

Hardness - Vickers
Health hazards
History of the element
Ionic radii (Shannon)
Ionic radius (Pauling)
Ionic radius (Pauling) of monocation
Isotope abundances

Isotope nuclear spins
Isotope nominal mass
Isotope nuclear magnetic moment
Lattice energies
Linear expansion coefficient
Meaning of name
Melting point
Mineralogical hardness
Molar volume
Names and symbols
NMR frequency
NMR isotopes
NMR magnetogyric ratio
NMR quadrupole moment
NMR receptivity

NMR relative sensitivity
Poisson's ratio
Properties of some compounds
Radius metallic
Radioactive isotopes
Reactions of elements
Reduction potential
Refractive index
Registry number
Rigidity modulus
Standard atomic weights
Standard state
Superconductivity temperature
Term symbol
Thermal conductivity
Thermodynamic properties
Valence orbital
Van der Waals radius
Velocity of sound

X-ray crystal structure
Young's modulus

Quantum Mechanics, Atomic Emission Spectra & The Periodic Table

In the first half of twentieth century, much effort was expended trying to make the periodic table of the elements axiomatic, in other words, trying to fully understand the Mendeleev system in terms of a deeper theory, that deeper theory being quantum mechanics.

Paul Dirac famously claimed this situation had been fully and completely achieved in principle:

"The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble." P.A.M. Dirac, Proc.R.Soc.Lond.Ser.A 123 (1929) 714

We certainly teach our school and university students that "the periodic table is fully explained in terms of electronic theory", and this line of reasoning is advanced elsewhere in this web book, here and the HyperPhysics site, here.

The argument is put forward that:

The pattern of spectral lines experientially obtained from a sample gas phase atoms can be "explained by" (mapped to) quantum mechanics in the form of the Schrödinger wave equation, and the spectral lines and quantum patterns obtained by experiment and theory can be mapped to the Mendeleev Periodic Table of the Elements.

  • We teach that there is a full, complete and beautiful correspondence. We say that the relationship between electronic theory and atomic spectra is linear in the sense that there is a one-to-one mapping between theory and experiment, like the one-to-one mapping between behaviour of a real gas in a piston and the ideal gas equation, PV = nRT.

But is there a 1-to-1-to-1 correspondence between QM, spectra and the periodic table?

Eric Scerri: Philosopher, Theorist, Chemist, Author

Eric Scerri disputes the full and complete axiomatic mapping between theory and the periodic table:

"Electronic configurations are not [fully] reduced to quantum mechanics nor can they be derived from any other theoretical approach. They are obtained by a mixture of spectroscopic observations and semi-empirical methods like Bohr's aufbau scheme".
Has The Periodic Table Been Fully Axiomatized? Erkenntnis
, 47, 229-243, 1997

Eric Scerri, The Periodic Table: Its Story and Its Significance, Oxford University Press, 2006. Read an interview with the author, here, and a review of the book here.

The reason for the discrepancy concerns multi-electron atoms, ions and molecules.

  • The Schrödinger wave equation can only be solved analytically for one electron systems like the hydrogen-atom, H, and other one electron systems: He+, Li2+, Be3+, etc., Wikipedia. For multielectron systems, approximations in the math have to be made to deal with electron-electron interactions and correlations. Multi-electron atoms are complex objects, in the systems sense. The mathematical techniques employed to describe chemical systems are usually pragmatic rather than rigorous, and they are often semiempirical: ie partially based on experimental data.

    From the Wikipedia: "For atoms with two or more electrons, the governing equations can only be solved with the use of methods of iterative approximation. Orbitals of multi-electron atoms are qualitatively similar to those of hydrogen, and in the simplest models, they are taken to have the same form. For more rigorous and precise analysis, numerical approximations must be used. Atomic orbitals are often expanded in a basis set of Slater-type orbitals which are orbitals of hydrogen-like atoms with arbitrary nuclear charge Z."

  • The effect is to produce nice fast computer code that efficiently predicts atomic & molecular energies, geometries and spectra, etc., but at the expense of the theory being fully axiomatic: formally the logic of the underlying theory becomes blurred. As a result, we get a useful model but not a mathematical proof.

On this page we identify three different periodic tables:

  • Periodic table of basic substances
  • Periodic table of gas phase atomic simple substances
  • Periodic table of standard state material simple substances

Are any of these periodic tables axiomatized with respect to theory?

Note that a distinction has been introduced between "quantum chemistry", the techniques, methodologies and computer software used by physical chemists and chemical physicists, and the underlying quantum mechanics in the form of quantum electrodynamics (QED), the most accurate and precise theory known to humankind. However, chemical problems are simply too involved [currently] to be studied by QED, although in principle they can be.

Q: Has The Periodic Table of Chemical Substances Under Standard Conditions Been Axiomatized?

The elements-as-chemicals periodic table – real simple material substances under standard conditions of temp & pressure – is certainly not axiomatized. Quantum chemistry calculations cannot predict the equation of state of an element.

The Schrödinger wave equation without mathematical approximation cannot be used to predict that sulfur exists in S8 rings, that copper is a reddish coloured metal or that mercury is a liquid at room temperature.

There is no axiomatic mapping between quantum mechanical theory and material properties:

Q: Has The Periodic Table of Gas Phase Atoms Been Axiomatized?

Multi-electron atoms, even as isolated gaseous atoms, are too complex to be understood fully and exactly... although modern quantum chemistry mathematical modelling techniques do give very good and useful answers.

The quantum chemistry methodologies are numerical models, and as a result the periodic table of even the simplest of simple substances – ground state gas phase atoms – is not fully axiomatized.

There is no formal proof of one-to-one-to-one correspondence between quantum chemistry methodology, atomic spectra and the periodic table of gas phase atomic simple substances, although the predictions are useful:

Q: Has The Periodic Table of Metaphysical Basic Elements Been Axiomatized?

The essential, metaphysical basic elements do not have properties other than atomic number, group and period... so we can can ignore spectra and simply concentrate on the pattern of the periodic table schema.

The question becomes: Does quantum theory predict/explain the patterns of the periodic table and the periodic law, even if the quantum chemistry mathematical techniques do not exist that enable us to crunch the numbers with absolute precision? Is the periodic table of basic substances axiomatic?

It is proposed here that yes, the periodic table of metaphysical basic substances is axiomatic with respect to theory, in that the pattern of the periodic table can be deduced from the quantum theory directly.

Quantum theory, sometimes called the old quantum theory, produced the four quantum numbers (principal, subsidiary, magnetic & spin), and four rules that are used to construct the periodic table: the Pauli exclusion principle, the aufbau principle, the Pauli exclusion principle, Hund's rule and Madelung's rule.

The pattern of the periodic table schema can be shown to directly arise from these rules followed by a couple of trivial mappings – as discussed on the previous page of this chemogenesis webbook here – and thus, the periodic table of the basic meta-physical elements is axiomatic with respect to the Old Quantum Theory.


The techniques of quantum chemistry cannot/do not completely – axiomatically – describe gas phase atoms (except H, He+, Li2+, etc.), because multi-electron systems are too complex to be described analytically.

However, the various quantum chemistry techniques/models are good enough for the periodicity of the metaphysical basic substances to be mapped to the periodic tables of gas phase and material simple substances.

The periodic table of metaphysical, essential, basic elemental substances can be reduced to quantum numbers and simple rules (the old quantum theory)... but Richard Feynman told us (here): "I think I can safely say that nobody understands quantum mechanics."

In other words, we cannot unpick quantum mechanics and ask "where it all comes from": it is simply how our world works.

The relationship between the old quantum theory and the Schrödinger wave equation is mysterious...

The Periodic Law, The Nature of Periodicity & Electronegativity

The periodic law is a property of the periodic table. As a consequence, periodicity and periodic trends get mapped to the element-the-basic-substance along with block, period & group.

Average atomic mass maps closely – but not exactly – to atomic number, and there are anomalies. However, most commentators would agree with the statement that "generally atomic mass increases with atomic number, Z" and this is a classic manifestation of the periodic law.

Electronegativity is a parameter of huge importance to understanding and predicting chemical structure and reactivity.

There is a clear electronegativity trend across the periodic table in its long form from the (Group 17) top-right where the most electronegative elements are found to the bottom-left where there are electropositive elements. This trend is a manifestation of the periodic law.

This author holds that while the actual elemental electronegativity data, for example (revised Pauling):

H   2.20
Li  1.00
F   3.98
O   3.44
Cl  3.16, etc.

is a property of the simple elemental substance, the periodic trend is a manifestation of the periodic law that is inherent to the periodic table.

Like atomic number, Z, electronegativity is an atomic property that is conserved in molecules and ionic substances.

It follows that relative electronegativity is a basic property and not a simple property.

From the underlying quantum patterns element Z = 9 [Period 2, Group 17, fluorine, F] is electronegative, and indeed *must* be the most electronegative element. It is just how quantum mechanics works (and we do not understand QM in terms of a deeper theory).

Thus, the relative electronegativity of element Z = 9, for example, is a basic (essential) elemental property that comes from periodicity and the periodic law, even though the absolute electronegativity of the simple (real) elemental substance is 3.98.

Basic & Simple Substance, Theoretical & Practical Chemistry: Does Any of This stuff Matter?

The reader may consider that the concept of element-as-basic-substance and element-as-simple-substance to be an arcane distraction limited to the study of the periodic table. However, the idea is actually rather general and has implications for how we understand and teach the subject of chemistry.

Beginning students of chemistry always have access to periodic tables, but unfortunately not the PT they actually need.

Students are expected to know that in all equations hydrogen is molecular should [nearly always] be written as H2. Likewise, nitrogen is N2, oxygen O2, fluorine F2, chlorine Cl2, bromine Br2 and iodine I2, and should always be written as the dimeric species. But somehow students are expected to know that molecular sulfur, S8, and phosphorus, P4, should be written as S and P.

These matters are explored further elsewhere in the Chemogenesis web book.

Quantum Numbers to Periodic Tables
Periodic Table Formulations

© Mark R. Leach 1999-

Queries, Suggestions, Bugs, Errors, Typos...

If you have any:

Suggestions or periodic table representations not shown on this page
Suggestions for links
Bug, typo or grammatical error reports about this page,

please contact Mark R. Leach, the author, using

This free, open access web book is an ongoing project and your input is appreciated.