Introduction

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

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 basic substance, that is: the element as a bearer of properties but not having any actual properties, except for atomic mass and atomic number Z. Chemical symbols, such as H and Cu, are assigned to the element as basic substance.

Secondly, there is the element as simple substance, for example, a 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 element the basic substance 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 characteristic 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.

The Elements As Metaphysical Basic Substances Periodic Table

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

For example, the well known element oxygen is show as O and not as the common molecular form O2 found in air. 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 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.

Basic Element Properties:

• Atomic number, Z, plus name & symbol (assigned to Z)
• Block of periodic table: s, p, d, f
• Group number: 1-18
• Period: 1-7
• Periodicity
• Electronic configuration (see below)

The Periodic Table of the Elements As Ground-State, Monoatomic, Gas Phase Entities

The chemical elements as simple substance (material stuff, chemicals in bottles) 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.
• Liquid 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 the same as the periodic table above, 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, mass spectrometers and atomic clocks. Many man-made transuranium elements are only known as isolated gas phase atoms.

It is sometimes suggested 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.

The Elements As Chemical Substances Periodic Table

Under standard conditions, 25°C & 1.0 atm, the material chemical elements, 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.

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.

Eric Scerri: Philosopher, Theorist, Chemist, Writer

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.

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

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.
• 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.

  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."

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?

Q: Has The Periodic Table of Standard State Material Simple Substances Been Axiomatized?

The elements-as-chemicals (simple material substances at room temp & pressure) periodic table, 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, copper is a reddish coloured metal or that mercury is a liquid at room temperature.

There is not axiomatic mapping between theory and material properties:

Has The Periodic Table of Gas Phase Atoms Been Axiomatized?

The ground-state, monoatomic, gas phase element periodic table is not fully axiomatized.

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

The methodologies are 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 not a one-to-one-to-one correspondence between quantum chemistry calculations, atomic spectra and the periodic table of gas phase atomic simple substances. There is not formal proof:

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 or quantum electrodynamics (QED), the most precise theory known. Chemical problems are too involved to be studied by QED, although in principle they could be.

Has The Periodic Table of Metaphysical Basic Substances Been Axiomatized?

Metaphysical basic substances 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 its patterns can be deduced from the quantum theory.

Quantum theory, sometimes called the old quantum theory, produced the four quantum numbers (principle, subsidiary, magnetic & spin), the Pauli exclusion principle, the aufbau principle, Hund's rule and Madelung's rule.

The pattern of the periodic table schema directly arises from these rules and is axiomatic with respect to them, as discussed here.

Richard Feynman told us (here): "I think I can safely say that nobody understands quantum mechanics."

This means that 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 mysterious...

Morphing Multi-PTs

It is clear that the periodic table is not one single thing as we identify at least three periodic tables:

• The periodic table of metaphysical basic substances with atomic number Z that arises from quantum theory.
• The periodic table of gas phase atoms with their associated spectra. The techniques of quantum chemistry cannot completely – axiomatically – describe gas phase atoms (except H^•, He⁺, Li²⁺, etc.) because multi-electron systems are too complex to be described analytically (which would be axiomatic). However, the models are good enough for the periodicity and periodic table of the metaphysical basic substances to be mapped to the periodic tables of gas phase and material simple substances.
• Then there is the periodic table of chemicals in bottles, the actual materials at room temperature and pressure.
• And there are others. For example, periodic tables of phase changes: melting points, boiling points, etc,, show the real materials at 1.0 atm but not at room temperature.
• When moving through these PTs the system complexity increases.

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

The Periodic Law, The Nature of Periodicity and 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 conserved in molecules and ionic substances, so relative electronegativity is a basic property and not a simple property.

It follows from the underlying quantum patterns that element Z = 9 [Period 2, Group 17, fluorine, F] is electronegative, and indeed *must* be the most electronegative element. It is just how QM works.

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

Basic & Simple Substance: Theoretical & Practical Chemistry

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.

We teach students how to do titration calculations in class, but in the lab they are confronted with running taps and water getting everywhere. They can find it difficult to see how the class work has anything to do with lab work.

Consider the drug 'morphine'. Does this refer to morphine in opium, morphine free base, morphine sulfate, a 5mL amplue of injectible drug in saline, or what?

In large part getting to grips with chemistry involves understanding that the idealised chemical entities we hear about in class and read about in books are often complicated, difficult to use material substances in real life.

© Mark R. Leach 1999-2008

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