Triads,Bonchev etc.FOCH.pdf

8/12/2019 Triads,Bonchev etc.FOCH.pdf

1/15

Explaining the periodic table, and the role of chemicaltriads

Eric Scerri

Published online: 28 January 2010 Springer Science+Business Media B.V. 2010

Abstract Some recent work in mathematical chemistry is discussed. It is claimed thatquantum mechanics does not provide a conclusive means of classifying certain elementslike hydrogen and helium into their appropriate groups. An alternative approach usingatomic number triads is proposed and the validity of this approach is defended in the lightof some predictions made via an information theoretic approach that suggests a connectionbetween nuclear structure and electronic structure of atoms.

Keywords Chemistry Quantum mechanics Periodic table Information theory Chemical triads

Introduction

Much work in the philosophy of chemistry has focused on the question of reduction(Bogaard 1978 ; Hendry 2008 ; Ramsey 2006 ; Scerri 1994 ). These authors consider thereduction of chemistry to quantum mechanics and seldom mention a broader sense of reduction to numerical regularities in the absence of a physical theory. In chemistry thediscovery of numerical regularities among groups of three elements such as lithium,sodium and potassium produced the concept of triads as rst discussed by Do bereiner(1829 ). Many triads were then assembled together to produce some early periodic systemsby Lothar Meyer and Mendeleev among others. At this point there was no explanatorytheory to account for the existence of chemical periodicity. Such a theory became availablesome time later, namely in the 1920s with the advent of quantum mechanics and the notionof electrons occupying shells around the nucleus of the atoms of the various elements.I suggest that the use of a triad approach may be considered as a form of reduction, broadly

construed, although I will postpone arguing the case for a later paper.

E. Scerri ( & )Department of Chemistry & Biochemistry, UCLA, Charles Young Drive East,Los Angeles, CA 90095, USAe-mail: [email protected]

1 3

Found Chem (2010) 12:6983DOI 10.1007/s10698-010-9082-9


2/15

Turning to current developments, the eld of mathematical chemistry has existed forseveral decades and yet it is never mentioned in the context of the reduction of chemistry. 1

This is an unfortunate omission in my view. In a later section of the present paper amathematical approach to the periodic table, due to Bonchev, will be discussed.

Reduction of the periodic table in particular

Attempts to explain the periodic table have included reduction to quantum mechanics aswell as approaches from mathematical chemistry. The latter work has been approachedthrough various group theoretical schemes (Novaro 1973 ; Kibler 1989 , 2007 ; Ostrovsky2005 ), similarity studies (Carbo-Dorca 1998 , 2000 ; Restrepo and Pacho n 2007 ; Sneath2000 ), and other mathematical approaches (Scerri et al. 1998 ).2

The periodic table is an absolutely central concept in the study of chemistry and deessimple categorization, while at the same time being the supreme example of a scienticsystem of classication. In addition to being described as a system of classication theperiodic table has been described as a model, a theory, a representation and many thingsbesides (Shapere 1974 ; Weisberg 2007 ). It is important theoretically as well as being of immense educational and practical value to working scientists.

A seldom mentioned attribute of the periodic table is that it concerns a rather holisticand all-inclusive aspect of chemistry since it embodies all the elements and in additionhighlights layer upon layer of relationships among the elements. I would like to suggestthat quantum mechanics provides a good means of reducing the properties of individual

elements but is not quite so effective when it comes to capturing some of the more globalaspects of the periodic table. By global property I intend, for example, the membership of elements to particular groups in the periodic table such as which elements belong togetherin the same group. This property connects together several elements in a way that isrecognized by the chemist but not always by physicists who concentrate on the morereductive aspects of the periodic table. For example, one can ask whether hydrogenbelongs in group 1 or 17 of the periodic table (Fig. 1). The traditional reductive approachusing the number of outer-shell electrons is unable to settle this question.

The individual properties of hydrogen, lithium and sodium may be predicted to anydegree of accuracy, subject to ones computational resources. However, when it comes toplacing the element hydrogen into its most appropriate group, the electronic congurationsleaves the question under-determined. Should hydrogen be placed in group-1 merelybecause it possesses one outer shell electron or should it be placed in group-17 because itlacks one outer-shell electron to produce a full outer shell? Similar issues arise for theelement helium. Does its possession of two outer-shell electrons require that it be placed ingroup-2 or does the fact that it does not lack any outer shell electrons require that it beplaced in group-18? It is interesting to note that in the traditional form of the periodic tableshown in Fig. 1, one criterion (number of outer electrons) is used in the placement of hydrogen, but the other criterion (number of electrons needed to complete the outer-shell)

is used in the placement of helium.

1 Several journals are devoted to the eld of mathematical chemistry including, MATCH Communicationsin Mathematical and in Computer Chemistry , published since 1975 and Journal of Mathematical Chemistrypublished since 1987.2 In addition many other branches of chemistry have been studied using these and other mathematicalapproaches which are not physical theories.

70 E. Scerri

1 3


3/15

I claim that quantum mechanics has not effected a full reduction when it comes todeciding on the placement of several elements within groups of the periodic table. Thisstate of affairs can also be seen in the current debate taking place over the elements thatshould be considered to be members of group-3 of the periodic table. Opinions aresomewhat divided between the group consisting of scandium, yttrium, lanthanum andactinium (Lavelle 2008 ), to others believing that the group should contain scandium,yttrium, lutetium and lawrencium (Scerri 2009 ). This can be considered as a failure of

quantum mechanics to predict or explain a global property or a relationship between sets of elements rather than any failure to predict precise numerical data on any of the elements.The frequently given account of the periodic table is that it results from analogies

between the electronic congurations of atoms. It is claimed that, broadly speaking, theelements that fall within a particular group do so because they share the same number of outer-shell electrons. But this account only serves as a rst approximation. Moreover, as Ihave argued previously, there is a semi-empirical aspect that is usually not conceded intextbook accounts of this alleged full reduction (Scerri 2007 ). Admittedly, the particularcombination of four quantum numbers that may be assigned to each electron in a many-electron atom can be derived from rst principles and this leads naturally to a prediction of the shell capacity for each electron shell, namely 2, 8, 18, and 32 for the rst four shells.However, it is not the closing of shells that is chemically relevant but the closing of periods . The latter takes place at the cumulative electron counts of 2, 10, 18, 36, 54, etc.(the noble gases) and not at the cumulative totals that one would expect if the shells werelled in a strictly sequential manner (2, 10, 28, 50, etc.).

One can summarize the situation by saying that the problem of the closing of the shells isdeductively solved by using the Pauli principle and the values that the four quantum numberscan adopt. But that the problem of the closing of the periods remains to be solved. The latterfollows an empirical rule called the Madelung, or n ? rule, whereby electron orbits are

lled in order of increasing n ?

quantum numbers rather than just an order of increasing nquantum numbersas would be the case if the shells were to ll in a strictlysequentialmanner. 3

The Madelung rule, which describes the closing of the periods, has not yet been theoreticallydeduced in spite of a number of claims to the contrary (Bent 2006 ; Ostrovsky 2005 ).

Fig. 1 Eighteen column or medium-long format periodic table. Hydrogen is placed in group 1 (alkalimetals) and helium in group 18 (noble gases). Groups are numbered 118 from left to right

3 The rule was rst discovered by C. Janet for reasons quite independent of quantum mechanics (Stewart2010 ).

Explaining the periodic table, and the role of chemical triads 71

1 3


4/15

Moreover, we reject Schwarzs recent suggestion that the n ? rule is irrelevant tochemistry because chemistry deals with bonded atoms rather than isolated ground stateatoms (Schwarz 2007 , 2009 ). We suggest that the elements, as represented in the periodictable, are as much a reection of bonded atoms as they are of isolated atoms that are

mainly of interest to physicists and astrophysicists. It is too easy to uphold the reduction of the periodic table to quantum mechanics, as Schwarz does, by merely declaring that theMadelung rule is of no importance to chemistry and that the more chemically relevant nrule has been successfully reduced to quantum mechanics. 4 Contrary to Schwarzs claim,I maintain that the periodic table is as much concerned with neutral atoms as it is withbonded atoms. After all, the periodic table is as much the province of physics as it is that of chemistry even if it originated historically in the eld of chemistry.

Nevertheless, quantum mechanical calculations which go well beyond the independentelectron approximation, on which electronic congurations are based, are quite capable of high accuracy in reproducing the periodic behavior in any number of properties displayedby a group of elements. For example, the characteristic plot of ionization energies againstatomic numbers for the elements shows maxima at elements in the alkali metals. Quantummechanical calculations can reproduce these graphs very accurately (Scerri 2007 ).5 In thissense the periodicity of the elements is captured from rst principles. But the problem of the closing of the periods, an example of what I take to be a global property of the periodicsystem, is not as I have argued above.

Triads

When Do bereiner discovered that lithium, sodium and potassium formed a triad he usedthe atomic weights of these elements. He found the atomic weight of sodium wasapproximately the mean of the atomic weights of lithium and potassium. Some 40 yearslater atomic weights became the basis on which the elements were ordered in the earlyperiodic systems of De Chancourtois, Newlands, Odling, Hinrichs, Lothar Meyer andMendeleev (Scerri 2007 ). These early periodic table had two main ingredients. Atomicweights were used to order the elements into a continuous sequence (primary classica-tion) and chemical properties such as valences were used to mark periodicity betweencertain elements (secondary classication) within the primary sequence of elements. 6 Thetriad of lithium, sodium and potassium represents such a case of periodicity in that theseelements have similar chemical properties including their valences of one unit.

But in a few isolated cases an interesting problem arose. If the elements were orderedstrictly according to increasing atomic weight, there would be a problem in achieving thedemands of chemical similarity. The best known of these cases involved the elements iodine

4 What is intended here by elements is elements as abstract or basic substances as they have been termedby Paneth. For example, the element sodium is the abstract element that underlies both sodium as a simpleuncombined substance and also sodium as it occurs when combined in sodium chloride for example. Thereis an already large literature in contemporary philosophy of chemistry which seeks to clarify this notion of element as basic or abstract substance (Hendry 2005 ; Earley 2009 ; Ruthenberg 2009 ; Scerri 2008c ).5 See p. 245 especially.6 The term secondary is being used here not in the sense of being less fundamental than primary classi-cation but simply to mean a classication that depends upon assuming and building further upon theprimary classication. One may think of an analogy with primary, secondary and tertiary structure in proteinchains.

72 E. Scerri

1 3


5/15

and tellurium. These elements should follow each other according to their atomic weightvalues and yet chemically speaking tellurium should be placed before iodine.

The solution to this conundrum remained elusive until well beyond the turn of thetwentieth century and was provided by van den Broek and Moseley (Scerri 2007 ).7 It

turned out that another property that is highly correlated with atomic weight, although notperfectly so, offered a superior ordering principle for the elements. This property was thenumber of positive charges in the nucleus of any particular atom, later termed atomicnumber. If this ordering parameter was used instead of atomic weight, there was no clashwith the chemical demands of secondary classication. Both tellurium and iodine fell intotheir natural chemical groups since the atomic number of tellurium, unlike its atomicweight, is less than that of iodine.

It is tempting to re-examine the notion of triads in terms of atomic number instead of atomic weight. Doing so reveals a remarkable outcome. Many of the triads of elements thatthe nineteenth century chemists had found to be approximately correct are now found to beexact. Sodium with atomic number 11 represents the exact mean between that of lithium(3) and potassium (19). Moreover the reason why atomic number triads are exact ratherthan approximate can be explained by recourse to long forms 8 of the periodic table ratherthan the short-form tables that the nineteenth century chemists discovered. 9

But atomic number triads have seldom been put to much use in chemistry or indeed inthe study of the periodic system. The rst mention of them in the literature, as far as I amaware, is a very short note in the Journal of Chemical Education in 1931, although it is notclear whether the author understood the full signicance of atomic number triads (Mont-gomery 1931 ). More recently they were used by Ibrahim to predict the atomic weights of

the transuranium elements (Ibrahim 2005 ).In papers published in 2008 I suggested that atomic number triads could be used in anattempt to resolve a question that quantum mechanics has not been able to settle, namelythe placement of hydrogen and helium in the periodic system (Scerri 2008a , b). It is clearlya requirement that triads of elements should be embedded in any correct periodic table.This embedding feature explains why they were so instrumental in the early historical stepstowards mature periodic systems as well as being important to pioneers of the system suchas Mendeleev even though he may have denied that he was focusing on triads speci-cally. 10 It is also the basis for suggesting that triads may be further exploited in contem-porary efforts to obtain a more denitive placement for elements such as hydrogen, heliumand the elements of group 3. Quantum mechanical calculations do not provide answers tothese questions even though they provide exceptionally accurate calculations for theproperties of the individual elements concerned.

However, any embedding relationship that may exist regarding triads is not asstraightforward as one might believe. Even if one uses atomic number triads, it is not clearthat all possible triads are valid. This is especially controversial in the case of triadsinvolving hydrogen and helium that some authors consider as not belonging to any triadswhatsoever (Bent 2006 ). To consider that triads are embedded in the periodic table in an

7

See pp. 165175.8 The left-step (Fig. 2) shown no exceptions in this respect whereas the medium-long form (Fig. 1), isanomalous in the case of groups 1 and 2.9 Nevertheless, some of Mendeleevs early tables were actually of a long-form.10 There is little doubt that even if Mendeleev tried to distance himself from numerological aspects of usingtriads, his reasoning when making predictions about yet undiscovered elements lay precisely in the use of the concept of triads, sometimes even using triads in two directions. See Scerri 2007 , p. 135.


1 3


6/15

invariable fashion would be to pre-judge the very question being addressed in parts of thepresent article, namely whether hydrogen and helium actually belong to genuine triads.

The reason for the recent neglect of triads is not difcult to discern and is relevant to thepresent issue. In spite of their early fruitful role, triads of elements fell into disrepute for a

number of reasons. While triads were still based on atomic weights, the relationshipbetween the elements remained approximate, thus suggesting the lack of any fundamentalstatus for triads. Secondly, many authors became so enamored with triads that theyattempted to create new triads even among elements, which did not show chemical sim-ilarities. Thirdly, Mendeleev, the leading discoverer of the periodic system took pains todistance himself from the use of triads although he was not averse to making interpolationsamong groups of three elements when in the course of predicting new elements.

Nevertheless, triads can retrospectively said to have made a comeback, a point rstmade explicitly by Imre Lakatos ( 1980 ). With the advent of atomic number as the keycriterion for identifying each element, atomic number triads become exact rather thanapproximate as they are when dealing with atomic weights. Furthermore, their existencefollows naturally from the expanded, or long-form of the periodic table, as compared withthe original short-form tables of Mendeleev and other pioneers of the periodic system.

But to the best of my knowledge, nobody has previously embraced atomic numbertriads in order to clear up any remaining issues concerning the placement of difcultelements like hydrogen and helium. This is because at the same time as atomic numbertriads were making a comeback, the attention had turned towards the electronic structure of atoms as the means to examine the classication of elements in the periodic table.

However, electronic congurations show many exceptions, whereas the property of

atomic number progresses smoothly and regularly as one moves through the elements.I therefore suggest that atomic number is a more fundamental means to nalize theclassication of the elements, a process begun by classical chemistry by concentrating onthe macroscopic properties of the elements and their compounds. Atomic number is alsomore fundamental in an ontological sense of referring to a deeper level of reality, namelythe nucleus, whose energy is orders of magnitude higher than those of electronic processes.

An analogy between the use of atomic number and electronic conguration to affectprimary and secondary classication, respectively

There is something of an analogy between the manner in which atomic number replacedatomic weight in affecting the ordering the elements (primary classication) and a similarreplacement of chemical properties by electronic conguration in affecting placement intogroups (secondary classication). These days, under the reductionist climate that prevailsin chemistry, secondary classication or the placement of elements into groups tends to becarried out primarily by recourse to electronic conguration. 11 It should also be quicklyemphasized that the analogy is far from perfect, given that electronic congurationssometimes fail to resolve questions of secondary classication, unlike the use of atomic

number in primary classication, which is exceptionless.There are many groups of the periodic table where outer-shell electronic congurationvaries such as group 10 consisting of nickel, palladium and platinum. The congurationsare, Ni [Ar] 3 d

8 4s2 , Pd [Kr] 4 d

10 5s0 , Pt [Xe] 4 f

14 5d 9 6s

1 . This feature strengthens the

11 There is a sense in which the periodic table of the elements has been replaced by the periodic table of the neutral gas-phase atoms.

74 E. Scerri

1 3


7/15

view that electronic conguration alone does not provide an unambiguous means of classifying elements into groups.

But there are some even more troublesome cases. Where for example should the ele-ment hydrogen be placed? It has been argued that, on the basis of its electronic congu-ration, it belongs in as many as four different places in the table. Should it be placed, as itmost frequently is, in group-1 of the periodic table because its atoms possess one outer-shell electron as do those of the alkali metals? Should it be placed in group-17 among thehalogens because hydrogen atoms can easily form singly charged negative ions as atoms of the halogen elements can? (Laing 2007 ; Sacks 2006 ). Should it be placed in group-14of the periodic table, underneath the element carbon since hydrogen atoms, like those of carbon, have a half-full shell? 12 (Cronyn 2003 ).

And nally, should hydrogen be removed altogether from the main body of the table, asis carried out by many textbook authors? These authors remove both hydrogen and heliumand allow these two elements to oat above the main body of the table on the grounds thathelium too is somewhat anomalous and does not truly belong to the noble gases or to thealkaline earths (Atkins and Kaesz 2003 ).

The more usual choice for helium, as shown in Fig. 1, is in group 18, or the noble gases,in view of its extreme inertness. However, there are some physicists and even chemists thatlike to emphasize that helium atoms have two outer-shell electrons which justies theelement being placed among the alkaline earth elements, as shown in Fig. 2, in spite of anyapparent chemical differences between helium and group-2 elements like beryllium,magnesium and calcium.

I propose that all of these issues can be settled conclusively by considering atomicnumber triads, preferably on a left-step table arrangement of the elements. 13 This displayalso explains the very existence of perfect atomic number triads. It emerges that if thesecond and third elements of a potential triad fall in periods of equal lengths, then the triadrelationship is exact, producing what I am calling a perfect triad.

The reason why perfect atomic number triads occur is the repetition in the lengths of allthe periods except for the very rst short period, which contains just two elements. 14

Fig. 2 Left-step periodic table, which places helium in the alkaline earth elements as well as relocatinggroups 1 and 2 to the right side of the table

12 In the case of hydrogen a half-lled shell represents one of a possible two electrons, while carbon hasfour out of a possible eight outer-shell electrons.13 This can also be achieved using a conventional or medium-long form display but matters are not so clear.Because of the peculiarities of placing the s-block elements at the left of the medium-long format, perfecttriads involving s-block elements require that the rst and second elements should be in periods of equallengths.14 This is true of the conventional medium-long form table but not the left-step format.


1 3


8/15

Consider for example the triad consisting of sulfur, selenium and tellurium. A glance at the

left-step periodic table, or even the medium-long form in this case, reveals that to movefrom sulfur to selenium requires an advance of 18 places. Moreover, in order to reach thenext element in this group, tellurium, one must advance a further 18 places. It is no surprisetherefore that selenium falls exactly mid-way between the elements sulfur and tellurium interms of atomic number values.

Needless to say, such triads appear to lack any explanatory signicance, at least at rstsight. 15 However, they do have signicance of a rather important kind since they areconcerned with atomic number which is the one characteristic property of elements asbasic substance, a topic that has received considerable attention in contemporary phi-losophy of chemistry and which also has a long history (Bensaude-Vincent 1986 ; Earley

2005 ; Scerri 2008c ; Ruthenberg 2009 ).

A new application of atomic number triads

Let us return to the question of where hydrogen should be placed in the periodic table. If atomic number triads are in some sense fundamental, as I have suggested previously, thenone should seek to maximize the number of atomic number triads that appear on a periodictable. 16 This aim can be easily achieved by moving hydrogen from its usual position at thetop of group 1 (alkali metals) to the top of group 17 (the halogens). This action results inthe formation of the following perfect atomic number triad, H (1), F (9), Cl (17). 17

Furthermore, the desire to maximize perfect atomic number triads would suggest thathelium should remain in its traditional place at the top of the noble gases and should not beplaced at the head of the alkaline earth elements as is carried out in the left-step tableformat. Leaving helium in its tradition place means that one retains the triad of He (2), Ne(10), Ar (18), as opposed to losing it, on moving to the left-step format He (2), Be (4), Mg(12). The proposed periodic table format, in which hydrogen is moved to the halogens, isshown in Fig. 3.

Fig. 3 Modied left-step periodic table which places hydrogen in group 17 (halogens) (Scerri 2008a , b). Numbers along the right-hand edge represent values of n ? , the sum of the rst two quantum numbers forthe differentiating electron. I thank Philip Stewart for pointing out to me that precisely this form of theperiodic table was proposed by Janet in 1928 but immediately discarded by this author (Janet 1928 )

15 This point will be taken up again shortly.16 Some authors object to this proposal and claim that H and He do not belong to triads in principle (Bent2006 ).17 The placement of H among the halogens has been proposed by many authors for a variety of chemicalreasons such as the fact that hydrogen forms a singly-charged negative ion as do the halogens and the factthat hydrogen occurs as a diatomic molecule H 2 as do all the halogens.

76 E. Scerri

1 3


9/15

The desire to maximize atomic number triads also provides a further argument for thegrouping of scandium, yttrium, lutetium and lawrencium into group 3 rather than thegrouping of scandium, yttrium, lanthanum and actinium as in many textbooks. 18 Such a re-placement of Lu and Lr has been suggested on chemical as well as physical grounds

(Jensen 1982 ) and has been adopted by many textbooks and designers of periodic tables,although there are still pockets of resistance (Clark and White 2008 ; Lavelle 2008 ).19

Possible objections to such use of triads

Some critics have objected to the manner in which I propose using atomic number triads.The objection is that triads are not fundamental, that they do not provide an underlyingmechanism for why certain elements should represent atomic number triads. It has evenbeen suggested that my proposal amounts to mere numerology. 20

Before proceeding any further I would like to ward off such objections. First of all,I claim that the use of atomic number triads, is capable of completing a global aspect of the periodic table which reference to electronic structure fails to do. 21 The proposedargument appeals to atomic number, the only property of elements as basic substances,which philosophers of chemistry and some chemists are increasingly accepting as the morecorrect basis for periodic classication than elements as simple substances (Bent 2006 ;Bent and Weinhold 2007 ; Schwarz 2007 ).22 Regarding the charge that the use of triadsdoes not provide any mechanistic or causal argument for its efcacy, I would counter bymaking a radical suggestion but one which will be supported by empirical data.

If nuclear structure in fact governs electronic structure this would provide the requiredcausal argument to lend greater validity to the use of atomic number triads. This idea isradical because it is not generally believed that nuclear structure, as such, has a directcausal inuence on the arrangement adopted by the electrons surrounding the nucleus. 23

What is undeniable, however, is that the physics governing the particles within thenucleus is analogous to the physics of the electronic motion. Both cases can be successfullyanalyzed using an independent-particle model which can subsequently be modied toobtain more accurate agreement with experimental data. Both cases can be approached viaHartreeFock approximations for the analysis of central-eld problems.

More to the point of the present study, both cases have periodic tables of sorts. Theyboth involve shell structures, the occupation of shells and the appearance of magic numbers

18 Yttrium (39), lutetium (71) and lawrencium (103), form a perfect atomic number triad, whereas yttrium(39), lanthanum (57), and actinium (89) do not.19 For a critique of Lavelles resistance see Scerri 2009 .20 Such an objection was made by Eugen Schwarz at the 2008 ISPC meeting in Coburg, Germany.21 Triads may be regarded as local in the literal sense of referring to a region of elements lying adjacent toeach other in the periodic table. But here I am using global to mean that the triads concern a relationshipbetween different elements and so capture the key aspect of chemical periodicity.22 Some of the philosophers who have written on this topic were listed above.23 At a conference in Leuven, Belgium, Robin Hendry objected to this notion and expressed the view thatthe nucleus does indeed govern the motion of electrons in the sense of energetic terms such as Coulombicattraction. But this is not what I am getting at. Of course it is trivially the case that electronic motion inatoms and molecules is mathematically governed by the nucleus. My claim is that it is not usually believedthat nuclear structure, per se, causes a particular electronic conguration to be adopted in any particularatom.


1 3


10/15

associated with enhanced stability (Goeppert Mayer and Jensen 1955 ). Needless to say thevalues for these magic numbers are not the same for the nuclear and electronic systems.

Information theoretic approach of Bonchev

I now turn to Bonchevs information theoretical approach, which I believe may lendsupport to the idea that atomic number triads represent a fundamental method for assigningelements to groups of the periodic table. This is because Bonchevs approach highlights apossible connection between nuclear structure and electronic structure.

Bonchevs approach draws directly from Shannon information theory (Shannon 1948 ).If we consider a system of N elements, which are distributed into k classes and where theith class has N i elements. We can now dene a nite probability, pi, for a randomly chosenelement to belong to class I ;

Groups of elements 1, 2, , k Number of elements N 1 , N 2 , , N k Probability distribution p1 , p2 , , pk

where pi = N i / N , P N i = N , and P pi = 1.The mean entropy, in the sense of information, H a is given by

H a Xk

i1 pi log pi X

k

i1 Ni= N log Ni= N 1

while the total entropy (information), H (a ) is given by

H a N log 2 N Xk

i1 N i log N i 2

Bonchev denes a third information function which he calls differential information D I x( z).This represents an increase in information content between two systems. It is immediatelyapplied in order to obtain the differential information on moving from an element withatomic number z to one with z - 1.

D I x D I x z D I x z 1For example, it can be used to calculate the differential information between the chromiumatom with atomic number 25 and its 25 electrons and the vanadium atom with atomicnumber 24 and 24 electrons. The electrons are distributed into four classes ( k ) or electronshells containing 2, 8, 13, and 1 electrons for chromium and 2, 8, 11, and 2 electrons forvanadium. 24

On substitution into the above equations one nds, 25

D I 25 log 2 25 13 log 2 13 1 log 2 1 24 log 2 24 11 log 2 11 2 log 2 2 2:0548

24 This comes about by simplifying the electronic conguration of chromium, namely, 1 s2 , 2s2 , 2 p6 , 3s2 ,3 p6 , 3d 5 , 4s1 and vanadium, which is taken to be 1 s2 , 2s2 , 2 p6 , 3s2 , 3 p6 , 3d 3 , 4s2 .25 The terms containing 2 and 8 cancel and have therefore been omitted.

78 E. Scerri

1 3


11/15

Bonchev has calculated such values of differential information for successive atoms in theperiodic table. These differential values were then plotted against atomic number. Theresult was to display, rather prominently, the so-called anomalous congurations that occurin atoms such as chromium and copper that show minimum values. Maxima are found onthe resumption of expected electronic congurations as in the case of manganese or zincfor example, that is atoms that follow directly after anomalous cases as shown in Fig. 4.

Bonchev then turns to the atomic nucleus and subjects it to a similar analysis in terms of information theory. A nucleus is characterized by a number of neutrons, n, and p protons,thus yielding an atomic number A = n ? p. Introducing these quantities into the infor-mation Eqs. 1 and 2, produces total and mean information entropy expressions for theproton-neutron composition of any particular isotope, I np .

I np A log 2 A z log 2 z n log 2 n 3

I np z= Alog 2 z= A n= Alog 2n= A 4

These equations in turn permits Bonchev to discuss periodic trends among isotopes of the elements. Finally, a comparative analysis is carried out between the periodic trends inatoms and nuclei. More specically the average spin information per electron in a chemicalelement is compared to the average information for proton-neutron composition pernucleon in any isotope using Eqs. 5 and 6,

I np 1 1=2 ln2 b2= A2 5

I ms 1 1=2 ln2 a2= z2 6

where b = n - z = A - 2 z and A = mass number = n ? z. The terms b / A may beinterpreted as number of unpaired electrons per total electrons while a / z represents theexcess neutrons per total protons.

Fig. 4 Differential information on the distribution of electrons over electron shells (Rouseva and Bonchev1980 )


1 3


12/15


13/15

On the basis of the analogy between these information expressions Bonchev was able toestablish a correlation between electronic energy and nuclear binding energy in 1980 andto use this empirically calibrated correlation to predict the binding energies for manynuclides of atomic numbers 101108 inclusive. Since that time many of these nuclideshave been synthesized and their experimental binding energies have been obtained. Thesepredictions are compared with experimental binding energies in Table 1. The standarddeviation is 1.9 MeV for binding energy range 1,8901,950 MeV, which means that therelative standard deviation is approximately 0.1% for the 41 isotopes in which experi-mental values are available (Rouseva and Bonchev 1980 ).26 Such results suggest that theremight be a causal role for nuclear structure on electronic structure or at least a correlationbetween nuclear and electronic structure.

It should be added that a 0.1% correlation may not be so signicant in view of the sizeof the binding energies involved. In addition it is not immediately clear whether Bonchevs

correlations may partly be the result of the manner in which the electronic calculationswere performed. For example the work that Bonchev draws upon makes the assumption of a nite nuclear density which is a function of mass number of any isotope A. The extent towhich this might inuence the claimed correlations is open to further examination.

If the nucleus of an atom does indeed govern electronic structure then the former ratherthan the latter, would provide the more fundamental reduction of the form of the periodictable and such questions as the placement of elements into groups of the table. It wouldfollow that a relationship involving atomic number, an attribute of the nucleus, couldprovide a more fundamental means of assigning elements to groups. The role of atomicnumber, as opposed to electronic structure, in the context of grouping of elements would be

somewhat akin to the role of atomic number as opposed to atomic weight when it comes toassigning an order to the elements. In both cases the two alternative classication criteriaare highly correlated but in both cases the rst named alternative provides the morefundamental criterion for the appropriate kind of classication or ordering in question. Itremains to be seen whether Bonchev and co-workers can extend their studies to showfurther cases in which nuclear and electronic structure are highly correlated or whether theformer causes the latter.

Another such connection between electronic and nuclear levels concerns attempts toexplain the predominance of L form-optical isomers among naturally occurring bio-molecules and parity violation in the weak nuclear force (Service 1999 ). As in the Bonchevstudy this is an example of the inuence of nuclear effects upon electronic structure.

Finally, to return to the question of reduction, we should note that if there is indeed anuclear explanation for electronic structure, this would represent a point in favor of the

Table 1 continued

b z A Eb pred E b exper.

57 101 259 1,918.088 1,917.837

102 261 1,928.501 1,928.317103 263 1,938.762 1,938.413

104 265 1,948.847 Unknown

105 267 1,958.840 Unknown

26 The predictions were actually made on 45 isotopes but four of these could not be veried as they had notyet been synthesized. See table for details.


1 3


14/15

reduction of chemistry, a reduction to an even more fundamental level than is usuallybelieved to operate. As I have suggested previously, philosophers of chemistry must wardagainst the denial of reduction as a matter of ideology. One of the benets of examining thereduction of chemistry is that it might point to issues where an ever greater degree of

reduction can be achieved. If this turns out to be the case, then so much better for the sub-discipline of philosophy of chemistry.

Conclusion

Several theses have been put forward in this article. First of all it has been proposed that thequestion of the explanation of the periodic table should be investigated in a wider arena thanmerely asking if chemistry reduces to quantum mechanics. Secondly it has been argued thatthe question of the secondary classication, or grouping of the elements in the periodictable, cannot be conclusively settled by appeal to electronic structure of atoms. It may bethat one has to appeal to nuclear structure. 27 It was proposed that an early hint of thispossibility might lie in the work of Bonchev who uses information theory to investigateanalogies between nuclear and electronic structure and who has successfully predicted thebinding energies of 41 isotopes to an average accuracy of approximately 0.1%. Thisapparent connection between nuclear and electronic structure was used to lend someempirical support for the assignment of elements to groups of the periodic table on the basisof atomic number triad relationships. The nal conclusion of this paper is that contrary tothe general trend a greater degree in the reduction of chemistry to physics should be sought

instead of making the usual arguments that chemistry has not or cannot be reduced.

Acknowledgment The author thanks reviewers for the careful reading of this paper and for many usefulsuggestions.

References

Atkins, P.W., Kaesz, H.: The placement of hydrogen in the periodic table. Chem. Int. 25, 14 (2003)Bensaude-Vincent, B.: Mendeleevs periodic system of chemical elements. Br. J. Hist. Sci. 19 , 317 (1986)Bent, H.A.: New Ideas in Chemistry from Fresh Energy for the Periodic Law. Author House, Bloomington

(2006)Bent, H.A., Weinhold, F.: News from the periodic table: an introduction to periodicity symbols, tables, and

models for higher-order valency and donoracceptor kinships. J. Chem. Educ. 84, 11451146 (2007)Bogaard, P.A.: The limitations of physics as a chemical reducing agent. Proc. Philos. Sci. Assoc., (PSA

1978) 2, 345356 (1978)Bonchev, D.: Periodicity of the chemical elements and nuclides: an information-theoretic analysis. In:

Rouvray, D.H., King, R.B. (eds.) The Mathematics of the Periodic Table, pp. 161188. Nova SciencePublishers, New York (2006)

Carbo-Dorca, R.D.: On the extension of quantum similarity to atomic nuclei. J. Math. Chem. 23 , 327351(1998)

Carbo-Dorca, R.D.: General trends in atomic and nuclear quantum similarity measures. Int. J. QuantumChem. 77 , 685692 (2000)

Clark, R.W., White, G.D.: The yleaf periodic table. J. Chem. Educ. 85, 497 (2008)Cronyn, M.W.: The proper place for hydrogen in the periodic table. J. Chem. Educ. 80 , 947951 (2003)

27 The governing physical theory could still be quantum mechanics although now applied to the study of thenucleus. However, early hints of this might come in the form of purely mathematical approaches such asBonchevs use of information theory.

82 E. Scerri

1 3


15/15

Dobereiner, J.: Versuch zu einer gruppirung der elementaren stoffe nach ihrer analogie. Poggendorfs Ann.Phys. Chem. (Leipzig) 15, 301307 (1829)

Earley, J.: Why there is no salt in the sea. Found. Chem. 7, 85102 (2005)Earley, J.: How chemistry shifts horizons: element, substance, and the essential. Found. Chem. 11 , 6777

(2009)

Goeppert Mayer, M., Jensen, H.: Elementary Theory of Nuclear Shell Structure. Wiley, New York (1955)Hendry, R.F.: Lavoisier and Mendeleev on the elements. Found. Chem. 7, 3148 (2005)Hendry, R.F.: Entry for chemistry. In: Psillos, S., Curd, M. (eds.) The Routledge Companion to Phi-

losophy of Science, pp. 520530. Routledge, London (2008)Ibrahim, S.A.: Predicting the atomic weights of the trans-lawrencium elements: a novel application of

Dobereiners triads. J. Chem. Educ. 82, 16581659 (2005)Janet, C.: La Classication he lico dale des e lements chimiques. Imprimerie De partmentale de lOise,

Bauvais (1928)Jensen, W.B.: The positions of lanthanum (actinium) and lutetium (lawrencium) in the periodic table.

J. Chem. Educ. 59 , 634636 (1982)Kibler, M.: The periodic system of chemical elements, old and new perspectives. J. Chem. Struct. (Theo-

chem) 187 , 8393 (1989)

Kibler, M.: From the Mendeleev table to particle physics and back. Found. Chem. 11 , 221234 (2007)Laing, M.: Where to put hydrogen in a periodic table? Found. Chem. 9, 127137 (2007)Lakatos, I.: The Methodology of Scientic Research Programmes: vol. 1, Philosophical Papers. In: Worrall,

J., Currie, G. (eds.), pp. 43, 5355, 118119, 223. Cambridge University Press, Cambridge (1980)Lavelle, L.: Lanthanum (La) and actinium (Ac) should remain in the d-block. J. Chem. Educ. 85, 14821483

(2008)Montgomery, J.P.: Dobereiners triads and atomic numbers. J. Chem. Educ. 8, 162 (1931)Novaro, O.: Comments on the group theoretical justication of the aufbau scheme. Int. J. Quantum Chem.

(Symposium) 7, 2333 (1973)Ostrovsky, V.: On recent discussion concerning quantum justication of the periodic table of the elements.

Found. Chem. 7, 235239 (2005)Ramsey, J.: Chemistry. In: Pfeifer, J., Sarkar, S. (eds.) Philosophy of Science, an Encyclopedia. Routledge,

London (2006)Restrepo, G., Pacho n, L.: Mathematical aspects of the periodic law. Found. Chem. 9, 189214 (2007)Rouseva, B., Bonchev, D.: Prediction of the nuclear binding energies of the nuclides of period VII.

Radiochem. Radioanal. Lett. 45, 341346 (1980)Ruthenberg, K.: Paneth, Kant and the philosophy of chemistry. Found. Chem. 11 , 6577 (2009)Sacks, L.: Concerning the position of hydrogen in the periodic table. Found. Chem. 8, 3135 (2006)Scerri, E.R.: Has chemistry been approximately reduced to quantum mechanics. Proc. Philos. Sci. Assoc.

(PSA), Hull, D., Forbes, M., Burian, R. (eds.), 160170 (1994)Scerri, E.R.: The Periodic Table, Its Story and Its Signicance. Oxford University Press, New York (2007)Scerri, E.R.: The Role of triads in the evolution of the periodic system. J. Chem. Educ. 85 , 585589 (2008a)Scerri, E.R.: The past and future of the periodic table. Am. Sci. 96, 5258 (2008b)Scerri, E.R.: The dual sense of the term element, attempts to derive the Madelung rule, and the optimal

form of the periodic table, if any. Int. J. Quantum Chem. 109 , 959971 (2008c)Scerri, E.R.: Which elements belong in group 3? J. Chem. Educ. 86, 1188 (2009)Scerri, E.R., Kreinovich, V., Wojciechowski, P., Yager, R.R.: Ordinal explanation of the periodic system of

chemical elements. Int. J. Uncertain. Fuzziness Knowl. Based Syst. 6, 387400 (1998)Schwarz, W.H.E.: Recommended questions on the road towards a scientic explanation of the periodic

system of chemical elements with the help of the concepts of quantum physics. Found. Chem. 9, 139188 (2007)

Schwarz, W.H.E.: Review of Eric Scerri, the periodic system, its story and its signicance. Angew. Chem.Int. Ed. 48, 33913392 (2009)

Service, R.: Does lifes handedness come from within? Science 286 , 12821283 (1999)Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27, 379423 (1948). (623656)Shapere, D.: Scientic theories and their domains. In: Suppe, F. (ed.) The Structure of Scientic Theories,

pp. 518599. Illinois University Press, Chicago (1974)Sneath, P.H.A.: Numerical classication of the elements and its relation to the periodic system. Found.

Chem. 2(3), 237263 (2000)Stewart, P.: Charles Janet: unrecognized genius of the periodic table. Found. Chem. 12 (2010). doi:

10.1007/s10698-008-9062-5Weisberg, M.: Who is a modeler? Br. J. Philos. Sci. 58, 207233 (2007)


1 3
http://dx.doi.org/10.1007/s10698-008-9062-5http://dx.doi.org/10.1007/s10698-008-9062-5

Triads,Bonchev etc.FOCH.pdf

Documents

Transcript of Triads,Bonchev etc.FOCH.pdf