Holistic Approach to the Periodic System of Elements

Transcription

1 Holistic Approach to the Periodic System of Elemets N.N.Truov * D.I.Medeleyev Istitute for Metrology Russia, St.Peterburg Moskovsky pr. 19 (Dated: February 20, 2009) Abstract: For studyig the objectivity ad the quality of a give form of the Periodic system as a sigle whole we compare plots of fuctios presetig properties of elemets i pairs of periods. Usig mathematical statistics we itroduce a dimesioless parameter which idicates high quality of the log form of the Periodic system. 1. Itroductio Sice the discovery of the Periodic System of Elemets (PSE) by Medeleyev i 1869, it has bee the object of extesive research by scietists of may specialties i its fudametal ad applied aspects. Chemists treat the PSE as the basic for makig ew substaces with preset properties. More tha 400 geometric forms of the PSE are proposed ad may suppositios are made but oly at the level of qualitative observatios ad treds, see review [1]. O the other had, a lot of partial coclusios ad suppositios were made i order to coect forms of the PSE with the electro structure of the atoms by meas of some qualitative use of the quatum mechaics. Ufortuately a great part of them is usatisfied, see sectio 4. Meawhile, there is a obvious iterest i solvig the iverse problem: begiig with the iformatio cotaied i the moder PSE to fid umerical * Electroic address: truov@viim.ru

2 parameters characterizig the PSE as a sigle whole. Such holistic approach was aouced ad realized at the microscopic level i [2]. I the preset paper we maily cocetrate attetio o the macroscopic level, takig ito accout kow physical ad chemical properties of the elemets. Note that holistic parameters fouded ad discussed below may be treated as metrological characteristics of the PSE, so that we uite ad cotiue two mai directios of Medeleyev s activity: o the oe had chemistry i geeral ad the PSE specifically, ad metrology i itself o the other had. 2. Parameters evaluatig the closeess of properties As it follows from its ame itself, the mai feature of the PSE is its periodicity. We iterpret it i such maer that plots of fuctios presetig each property must be similar i differet periods. Hereafter we restrict ourselves to the eighborig pairs of periods sice we suppose they have maximum similarity. Now we have a) to itroduce umerical parameters evaluatig this similarity; b) to calculate these parameters for a give subset of properties ad c) to exclude the possibility of radom, but erroeous evidece of the closeess of properties. We deote umerical values of properties as p ( Z ), where is property umber ad Z is the ordial umber of elemet. We ca get rid of propertyfuctios by icludig i the set p coefficiets for the chage i properties with a chage of exteral coditios (pressure, temperature ad so o). The chose subset of properties we deote as Π= { p }. Differet properties correspod to differet measuremet scales [3], i.e. have differet permissible trasformatios of uits measure. All the itroduced parameters must be ivariat uder such trasformatios. Most popular are scales of itervals ad of relatios. The first oe permits liear trasformatios p p = ap+ b (1)

3 with ay a > 0 ad b(e.g. measurig of time i differet uits ad with a shift of begiig). The secod oe oly permits (1) with b = 0 like the absolute temperature. We require that a may chage idepedetly i each of two compared periods. Let s deote Z( N, M ) a elemet of the log form of the PSE which is placed i the N th period ad i the M -th colum. Hereafter we shall compare p ( Z) for each ad pair of elemets Z( N, M ) ad Z( N, M) i differet periods but i the same colum. Note that differet periods may have differet umber of elemets L( N ), so that for such periods we oly ca compare ( ( ) ( )) mi L N, L N pairs ad some places i the PSE remai empty. Hereafter we use the followig parameter, which is ivariat uder trasformatios (1): (, ) (, ) (, ) (, ) R N N = R N N = t N M t N M ; (2) M t ( N, M) (, ) ( ) D ( N) p Z N M p N =, (3) 2 M D ( N) = p ( N, M) p ( N) 2, (4) where p ( N ) is the average value of p over all M i (2) or (4). Sice t (, ) N M are ormalized vectors, 1 R 1, (5) a arbitrary fuctio of R also remais ivariat uder the trasformatio (1), but amely (2) is the most coveiet oe sice it coicides with the well kow coefficiet of correlatio ad may statistical methods for its evaluatio are kow, see below. Note that besides two discussed scales (1) there also exists the absolute scale with o trasformatio ad all the sequetial scales which permit ay trasformatios maitaiig orderig. For both these scales A part of elemets may ot have a property p are dimesioless. p, so that i this case we get a partly

4 orderig set. Hereafter we do ot deal with these scales (though R ad F (6) themselves belog to the absolute scale). 3. Parameters evaluatig the quality of the PSE as a whole Values of R (2) - (4) for pairs of the eighbourig periods ad for the idicated subset Π of properties [4,5] are preseted i the Table: Table: Correlatio coefficiets R (per cet) for the eighbourig pairs of periods Properties Ioizatio eergy Eergy of atom formatio Ethalpy of formatio i the gas phase Ethalpy of evaporatio Desity Boilig temperature Meltig temperature Mi L=8 R 0.82 α 0.01 α L=18 < 10 4 Now we have to exclude the possibility of radom, but erroeous evidece of the closeess of properties i differet periods. Several methods of the mathematical statistics [6] give close sigificace levels α of the hypothesis for the absece or presece of correlatio at give R ad mi L. As it follows from the Table, the probability α of the absece of correlatio is egligibly small for almost all calculated R. If we eed to have a sole parameter evaluatig the objectivity of a give structure of the PSE with respect to a give subset Π of properties it is meaigful to itroduce the fractio F ( ; α ) Π of all values R i the Table for which the correlatio does exist with the probability ot less tha 1 α. Evidetly 0 F 1. Thus for the Table :

5 ( ) 26 F Π ;0.01 = = (6) 28 It is iterestig to compare F (6) with the same parameter for the same properties, but calculated for all pairs of periods 2 N 6. Usig results of [5], we get: ( ) 55 F Π ;0.01 = = Some chage for the worse is maily coected with irregularities of meltig temperature. I a similar way we ca certaily compare other subsets of elemets, e.g. the eighbourig groups at a fixed umber of periods. 4. Discussio The followig fact attracts our attetio i the Table: correlatio coefficiets R are especially close to uity ad respectively α is very small for the microscopic properties ioizatio eergy ad eergy of atom formatio. It makes us to suppose a close coectio betwee the structure of the PSE ad the atomic structure. This suppositio is really correct, amely a ew effective quatum umber [7] ( 1 ) ϕ ( 1 ) T = + + l+ (7) 2 2 may be itroduced, where ad l are the radial ad orbital quatum umbers respectively. New shells ope ad could be filled i order of icreasig T at the eergy ε = 0. The ew parameter ϕ does ot chage with icreasig amplitude of the self cosisted potetial with icreasig Z. I such a way the geuie structure of the electro shells i atoms ad thus the geuie structure of the PSE arises whe the value of ϕ is i the followig iterval [7]: < ϕ <, but a umber of argumets allows to estimate ϕ as: ϕ = 1.72 ± Sice ϕ determies both the electro structure of each atom ad of the PSE as a sigle whole we ca imagie a abstract object a superatom whose

6 structure is govered by the oly costat ϕ. The states of this superatom distiguished by Z are the groud states of a atom with Z electros. ( Note that usual explaatios of the PSE remai still valid outside this iterval, e.g. for ϕ=1.6, while shells orderig ad the whole structure of the PSE chages drastically. That is why these qualitative argumets metioed i hadbooks are usatisfied.) Thus we have a pair of quite differet holistic parameters ( F, ϕ ) with F (6) determiig the PSE as a sigle whole from the macroscopic ad microscopic poits of view respectively. It is however extremely difficult to trace how a give shells orderig (ϕ) results i a very high similarity of macroscopic properties (F) by meas of atomic physics, electrodyamics, statistical physics ad so o. For the completeess of our approach it ll be iterestig to calculate R ad Π for each deformed structure of the PSE ad the prove that the maximum value of Π is reached for the geuie log form of the PSE. Oe may expect that this really takes place sice ay deformatio of the PSE disturbs extremely good correlatio of the microscopic properties coected with the electro shells. It should be metioed i coclusio that similar holistic (or system) approach is also uavoidable for may ew objects of great iterest: clusters ad other ao-objects ad ao-systems, logic of the future quatum computers ad the swarm calculatios ad so o. May algebraic objects (graphs, matrices etc) may be used to describe orderig ad other details of such systems. As to the future developmet of metrology, though precise ad reliable measuremets still remai actual, its maistream i the XXI-st cetury will be coected amely with holistic or system approach ad with ew types of metrological quatities. 1. V.N.Ostrovsky. Foudatios of Chemistry. V.3, N2, p.145 (2001).

7 2. Yu.V.Tarbeev, N.N.Truov, A.A.Lobashev ad V.V.Kukhar. J. Exp. Theor. Phys. V. 85, p.667 (1997). 3. P.Pfazagl. Theory of Measuremet. Physica-Verlag, Würzburg - Wie, J.Emsly. The Elemets. Claredo Press. Oxford, V.S.Aleksadrov, N.N.Truov ad A.A.Lobashev. Measuremet Techiques. V.51, N.4, p. 345 (2008). 6. G.A.Kor, T.M.Kor. Mathematical Hadbook. McGraw-Hill Comp. N.Y., Toroto, Lodo, A.A.Lobashev, N.N.Truov. arxiv: 0710, 5400 [math.ph.] (2007).