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Chemical Education Today

JChemEd.chem.wisc.edu Vol. 75 No. 7 July 1998 Journal of Chemical Education 817

The following is the second of three invited keynote lecturesgiven at the 57th annual summer conference of the New En-gland Association of Chemistry Teachers held in August of 1995at Sacred Heart University in Fairfield, Connecticut. Lecture Ihas appeared in an earlier issue of theJournal (1).

Our interest in the classification of the various theoreti-cal concepts and models of modern chemistry, presented inLecture I, lies not only in our desire to better understand theirinterrelationships and underlying assumptions, but also in theuse of this classification to aid us in detecting and clarifying

errors and ambiguities in our presentation of these conceptsand models and in the consequences which this classificationhas for how we teach these concepts and models to our students.

Relative to both of these functions, the most importantpedagogical lesson to be extracted from Lecture I is the logi-cal necessity of carefully distinguishing between the molar,molecular, and electrical levels of discourse in chemistry.Unhappily, this is also the point on which most modern text-books falter, as not only do they generally fail to explicitlypoint out the existence of these three levels, they normallyproceed to randomly mix them together throughout the book.Take, for example, the three textbook descriptions of dioxygengas given in Table 1, each of which corresponds to one of

our three levels of discourse. Not only is the sequence listedin the table the most logical order for developing a progres-sively more abstract description of dioxygen gas, it also hap-pens to correspond, as we will see in Lecture III, to the ac-tual historical evolution of our views concerning this mostimportant of chemical substances. Yet how frequently is thisorder scrambled in the average textbook and the studentsshown a Lewis diagram or a molecular orbital configurationfor dioxygen gas before they are told anything about its ap-pearance, molar properties, natural occurrence, preparation,or chemical reactivity?

To further illustrate these problems, we will first look atsome examples, taken from both the introductory and theadvanced inorganic textbook literature, in which the randommixing of concepts from each of these distinct levels of dis-course leads to illogical and, at times, even incorrect results.This will be followed by examples of how a failure to explic-itly identify concepts and definitions with their correspond-ing levels of discourse has resulted in the propagation of his-torically outdated definitions and vocabulary in the textbookand has led to a proliferation of conflicting claims about the

nature and sphere of application of certain models and teach-ing diagrams in the current chemical and educational literature.Some of you may be asking yourselves why, given that I

am addressing a room full of chemistry teachers, I just dontoutline an introductory chemistry course based on the logictable given in Lecture I? Of course, it goes without sayingthat I have been progressively modifying my own introduc-tory chemistry course along these lines, but this is still verymuch a work in progress. In addition, I am loath to tellothers how they must teach chemistry. My job, as I indicatedin Lecture I, is rather to enhance and stimulate your ownunderstanding of chemistry. How you translate that enrich-ment into specific classroom activities for your students will

naturally vary from teacher to teacher, depending on the levelof the course being taught and on the degree of control whichyou are given over both the content and mode of presentation.

Detection of Logically Flawed Definitionsand Concepts

As our first example, consider the following typical defi-nitions of physical versus chemical change taken from thethird edition of a well-known introductory chemistry text forhealth science majors (2):

In a physical change, the form of the matter is changed,but not its chemical properties. The tearing of paper,melting of ice, or dissolving of salt and sugar in water

are all examples of physical changes ... A chemical changeinvolves electron cloud interactions between the atomsof the matter involved.

Setting aside some very real doubts about whether the pro-cesses cited in this quotation are really chemical or physical,the important point is that these two definitions are incom-mensurate with one anotheror, in common parlance, theyamount to an attempt to compare apples with oranges. Theprocesses cited to illustrate physical change are all describedat the molar level, whereas the definition given for chemicalchange is at the electrical level. The problem of how to dis-tinguish a chemical versus a physical change at either themolar or at the electrical level is simply not addressed, mak-

ing the above definitions operationally useless to a student

Logic, History, and the Chemistry Textbook

II. Can We Unmuddle the Chemistry Textbook?

William B. JensenDepartment of Chemistry, University of Cincinnati, Cincinnati, OH 45221-0172

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818 Journal of Chemical Education Vol. 75 No. 7 July 1998 JChemEd.chem.wisc.edu

who is asked decide whether the rusting of iron (a molar de-scription) or the van der Waals attraction between two xe-non atoms (an electron cloud interaction) are chemical orphysical in nature.

The supposed distinction between chemical and physi-cal changes has formed a part of the opening chapters of vir-tually every introductory chemistry text since the late 18th-century and you would think that, after 225 years of prac-

tice, we would have finally gotten it right. The complication,of course, is that these 225 years also happen to correspondto the very time period which saw the elaboration of the molarchemistry of the 18th century via the addition of both themolecular and electrical levels of discourse. Though modernchemistry is certainly more complete and more reductionis-tic than late 18th-century chemistry, it is also more compli-cated in that it offers many more levels of discourse and hencemany more alternative ways of viewing a given phenomena.And with this conceptual diversity also comes an increasedopportunity for confusion via an incompatible mixing ofthese alternatives.

A slightly different twist to this question can be found

in a debate which took place in theJournal of Chemical Edu-cation in 1970. In the February issue of that year, WalterGensler of Boston University published an article entitledPhysical versus Chemical Change in which he argued thatsuch a distinction is not really meaningful [and that] in fact,as now presented, it is frequently indefensible and confus-ing (3). Genslers critique of these concepts was made at themolecular and, to a lesser degree, at the electrical level, andessentially involved an analysis of various supposed examplesof chemical versus physical change in terms of the question:Is a chemical bond made or broken in the process?

In the October issue, Laurence Strong of Earlham Col-lege wrote a rejoinder entitled Differentiating Physical andChemical Changes in which he argued that these were in-deed legitimate and useful distinctions which could be speci-fied at the molar level in terms of four operational criteria(Table 2), which he called identity, mixing, discontinuity, andinvariance (4). Strong was quite explicit about the fact thathe and Gensler were using two distinct levels of discourse(which Strong labeled theoretical and experimental ratherthan molecular and molar), so there is no question here ofconfusing levels as there was in the case of our textbook ex-

ample. Nor do I think that there is a question of whether oneof these authors is right and the other wrong. Both presentedcogent and, in many ways, equally convincing arguments.

What is of most importance is that this debate raises thevery real question of whether the same distinction can bemade with equal force at all three levels of discourseor, putanother way, are there perfectly valid distinctions at one levelof discourse (e.g., the molar) which disappear at another level(e.g., the molecular)? I think the answer to this question is adefinite yes, and it further underscores the importance ofavoiding such potential confusion by carefully characteriz-ing the models and concepts that we use in chemistry in termsof their location in the logic table introduced in Lecture I.

As a second example, this time at the advanced inor-ganic level, consider the use of the term polymorphism,briefly mentioned in Lecture I. According to a well-known

dictionary of chemistry, this term refers to (5):A phenomenon in which a substance exhibits two ormore different crystal forms ... Polymorphism is restrictedto the solid state. Polymorphs yield identical solutions,liquids, and vapors.

At first glance this would appear to be a molar definition,though in fact, like our previous example, it involves an in-termixingalbeit a much less obvious oneof both molarand molecular criteria. The key to disentangling this admix-ture is to note that its use of the term substance violates thestandard molar definition of a pure substance as a materialhaving a fixed and reproducible set of specific properties at agiven temperature and pressure by asserting that in the solid

state, at least, the same substance can have two or more dif-ferent sets of specific molar properties (i.e., not only differentcrystal forms but also, as a result, different densities and solu-bilities, and, at times, even different optical, electrical, andmagnetic properties). Thus no one would ever define isomersas different forms of the same substance. Rather they are de-fined as different substances which happen to have identicalcomposition, but otherwise totally distinct properties.

This curious use of the word substance in our defini-tion of polymorphism is in fact based on an outdated 19th-century molecular interpretation of the phenomenonan in-terpretation which, in turn, assumes that all materials,whether in the solid, liquid, or gaseous state, are composed

of discrete molecules (6). Isomers were thought to correspond

Figure 1. Top (left) and side (right) views of the structure of theoctasulfur rings found in both the rhombic and monoclinic forms ofsulfur.

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to substances having discrete molecules of identical compo-sition but differing intramolecular structure. Because thesedifferences resided within the molecule itself, each isomercorresponded to a different substance. Furthermore, thesedifferences persisted as long as the molecules remained in-

tact. As a result, the distinctions between one isomer and an-other frequently survived such processes as melting, vapor-ization, and dissolution.

Polymorphs, on the other hand, were assumed to con-tain molecules of identical composition and identical in-tramolecular structure, their differences being due instead todifferences in the intermolecular packing of these moleculesin the solid state. Since the identical molecule was present ineach polymorph, they were considered to be merely differ-ent forms of the same substance. Furthermore, once thesedifferences in intermolecular structure were destroyed viamelting, vaporization, or dissolution, all differences betweenthe polymorphs likewise disappeared (Table 3).

The classic example of polymorphism corresponding tothese definitions is the case of rhombic versus monoclinicsulfur, both of which contain the identical S8 molecule inthe form of a puckered eight-member ring (Fig. 1), but dif-fering packing arrangements of these molecules within thecrystal lattice (7). Indeed, the distinction which these defi-nitions make between isomers, on the one hand, and poly-morphs, on the other, is not unlike the classic distinction be-tween chemical and physical changes discussed earlier.Interconversion of isomers involved a change in intramolecu-lar structure and thus corresponded to a transformation ofone substance into another or to a chemical change.Interconversion of polymorphs involved a change in inter-

molecular structure only and thus corresponded to a changein form rather than substance or to a physical change.The problem with these interpretations is that their un-

derlying premisethat all materials must contain discretemoleculeshas been known to be incorrect for the last 80years. Indeed, X-ray crystallography has shown that the vast

majority of inorganic solids do not contain discrete moleculesat all but rather infinitely extended framework, layer, andchain structures. As a consequence, it is now known that thedifferences between many classic examples of polymorphismactually reside in a fundamental difference in the structures

of these infinitely extended molecules, rather than in a dif-ference in the intermolecular packing of discrete, but other-wise identical, molecular units.

Thus the differences between the - and -polymorphsof tin are ultimately traceable, as we saw in Lecture I, to adifference in their bonding topologies, -tin being composedof an infinitely extended framework structure with 4/4 co-ordination and -tin being composed of an infinitely ex-tended framework structure with distorted 6/6 coordination(Fig. 2). Hence these two polymorphs are, in effect, topo-logical isomers of one another.

Likewise, the difference between wurtzite and zincblende, the two most common polymorphs of zinc sulfide,

is also ultimately traceable to a difference in molecular struc-ture. Both substances contain infinitely extended frameworkstructures with 4/4 coordination and hence have identicalbonding topologies (Fig. 3). However, their structures areneither superimposable nor are they mirror images of oneanother. Thus, in effect, they correspond to diastereomers orgeometric isomers of one another.

Finally, it is of interest to note that quartz crystals canexist in both left- and right-handed forms (Fig. 4). I havenever seen these listed as polymorphs, which suggests thatthe traditional definitions of just what constitutes a differ-ence in crystal structure fail to take chirality into account.Needless to say, the underlying molecular structures of these

two forms have identical bonding topologies, correspondingto an infinitely extended framework structure with 4/2 co-ordination, but are, otherwise, nonsuperimposable mirrorimages of one another (Fig. 5). Hence they correspond toenantiomers or chiral isomers of one another. In all three ofthese examples, the differences observed in the solid state fail to

Figure 2. The structures of -tin (left) and -tin (right).

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persist after melting, vaporization, or dissolution, not becausethese processes have destroyed an intermolecular structure uniqueto solids, but because they have destroyed an infinitely extendedintramolecular structure which is also unique to solids(8).

These considerations further suggest that polymorphism(an observable difference in crystal form) should be regardedas a molar phenomenon, whose underlying molecular ratio-nale is ultimately traceable to one of the above forms of isom-

erism and that, as a consequence, those few cases which ac-tually correspond to the classic rationale of this phenomenon,as personified by the case of rhombic versus monoclinic sul-fur, should also be given a distinct name at the molecularlevel (Table 4). As mentioned in Lecture I, this logical ne-cessity was already recognized by 19th-century textbooks, which usually employed the term physical isomerism todenote alternative intermolecular arrangements of otherwiseidentical discrete molecules, though I think that the termpacking isomerism is more descriptive (6).

In summary, we may now restate our conclusions as fol-lows:

Polymorphism: At the molar level, a phenomenon in

which two or more substances of identical compositionexhibit different crystal forms in the solid state, these dif-ferences disappearing upon melting, vaporization, or dis-solution. At the molecular level, these differences are alltraceable to examples of topological, geometric, chiral,or packing isomerism or to examples of polymerismwhich fail to persist beyond the solid state.

In more general terms, we can state that there is no singleunique molecular rationale for the phenomenon of allomor-phism at the molar level, whether the allomorphs in ques-tion correspond to differences in state (as in our discussionof chemical versus physical change), to differences in crystalform (as in our discussion of polymorphism), or to differ-ences in liquid crystal mesophases. In each instance, it is pos-

sible to find examples which correspond either to cases ofpolymerism or to cases of topological, geometric, chiral, orpacking isomerism at the molecular level.

Detection of Historically Outdated Definitionsand Concepts

As our previous example has shown, a knowledge of thehistorical evolution of chemical concepts and models can of-ten play a key role in disentangling illogical admixtures. Theproblems with most textbook definitions of polymorphismwere the result of both a failure to correct outdated 19th-century assumptions about the molecular structure of solidsand a failure to generalize the terminology of isomerism so

as to subsume not only traditional finite molecular structures,but infinitely extended framework, layer, and chain structuresas well. In this section we will analyze an even more blatantexample of how a failure to correct outdated ideas can leadto the generation of illogical and self-contradictory defini-tions, this time involving the terms atom, molecule, andelementthree concepts which lie at the very core ofchemistry.

Even the most cursory survey of chemistry texts rapidlyreveals that all of them adopt, either explicitly or implicitly,the hierarchical arrangement given in Figure 6 when discuss-ing the composition/structure dimension of chemistry. Sub-stances are made of molecules; molecules, in turn, are madeof atoms; atoms are made of electrons and nuclei; and nucleiare made of protons and neutrons. We talk about diatomicandpolyatomicmolecules and about characterizing individual mol-ecules in terms of the kind, number, and arrangement of theircomponent atoms. You may recall that this was precisely theterminology that I used in Lecture I when defining molecularcomposition and structure. The relation between molecule andatom assumed by this hierarchy is in fact the classic 19th-cen-

tury relationship introduced by Cannizzaro in 1858. However,it has been known for nearly a century that this relation is in-correct and does not properly merge with either the molar orthe electrical levels of chemical discourse.

Figure 5. The structure of -quartz. The triangular channels are dueto helixes of tetrahedral SiO4/2 units that repeat every four tetrahe-dra. In the left- versus the right-handed forms of -quartz these he-lixes spiral in opposite directions and thus account for the observeddifferences in optical activity.

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Figure 4. Left- and right-handed crystals of -quartz.

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At the molar/molecular interface, we are now aware ofsubstances that are composed of discrete atoms rather thanpolyatomic molecules. These include, of course, the noblegases, and, if we divest ourselves of the parochialism that STPis somehow the natural state of things, rather than just anarbitrary reference state, it also includes most metals in thegas phase. At the electrical/molecular interface, we haveknown, almost since the turn of the century, that the par-

ticular electronic arrangement characteristic of isolated, neu-tral atoms is not preserved as such when atoms unite withone another to form molecules. An atom, according to theelectrical theory of matter, corresponds to an isolated, non-interacting unit composed of a single nucleus and sufficientelectrons to give the overall structure a zero net charge. Whenthese atoms form ions or interact to form molecules, the num-ber and/or the specific structural arrangement of these elec-trons is destroyed. Indeed, this electronic rearrangement isthe very essence of chemical change, and it follows as a nec-essary corollary of this fact that neutral atoms cannot serveas structural units within polyatomic molecules and ions. AsMulliken and other proponents of molecular orbital theory

have repeatedly emphasized over the last 60 years, moleculesare made of nuclei and electrons and not of neutral atoms.

These considerations suggest two important conclusions:

1. We should drop the use of such terms as diatomic,polyatomic, etc. and instead talk about dinuclear andpolynuclear molecules. Likewise, we should talk aboutcharacterizing molecular composition and structure interms of the kind, number, and arrangement of thecomponent nuclei rather than the component atoms.

2. We should subsume isolated neutral atoms under therubric of mononuclear molecules. In other words, theterm molecule should be divorced from the late 19th-century requirement that it must of necessity be poly-nuclear and instead should be used as a general termfor any neutral, kinetically independent, submicro-scopic unit.

Adoption of these suggestions leads to the revised hierarchyshown in Figure 7, in which the term molecule now sub-sumes any of the following six classes: mononuclear moleculesor atoms, discrete polynuclear molecules, discrete macromol-ecules, infinitely extended framework molecules, infinitelyextended layer molecules, and infinitely extended chain mol-ecules (Fig. 8).

Closely related to this problem is the pervasive use of theterm element as both a synonym for the term simple sub-stance at the molar level and as a synonym for the termatom at the molecular level. However, if we critically ana-lyze how we use this term, we quickly conclude that it be-longs to neither the molar nor the molecular level of discoursebut to the electrical level. All chemists would agree that theelement copper is present not only in copper metal, but incopper dichloride; in an isolated, neutral, gaseous copper atom;in a Cu2+ ion; and in a bare Cu29+ nucleus found in the coreof a high-temperature star. All of these examples correspondto very different electronic environments, and the only fea-ture they have in common which would justify the assertion

that the element copper is present in all cases is the presenceof the copper nucleus. In other words, neither metallic cop-per nor the neutral copper atom correspond to the elementcopper, any more than does copper dichloride, though all threecontain the element copper. This admission leads, in turn, tothe logical conclusion that the term element is in fact a de-scriptor for a particular kind of nucleus, or more accurately,for a particular class of nuclei, all of which have the sameatomic number. Hence the formal definition:

Element:A class of nuclei, all of which have the sameatomic number.

The caveat in this definition is, of course, necessitated

by the existence of isotopes. The introduction of this con-cept by British radiochemist Frederick Soddy in 1913 led toan intensive debate in the chemical literature of the periodamong Soddy, Kasimir Fajans, and Fritz Paneth as to howthe term element should be redefined in light of our newknowledge of the electrical structure of matter, in whichSoddys proposed terminology proved the ultimate victor (911). Each box of the periodic table corresponds to a singleelement or class of nuclei having identical atomic numbers.Since the individual varieties corresponding to each class alloccupy the same location in the table, Soddy suggested theterm isotope (from the Greek for equal place) to describethem (12). Thus, our final hierarchy is as follows: substancesare made of molecules; molecules are made of electrons and

Figure 6. The normal textbook version of chemistrys composition/

structure hierarchy.

Figure 7. A corrected composition/structure hierarchy.

Figure 8. A modern structural classification of molecules.

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elements, elements are resolvable into individual isotopes, andisotopes are made of protons and neutrons (Fig. 9). Contraryto Lavoisier, isotopes rather than elements now represent thefinal stage of chemical (i.e., non-nuclear) analysis (13).

Just to make life even more complex, it is worth notingthat chemists customarily introduce at least one additionalstage into this hierarchy at the electrical level via the time-honored practice of distinguishing between valence electronsand core electrons (Fig. 10). Nevertheless, however we slice

things, we cannot evade the conclusion that the isolated neutralatom no longer occupies a fundamental position in the struc-tural hierarchy of modern chemistry, but is instead but one ofmany possible productsalbeit the simplestwhich result fromthe interaction of atomic cores and valence electrons or, at a more fundamental level, from the interaction of elements and elec-trons.The specific structural entity we now identify with the termatom does not persist within ions and molecules. What does per-sist are the electrons and nuclei. The fact that most introduc-tory textbooks continue to use what are, in essence, residual19th-century definitions of the concepts of atom, elementand molecule, while simultaneously describing modern 20th-

Figure 9. A composition/structure hierarchy incorporating theconcept of element.

Figure 10. A composition/structure hierarchy incorporating theconcept of core and valence shell.

Figure 11. Jollys 1984 bond-type triangle.

Figure 12. Fernelius and Robeys 1935 bond-type triangle.

Figure 13. Grimms 1928 Dreieckeschema produced by construct-ing intra- and inter-row binary combination matrices for the ele-ments in the periodic table. Each square represents a binary com-pound formed from the elements having the group numbers listedalong the vertical and horizontal edges. The letters indicate thegeneral character of the resulting compounds: A = Atommolekle,

D = Diamantartige Stoffe, M = Metalle, and S = Salze.

century developments, is an apparent contradiction madepossible only by the pervasive compartmentalization of top-ics characteristic of most of these textbooks (14).

Before leaving this subject, it should perhaps be empha-sized that the assertions made in this section about the sta-tus of neutral atoms in the current composition/structure hi-erarchy of chemistry are not meant to imply that atoms areunimportant in modern chemistry. As the simplest possiblemononuclear molecules, they display simple and easily ra-

tionalized periodic trends for such properties as their ioniza-tion potentials, electron affinities, electronegativities, radii,polarizabilities, and characteristic X-ray absorption frequen-cies. Much of introductory chemistry can be viewed as anattempt to establish correlations between these simple atomicproperties and the more complex properties of polynuclearmolecules and infinitely extended solids. However, this useof neutral atoms as reference standards in establishing prop-ertycomposition correlations with more complex systems isquite independent of their status (or rather nonstatus) as fun-damental units in the current composition/structure hierar-chy of modern chemistry.

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Resolution of Contradictory Claims

As a final example of how our logic diagram can be usedto clarify our use of chemical concepts and models, I wouldlike to briefly illustrate how it may be used to resolve a de-bate in the current chemical education literature over the bestway of representing the concept of intermediate bond types.

Most introductory textbooks discuss the idea of three

separate mechanisms for chemical bonding: the ionic bond,based on electron transfer; the covalent bond, based on thesharing of electron pairs; and the metallic bond, based onthe sharing of a delocalized electron gas. More advanced textsfurther point out that these three bonding mechanisms cor-respond to idealized limiting-cases, and that most chemicalsubstances actually contain bonds that are intermediate incharacter. Often this concept is illustrated by means of a tri-angular diagram, similar to the example in Figure 11, whichis taken from the 1984 edition of William Jollys text, Mod-ern Inorganic Chemistry(15). I have been able to trace these dia-grams back to a 1935 article in theJournal of Chemical Educa-tion by the American chemists Conard Fernelius and RichardRobey (Fig. 12) and, in a more preliminary form (Fig. 13), to a1928 article by the German chemist Hans Georg Grimm (16,17). However, the version of the diagram (Fig. 14) which ap-pears in most textbooks is actually based on a 1941 text by theDutch chemist Anton Eduard van Arkel (18) or, even more com-monly, on a 1947 text (19) by van Arkels colleague, the Dutchchemist Jan Arnold Albert Ketelaar (Fig. 15). For this reasonthese diagrams are often referred to in the chemical literature asvan ArkelKetelaar triangles, though this is certainly unfair withrespect to the earlier contributions of Fernelius and Grimm.

In all of these examples, the placement of the examplecompounds on the triangle was qualitative and, as such, wasa function of each authors chemical intuition. However, morerecently, several proposals (2022) have been made for quan-

tifying these diagrams, one of the simplest involving a plotof the difference in the electronegativities of the two bondedatoms versus their average electronegativity (Fig. 16):

Ionicity = I= EN = (ENB - ENA) (1)

Covalency = C= ENav= (ENA+ ENB)/2 (2)

The first of these parameters serves as a measure of the asym-metry or ionicity of the AB bond and the second as a mea-sure of its localization or covalency.

But even as these apparent advances were being made, anumber of articles began to appear in the educational litera-ture criticizing the use of such diagrams and proposing in-stead alternative approaches to this subject. Thus, in an ar-

ticle which appeared in the November 1993 issue ofEduca-tion in Chemistry, the South African chemist Michael Laingcriticized the traditional Ketelaar triangle for having misledgenerations of students into wrongly associating a low melt-ing point with covalent bonding within the solid, when infact the reverse is true. It is the weak van der Waals forcesthat must be associated with low melting points, while a net-work of covalent bonds leads to a high melting point (23).To remedy this defect, Laing proposed replacing the Ketelaartriangle with a bonding tetrahedron which explicitly takesvan der Waals attractions into account as a fourth type ofbonding interaction (Fig. 17). Through a consideration ofmelting points, conductivities of the solid and liquid states,

bond distances, and coordination numbers, Laing then pro-

Figure 14. van Arkels 1941 bond-type triangle. Formulas inbrackets indicate infinitely extended solids.

Figure 15. Ketelaars 1947 bond-type triangle.

Figure 16. A quantified bond-type triangle.

Figure 17. Laings bond-type tetrahedron.

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ceeded to discuss the qualitative placement of several binarycompounds and simple substances on his diagram.

Two months later, in the January 1994 issue of theJour-nal of Chemical Education, the British chemist Peter Nelsonpublished yet another paper criticizing the traditional classi-fication of substances in terms of bond type (24). In this ar-ticle Nelson argued that bond type was a purely theoreti-cal concept which could not be directly measured, and which

lacked a precise theoretical definition (as reflected by a lackof agreement as to how it should be quantified). As an alter-native, he proposed a triangular diagram which classified sub-stances in terms of the magnitude and nature (i.e., electronicversus ionic) of their bulk conductivities (Fig. 18). This re-placed the traditional classes of covalent, ionic, and metallicsubstances with insulators, electrolytes, and metals, respec-tively, and included semiconductors, semi-electrolytes, andmixed-conductors as intermediate cases.

The key to unraveling these apparently conflicting claimslies in the recognition that the terms ionic, covalent, andmetallic represent a classification of bonds at the electricallevel. The quantified van Arkel triangle, shown in Figure 16,

is designed to graphically illustrate the gradual transition be-tween these various limiting-case models at the electrical leveland does so in terms of atomic electronegativities, which arealso defined at the electrical level. Though most presentationsof the triangle have used examples of binary compounds andsimple substances to illustrate these transitions, the trianglereally refers only to the AB bonds present in these materi-als. It has nothing to say about the AA and BB interac-tions which are also present and hence nothing to say abouteither the molecular structures of these example compoundsor their bulk molar properties, such as melting points, boil-ing points, electrical conductivities, or solubilities. To avoidpotential misunderstandings and misuse of the diagram, itwould probably be best to discontinue the practice of usingactual compounds as examples and instead simply indicatethe nature of the two interacting atoms in the correspond-ing bond (Fig. 19).

The diagram proposed by Nelson, on the other hand,corresponds to a classification of pure substances at the mo-lar level. In keeping with this premise, he has recently sug-gested that the diagram might be quantified by plotting atemperature-weighted function of the bulk ionic and bulkelectronic conductivities of a material, both of which are ca-pable of being defined and measured at the molar level,

though the necessary data are still lacking for many substances

(25). In any case, the bottom line is that the terms ionic,covalent, and metallic should be eliminated from the molarlevel of discourse. They describe bonds at the electrical leveland not substances at the molar level, where they should bereplaced instead by a terminology similar to that proposedby Nelson.

The diagram proposed by Laing is more problematic innature, in part because of the absence of any quantitative co-ordinates and, in part because it appears to be simultaneouslyintermixing two separate classification criteria. Indeed, amuch more precise version of the identical diagram was pro-posed by the German chemist Hans Georg Grimm nearly70 years ago. This is shown in Figure 20 as a three-dimen-sional tetrahedron, taken from a 1934 paper by UlrichDehlinger, and in Figure 21 as a two-dimensional projectionof the tetrahedron, taken from a paper published by Grimmthe same year, but redrawn for greater clarity using correctedand updated examples (26, 27).

As can been seen, Grimm attempted to simultaneouslyclassify binary compounds and simple substances in termsof both their structure type and bond type. Three of the ver-tices of the tetrahedron correspond to infinitely extendedframework structures containing ionic (i = Ionenbindung),covalent (a = Atombindung), and metallic (m = metallische

Figure 20. Dehlingers 1934 drawing of Grimms bond-type tetra-hedron.

Figure 19. A relabeled version of Figure 16 showing bonds rather

than compounds and simple substances.

Figure 18. Nelsons alternative classification triangle based on bulk

conductivity mechanisms.

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interactions and the third to the nature of bonding withinthe chain itself. Thus, selenium corresponds to a covalentlybonded (1a,2z) chain structure and beryllium dichloride toan ionically bonded (1i,2z) chain structure (Table 5, third

section). The other edges of the triangle correspond to struc-tures with mixed bonding. Thus, calcium polydisilicide,which contains infinitely extended, covalently bonded 3/3silicon layer anions held together by calcium cations, corre-sponds to a (1i,2a) structure and is located two-thirds of thedistance along the edge of the tetrahedron connecting the 3iand 3a vertices, whereas zinc polydiphosphide, which con-tains infinitely extended, covalently bonded 2/2 phosphoruschain anions held together by zinc cations, corresponds to a(2i,1a) structure and is located at one-third of the distancealong the same edge (Table 5, fourth section).

Note that Grimm failed to provide examples for mostof the points on the tetrahedron corresponding to structures

containing infinite one- and two-dimensional metallic bond-ing arrays, though the so-called condensed cluster compoundsstudied in recent years by the German chemist Arndt Simonand others probably correspond to these cases, as do severalmore classic compounds, such as nickel arsenide (28). Thislatter substance would appear to contain infinitely extended

nickel chains held together by weak, me-tallic bonding and cross-linked by meansof idealized, ionic NiAs bonds into an in-finitely extended (2i,1m) framework struc-ture (Fig. 22).

As appealing as Grimms classificationappears, it does not take long to uncover

some fundamental limitations. Take, forexample, the monoxides, mononitrides andmonocarbides of the D- and F-block ele-ments (Fig. 23), often referred to as so-called interstitial compounds (29). At firstglance, these all appear to have a simple 6/6infinitely extended (3i), ionic AB frame-work structure like sodium chloride. How-ever, their optical and electrical properties,the metal-metal distances in their struc-tures, and their metallic appearance are allsuggestive of the simultaneous existence ofan infinitely extended (3m), metallic AAbonding network, and it is not apparent

sdnuopmoCyraniBdnasecnatsbuSelpmiSfoselpmaxE.5elbaTgnidnoBs'mmirGnidnuoFsessalCsuoiraVehtotgnidnopserroC

12erugiFninwohSsanordeharteT

elpmaxE ssalCerutcurtS C alumroFN mmirG

edirolhcmuidoS krowemarfcinoi i3

dnomaiD krowemarftnelavoc a3

netsgnuT krowemarfcillatem m3

nogradiloS raelcunonometinifselucelom

z3

etihparG reyaltnelavoc z1,a2

edirolhcidmuimdaC reyalcinoi z1,i2

muineleS niahctnelavoc z2,a1

edirolhcidmuillyreB niahccinoi z2,i1

edicilisidmuiclaC noinareyaltnelavoc a2,i1

edihpsohpidcniZ noinaniahctnelavoc a1,i2

3[NaCl

6/6]

3[C4/4]

3[W

8/8]

0[Ar

12/12]

2[C

3/3]

2[CdCl

6/3]

1[Se

2/2]

1[BeCl

4/2]

2Ca[Si

3/3]2

1Zn[P

2/2]2

Figure 21. Grimms two-dimensional projection of his 1934 bond-type tetrahedron. Redrawn using corrected and updated examples(i = Ionenbindung. a = Atombindung, m = metallische Bindung,and z = zwischenmolekulare Krfte). Figure 22. The structure of nickel arsenide.

Bindung) bonds, respectively, whereas the fourth vertex cor-responds to discrete atoms and molecules held together inthe crystal lattice by means of weak van der Waals interac-tions (z = zwischenmolekulare Krfte). In each of these cases,

the particular type of bonding network present in the crystal isenvisioned as extending indefinitely in three dimensions, whencethe use of the symbols, 3i, 3a, 3m and 3z. Thus, sodium chlo-ride corresponds to an ionically bonded (3i) framework struc-ture, diamond to a covalently bonded (3a) framework structure,tungsten to a metallically bonded (3m) framework structure, andargon to a (3z) discrete molecular structure. (Table 5, first sec-tion.)

At one-third of the distance along the edges connectingthe (3z) van der Waals vertex with the other vertices, one findsinfinitely extended layer structures in which one dimensioncorresponds to a van der Waals interaction between the lay-ers and the other two dimensions correspond to one of the

other three bond types. Thus, graphite corresponds to a co-valently bonded (2a,1z) layer structure and cadmium dichlo-ride to an ionically bonded (2i,1z) layer structure (Table 5,second section). At two-thirds of the distance along theseedges, one encounters infinitely extended chain structures inwhich two of the dimensions correspond to van der Waals

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where one would locate the resulting (3i,3m) structures onGrimms tetrahedron. A somewhat similar problem also arises with the nickel arsenide and condensed cluster structuresmentioned earlier, as in all cases the AB bonding appearsnot only to cross-link the metallic chains and layers, but alsoto reinforce the bonding within the chains and layers. Ortake the case of calcium polydicarbide, which contains notonly an infinitely extended 6/6 framework structure held to-gether by ionic CaC bonds, but discrete, dinuclear C22-

anions held together by covalent CC bonds (Fig. 24). Since

the bonding tetrahedron explicitly indicates only infinitelyextended bonding networks, it cannot represent the existenceof the isolated covalent bond within the polydicarbide an-ion.

Though Laings version of the tetrahedron lacks the pre-cision of Grimms diagram, he appears to be groping towarda similar classification. His point about the neglect of vander Waals interactions and their role in determining meltingpoint is really a way of saying that melting point dependsmore on molecular structure type than on bond type. Thusthe fact that magnesium oxide, diamond, and tungsten metalall have extremely high melting points has more to do withthe fact that all three contain infinitely extended frameworkstructures than it does with the fact that the first containsionic bonds, the second contains covalent bonds, and thirdcontains metallic bonds (Table 6). Likewise, the fact that bothwhite tetraphosphorus and sulfur hexafluoride have low melt-ing points (Table 7) has more to do with the fact that theyare both composed of discrete molecules than with the factthat the first contains covalent bonds and the second con-tains relatively ionic bonds (30). The observation that mostmolar properties correlate with molecular structurerather than with bond type has been emphasizedmany times, most notably by Kossel in 1920, byPauling in 1932, by Stillwell in 1936, and mostrecently by Lingafelter in 1993; yet as Laing em-phasized, it is almost impossible to pick up an in-

troductory textbook which does not claim that highmelting points indicate ionic bonding and lowmelting points indicate covalent bonding (3135).

It seems to me that the best way to eliminatethis incorrect association is to completely removethe bond type criterion from the diagrams sug-gested by Laing and Grimm and to develop insteada purely structural classification at the molecularlevel of discourse. The result is shown in Figure 25(36). As can be seen, we still have a tetrahedron,but the interpretation of the vertices and edges isvery different from that found in the originalGrimmLaing diagram. In our case, each vertex

corresponds to one of the four major classes of moleculesmentioned in the previous section. The 0 vertex correspondsto discrete molecules, including isolated atoms; the 1 vertexcorresponds to infinitely extended chain structures; the 2 ver-tex corresponds to infinitely extended layer structures; andthe 3 vertex corresponds to infinitely extended frameworkstructures. No reference is made to the nature of the bond-ing which gives rise to these structures.

Thus, materials with high melting points, such as mag-nesium oxide, diamond, and tungsten, are all found at the 3vertex by virtue of their common framework structures, ir-respective of the fact that the first is held together by ionicbonds, the second by covalent bonds, and the third by me-tallic bonds.1 Likewise, low-melting materials, such as whitetetraphosphorus and sulfur hexafluoride, are all found at the

0 vertex by virtue of their discrete molecular structures, irre-spective of the nature of their intramolecular bonds. Theedges and faces of this tetrahedron now correspond to inter-mediate structures containing strong intermolecular attrac-

3[MgO6/6]

3[C4/4]

3[W

8/8]

gnitleMhgiHhtiwserutcurtSkrowemarFfoselpmaxE.6elbaTcillateMdna,tnelavoC,cinoIehtotgnidnopserroCstnioP

semertxEgnidnoB

ecnatsbuS erutcurtS (tnioPgnitleM )C gnidnoB

edixomuisengaM 2082 cinoI

dnomaiD 7453 tnelavoC

netsgnuT 7043 cillateM

0[SF

6]

0[P4]

woLhtiwserutcurtSraluceloMetercsiDfoselpmaxE.7elbaTtnelavoCdnacinoIehtotgnidnopserroCstnioPgnitleM

semertxEgnidnoB

ecnatsbuS erutcurtS (tnioPgnitleM )C gnidnoB

ediroulfaxehrufluS 7.05 cinoI

suorohpsohpetihW 3.44 tnelavoC

Figure 23. The structure of the so-called D- and F-block interstitialoxides, nitrides, and carbides.

Figure 24. The structure of calcium polydicarbide.

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tions, such as hydrogen bonding (Fig. 26). Thus the H-bonded framework structure of ice is found along the edge

joining the 0 vertex with the 3

vertex; the H-bonded layerstructure of boric acid is located along the edge joining the

0 vertex with the 2 vertex; aluminum trihydroxide, whichcontains an infinitely extended layer structure held togetherby strong interlayer H-bonds, lies along the edge joining the

2 vertex with the

3 vertex, etc.These results reinforce the conclusion that we reached

earlier with respect to Nelsons work. The terms ionic, cova-lent, and metallic should be eliminated, not only from themolar level of discourse, but from molecular level as well.They describe bonds at the electrical level and not substancesat the molar level nor structures at the molecular level. Ouranalysis also leads to the further conclusion that we are notdealing with three competing versions of the same thing, butrather with three distinct diagrams, each of which refers to aseparate level of chemical discourse: the quantified van Arkeltriangle, which deals with the classification of bonds at theelectrical level; the modified Grimm tetrahedron, which dealswith the classification of structure at the molecular level; andthe Nelson triangle, which deals with the electrical classifi-cation of substances at the molar level (Table 8).

These diagrams cannot be mutually interchanged be-cause they each deal with a different subject and do so atvery different levels of discourse. Thus, though I agree withNelsons suggestion that it would be better in an introduc-tory course to begin with a molar classification of chemicalsubstances, similar to that in his proposed diagram, rather

than with a more abstract bond classification, such as the vanArkel diagram, it is apparent that Nelsons diagram can onlyfunction as a supplement, rather than as an alternative, tothe van Arkel triangle, since in a textbook chapter dealingwith the nature of the chemical bond, one needs to use a

diagram that deals directly with the bonds themselves, how-ever theoretically abstract the parameters may be, rather than

with a diagram that deals with the molar electrical proper-ties of pure substances.This is not to say that these three diagrams cannot be

interconnected. However, this interfacing requires the use ofthe two fundamental postulates given in Lecture I. Theseshow that properties at one level of discourse are usually afunction of several factors at the next lower level. Only bycarefully selecting our examples so that all but one or two ofthese factors are held constant, is it possible to correlate onelevel with another using a two-dimensional diagram. Thus,the structure of binary compounds at the molecular level de-pends not only on the nature of their AB bonds but on theratio in which A and B are combined, and thus ultimatelyon the ratio of valence electrons to atomic cores or the VEC(valence electron concentration), as it is commonly called inthe solid-state literature:

Structure of AaBb =f(ENA, ENB, VEC) (3)

This is really another way of saying that structure at the mo-lecular level depends on both the kind (as expressed by theEN values of the cores) and relative number (as expressed bythe VEC value) of the component particles at the electricallevel. If we suppress this second factor by plotting only com-pounds with identical VEC values, we then find that there is

Figure 26. An elaboration of the structure-type tetrahedron in Fig-ure 25 showing examples of intermediate, H-bonded structuresalong the edges.

Figure 25. A tetrahedron for the structural classification of puresubstances at the molecular level.

smargaiDeerhTehtfoyrammuSA.8elbaT

margaiD tcejbuS leveL

elgnairtlekrAnav epytdnoB lacirtcelE

nordehartetmmirGdeifidoM erutcurtS raluceloM

elgnairtnosleN ytivitcudnockluB raloMFigure 27. A structure-sorting map for 1:1 binary compounds and

simple substances with a VEC = 4.

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a correlation between molecular structure at level 2 and bondtype at level 1 that can be displayed on the van Arkel dia-gram, and that is shown in Figure 27 for 1:1 AB compoundswith VEC values of four.2

Notes

1. Even after eliminating bond type, the correlation between struc-ture type and melting point is not as simple as suggested here, since italso depends on bond strength. Thus, despite its infinitely extendedframework structure, cesium metal has a low melting point because ofthe low bond energy per CsCs pair.

2. See reference20. Reference21 provides examples of an incor-rect mixing of compounds with variable VEC values on the same bond-type diagram, as well as an incorrect use of the diagram to maintainthe idea that there are sharp boundaries separating substances contain-ing metallic, covalent, and ionic bonds. Unfortunately, all of these mis-interpretations are repeated in the recently published text: Bodner, G.M.; Rickard, L. H.; Spencer, J. N. Chemistry: Structure and Dynamics;Wiley: New York, NY, 1996; pp 184189.

Literature Cited

1. Jensen, W. B. Does Chemistry Have a Logical Structure?J. Chem.

Educ.1998, 75, 679687.2. Baker, J. J. W. and Allen, G. E.Matter, Energy, and Life, 3rd ed.;AddisonWesley: Reading, MA, 1974; p 33.

3. Gensler, W. J. Physical versus Chemical Change.J. Chem. Educ.1970, 47, 154155.

4. Strong, L. E. Differentiating Physical versus Chemical Change.J. Chem. Educ.1970, 47, 689690.

5. The Van Nostrand Chemists Dictionary; Honig, J. M. et. al., Eds.;Van Nostrand: Princeton, NJ, 1953; p 550.

6. vant Hoff, J. H. Lectures on Theoretical and Physical Chemistry,Part II: Chemical Statics; Arnold: London, 1899; pp 132133.

7. RaynerCanham, G. and Kettle, J. The True Allotropes of Sul-phur. Educ. Chem.1991,28, 4951.

8. Wells, A. F. Structural Inorganic Chemistry, 4th ed.; ClarendonPress: Oxford, 1975.

9. Soddy, F. The Complexity of the Chemical Elements. Chem. News1917, 116, 7375, 8587, 98101; The Conception of theChemical Element as Enlarged by the Study of RadioactiveChange. Chem. News1919,118, 8587, 9799, 109112; TheOrigins of the Conception of Isotopes. Chem. News1923,127,9295, 103106.

10. Fajans, K. Der Begriff des chemischen Elementes und dieErscheinung der Isotopie.Jahrbuch Radioakt. Elek. 1917, 14, 314352; Radioactivity and the Latest Developments in the Study of theChemical Elements; Dutton: New York, NY, 1922; p xi and Chap-ter 11.

11. Paneth, F. ber den Element- und Atombegriff in Chemie undRadiologieZ. physik. Chem.1916, 91, 171198; Radio-Elementsas Indicators and Other Selected Topics in Inorganic Chemistry;McGraw-Hill: New York, NY, 1928; Chapter 15; The Epistemo-

logical Status of the Chemical Concept of Element. Brit J. Phil.Sci. 1962, 13, 114, 144160.12. Soddy, F. Intra-atomic Charge. Nature1913, 92, 399400.13. This particular use of the term element was partly anticipated by

William Ramsay at the turn of the century. See, Ramsay, W. Ele-ments and Electrons.J. Chem. Soc. 1909, 95, 624637.

14. In the past, several of the problems outlined in this section havebeen brought to the attention of chemical educators, but withlittle apparent impact. See, for example, Roundry, W. H. What isa Chemical Element?J. Chem. Educ.1989, 66, 729730.

15. Jolly, W. L.Modern Inorganic Chemistry; McGraw Hill: New York,NY, 1984; p 265.

16. Fernelius, W. C. and Robey, R. F. The Nature of the Metallic State.J. Chem. Educ.1935, 12, 5368.

17. Grimm, H. G. Allgemeines ber die verschiedenen Bindungsarten. Z. Elekrochem. 1928, 34, 430435; Zur Systematik derchemischen Verbindungen vom Standpunkt der Atomforschung,zugleich ber einige Aufgaben der Experimentalchemie. Naturwiss.1929, 17, 535540 and 557564; Das Periodische System derchemischen Verbindungen von Typ AmBn. Angew. Chem.1934,47, 5358; Die energetischen Verhltnisse im Periodischen Sys-tem der chemischen Verbindungen vom Typ AmBn. Angew. Chem.1934, 47, 593561; Wesen und Bedeutung der chemischen

Bindung.Angew. Chem. 1940, 53, 288292; Grimm, H. G. andWolff, H. Atombau und Chemie. In Handbuch der Physik, 2nded.; Geiger, H. and Scheel, K., Eds.; Springer: Berlin, 1933; pp10221057.

18. van Arkel, A. E. Molecules and Crystals; Butterworths: London,1949; p 205. The first Dutch edition appeared in 1941.

19. Ketelaar, J. A. A. Chemical Constitution, 2nd ed.; Elsevier:Amsterdam, 1958; p 21. The first Dutch edition was publishedin 1947.

20. Jensen, W. B. The Historical Development of the van Arkel Bond-Type Triangle. Bull. Hist. Chem.1992-93, 13-14, 4759; Quan-tity or Quality? Educ. Chem.1994, 31, p 10; A Quantitative vanArkel Diagram. J. Chem. Educ.1995, 72, 395398.

21. Sproul, G. D. Electronegativity and Bond Type: I. Tripartite Sepa-ration.J. Chem. Educ.1993, 70, 531534; Electronegativity and

Bond Type: II. Evaluation of Electronegativity Scales. J. Phys.Chem.1994, 98, 66996703; Electronegativity and Bond Type:III. Origins of Bond Type. J. Phys. Chem. 1994, 98, 1322113224; Allen, L. C.; Capitani, J. F.; Kolks, G. A.; Sproul, G. D.Van ArkelKetelaar Triangles.J. Mol. Sci.1993,300, 647655.

22. Alcock, N. W. Bonding and Structure: Structural Principles in In-organic and Organic Chemistry; Horwood: Chichester, 1990; pp1824.

23. Laing, M. A Tetrahedron of Bonding. Educ. Chem. 1993, 30,160163.

24. Nelson, P. G. Classifying Substances by Electrical Character: AnAlternative to Classifying by Bond Type. J. Chem. Educ. 1994,71, 2426.

25. Nelson, P. G. Quantifying Electrical Character. J. Chem. Educ.1997, 74, 10841086.

26. Dehlinger, U. Das Wesen der metallischen Mischkristalle undVerbindungen.Z. Metallkunde1934,26, 227232.27. For Grimms tetrahedron, see reference 17.28. Simon, A. Condensed Metal Clusters. Angew. Chem., Int. Ed.

Engl., 1981,20, 122; Lee, S. C. and Holm, R. H. NonmolecularChalcogenide/Halide Solids and Their Molecular Cluster Ana-logs.Angew. Chem., Int. Ed. Engl., 1990,29, 840856.

29. Rundle, R. E. A New Interpretation of Interstitial CompoundsMetallic Carbides, Nitrides and Oxides of Composition MX.Acta Cryst.1948, 1, 180187.

30. For a recent discussion of the ionic character of the bonds in SF6,see Reed, A. E. and Weinhold, F. On the Role of d-Orbitals inSF6. J. Amer. Chem. Soc.1986, 108, 35863593.

31. Kossel, W. Bemerkungen ber Atomkrfte.Z. Phys. 1920, 1, 395415.

32. Pauling, L. The Nature of the Chemical Bond. III. The Transi-tion From One Extreme Bond to Another. J. Amer. Chem. Soc.1932, 54, 9881003.

33. Stillwell, C. W. Crystal Chemistry; McGrawHill: New York, NY,1938, pp 165177.

34. Lingafelter, E. C. Why Low Melting Does Not Indicate Cova-lency in MX4 Compounds. J. Chem. Educ.1993, 70, 9899.

35. Jensen, W. B. Bond Type versus Structure Type. Educ. Chem.1994,31, 94.

36. This tetrahedron is a graphical representation of the classifica-tion scheme used in Jensen, W. B. Crystal Coordination Formu-las: A Flexible Notation for the Interpretation of Solid-State Struc-tures. In The Structures of Binary Compounds; de Boer, F. R. andPettifor, D. G., Eds.;Elsevier: New York, NY, 1989; Chapter 2.

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