2308 BARLOW AND POPE : THE RELATION BETWEEN THE CRYSTALCCXLII.-TlZc Relation between the Crystal St uwct tireand the Chemical Composition, Constitution, andConjguration o f Organic Suhstan ces.By WILLIAM BARLOW and WILLIAM JACKSON POPE.DURING the last few years the authors have investigated a novelmethod of studying the relations between crystalline and molecularstructure, and have demonstrated the existence of a very simplerelation between the two species of structure in a. great variety ofcases (Trans., 1906, 89, 1675; 1907, 91, 1150; 1908, 93, 1528);the principles involved in the method referred to were brieflysummarised in the introduction of the last-mentioned communi-cation. One of the chief results of this work has been to demonstratSTRUCTURE AND THE CHEMICAL COMPOSITlON, ETC, 2309that, in a given crystalline substance, the volumes appropriatedby the spheres of influence of the different atoms contained in theniolecule are approximately proportional to the numbers represent#-iiig the respective fundamental valencies ; this conclusion has beenindependently verified for hydrocarbons and their simple derivativescontaining oxygen or nitrogen in the liquid state by Le Bas (Trans.,1907, 91, 112; Phil.Mag., 1907, [vi], 14, 324; 1908, 16, 60). Thelatter author, indeed, carries the valency law a step further byshowing that throughout a series of liquid hydrocarbons, undercorresponding conditions, the atomic volumes are directly pro-portional to the numbers representing the fundamental valencies ofthe elements carbon and hydrogen.I n view of the close relation which has been shown to existbetween the sum of the fundamental valencies of the atoms com-posing the molecule-the valency volume-and the crystallinestructure affected by the substance, it is convenient to deriveconstants for related series of substances which are simple functionsof the valency volume and of the crystalline structure as expressedby the geometrical data.We have therefore introduced the swalled“ equivalence parameters,” x, y, and z , which are the lengths of theedges of a parallelepidon, of which the volume is the valency volume,TV, and of which the relative linear and angular dimensions accordwith the axial ratios and the interaxial angles (Trans., 1906, 89,1681) ; the equivalence parameters are calculated as follows :.l: = y76’ W y = X/CL and z = cy.c sin A sin p sin y’The important nature of the information t o be obtained by theaid of the equivalence parameters has been fully demonstrated inour previous papers, and by Jaeger (Trans., 1908, 93, 517),Jerusalem (Trans., 1909, 95, 1275), and Armstrong (this vol.,p.1578).I n the present paper we propose to discuss the close-packedassemblages representing the molecular composition, constitution,and configuration of the paraffinoid, ethylenic, and acetylenic hydro-carbons. As a result of this investigation we shall be able t o showthat, adopting the same principles as have been previously laiddown, each hydrocarbon has its own specific kind of structural unit,and that geometrical peculiarities are distinguishable in theappropriate assemblages corresponding with the presence in themolecule of single, double, and triple bonds between carbon atoms.It will further be shown that the configurations derived for thevarious hydrocarbons by closely packing spheres of magnitudesappropriate for representing the spheres of influence of their atomsare in accordance with the conclusions of van’t Hoff and Le Be2310 BARLOW AND POPE : THE RELATION BETWEEN THE CRYSTALconcerning the environment of a methane carbon atom.Finally,it will be shown that a process of simple adjustment furnishes ageometrical interpretation of polymerisation and isomeric change,such, for instance, as the conversion of acetylene into benzene. Asa preliminary t o the main argument, and in justification of themethods employed, a passing reference may be made to one or twosimple considerations and the data supporting them.Concerning the legitimacy of attributing to carbon a sphere ofatomic influence four times as large as that of hydrogen, little morenow remains to be said.Since we first drew this conclusion, Le Bashas conclusively proved the atomic volume of carbon to be fourtimes that of hydrogen, and Jerusalem has shown the same relationto hold approximately as between crystalline substances which arenot examined under strictly corresponding conditions. Most of thehydrocarbons of the series with which we have now to deal are,however, either liquid or gaseous under ordinary conditions, andtherefore yield no crystallographic data for employment as a directexperimental check.For our present purpose it is consequentlynecessary to use crystallographic data referring to the halogenderivatives of hydrocarbons, and to rely on them to furnish thenecessary check on the dimensions of the hydrocarbon assemblagesdscribed. The legitimacy of the use of these derivatives for thispurpose depends on our previous conclusion that the spheres ofatomic influence of hydrogen and the halogens differ but slightlyin volume when contained in the same molecular complex (Trans.,1906,89,1679), although the sphere of atomic influence of hydrogenis somewhat smaller than those of the halogens (Trans., 1907, 91,1197).That the spheres of atomic influence of hydrogen and thehalogens have approximately the same valency volume may beconveniently demonstrated by showing that the chemical substitutionof a halogen atom for one of hydrogen in a crystalline substance isfrequently not accompanied by it profound change in axial dimen-sions; in the instances quoted below, it will be seen that thegeometrical change accompanying the substitution in question isin general greater than that ordinarily observed in cases of iso-morphism, but not so great as to obscure the obvious morphotropicrelationship. The comparatively large change in axial dimensionswhich is in general thus presented, and also the rarity of suchinstlances, must be attributed to the sphere of atomic influence ofhydrogen differing appreciably in magnitude from those of chlorine,bromine, or iodine, the latter being much more nearly of the samesize; the discrepancy in volume between the spheres of hydrogenand of the halogens is, however, not sufficient to necessitate theemployment of different sizes of spheres of influence for thosSTRUCTURE AND THE CHEMICAL COMPOSITION, ETC.2311elements in the construction of the close-packed assemblagesdescribed below.The substitution of hydrogen by bromine, unaccompanied byconsiderable changes in axial dimensions, is illustrated by the datafor the monosymmetric pentabromoethane and the orthorhombichexabromoethane (Trans., 1906, 89, 1682) :CHBr,*CBr, ...........CBr,*CBr, ...............a : b : c=0.5650 : 1 : 0'3118 ; 8=91"19'n : b : c=0*5639 : 1 : 0'3142 ; /3=90°A similar case is presented by the orthorhombic Ir-sulphonylchlorides and bromides of camphor and of a-bromo- and a-chloro-camphor (Kipping and Pope, Trans,, 1893, 68, 548; 1895, 67,367) :d-CIoH1,O*SO,C1 ..............d-C,oH,,0*S02Br ...............d-C,oH,,OBr~SO,Cl ..........d-C,,H,,OCl~SO,Br ..........a : h : C = 0 9980 : 1 : 1.0368a : b : c=0.9816 : 1 : 1.0249cc : b : c=0.8912 : 1 : 1.0518cc : b : c=0*8795 : 1 : 1.0494The axial dimensions of the monosymmetric p-azoxytoluene andits monobromo-derivative are almost identical (v.Zepharovitch,Zeitsch. Xryst. Min., 1889, 15, 214)) and a similar relationshipholds between the values for the orthorhombic ptolyl-mono- anddi-chloro-methylsulphones (Brugnatelli, Zeitsch. Kryst.Min., 1892,20, 604-605) :p-Azoxytolnene .............................Rromo-~-azoxytoluene ....................p-Tolylmonocliloromethylsul~~honr. ...11-Tolyldichlorornethylsulphone .........CL : h : c=1'4971 : 1 : 1.0196 ; 8=75"30'n : h : c=1'5194 : 1 : 1 01 ;ci : b : c-0'6070 : 1 : 0.7865n : b : c=0'5324 : 1 : 0-79138=75"28'30"Acetamide is rhombohedra1 with a : c = 1 : 0.5916 (Kahrs,Zeitsch. Kryst. Min., 1905, 40, 476); on referring the substance torectangular axes by changing the indices (loo}, { 101 ), and f 1 1 O}to { l l O } , (301}, and (301) respectively, the values are obtained as :c6 : b : c = 1 6904 : 1 : 0.9759 ; /3=90".Dibromoacetamide is monosymmetric with a : B : c=1.6887 : 1 : 1.2785, @ = 87O2' (Fock, Zeitsch.R?yst. illin., 1888, 14,538); when the transposition involved in changing the indices of{ 203) t o { 101 } is made, the axial ratios are obtained as a : b : c =1.6887 : 1 : 0.8625, @=87O2'. The change of indices here made islegitimate, because the form { 203} is actually observed. Tribromo-and trichloro-acetamide are also monosymmetric, and exhibitthe axial ratios a: b : c = 1.7339 : 1 : 0.8636, /3=79O37/, and1-7485 : 1 : 0.8490, @=78O36' respectively. The four sets of axialratios show a fairly close agreement.The orthorhombic monochloro-pbenzoqu.inone exhibits the axialratios, a : B : c = 1.7461 : 1 : 0.9619 (Fels, Zeitsch. Kryst. Min., 1903,37, 479); these ratios, expressed in the form b : c : u2312 BARLOW AND POPE : THE RELATION BETWEEN THE CRYSTAL1.0396: 1 : 1.8153, closely approximate to those of the mono-symmetric dichloro-p-benzoquinone (Fock, Z ~ i t s e l ~ .Kryst. Min.,1883, 7, 40), namely, 0 : 7,: c=1.0920: I : 1.8354, P=89O11',and dibromo-y-benzquinone (Fels, loc. cit.), which exhibitsa : b : c = 1.0941 : 1 : 1.8229, /3=92O32/. The three substancesare, however, pseudohexagonal, a.nd the morphotropic relationbet'ween them is probably even closer than is indicatedby the above axial ratios. Thus, on changing the forms(lor), { l O l } , {loo}, and { 103) observed on dibromo-p-benzoquinoneto {OOl}., {101}, {103}, and (100) respectively, the axial ratiosbecome a: b : c =1.7416 : 1 : 0-9491, /3=90°41'. These valuesapproximate much more closely to the original ones given abovefor monochloro-p-benzoquinone than do those stated by Fels.It isin any case clear that, contrary to the views of Griinling (Zeitscli.Kryst. illin., 1883, 7, 582) and of Fels, very little change in axialdimensions attends the passage from monochloro-p-benzoquinone todichloro- or dibromo-p-benzoquinone.In the instances quoted above, the replacement of hydrogen bya halogen atom leads to no very profound change in crystallographicdimensions. The same kind of relation its is thus expressed mustbe looked for amongst halogen derivatives which are positionisomerides, and several instances from amongst such substances maynext be quoted.The di- and tri-halogen derivatives of camphor have been verycompletely examined by (1) v.Zepharovitch (Zeitsclh. Kryst. Min.,1883, 7, 588), (2) Cazeneuve and Morel (ibid., 1888, 14, 267), (3)Kipping and Pope (Trans., 1895, 67, 371), and (4) Armstrong andLowry (ibid., 1898, 73, 579). The close morphotropic relationshipbetween these orthorhombic substances becomes evident on inter-changing the dimensions b and c in the data (1) and (2), dividingdimension b by two and writing 7, for a, c for b, and a for c in thedata (3), and leaving data (4) as stated by Armstrong and Lowry;the following values are thus obtained:(1) aa-Dibromocamphor . . , . . . . . . . . ,(2) aa-Dichlorocamphor .. . . . . . . . . . .aa-Brornochlorocamphor , . . . . .(3) ax-Dichlorocarriphor ............ax-Dibromocamphor .. . . . . . . . . . .ax- Chlorobromocamphor . . . . . .a*-Bromochlorocamphor . . . . . ,(4) aa-Chlorobromocamphor . . . . . .Baa-Dibromochlorocainphor . . .Original.n : b : c .05'925 : 1 : 0-51430-8074 : 1 : 0'54480*8040 : 1 : 0'52280.6933 : 1 : 0.32970.6860 : 1 : 0.33230.6884 : 1 : 0.83010'6861 : 1 : 0'331715338 : 1 : 1'90201'4627 : 1 : 2.1332Transposed.a : b : c .1.5409 : 1 : 1.99431.4830 : 1 : 1.83651.5379 : 1 : 1.83551'5160 : 1 : 2.102915148 : 1 : 2.08561.5074 : 1 : 2'06421.5045 : 1 : 2.06841.5338 : 1 : 1.90201.4627 : 1 : 2.1332Jaeger has shown (Zeitsch. 2i'ryst. Min., 1904, 38, 570) that themonosymmetric position isomerides, the 1 : 2 : 4- and the 1 : 3 : 4-triSTRUCTURE AND THE CHEMICAL COMPOSITION, ETC.2313bromotoluenes, have almost identical axial ratios, namely, a : b : c =3.5283 : 1 : 4.1958, and a : b : c =3*5470 : 1 : 4.2603,I3 = 58O551, respectively.A large number of instances similar to those quoted above mightbe selected from the crystallographic literature, but the above willsuffice to confirm our previous conclusion that the sphere of atomicinfluence of hydrogen differs but slightly in volume from those ofthe halogen elements, and consequently that they are all representedin the close-packed, homogeneous assemblage with sufficientexactness by spheres of the same size. I n the following pages weshall therefore assume that the crystallographic configuration ofany hydrocarbon can be presented under some conditions by itshalogen derivatives, and, when crystallographic data are availablefor any of the latter, shall directly employ those data for checkingthe correctness of the assemblage derived for the hydrocarbon itself.I n connexion with the concluding portions of this communication,in which the occurrence of polymerisation and isomeric change istreated, it may possibly be suggested that no method of discussioninvolving considerations connected with crystal structure can bejustified, inasmuch as such changes occur in general in the liquidor even in the gaseous state. To this objection the reply is madethat the great mass of work done during recent years on secalledliquid crystals has greatly extended the domain of crystal structure,It is now known that in those liquid substances which exist in theliquid crystalline condition, tracts, so large as to be readily discernedmicroscopically, exist in which the regularity of arrangementexhibited by solid, crystalline structures is present.These tractaare continually forming and disappearing, and their Occurrenceindicates clearly that in these mobile liquids the particles aggregatethemselves together in masses which, measured on a molecular scale,are of enormous extent, and in which very complete regularity ofstructure prevails. Since, in such instances as these, the eye candiscern the existence of a liquid, crystalline structure, it is legitimateto assume that in liquids generally, arrangements of parts, com-parable in regularity with crystalline structures, are being con-tinually formed and dissolved, although possibly not to such anextent as in the cases of known liquid crystals.The occasionaljuxtaposition of parts in orderly close-packed arrangement thuspremised is all that is required to legitimise the discussion ofisomeric change in connexion with crystalline structure./3=58O47f,Methane.As a preliminary to an attempt to apply the methods which wehave previously described to the elucidation of the configurations'VOL. XCVlI. 7 2314 BARLOW AND POPE : THE RELATION BETWEEN THE CRYSTALand properties of the paraffins, it is necessary briefly to enumeratethe available chemical and crystallographic facts and conclusionsbearing on the configuration of the simplest paraffin, methane.The following may be quoted as sufficient to lead to the constructionof the homogeneous close-packed assemblage of spheres which repre-sents this hydrocarbon.(1) I n accordance with the conclusions respecting valency whichwe have previously drawn (Trans., 1906, 89, 1723), the space appro-priated in the methane assemblage by each carbon atom shouldbe four times as large as that appropriated by each hydrogen atom.(2) Carbon tetrabromide, CBr,, possesses the same configurationas methane, and its assemblage will be represented by the samespheres.The halogen derivative is dimorphous, crystallising above47O in the cubic system (Rothmund, Zeitsch. physikal. Chem., 1897,24, 712) and at the ordinary temperature in the monosymmetricsystem. Carbon tetrachloride and tetraiodide crystallise in thecubic system.(3) Stereochemical facts indicate that in the free methanemolecule the four hydrogen atoms are situated at the apices of aregular tetrahedron described about the carbon atom, and that thistetrahedral environment of the methane carbon atom must beregarded as surviving a substitution of one or more of the fourhydrogen atoms by other atoms or radicles.(4) The assemblage representing methane, built up in accordancewith the principles laid down in previous papers, should be capableof geometrical modification so as to yield assemblages representingother paraffins ; the geometrical process thus involved should bestrictly illustrative of the practical methods by which methane canbe converted into these homologous paraffins.It should thus bepossible to derive one assemblage corresponding in composition,constitution, and configuration with each paraffinoid hydrocarbon.An extension of the same method should lead to the derivation ofcharacteristic assemblages for other aliphatic hydrocarbons andcompounds other than the paraffins ; the applications should embraceall the varieties of isomerism, and express the facts that have ledto the conception of the asymmetric carbon atom.An assemblage which, both as a whole and when partitioned,fulfils the above and. other conditions concerning methane isarrived at in the following manner. Alternate layers are removedfrom a cubic closest-packed assemblage of equal incompressible butdeformable spheres (Trans., 1907, 91, 1152), regarded as composedof layers of square arrangement (Fig.l), the remaining layers beingcaused to retain their original positions. The resulting skeletonassemblage, which has tetragonal symmetry, is shown in plan anSTRUCTUICE AND THE CHEIdtCAL COMPOSITION, ETC. 2315elevation in Figs. 2 and 3 ; the dotted lines which join the centresof nearest spheres inpartitioning of spaceinto equal rightsquare prisms.The next step con-sists in distorting theskeleton assemblageby a contractionalong its fourfoldaxis, accompanied bya compensatory ex-pansion in directionstransverse to thisaxis, so that thesphere centres finallylie a t the corners ofcubes equal in con-tent or volume to theoriginal right squareprisms.The systemthus derived possessesholohedral cubicsymmetry, and iscomposed of sphereswhich do not quitetouch one another;its projection parallelto any cube plane isshown in Fig. 4.This cubic system,like the tetragonalsystem from which itis derived, possessesone-half the densityof packing of theparent assemblage;if, therefore, smallspheres of the samedef ormable material,four times as numer-the three principal directions outline aFIG. 1.FIQ. 2.ous and one-fourth the volume of the original large ones, are forcedinto its cavities, and the whole system is then subjected to com-pression so as to eliminate the interstitial space, the polyhedraproduced from the large spheres will be about four times a.a large ils7 M 2316 BARLOW AND POPE : THE RELATION BETWEEN THE CRYSTALthose produced from the small ones.I n the skeleton assemblage ofFig. 4, the cavities are as numerous as the spheres; if, therefore, eachFIG. 3.cavity bounded byeight neighbouringspheres can be madeto accommodate agroup of four of thesmall spheres in sucha manner as to givestable equilibriumand to be compatiblewith cubic symmetry,several of the moreessential conditionsfor methane will beobeyed .by the assem-blage.Each cavity of theskeleton assemblagedescribed exhibits sixidentical four-sidedhollows, the centresof which lie on threerectangular a x e sdrawn through thecentre of the cavity,and, in placing atetrahedral group ofthe small sphereswithin the latter, anythree of the hollowswhich lie nearesttogether are selectedfor the reception ofthree out of the foursmall spheres of thegroup, one juttinginto each of theselected hollows.Thefourth sphere of thetetrahedral g r o u pwill then touch thatsphere of the eight of volume four which does not border either ofthe selected hollows, the point of contact being on the cube diagonaSTRUCTURE AND THE CHEMICAL COMPOSITION, ETC. 2317which passes through the centre of this large sphere. The mar-shalling thus arrived at for a cubic unit of the assemblage is theone required; it has yet to be shown what relative orientations ofthe contents of the different cube cells are consistent with cubicsymmetry and what adjustment of the arrangement described willrestore the close-packing which has been impaired by substitutingthe tetrahedral groups of small spheres for one-half of the largerspheres of the closest-packed assemblage.The introduction of the tetrahedral group into the cubic cell inthe manner described lowers the symmetry by destroying three ofthe four trigonal axes of the cell; if cubic symmetry is to survivethe introduction of such a tetrahedral group into each cavity, thoarrangement of the completed assemblage must consequently be ofone of the types in which the trigonal axes do not intersect.TheFIG. 5.mode of ascertaining the relative positions of the non-intersectingtrigonal axes has been already described (Trans., 1907, 91, 1183);its application to the present case leads, in the following manner,to the production of the appropriate type of symmetry for themethane assemblage.In the cubic partitioning of space shown in Fig.4, one trigonalaxis, a, of one cube of the partitioning is drawn and produced inboth directions, so as to pass through a st'ring of cubic cells whichare in contact at their corners (Fig. 5), the latter being centres ofcarbon spheres; in the first selected cube of the part,itioning, thegroup of four small or hydrogen spheres is inserted in its appropriateposition with respect to this trigonal axis. I n any one of the sixcubic cells which make face contact with the first selected cubecell, a single diagonal is drawn, the position chosen being suc2318 BAHLOW AND POPE : THE RELATION BETWEEN THE CRYSTALthat, like c in Fig.6, it is not parallel to the trigonal axis a,already located and does not intersect it. This last drawn diagonalis used as a trigonal axis, and by rotations about it through 120°, theexisting trigonal axis and group of small spheres are transferredto two new positions, so as to locate other trigonal axes and groupsof hydrogen spheres in the system. The latter process is repeatedabout the axes thus located and about subsequently located axes,until all the situations for trigonal axes in their four orientationsand all the positions for groups of small spheres derivable in thismanner have been ascertained ; the minimum distance separatingtrigonal axes of different orientations is that separating the two firstlocated. A diagram showing the relative situations of the axes hasFIG. 6 .been already given(Trans., 1907,91, 1183).As a result of t,hisseries of operations, onetrigonal axis becomeslocated in each cubecell of the cubic par-titioning of space, butthe original tetrahedralgroup of small spheresbecomes transferred tobut one-half of thesecube cells.The cubecells forming the halfsystem, distinguished byeach cell containing atetrahedral group ofsnia.11 spheres, are incontact at their edges only; they have the arrangement of the lightor the dark cubes of the previously described stack of cubes of twokinds (Trans., 1908, 9 3 , 1533, Fig. 1). The skeleton assemblagethus derived has the symmetry of Barlow’s type 1.Only one kind of arrangement possessing cubic symmetry can bearrived at in the manner just. described, but there are two alternativeways in which to complete the assemblage homogeneously by fillingthe unoccupied cavities, which are equal in- number to thoseoccupied, with the tetrahedral groups of small spheres in a mannercompatible with cubic symmetry ; both of these involve slightadjustment of the skeleton assemblage, but no re-marshalling.Thecompletion is in both cases effected by bringing the one-half systemof cubes to the place of the other half system. One of the twoalternative operations consists in rotating the system through 180STRUCTURE AND THE CHEMICAL COMPOSITION, ETC. 2319about an axis drawn perpendicular to a cube face and passingthrough the centre of a cube edge, such perpendicular not being adigonal screw axis of the skeleton assemblage; this involves theaddition of digonal rotatioii axes to the original system of trigonalaxes and digonal screw axes, and yields a completed assemblagehaving the symmetry of Barlow’s type 2.The other operation isone performed about a centre of symmetry situated at a cube angle,arid leads to the production of a completed assemblage having thesymmetry of Barlow’s type la (Zeitsch. Krgst. Min., 1894, 23, 10,44). Both assemblages thus derived become very closely packedas the result of a slight adjustment, but the assemblage of type 2,which displays tetartohedral cubic symmetry, appears to be capableby modification of closer packing than the other.It is, moreover,the assemblage indicated by the facts as representing methane;each of the large spheres in it is similarly situated with respect tothe groups of small spheres, whilst in the assemblage of type lathe large spheres form two seh, the members of one of which differin environment from those of the other. The latter type ofassemblage probably has a practical application, although not inthe present connexion.With respect to the relative orientation of the tetrahedral groupsof small spheres in the assemblage of type 2, it is to be noted thatthe groups contained within the one half set of the cubes of thepartitioning are related by a simple operation, besides that ofrotation about a digonal axis, to those contained within the otherhalf set.The relation consists in the existence of four similartranslations having the four directions of the sets of trigonal axes.Either of these operates to bring a cubic cell to the place of aneighbouring cubic cell, which is in contact with the first at one ofits corners. I n addition to being identical, the two half systems ofcubes with their contained groups of four small spheres consequentlyhave the same orientation, and the wsemblage as a whole ishemimorphous, like the assemblage of type 1 from which it is derived.It has been already noticed that in order to render the packingclose, a modification or deformation of the whole assemblage mustoccur. The eight large or carbon spheres enclosing a single cavitymay be regarded as forming six indivisible quartettes, one for eachof the six faces of the cubic cell containing the cavity; the fourspheres composing a quartette form two square hollows, one in eachof its opposite faces, and these two hollows communicate with eachother at the centre of the quartette.Where a small sphere occupiesthe hollow on one face, the existence of a digonal axis bisecting thecell face involves the presence of another small sphere in the hollowon the other face of the same quartette, and therefore one half o2320 BARLOW AND POPE : THE RELATION BETWEEN THE CRYSTALthe quartettes of large spheres in the assemblage are occupied, andthe other half unoccupied, by the smaller spheres. It follows thatsome increase in the closeness of the packing will be likely tosupervene if it is possible symmetrically to adjust the arrangement ofthe larger spheres, without altering the marshalling, in such a wayas similarly to diminish the size of onehalf of the hollows-theunoccupied ones-while slightly increasing the size of the rest-theoccupied ones.Three of the six hollows present in each cavity,namely, the unoccupied ones, will in this event become contracted.Such an adjustment of the larger spheres, which does not alter thetype of symmetry, consists in a slight equal shift of each largosphere along its trigonal axis in either direction; the choice made ofFIG. 7.the direction of shift for any one sphere necessarily determines thedirections for all if the assemblage is to remain compatible withthe coincidence movements of type 2.The amount of shift islimited by the approximation of the large spheres, causing them tocome into contact at points lying on the digonal axes of rotationwhich characterise type 2. An important feature of the change isthat the large or carbon spheres, in shifting, close in around thathydrogen sphere of each tetrahedral group the centre of which lieson the trigonal axis; the position of the tetrahedral groups under-goes slight adjustment during the process.A projection of the resulting assemblage, showing the carbonspherea alone, is given in Fig. 7; the centres of these spheres lie iSTRUCTURE AND THE CHEMICAL COMPOSITION, ETC. 2321four different planes parallel to the plane of projection, and aretherefore distinguished by circles drawn in heavy or light, con-tinuous or broken lines.The adjustment of the positions of thetetrahedral groups which accompanies the shifting of the carbonspheres, and indeed the entire process, is compatible with themaintenance of cubic symmetry ; the existence of the coincidencemovements of the system involves that all the cavities for thereception of the tetrahedral groups remain identical with oneanother.One of the surest indicakions of closepacking is obtained whencach sphere is in contact with, or in very close proximity to, sucha number of surrounding spheres as approaches the maximum. Thenumber of contacts and near proximities in the assemblage underconsideration is as follows: for each of the carbon spheres, 19,namely, 6 with carbon spheres and 13 with hydrogen spheres.Foreach of three-fourths of the hydrogen spheres, 8, namely, 4 withcarbon spheres and 4 with hydrogen spheres; for each of one-fourthof the hydrogen spheres, 7, namely, 4 with carbon spheres and 3with hydrogen spheres. These numbers of ccjntacts approach themaxima, taking into account the different sizes of the componentspheres; they thus afford a proof that the marshalling of theassemblage is compatible with very close-packing.In connexion with the partitioning of the assemblage into identicalmolecular units of the composition CH,, it should be noted thatfour of the thirteen contacts of hydrogen spheres with a carbonsphere are nearly symmetrically distributed over the surface of thelatter; the four hydrogen spheres concerned are thus situated atthe apices of an approximately regular tetrahedron, of which thecentre is the centre of the carbon sphere.The four hydrogenspheres referred to may be identified as follows. In any pseudo-cubic group of eight carbon spheres in the assemblage, the singletrigonal axis intersects two of the eight; one of these makes contactwith a single hydrogen sphere of the enclosed group at its point ofintersection with the trigonal axis. Regarding this carbon sphereas that of the molecular unit, CH,, to be picked out, it is to benoted that the three contacts with it of hydrogen spheres, which,with the one on the trigonal axis, make up the four referred to, arethose of the hydrogen spheres lying in three of the outside hollowsof those faces of the cubic group which have as their commonangular point the centre of the selected carbon sphere.The fourcontact3 of the unit molecular group CH, thus derived do notprecisely mark the angular points of a regular tetrahedron, butthe arrangement of the four hydrogen spheres about the carbonsphere approximates so closely to the regular tetrahedral dispositio2322 BARLOW AND POPE : THE RELATION BETWEEN THE CRYSTALpremised by the theory of van't Hoff and Le Be1 (Figs. 8 and 9) thatits departure from the latter cannot be clearly indicated in a,diagram; the assemblage is divisible into identical units of the formdepicted. The result of the close approximation to regularity ofthe tetrahedra marked out by the hydrogen sphere centres thusselected is that different assemblages produced by fitting togetherthe molecular units in different orientations will be so nearlyidentical that the equilibrium arrangements to which they pass willbe actually identical.I n this connexion it is instructive to observethat the tetrahedral arrangement is indicated in another manner ;each carbon sphere, before the adjustment, is similarly related toeight cavities, of which the relative positions are those of the angularpoints of a, cube, and the greatest number of these cavities whichcan participate in containing the hydrogen spheres attached to thecarbon sphere is four. Consequently, the most symmetrical modeFIG.8. FIG. 9.of allotment of the hydrogen spheres is for each carbon splierc t oattach to itself four hydrogen spheres contained in four out ofthe eight cavities surrounding it, and for these four cavities to beselected with a, regular tetrahedral disposition. Thus, like thehydrogen atoms in the usual graphic formula of methane, the fourcavities concerned have interchangeable positions with respect tothe carbon sphere to which they relate.It is thus to be finally concluded tha.t the investigation of theclose-packed arrangement of the methane assemblage indicates thatthe molecular units can be so chosen as to have t'he tetrahedralconfiguration depicted in Figs. 8 and 9.The relation thus established between the theory of the tetra-hedral arrangement of the links within the molecule, based on thechemical behaviour of methane and its derivatives, and the concretegeometrical properties of the corresponding close-packed arrange-ment of the spheres of influence of the component atoms is of funda-mental importance. It is worth recapitulating in precise language,because it will subsequently be shown that a relation of the samSTRUCTURE AND THE CHEMICAL COMPOSITION, ETC.2323nature obtains for the carbon compounds generally; in other words,that a tetrahedral arrangement of the contacts of a carbon sphere ofinfluence with its companion spheres persists after substitution hastaken place. The relation for methane may be thus stated.Represent the carbon and hydrogen atoms of a methane molecule byspheres of the valency volumes 4 and 1 respectively, and form thespheres into groups of five according to the van’t Hoff-Le Be1 theory,each sphere of volume 4 being in contact with four spheres, eachof volume 1, placed around it symmetrically, so that their centresmark the angular points of a regular tetrahedron; it is then foundthat, while preserving the marshalling of the spheres of eachindividual molecular unit, a close-packed assemblage can be formedby fitting the groups together symmetrically, of such a nature thatits geometrical properties are those of the crystalline tetrahalogenderivatives of methane.The following crystallographic data ar0 available i ~ s bearing onthe symmetry and dimensions of the methane assemblage.Carbontetraiodide, CI,, is cubic, and carbon tetrabromide, CBr,, crys-tallises above 46*7O in the cubic system, the crystal class beingknown in neither case. Below 46*7O, carbon tetrabromide crys-tallises in the monosymmetric system, but, as previously pointedout (Trans., 1908, 93, 1530), this modification is referable tothe pseudocubic axial system, a : b : c=1.0260 : 1 : 1, a=90°16‘,P = y = 90°33’ ; the monosymmetric form thus scarcely differs indimensions from the truly cubic one, and both indicate the cubicmarshalling of the assemblage. On replacing each hydrogen spherein the methane assemblage by the group CH,Br, in accordance withthe second geometrical property of close-packed assemblages (Trans.,1907, 91, 1204), tetrabromo-PB-dimethylpropane, C(CH2Br)*, isobtained; as Jaeger has found (Trans., 1908, 93, SZO), this sub-stance may be regarded as pseudocubic, with the axial ratiosa : b : c=1.0484 : 1 : 0‘9472, B=9Oo45’.The cubic marshallingof the methane or carbon tetrabromide assemblage thus survivesthe symmetrical introduction of four methylene groups, CH,, intoeach molecular unit, CBr,, in accordance with the second geometricalproperty.I n connexion with the assemblage attributed above to methaneand to its fully substituted halogen derivatives, it may be noted thatiodoform, CHI,, is described as hexagonal with a: c = l : 1.1084(Pope, Trans., 1899, 75, 46). It is evident that the symmetry ofthe space arrangement of the methane assemblage may be loweredwithout any appreciable alteration of the relaOive situations of thespheres by a partial substitution of the spheres representinghydrogen atoms which leads to the production of an arrangemen2324 BARLOW AND POPE : THE RELATION BETWEEN THE CRYSTALappropriate for iodoform.I n order to trace the probable effect ofsuch a substitution, it is convenient to work with an ideal less closely-packed assemblage of higher symmetry, from which the methaneassemblage may be regarded as derived. Let the centres of thecarbon spheres occupy precisely the points of a cubic space-lattice(Fig. 3), and let each of the tetrahedral groups, CH,, which arenow to be of completely regular configuration, be rotated fromthe orient-ations which they present in the closest-packed assemblage,so that the centres of the small spheres all lie on trigonal axes;the system thus consists of units of the composition CH,, less closelypacked, but all similarly orientated.Next substitute iodine spheresfor three of the four hydrogen spheres of each unit, without alteringthe positions of the centres, in such a way that the new units, CHI,,are all similarly orientated; the result is to destroy threefourth ofthe trigonal axes, and to leave only those which contain the centresof the unsubstituted hydrogen spheres. On performing finally therotations and adjustments requisite to restore the closest-packedcondition prevailing in the methane assemblage, threefourths of thesurviving trigonal axes are destroyed. The closest-packed wem-blage thus arrived at has rhombohedral symmetry and is pseudo-cubic.In the assemblage just derived let the dimension c be three timesthe distance between the centres of carbon spheres lying on the sametrigonal axis ; the distance separating these centres along directionsperpendicular both to this axis and to a face diagonal of a cube ofthe pseudocubic partitioning will be approximately J 2 .c / 3 . I f thelatter distance is taken as a/2, the axial ratio is obtained as :a:c=2J3:3=1:3/2JZ=l :1*0606.This ratio is not far removed from that of iodoform, and it istherefore established that the rhombohedral form displayed by thecrystalline substance may, like the rhombohedral assemblage sug-gested, be pseudocubic.The Normal Homologues of Methane.The most obvious method of constructing assemblages representinghydrocarbons homologous with methane consists in symmetricallyremoving one or more hydrogen spheres from the groups of fourcontained in the assemblage of the parent hydrocarbon, and then,by appropriate adjustment of the spheres remaining, to close upthe gaps which have been produced.Thus, an assemblage of the empirical composition CH3 may bederived by symmetrically removing a hydrogen sphere from eachgroup of four in the methane assemblage, and then adjusting sothat with the same number of cavities each cavity among the carboSTRUCTURE AND THE CHEMICAL COMPOSITION, ETC.2325spheres shall be as closely packed as possible, although nowcontaining but three hydrogen spheres instead of four.Suchan operation corresponds with the removal of the iodine atomfrom methyl iodide; the observed fact that in this reaction, asin all similar ones, the condensation of two hydrocarbon radiclesyields one molecule, finds expression in the way in which theassemblage undergoes contraction during the adjustment necessaryfor closing up the produced gaps. The fact that the methyl iodideassemblage, which has the same marshalling as that of methane,yields ethane on treatment with sodium, can be represented asfollows. In the methane assemblage, the carbon spheres are pre-vented from making intimate contact with one another by thepresence of hydrogen spheres packed around them, but when thenumber of the latter is reduced by each group of four becoming agroup of three, the carbon spheres necessarily draw nearer together ;it is conceivable that equilibrium, represented by close-packing,requires them to come into closer contact, and to press on eachother two by two, and that the intimate relationship thus establishedbetween the individuals of a pair corresponds with the linkingbetween the two methyl carbon atoms in the ethane molecule.Itwill be shown in connexion with the assemblage described belowthat the condensation of the assemblage following elimination ofhydrogen spheres and the adjustment which restores closepacking,lead to close contact of the kind referred to between carbon spheres;such contact is thus representative of the formation of a linkbetween carbon atoms such it9 that present in the ethane molecule.The production of the ethane assemblage from that of methanemay also be regarded as resulting from the replacement of one-fourth of the hydrogen spheres, each by one carbon sphere, when,in accordance with the second geometrical property of close-packedassemblages, the introduction of three hydrogen spheres with eachnew carbon sphere suffices for the preservation of close-packing.The alternative ways in which the paraffins may be regarded, such,for instance, as the possibility of considering propane as dimethyl-methane and as ethylmethane, also find expression in the geometricalmode of regarding these substances now advanced.The mmtgeneral method of formulating the normal parafiirs consist8 inassigning to them the constitution H*[CH,],.H, in which an openchain of n-carbon atoms forms the backbone of the molecule, andis isolated from other similar chains in front and rear by theaddition of a hydrogen atom to each of the end methylene groups.For the present purpose it will therefore be convenient to derivefirst an assemblage of the empirical composition CH2, correspondingwith the radicle methylene; it will then be shown how this assem2326 BARLOW AND POPE : THE RELATION BETWEEN THE CRYSTALblage, composed of strings of methylene groups the carbon spheresof which are in close contact throughout the length of the string,is related to that of methane, and in what manner hydrogen spheresFIG.10.can be homogeneously intercalated so as t o divide the methylenestrings of indefinite length into definite molecular groups torepresent any individual normal p a r f i .FIG.11.The general methylene msemblage may be constructed in thefollowing manner. Space IS divided into endless hexagond prisms,each of which is divided into identical hexagonal cells by describinSTRUCTURE AND THE CHEMICAL COMPOSITION, ETC, 2327a series of parallel planes perpendicular t o the prism axes at adistance apart equal to the smaller diameter of the prisms. I neach prismatic cell thus obtained is inscribed a sphere; the diameterof the latter will be the smaller diameter, and also the height, ofthe hexagonal cell. In t,he system produced, each cell corner marksthe centre of a cavity between adjoining spheres, and about eachmeeting point of cell corners a small sphereis now described of such diameter as justto touch the six surrounding large spheres.The resulting system is shown in plan inFig.10, and in elevation in Fig. 11,and possesses a general arrangementwhich may be visualised by the perspec-tive view of a fragment shown in .Fig. 12.Each small sphere of the assemblage, inaddition to making contact with sixlarge spheres, is nearly in contact with three other small spheres,and each large sphere is in contact with twelve small spheres andeight large ones. If t,he large spheres represent carbon, and thesmaller ones hydrogen atoms, the assemblage has the empiricalcomposition CH,; since, however, the volume of the smaller spheresis appreciably less than one-fourth that of the larger, the valencyFIG.12.FIG. 13.relation of the volumes requires the smaller to increase until thevolumes of small and large spheres, with the addition of theappropriate proportions of interstitial space, are in the ratio of 1 : 4.This expansion of the small spheres necessarily forces the largerspheres apart, and for this to occur in such a manner that themodified system possesses maximum closeness of packing, it musttake place so as to break as few of the contacts as possible in 2328 BARLOW AND POPE : THE RELATION BETWEEN THE CRYSTALsymmetrical manner. The most symmetrical expansion of thekind which can occur is one which breaks all the contacts betweenlarge spheres and converts the assemblage of Figs.10 and 11 intothat represented in Figs. 13 and 14; the smaller number of contactsin the modified system is indicative of looseness of packing, andin order to reproduce close-packing as many of the original contactsits possible must be reestablished in a symmetrical manner.A consideration of the assemblage of Figs. 13 and 14 in con-nexion with the cubic disposition of large spheres shown in Fig. 4,from which the methane assemblage was derived, shows that t4heplane arrangement of the large spheres in their layers, shown bythe continuous line circles of Fig. 14, is approximately that obtain-ing in the layers of the assemblage of Fig. 4, which are parallel t oFIG. 14.the plane the trace of which is the diagonal line C in Fig. 15, asshown by the continuous line circles of Fig.16. Whilst, powever,in Figs. 15 and 16 each cavity between the large spheres is destinedand is sufficient for the accommodation of a tetrahedral group offour small spheres, the corresponding space in Figs. 13 and 14 hasbeen reduced so that it can accommodate but two small spheres;this has been effected by somewhat increasing the distance betweenthe large sphere centres in the direction of the diagonal C inFig. 15, and considerably diminishing the distance between thelayers of sphere centres in the direction perpendicular thereto.Each cavity which suffices to contain four small spheres, such as isenclosed by the eight spheres, four, p, q, r, and s, of one plane ofFig. 15, and four, t, u, v, and w, of the plane immediately below, aSTRUCTURE AND THE CHEMICAT, COMPOSITION, ETC.2329shown in Figs. 15 and 16, has by the process just described beenconverted into two cavities, namely, one enclosed by the correspond-FIG. 15.ing spheres, p, r, s, t, v, and w, of Figs. 13 and 14, the other by thespheres, q, r, s, u, v, and w. The two small cavities thus derivedFIG. 16.from the original large one each suffices for the accommodation ofone small sphere; these are marked a and b in Figs. 13 and 14.VOL. XCVII. 7 2330 BARLOW AND POPE : THE RELATION BETWEEN THE CRYGTALThe process by which the present assemblage can be derived fromthat, of methane, and also the converse, by which t'he former canbe converted into the latter, are applications of the secmdgeometrical property of close-packed homogeneous assemblages.It remains to indicate the manner in which close-packing can beestablished in the assemblage of Figs. 13 and 14, that is to say, theway in which the assemblage can be caused to occupy the minimumspace as the result of an adjustment which does not involve re-marshalling.The requisite deformation will be understood by con-sidering its effects on the system; these are indicated in Fig. 17,which represents one double layer of the two kinds of spheres, andin Figs. 18 and 19, which are projections of the altered assemblageFIG. 17.on two planes at right angles to one another. For the sake ofclearness, the hydrogen spheres are omitted from Fig. 18.The symmetrical adjustment which increases the closeness of thepacking brings the members of the rows of carbon spheres shownin Fig. 13 alternately into contact and further apart, as indicatedin Figs.17 and 18; thus a carbon sphere, such as p, makes contactonly with m and n, and draws away from r and s. The sequenceof making contact and moving further apart alternates in con-secutive layers of the form shown in Fig. 17, so that these layersnow have two distinct projections on the same area of Fig. 18;the latter diagram thus shows two alternating sets of carbon spheres,those indicated in continuous lines, and those in dotted circles, inplace of the one set shown in Fig. 13. This alternation results inthe formation, in each of the planes projected on Fig. 18, of zigzaSTRUCTURE AND THE CHEMICAL COMPOSITION, ETC.2331strings of carbon spheres in contact and of indefinite length, thezigzag strings in one plane of the assemblage being located fromthe positions of others in the same plane or of those in the nextneighbouring planes by some simple symmetrical operation sucli asthat about a centre of symmetry; the zigzag strings in one planedo not lie immediately beneath or above those in the next plane.The assemblage of Figs. 17, 18, and 19 represents the generalmethylene assemblage, and is to be regarded as an arrangementhaving the empirical composition CH,, which constitutes tlie open-chain portion of a normal paraffin. By dividing the zigzag stringsinto fragments of suitable length by the introduction of pairs ofhydrogen spheres at appropriate intervals, it may be converted, asis shown below, into an assemblage of molecular aggregates repre-FIG.18.sentative of any particular normal paraffin. The existence of thiscorrespondence between the feature of close-packed assemblages justdescribed and the observed fact that, in the normal paraffins, thechains of methylene radicles connecting the terminal methyl groupsexhibit behaviour which warrants the representation of the normalparaffins by the general formula, CH,*[CH,],*CH,, is worthy of note.It has been shown in previous papers that the configurationsassignable, in accordance with the crystallographic evidence andwith the theory of homogeneous close-packing, to numerous organicsubstances is in entire accord with some features of the chemicalbehaviour of such compounds.Before proceeding to employ theconfiguration arrived at for the general methylem chain, *[UH2In*in the production of assemblages representing the normal paraffins7 N 2332 BARLOW AND POPE : THE RELATION BETWEEN THE CRYSTALfor comparison with the chemical facts and crystallographic evidence,it is therefore desirable to consider stereochemical features of thechain, *[CH,],*, as now presented. Any such continuous chainseparated from the whole msembhge presents the plan and elevationFru. 19.shown in Figs. 20 and 21 ; a rough perspective view of a fragmentof the indefinitely prolonged chain is given in Fig. 22. It will beseen that each carbon sphere is directly attached to two other carbonFIG.20.spheres and to two hydrogen spheres, and that the plane containingthe centres of the three carbon spheres is perpendicular to theplane drawn through the centres of the two hydrogen spheres andthat of the carbon sphere which they touch. Further, it wiIl bBTRUCTURE AKD THE CHEMICAL COMPOSITION, ETC. 2338seen that by joining the four points of contact made on each carbonsphere, two by hydrogen and two by carbon spheres, a tetrahedronresults. Since these are the essential features of the environmentof any carbon atom of the chain in a normal paraffin, ils summarisedby the theory of va.n’t Hoff and Le Bel, it follows that the con-figuration for the chain deduced above is in accordance with thechemical facts.In this connexion, it is interesting to recall theinterpretation usually put on the important fact of the persistenceof the tetrahedral arrangement of links from term to term of theseries of assemblages representing the normal paraffins. AdoptingFIG 22.the method employed by van’t Hoff and Le Bel, the configuration ofa string of methylene complexes which forms the backbone of anormal paraffin moIecule is derived by first substituting carbonatoms for two of the tetrahedrally disposed hydrogen atoms of amethane molecule, preserving the tetrahedral disposition of thelinks, and then attaching two hydrogen atoms to each added carbo2334 BARLOW AND POPE : THE RELATION BETWEEN THE CRYSTALatom in such a way that the two outer methylene complexes thusformed are identical with the central one and identically related toit, while having the opposite orientation.The central portion,-CH,*CH2*CH,*, of the propane molecule is thus arrived at.Arrangements proper for the representation of succeeding termsof the homologous series of paraffins are derived by repetitions of thesame process.The form of a group of methylene complexes reached in this wayis quite definite and is that shown in Fig. 22; as the precedingargument has established, a number of the groups representing thesame term of the series can be packed closely together so that thepassage to closest-packed equilibrium involves but a quite trivialadjustment. When additional hydrogen spheres are inserted appro-priately to complete the representation of a given paraffin, anassemblage results, as will be shown immediately, which displaysthe geometrical and dimensional properties appropriate to the crystalof the substance concerned.It is easy to demonstrate that the persistence of the tetrahedraltype of arrangement is a geometrical consequence of substitutioneffected in accordance with the second geometrical property. Forin carrying out such a substitution in a, methane assemblage, theadded carbon spheres are deposited in the hollows on the faces ofthe layers of the assemblage left vacant by the removal of hydrogenspheres, and consequently the incoming large spheres occupy prac-tically the same situations with respect to the unsubstituted portionsrof the assemblage as were previously occupied by outgoing smallspheres.Consequently, since the situations of the paraffin spheresgive a tetrahedral arrangement of the contacts within a moleculargroup, this tetrahedral disposition of the contacts still obtains afterthe substitution. It is not suggested that the tetrahedral arrange-ment of the contacts will remain precisely regular.The Ethane Assemblage.The unit, shown in Fig. 22, of the general methylene assemblageof Figs. 17, 18, and 19 possesses the constitution of an indefinitelylong string of attached methyIene groups,- - -CH2*CH2*CH2*CH2- - -,and is represented by the graphic formula:H H H 13 II Er H H - -c.(-J.c.c.c.(-J.c.c- -H H H H H H I3 HThe comparison which has been made between the generalmethylene assembla-ge and the methane assemblage sliows that ifextra, pairs of hydrogen spheres are introduced between succeedinSTRUCTURE AND THE CHEMiCAL COMPOSITION, ETC.2335carbon spheres, the resulting assemblage assumes the compositionand constitution of methane, thus :H H H H H HHCH HCH HCH HCH BCH HCH.H H H H H HIf, however, such pairs of hydrogen spheres are intercalated, noteverywhere between succeeding carbon spheres, but intermittentlyat points homogeneously selected, the resulting assemblage shouldrepresent a normal paraffin homologous with methane. On intro-ducing pairs of hydrogen spheres symmetrically at half the pointsindicated, the assemblage representing ethane should be produced,thus :H H H H H H H H HITHC-CH HC*CH HC*CH HC-CH HC*C€I.H H H H H H HE3 H HIt is desirable to confirm this deduction by an examination of theethane assemblage, thus derived, in the light of the principalcrystallographic evidence available; this is found in the dataobtained by Gossner for the hexahalogen derivatives of ethane andfor pentabromoethane (Trans., 1906, 89, 1682).I n the tabulateddata for these substances it is convenient t o double the ratio ofc / b , and to state the equivalence parameters and axial ratios as inthe appended table; the valency volume, Iy=14, is regarded asthe molecular space unit, so that the linear unit employed for theequivalence parameters is the edge of a cube of unit valency volume.The closeness of the packing of the spheres is taken to be the sameas that of the closest-packed assemblage of equal spheres:n : b : c.B. x. : y : 2.CCl,'CCl, ............... 0.5677 : 1 : 0.6320 90" 1.9255 : 3.3917 : 2.1435CHI,Cl'CCI, ........ 0 5612 : 1 : 0.6342 ,, 1.9086 : 3 4009 : 2.1520CBtCI,'CBrCl, ...... 0'5646 : 1 : 0'6384 ,, 1.9120 : 3.3567 : 2.1620CHr,*CBr, ............ 0.5639 : 1 : 0 6284 1.9205 : 3'4058 : 2.1403CHBr,*CBr, ........ 0'5650 : 1 : 0'6286 91bi9' 1.9282 : 3'4126 : 2.12811.9166 : 3.3963 : 2'1494 Mean for fir& four substances :In calculating the mean equivalence parameters, the four ortho-rhombic substances only .have been considered, the monosymmetricpentabromoethane being excluded from the calculation.It has now to be considered how the general methylene assemblageof Figs.17, 18, and 19 can be converted into a close-packed assem-blage of the dimensions represented by the above mean equivalenceparameters, z: y : z=1'917: 3,396: 2.149, by the intercalation ofpairs of hydrogen spheres in the manner already indicated. Thediameter of a univalent sphere is obtained in terms of the linearunit from the consideration that it is the face-diagonal of the cubeoutlined by joining the obtuse solid angles of the unit dodecahedro2336 BARLOW AND POPE : THE RELATION BEI'WEEN THE CRYSTALof a closest-packed assemblage of these spheres (Trans., 1907, 91,1181). Thus, if a be the diameter in question, a / .\/Z is the edge ofthe cube inscribed in the unit dodecahedron; the content of thiscube is a3/ZJ2, and that of the dodecahedron is equal to a 3 1 d2,which is taken as unity. Consequently, a=2h =1*1225, and sincethe volume of a quadrivalent.sphere is four times that of a univalentone, the diameter of the former is 23 x 2: = 2." = 1 *781@.The sphere projections in the general metliylene assemblage ofFigs. 17, 18, and 19 are drawn to the scale thus indicated, and thedimensions indicated in these figures are, two of them, the values,x=1*917 and y=3*396, of the mean equivalence parameters in thetable last given. On introducing between each pair of layers ofthe general methylene assemblage extra hydrogen spheres equal innumber to the carbon spheres already present, the preservation ofPIG. 23a.close-packing demands that the one pair shall shift upon the nextpair, so that the projection of the two pairs now consists of foursuperposed sections, as depicted in Figs, 23, a and b .I n these diagrams, which, taken together, give a projection of theethane assemblage, some of the intercalated hydrogen spheres aremarked a; the dimensions, x = 1.917 and y = 3.396, are shown in theplane of the section.The packing is about as close as in the methaneassemblage described above, and since the closeness of the packingis thus adhered to and the composition is that corresponding withethane, the translation perpendicular to the plane of the sectionwill necessarily have the corresponding value of z = 2.149. Sincethe valency volume of the molecular unit is 14, and that, of theterminal hydrogen spheres is 2, the dimension 2: of the methyleneassemblage, as shown in Figs.17 and 18, is six-sevenths of the z valuSTRUCTUKE AND THE CHEMICAL COMPOSITION, ETC. 2331just stated, and therefore equals 1.842; this is the value of a used inthese diagrams. It is concluded from the above that the assemblagedepicted in Fig. 23 is related to the general methylene assemblagein the appropriate manner, and has the dimensions indicated forethane by the crystallographic data; the crystalline symmetry ofthe assemblage, when all the smaller spheres are identical in kind,is the orthorhombic symmetry exhibited by the liex&hdogenderivatives of ethane named in the table. It is, however, obviousthat differences in kind occurring among the smaller spheres mighthave the effect of reducing the symmetry of the assemblage in themanner indicated by the existence of the monosymmetric penta-bromoethane.Lehmann has shown (MoZel~uZal.-F~ysil., 1888, 1, 178) that hexa-FIG.23b.chloroethane, C,C16, crystallises in an anorthic and a cubic formas well as in the orthorhombic form dealt with above; no measure-ments are available for the former modification, but it is instructiveto deduce the assemblage representing the cubic form of the sub-stance. In view of the close relationship which must exist betweenthe orthorhombic and the cubic modifications of hexachloroethaue,it is convenient to derive tlhe assemblage for the latter from thatof the former. The orthorhombic assemblage may, for purelycrystallographic purposes only, be regarded as built up from aunit of the form shown in Fig.24, a, b , c, and d, and consisting oftwo carbon spheres in contact having a circlet of six chlorine spheresplaced round the neck produced between the two large spheres;the volumes of the two kinds of spheres, namely, 4 : 1, are suchthat when all the six small spheres touch the two larger ones, theyvery approximately form a continuous ring of small spheres i2338 BARLOW AND POPE : THE BELATION BETWEEN THE CRYSTALcontact as shown in the diagrams. A geometrical unit of this kindis marked ABcdefgh in Fig. 23u, and presents in that diagram theC.FIG. 24.b.d.aspect depicted in Figs. 25 a and b ; it can be used in the mannerdescribed below for the construction of the assemblage representingthe cubic modification of hexachloroethane.FIG.25.a. b.The geometrical units referred to and figured occupy the valencyvolume, W=14, and can be fitted together in cubic symmetry sS'l'HUCTURE BND THE CHEMICAL COMPOSITION, ETC. 2339that their centre points lie at the centres of the cube cells of acubic partitioning of space provided that the cube cells have thevolume 14; the length of tihe cell edge should therefore be 3d14,t8he scale previously used being adopted. The units are fitted intoa system of non-intersecting trigonal axes of the kind alreadydescribed (p. 2317), and in the following manner. In one cubecell the trigonal axis of which has the direction indicated by u(Fig. 5), place a geometrical unit group so that its centre is at thecentre of the cube cell, and so that the centres of its two largespheres lie on the single trigonal axis of the cell; whatever theposition of the small spheres, it is evident that their centres lie ona circle the centre of which is the point of contact of the two largespheres, and the plane of which is perpendicular to the trigonalaxis of the unit.This circle is projected on one of the three facedirections of the cube cells as an ellipse, as indicated in Fig. 25b.Geometrical units are now fitted in similar manner into the othercube cells of the system, due regard being paid to the preservationof the respective trigonal axes, a, b, c, and d, of the different cellsof the partitioning.A single layer of the resulting system of cells with their contentsis depicted in Fig.26 as a projection on a cube plane; the projectionsof the trigonal axes are shown a,s continuous straight lines, andare lettered a, b, c, and d, in accordance with the conventionpreviously adopted (Trans., 1907, 91, 1183). Digonal axes ofrotation pass through the assemblage' perpendicular to the plane ofprojection at the points S, T, U, and V.The precise position of the small spheres in the assemblage isdeduced by reference to the digmal axes of symmetry. Thus thegeometrical unit is so placed in the cube cell of the partitioningthat the distance of the centre of one of its small spheres from adigonal axis is equal to the radius of the small sphere; thiscondition is practically fulfilled if the position of the circlet ofsmall spheres is such that the centre of one of them lies at thehighest point of the circular locus, the projection of this centretherefore falling a t one extremity of the minor axis of the ellipsein which the circular locus is projected on the plane of a cube face.When one geometrical unit has been placed in position in themanner indicated, others can be similarly located with their centresa t the remaining cell centres by carrying out the coincidencemovements and operations with respect to the axes of the firstselected cell. The type of symmetry is that numbered Za, inBarlow's list (Zeitsch.Kryst. Min., 1894, 23, 44).It is evident from Fig. 36 that the spheres of the single layerof complexes fit closely together in the marshalling indicated, and2340 BARLOW AND POPE : THE RELATION BETWEEN THE CRYSTALas the assemblage can be regarded as made up of such layers parallelto either of the three directions of the cube faces, it follows thatthe georretrical units employed can be fitted together in space inthe manner indicated, and that the packing is very close.The geometrical unit which has been used in building up theorthorhombic and the cubic crystalline assemblage of hexachloro-ethane is, as before mentioned, merely used for constructionalFIG.26.purposes, and is not to be regarded as possessing the configurationof the chemical unit or molecule. The possession of a larger massof crystallographic data than is at present available should enablethe configuration of the chemical molecule to be determined by aprocess of elimination.The various polymorphous forms of thedifferent halogen derivatives of ethane must all consist of packedarrangements of units having the configuration of the ethanemolecule; further, the latter must be derivable from the generaSTRUCTURE AND THE CHEMICAL COMPOSITION, ETC. 2341inethylene assemblage by the sy’inmetrical intercalation of spheresof unit valency volume in this assemblage, as already described.These conditions are fulfilled, not only by the geometrical unit usedabove, but also by groups of the composition C,CI, possessing aconfiguration such t,hat the eight component spheres are centredat the apices of two tetrahedra so placed that an apex of the oneis directed towards an apex of the other.The two kinds of unitof the structure thus distinguished possess the configuration of theethane molecule as it has been deduced from the principles laiddown by van’t Hoff and Le Bel; rough perspective views of thecliemicnl nnit or molecule thus derived are given in Figs. 27 a and h .FIG. 27.na. b.It will be seen that the one may be derived from the other byrotating onehalf of the unit through 180° with respect to theother half. The fact that these two configurations of unit, closelyrelated by the mode in which one is convertible into the other, canbe traced in the assemblage as depicted in Fig. 26, is of interest inconnexion with van’t Hoff’s doctrine of the free rotation of a singlyhound carbon atom.An Alternative General Met?&iylene Assemblage.A simple method has been given above (p.2333) for derivingan aasemblage which can be geometrically partitioned into endlessstrings of the general form n(CH,), and it has been shown howthe assemblages representative of the normal paraffins can be derivedfrom this general methylene assemblage by the intercalation ofhydrogen spheres. Examination shows, however, that by modifyingthe assemblage referred to by means of a particular kind ofdistortion, an alternative series of assemblages is obtained , in whichthe arrangement of the carbon and hydrogen spheres which formthe methylene fragments is very nearly the same as before: thi2342 BARLOW AND POPE : THE RELATION BETWEEN THE CRYSTALarrangement, like the first, is related to a number of crystallographicfacts. The new kind of arrangement can be derived from thefirst by an adjustment or deformation which leaves each sphere wit,liFrc.28.practically the same surroundings but which changes the generalsymmetry: the nature of the adjustment is a-s follows.The large spheres of one layer (Fig. 28) in the unadjusted methyleheassemblage of Fig. 14, when pressed together in the direction whichFIG. 39.U b.is horizontal in the diagram, fall into a square arrangement;simultaneously, the smaller spheres, by movement on each otherand on the large spheres with which they are in contact, are ableto accommodate themselves to the altered form of the layer, anSTRUCTURE AND THE CHEMICAT. COMPOSITTON, ETC.2343can pack very closely into the hollows remaining after the changeis made. The section of the assemblage shown in Fig. 28 thusbecomes that shown in Fig. 29 a and 6; the modified layer consistsof a plane of the larger spheres in square arrangement with thesmaller spheres sunk in the hollows on both of its faces; the smallspheres touch each other in the plane drawn through the centresof the large spheres, as shown in Fig. 29b. Layers produced in thismanner can be fitted closely together in such a way that theresulting assemblage is practically identical with that previouslyreached by compounding the layers in their other shape. I n otherwords, the layers depicted in Fig. 29 are obtained from the generalmethylene assemblage of Fig.14 if, instead of making the separationinto layers parallel t o the plane of Fig. 14, it is made parallel tothe plane of projection of thesame assemblage shown inFig. 30. The plane of projec-tion of Fig. 30 is at rightangles to those of both Figs. 13and 14; thus, in Fig. 31a, inwhich the arrangement isidentical with that in Fig. 13,a plane perpendicular to theplane of the diagram, drawnthrough AB, gives the projec-tion shown in Fig. 14, whilsta plane drawn through CD,also perpendicular to the planeof Fig. 31a, gives the sectiondepicted in Fig. 31b.FIG. 30.The conversion of the general methylene assemblage depicted inFig. 30 into that representing a normal paraffin is, as before, effectedby intercalating hydrogen spheres, twice as numerous as the carbonspheres in a single layer, between consecutive layers of carbonspheres appropriately selected, the planes of these layers beingparallel to the plane of projection of Fig.30. It is seen fromFig. 29a that the principal hollows, which are of the kind markedA, in one of the surfaces of a layer are twice as numerous as arethe carbon spheres of the layer; if therefore two such layers areappropriately placed together, a layer of hydrogen spheres twiceas numerous as the carbon spheres of a layer can be closely fittedbetween them, each sphere occupying a principal hollow, such aa A,in both the opposing faces. The combination of two layers of thecomposition CH2 with the layer of hydrogen spheres thus fittedin between them, is shown projected in Fig.32: the small sphere2344 EARLOW AND POPE : THE RELATTON RETWEEN THE CRYSTALof the intercalated layer are indicated-by double circles. In theassemblage representing a normal paraffin formed in this manner thePro. 31a.FIG. 31b.hydrogen spheres added to aterminal layer of the formCH,, and allotted to this layer,occupy the same positions inthe face of the layer as theywould i f an additional CH,layer, of which they formedpart, were added; this can beseen on inspection of the pro-jection of a stratum of aparaffin assemblage of theform under consideration. Thestratum represented in Figs.31 a and h is that appropriateto normal butane,CH,- CH2*C H,* C H, ;corresponding with the fourmethylene radicles, CH,, thereare present four layers of largespheres in each stratum, asshown in Fig.31a.The centres of the terminalsmall spheres which have beenintroduced lie on two similarsets of digonal axes of theassemblage having two direc-tions perpendicular to oneanother as indicated by thediagonal broken lines of Fig.32; the identity of these di-gonal axes in the two directionsinvolves the presence of screwtetragonal axes perpendicularto the planes containing thedigonal axes. Thus, the assem-blage of a, normal p a r d n inthe modified form now de-scribed can present tetragonalsymmetry ; the orientations ofthe succeeding strata, each ofwhich is composed of a certain number of layers of the compositionCH, with the terminal hydrogen spheres added, will then differ bSTRUCTURE AND THE CHEMICAL COMPOSITION, ETC.234590°. As remarked above, a section of a single stratum present ina, butane assemblage is represented in Fig. 31.That the marshalling of the assemblage of a normal paraffin,when of this altered form, is compatible with very close packing,is evidenced as before by the approach to a maximum of the numberof contacts or close proximities round each sphere. Thus, each endcarbon sphere of a chain is in contact with, or in close proximity to,six large spheres and fifteen small ones, together twenty-one, andeach of the other large spheres is similarly environed by eight largeand twelve small spheres, together twenty. Each terminal smallsphere is environed by four large and four small spheres, and nextto the terminal ones occur other small spheres immediatelysurrounded by five large and five small spheres: in the interiorof the assemblage each small sphere is environed by six large andthree small spheres. As before, the carbon spheres of a moleculecontaining several atoms forma zigzag string; the angles ofthe zigzag are, however, muchmore obtuse in the form ofassemblage now under con-sideration.The relation be-tween the latter, which may becalled the tetragonal assem-blage, and the orthorhonibicform of assemblage previouslydescribed, is indicated bystating that whilst the mar-shalling of the methylene por-tion is the same in both, theone is obtainable from theother by a general distortionFIG.32.which alters the angles of the zigzag formed by the chain of carbonspheres, but does not appreciably alter the environment of thedifferent spheres or the closeness of the packing; the molecular unitsare of a slightly altered form, although they retain much the samegeneral configuration. I n view of the indications obtained of theexistence of alternative modes of partitioning, which do not giverise to observable tautomerism, in connexion with benzene (Trans.,1906, 89, 1696), and of such alternative modes which furnish amechanism for the occurrence of tautomeric and isomeric change,it is very possible that a paraffin derivative which occurs in oneform of assemblage throughout one range of temperature wouldundergo conversion into the alternative form on entering a differentrange of conditions.VOL.XCVII. 7 2346 BARLOW AND POPE : THE RELATION BETWEEN THE CRYSTALCrystalline Form of Iialogen Derivatives of IIomologues of Ethane.It will be convenient now to discuss the rather sparse crystallegraphic data available for the halogen derivatives of the homologuesof ethane, and to show that these data are very closely and verysimply related t o the two forms of assemblage described above.The whole of the available goniometric data are dealt with underthis heading.@By y-Tetrabromobutane, CH,*CBr2*CBr2*CH,, is dimorphous, andexists as a tetragonal and an orthorhombic modification, whichhave been measured by Fedoroff (J. p. Chem., 1890, [ii], 42, 145).The tetragonal form has the axial ratio a : c =1: 1-28; on statingthis in the alternative tetragonal form of a : c = d 2 : 1.28, andmultiplying the value of cla by four, the number of carbon atomsin the open chain, the ratio becomes, when stated in the moreconvenient orthorhombic form :a : b : c=1'414 : 1.414 : 5.120.The valency volume of this butane derivative is W = 2 6 , and theequivalence parameters are thence calculated as :x : y : ~ = 1 ' 9 2 9 : 1.929 : 6.985.Since, in the tetragonal type of assemblage, the strings of carbonspheres which form the backbone of the molecules all have the samemean direction, symmetry would indicate that this is the direction ofthe axis c in the tetragonal crystal form now under discussion.The longer direction of the molecule having thus the direction ofthe parameter z, the dimensions of the fragment, CH,, in the crystalstructure should be the above x and y and the fraction, 6/26, ofthe above length, z .The equivalence parameters of the fragment,CH,, of the normal butane assemblage are thus calculated fromthe crystal form of the tetragonal modification of PBy y-tetrabromo-butane asx : 9 : ~ = 1 * 9 2 9 : 1'929 : 1.612.These values should represent translations in the tetragonal formof the general methylene assemblage; that they do represent suchtranslations is shown by the manner in which they adapt themselvesto the description of Fig. 31, in which they are marked. Theorhhorhombic modification of the substance is dealt with later(p. 2347).The isomeric aSy6-t etrabromobut ane,C H,B re CH B r C HB r *CH,B r,is described by La Valle (Ber., 1886, 19, 572) m orthorhombic witha : b : c =0*9776 : 1 : 1.6820, and is thus pseudotetragonal.MultiSTRUCTURE AND THE CHEMICAL COMPOSLTION, ETC. 2347plying the ratio, c / b , by two, and ca-lculating the equivalenceparameters for bhe whole molecule with ST'= 26, and for the fragmentwith W I-: 6 in the same manner as before, the values are obtained as :x : y : :=1'947 : 1'992 : 6.701.x : : :=1'917 : 1'992 : 1.546. ,, W=6.With W = 2 6 .These values are not far removed from those obtained with thetetragonal isomeride ; they suggest a slight spreading of the layersin the present instance as compared with the previous one, and aslight compensatory approximation of succeeding layers in thedirection of the axis c.A stereoisomeride of this substance isconsidered below (p. 2348).The tetrabromohexane of the constitutionCH,Br*CHBr*CH,*CH2*CHBr .CH,Bris described by Negri (Ber., 1889, 22, 2498) as orthorhombic witha : b : c=O.3641: 1 : 0.3788, and is also pseudotetragonal. Onmultiplying the length, 6 , by two, interchanging b and c, andcalculating just as before the equivalence parameters for the wholemolecule, with TV=38, and for the fragment, CH,, with W=6,the following values are obtained :x ; y : 2=1*880 : 1.956 : 10.329. With TV=38.x : y : ~=1'88O : 1.956 : 1'631. ,, W= 6.These values for the methylene fragment approximate closelyto those derived from the two previous cases.The three halogen derivatives just above discussed thus presentthe tetragonal type of assemblage; the following appear to exhibitthe alternative orthorhombic type first described, of which thehalogen derivatives of ethane previously referred to afford examples.As already noted, j?By y-tetrabromobutane is dimorphous, andfrom Fedoroff's data for the orthorhombic modificakion Jaeger hascalculated (Trans., 1908, 93, 521) the axial ratios as a : b : c =1.8671: 1 : 3-478.On multiplying the length b by four, thenumber of carbon atoms in the chain, interchanging b and c, andcalculating the equivalence parameters for the whole molecule, withTV=26, and for the methylene fragment, CH,, with W=6, thefollowing values result. :x : IJ : x=1*868 : 3.479 : 4.000.x : 2/ : z=1'868 : 3.479 : 0.923.With W=26.,, w== 6.From the mean values of the equivalence parameters for thehalogen derivatives of ethane, namely, x : y : z = 1.967 : 3.396 : 2.149,with the valency volume, W=14 (p. 2335), we obtain for themethylene fragment, CH,, with W = 6 , the values x: y: z =1.917 : 3.396 : 0.921; the value of z here is half that of the z ofFigs. 17 and 18.This set of values approximates closely tothat calculated from the data for the orthorhombic Wyy-tetra-7 0 2348 BARLOW AND POPE: THE RELATION BETWEEN THE CRYSTALbromobutane, and indicates that the latter substance affects a formof assemblage identical in type with the orthorhombic ethanederivatives.An aP y 8-tet r ab r omobutane, CH,B r CHB r CHB r CHzB r, st er e eisomeric with that discussed above, has been described byPanebianco (Ber., 1888, 21, 1432) as crystallising in the mono-symmetric system with a : b : c = 2.6348 : 1 : 2.3335, p =80°55/.Ontransposing these axial ratios so that (101) becomes {loo}, and(101) becomes {OOl), the axial ratios are obtained in the forma : b : c=1'6198 : 1 : 1.8678, /3=8Z058/30//; in these values a isdoubled and taken as b , c is taken as u, and b is multiplied by fourand taken as c. The axial ratios are thus obtained in the forma : b : c = 1.8978 : 3-2396 : 4.000, The equivalenceparameters for the whole molecule, with W =26, and for the frag-ment, CH,, with W=6, are now calculated as before; the valuesobtained are :a = 82O58/30/'.a: : y : z=1'938 : 3.316 : 4.085.With lY=26.x : y : ~=1'938 : 3'316 : 0'943. ,, ll'= 6 .The latter set of values also agrees well with that derived fromthe halogen derivatives of ethane, namely, with x: y: z =1.917 : 3.396 : 0.921.The Secondary and Tertiary Paraffins.The discussion of the configurations of the normal paraffins inthe previous pages has revealed a singularly close correspondencebetween the customary method of representing the constitution ofsuch substances and the conception of their configurations derivedfrom the geometrical application of close-packing to assemblages ofspheres of two volumes in the ratio of 4 : 1. It has yet to be shownthat the correspondence extends to the secondary and tertiaryhydrocarbons of the same series.Tetramet h ylmet hane (PB-Dimet hylpropne), C(CH,),.The most obvious method of arriving at the assemblage represent-ing tetramethylmethane consists in replacing each hydrogen spherein the methane assemblage by the methyl radicle, CH,, in accordancewith the second geometrical property; the discovery of the precisearrangement of the assemblage is, however, attended with muchdifficulty if this mode of procedure is adopted.Another method,which is more readily traceable, depends on the application of thefirst geometrical property to the methane assemblage, and may bethus described.It has been pointed out that the four hydrogen spheres associatedto form a close group in a methane assemblage belong to fouSTRUCTURE AND THE CHEMICAL COMPOSITION, ETC.2349different molecular groups, CH, ; if t,herefore a carbon sphere, whichis quadrivalent, be substituted for .a single close group of fourhydrogen spheres, it will belong to, and will connect, four partialgroups or radicles, CH,, and thus give a composite group of therequired composition, C( CH,),. Consequently, if throughout themethane assemblage every fourth close group of the composition H,,selected symmetrically, is removed, and a carbon sphere substituted,this being done in such a manner as to make the relation of theunits so obtained to the assemblage as a whole identical, such anassemblage will furnish a possible solution. There are two waysof accomplishing this in a highly symmetrical manner, eitherof which would appear to be in harmony with the ascertainedfacts.When the structure of the methane assemblage was under con-sideration, it was pointed out that the shape and orientation of thegroups, H4, influence the form of the skeleton framework composedof the carbon spheres, the reason of this being that the arrangementaffected by these spheres must be such as gives the closest-packingof the groups in the cavities containing them; the substitution ofsingle carbon spheres for some of the H, groups will, on the sameprinciple, involve some sIight modification of the skeleton frame-work of carbon spheres.The precise nature of this change isdifficult to trace, especially in the absence of crystallographic data;for diagrammatic purposes it is therefore better, in each of thetwo solutions of the problem, to employ the simple arrangement ofthe carbon spheres in ax high it symmetry as the marshalling whichthey present is capabIe of, without attempting to depict the exactequilibrium conditions ultimately attained.The simpler of the two arrangements possible for the substitutedcarbon spheres has cubic symmetry. Thus, let the points of acertain cubic space-lattice indicate the centres of the carbon spheresof a methane assemblage; the centres of the cubes outlined by thesystem form a second similar space-lattice and mark the positionsof the tetrahedral hydrogen groups.One-fourth of the groups canbe selected for removal and substitution by additional carbonspheres in such a way that their arrangement is that of the un-hatched cubes indicated in Fig.33 a and 5 , which gives the twoprojections of the two sets of alternate layers. The alternativearrangement is a simple tetragonal one, and is shown in Fig. 34;this diagram is identical with Fig. 4, with the exception that everyfourth cavity, symmetrically selected in tetragonal symmetry, isoccupied by a carbon sphere. The newly introduced carbon spheresare shown as broken line circles, and are arranged contiguously inone of the three axial directions, namely, that perpendicular to th2350 BARLOW AND POPE : THE RELATION BETWEEN THE CRYSTALplane of the figure; the groups of hydrogen spheres, H,, fill thestrings of cavities marked A, which are not occupied by the addedcarbon spheres. Thus, the complete assemblage, much as in thecase of benzene previously described (Trans., 1906, 89, 1693),consists of continuouscolumns of c a r b o nspheres in contact, theinterstices b e t w e e nwhich are filled withgroups of hydrogenspheres so arranged asto produce close-pack-ing; the columns coii-sist of square groups offour separated by singlecarbon spheres through-out.In both the cubic andthe tetragonal assem-blage described, themost symmetrical way ofpartitioning the systemof carbon spheres intogroups of five is to makethese groups tetrahedralwith the substitutedspheres a t the centres.Thus, in Fig.34, thesphere R can be asso-ciated with P and Q ofthe four above, and withS and T of the fourbelow, the four spheresP, Q, S, and T thus pre-senting a tetrahedralarrangement about thesphere R.The disposi-tion of the carbonspheres in the moleculeof tetramethylmethane as thus derived is identical with thatindicated by ths theory of van’t Hoff and Le Bel. The tlirechydrogen spheres attached to each methyl carbon atom will lie, asin methane, one in each of three out of four tetrahedrally situatedcavities surrounding each methyl carbon sphere ; the hydrogeSTRUCTURE AND THE CHEMICAL COMPOSITION, ETC. 2351spheres will follaw, as closely as possible, the original arrangementprevailing in the methane assemblage.Inspection of Fig. 34 shows that the tetragonal assemblage fortetramethylmethane, like the first described, must approximateclosely to cubic symmetry.The correctness of the mode of arrivingat the arrangement which is here adopted is confirmed by Jaeger’sdetermination of the crystal form of the tetrabromo-derivative ofthe hydrocarbon, namely, C(CH2Br),; this author has shown, mFIG. 34.already indicated, that the substance is pseudo-cubic, being monesymmetric with the axial ratios, a : b : c = 1.0484 : 1 : 0.9472,B=9Oo45’ (Trans., 1908, 93, 520).Trim e t hglme t hane (isoBu t an e), CH (CH,),.It has been seen that the appropriate symmetrical intercalation orexcision of methylene layers, CH,, effected in the case of a givennormal paraffin assemblage produces some other normal paraffinassemblage ; similar operations applied singly or in succession tothe tetramethylmethane assemblage just described are productiveof other assemblages appropriate to secondary or tertiary paraffins.The principle involved in such operations may be stated as follows.A regular layer of spheres, so constituted as to form the unit layerof a, closest-packed assemblage of spheres, for example, a methylenelayer, forms a, constituent of some closestrpacked assemblage.It isthen found (a) that the two parts of this assemblage obtained b2352 BARLOW AND POPE : THE RELATION BETWEEN THE CRYSTALthe excision of this layer can close up and become closesbpackedwithout any material rearrangement., and ( b ) that if, instead ofremoving this layer, the assemblage is divided so a to expose oneof the faces of the layer, a second similar layer can be fitted onto this face and then the parts fitted up so as to form a closest-packed assemblage in which the added layer is intercalated.The application of this principle depends on the property of anassemblage composed solely of such identical layers that sometranslation brings the contour of the adjoining portion of theassemblage which is fitted against one side of a single layer tocoincidence with the contour of the other side of this layer, coupledwith the fact that close-packing involves a close similarity of contourbetween all the surfaces of different sphere combinations whichdisplay the common property of fitting closely on to the samesurface.For the present purpose, isobutane or trimethylmethane,CH(CH,),, may be regarded as derived by the removal of methylene,CH,, from the tetramethylmethane molecule.The possibility ofperforming this operation symmetrically on the tetramethylmethaneassemblage, whether of the cubic form or of the tetragonal formrepresented in Fig. 34, constitutes a parallel between our methodof formulation and the chemical relationship subsisting between thetwo hydrocarbons. The process is rather simpler as applied to thetetragonal form; this case may be described as follows.One-fifth of the total number of carbon spheres in the tetriLmethylmethane assemblage are symmetrically removed, together withtwice the number of hydrogen spheres, by withdrawing every fourthlayer of the original carbon spheres taken parallel to a planeperpendicular to the diagram through a line DE in Fig. 34, togetherwith the accompanying hydrogen spheres, and closing up thestructure by bringing the exposed surfaces together.As the resultof this operation, the skeleton assemblage depicted in Fig. 35 isobtained; it will be seen that, of the carbon spheres P, Q, S, and T,and the set of four, PI, Ql, S,, and TI, making up the eight carbonspheres which together enclose a substituted carbon sphere, onlyP, PI, Q, Q1, S, and S, survive, and that the new groups form twosets oppositely orientated in the resulting assemblage. In closingup the structure after removal of the methylene layer, a lateralshift is made, such a relative disposition of the opposing boundariesbrought together being selected as brings the columns of carbonspherw at one boundary opposite to the strings of hydrogen spheresin the opposing boundary; this is possible because the central planeof the methylene layer in the original assemblage and the plane ofthe hydrogen spheres which becomes central in the modifieSTRUCTURE AND THE CHEMICAL COMPOSITION, ETC.2353assemblage are alike capable of functioning as planes of glidingsymmetry. The approximation of the portions fitted together, orthe space gained by the excision, is treated as dependent on thepostulate that the group of four hydrogen spheres occupies thesame space in the assemblage as one carbon sphere; it follows fromthis that the CH, layer occupies three-fourths of the space requiredby a, CH, layer.The method described indicates roughly the relation of theF I G .35.required assemblage to that of t,etramethylinethane, but thecharacter of the marshalling in the arrangement derived is imper-fectly defined; still less is the precise nature of the crystallinesymmetry exhibited. The absence of crystallographic data for thehalogen derivatives of isobutane leaves the aymmetry in doubt, butit is possible to assign to the marshalling of tetramethylmethane avery simple form, from which an equally simple one for trimethyl-methane can be derived.I n the absence of crystal data, much latitude is presented forthe shape taken by a given marshalling, and naturally the mar2354 BARLOW AND POPE : THE RELATION BETWEEN THE CRYSTALshalling of an assemblage can be most readily investigated whenin its simplest form.Although this form will not, in general, beFIG. 36n.FIG. 36b.FIG. 36c.the closest-packed one, itwill approximate to theclose-packed condition,and will display the pro-perty that every spherewill be in contact with,or in close proximity to,a large number of sur-rounding spheres. Now,the methane assemblage,without changing itsgeneral marshalling, cantake the form depictedin Fig. 36 a and b ; thesimplest shape of themethylene layer, CIA,, asdepicted in Fig. 11, canhere be recognised. Eachlayer of molecules, CH,,consists of the simplemethylene layer withthe additional hydrogenspheres symmetrically dis-posed in the same manneron both sides of it, asshown in the section f ;the manner in which suc-ceeding layers are fittedtogether in this simplemarshalling is indicatedby superposing b on a ofFig.36. It must beclearly understood thatthe assemblage t'hus pre-sented is not in itsclosest-packed form; it isa distortion of the closest-packed assemblage abovedescribed of such anature as to simplify t.he internal symmetry without changingthe marshalling. The methane assemblage, thus regarded,gives a configuration of the important radicle CH,, which iSTRUCTURE AND THE CHEMICAL COMPOSITION, ETC. 2355entirely in harmony with the symmetrical properties expressed bythe graphic formulae of the hydrocarbons. When it is employed asthe root methane assemblage, the marshalling of the compoundunder consideration, and that of other kindred compounds, canbe readily traced, and will be seen to be in accordance with thegraphic formulae.The first step is toderive the correspond-ing simple form ofmarshalling for tetra-methylmethane; t h i scorresponds with thetetragonal type of thecompound above indi-cated, and may beregarded as producedfrom it by a distortion.The process of deriv-ation from the methaneassemblage just de-scribed consists in sub-stihting strings ofcarbon spheres for one-fourth of the strings ofgroups of the composi-tion H,; as a result ofthis change, the arrange-ment shown by super-posing b on a of Fig.36becomes that obtainedby superposing c on aof this figure. Thelayers shown on theplane of the dia.gramare the same as thosefound parallel to aplane drawn throughDE perpendicular tothe plane of Fig.34.FTG. 36d.>ooooq)ooooq\ n n n n dFIG. 36e.FIG. 36f.Tlie marshalling for trimethylmethane is obtained directly fromthat for tetramethylmetha.ne by removing the central methyleneportion from half the unsubstituted methane layers selected sym-metrically and closing up the gaps, using the remnant hydrogenspheres in filling in between ihe parts of the assemblage which haveto be fitted together after the excision. I n the tetramethylmethan2356 BARLOW AND POPE: THE RELATION BETWEEN THE CRYSTALassemblage, layers of the composition CC + CH, (Fig. 36c) alternatewith layers of the composition ZCH, (Fig. 36a); the result of theexcision described is to give a succession of sets of layers,2CH, : CC + CH, : 4H : CC + CH,, etc., as represented roughly bysuperposing a2 c, d, and e of Fig. 36.The corresponding molecular unit is a combination of one carbonsphere from a layer a with the three carbon spheres of one of thetwo adjoining layers together with a due proportion of hydrogenspheres; the units are so constituted as to be all alike.Withregard to the positions of the planes of gliding symmetry mentionedabove, it is to be noted that in the tetramethylmethane assemblagethe gliding plane is the median plane of layer a, and in the trimethyl-methane assemblage it is the median plane of layer a or d ; theseplanes of gliding symmetry can be traced in Figs. 34 and 35respectively. As already intimated, the nature of the adjustmentof thia marshalling which would be productive of the precisecrystalline form remains unidentified owing to the absence ofcrystal data.‘I he geometrical process by means of which the trimethylmethaneassemblage can be converted into that of tetramethylmethane isanalogous to the process of preparing tetramethylmethane by theaction of zinc methyl on ten!.-butyl iodide.Dimethylcthylmethanfe (isopentune), (CH3),CH*CH,*CHI.If, in the derivation of the isobutane assemblage from that oftetramethylmethane, a layer of the general methylene composition,CH,, such as is excised from one side of the layer of tetramethyl-methane complexes, is inserted in the symmetrical position on theother side of the layer, and the requisite shift of one layer onanother made to close up the packing, the assemblage appropriateto isopentane or dimethylethylmethane results.The added layerhas thus t o be inserted on one side of each layer marked DE inFig. 35. The relative situations of the two layers, CH,, thus placedtogether are indicated in Fig. 14; the relation of the compositelayer formed to the remaining portion of the assemblage is shownin Figs. 34 and 35. The marshalling, as before, is represented in itssimplest form.Propane, CH,*CH,*CH,.I f , in addition to the excision of the layers, DE, from the tetra-methylmethane assemblage of Fig. 34 a set of layers, BAAC, parallelto them and symmetrically situated, is similarly removed and theexposed surfaces brought together as before, an assemblage isobtained which has the composition of propane. It is possible soto partition the assemblage thus obtained as to derive moleculaSTRUCTURE AND THE CHEMICAL COMPOSITION, ETC.2357units of the form already indicated for propane (p. 2334); this ismost readily shown in connexion with the original tetramethyl-methane assemblage of Fig. 34.In the propane unit described in connexion with the normalparaffins, the central carbon sphere makes four tetrahedrallysituated contacts with surrounding spheres, two with carbon and twowith hydrogen spheres, and of the four tetrahedrally situated con-tacts of each end carbon sphere, one is with a carbon and three withhydrogen spheres. Now in the tetramethylmethane assemblagereferred to, each central carbon sphere makes four tetrahedrallysituated contacts with carbon spheres, and when the withdrawal ofparallel strata occurs, two of the four outer carbon spheres and fourhydrogen spheres are removed from each molecular group.And asthe removal of the carbon spheres reveals hollows on the surfacesexposed, formerly occupied by these spheres, which, when the closingup takes place, are occupied by hydrogen spheres projecting fromthe opposing similar surfaces, it i s evident that the central carbonsphere of each group is, after the process, surrounded by fourspheres, two of each kind tetrahedrally arranged. Further, it canbe shown that each end carbon sphere, as in the tetramethylmethaneassemblage, has tetrahedrally situated contacts with a carbonsphere and three hydrogen spheres.Thus, the hydrogen spheresremaining of a methane stratum, from which the central CH, layerhas been removed, are left embedded in the two faces exposed, halfin each; they consequently retain the same positions relatively tothe end carbon spheres of the group found in the stratum to whichthey are attached. The same is true of the hydrogen spherescentrally placed in the stra.tum containing the end carbon spheres.Consequently, each end sphere of a group has the same tetrahedrallyarranged contacts with a single carbon sphere and three hydrogenspheres after the excisions are made, just as it had before. It istherefore established as above stated that the two propaneassemblages, that of f he ordinary paraffin structure and that derivedfrom the tetramethylmethane assemblage, can be partitioned intounit groups of the same form ; in other words, they are polymorphousarrangements of the same molecular groups.The geometricalprocess, inverse to that indicated above, by means of which thesecond kind of propane assemblage can be converted into that oftetramethylmethane, is analogous to Friedel and Ladenburg’sconversion of Wdichloropropane, (CH,),CCI,, into tetramethyl-methane by the action of zinc methyl.The graphic formulz for all the hydrocarbons of the generalmolecular composition CnHp12+2 can be derived from those ofmethane, trimethylmethane, and tetramethylmethane by the i n t 2358 BARLOW AND POPE : THE RELATION BETWEEN THE CRYSTALduction of methylene groups, CH,, into the formulae in all the waysconsistent with the quadrivalency of carbon.It has been shown inthe foregoing pages that close-packed assemblages of the generalcomposition of the parailins can be constructed which correspondin constitution and configuration with the normal hydrocarbons ofthe series; it has also been shown that other assemblages may bederived which possess geometrical properties exactly representativeof the simple secondary and tertiary paraffins by means of simplesubstitution processes which closely parallel the modes of preparationof these hydrocarbons. The mode in which the geometrical substi-tutions are made renders it clear that similar operations appropri-ately performed will lead to the productioii of an assemblagecorresponding in constitution with any primary, secondary, ortertiary paraffin.The conclusion must thus be drawn that thecontinued prosecution of the method described for derivingassemblages representing the paraffins must lead to a completeparallel between the possibilities of our geomet-rical method forinterpreting atomic space arrangement and the variety of chemicalprocesses of derivation which are so completely pictured with theaid of the ordinary graphic formulze:.The Olefinic Hydrocarbons.The results obtained by means of the above method of derivingclose-packed assemblages, which represent in composition, con-stitution, and configuration all the primary, secondary, and tertiaryparaffins, have been shown to accord with all the available gonio-metric data; although this evidence is small in amount, it appears tobe of a very direct character.The assemblages for the normalparaffins are characterised by being built up wholly from the generalmethylene assemblage by the intercalation of a,dditional hydrogenspheres in appropriate ways. It will now be shown that the reverseprocess, namely, the removal of hydrogen spheres from the generalmethylene assemblage, gives rise to a geometrical feature corre-sponding with the element. of chemical constitution described as anethylenic double bond. By the application of this process ofexcision to paraffinoid assemblages, fresh assemblages can be derivedrepresenting all the open-chain olefines of the general compositionCnHZn, and it will further be demonstrated that peculiarities ofconfiguration, which arise naturally during the process, representthe properties associated with the cis- and trans-isomerism of certainethylene derivatives.The formakion of an olefine may be represented by the chemicaloperation of removing two hydrogen atoms from one t,erminal carbonatom of each of two paraffin molecules, and allowing the t,wSTRUCTURE AND THE CHEMICAL COMPOSITION, EI'C.2359bivalent radicles thus obtained to condense, forming a hydrocarbonmolecule containing an ethylenic double bond ; this correspondswith the product,ion of ethylene by the action of copper on methyleneiodide, and may be thus formulated:CH$, - 21 + CH& - 21 = CH,:CH,.It has been shown that, the orthorhombic and the tetragonal formsof the general methylene assemblage are capable of interconversionFIG. 37n.FIG. 37b.by means of simple adjustment without any violent rearrangement.The described process of excision may be applied to either form,but the tetragonal one lends itself the more readily to its appli-cation; it is the one the employment of which leads to a resultthat can be checked by crystallographic data, whilst that of theorthorhombic form at present does not. The application of theprocess to the tetragonal form of the general methylene assemblagealone is given here; the treatment of the other form is notattempted, first, because it is not at the moment of practicalimportance, and secondly, because the first step in the derivationof an olefinic form from an orthorhombic paraffinoid form maypossibly consist in the passage of the latter t o the tetragonal form.The configuration of the ethylenic grouping, as it presents itselfin a homogeneous close-packed assemblage, is deduced by removingfrom a, face of each of two composite layers, CH,, of the generalmethylene assemblage of the tetragonal form, the small sphereslying in the hollows of the face, and by then keying together thetwo faces thus laid bare. The process itself parallels that whichmay be thus formulated:H*F= + ZC*H = H*C:C*H, I I 2360 BARLOW AND POPE : THE RELATION BETWEEN THE CRYSTALand is depicted in Fig.37 a and b by superposing b on a ; the sectionof the resulting assemblage of radicles through a line AB is givenin Fig.37c; it is concluded that the presence of such a doublestratum as this is characteristic of an olefine. The only additionrequisite to t,he stratum just described to produce from it theFIG. 37r.arrangement for ethylene is the appropriate insertion of spheres ofvalency volume 1 at each of its faces in the manner alreadydescribed in connexion with the normal paraffins ; the close-packedassemblage composed of the strata thus completed is obtained byarranging a succession of the strata of Fig. 37c with a layer of theF I G . 376.small spheres between each ofthem, such as is shown inFig. 37d (compare Fig. 32).In view of the comparativesimplicity of this process, itwill be convenient at once todemonstrate its application toa specific case in which aslight complicating adjustmentaccompanies the formation ofthe assemblage.Jaeger has described tetra-mdoethylene, C,I,, as crys-tallising in the monosymmetricsystem witha : b : c=2*9442 : 1 : 3.4387,/3=7O044/3O/’ (Trans., 1908, 93, 523).I n this description it isconvenient to change the indices 001, TOl, 501, i l l , and 100to 100, 203, 001, 263, and 203 respectively; the introduction ofthe factor three in this connexion seems permissible in view ofthe pronounced pseudohexagonal character of the compound, andas a result of the change the indices become more symmetricallSTRUCTURE AND THE CHEMICAL COMPOSITION, ETC. 236 Ldistributed, although numerically somewhat more complicated.' Theaxial ratios are t,hen obtained in the form:n : b : c =1.0891 : 1 : 0.7360 ; B=81"3';whence the equivalence parameters are calculated, with TV = 12, as :The assemblage appropriate to tetraiodoethylene, and also there-3.- : ?/ : z=2'689 : 2'469 : 1 817 ; /3=84"3'.FIG.38n.Fra. 3%.fore to ethylene itself, is constructed on the basis of these values inthe following manner. Carbon spheres are arranged in squareorder so as to be equidistant in t.he same plane, and nearly in contact,as in the general methylene assemblage of tetragonal form depicted inVOL. XCVII. 7 Fig. 29 ; the squares are then converted by a distortion into rhombs,the diagonals of which are in the ratio of x : y. The arrangementand dimensions of the system thus produced are indicated by thelarge circles of Fig.38a. On one face of the layer of carbon spheresare-now placed spheres of volume 1, representing hydrogenFJG. 39a.or iodine,FIG. 39b.one in each of the hollows. If the arrangement of the large sphereswere a square one, each small sphere would touch four large ones, rnshown in Fig. 37 ; as it is, each small sphere makes but three contacts,and symmetry requires that these shall be such that the centres oSTRUCTURE AND THE CHEMICAL COMPOSITION, ETC. 2363the small spheres display the same relative arrangement as the largeones, and are thus equidistant. Their situations are indicated bythe circles marked a in Fig. 38a. The principal hollows now presenton the face of the composite layer to which the spheres a have beenadded are just twice as numerous as the carbon spheres; they arenext occupied symmetrically by other spheres of volume 1, as shownby the circles b and c.The block thus obtained can be regardedas consisting of three layers of spheres, one of large and two ofsmall ones.A second block of spheres similar to the first is now formed andthe two blocks put together, so that the faces on which no smallspheres have been placed fall together and key into one another, eachlarge sphere of one block making contacts with three large spheres ofthe other block. Three contacts of large spheres are so made thatthe shift of the large spheres on the face of the block has theopposite direction to the shift of the small spheres, a, first placed onthe other face.The relative situations of the two blocks are shownby superposing Figs. 38a and 39a, when it is seen that the systemformed has digonal axes parallel to the direction y passing throughpoints of contact of carbon spheres; sections of the double layerof two blocks by two planes perpendicular to those of Figs. 38nand 39a, through the lines AB, CD and EF, GH respectively, areshown in Figs. 38b and 39b. Double blocks of the form thusobtained are fitted together in such a manner that the end layer ofsmall spheres, b and c, of one block serve ils the end layer of thonext double block; thus, if the spheres b are allotted to one doubleblock, those marked c are t o be allotted to the adjoining one.The composition of the completed assemblage corresponds with thatof ethylene or tetraiodoethylene, and its dimensions, x, y and z, arethe equivalence parameters above derived for the latter substance.It will be seen that the assemblage differs merely by it slight adjust-ment and a sheer equivalent to the angle B = 84O3' from the moresymmetrical one constructed to represent ethylene in which thecarbon and hydrogen spheres are in square arrangement.It isinteresting to note that the plane directions with the mmewha.tcomplex indices given above are found to be very important direc-tions in the assemblage when their traces are drawn on the planand section here given.The fact of the limitation of the number of derivatives of thenormal paraffins led van't Hoff to ascribe freedom of rotation to asingly bound carbon atom; a limitation of this kind does not obtainin the case of a doubly bound carbon atom, although the acceptedgraphic representation of the altered molecule and its derivativesare capable of a digonal rotation by means of which it could be7 r 2364 BARLOW AND POPE : TBE RELXTZON BETWEEN THE CRYSTALrepresented.This discrepancy points to the loss of some sym-metrical feature due to the presence of the double bond. Anexamination of the effect on a normal methylene assemblage of theremoval of the double layer of hydrogen spheres and the closing ofthe gap, which, it is suggested, expresses the change referred to,shows that a deterioration of symmetry has supervened ; strings ofcarbon spheres are not continued in the same planes across theplane at which the modification occurs, but a side shift or ‘‘ fault ”is necessitated in order that the two denuded surfaces may fitclosely together.It is suggested that the existence of this breakin the regularity of the strings of carbon spheres makes two differentorientations equally available for the portion of an assemblage lyingbeyond the surface of modification; in other words, whilst in thecase of a normal paraffin assemblage of the tetragonal form theaddition of a, layer, in order to give equilibrium, must be so per-formed that the joined layers have the same orientation, in thecase where a double or ethylenic bond is present, it is equallyf avourable to equilibrium whether the two layers coming togetherhave the same orientation or differ in orientation by 90°.Theparallel which thus exists between the properties of close-packedassemblages and the occurrence of cis- and trans-isomerides amongstthe derivatives of ethylene will be treated immediately in connexionwith the isomerides of the composition C,H,.It should be remarked that the existence of the alternative re-ferred to for the attachment of an added layer is inoperative in thecase of ethylene, and in the cases of compounds equally symmetrical,such as tetraiodoethylene; in all such cases the two resulting formsbecome identical, and thus indicate, as they should do, that thereis but one kind of molecule. The alternative becomes operative,however, in some derivatives of this simple form, since partial sub-stitution of the hydrogen spheres allows two kinds of partitioningto be discriminated which are not interchangeable without re-marshalling.The effects of the presence of the structure repre-sented by the ethylenic bond, as exhibited in cases of homologues ofethylene, will now be traced.The Honaologues of Ethylene.After having traced a peculiarity of geometrical structure in aclose-packed assemblage which corresponds with the double bondpresent in the ethylene molecule, it is desirable to ascertain whetherthe introduction of this geometrical feature into a paraffinoidassemblage leads to the production of the type of assemblagespecifically associated with the presence of an ethylenic bond, Forthis purpose, i t is convenient to consider the four butylenes, C,H,STRUCTURE AND THE CHEMICAL COMPOSITION, ETC.2365which may be regarded as derived from butane and isobutane bythe removal of two hydrogen atoms from the molecule; thesesubstances are (1) cis-s-dimethylethylene, c%>C:C<&H$, (2) trans-Hs-dimethylethylene, c%>":c<cH , (3) ethylethylene (butylene),CH,*CH,*CH:CH,, and (4) us-dimethylethylene (isobutylene),(CH,),C:CH,. Further, in order that the possibilities of close-packing under the specified condition of unit composition may beexhausted, as also those of isomerism of the molecular compositionC,H,, it will be convenient to discuss the assemblages representativeof the remaining two hydrocarbons of the latter composition,3namely, (5) tetramethylene (cyclobutane), CH,<CH2>CH,, and (6)c=2inethyltrimethylene (methylcyclopropane), CH,*CH<FH2CH,'It will now be shown that the process of excising hydrogen sphereswhich has been applied above for the purpose of deriving theassemblage representing ethylene can be applied in three differentways to the normal butane assemblage; these lead to the productionof three distinct assemblages, which represent the butylenes, num-bered (l), (Z), and (3).A similar process applied to the isobutaneassemblage leads to the formation of an assemblage, which representsthe asymmetrical dimethylethylene, numbered (4). I n connexionwith the analogy existing between the geometrical mode of repre-sentation employed and the possibilities of chemical isomerism, it willbe shown that these four methods of applying the process of excisionare the only ones that lead to the production of the geometricalpeculiarity of structure corresponding with the presence of anethenoid double bond ; three other modes of excision are, however,also applicable, two to the butane assemblage, and one to that ofiaobutane; all of these result in the formation of assemblages whichdo not contain the ethenoid peculiarity of geometrical structure.Of the latter assemblages, two are identical and represent methyl-trimethylene (6), which is derived both from the butane andthe isobutane assemblage; the remaining assemblage is that of (5),tetramethylene, and is derived only from the butane assemblage.Since assemblages representative of the four homologues of ethyleneand the two polymethylenes, which constitute all the isomerides ofthe composition C,H,, are derivable from those representing theonly two hydrocarbons of the composition C4Hlo by processes entirelyanalogous to the chemicaI methods of preparing the former hydro-carbons, strong confirmation is afforded of the general accuracy ofthe mode of formulation now put forward.These cases illustratein a striking manner the geometrical property to which has bee2366 BARLOW AND POPE: TEE RELATION BETWEEN TEE CRYSTALascribed the persistence of the tetrahedral disposition of the atomiclinks when substitution occurs ; they consequently throw light onthe precision with which the ordinary formulz indicate the numberand nature of the isomerides obtainable in any particular case,The three modes of excision applicable to the assemblagerepresenting normal butane for the purpose of derivingassemblages representative of the butylenes (I), (a), and (3)are the following.I n a stratum of the normal butaneassemblage of the tetragonal form, four attached layers ofthe composition CH,, and of the square configuration, are present ;to each of the terminal faces of the block an appropriateset of hydrogen spheres has been added, as already explained(p. 2336). The stratum indicated is first divided at the median plane,so that each half consists of t*o layers of the form CH,, to one ofFIG. 40n. FIG. 40b.which an additional set of hydrogen spheres has been added.Fromthe surface of each of the two halves exposed by the separationthe small spheres are now removed, and the two halves are thenrefitted together, making the large spheres of one face fit into thehollows of the other ; this operation may be performed in two ways,one being represented by superposing b on a, and the other bysuperposing c on a of Fig. 40. In these diagrams the groups fittedtogether are of the form H, all the hydrogen spheres attachedto the outer layer of carbon spheres-those representing the methylhydrogen atoms-being omitted for the sake of clearness. Thesuperposition of b on a gives the assemblage corresponding withcis-s-dimethylethylene, and that of c on a the assemblage for thetrans-isomeride.No hydrogen spheres lie between the two denudedlayers of carbon spheres fitted together as described, and the hollows-c-STRUCTURE AND THE CHEMICAL COMPOSITION, ETC. 2367on the denuded surfaces from which the hydrogen spheres have beenremoved are now occupied by the carbon spheres of the opposingface. The geometrical operation which has thus been performedupon the normal butane assemblage may be roughly representeddiagrammatically by the following scheme :The assemblage of (3), ethylethylene, is derived from that ofiiormal butane by dividing the latter at the place of either the firstor third linking so as to give as one segment a single layer of theform CH,, with its a.dditiona1 small spheres attached to one faceonly, and, as the other seg-ment, a block or stratum ofthree layers, CH,, with theadditional small spheresattached to one of its boundaryfaces.As before, the smallspheres are removed from allthe hollows of t.he two facesexposed by the separation, andthe surfaces are then refitted;this operation can only be per-formed in one way withoutdestroying symmetry, or rather,the two most symmetrical waysof fitting the strata closelytogether to produce a con-FIG. 40e.tinuous assemblage give identical results.The assemblage for as-dimethylethylene (4) can be obtained fromthat of isobutane, but is most conveniently derived from that oftetramethylmethane, C(CH,),, already described (Fig. 34), byremoving from the latter one-half of the layers which have thecomposition H4C, namely, either those whose median planes passperpendicularly to the plane of the diagram through all the diagonallines BC, or those whose median planes pass through all the linesDE.On removing the strata thus indicated and then refittingthe denuded surfaces, a general arrangement is attained of whichthe large spheres alone are represented in Fig. 41; in this figure theunremoved layers of the composition H4C are marked DE.The three modes of excision which do not lead to the productionof the geometrical feature corresponding with the ethenoid doublebond remain to be dealt with; that which yields the assemblag2368 BARLOW AND POPE : THE RELATION BETWEEN THE CRYSTALCH corresponding with (5), tetramethylene, CH2<CH2>CH2, is next2described.The assemblage for this hydrocarbon i s derived fromthat of normal butane of the tetragonal form (Fig. 31) by removingthe layers of hydrogen spheres which correspond with the terminalatoms in the paraffin chain, thus obtaining strata of the methyleneform of assemblage, each consisting of four tetragonal layers ofcarbon spheres and the accompanying hydrogen spheres. Eachstratum is next adjusted so as to bring its constituent carbon spheresclose together in fours, as indicated diagrammatically in Fig. 42a and b ; the planes of these sections are perpendicular to those ofFIG. 41.the tetragonal layers. The grouping is so performed that each fourtetragonal layers of carbon spheres and attendant hydrogen spheresfurnish two layers of tetramethylene complexes, as indicated bya and b.A single tetramethylene complex, plan and elevation ofwhich are shown in Fig, 43 a and b , consists of four large spheres inthe same plane and eight small spheres, four in each of two planesparallel to and equidistant from the plane of the carbon spheres.The two remaining modes of excision applicable as already inti-mated (a) to the butane and ( h ) to the isobtitane assemblage, leadto the production of the same molecular unit, that which representsmethyltrimethylene ; the former is applied t.0 the normal butaneassemblage of the orthorhombic form in the following manner.( a ) Alternate layers of the small terminal spheres of the normaSTRUCTURE AND THE CHEMICAL COMPOSITION, ETC.2369butane assemblage are removed as if for the purpose of condensingthe strata, in pairs so as to form strata of the normal octaneFIG. 42b.assemblage, and each mutilated butane stratum is divided sym-metrically into blocks of the composition CIA, and 3(CH,) respec2370 BARLOW AND POPE: THE RELATION RETWEEN THE CRYSTALtively. The latter blocks are next so adjusted that, as condensedtogether two by two, they form aggregates of trimethylene complexesarranged as is indicated by superposing a and b of Fig. 44; eachstratum consists of six layers of large spheres as shown in thefigure, and, as in the previous case, each small circle represents twospheres. Finally, one-half of the hydrogen spheres are removedsymmet.rically from the outer layers of the double stratum justdescribed, and the outer layers are refitted to the CH, layers fromwhich they were separated, the projecting carbon spheres of oneface of the latter being allowed to fall into the hollows vacated bythe hydrogen spheres last removed ; simultaneously, the small spheresof the added CR, layers are adjusted for close-packing.The preciseeffect of the last process is difficult to trace, but its practicability isindicated by the facts that when the refitting has taken place, eachsphere of the altered trimethylene layers is still immediately sur-FIG. 43a. PIG. 43b.rounded by the large number of spheres appropriate for very close-packing, and that no destructive deformation of the original CH,arrangement is necessitated. The refitting of the layers is repre-sented by the equation : 3(CH,) - H + CH, = CH,.C€I<ctl 9 andthe assemblage attained is, as indicated by its constitution, composedof trimethylene complexes, from each of which a hydrogen spherehas been removed, grafted on to modified methyl complexes.( b ) A different assemblage of the same trimethylene complexes isderived from that of trimethylene by an appropriate excision ofhydrogen spheres in the manner next described. I n Fig. 35, repre-senting the arrangement of the ca.rbon spheres in the trimethyl-methane assemblage, cavities and portions of cavities, which liebetween two planes of which the traces are marked FG, IIK, areavailable for the reception of hydrogen spheres; the central entirecavities are each of the magnitude requisite for the reception ofy I€,STRU OTURE AND THE CHEMICAL COMPOSITION, ETC.2371four hydrogen spheres, and the portions or half cavities have halfthis magnitude. Whilst maintaining the same configuration of the -groups of four largespheres as is indicated inthe figure, the two OPPD-site sets are now allowedto approach until thecavities and portions ofcavities afford only one-half the previous accom-modation for hydrogenspheres; the effect ofdiminishing the accom-modation offered by thecavities and portions ofcavities to one-half theoriginal amount is indi-cated in Fig. 45. Finally,without re-marshalling,such mutual adjustmentsare conceived as willadapt the various cavitiesbetween the large spheresto the close-packing ofthe appropriate numbersof hydrogen spheres ;these adjustments willresemble, in general,those previously de-scribed in connexion withsimpler cases.The unaltered mar-shalling of the largespheres in each moleculargroup which is thus pre-scribed, and the advance-ment of large spheresto occupy cavities leftvacant by the removal ofthe small ones, is fittinglyrepresented by the transi-tion from the graphicFIG. 44n.FIG.44b.formula of triniethylmethane to that of methyltrimethyleneimmediately accomplished by removing a hydrogen atom from eac2372 BARLOW AND POPE : THE RELATION BETWEEN THE CRYSTALof two neighbouring branches of the molecule of t,he former, andrestoring the linkage equilibrium by adding a link between thecarbon atoms concerned, thus :I n connexion with the two methods just described for derivingFIG.45.an assemblage representing methyltrimethylene, it is suggested thatthe course followed involves the initial formation and the subsequentpreservation of an approximately regular tetrahedral dispositionof the contacts of the carbon spheres with other spheres of the samemolecular unit ; the functional identity of the molecular units hereindicated will necessitate that the two assemblages should beregarded as an example of polymorphism.The arguments stated and the data given in the previous pagesindicate that an ethenoid compound is derived from the correspond-ing paraffinoid compound by the removal of the requisite proportionof hydrogen spheres from the assemblage, and the subsequent con-traction necessary to restore close-packing to the assemblage.Itwould consequently be anticipated that pairs of substances ofcomplex molecular constitution, the members of which differ in thata paraffinoid element, *CH,-CH,*, in the one is substituted by aSTRUCTURE AND THE CHEMICAL COMPOSITION, ETC. 2373ethenoid element, *CH:CH-, in the other, might afford frequentinstances of intimate crystallographic relationship ; in such cases,the substitution just mentioned should occur in such a manner thatthe closing up of the assemblage would be manifested by a contrac-tion in the directicn of one of the properly selected equivalenceparameters. Examples of this kind are not unconimon, and onetnav be here auoted.J aAcenaphthene, CloH6<XE: is morphotropically related toacenaphthylene, (Billows, Zeitsclt, Kryst. Mirt., 1903,37, 396); both belong to the orthorhombic system, and t,he axialratios are respectively :Acenaphtheue, n : b : c-0'5903 : 1 : 0.5161.Acenaphthylene, n : Z, : c=0*5926 : 1 : 0.4996.The equivalence parameters corresponding with these axial ratliosm e :Acenaphtiieno, x : y : 2=3'3959 : 5.7527 : 2.9689.Acenaphthylene, 2 : y : z=3'4017 : 5'7404 : 2.8678.Differences - 0'0058+0~0123+0'1011.It is observable that the differences between correspondingequivalence parameters for the two substances are negligibly smallin the directions x and y, but appreciably large in the directionof z ; in this case the replacement of the para.ffinoid by the ethenoidstructural element has involved a closing up of the crystallineassemblage, which is only experimentally appreciable in the directionof the c or z axis.An equally striking example is afforded by themorphotropic relationship observed between dibenzyl and stilbene,which is referred to later (p. 2379).W = 5 8117=56.The Acetylene Series, C1LH2n-2.The space formation of spheres, which corresponds with thecharacteristic acetylene grouping, -CiC*, is rather more difficult totrace than that representing the double linking present in ethylene,but owing to the extreme simplicity of the molecular compositionof the first member of the acetylene series, the possibilities to besubmitted to consideration are so limited in number thatexamination of the arrangement to be described, which fulfils someof the principal conditions requisite, leaves practically no doubtthat it is the one sought.The assemblage representing acetylene itself is derived from aclosest-packed cubic assemblage of spheres, each of the volume 4(Fig. l), in the following manner.Each of the large cavities inthe assemblage, which are found at the centres of groups of si2374 BARLOW AND POPE: THE RELATION BETWEEN THE CRYSTALspheres and are just its numerous as the spheres, is occupied bya smaller sphere of such magnitude as just to touch the six spheresenclosing the cavity; the assemblage represented in Fig. 46 is theresult. The smallerspheres which can thusbe inserted are of amagnitude less thancorresponds with thevalency volume 1, andaxe next to be ex-panded until theyattain this volume.I f the relativearrangement of theFIG.46.FIG. 47.centres of both kindsremained unchangedduring t,he expansion,i t is evident that allthe contacts betweenthe larger sphereswould be broken, asshown in Fig. 47, butit is possible for thelarge spheres to pre-serve some of theircontacts, notwithstand-ing the presence ofthe smaller spheres, ifslight mutual adjust-ment occurs. Theassemblage can be re-garded as composed oflayers the planes ofwhich are parallel to aplane drawn throughopposite edges of acube of the fundacmental space-lattice ;such a layer, viewedprior t o the adjust-ment, is representeddiagrammatically by the continuous lines in Fig.48. Let thereforeeach larger sphere continue in closest contact with one other sphereof its own size, and let the partners be so selected that the contactSTRUCTURE AND THE CHEMCAL COMPOSITION, ETC. 2375of sphere with sphere are distributed through space as evenly aspossible. The latter condition is fulfilled if the points of contactare approximately at the centres and angles of a cubic partitioningof space such that the length of a cube edge equals the translationof the assemblage along the directions in which carbon and hydrogenspheres are found placed alternately in contact. The assemblagethus modified consists of groups of a composition correspondingwith acetylene, C,H,; the projections of the points at which thelarge spheres come into cont.act when the adjustment occurs aremarked A in the diagram.The geometrical properties of theadjusted assemblage thus constituted parallel the chemicalbehaviour of acetylene, as will be perceived below in connexion withFIG. 48.the homologues of the hydrocarbon. Owing to the fact that nocrystallographic data are available for directly checking thegeometrical results, the exact nature of the processes by which thesucceeding terms of the acetylene series are arrived at, and theprecise forms of the assemblages, are difficult to determine; thefollowing is put forward, however, as the probable mode of derivingthe correct form of assemblage for methylacetylene (allylene),C H, C i CH .Methylacetylene ( A llylene).From one of the two similar faces of a layer of complexes suchas is depicted in Fig.48, the small spheres axe removed, so that, theresidue represents the radicle, *CiCH ; the stratum is then slightlydistorted by diminishing its thickness and consequently increasingits face dimensions so its to make the latter equal to those of a laye2376 BARLOW AND POPE : THE RELATION BETWEEN THE CRYSTALof the unadjusted orthorhombic ethane assemblage, Fig. 49b. Onthe denuded face of a stratum of this kind, which is shown inFig. 49a, is next grafted a stratum of methyl complexes (Fig. 49b),namely, a half stratum of the ethane assemblage (Fig. 23); theprocess is represented diagrammatically hy superposing Fig.497,on Fig. 49u. The composition of the resulting compound stratumFIG. 49b.is represented by the formula CH,*CiCH, and the correspondingequilibrium assemblage will consist of a number of such strata keyedinto one another ; succeeding strata will be oppositely orientated,and the contacts between them will consequently be of two differentkinds occurring alternately. *One kind will resemble the terminalcontacts found in the acetylene assemblage, as above presentedSTRUCTURE AND THE CHEMICAL COMPOSITION, ETC. 2377and the other, those found in the normal paraffin assemblage of theorthorhombic form. As in most. of the preceding cases, themarshalling described is represented in a simple form, and theFIG. 50a.FIG.50b.assemblage must therefore be conceived to be subjected to suchadjustmentl as is requisite t o produce closest-packing of this mar-shalling. It is important to observe that the assemblage justVOL. XCVII. T 2378 BARLOW AND POPE : THE RELATION BETWEEN THE CRYSTALdescribed can be distorted so as to assume a tetragonal form withoutany considerable alteration of the environment of the constituentatoms; perhaps the simplest way of describing the result is to buildup an assemblage from two kinds of tetragonal strata representingrespectively the acetylenic remnant, *C ICH, and the methyl radicle,*CH,. The former is obtained from a tetragonal acetylenic layer(Fig. 47) by removing the hydrogen spheres from one of its faces,as shown by Fig.500, whilst the layer representing methyl is derivedfrom the tetragonal ethane assemblage (Figs. 29 and 30), and itsprojection is represented diagrammatically in Fig. 50b. It isremarkable that in putting the two layers together the terminalface of the ethane stratum has in this case to be turned towardsthe denuded a.cetylene face, so that what was the inner face is nowoutwards; this is represented by superposing b on a of Fig. 50. TheFIG. 51. FIG. 5 2 .methylacetylene strata, CH,*C i CH, its before, present oppositeorientations in succeeding layers ; in the tetragonal asemblage nowbeing described, junctions, such as occur in acetylene, alternate withjunctions having much the same marshalling as those of tetr%methylmethane strata (p. 2348).The diagrams merely indicate themarshalling, and minet be supposed subjected to adjustment whichrenders the packing closer.The assemblage representing the next homologue of the series,ethylacetylene, CH,*CH,*CiCH, results from employing a butanestratum instead of an ethane one in the geometrical process describedabove.It has been shown in the foregoing pages that the geometricalmethod indicates that the conversion of a paraffin assemblage intothat of the corresponding olefinic and acetylenic hydrocarbon occursby the excision of lrtyers of hydrogen spheres, appropriately selected,in such a manner that the dimensions of the residual paraffinoidradicle suffer but little change. The paraffinoid, olefinic, oSTRUCTURE AND THE CHEMICAL COMPOSITION, ETC.2379acetylenic assemblage can, in fact, be regarded as consisting of aparaffinoid assemblage into which have been appropriately inter-calated layers of the composition *CH,*CH,*, *CHiCH*, or *C:C*respectively, these layers being dimensionally compatible with theparaffinoid part of the assem-blage. I n this dimensional com- FIG. 53.patibility between the threeelements of constitution justmentioned is found a completeexplanation of the frequentoccurrence of close morphGtropimc relationships betweencorresponding paraffinoid, olefinic,and acetylenic compounds, andfor which no cause has hithertobeen traced. For the presentit will suffice to quote theaxial ratios for the monosymmetric dibenzyl, stilbene, and tolane asillustrating the relationship referred to (Boeris, Zeitsch.X;_'?.gst. illin..,1901, 34, 298):a : b : c . P.Dibenzyl, Ph'CH;CII2*P1i ............ 2.0806 : 1 : 1.2522 115"54'Stilbeiie, Ph*CH:CEI*Ph.., ........... 2 1701 : 1 : 1'4003 114 6Tolane, Ph*CiC'l'h .................... 2.2108 : 1 : 1.3549 115 1T h e Conversion of 9 cetylene De?.ivatiues i n t o AromaticHydrocarbons.The assemblages just put forward as representing acetylene andits homologues have been constructed in accordance with the prin-ciples of close-packing applied in numerous other cases, the existenceof this condition consisting in the large number of spheres in contactwith, or close proximity to, each sphere. Although uncorroboratedto any considerable extent by crystallographic evidence, the correct-ness of the representation is strongly supported by the fact thatthe tetragonal assemblages described immediately provide amechanism which can be shown to illustrate the well-known con-version of acetylene derivatives into aromatic hydrocarbons.The acetylene assemblage has been derived from the closest-packed'assemblage of equal spheres of the valency volume 4 by forcingspheres of valency volume 1 into the interstices which occur at thecentres of close octahedral groups of six spheres and are as numerousas the large spheres.Before the insertion of the smaller spheres,the assemblage can be regarded as composed of identical groups ofsix spheres, and similarly, after the insertion of the hydrogen spheres,as built up of t,he composite groups of twelve spheres, six large and7 Q 2380 BARLOW AND POPE : THE RELATION BETWEEN THE CRYSTALsix small, depicted in Fig.51. The latter groups, however, can bereadily distorted into groups of the benzene configuration shownin Fig. 52, which has been previously described (Trans., 1906, 89,1693), and is shown in rough perspective in Fig. 53; this is easilyseen by compa,ring Figs. 51 and 52. The acetylene assemblage ofFig, 48 can, consequently, be regarded as built up of units of thecomposition C,H,, each representing an acetylene molecule, or asconsisting of units of the composition C,H,, each of which representsa distorted benzene complex.The choice between the two kinds of partitioning can be expressedby the number of large spheres in immediate contact; if these touchtwo by two, while the pairs are not in contact but are kept apart bythe smaller spheres, the acetylene grouping is indicated, but if thelarger spheres form groups, each containing six carbon spheres inring contact, each sphere of a layer of three being attached to twospheres of the neighbouring layer of three, the benzene configurationis portra.yed.It is conceivable that the simpler form of groupingmay give closest-packing when the ratio of the sphere magnitudes lieswithin certain narrow limits, and that when these limits areexceeded in one of the two directions, the other form of groupingmay be brought about; the polymorphous forms of assemblage forbenzene previously described would then follow from further changeof the conditions.The passage from one grouping to the otherabove indicated will thus occur at some critical condition of tem-perature or the like; it involves no re-marshalling but some slightadjustment of the relative positions of the two sizes of spheres.The slight adjustment which is thus requisite to the conversion ofthe acetylene assemblage into an aggregate of units having thebenzene Configuration is the geometrical analogue of the conversionof acetylene into benzene by heat.The acetylene assemblage, in accordance with our previous results,must be regarded as practically identical in form and relativedimensions with the assemblages of the halogen derivatives of thehydrocarbon; by replacing each hydrogen sphere in it by an iodinesphere of approximately the same valency volume, the assemblagerepresenting the crystalline di-iodoacetylene would be obtained.Von Baeyer's observation (Ber., 1885, 18, 2269), that di-iodo-acetylene, C,I,, is converted into hexaiodobenzene by slight warmingor the action of light, is in complete accordance with this.A similar kind of mechanism elucidates t'he polymerisation whichoccurs when monoido- or monobromo-acetylene is preserved in thecrystalline or dissolved state, and which leads to the production ofthe 1 : 3 : 5-tri-iodo- or tribromo-benzene respectively (von Baeyer,Zoc. c i t .) . For the representation of these changes, the hydrogeSTRUCTURE AND THE CHEMICAL COMPOSITION, ETC. 238 1spheres on one side only of the stratum of acetylene compIexes arereplaced by iodine or bromine spheres of approximately the samevalency volume as hydrogen; the distortion of the assemblage formonoiodo- or bromo-acetylene thus obtained leads to the re-partitioning in the sense of the equation:3CH iC!I = C,H,T, (1 : 3 : 5 ) .The compound tetragonal assemblage for allylene, CH,*C;CH,depicted in Pig.49, was obtained by the intercalation of two kindsof strata, those, namely, of the acetylene and methane radicles.Polymerisation strictly corresponding with that observed is broughtabout geometrica.lly by altering the acetylene stratum just as in theprevious case; the three carbon spheres of a newly constitutedbenzene complex thus formed, which lie in one of the two planes,become respectively attached to three methyl groups of the adjoiningstratum so as to give the symmetrical constitution of mesitylene.I n the derivation of the assemblage for allylene, the hydrogenspheres on one side only of the acetylene layers were replaced by alayer of methyl complexes; if the hydrogen spheres on both sides ofeach acetylene layer are replaced by layers of methyl complexes, theassemblage produced represents that of the symmetrical dimethyl-acetylene (crotonylene), CH,*C iC-CH,.The passage by slight dis-tortion of the acetylene layers to the benzene configuration involvesthe conversion of each set of three dimethylacetylene complexes intoone molecular complex of hexamethylbenzene in a manner preciselyparalleled by the observed facts.If oxygen spheres are introduced into the allylene assemblagedescribed above in the proportion of one for each acetylene unitpresent, and two hydrogen spheres are simultaneously introducedfor each oxygen sphere, in accordance with the second geometricalproperty, an assemblage is obtained which has a constitution corre-sponding with that of acetone, CH,*CO*CH,.The removal of theelements of water from this assemblage, so as to convert it into theallylene assemblage, accompanied by the slight distortion whichcauses the assemblage to pass to the benzene form, correspondsgeometrically with the observed conversion of acetone into mesityleneby the dehydrating action of sulphuric acid. The following methodmay be applied to the production of an assemblage for acetone, inwhich the process referred to can be readily traced.The general methylene assemblage described above (p.2326) canbe distorted to an acute rhombohedral form without sensibly affect-ing the closeness of the packing; the volume of each rhombohedralunit cell, like that of the orthorhombic cell before described, isthat proper for one radicle unit, CH2. The corners of the cell areoccupied by the centres of carbon spheres, and the cell contains a2382 BARLOW AND POPE : THE RELATION BELWEEN THE CRYSTALits centre a pair of hydrogen spheres, the pairs being similarlyorientated. The resulting assemblage can be traced in the followingsimple manner. The assemblage may be regarded as consisting ofsimilar layers of the same composition as the layers depicted inFig.19, the planes of centres being parallel to a plane drawnthrough the axis of a cell to contain one of the rhombohedra1 edgeswhich intersect this axis; a constituent layer can be derived ifthe configuration of a single unit cell is ascertained in the followingmanner .Two circles, centred a t A and C, of which the diameters are inthe ratios of those of the carbon and hydrogen spheres, are drawnin contact (Fig. 54); on the line joining the centres, AC, a semi-circle, AVC, is erected, and AC is produced to cut the smallercircle in L. The line LH is drawn perpendicular to AL, AL istrisected in D, and DV isdrawn perpendicular to AL tointersect the semicircle in V,A and V are joined, and thejoin produced to cut LH in 13;VH is then bisected in B, andwith B as centre a circle equalto circle A is drawn.Since AD=1/3 AL, AV=1 13 AH = VB ; theref ore, sinceAVC is a right angle, AC=CB, and circle C, whicht,ouches circle A, dso touchescircle B.L is now used as acentre of symmetry, aboutwhich points, lines, and circlesFIG. 54.W are symmetrically repeated, asahown in the figure; theparallelogram ABFG is the section of a rhombohedron ofthe form required, and the circles centred a t A, By F, andG are the sections of carbon spheres of which the centres lie atthose angular points of the rhombohedron which are intersectedby the plane of the section depicted; the small circles indicated arethe sections of hydrogen spheres enclosed, each having contactwith the other and with three carbon spheres lying on one sideof L in a plane perpendicular to AL as well as with one carbonsphere of which the centre lies on this line.For, if the line ALFbe made a trigonal axis, and by successive rotation through 120°about it, four other points are located from the points B and G,while points A and F remain unmoved, the eight points thuSTR UCTURE AND THE CHEMICAL COMPOSITION, ETC.FIG. 55.2383FIU. 56a2384 BARLOW AND POPE : THE RELATION BETWEEN THE CRYSTALFIG. 56b.FCG. 56c--------00I 1 STRUCTURE AND THE CHEMICAL COMPOSITION, ETC. 2385indicated are connected by the property that the six points noton the axis are equidistant from it and lie three and three ontwo planes which trisect the axis AF, a property characteristic ofthe angular points of a rhombohedron.The proof of the existenceof this property lies in the fact that the traces of the planes referredto, being the horizontal lines through B and G, respectively trisectthe semi-axes AL and FL.When space has been partitioned by means of three sets of parallelequidistant planes into rhombohedra of the form indicated, andcarbon spheres have been placed with centres at all the angles,while pairs of hydrogen spheres occupy the cell centres, each largesphere is in contact with eight small spheres and almost in contactwith six large spheres, while each small sphere touches four largespheres and oneasmall one. This kind of arrangement can betraced in Fig.55, which shows layers comtructed according to thecell conditions derived above ; the continuous and discontinuouslines respectively depict projections of succeeding layers.I n order to derive from the methylene assemblage of the formthus described an assemblage for acetone, CH,*CO*CH,, three suc-ceeding layers are taken, and an oxygen sphere substituted for eachpair of hydrogen spheres of the central layer pf the three; thiscan be done without any sensible change in the form of therhombohedra1 cell. A number of sets of three layers thus modifiedmust then have hydrogen spheres added, just as in the case of thepreviously described methylene assemblage, so as to produce stratawhich appropriately represent the formula of acetone ; the resultingassemblage in its most symmetrical condition is indicated by super-posing b on a of Fig. 56, a representing the two layers, CH, andCO respectively, with the terminal hydrogen spheres added, and Z,representing a single layer, CH,, with its terminal hydrogenspheres.Geometrical complexes of the composition OH, can be readilyrecognised in the assemblage described, each oxygen sphere beingnearly in contact with four hydrogen spheres, two above and twobelow the plane of the layer. The withdrawal of the pairs ofhydrogen spheres of one of the layers simultaneously with the oxygenspheres of the next layer, with which they are nearly in contact,deprives the middle carbon layer of all its smaller spheres, andleaves but one hydrogen sphere, namely, the end one, to be allottedto each of the carbon spheres of an end layer; this, when theassemblage is closed up after the extraction of the OH, complexes,leads to the marshalling above deduced for allylene, CH,*CiCH, andsubsequently the deformatioii already referred to converts theallylene assemblage into that proper for meeitylene (p. 2381)2386 BAKLOW AND POPE : THE RELATION BEI'WEEN THE CRYSTALA mechanism similar in kind to that explained and illustratedabove is apparently applicable to the more complex cases presentedby the various pyridine, quinoline, and quinaldine syntheses.The geometrical simplicity of the operation by which the elementsof water can be introduced into an acetylenic assemblage is com-pletely paralleled by the ease with which acetylene, CIiIiUH, reactswith water under the influence of a catalyst to yield acetaldehyde,CH,*CHO. I n order to represent the assemblage correspondingwith the latter substance, two of the layers of a methylene assem-blage of rhombohedra1 form, as just described, are requisite, thepairs of hydrogen spheres in one of the two layers being exchangedfor single oxygen spheres. The appropriate addition of hydrogenspheres at the opposite faces completes a stratum correspondingwith the constitution of acetaldehyde; the latter is represented bysuperposing c on u of Fig. 56.I n connexion with the distorted configuration (Fig. 51) justdescribed as an intermediate form displayed by the benzene complexduring ibs production from acetylene, it is significant that unitshaving this configuration can be fitted together to make an a.ssem-blage which is compatible with the axial ratios and crystalline formof benzene itself, but which is not identical with the benzene assem-blage previously described (Trans., 1906, 89, 1694, Fig. 3). ThSTRUCTURE AND THE CHEMICAL COMPOSlTION, E'L'C. 2387somewhat remarkable fact that the same spheres present in thesame proportions are capable of two widely different arrangemeiitspresenting almost the same crystalline form and axial ratios istraceable to the spheres occurring in continuous strings in two ofthe three axial directions; thus, in the earlier assemblage justreferred to, the large spheres range in contact in the direction ofthe axis 9, and large and small spheres alternately range in contactalong the direction of the axis z . The dimensions y and z closelycorrespond, in fact, with twice the diameter of a carbon sphere andto the sum of the diameters of a carbon and a hydrogen sphererespectively ; any assemblage in which carbon and hydrogen spheresrange in this manner along directions of translations will con-sequently be morphotropically related to the benzene assemblagepreviously described. The assemblage of the distorted units ofFig. 51 is shown diagrammatically in Fig. 57, in which each ofthe parallelograms indicated marks the projection of the centreportion of a column of benzene complexes of which the axis isperpendicular to the plane of the figure; the sphere centres of eachsingle molecular unit are projected on the outline of some one suchparallelogram. The partitioning of this diagrammatic assemblageinto molecular complexes can occur in several different ways, whichproduce identical results ; the parallelograms are drawn appro-priately for two of these ways, the column indicated by a singleparallelogram being divisible into molecular units in two ways(compare Trans., 1906, 89, 1696). The values of z and y used inthe construction of Fig. 57 are those calculated for crystallinebenzene, in whichx : y : z=3.101 : 3.480 : 2.780.The assemblage shown in Fig. 57, being unadjusted, is equallyapplicable to acetylene and benzene; it shows neither of the kindsof coalition of spheres to form a complex which have respectivelybeen described as productive of molecular units proper to thesecompounds. This accounts for the marked flattening of the sphereswhich is found to be necessary in order that they may be packedinto the space accorded by the benzene valency volume of IY=30.I n three directions in the assemblage, namely, one perpendicular tothe plane of the diagram, and two, those of the diagonals MN, PQ,the large and small spheres alternate. I,n the diagram they arerepresented as precisely in line, but this will not be strictly thecase, especially along the directions of the diagonals ; increasedcloseness of packing, and therefore less flattening, will result froma slight zigzagging of the positions of the two sets of centres of thesame string.The conclusion referred to, that continuous strings of spheres ar2388 GREEN AND WOODHEAD: ANILINE-BLACK BNDpresent in which the two sizes alternate, throws light on thenumerous cases in which the value of the 2; axis for benzene is veryapproximately presented as one of the equivalence parameters ofbenzene derivatives ; many such instances have been recorded byJerusalem (Trans., 1909, 95, 1275) and by Armstrong (this vol.,p. 1578).UNIVELSITY C H EM IUA L LA BOILATOIIY,C A JI B HI DG E