T H E J O U R N A L O F
`
`P H Y S I C A L C H E M I S T R Y
`
`Registered in U . S. Patent Ofice @ Copyright, 1964, b y the American Chemical Society
`
`VOLUME 68, NUMBER 3 MARCH 16, 1964
`
`van der Waals Volumes and Radii
`
`by A. Bondi
`
`Shell Development Company, Emeryville, California
`
`(Received August 6 , 1965)
`
`Intermolecular van der Waals radii of the nonmetallic elements have been assembled into
`a list of “recommended” values for volume calculations. These values have been arrived
`at by selecting from the most reliable X-ray diffraction data those which could be reconciled
`with crystal density a t OOK. (to give reasonable packing density), gas kinetic collision cross
`section, critical density, and liquid state properties. A qualitative understanding of
`the nature of van der Waals radii is provided by correlation with the de Broglie wave length
`of the outermost valence electron. Tentative values for the van der Waals radii of metallic
`proposed. The paper concludes with a list
`elements-in metal organic compounds-are
`of increments for the volume of molecules impenetrable to thermal collision, the so-called
`van der Waals volume, and of the corresponding increments in area per molecule.
`
`Table of Nomenclature
`Surface area of molecules (based on model of Fig. 3) per
`mole (em. */mole)
`Covalent bond radius, d.
`Kormalization constant in eq. 1
`Nonbonded internuclear distance between atoms of neigh-
`boring molecules
`Standard energy of vaporization, defined in ref. 2
`Planck constant
`First ionization gotential
`Bond distance, A.
`Rest mass of an electron
`Avogadro number
`Ilistance from atom nucleus
`Ilistance between like nonbonded atoms (molecules) at
`potential energy minimum
`tion, rb = b + const.
`van der Waals radius estimated by Pauling’s approxima-
`
`van der Waals radius derived from nonbonded contact dis-
`tanre in crystals
`van der Waals radius
`AMean van der Waals radius for volume calculations in-
`volving anisometric atoms
`Molal volume
`
`Vo Molal volume a t OOK.
`V , Molal volume at critical temperature and pressure
`V , van der Waals volume (ralculated on the basis of Fig. 3)
`Number of nearest neighbors of a molecaule
`Z
`6( ) Decrement of V w or A , (as indicated) due to intramolecular
`crowding or hydrogen bonding
`AB E hd/m,lo = de Broglie wave length of outermost valence
`electron
`PO* E V,,/V, = parking density at 0°K.
`pe* E V,”/V, = packing density at critical point
`Distance between like atoms (molecules) at steepest ascent
`u
`of repulsion branch of potential energy well (as obtained
`from gas properties by means of Lennard-Jones and
`Devonshire theory)
`Charge density (probability of finding an elertron) a t
`distance r from the atom nucleus
`Purpose and Scope
`The primary purpose of the present investigation is
`the calculation of the volume occupied by a molecule,
`i.e., impenetrable for other molecules with thermal
`energies a t ordinary
`temperatures. This volume,
`called here the van der Waals volume (V,) , is to serve
`
`$z
`
`44 1
`
`Pfizer v. Genentech
`IPR2017-01488
`Genentech Exhibit 2045
`
`

`

`442
`
`A. BOXDI
`
`as reducing parameter in the study of the physical
`properties of condensed
`The calculation of
`V , assumes a knowledge of bond distances, bond angles,
`and the contact distances [Le., intermolecular van der
`Waals radii (r,)] and shapes characteristic of atoms in
`various molecular configurations. While most of the
`important bond distances and angles in organic mole-
`cules are reasonably well known and recorded, 3,4 only
`a few and semiquantitative contact (= nonbonded
`intermolecular) distances in crystals have been col-
`lected.
`The first task at hand was therefore the collection of
`contact distances from reliable X-ray diffraction data
`in the literature. Selection of the “best” values from
`the resulting mass of data is of necessity an arbitrary
`procedure, which will be discussed in the body of this
`paper. Two important approximations have been
`made which can only be excused by the desire to ob-
`tain results now; namely, the contact distances have
`not been corrected6 to OOK. and all atoms have been
`treated as spheres and sphere segments, aIthough it
`is well known that many are more nearly pear-shaped.
`The reason for the first approximation is the absence
`of data regarding the often anisotropic thermal expan-
`sion coefficients of the crystals from which the contact
`distance data were obtained. Spherical shapes have
`been assumed because of
`the absence of generally
`agreed pear shapes for the various atoms and the mathe-
`matical complexity of
`testing various alternative
`shapes for internal consistency. The values of V ,
`presented in this paper are therefore subject to further
`improvement, However, they have proven sufficiently
`useful, even in their present state, so that their publi-
`cation appears justified.
`General Principles
`Assumption of the existence of a defined spatial ex-
`tent of atoms is common to kinetic gas theory and X-
`ray crystallography. The extent of agreement be-
`tween the dimensions of the rare gas atoms produced
`by both approaches can be considered as a measure of
`the status of kinetic gas theory.6 However, this test
`is restricted to the rare gases because the crystals of
`metals, the only other monatomic species, are held
`together by covalent bonds and therefore do not per-
`mit comparison of the interatomic distances during
`thermal collisions of gas atoms with those prevailing
`in the solid.
`This observation brings us to one of the tacit assump-
`tions of this inquiry, the invariance of the van der
`Waals radius of an atom under the most drastic en-
`vironmental changes, Le., irrespective of its chemical
`combination and of its nearest nonbonded neighbors
`
`The Journal of Phusical Chemistry
`
`as well as of the phase state in which it is found. Closer
`examination shows that this assumption is surprisingly
`valid for heavy atoms, but is not very good with atoms
`containing only a few electrons, such as hydrogen,
`fluorine, etc. It may, therefore, be worthwhile to look
`a t the nature of the van der Waals radius from the point
`of view of the electron density distribution around an
`atom.
`The electron density +z at distance r from the core of
`a hydrogenic one-electron atom is given by the well
`known relation
`
`where me is the rest mass of the electron, lo the first
`ionization potential of the atom, and C is a normaliza-
`tion constant chosen such that f $2do = 1 when the
`integration is carried out over the whole of space ( w ) .
`The shape of the electron density distribution for typical
`one and multielectron atoms is shown on Fig. 1. -4s
`two atoms approach each other from r = m their elec-
`tron clouds interpenetrate more and more. The Pauli
`exclusion principle then causes a repulsion’ of the two
`atoms in direct proportion8 to the electron density in
`the region of interpenetration. One might define the
`van der Waals radius in terms of that distance r a t
`which this repulsion just balances the attraction forces
`between the two atoms. Comparison of the abscissa
`of Fig. 1 with the empirically known van der Waals
`radii shows that this distance correeponds to a very
`low electron density-so
`low, in fact, that one cannot
`formulate a “critical” electron density and hope to
`calculate T, from it because of the low degree of ac-
`curacy of the known calculating schemes in that part
`of the energy spectrum.
`However, eq, 1 contains a parameter h / d Z > the
`de Broglie wave length AB of the outermost valence
`
`(1) A. Bondi. -4.I.Ch.E. J.. 8 , 610 (1962).
`(2) A. Bondi and D. J, Simkin, ibid., 6 , 191 (1960).
`(3) Landolt-Bornstein, “Zahlenwerte and Funktionen I /4. Kristalle,”
`Springer, Berlin, 1955.
`(4) L. Pauling, “The Nature of the Chemical Bond,” Cornel1 Uni-
`versity Press, Ithaca, N. I-., 1942, p. 192
`(5) Since the volume increase of most solids between O’K. and the
`melting point is about lo%, the van der Waals radii at O’K. differ
`from those given here, probably by less than 37,.
`(6) J. 0. Hirschfelder and R. B. Bird, “Molecular Theory of Gases
`and Liquids,” New York, N. Y . , 1959.
`(7) See J. A. A. Ketelaar, “Chemical Constitution,” Elsevier Pub-
`lishing Co., Amsterdam, 1953. p. 146 ff. for a more detailed discus-
`sion; a somewhat different treatment is presented by K. S. Pitzer,
`“Quantum Chemistry,” Prentice-Hall, Ind., New York, S. Y-.. 1953.
`p. 201.
`
`(8) The repulsion potential E,% - exp(r /p) ; henp, at the distances
`of the order rlV, considering eq. 1, E, - $2, the electron densit3 in the
`
`region of interpenetration.
`
`

`

`VAN DER WAALS J’OLUMES AND RADII
`
`443
`
`electron of an atom, which might be related to the van
`der Waals radius ( T ~ ) . A few years ago Morrison9
`suggested that perhaps rw = (const.) AB. Our examina-
`tion of this suggestion brought to light that this simple
`correlation holds surprisingly well. One finds that the
`constant is 0.61 for the rare gas atoms, and is-as one
`might expect-appreciably
`smaller for bound atoms,
`ca. 0.53 for the halogens and about 0.48 for the re-
`mainder of
`the nonmetallic elements. Deviations
`from this correlation (shown in Fig. 2) are in the di-
`rection of predicting too large a diameter for the
`lightest elements. As the valence electrons of a
`covalently bound atom are concentrated between it
`and its bound neighbor to a degree that depends on
`the nature of the bond, it is surprising that rw retains
`enough individuality to be correlatable in terms of the
`ionization potential of the free atom.
`
`Comparison ofoCorrelation with ‘‘ Observed” Mean
`Table I :
`van der Waals Radii (in A.)
`
`Broken Lines are at
`Van der Waals Radii
`
`I
`
`0.1
`
`0.01
`
`?/so
`
`Figure 1.
`Electron density distribution near the atom
`“surface,” for hydrogen (I) and argon (11) [D. R. and X7
`Hartree, Proc. Roy. SOC. (London), A166, 450 (1938)].
`
`0
`
`0
`
`0 Rare Gases
`v Group VI1 Elements
`A Group VI Elements
`0 Croup V Elements
`0 Group IV Elements
`
`B
`
`x
`
`x
`
`0
`
`0
`
`0
`
`V
`
`0
`0
`A
`
`o
`
`I I
`
`2
`
`3
`
`4
`
`~
`
`~~~
`
`Row NO
`Figure 2.
`Relation of ru (single bond value) to de Broglie
`wave length AB of outer valence electron for
`nonmetallic elements.
`
`H
`1 .OB
`1.67
`1.20
`F
`1.40
`1.47
`1.47
`C1
`(1 .75Id
`1 . 7 0
`1.75
`Br
`1.87
`1.79
`1.85
`\
`I
`2.04
`1.90
`1.98
`
`He
`. .
`1.24
`1.40
`Ne
`. .
`1 . 3 2
`1.54
`Ar
`.
`.
`
`1 . 5 5
`1.88
`Kr
`. .
`1.64
`2.02
`Xe
`2.05
`1.76
`2.16
`
`B
`1 . 6 5
`2.13
`. .
`A1
`. .
`2.51
`. .
`Ga
`. .
`2.51
`. .
`
`C
`1 . 5 3
`1.82
`1 . 7 0
`Si
`1.93
`2.15
`2.10
`
`Ge
`1.98
`2.19
`. .
`
`S D
`2.16
`2.27
`
`iY
`1.46
`1 . 6 1
`1 . 5 5
`P
`1.86
`1.87
`1.80
`
`As
`1.94
`1.96
`1 . 8 5
`Sb
`2.12
`2.09
`
`0
`1.42
`1.66
`1.62
`S
`1.80
`1 . 9 1
`1.80
`Se
`1.90
`1 . 0 7
`1.90
`T e
`In
`. .
`2.08
`2.05
`2.55
`. .
`2.06
`b + 0.76.
`* ~n = 6.13 x 1 0 - 8 / 6 1 .
`Only the most
`Tb
`frequently used values for single bonded forms of the elements are
`Reference point cnhosen for the system ?‘b =
`quoted here.
`b + constant.
`
`Compatibility with Physical Proper ties
`The range of numerical values of T , for a given atom
`obtained from X-ray crystallographic contact distances
`is usually too wide for the direct application to a mean-
`ingful calculation of rw. The “best” value of r,, must
`
`5
`
`(9) J. D. Morrison, Rm. Pure A p p l . Chem., 5, 40 (1955).
`
`Volume 68, Number 3
`
`:March, 1964
`
`

`

`444
`
`A. BOSDI
`
`atomic molecules calculated from liquid density and
`energy of vaporization ( E o ) data, as 0.391EOIRT.
`(1.8) = 6 * 0.2.2 The application of this criterium
`requires the availability of good liquid density and
`vapor pressure or calorimetric heat of vaporization
`data and great faith in the form of the corresponding
`states principle adopted for this purpose.
`
`Experimental Data for Nonmetallic Elements
`
`‘--’
`
`then be obtained by appeal to extraneous information.
`In the ideal case enough crystal structure and zero point
`density data are available to fix through calculation
`that value of rw which yields the correct packing den-
`sity PO* a t 0°K.
`,This has been done for the simple
`tetrahalides by Sackmann. I n
`In general, when a detailed analysis is either too
`laborious or not possible, the “plausibility” of
`the
`packing density calculated from
`the experimental
`zero point volume” and V , will fix at least the upper
`In this account the long form of the periodic table is
`limit of ra. The empirical observation that for molec-
`followed from right to left.
`ular crystals PO* always >0.6 may be used to fix a
`The Halides. Fluorine. The data of Table I1 show
`lower limit on V,,. The method used to estimate
`that Pauling’s suggestion to set the ionic radius (1.35
`from bond distance 1 and from rn is shown on Fig. 3.
`Ti,
`A.) = r , yields rather improbable values for pb* of
`A lower limit is set for r , by the requirement that
`well studied fluorine compounds. Crystallographic
`the packing density at the critical temperature must
`data for SH,.BF,,14 SiF4,3,15 phosphonitrilic fluoride,16
`be sufficiently high to permit the existence of a con-
`yield r,(F) = 1-50 A., a value which,
`and T e f l ~ n I ~ , ’ ~
`tinuous three-dimensional network, i.e., the number of
`according to Table 11, is compatible with po, pc, and P.
`nearest neighbors Z >_ 3.12 A related criterium is the
`The liquid state criterium led to the very similar value
`compatibility with the size of the equivalent sphere
`r,(F) = 1.47 A. for perfluoroalkanes, aryl fluorides,
`obtained by application of kinetic gas theory.13 The
`and for secondary and
`tertiary alkane fluorides.
`somewhat surprising regularity that V , = N ~ ( a / 6 ) g ~
`Fowever, for primary alkane fluorides r,(F) = 1.40
`which we observed whenever r, met the other two
`A. was found more compatible with the data. In
`criteria, is useful when neither po nor pc but only gas
`view of the exactly opposite trend of the C-F bond
`properties have been measured.
`length, this is an unexpected result, for which neither
`A particularly arbitrary criterium developed in the
`conflicting nor confirmatory evidence could be found in
`course of the present work is that the number of ex-
`X-ray diffraction data.
`ternal degrees of freedom of rigid (nonlinear) poly-
`Iodine. The previously
`Chlorine, Bromine, and
`mentioned analysis of the crystallographic and density
`data of the tetrahalides of the group IV elements by
`SackmannIO yielded r,(Cl) = 1.76 A., r,(Br) = 1.85
`A,, rw(I) = 1.96 K., in good agreement with many
`other X-ray diffraction data, shown in Table 111.
`The data of Table IV indicate that the resulting values
`of V , for various halogen compounds are more com-
`patible with experimental density data than are the
`frequently quoted van der Waals radii based on I’aul-
`ing’s approximation r, = r,. The radii and volume
`increments derived from the liquid state criteria have
`been assembled in Table IV. They all show the same
`value for the primary alkanes previously noted for
`flu0 r in e.
`
`(IO) H. Sackmann, 2. physik. Chem., 208, 235 (1958).
`(11) W. Bilts, “Raumchemie der festen Stoffe,” Leipeig, 1934.
`(12) This point will be discussed in detail in a subsequent article.
`(13) A. Bondi, J . Phys. Chem., 5 8 , 929 (1954).
`(14) J. L. Hoard, et al., Acta Crust.. 4, 396 (1951).
`(15) M . Atoji and W. N. Lipscomb, ibid., 7, 173 (1954).
`(16) H. A4cGeac:hin and F. Tromans, J . Chem. Soc., 4777 (1961).
`(17) C. 1%’. Bunn and E. R. Howells. .Vatwe, 174, 548 (1954).
`(18) H. G. Killiam and E. Jenckel, 2. Ekktrochem.. 63, 308 (1959).
`
`rl, rz = van der Waals radii
`1 = covalent bond distance
`m = auxiliary parameter
`h,, h, = height of sphere segments
`r:
`m =
`
`2 1
`
`’ ” ; h, = r, + 1 - m; h, : rL - m
`vf = nhf (rl - k); AVz-l = rh: (r, - $): V, = 4“ .:
`
`3
`
`h
`
`Example :
`van der Waals volume of diatomic molecule:
`V w = NA [V: + VI - AVz.l] cm’/mole
`where NA = 6.02 x 10’’ molecules/mole
`and r’s are given in Angstrom units
`Volume of center atom = Total volume of atom
`Surface area = Zmh
`Figure 3. hlethod of calculation
`
`The Journal of Physical Chemistry
`
`

`

`Table I1 :
`Compatibility of Proposed van der Waals Radii
`of Fluorine with Physical Properties
`v,. a
`
`Sub-
`stance
`F2
`
`CF,
`
`n-CiFi6
`
`I
`
`rw
`A.
`1.35
`1.50
`1.35
`1.50
`
`V W .
`cm.gjmoie
`1 0 . 5
`14.2
`
`16.5
`29.5
`
`I V ~ ( ~ ~ / G ) ~ ~
`pa**,c
`0.88
`0.42
`1.19
`0.57
`, . .
`0.56
`0,161
`0.51
`0.200
`0.695
`0.92
`. . .
`. .
`0,160
`1.35
`105.8
`0.193
`1.50
`127.7
`. . .
`. .
`* See ref. 11.
`G. A. Miller and R. B. Bernstein,
`See ref. 6.
`J . Phys. (?hem., 63, 710 (1959). K. A. Kobe and R. E. Lynn,
`Chem. Rev., 52, 121 (1953), and Miller and Bernstein, preceding
`reference.
`
`V A N D E R WA.4LS VOLUMES A N D RADII
`
`445
`
`Table IV:
`Compatibility of Proposed van der Waals Radii
`of C1, Br, and I with Physical Properties
`
`P
`
`Y
`
`
`
`. . .
`
`CL
`Brr
`
`I2
`
`I .09
`1.16
`1.42
`0.96
`1.17
`0 . 8 -
`0.84
`
`,
`
`.
`
`
`
`0.195
`0.200
`0.240
`. . .
`. . .
`0.186
`. . .
`,
`.
`.
`
`
`24.1
`0.74
`1.75
`I ,85
`28.9
`0.75
`35.6
`2.00
`0 . 9 3
`37.6
`1.96
`0.76
`46.0
`2.16
`0.93
`51 .4d
`1 . 7 5
`CCI4
`0.69
`60.Sd
`1.85
`CBr4
`0.69
`CI,
`74
`1 97
`0.68
`a See ref. 6 and R. A. Svela, NASA Technical Report R-132
`* See ref. 11.
`No allowance
`See ref. d in Table 11.
`(1962).
`has been msde for possible volume reduction due to overlap of thc
`halogen atoms or for their probable pear shape. Such corrw-
`tion would reduce Tiw and the other reduced densities, but would
`leave the constancy in the series unchanged.
`
`Table 111 :
`Rerent Intermolecular Contact Radii r' for
`Chlorine, Bromine, and Iodine in Molecular Crystals
`
`rw'(X), A.
`Compound
`(Cl) 1.70"
`X-C6H6C16 (Gammexane)
`(Cl) 1.74b
`Perchlorofulvalene
`(C1) 1.78"
`Tetrachloro-p-benzoquinone
`(C1) (1.77), 1 .82d
`1,3,3-Trichlorobenzene (SoOK.), 293°K.
`(Br) 1.80"
`1,4-Ilibromocyclo [3.2.2]azine
`(Br) 1.88'
`p-Di bromobenzene
`(Br) 1.88d
`1,3,5-Tribromobenzene
`Iodoform, 0°K.
`1.99*
`1.95*
`Phosphorus thioiodide
`2.07<
`p-Diiodobenzene
`* See
`a G. W. Van Z'later, et al., Acta Cryst., 3, 139 (1950).
`S. Chu, el al., Acta Cryst., 15, 661 (1962). H. J.
`ref. 21.
`Milledge and L. hl. Pant, ibid., 13, 285 (1960).
`ibid., 14, 124 (1961). ' U. Croatto, et al., zbid., 5, 825 (1952).
`e A. Hanson,
`See ref. 30. D. A. Wright and B. R. Penfold, Acta Cryst.,
`' S . B. Hendricks, et al., J . Chem. Phys., 1 ,
`12, 455 (1959).
`549 (1933).
`
`Table V :
`Effective van der Wads Radii of Bound Halogen
`Atoms from Liquid State p-v-l Property Calculations
`
`X
`F
`
`C1
`
`Br
`
`I
`
`Attachment
`Primary alkyl
`sec, tert, alkyl, perfluous alkane, phenyl
`
`Primary alkyl
`Vinyl
`sec, terl, alkyl, polychloroalkyl, phenyl
`
`Primary alkyl
`teri, polybromoalkyl
`Phenyl
`
`Primary alkyl
`tert, polyiodoalkyl, phenyl
`
`T w ( X ) , b
`1 40
`1.47
`
`1.73
`1 . 7 5
`1 . 7 7
`
`1.84
`1.85
`1 . 9 2
`
`2.01
`2.06
`
`Oxygen, Sulfur, Selenium, and Tellurium. As the
`diameter of singly bonded oxygen and sulfur is smaller
`than that of methylene or other functional groups to
`which they might be attached, one finds only few good
`contact distance data for them. AIoreover, in the
`present context all contacts involving hydrogen bonds
`must be excluded. As a consequence of being too
`deeply "buried" within the molecule for frequent (or
`any) collision with neighboring molecules, the "ef-
`fective volume" occupied by an ether oxygen atom
`differs from molecule to molecule. The upper limit on
`r,"(>O) set by the few available X-ray data (Table VI),
`1.32 A., corresponds to V,(>O) = 5.5 cm.3/mole.
`Only in ethylene oxide is the ether oxygen sufficiently
`
`exposed to yield this value of V,(>O).
`I n alkyl ethers,
`especially polyethers, V,(>O) = 3.7 ~rn.~/rnole and i n
`polyphenyl ethers it is 3.2 ~m.~/rnole. These values
`were obtained by fitting polyether data to the general-
`ized density correlation for liquids.
`For single-bonded suJfur, X-ray diffraction data
`point to r,"(>S) = 1.83 'I., while 1.80 8. is compatible
`with various physical properties (Table VII) and with
`I o r single-bonded oselenium,
`liquid phase densil y.
`a single set of datalg yields :,"(>Se) = 1.87 A. and for
`tellurium V , (>Te) = 2.06 A.
`The geometry of double-bonded oxygen and sulfur
`In the direction parallel
`is probably quitc anisometric.
`
`(19) R. E. Mush, Acta Cryst., 5 , 458 (1952).
`
`VQlUme 68, >Vumber 3 March, 1964
`
`

`

`446
`
`A. RONDI
`
`Table VI : Intermolecular Contact Radii for Oxygen"
`
`Table VI11 :
`Compatibility of van der Waals Radius of
`Double-Bonded Oxygen with Physical Properties
`
`Single-bonded (ether) oxygen
`Calcium peroxide
`1,3,6-Trioxane
`Bis( 1,3-dioxacyclopentyI)
`Hydroxy 1.-proline
`
`Double-bonded (carbonyl or nitro) oxygen
`Tetrachloro-p-benzoquinone, parallel arrangement
`Tetrachloro-p-benzoquinone, vertical arrangement
`Butyne-2, iron octacarbonyl, parallel arrangement
`Acrylonitrile iron tetracarbonyl
`
`1 468
`5b
`
`-1
`1
`1 52c
`
`1 . 3 j d
`1 . 63d
`1 .47"
`1 , j 3 a 8 f
`
`Only 0 . . .O contact are quoted here.
`The recommended
`van der Waals radii also take into account the intermolecular
` contact^ distances between oxygen and other atoms. ' See ref. 3.
`J. Donohue and IC. K. Trueblood, Acta Cryst., 5 , 419 (1952).
`See ref. c in Table 111. e ,4. Hock and 0. Mills, Acta Cryst.,
`iZ. Luxmoore and M. Truter, ibid., 15, 1117
`14, 139 (1961).
`N. G. S'annenberg, Proyr. Inorg. Chem., 4, 125 (1962).
`(1962).
`
`Table VII:
`Compatibility of van der Waals Radii of Sulfur
`and Selenium with Physical Properties
`v,, 2- a
`crn.>/'mole ~ V A ( ~ : ' G ) U S
`1 8 . 0
`1.21
`92.0
`. .
`
`Sub-
`;w, A.
`stance
`HzS
`1.8O(S<)
`1,75(-S-)
`Ss
`I
`cs2
`l
`i
`
`1.75 (S)
`0.69
`1 . 0 9
`31.2
`H2Se
`0 . 6 8
`20.9
`1 . 9 0 (Se<)
`. .
`See ref. c in Table TI.
`See ref. 11.
`a See ref. 6.
`
`0,675
`0.766
`
`PO*<
`0.186
`. . ,
`
`0,184
`. . .
`
`to the double bond. one finds r,(O) = 1.40 i. (about
`1.39 8. is the cut-off distance of the neutronfiffraction
`pattern of liquid oxygen,20 ai>d 1.43 =t 0.03 A. for simi-
`lar dimensions in various carbonyl compounds. Sormal
`to the double bond :xis, one finds dimensions of the
`order of 1.6 to 1.7 &4., perhaps characteristic for the
`diameter of a n-electron cloud. Owing to the uncer-
`tainty of the geometry of the n-elec5ron cloud, the
`arbitrary compromise r,(=O) = 1.50 A. was adopted,
`which is compatible with the density data, as shown
`in Table YIII, and meets the requirements of the liquid
`state correlation.
`The contact distance involving double-bonded sulfur,
`as in rhodanin2' [i,?(S) = 1.74 A.1 is snialler than in
`single-bonded sulfur. Similarly, the contact distance
`between sulfur atoms in polysulfides and in crystalsne
`sulfurz2 is comparatiT-ely short ( T , = 1.76 i 0.04 A.),
`probably connected with
`the pronounced double-
`bond character of the s-S bond.
`
`The Journal of Phvsical Chemistry
`
`I'
`
`VW,
`sutiatance GW(o), 1. cm.~:'molo
`13.0
`1 . 5 0
`0 2
`C=O
`1.50
`1 6 . 2
`co2
`1.50
`19.7
`See ref. 11.
`a See ref. 6.
`
`Q
`. ~ A ( r , ' ~ ) o t
`pr*b
`1.03
`0.60
`0.60
`1.01
`0.98
`0.76
`'See ref. c in Table IT.
`
`Pc *e
`0.167
`0.174
`0 . 2 1
`
`~~~
`
`~
`
`Xztrogen, Phosphorus, and drsenzc. The contact
`distances in a wide T ariety of nitrogen compounds,
`assembled in Table IX, euggest that for single- and
`double-bonded nitrogen T * ( X ) = 1.55 =t 0.04 K.
`and in cyano groups,
`Triple-bonded nitrogen. as in ?;2
`appears to be quite anisometric with (T,,) = 1.40
`parallel to the bond and up to 1.7
`in the direction
`normal to bonds. Again the "effective" vajue was
`somewhat arbitrarily set a t r , ( S ) = 1.60 -4. The
`data of Table X show that the adopt2d radii for nitro-
`gen are compatible with the density data They
`also yield the plausible packing density ps* = 0.706
`for the carefully investigated (tightly packed) crystal
`tetra~yanoethylene.~~ Smaller values for rM(X)
`of
`would not have given a packing density in keeping
`ith X-ray observations. Further confirmation of the
`proposed radii comes from the liquid state correlation.
`Only very few contact distance data involvilig phos-
`Their average, rM(P) =
`phorus are a ~ a i 1 a b l e . l ~ ~ ~ ~
`
`1.80 a., is of the order expected from the ionization
`
`potential a i d from the coyalent bond radius of phos-
`phorus. Only one set of crystallographic data (bromo-
`diphenylarsine) could be founi from which r , (-4s)
`could be estimated as = 1.85 A. This value agrees
`with the physical property data (Table XI).
`Carbon and Hydrogen.
`The available data for the
`van der Waals radii of carbon in aliphatic and aro-
`matic compounds have been assembled in Table XII.
`Olefinic carbon is not found in the list because only
`the so-far unexamiiied cuniulenes, such as allene,
`could yield the desired contact distances, not being
`spaced apart by the a h a y s larger methyl or functional
`groups. The data obtained on the various acids and
`amides on the list could be considered as meaaures of
`the width of a double-bonded carbon atom, because
`the reported spacings are those characteristic of the
`
`92, 1229 (1953). 119, 22
`
`(20) D G Henshaw, et al Phys Rei
`(1960).
`(21) P Yheatley, d Chem Snc , 4936 (1961)
`(22) J Donohue In Orgainc Sulfiir Compounds," N Kharasch,
`E d , Ne%% I-ork N Y 1961, p 1.
`(23) D A Bekoe and K pi Trueblood, 2 Krzd , 113, 1 (1960)
`(24) E Ke-der and .i Vos, Acta C r y s t , 12, 323 (1959)
`
`

`

`\'AX DER \vAAIS VOLUMES AND R A D I I
`--
`
`447
`-
`
`Table IX : Intermolecular Contact Radii for Nitrogen"
`T w ' ( X ) , a.
`
`Compound
`
`carboxylic carbonyl group. The single datum for the
`radius of the acetylenic triple bond is supported by the
`volume data in Table XIII.
`
`1 .48b
`1 .58'
`
`1 .SOd
`1 .soe
`1.54'
`1 .56Q
`1.58'
`1 .68h
`1 .63'ji
`
`Single-bonded
`Xanthaml
`S4?44&
`Double-bonded
`4-hIethylimidazole
`Glyoxinne
`Guanine
`Pyrimidine
`Adenine hydrochloride
`I>iazoarninobenzene copper
`Tetrazole
`Triple-bonded
`Tetracyanoethylene (N , , , X contact, normal to
`bond)
`Tetracyanoethylene (S . . . C contact, parallel a-ith
`1 .3gk
`bond)
`a Only IC, , , N contacts are quoted here, except where noted
`otherwise. The recommended van der Waals radii also take int'o
`account the intermolecular contact distances between nitrogen
`and other at>oms. ' W, Nowacki and H. Burki, 2. Krist., 106,
`339 (195.5). R. L. Sass and J. Donohue, Acta Cryst., 11, 49:7
`(1958). H. Zimmerman, Ann., 612, 193 (1958).
`ton, Acta Cryst., 14, 95 (1961). ' J. M. Broomhead, ibid.,
`e W. Hamil-
`1, 324 (1948); 4, 92 (1951). K. E. White and C. J. B. Clews,
`J. H. Bryden, ibid., 8,211 (1955).
`ibid., 9,586 (1956).
`R. IT.
`' Y. D. Kondenshev, Soviet Phys.
`Brown, ibid., 9, 163 (1956).
`See ref. 23.
`Cryst., 6 , 413 (1962).
`-.
`
`1 .70k
`
`Table X :
`Compatibility of van der Waals Radii of Nitrogen
`with Physical Properties of Its Compounds
`
`N F N
`0 .5gd
`1.01
`1 5 . 8
`1.60
`(CsS), 1.60
`1.10
`29.4
`. .
`N=O
`1 . 5 5
`0.73
`13.9
`1 . O i
`0.70
`?;=S=O
`1 . 0 3
`18.9
`1 . 5 5
`KH3
`0 . 7 0
`1 3 . 8
`1.55
`. .
`See ref. c in Table 11.
`See ref. 11.
`See ref. 6.
`Bolz, et al., Acta Cryst.: 12, 247 (1959).
`
`0.176
`
`I
`
`.
`
`.
`
`
`
`. . .
`0.196
`0,190
`L. H.
`
`Table XII:
`Hydrogen
`
`Observed Contact Radii of Carbon and
`
`Ref.
`
`a
`b
`C
`e
`f
`9
`h
`
`i
`
`j
`IC
`
`1
`
`rw(H), A
`. .
`.
`.
`
`
`, .
`
`. .
`. .
`. .
`. .
`1.17
`1.18
`
`1.18
`1.17
`
`. .
`
`. .
`
`, .
`
`, .
`
`1 . 0 1
`1.01
`
`I
`
`,
`
`
`
`r w ' ( C ) , 4.
`1.66
`1.67
`1 . 6 7
`1 .6gd
`1.69; 1 . 7 0
`1.70
`1.70
`. .
`. .
`
`. .
`
`,
`
`.
`
`
`
`1.78
`
`Aliphatic compounds
`1,3-Dimethyliminotetrazole~HCl
`Uracil
`Sodium tropolonate
`Octatriyne
`Succinamide
`Isocyanic acid
`Succinamic acid
`n-Triacontane at 0°K.
`n-Hexatriacontane
`Triethylenediaminenickel( 11)
`nitrate
`Potassium methylene disulfonate
`Acetylenic triple bonds
`Carbides of Ca, Ba, and Si-
`Aromatic ring systems
`Coronene
`m
`1 . 7 2
`Ovalene, perylene
`1 . 7 5
`n
`1.77
`Anthraquinone
`0
`P
`Naphthalene
`1.77
`Anthracene
`1 . 7 7
`P
`r
`1 . 7 8
`Graphite
`Dibenzocoronene
`1 . 8 0
`S
`1,01;1,05 t
`Benzene
`. .
`Pyridine.BFa
`1 . 0
`u
`, ,
`a See ref. h in Table IX. G. 8. Parry, Acta Cryst., 7, 313
`' R. A. Pasternak, Acta Cryst.,
`(1964). R. Shiono, ibid., 14, 42 (1961). Head-on approach
`of methyl groups.
`e See ref. 3.
`6, 808 (1953); 9, 334 (1946). ' See ref. 3. * R. A. Pasternak,
`Acta Cryst., 7, 225 (1954). ' P. W. Teare, ibid., 12, 294 (1959).
`' L. Swink and M. Atoji, ibid., 13, 639 (1960).
`ie M. Truter, J.
`' M. Atoji and R. C. Medrud, J . Chem.
`Chem. Xoc., 3393 (1962).
`J. ill. Robertson, et al., J . Chem. Soc.,
`Phys., 31, 332 (1959).
`See ref. 28. ' S. N. Sen, Zndia,n J . Phys., 22,
`925 (1956).
`347(1948). D. W. T. Cruikshank, Acta Cryst., 10, 504 (1957).
`* D. NT, T. Cruikshanlr, ibid., 9, 915 (1956). ' J. .4. Ibers,
`' E. G. Cox, Rev. M o d .
`privat,e communication.
`* See ref. 29.
`Phys., 30, 159 (1958). ' 8. V. Svenkova, Kristallografiya, 2 ,
`408 (1967).
`
`Table XI:
`Compatibility of van der Waals Radii of
`Phosphorus and Arsenic with Physical Properties of Their
`Compounds
`
`I
`
`v,
`Substance r L v ( X ) , .i. cm.ximole
`Pa
`1.80
`41.6
`PH3
`1 . 8 0
`1 9 . 7
` ASH^
`1.85
`21.3
`* See ref. 11.
`a See ref., 6.
`
`
`I
`I
`,
`v
`. ~ . 4 ( n / 6 ) v 9
`PO**
`( 0 81)
`0.676
`0 . 5 5
`0,986
`1 .oo
`0.56
`See ref. c in Table 11.
`
`PD*e
`. . .
`0.174
`, . .
`
`the methyl groups on
`The head-on approach of
`octatriyne provides the most direct measure of the
`dimensions of an aliphatic carbon atom. However,
`adoption of r,(C) 2 1.69 A. would have inspired a
`false sense of precision regarding this central factor in
`the entire schemeo of volume calculations. Hence
`r,(C,liph) = 1.70 A. is, in agreement with Briegleb25
`and others, 26 the preferred choice at present.
`
`Volume 68, Sumber 3 March, 1964
`
`

`

`448
`
`A. Roiv-Dr
`
`Table XI11 :
`Compatibility of van der Waals Radii of Carbon
`and Hydrogen with Physical Properties of Simple
`Hydrocarbons
`
`I,
`
`.-!!!-*
`V,",
`rbV!H).
`rw(C),
`h. cm.3/1nok
`A.
`Substance
`pe*c
`. \ r ~ ( r r / ~ ) 0 3
`0.173
`0.67
`1.00
`1.20
`17.1
`1.70
`Methane
`0.684 0,184
`1.00
`1.20
`27.3
`1.70
`Ethane
`1 .OO
`1.20
`23.9
`1 . 7 0
`Ethylene
`0 . 6 7 0.193
`0.204
`0 . 9 8
`1 . 2 0
`23.1
`Acetylene 1.78
`. .
`1 .OO
`Benzene
`1 . 7 7
`0 . 7 0 0.186
`1.04
`48.4
`* See ref. 11.
`See ref. c in Table 11.
`
`a See ref. 6.
`
`The van der Waals radius of aromatic carbon atoms
`is one-half of the contact distance between nonbonded
`carbon atoms in adjacent ring planes and is distinctly
`different from the often quoted interplanar distances.
`A few shorter distances (as low as 3.20 8.) bekeen
`ring carbons in neighboring rings disposed at steep
`
`angles have been r e p ~ r t e d ~ ' - ~ ~ but have not bcen given
`any weight in the final estimate of rw(Cw). The
`value r,(C,) = 1.77 A. has been adopted as bcing
`authenticated by the most careful measurements on
`the list.
`'I'he supposition is then, that the benzene
`ring has a slightly humpy top (likc the top of an old
`&44rab house). The smooth r-ele~t~ron model of Briegleb
`does not seem entirely compatible with the observed
`X-ray diff ractiori patterns of condensed ring systems.
`However, the difference in volume between the two
`models is small
`the density of graphite
`Seither model predicts
`correctly, Our model yields a density that is about 5%
`too high and that of the Rriegleb model is about 5%
`too low. The model predicting t'oo high a density
`is probahly more realistic because it mctlns that there
`is still some space for thermal motion between the
`graphite planes. From Bridgman's experiments we
`know that, ahout 57' ol graphite volume is squeezed
`out by compression t o about 10,000 atm.
`Thc contact distances of hydrogen are difficult to
`ascertain. The best autheriticatcd contacts, by Kitai-
`gorodskii's measurements on n-paraffins cxtrapolated
`to 0°K.26~30 and also observ:d
`in several other crystals
`lead to rw(HRliph) = 1.17 A. This k so close t'o tbe
`r,(Ha~iph) = 1.20 A.
`commoiily accepted value of
`that the latter value was retained for the present work.
`This choice of aliphatic radii is compatible with the
`shortest observed contact' distance of methyl groups,
`3.57 K., in gear-like packing of aliphatic compounds,
`as well as with densities and gas properties of small
`hydrocarbon molecules, as shown by the data of Table
`XIII. The observed hydrogen-hydrogen contact dis-
`
`The Journal of Physical Chemistry
`
`tances between aromatic rings in the ring plans of
`several very careoful measurements
`lead uniformly
`to r w ( H w ) = 1.0 A. The choice of aromatic dimen-
`sions is also compatible wit,h density da