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Atomic Radii Dnd Interatomic Distances in Metals' 542 LINUSPAULING Vol. 69 The values deduced from van der Waals equation than those deduced from van der Waals equation. are, respectively, 2.67 for the critical ratio, 3.38 Summary for the reduced Boyle temperature and 6.75 for the reduced maximum inversion temperature. An equation of state for gases, equation (4) is The average experimental value for the critical proposed, in which all the constants can be calcu- ratio is 3.7 and that for the reduced Boyle lated from the critical values of the temperature temperature is near 2.5. The maximum inver- and pressure in a manner analogous to the evalua- sion temperature has been established for only tion of van der Waals constants. This equation a few substances. The reduced value for ni- is believed to reproduce the behavior of real gases trogeng is 4.93, that for airlo is 4.55. It is seen with a higher degree of accuracy than the van der that the values predicted by equation (4) are Waals equation. In the reduced form (5) it repre- considerably closer to the experimental ones sents an analytical expression of the law of cor- responding states. (9) J. R. Roebuck and H. Osterberg, Phys. Rev., 48,450 (1935). (10) J. R. Roebuck and H. Osterberg, ibid., 43, 60 (1933). RECEIVEDOCTOBER 21, 1946 [CONTRIBUTION FROM THE GATESAND CRELLIN LABORATORIESOF CHEMISTRY, CALIFORNIA INSTITUTE OF TECHNOLOGY, No. 10991 Atomic Radii dnd Interatomic Distances in Metals’ BY LINUSPAULING The problem of the nature of the interatomic mulation of a system of atomic radii which can be farces in the elementary metals and in interme- used for the calculation of interatomic distances tallic compounds and other alloys continues to be in metals and intermetallic compounds and for the puzzling, despite the clarification of some ques- interpretation of observed interatomic distances tions which has been provided by quantum me- in terms of the electronic structure of the crystals. chanical considerations.2 It has been my opin- These atomic radii (which may be called metallic contrary to that of other investigator^,^^^ radii) are found, as expected, to show an intimate that the metallic bond is very closely related to relation to the covalent radii of the atoms-a the covalent (shared-electron-pair) bond, and that relation which, in its general nature, permitted each atom in a metal may be considered as forming Goldschmidt6 over twenty years ago to use data covalent bonds with neighburing atoms, the co- taken from both metals and ordinary covalent valent bonds resonating among the available in- crystals in formulating a table of atomic radii, teratomic positions. It was shown in the first and which has been recognized3 as providing paper of this series‘ that the number of covalent very strong support for the concept that metallic bonds resonating among the available positions bonds are essentially resonating covalent bonds. about an atom (the metallic valence of the atom) The Relation of Atomic Radius to Bond Type increases from one to nearly six (5.78) in the se- quence K, Ca, Sc, Ti, V, Cr in the first long period In the discussion of metallic radii we may make of the periodic table, remains nearly constant a choice between two immediate alternative pro- from Cr to Ni, and begins to decrease with Cu. cedures. The first, which I shall adopt, is to con- This concept, which is substantiated by the mag- sider the dependence of the radius on the type of netic properties of the metals and their alloys, the bond, defined as the number (which may be provides a qualitative explanation of many prop- fractional) of shared electron, pairs involved erties of the transition metals (including those of (corresponding to the single, double, and triple the palladium and platinum groups), such as bonds in ordinary covalent molecules and crys- tals), and then to consider separately the effect of characteristic temperature (heat capacity at low resonance in stabilizing the crystal and decreasing temperatures), hardness, compressibility, coeffi- the interatomic distance. This procedure is simi- cimt of thermal expansion, and the general lar to that which we have used in the discussion trend of interatomic distances. It will be shown of interatomic distances in resonating molecules.’a in the following pages that it also permits the for- The alternative procedure would be to assign to (1) “The Nature of the Interatomic Forces in Metals. 11.” each bond a number, the bond order, to represent For paper I of this series see L. Pauling, Phys. Rev., 64, 899 (1938). .the strength of the bond with inclusion of the (2) A. Somnierfeld and H. Bethe, “Handbuch der Physik,” VOI.24, second edition; N. F. Mott and H. Jones, “The Theory of resonance effect as well as of the bond type.8 the Properties of Metals and Alloys,” Oxford University Press, (6) V. M. Goldschmidt, “Geochemische Verteilungsgesetze der Oxford, 1936; F.Seitz, “The Modern Theory of Solids.” McGiaw- Elemente,” VoJ. VII, Oslo, 1926. Hill Book Co., New York, N. Y., 1940. (7) L. Pauling, Proc. Nall. Acad. Sci., 18, 293 (1932); L. Pauling, (3) L. Pauling, “The Nature of the Chemical Bond.” 2nd ed., L. 0. Brockway and J. Y. Beach, THISJOURKAL, 67, 2705 (1935); Cornel1 University Press, 1940, chap. XI. L. Pauling and L. 0. Brockway, ibid., 59, 1223 (1037); ref. 3, Chap. (4) J. D. Bernal, Trans. Faraday Soc., 26, 367 (1929). V. (51 W. L. Bragg, J. Roy. Soc. Arts, 86, 430 (1937). (8) W. G. Penney, Proc. Roy. SOC.(London), 8168, 306 (1937). March, 1947 ATOMICRADII AND INTERATOMICDISTANCES IN METALS 543 Because the empirical information about the simply by decreasing the value of the constant effect on interatomic distances of change in cobr- to about 0.31. dination number is not sufficiently extensive for I have chosen to use the equation our purpose, we make use of our knowledge of normal covalent radii. The available evidence R(1) - R(n) = 0.300 log n (3) indicates that the differences between single-bond, in the following considerations. This equation, double-bond, and triple-bond radii are very nearly which is based upon the study of interatomic dis- the same for all a.t~ms,~and hence that an expres- tances for non-resonating and resonating covalent sion can be found for a term to be added to the bonds in simple non-metallic substances of known single-bond covalent radius for any atom to give structure, is found to agree reasonably well with approximately the radius corresponding to an- those data for metallic crystals which are suited other bond type. The carbon-carkon bond dis- to a check on its validity, and its use permits a tances are 1.542, 1.330, and 1.204 A. for a single penetrating analysis of the structure of metals and bond, double bond, and triple bond, respectively, intermetallic compounds to be made. There is the double-bond and triple-bond radius correc- some evidence that the constant (taken as 0.300) tions hence being -0.106 and -0.169 8. The varies with the kind of atom and with the type of ratio of these numbers is just equal to log 2/log 3; bond; but the evidence is not sufficiently exten- accordingly the bond-type correction may be sive to lead to the determination of the nature of taken proportional to the logarithm of the bond this variation. number, n Correction from the A2 Structure to Coordi- - AR(n) = 0.353 log ?Z The (1) nation Number 12 Here AR 1s R(n) - R(1), in 8., and n is the number of shared electron pairs involved in the In the A2 structurelo (with atoms at the points bond. This logarithmic relation is, of course, to be of a cubic body-centered lattice) each atom has expected in consequence of the exponential char- eight neighbors at the distance a0d3/2 and six acter of interatomic forces. neighbors at the larger distance ao. If the valence Atomic radii calculated with this expression of the atom were used only for bonds to the8eight need to be corrected for the stabilizing (bond- nearest neighbors, the radius for the A2 structure shortening) effect of resonance energy for use in would be that for coordination nymber 8 (CN8), resonating molecules or crystals. For example, which by equation 3 is 0.053 A. less than the the interatomic distance in benzene is not 1.4188.) radius for CN12. But the bond number n of the as given by equation 1 for n = 3/2, but is 0.028 A. eight stronger bonds is somewhat smaller than v/8, smaller, because of the effect of the resonance of because part of the bond-forming power of each the double bonds between two alternative posi- atom is used for the longer bonds, and accord- tions. Of the several simple expressions which ingly the correction from A2 to CN12 is expected might be selected to represent empirically the to be less than 0.053 8. effect of resonance on atomic radius, I have chosen The amount of this correction depends upon the loga~thmof the number of alternative bond the value of the atomic radius. Zirconium, for arrangements among which resonance occurs, example, has a. = 3.61 8. for the A2 structure, divided by the number of resonating bonds. The the effective radii for th? shorter and longer bonds observed decrease in radius for benzene, corre- being 1.563 and 1.505 A., respectively. rlpplica- sponding to the resonance of a double bond be- tion of equation 3 leads to v/8.94 for the bond tween two positions, is 0.047 log 2; a similar ex- number of the shorter bonds, and hence to 0.035 pression, 0.017 = 0.036 log 3, is valid for graphite.
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