Molarity (Atomic Density) of the Elements as Pure Crystals Linus Pauling and Zelek S. Herman Linus Pauling Institute of Science and Medicine, 440 Page Mill Road, Palo Alto, CA 94306 In 1869 Lothar Meyer, then in the Karlsruhe Polytechnicum, published an article (1,2)about the periodic law illustrated by a figure showing the atomic volumes of the elements plotted with the atomic weights along the x coordinate (Figure 1). This figure clearly shows the two short periods, the two long periods, and the very long period of the periodic table. Ever since then this figure, revised some 40 years later (3)by replacing atomic weight by atomic number, has been one of the best ways of illustrating the periodicity of the elements, most recently in the careful study by Singman ( 4 ) . Some 30 years ago one of us observed that the reciprocal of the atomic volume, which may he called the atomic density, the atomic concentration, or the molarity of the element in the crystalline state, has the advantage of indicating in a clear way the strength of the interatomic forces, as well as the periodicity (5).We have accordingly made use of the atomic-volume values given by Singman, supplemented by additional values for some allotropes, in preparing the atomic-density diagrams shown in Figures 2, 3,4, and 5. Figure 2 shows the values for the principal allotropes of all of the elements. The unit for the atomic density is moles per liter, so that the quantity can be described as the molarity of the element in the crystalline state. The values are obtained by dividing 1000 by the atomic volume in units cm3 mol-1. For over half of the elements shown in Figure 2 the crystal structure is cubic or hexagonal closest packing, in which each atom has ligancy 12. The cubic body-centered structure is the stable one for 15 metals; for this structure the ligancy is somewhere between 8 and 14 (each atom has 8 nearest neighbors and 6 a t a distance 15% greater). Several metals have both structures, and for them the molar concentrations usually differ by less than 1%. The dependence of the molar concentration on the structure may he illustrated by dia-
mond and graphite. In each of these allotropic forms carhon is quadricovalent, but the loose packing of the layers in graphite causes its molarity to he 35% less than that for diamond. All of the nonmetallic elements except the noble gases have structures in which each atom is bonded to one or more others by covalent bonds and also has several other neighbors a t larger distances. As with carhon, the molarity depends on the structure. Over 50 allotropes of sulfur are discussed by Donohue in his book on the structures of the elements, and several of them have been well characterized (6).The densest form is rhombohedra1 sulfur, with Sg rings and molarity 69.03 mol IF1. The next dense is S7, with molarity 67.0 moll-', then the stable form, orthorhomhic sulfur, Ss. with molarity 64.5 mol I-'. Fibrous sulfur, S,, with helical chains, has molarity 62.5 mol I-'. These values suggest that the variation caused by a difference in structures is about 10%. It is seen from Figure 2 that the values of the molarity for elements of successive groups decrease with increase in the atomic number Z. This decrease corresponds to the wellknown increase in the size of atoms with increase in Z. On the other hand, there is an increase for the tricovalent rare earth metals, from La to Lu, corresponding to the "lanthanide contraction" caused by the only partial shielding of the increasing nuclear charge by the successive 4f electrons. A similar increase is shown for the bicovalent metals Ba, Eu, and Yb. It is also seen from Figure 2 that an analogous increase occurs in the molarity of the tricovalent actinide elements from Ac to Cf and of the bicovalent elements Ra and Es. The most striking feature is the nearly linear increase in the atomic density observed for the sequences K to Cr, Rh to Mo, and Cs to W, as seen in Figures 3 to 5. In these sequences
Figure 1. Lothar Meyer's periodic curve of atomic volume as a function of atomic weight (reproduce3 from Ref.( I)).
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Journal of Chemical Education
Figure 2. The atomic density, or molarity, of the principal aliobapes of the elements as a function of atomic number Z me values for the noble gases have been extrapolated to STP. For at r w m temperature and atmospheric sulfur, the values are shown with increasing molarlty in the order Sm, c-Sa, ST, and c-S8. The values for P correspond to white and black phosphaus, respectively. in the order of increasing molarily. Lines correspondingto covalence 2 and 3 have been drawn far the rare earth metals and for the actinide elements.
A similar effect, with metallic valence about 2.6 attributed to an 80%contrihution of the trivalent configuration sp2 and a 20% contrihution of the univalent . obnormal confieuration s 2. ~. is served for S C ~ Yand , La. The values 5.56 for Cu. 4.56 for Zn, and 3.56 for Ga and the same values for their congeners, given by the resonating-valence-bond theory of metals (7-9), are seen to be reasonably well supported by the atomic densities. T h e values of the molarity shown on the right sides in Figures 3 to 5 are slightly lower than where they should be according to the metallic valence values indicated by the lines. The reason for this is that the hvperelectronic metals have &shared electron pairs as well as valence electrons in t h e outer shell. These unshared electron pairs on adjacent atoms repel one another strongly, and accordingly reduce the observed molarity.
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Fioure 4. Molaritv versus atomic number Z for Me second long row of the periodic table.
Figure 3. Molarity versus atomic number Zfor the first long row of the periodic table. Lines have bean drawn to Indicate in an approximate way the metallic cavalencss vfrom 1 to 6.
the metallic valence (resonating covalence) increases from 1 to 6. We conclude that the molarity in the solid state is a reasonahlv reliable indicator of the metallic valence. I t is seen that the values of the molarity of Ca, Sr, and Ba lie about 20%below the lines for valence 2. We think that the low values of the molarity of these elements is explained by the stability of the completed subshell of two s electrons, corresponding to zero valence. Bivalence for these elements rewires excitation of anelectron to the next subshell, 4p, 5p, or 6 p . Accordingly, we assign t o these elements in thesolid state the metallic valence 1.6, corresponding to an 80%contribution 01 the bivalent s p configuration and a 20ro rmtriI,ution of the zero-vnlent normal coniiguration. 3.'.
Figure 5. Molarity versus atomic number Zfor the Mird long row of the periodic table.
Volume 62
Number I 2
December 1985
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