R. T Sanderson
Arizona State University Tempe, Arizona
The Nature of "Ionic" Solids The coordinated polymeric model
In the teaching of inorganic chemistry, the "ionic" concept of many solids is so firmly entrenched, and so convenient in the classroom, that it has become part of the "unquest,ionable." For example, recent textbooks of inorganic chemistry state: There are a. vast number of solid compounds which can be considered as very closely approximrating aggregates of positive and negative ions interacting in a purely electrostatic manner ( 1 ) . Crystals of simple inorganic salts consist of positive and negative ions packed in such a. way as to allow a minimum distance between cation and anion, and to prov~demaximum shielding of ions of like chargefromoneanother ( 8 ) .
Many similar statements are to he found in other modern textbooks. On the other hand, every well informed chemist realizes that some "covalent character" is probably a part of even the most "ionic" bonding, and most textbooks, including those quoted above, disclose or imply elsewhere that most bonds are intermediate between purely covalent and purely ionic. Nevertheless it seems much easier especially at the introductory level hut also at advanced levels, to establish a dichotomy of "covalent bonds" and "ionic bonds," or "covalent compounds" and "ionic compounds" as though these were the principal types. This may be quite satisfactory for the sophisticated inorganic chemist, who is well aware of the approximate nature and the limitations of such terminology. But it may be very misleading to the student as well as to the professional scientist specializing in other areas, such as geology and physics. Indeed, it even seems misleading to those who know better. Curiously, the very chemists who willingly agree as to the intermediate quality of most bonds are likely also to use "ionic radii" as though they were real, apply "radius ratio" rules based on these radii, and refer to crystals as "anion lattices" containing "tetrahedral holes" and "octahedral holes" capable of accommodating "cations." Great interest continues to be shown in attempts to modify or "correct" the Born-Mayer equation for ionic crystal energy so that it will give values in better agreement with experiment. In short, while paying "lip service" to the view of most bonding as intermediate in nature many chemists go on thinking of actual solid compounds as aggregates of ions. If often proves instructive to examine critically and to question the 'Lunquestionables" of science, and here is a good example. Let us take a close look at the "ionic" model of solids. First, why do we believe in the ionic concept? Perhaps the chief arguments can be summarized as follows: (1) The stoichiometry of many solids is consistent with the mutual "stabilization" of atomic outer shells through transfer of electrons to form ions, initially incomplete electron groups being either removed or completed.
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(2) All structural studies show these oompounds to be nonmoleoular. (3) Such solids commonly dissolve in polar solvents, especially water, giving solutions that by their conductance indicate the abundance of ions. The charges on the ions sre confirmed by conformancewith Faraday's laws. (4) I n the fused state, such substances readily conduct an electric current, with simultaneous deposit of elements a t the electrodes that appear to result from the discharge of their ions, again conforming to Faradsy's laws. (5) Ionic radii determined by reasonable procedures are found to be nearly invariant, and additive to give nearly oorrect internuclear distances in the crystals. (6) The Born-Mayer equation (to he discussed in a later section of this article), based directly on the ionic model, permits quite accurate oaloulation of the crystal energy for a number of salts, principally the alkali halides. (7) X-ray diffraction of certain alkdi hdidea has been interpreted as implying very close to the ionic numbel. of electrons around each nucleus.
Let us consider these arguments one by one. The first argument is correct to the extent that atoms of differentelements do tend to combine in relative proportions consistent with ion formation. For example, sodium reacts with chlorine 1:1, as it would if one electron from each sodium atom were transferred to a chlorine atom. Such a transfer would leave Ka+ ions isoelectronic with neon and C1- ions isoelectronic with argon. But the same stoichiometry would be predicted from the viewpoint of covalence, so ionicity is not established by this argument. Furthermore, the idea that structures isoelectronic with the M8 elements (helium group, (3)) necessarily have the special "stahility" of M8 is in direct conflict with the facts that all such ions have far greater chemical reactivity than the 1118 atoms, and that all cations, and some simple anions, are formed from the corresponding atoms only endothermically. The second argument is certainly consistent with the ionic model in that several close neighbors appear equivalent in their relationship to a given atom, and no evidence for discrete molecules has been observed. This does not rule out the possibility of the components being better described as atoms, however, rather than ions, as will he discussed below in much detail. The third argument is again consistent with the ionic model but by no means exclusive of other possibilities. One needs only to think of hydrogen chloride and hydrochloric acid to recognize that the formation of ions in solution does not prove their existence in the pure compound. With regard to the fourth argument, it is tempting to regard the conductance of fused salts and the electrodeposition of their component elements as convincing evidence of their existence as ions, even in the solid state. Yet a little salt will permit conductance of water and the electrodeposition of hydrogen and oxygen.
Does this prove that water consists of ions? We really do not know much about reactions at electrodes, or whether the old established criterion of conductance in the fused state is a valid proof of its ionic nature. Even if it were, a fused salt is ordinarily at a much higher temperature and therefore contains much greater energy. Can we say that ious in the melt at 1000°K mould prove the presence of ious in the crystal a t 29S°K? The fifth argument has never been very convincing because the nonpolar covalent radius sums and ionic radius sums for many compounds are nearly equal. It has been greatly weakened by very careful X-ray studies of the electron density from one nucleus to the next in a crystalline solid. These show a minimum (not zero) density at a point iutermediate between the nuclei. If ions as such have any individual identity within the crystal, one would expect that the density minimum would occur where one ion comes in contact with the other, owing to the n~ntualrepulsion of the electron clouds. Indeed, it is not easy to imagine any other interpretation of such a density minimum. Yet in fact, these minima, as they have been measured for several different common crystalline substances (4), do not correspond to the coriventional ionic radii (5, 6). Some typical data are given in Table 1. It will be Table 1.
Comporison of Some "lonic" Radii with Radii Estimated from Electron Density M a ~ s
Comoomd LiF
CaR CuCl NaCl KC1 CuBr MgO LiF
NaCl KC1 CuCl CuBr MgO CaF,
-Calc. Charge- -Radi~a, (HIAtom in ComooundY X-rav' " I ~ n i o " ~ Ian
F F
CI C1 CI Br 0 Li Na K Cu Cu
-Mg Ca
-0.74 -0.47 -0.29 -06i -0.76 -0.25 -0.42 0.74 0.67 0.76 0.29 0.25 0.42 0.94
1 .09 1.10 1.25 1.64 1.70 1.36 1.09 11.92 1.18 1.43 1.10 1.10 10 2 1.26
" Reference (19). Reference (4, 7b). Reference (6).
observed that the radius of the metal atom is always appreciably larger than the "ionic radius" of the cation, and the radius of the nonmetal atom is always appreciably smaller than that of the anion. If these electron density minima actually do exist and do represent the point of contact between separate atoms, then we must conclude that the conventional "ionic radii" are merely parameters which may be added to give approximate internuclear distances within crystals but do not indicate the true size of the atom or ion. Alternatively, if the conventional "ionic radii" are approximately correct, then the atoms in the crystals do not appear to be present as ious. We shall return to this point later. The Born-Mayer equation does give the crystal energies of the alkali halides with striking success. However, it become decreasingly satisfactory when applied to other salts, and for those compounds generally recognized as having considerable "covalent char-
acter" it fails. If these other compounds are truly ionic, then factors not yet fully accounted for need to be recognized and incorporated in the equation. Finally, the X-ray evideuce for the electron popula tion around the separate nuclei is open to some question, because it is not sufficiently accurate to distinguish between complete and partial ionicity, and because the electron density even a t the minimum is appreciable (4a, 7a). Furthermore, it has been pointed out by J. C. Slater (7b) that even an exact knowledge of the electron density distribution within a crystal would not serve to establish the degree of ionicity. I n summary, the evidence for the existence of ions in solids seems reasonable, appealing, and persuasive. It is also indirect, incomplete, ambiguous, and inconclusive. Is the lonic M o d e l Realistic?
One of the rationalizations of ionic bonding frequently found in textbooks concerns the relative influence of ionization energy, electron affinity, and coulomb energy of the ionic lattice. For example, it is recognized that the formation of a double charged magnesium ion requires a large amount of energy, the sum of the first two ionization energies or 526 kcal/mole, in addition to the 36 kcal needed to atomize the magnesium. Furthermore, the production of gaseous oxide ions requires not only the 59 kcal/g-atom dissociation energy but also about 170 kcal/g-atom to force the oxygen atoms to take on two electrons each. To rationalize the high exothermicity of the process by which magnesium metal combines with gaseous oxygen to form the extremely stable compound, MgO, it is pointed out that the electrostatic energy evolved by bringing together the gaseous ions into the rock salt structure is more than ample to compensate for the endothermicity of the initial processes. This is quite true, but also quite irrelevant. One reason for its irrelevance is that in magnesium oxide, as in many other such solids, the internuclear distance would be practically the same whether the bonds were single nonpolar covalent or completely ionic. I n other words, the sums of the ionic radii and the covalent radii are nearly equal. When we try to apportion the valence electrons between the magnesium atom and the oxygen atom, what we are really doing is deciding on the title to part of the territory which these electrons occupy. Except for their very important penetration toward the nucleus of one atom or the other, they occupy almost the same external region whether they belong to the magnesium or to the oxygen. This is represented schematically in the figure. Ionization energy is defined as the energy required to remove an electron from an atom to infinity, and electron affinity is the energy released when an electron comes from infinity to an atom. Such a transfer of title as would be involved in forming ions from the atoms in adjacent positions cannot involve either the ionization energy or the electron affinity. There is nothing wrong with following a Born-Haber cycle of hypothetical processes in this way; the error comes in taking the processes too literally. The calculation of crystal energy will be discussed in greater detail later, but at this point it may be instructive to point out that the estimation of lattice euergy from a Born-Haber cycle requires no hypothesis Volume 44, Number 9, September 1967
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If this should happen, as surely must happen under such
Figure 1.
Representationof MgO.
Solid liner are ions; broken liner, atoms
as to the exact condition of the combined atomsin the solid. It is simply the energy which would be released if the gaseous separate ions could be brought together to form the solid as it exists. It is necessarily the same as the ionic lattice energy, such as we might calculate by the Born-Mayer equation which assumes the ionic model, only if the solid actually exists as an aggregation of ions. Thermodynamic evidence that an ionic MgO is feasible does not by any means prove that R4gO is ionic. From the viewpoint of understanding and teaching chemistry, the chief objection to the ionic model is that it is too rigid, too restrictive. A compound must be either ionic or not.. There is no room, for instance, for the observed differences between the strongly basic CaO and the amphoteric ZnO, if both contain the same oxide ions. In general, what is the advantage of recognizing a wide range in the fundamental properties of the metallic elements if we intend to claim that they are all alike in converting the nonmetals to negative ions? How shall we explain the well known differences among solid fluorides, or chlorides, or oxides, or sulfides, in bond strength, in volatility, in solubility, in oxidizing power, in acid.-base properties, if all contain the same ions? The properties of ions, furthermore, are sometimes such as to stretch our credulity regarding the ionic model. An outstanding example is the oxide ion. As an isolated species, oxide ions would certainly be strongly basic, as probably everyone would agree. They should also he powerful reducing agents, for thermochemical cycles involving oxides suggest that the reaction 0%-t O e- should be exothermic by about 210 kcal/mole. We know of no ordinary binary oxide in which oxygen acts as a reducing agent a t all. To maintain the concept of the existence of true 02ions in solid oxides, we would be obliged to claim that the oxide ion is somehow "stabilized" by its environment. Its environment consists of cations on all sides, in intimate contact with it. All cations are oxidizing agents. Does it seem reasonable that a powerful reducing agent should be "stabilized" by being surrounded by oxidizing agents? (Or that a strong base should he "stabilized" by being in contact on all sides by electron pair acceptors?) The only way t,o remove the reducing power of a chemical reducing agent is by taking away some of its surplus of electrons.
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circumstances, the oxygen atom could then no longer be regarded as an 02ion. It would seem far more reasonable to admit the absurdity of O 2ions in crystals than to attempt to just,ify their imagined existence. Yet oxide ions are a common species in the thought and writings of students of solid oxides and conspicuous in applications of the ionic model of nonmolecular solids. The greatest defect of the ionic model arises from its attempt to treat the anionic particle as a point negative charge locat,ed a t the nucleus. In fact, of course, the nucleus of an anion, like the nucleus of a cation or of a neutral atom, bears a positive charge. If the anion were far removed from the cation we could treat it successfully as a point negative charge. But in fact it is not far removed or inaccessible; it is adjacent. The force of attraction is not between cation center and anion center, but between cation center and anion electrons a t its periphery. In other words, the ionic model is defective mainly because it is unrealistic. I t fails to accord recognition to the special properties of its alleged con~ponent particles. For every cation is an atom with an outer shell of vacant orbitals, which are not effectively shielded from its nucleus. Every simple cation is therefore a potential electron acceptor. Every simple anion is an at,om with an outer shell of filled orbitallone pairs of electrons made more available for coordination by the negative charge furnished by the excess electron(s). Therefore every such anion is a potential electron donor. How could a crystal be assembled in which each elcctron donor is surrounded by electron acceptors, and each acceptor surrounded by electron donors, without appreciable coordination? Surely no metal-nonmetal compound could be assembled of metal cations and nonmetal anions without substantial contribution to the total bonding energy made bv coordinate covalence. This would necessarily result in a reduction in the ionic energy contribution because the sharing of electrons of the anion by the cations would reduce the electrostatic charge on each atom. For this reason, even though we can imagine the crystal to be formed, hypothetically, by a condensation of gaseous cat,ions with gaseous anions, t,heir donor-acceptor interaction in close proximity must destroy their identity as ions in the crystal. A better description is needed. How might "ionic" crystals be better described? We need a model consistent with the experimental evidence that up to now has been interpreted as supporting the ionic model, yet free of the defects and limitations of the latter. We need a model that is applicable over a complete range of bond polarity. It should permit the calculation of the total bonding force whether largely ionic or largely covalent. I t should permit ready distinction among different conditions of any given nonmetal, consistent with the very useful interpretations based on partial charge (8). Such a model has recently been proposed, tested on more than a hundred compounds, and found remarkably satisfactory, although certain questions about the nature of the bonding in nonmolecular solids still remain unanswered.
The Coordinated Polymeric Model of Nonmolecular Solids
The alternative to the ionic model of nonmolecular solids recently described (9) is based on the hypothesis that such solids consist not of ions, but of atoms sharing electrons in accordance with their relative electronegativities and the principle of electronegativity equalization. The simplest molecule that might form by combination of metal with nonmetal would have two requisites for further condensation. Each metal atom would have unoccupied low energy outer shell orbitals made more available to donor atoms by electron withdrawal by the nonmetal. Each nonmetal atom would have outer shell lone pairs of electrons made more readily available to acceptor atoms by the presence of excess negative charge. An important principle of bonding is that chemical combination tends to continue in the direction of making maximum use of all available orbitals and electrons. Consequently the simplest molecules of metal-nonmetal compounds do not remain separate. They condense together by coordination to aggregates limited in size only by the number of available molecules. Because the bonds between one atom and its equidistant close neighbors must all be equivalent, the individuality of the molecules completely disappears as the coordination progresses. The aggregate may be termed a "coordination polymer." The proposed model is called the "coordinated polymeric" model of nonmolecular solids. The total bonding energy in such a compound can he calculated as the atomization energy, which is the sum of the standard heats of atomization of the separate elements at 29S01C, minus the standard heat of formation of the compound. This can be called the "experimental" value. The atomization energy also can be calculated by an extension of a new method used for bond energies of molecular binary compounds (10). This method (9) evaluates the total bonding energy as the weizhted sum of a covalent energy -. and an ionic energy. The covalent energy is assumed, following Pauling ( 5 ) ,to be the geometric mean of the homonuclear single covalent bond energies. This must be corrected for the reduction in h&d length that commonly accompanies ~olarity,by multiplying by the factor RJR,, the nonpolar covalent radius sum divided by the observed bond length: E,
=
nR, ~ E I * E ~ B
R.
is selected easily in this way for about 60%. The remaining 40'% include the compounds of the smaller metal atoms, Li, Na, Be, Rlg, Al, Zn, Cd. For reasons not yet understood, n appears to be 3 in these, rather than 4. Yet the results agree so well with the experimental atomization energies of these compounds that a fundamental significance seems quite reasonable. For example, the average difference between calculated and experimental atomization energies for 18 halides of these elements is less than 4 kcal/mole compared to 28 kcal using n = 4. The homonuclear single covalent bond energies when possible are those experinlentally determined. Other methods of empirical evaluation have been used (10) where experimental values are unavailable. The ionic energy is calculated as the Bm-R'layer crystal energy. The Born-Rlayer equation (411) is based on the assumption that a nonmolecular solid can be regarded as an aggregate of oppositely charged ions, essentially hard spheres with center of charge at the nucleus. The potential energy is evaluated as the electrostatic energy over the entire crystal, giving a net attractive energy which is the conventional coulomhic energy multiplied by a geometric factor based on the crystal structure, called the 1,fadelung constant. This attractive energy must be exactly balanced at equilibrium by repulsions among the adjacent electron clouds. These repulsions reduce the attractive energy in most simple crystals by about 1&1570. They can be represented most simply by a "repulsion coefficient,," k, dimensionless, which is equal to a factor (1 - p/RJ where p, having a value close to 0.3 A for most such salts, is determinable from studies of the compressibility of the crystal. As suggested by Baughan (11), the values of the repulsion coefficient k were taken as (1 - l/g,R,) where g, is chosen (I$), from a consideration of compressihilities, lattice energies, and spectroscopic data, to be 3.09 for NaCl type compounds, 2.68 for CaO type compounds, 3.20 for CaF, type compounds, and 2.85 for LizO type compounds. Observe that g, = l/p, and if p is 0.311 as is usually accepted for alkali halides and often other salts, g. = 3.22. If a typical bond length R, of 3 A is assumed, then k might vary from 0.876 for g, = 2.68 to 0.896forgO= 3.20. Uncertainties in k might in some compounds make an error as much as 5-6 kcal/mole. The ionic energy is then
(1)
E , is the covalent bond energy and n is the number of single covalent bonds or their equivalent that must be broken per formula unit in the process of atomization. For example, in copper(1) chloride, each atom is joined to four of the other kind. With eight valence electrons per atom pair, this means that four covalent bonds must be broken per atom pair for atomization, or, n is 4. In potassium chloride, each atom is bonded to six of the other kind, but with only eight valence electrons per atom pair, each bond can only average ?/$ of an electron pair bond. Six such bonds are the equivalent of four 2-electron bonds, so n is taken as 4. Of the more than one hundred solid binary compounds whose energies have been calculated to date (9), n
When the charge on an electron is taken as 4.8029 X 10-lo statcoulomb and Avogadro's number is 6.023 X loZS,the coulomb energy of attraction between one unit negative charge and one unit positive charge separated by a distance of 1 A is 138.94 X 10" ergs/mole. One erg = 2.3889 X lo-" kcal, so the product is 332 kcal per mole. This is the factor to convert the coulomb energy to kcal per mole. I n eqn. (2) also, z+ and zare the charges on the ions, M is the Madelung constant, and a is an empirical constant equal to 1 for all halides, but 0.63 for oxides and 0.60 for sulfides. Except for a, eqn. (2) is the Born-Mayer equation for ionic lattice energy. The most crucial part of this calculation is t,he assignVolume 44, Number 9, September 1967
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ment of weighting factors to determine the relative contributions of the covalent and ionic energies. The ionic weighting factor, t,, is sinlply the partial charge divided by the oxidation number, or in other words, the fraction of ionicity calculated for the compound. The covalent weighting factor is simply 1.00 - t , = t,. The final equation for the atomization energy of a binary solid is then E
ICE,
+ t,E. = 1,R.n R.EX
+
332t.kaMz+ z_ e2
(3) R, By use of this equation, the atomization energies of well over a hundred binary nonn~olecularhalides and chalcides have been calculated. Table 2 gives some representative results. For these calculations, all structural (15-15) and thermodynan~icdata (16-18) are taken from st,andard references. =
Table 2 . Calculated and Experimental Atomization Energies of Same Nonmolecular Solids (kcal/mole, 25°C)
Formda
ti
Covalent teE.
Ionic tiE,
E Calc.'
E. Exper.'
NaF TIF CaF* LiCl AgCl hlgC1. KBr CuBr ZnBn llbI BeI. JInL K10
nao
TlO* CsB CaS hInS MgSe CdSe * Mrnor d~screpancies between these values and those of refwences ( 9 ) and (19) are the resrdt of using the oonversion factor 330 insbead of 332 in the earlier wo1.k. b Reference (19), pp. 125-129, gives experimental values for more than 600 hinarv compounds, ralculated from the data of references (16-18).
Examples
In order to be sure that the method is clearly understood, the following calculatious are given in detail, for three fairly typical solids. First, however, a brief review of the method of calculating partial charge (8, 19) may be helpful. I proposed the principle of electronegativity equalization, that when t,wo or more atoms initially different in electronegativity combine chemically, they all become adjusted to the same intermediate electronegativity within the compound. When the initial atomic electronegativities are evaluated by the "relative compaotness" method, the electronegativity of the compound is taken as the geometric mean of the initial values of all of the atoms in the compound. The process of equalization is pictured as involving acquisition of partial charges through uneven sharing of the bonding electrons. An atom initially relatively high in electronegativity acquires more thau half share of the bonding 520
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electrons, thus becoming partially negative, larger, and less electronegative. An atom initially lower in electronegativity retains less than half share of the bonding electrons, causing it to become partially positive, smaller, and higher in electronegativity. The atoms are able to be equal in electronegativity within the compound because they share the bonding electrons unevenly and thus bear charge. To calculate what this charge is, we need to make two assumptions. One is that electronegativity changes linearly with charge. The other is that NaF is 75% ionic. The electronegativities are 5.75 for fluorine and 0.70 for sodium. The electronegativity in the molecule of NaF is therefore 4 0 . 7 0 X 5.75 = 2.01. The electronegativity of fluorine has changed from 5.75 to 2.01, or by 3.74, in acquiring a charge of -0.75. If it had acquired a charge of -1.00, its electronegativity change would have been 3.74/0.75 = 4.99. Similarly, the electronegativity of sodium has increased from 0.70 to 2.01, or by 1.31, in losing 0.75 electron. Had it lost the electron completely, its gain in electronegativity would have been 1.31/0.75 = 1.74. We define the partial charge on any combined atom as the ratio of its change i n electronegativity i n forming the compound to the change in electronegativi% correspading to unit charge. Having obtained the values of 4.99 and 1.74 for fluorine and sodium, one can use these to determine, in similar manner, the corresponding values for other elements. For example, to obtain the electronegativity change for chlorine that would correspond to formation of choride ion, we first calculate the electronegativity of NaC1. The electronegativity of sodium is 0.70, and that of chlorine 4.93. The electronegativity of NaCl is then the geometric mean of 0.70 and 4.93 = 4 0 . 7 0 X 4.93 = 1.86. Theelectronegativity ofsodium has thus changed by 1.86 - 0.70 = 1.16. If sodium had become Na+, we earlier determined that its electronegativity change would have been 1.74. Therefore in NaCl the charge on sodium must be 1.16/1.74 = 0.67. Therefore the charge on chlorine must he -0.67. This corresponds to an electronegativity change of 4.93 - 1.86 = 3.07. The electronegativity change for chlorine to become chloride ion is therefore 3.07/0.67 = 4.62. The partial charge on chlorine in any compound is then equal to the change in electronegativity of chlorine in forming the compound, divided by 4.62. These values, the electronegativity changes corresponding to the acquisition of unit charge, have thus been computed for all the elements for which electronegativity values are known or estimated (19). They are used in all calculations of charge in all compounds. By determining the electronegativity of the compound, one learns how much each atom has changed in electronegativity. The partial charge on each atom is the ratio of the actual electronegativity change to that correspondingto acquisition of unit charge.
*
*
Sodium Fluoride
At the time NaF was selected as the standard of bond polarity, an "isolated molecule" was deliberately specified because the condensed crystalline state was then understood only in terms of a completely ionic model.
I n the development of the concept of partial charge, relative values were deemed of greatest utility and their possible diierence from "absolute" values was not of great concern (8). The recent work on bond energies (9, lo), however, required absolute partial charge values. It was assumed that preliminary calculations using relative partial charges would indicate the magnitude of any correction factor needed. I n fact, they indicated that the intuitive choice of 75% for the ionicity of NaF was remarkably accurate, and that no correction from relative to absolute partial charge is needed. We therefore begin with weighting coefficients of t , = 0.75 and t, = 0.25. For the covalent contribution, the Na-Na bond energy is 18.0 and the F-F energy is 37.8 kcal/mole. The geometric mean of these is 26.0. The covalent radiusosum is 2.26 A and the observed bond length 2.31 A. As previously discussed, for most salts of metals of the smaller atoms, Li, Na, Be, Mg, and Zn, the equivalent number, n, of covalent bonds to he broken per formula unit is 3 instead of the expected 4 (9). The covalent energy contribution, t 8 , , is then, by eqns. (1) and (3): 1.E. =
0.25 X 3 X 2.26 X 26.0 2.31
=
lg kc
ole
The ionic contribution, t'Et, is calculated (equ. (2)) by multiplying the weighting coefficientti = 0.75 times the factor 332 to convert to kcal/mole, times the Madelung constant which is 1.75 for the rock salt lattice, times the product of the charges = 1, times the repulsion coefficient k = (1 - 1/3.09 X 2.31) = 0.86, divided by the internuclear distance 2.31A: tiE. =
0.75 X 332 X 1.75 X 1 X 0.86 2.31
=
negativity of oxygen has decreased by 5.21 - 2.32 = 2.69. If oxygen had acquired an electron completely, its electronegativity from the tabulated data for unit charge (8, 19) would have decreased by 4.75. The partial charge on oxygen, by definition, is therefore 2.69/4.75 = 0.57, but -0.57 because the electronegativity decreased. Similarly, the electronegativity of calcium increased from 1.22 to 2.52, or by 1.30. I t would have changed by 2.30 if an electron had been lost completely, as determined from the tabulated data (8, 19). Therefore its partial charge is 1.30/2.30 = 0.57. But since the ionic form would involve charges of +2, the value of t, is 0.57/2 or 0.29, making t, = 1 - 0.29 = 0.71. Calcium oxide has the rocksalt structure with each atom surrounded ~ctahedrally~by 6 equidistant atoms of the other element, a t 2.40 A. With only 8 valence electrons per pair of atoms (2 from calcium and 6 from oxygen) to form 6 bonds, each bond can average only = 4 / S electron instead of 6/a, or, each bond is only of a normal covalent bond. The equivalent number, n, of covalent bonds per atom pair that must be broken for atomization is therefore 6 X 2/a = 4. The homonuclear single covalent bond energy of calcium was estimated (10) as 37.8 kcal/mole, and the oxygen value is 33.2 kcal/mole. The geometric mean is 35.4 kcal. The covalent contribution, t s , , to the bond energy of CaO is calculated by multiplying together the weighting factor t, = 0.71, the value n = 4, 35.4 kcal, and the nonpolar radius sum 2.47-A, and dividing by the experimental bond length 2.40 A:
162 kc
The total atomization energy [eqn. (3) ] is the sum of the covalent and ionic contributions: 19 162 = 181 kcal per mole. The experimental value is obtained by adding the standard heats of atomization of sodium and fluorine, 25.9 18.9 = 44.8, and subtracting the standard heat of formation of XaF, -136.3 kcal per mole: 44.84- 136.3) = 181.1 kcal per mole. The Born-Mayer crystal energy is the same as t,EJt,, or 162/0.75 = 216, but -216 because this is the energy not of separation of ions but evolved when the gaseous ions form the crystal. To form gaseous atoms from the crystal, we may form gaseous ions, then remove the electron from fluoride ion and restore it to sodium ion. Therefore, the atomization energy is calculated by adding to the crystal energy the electron affinity of fluorine, -81, and the ionization energy of sodium, 118, and changing the sign: -(216 - 81 f 118) = 179 kcal per mole. Thus for this particular conlpound, the ionic description appears to be about as satisfactory as the coordinated polymeric description, for determining total bonding eoery. This is true, however, only for salts of relatively small covalent energy, as can be seen in the following examples.
+
+
For calcium oxide, the AIadelung constant is 1.75, and the repulsion coefficientk = [l - 1/(2.68 X 2.40)] = 0.84. In this as in all nonmolecular oxides the factor a (eqn. (2)) is 0.63. This holds consistently for the 20 solid oxides for which data have been available for calculation of atomization energies. As discussed elsewhere (9),the fundamental significance of this factor is not yet understood, but one may speculate that it corrects the repulsion coefficient for failing to evaluate the true repulsion associated with a hypothetical dinegative ion. If as partial charge calculation^ indicate, no actual oxide contains anything closely approaching 02- ions, then observations of compressibility of actual oxides must be misleading as to the correct value of the repulsion coefficient for a hypothetical completely ionic oxide. Whatever the true explanation may be, the fact that ionic energy contributions of all nonmolecular oxides studied have been consistently too high by a constant factor suggests that we may make the necessary correction with reasonable confidence. The ionic contribution, t,E,, is then calculated by multiplying together the weighting coefficient tl = 0.29, the conversion factor 332, the Madelung constant 1.75, the numbers of charges on each ion, 2 and 2, the repulsion coefficient k = 0.84, and the corr$ction factor 0.63, and dividing by the bond length, 2.40 A:
Calcium Oxide
The electronegativity of calcium is 1.22 and that of oxygen is 5.21. The geometric mean is 2.52, which represents the electronegativity in CaO. The electro-
The total atomization energy is then 103 Volume 44, Number 9, September 7967
+ 149 /
521
= 252 kcal/mole. The experimental value is obtained by adding the standard heats of atomization of calcium 59.6 = 103.6, and subtracting the and oxygen, 44 standard heat of formation of CaO, -151.9; 103.6(- 151.9) = 255.5 kcal/mole. The Born-Mayer crystal energy is obtained by the same equation without t , or a. In other words, it is equal to -149/0.29 X 0.63 = -815 kcal/mole. To change this to atomization energy, one changes the sign and subtracts the electron afiinity of oxygen, about 170, and the ionization energy of 418 kcal/mole to produce Ca?+ ions; 815 - 170 - 418 = 227 kcal/mole. This is in error by 28 kcal/mole.
+
Mongonese (11) Iodide
Manganese has been assigned the electronegativity value of 2.07, from which it has been determined that loss of one electron would increase its electronegativity by 2.99 (19). The electronegativity of iodine is 3.84, which leads to a value for MnIz of 42.07 X 3.842 = 3.13. Manganese electronegativity has risen from 2.07 to 3.13 or by 1.06. The partial charge on manganese is therefore, by definition, 1.06/2.99 = 0.35. The iodine electronegativity has decreased by 3.84 - 3.13 = 0.71. By becoming iodide ion its electronegativity decrease would have been 4.08, so the partial charge on iodine is 0.71/4.08 = 0.17. Thus t , = 0.17 and t, = 1 - 0.17 = 0.83. The homonuclear single covalent bond energy for manganese has been estimated as 33.6 kcal/mole (19), and that for iodine is 36.1, giving a geometric mean of 34.7 kcal/mole. Manganese diiodide has a cadmium iodide structure consisting of layers of condensed octahedra with each manganese atom octahedrally surrounded by six iodine atoms. Since there are only four available orbitals in the manganese outer shell, only an average of four normal covalent bonds can be formed per manganese, and n = 4. The covalent contribution to the total atomization energy, t&, can be calculated by multiplying together the weighting factor t, = 0.83, the equivalent number of covalent bonds n = 4, the geometric mean bond energy 34.7 kcal per mole, and the covalent radius sum 2.50 .k, and dividing by the observed bond length 2.96 A:
The Born-Mayer crystal energy is t,E,/ti = 75/0.17 -441 kcal/mole. This would lead to an atomization energy by changing sign, subtracting the ionization energy to produce Ah2+, 535 kcal, and adding twice the electron affinity, 2 X -72 = -144; 441 - 535 144 = 50 kcal/mole, in error by 12.5 kcal. This is of course consistent with the fact that here the covalent contribution is major. The successful application of this coordinated polymeric model of bonding in nonmolecular solids to more than a hundred compounds ranging in calculated ionicity from 8-89yo (9) certainly suggests not only that the concepts of electronegativity equalization and partial charge are essentially correct. It also shows that the dichotomy of covalent and ionic compounds is largely unnecessary, for the bond energies in molecular compounds have been calculated (10) in a very similar manner. We may recognize the following advantages of the coordinated polymeric model over the ionic model : =
+
..
(1) a. uniform c o n c e ~ of t bondinz aoolic~hleto . . I t Drovides . nonmolecular comoounds over a com~leteranw of hond ~~,~oolaritv. \ 2 1 1 pernln; rlwmtit~rivrP Y H I ~ H ~01 ~ O the ~ m~pg.~tu
