Electronegativity in Two Dimensions Reassessment and Resolution of the Pearson-Pauling Paradox Derek W. Smith University of Waikato, Hamilton, New Zealand The thermochemical basis for Pauling's scale of electronegativity (I) should make i t useful in semiquantitative thermochemical arguments. Pauling (2) has recently recalled the origins of his electronegativity scale. The familiar to the eq 1 relates the bond energies E A 4 ,EB-B,and EA-B electronegativities X A and x s of the atoms A and B: EA-B= 'IZWA-A+ E d + &A
-x d 2
(1)
ativity eonsiderations indicate that a hydride EH,-where E is of lower electronegativity than Si-should react exothermically with SiOz or SiCl4 to give SiH4.Thus reagents such as AlHa, MgH2, or NaH should come to mind. All of these can, in principle, he used in the preparation of SiH4. That the reaction 6 is exothermic can he confidently predicted from electronegativity considerations. HSAB arguments are much less helpful.
-
The constant a is equal to 1 eV, i.e., 23.1 kcal mol-' or 96.5 kJ mol-l. Equation 1 is most appropriate for single-bonded, gaseous, molecular species; the more general expression eq 2 is often invoked for any binary compound ApBT
SjIa(s) + ZMgHds) S i H k ) + 2MgI.h) AH' = -354 kJ mol-I
The divisor n is equal to pnA (= -qnd where nA and ne are respectively the oxidation numbers of atoms A and B. Any metathesis involving binary compounds can he written in the general form of eq 3:
2KF(s)
(6)
There are cases where neither electronegativity nor HSAB arguments correctly predict the sign of AHofor a methathesis. Examples include reactions 7 and 8.
-
+ CdCln(s)
CeS(s) + FeOM
-
2KCl(s) + CdF2(s)
AHo = -50 kJ mol-' (7)
CaO(s) f FeSW
AHD= +19 kJmd-' (8)
where (A, X) represents a binary compound ApX,, etc. I t is then easy to derive eq 4: Clearly, such a metathesis will he exothermic if it brings into chemical combination the atoms of lowest and highest electronegativity among A, B, X, and Y. The validity of eq 4 is implicity in a host of chemical arguments and strategies. For example, the use of an alkyllithium reagent R'Li to effect the preparation of R,ER1 from R,EC1 can he rationalized if E is more electronegative than Li. The highly negative heat of formation of LiCl provides the thermodynamic driving force. In 1968 Pearson (3)drew attention to a considerable number of cases where eq 4 fails to account for the sign-let alone provide a reasonahle estimate for the magnitude-of AHo for metathetic reactions. For example, the reaction 5 is expected from eq 4 to he endothermic: However, it is found from standard tabulations of heats of formation (4) that AH" for eq 5 is -52 kJ mol-l. Pearson argued that the thermodynamic feasibility or otherwise of a metathesis was far better assessed in terms of the principle of hard and soft acids and bases (5) (HSAB).In the case of eq 5, Cs+ is a much softer acid than Li+, and the metathesis proceeds in order to produce the "hard-hard" pair LitFand the "soft-soft" pair Cs+I-. Thus we have two undeniably valuable concepts; one stresses the attraction of opposites, the other suggests that "like goes with like". Often the two are in agreement in their predictions regarding metatheses; but just as often they come into conflict. This state of affairs has been called the "Pearson-Pauling paradox" (6). Although in such cases of conflict the HSAB prediction is usuallv correct, Pauline sometimes prevails over Pearson. ~ o r e x a m ~ lconsider e. the preparation of SiH,,starting from a readily available reagrnr such as SiOzor SiCI,. Elecrroneg-
Pearson (7-9) has recently discussed the relationship hetween electroneeativitr and hardness in terms of density functional theor),. while the meaning of hardness has bee" clarified, the scale of "ahsolure hardness" is not altogether satisfactory for quantitative purposes, and the "abSsolute e1ectronegativities"-derived by Mulliken's method (10)are not (unlike Pauling electronegativities) directly useful in thermochemical arguments. I propose that the two approaches can he reconciledoffering some insights into the nature of hardnesslsoftness-by asimple extension of eq 2. The most ohvious way of combining the principle that "like goes with like" and the concept of electronegativity is t o write eq 9:
The factor of 96.5 kJ mol-I as in eqs 1 and 2 has been rounded to 100 for obvious reasons. Values of x and y-taking care t o distinguish between different oxidation states of an element-were obtained by least-squares analysis of the heats of formation of 540 binary hydrides, halides, chalcogenides, and nitrides, all in their standard states in 25 OC/1 atm. Results and Dlscusslon Since eq 9 only gives differences (XA- XB)and CYA YB), the scales were established by setting x for H(1) equal to the Pauling value of 2.1, with y for H(I) arbitrarily set equal to zero. The resulting values of x and y are collected in the table and are rounded to the nearest 0.05 of a unit. With a few exceptions, the calculated heats of formation agree with experiment within the limits arising from the uncertainty of f 0.025 in each x or y value. The worst discrepancies were found for oxides of the most electronositive elements (x < 1.2) and hydrides. The x values are very similar to Pauling electronegativities ( I ) and to the revised values of Allred (11); they may therefore he described as such and used in the same way.
-
Volume 67 Number 11 November 1990
911
Values of xand y from Eq 9 (Rounded to Nearest 0.05) Element
x
Y
Element
x
Y
(a) Main Group Elements
(b) d- aml +Block Elements
The "middle element anomaly7'-the apparently higher electronegativities of Ga, Ge, and As compared with Al, Si, and P, respectively-emerges quite strongly, as in the scales of Allred (11) and Allred and Rochow (12). In most cases, x increases with increasing oxidation number as might he expected. There are some exceptions among the heavier p block elements, compare, for example, S(-II), S (IV), and S(V1). A possible explanation is the increasing participation of sulfur 3d orbitals in bonding with increasing oxidation state. From the standpoint of Mulliken's interpretation of electronegativity (lo), the valence state ionization potential 912
Journal of Chemical Education
of the S atom is expected to be lowered by involvement of these high-lying orbitals. What is the significance of the quantity y? Note that a scale of y-values ranging from -1.3 for Cs to +1.15 for O(I1)-could just as well have been chosen; the convention adopted here ensures that most of the values are positive. There is clearly a strong relationship between the x and y parameters. For the elements of groups 1, 2, 3d, 16 (-11 state), and 17 (-I state), there are good linear correlations between x and y. Evidently y tends to increase as we go down agroup and decrease as we traverse a period. y nearly always decreases with increasing oxidation number. Notice the abruot increase in v (accompanied by a decrease in x ) betweeh Cu(I1) and z ~ ( I I ) ;this is no doubt related to the fact that the chemistniof zincis not to he predicted by extrapolation along the 3dseries, and zinc behaves more like a p block element in many respects. An interesting "anomaly" in the 3d series is the fact that Cr(I1) and Cr(II1) have virtually identical values for both x and y. This is probably to be attributed to crystal field stabilization energy, which contrihutes a significant amount to the stabilities of Cr(II1) compounds; the enhanced values of -AHor require a lower electronegativity than might have been expected. At first glace, we might discern a correlation between y and the atomiclionic radius. Hvdroeen-not unexpected- ly-fails to fit into such a pattern. The isoelectronic series Au(1). He(I1). Tl(II1). and Pb(1V) all exhibit lower values of j than their rbunter~artsimmediatelv above in the periodic table. It is beine increasingly recognized than relativistic effects are important in Griderstaiding the atomic and chemical properties of these heavy elements (13). tentatively that y is t6 he regarded as a measure I of the diffuseness of an atom's valence orbitals--this of course is closelv related tosize. and to factors such as the ionization potential which, in part, determine electrouegativity. The combination of a very diffuse atomic orbital with a very compact one usually leads to a very small overlap internal. , because of cancellations between oositive and neaative overlaps. Consider, for example, the LiF molecule. We exoect that covalencv here will arise mainlv from overlap be'tween the atomic orbitals 2s(Li) and Zp,(F). The overlap internal-calculated from Slater functions-is only about 0.06;much smaller than the integrals S(3s, 3s) or ~ ( 2 p , ,Zp,) in Liz and F2, which are calculated to be, respectively, 0.58 and 0.17. Given the large difference in energy between 281Li) and 2p(F), even a crude MO calculation results in negligible covalency for LiF. In the case of the NaF molecule, the overlap integral between 3s(Na) and Zp,(F) is only 0.03. The reason for this poor overlap is obvious from Figure 1, which shows the overlap between 2s(Li) and Zp,(F). More elaborate calculations (14) confirm the view that the diatom-
Figure 1. Schematic illustratlan of Me poor overlap between a compact fluorine 20 orbital and a diffuse llthlum 2s orbital.
ic group 1fluorides are t o be viewed as ion pairs, and we may suppose that the crystalline solids also are purely ionic. I t will be noted from the table that t h e y values for S(VI), Se(VI),and Te(V1) are allgreater than for thecorresponding IV states, against the general trend. This is consistent with the interpretation just offered of the meaning of y. In going from, e.g., S(IV) to S(VI), there is expected to be a considerable increase in the extent of 3d orbital involvement; the contribution made by these rather diffuse orbitals will tend to raise the magnitude of y. The steep increase in y, and drop in x, in going from Cu(I1) to Zn(I1) has already been noted, and may be attributed to the lack of involvement of 3d orbitals in bonding in zinc compounds. The numbers collected in the table encapsulate a vast amount of inorganic chemistry. Some examples now follow. Stabilkation of Oxidation States in Binary Compounds
Among elements which exhibit variability of oxidation number, the highest states are respresented mainly by fluorides, oxides, oxofluorides, etc., while iodide tends to stabilize low oxidation states; lower fluorides and oxides may be susceptible to disproportionation. This can he quantitatively explained from eq 9 and the results of the tahle. For example, for a fluoride of an element A we can write eq 10: -AIfor(AF,) = 100n[(4.05 - x d 2 - ( y + ~1.05)2]kJ mol-I (10)
+
+
Now if we increase the oxidation state of A to n 1or n 2, the effect of an increase in XA will usually be at least partly offset bya decrease inyA.But in thecase of an iodide AI,, we have eq 11: -m't(AI,)
= 100n[(2.65 - x d Z
- (0.75 - yd2]kJ mo1-'
(11)
In practically all casesof interest, y~ will be less than 0.75 for all stable oxidation states of A. An increase in n will always lead to a decrease in -AHof/n, and this will restrict the stabilities of iodides, toward decomposition to a lower iodide or to the elemental substances. I t is easy to show from eq 11 that the heat of formation of AI, will be positive if (XA- y ~ ) is greater than 1.9. Thus we can immediately predict from the table that TIIS, Pb14, Sb15, SI4, SeIs are unstable with respect to the elemental substances. Others such as P k , CrIa, Fe13, and Cu12 are predicted to have marginally negative heats of formation but can be shown to be unstable with respect to lower iodides. However, we would also p r e d i c t incorrectly-that iodides of the soft elements (soft in the HSAB sense) P t and Au should be unstable. These and the corresponding sulfides do appear to have some "extra" stability, in agreement with HSAB principles.
close to the borderline but is usually classed as a hard acid. As3+, with (x y) = 2.15 should also be borderline but is likewise classed as a hard acid. I t must be remembered that perceptions of hardness/softness are largely based on chemical reactions other than those involvine- binarv com~ounds: in particular, complex formation in solution is an important source of information. Aside from these anomalies, the (x y) measure is quite reliable in assessine the relative softnesses of cationic acids. For example, pearion (17)has noted that, although hardness usually increases with increasing charge for cations formed by a given element, T1 and P b seem to behave perversely. Thus Pearson states that T13+ is definitely softer than Tl+ and that Pb4+mav be softer than Pb2+. The x and v values in the table support these assertions. Large increases in x evidentlv accomoanv - " the removal of the tiehtlv" held 6s electrons. An alternative criterion for hardness might be the formation of an unstable iodide, in which case eq 11 yields the difference (x - y) as a quantitative measure. But this is a poor criterion because hard-hard attraction and hard-soft enmity are much more marked than soft+oft attraction. Thus the hard base 0'- discriminates far hetter among hard and soft acids than does the soft base I-. On the other hand, (x - y) seems appropriate as a measure of hardness among bases; it was shown from eq 11 that the formation of AI, would he endothermic if (XA- YA)> (XI- YI).This gives the order of increasing hardness among simple anions as:
+
+
~.
which seems reasonable. Note that this is not quite the same as the sequence of electronegativities x. Thus we can say t h a t w i t h i n the context of binary com~ounds-softness amone cationic acids is associated with a combination of high eleGronegativity and relatively diffuse valence orbitals, which ia rouehlv related to size: hardness is associated withlow electron&aiivity and compact valence orhitals. The somewhat ambi\,alent character of Cs7 is related to the fact that Cs has both the lowest x and the highest y among the elements. I.!
0 X
so11 Borderline Hard
A Ouantitatlve Measure of Hardnes/Softness
I.(
One of the hallmarks of a soft acid (such as Ag+, Hg2+)is the formation of an oxide that is thermally rather unstable, having a low value of -AHof. By substituing the x and y values for 0(-11) into eq 9, it is easy to show that the heat of formation of an oxide AO,,z or A20, will be positive if (xA YA)is greater than 2.5. Among the elements listed in the tahle, only N(III), CI(I), Br(I), I(I), A N ) , and Au(II1) meet this requirement. The corresponding oxides are indeed unstable under ordinary conditions. The limit of (x y) = 2.5 is approached closely by Ag(I), Hg(II), Pt(II), and Pt(1V). Is the sum (x + y) a measure of softness? In Figure 2 y is plotted against x for a number of cations that have been classified by Pearson (15) as soft, hard, or borderline. The classification in Figure 2 is the same as that of Pearson except that Cs+ has been reclassified as borderline; i t was originally (15) classed as soft, but in more recent tabulations (16) i t often appears in the "hard" column, though sometimes in brackets to indicate that i t is close to the borderline. It will be apparent that Figure 2 presents quite a neat demarcation between hard and soft acids, with the borderline cases close to a straight line whose equation is (x y) = 2.15. There are some anomalies, of course. H+ is apparently
+
Y O.!
+
+
I
0.:
Figure 2. Plot of yvs. x, showing the demarcation of son, hard, and borderline cationic acids; the straight line has a slops of -1. Volume 67
Number 11 November 1990
913
of crystalline solids like AgF, ZnF2 or PbF2, which are more convenient than oxidative fluorinating agents such as BrF3 or F9 itself. An obvious wav of assessine the relative ~trengthsof binary fluwides as metathetic fluorinating aeents is to list them in order of the differences IVlariAF~. . .. --AHof(~~1,)].I now show how this can be achieved graph< cally, using aplot of y against x for elements of interest. If we insert thevalues of x and y for F(-I) and CI(-I) into eq 12, it is easy to show that the metathesis:
-
(A, F) + (B, C1)
-
(A, C1) + (B, F)
+
(13)
+
~ (XB ) 1.32yd;the factor will be exothermic if (XA 1 . 3 2 ~> - YF).Thus the of 1.32 is equal to the ratio (XF- x~)/(.YcI quantity (xa 1 . 3 2 ~is~ a) measure of the strength of AF, as a fluorinating agent by metathesis with chlorides. If we plot y against x for a number of elements and draw a straight line of slope -0.76 (i.e., the reciprocal of 1.32) passing through the point representing the element A, then we can a t aglance identify those chlorides that can undergo metathesis by reaction with AF,; all other elements B whichlie above the line will form fluorides that are stronger fluorinating agents than AF,. An example is shown in Figure 3, where we see a t a glance the scope of PbF2 as a fluorinating agent. I t can be used to prepare BF3, PF3, BeF2, LiF, etc., from the respective chlorides: but i t will not react exothermicallv with CsC1. TICI, or Agc1under standard conditions. The value of A ~ F as a fluorinating agent will be apparent from Figure 3. If the tendency to undergo the metathesis represented by eq 13 were to be used to construct a scale of hardness/ softness, our index of hardness for acationicacid would he (x 1.32y), which will give a rather different scale from the (x V) index sueeested above. This serves to em~hasizethe poiit that the Fllative positions of two species onsuch a scale is likelv to denend on what criterion is beine" aoolied. .~ l o t like s 'those shown in Figures 2 and 3 extend the conceot of electroneeativitv into two dimensions. As in twodimensional chroma~ography,points that cannot be distinguished on the conventional scale are readily resolved. On the usual scale, we have in the intermediate range 1.5-2.5 a remarkable variety of species; for example, Ag, B, and Sb are usually quoted as having about equal electronegativities (1.9-2.0). but these are very different elements chemically. However, they are well separated on the two-dimensional scale. Distinctions between metallic and nonmetallic elements. the formation of basic. acidic. or amnhoteric oxides. and other features become better ddmarcated than on the conventional one-dimensional scale.
+
Figure 3. Plot of y vs. x for selected elements; the straight llne has a slope of -0.76.
+ +
Among anionic bases, softness is associated with a combination of low electronegativity x, and diffuse valence orbitals leading to a large y; hard bases have high electronegativity and compact valence orbitals. I t is obvious from the (x v) and (x ". . - "v .) criteria and from ea 9. that the hard-hard combination will lead to highly exoihe*mic heats of formation. while the hard-soft combination is less exothermic. I t is not immediately obvious that there is any special stability associated with the sofbsoft combination. Although, as already noted, compounds like PtS and HgI2 do have more negative heats of formation than would be calculated from eq 9 and their x and y values from the table, they are not particulary stable in an absolute sense.
+
Metathetical Reactions between Binary Compounds For a metathesis such as that represented by eq 3, we can derive from eq 9 an equation analogous t o eq 4: -AHoln = 200[(xA- r,)(xx
- xy) - CYA - YB)CYX - yy)lkJ mol-' (12)
From eq 4 it is asserted that the metathesis will be exothermic if the products include the most and least electronegative species in combination. The analogous prediction based on eq 12 is less straightforward. However, we can say that the metathesis will he particularly favorable if one of the products brings together the atoms of highest and lowest x and if the ~-~~ elements constitutine one of the reactants have the greatest disparity in y. In other words, there is a tendency to bring together atoms whose electronegativity difference Ax is greatest and to keep apart atoms where Ay is greatest. These reouirements mav come into conflict: this accounts for the earson on-padkg paradox" and for those cases where both aooroaches aooear to fail. One of themost impohant types of metathesis between hinarv compounds is the preparation of fluorides, by means ~
914
Journal of Chemical Education
Acknowledgment I am grateful to Lisa-Marie Costley, who performed most of the numerical computations of x and y values. Literature Clted 1. Psuling, L. T h r Noturea(Ihe Chemical Bond,3rd ed.:Cornell University Ithaes. NY, 1960 p91. 2. Peuling, L. J Chem. Educ. 1988.65. 375. 3. Pearson,R.G. Chsm. Commun. 1368.6547. 4. The primary source of thermahernieai data for this paper is Wagman, D. D. etal. J. Phys. Chom. Re(. Dots 1962.11, Supplement No. 2. 5. Pearson. R. G., ed. Hard ond Soft Acids ond Barer; Dowden, Hutchinson and Ross: Stroudsburg, PA, 1973. 6. Huheey, J. E. Inorganic Chemistry. 3rd. ad.:Harper and Raw: New Yorh. 1983:p 320. 7. Pearaon, R. G. J. Cham. E d u r 1987.64.561-567. 8. Pears0n.R. G.lnorg.Chm~1988.27.737-740. 9. P~arann,R. G. J. Am.Chem.Soc. 1988.110.76647690. 10. BrsWh,S.J. Chsm.Edue. 1988,65,34,223. 11. Allrod, A.L. J.lnargNuc1.Chrm. 1961,17,215-221. 12. Al1red.A. L.; Rochow, E. G.J.lnorg. N u c l Chsm. 1958.6.26P270. 13. Pyvhho, P. Chsm.Rau. 1988.88.563494. 14. Jordan. K. D. A l k o l i H o l i d ~Vapors;Oavidouit~.P.;McFadden, D. L.,Eds.;Aeademie: New Yorh, 1979;pp 479-534. 15. Pesrson, R. G. Seianca 1966,151,172-177. 16. Ref 6, pp 31"315. 17. Pesrson, R. G. J. C h ~ mEduc. . 1968.15,581687.
