914
J. Phys. Chem. 1990, 94, 914-911
estimate of the number of “condensed” counterions by calculating the coordination number n32, that is, the number of cations 2 around the large anion 3, according to the formula
The results of such a calculation are reported in Figure 6 . We see that in the case of small a a substantial number of counterions (25 or more) coordinates around each micelle in a shell of thickness equal to their own diameter, 0.1R3. These particles can be considered as bonded to the surface of the large anions, thus reducing the “micellar” charge by a factor of 2. This same picture of the short-range structural arrangement emerges from g32. In fact there is almost an order of magnitude drop in the value of this rdf within a distance 0.2R3 from the contact (see Table I1 and Figure 2 ) . A similarly rapid drop in the distribution of counterions around micelles is observed in the e~periment.~~ The MSA estimate of n32 (not reported in the figure) is in remarkable good agreement with the HNC; at I’ = 0.14, a = 0.1, and r / R 3 = 0.65 one obtains n32 = 24; at the same distance the H N C yields n32 = 26. Other H N C calculations yield similar results for t 1 3 2 . ~ From Figure 6 one can also see that the effect of r on the counterion condensation is a minor one. By converse, the effect of a looks substantial. In fact, screening of the micellar charge is much less effective at a = 0.4than at a = 0.1. This can be seen by comparing the distances where n32 40 at a = 0.1 and at a = 0.4, respectively; in the latter case this distance is almost twice than in the former. These last features of the H N C results are also fairly well reproduced by the MSA. As far as the effect of the addition of the salt on n32 is concerned, we see from Figure 6 that the counterion coordination number is sensitive to such a variation. However, by taking into account that in the added salt case the number of available counterions is twice that of the salt-free solution (see section 111), this effect can be considered a moderate one. The SANS data yielded a relative insensitivity of the counterion condensation to the addition of the salt.33
We note that in the course of this paper we have investigated the simplest possible configuration of a solution plus an added salt, by considering a mixture formed of only three ionic species. However, the MSA solution scheme is valid for a generic multicomponent mixture and the HNC can easily be extended to deal with four or more ionic components. This would allow us to treat, e.g., the interesting case of a solution containing counterions of different charge, and to study their competition in the condensation process around micelles. Calculations of this kind will be undertaken in the immediate future. It is also worth noting that the H N C is assumed to work satisfactorily for polyelectrolytes mainly on the basis of its previous good performances for the primitive model. However, computer simulations have been performed quite recently for the two-component (micelles and counterions) case, at not too high charges, by Linse and Jonsson4and Vlachy et al? These authors have found that the H N C is, on the whole, in good agreement with the simulation results, although its predictions tend to worsen with the increase of the asymmetry of the ionic charges. While some more investigation is probably necessary in order to assess the overall performances of the HNC, we believe this theory can still be considered semiquantitatively accurate and therefore used, as we have done in this paper, as a benchmark for the accurateness of the more simple MSA. This latter theory shows well-known faults and, sometimes, discrepancies with respect to the HNC; this is particularly true for systems formed only by micelles and counterions, as found el~ewhere.~ However, as it has been shown above, the theory approximates rather closely the HNC predictions in the case of micellar solutions with an added salt and in the limit of very unequal sizes. This is particularly true for not too high charges (e.g., case C of Table I). It seems then possible to conclude that the MSA can be used, with the above limitations, as a rapid tool for investigating micellar solutions and charged colloids which can be realistically modeled as charged hard-sphere mixtures. Acknowledgment. This work has been supported by the Centro Interuniversitario Struttura della Materia of Minister0 Italian0 Pubblica Istruzione, and by the Gruppo Nazionale Struttura della Materia of Consiglio Nazionale delle Ricerche.
New Electronegativlty Scale. 11. Comparison with Other Scales in Correlating Heats of Formation Yu-Ran Luo and Sidney W. Benson* Donald P. and Katherine B. Loker Hydrocarbon Research Institute, Department of Chemistry, University of Southern California, University Park, Los Angeles, California 90089- I661 (Received: April 3, 1989; In Final Form: July 20, 1989) Heats of formation [AfHo(RX)] for compounds of the type CH,(CH3),..,X and differences in heats of formation [AfHo(CH,(CH3)3-,X) - AfHo(CH3X)],with m = 1-3, are plotted against values of electronegativity obtained from 17 different scales. It is found that our newly described electronegativity Vx gives the least scatter and the highest correlation coefficients as compared to the other scales. The largest deviations in the other scales are always observed when X = H, the “hydrogen anomaly”. This is discussed in terms of the definition of covalent radius, which is used in many of the scales including our own. The V, scale correlates the data linearly within experimental accuracy, which ranges from 0.3 to 1.0 kcal. It is found that rXHbond lengths, X-H differ from the covalent sum r H o+ rxo by an amount that correlates well with IVX - VHI and less well with Ixp(X) - xp(H)I.
Introduction The concept of electronegativity as the attraction of an atom for electrons has been an important qualitative tool for chemists almost since the earliest days of quantum theory. Pauling’ was one of the first to try to provide a quantitative scale for the
elements. On the basis of heats of formation of gas-phase species and average bond energies derived from heats of formation, it provided a semiquantitative measure of the stability of chemical bonds. Since this initial effort, many proposals have been made in an attempt to improve the concept of electronegativityz4 and
( I ) Pauling, L. The Nature of the Chemical Bond, 3rd ed.; Cornell University Press: Ithaca, N Y , 1960.
(2) Huhey, J. E. Inorganic Chemistry: Harper and Row: New York,1975.
0022-3654/90/2094-0914$02.50/0
0 1990 American Chemical Society
The Journal of Physical Chemistry, Vol. 94, No. 2, 1990 915
Heats of Formation vs Electronegativity Scales
TABLE I: Correlation of Heats of Formation with Various Electronegativity Scales and Comparison of Electronegativity Values from Various Scales (a) AAfHo(RX/CH3X)(kcal mol-') and Various Electronegativity Scales
X F
hAfHo(C2H~X/CHjX) AAfHo(i-C,H7X/CHIX) AAfHo(r-C,H,X/CH,X)
OH
CI
NH,
Br
SH
I
CHI
H
-8.1 -16.9 -26.5
-7.2 -15.0 -24.0
-5.8 -14.5 -23.4
-6.3 -15.3 -23.1
-5.6 -12.7 -20.7
-5.3 -13.1 -20.7
-5.0 -12.1 -20.2
-2.2 -7.2 -14.3
(b) Electronegativity Values for Atom or Group X in Various Scales
X Pauling' Mullik& Gordy8v9 Allred-Ro~how"-~ Li" Sander~on'~.'' Gaolo Wells'7
XP
FSG0I6 In a m o t ~ ' ~ . ~ ~
XF
St. John-Bloch15 Mande18 Boyd-MarkusZ1 ZhangZ2 William2' Yuan24 revised Pauling scale' 'See text.
XU XG XA XL XS XGA
XE
XI XI XMA
XB XZ
xw VS
F 4.0 3.91 3.95 4.10 4.0 3.92 4.44 3.95 4.00 3.05 4.01 4.34 4.00 4.188 0.517 9.915 (4.04)
OH 3.5 3.17 3.45 3.50 3.5 3.46 3.61 3.7 3.52 3.02 3.51 3.71 3.53 3.642 0.434 8.11 (3.43)
CI 3.0 3.0
3.50 2.83 2.93 3.28 3.07 3.03 2.84 2.37 2.35 2.97 3.14 2.835 0.245 7.04 (3.07)
NH2
Br
SH
I
CH3
H
3.0 2.33 2.98 3.07 3.0 2.93 2.72 3.35 3.16 2.71 3.01 3.26 3.23 3.062 0.332 6.67 (2.95)
2.8 2.76 2.75 2.14 2.79 2.96 2.83 2.80 2.52 2.32
2.5 1.41 2.58 2.44 2.68 2.66 2.58 2.8 2.52
2.5 1.56 2.50 2.21 2.41 2.50 2.68 2.41
2.5 1.63 2.52 2.50 3.5 2.41 2.43 2.3 2.35 2.21 2.50 2.73 2.53 2.536 0.165 5.19 (2.45)
2.1 2.28 2.17 (2.20)b
2.89 2.78 2.529 6.13 (2.77)
2.15 2.12 2.63 2.65 2.479 0.109 5.71 (2.65)
2.56 2.48 2.142 5.25 (2.47)
2.3 1 2.28 2.00 1.94 2.271 0.0 2.70 (1.61)
From thermochemical data, see ref 7.
give it a more general applicability. This has included attempts to assign values not just to bonded atoms but also to functional group^.'*^-^^ This has given rise to a proliferation of such scales based on different properties of atoms or bonds. The Pauling scale,' xp, is based on thermochemical data; the Mulliken scale: xM,on ionization potentials and electron affinities; the Allred-Rochow ~ c a l e ,xA, ~ . ~on estimated effective nuclear potentials and covalent radii; the Gordy scale,89 xG,on the number of electrons in the valence shell of the atom and the covalent radius; the Sanderson ~ c a l e , ' ~xs, + 'on ~ covalent radii; the Phillips scale,I4 xh,on dielectric properties; the St. John-Bloch scale,'5 xJe,on quantum defects; and the Zhang scale,22xz,on ionization energies and covalent radii.
(3) Marriott, S.; Reynolds, W. F.; Taft, R. W.; Topsom, R. D. J. Org. Chem. 1984, 49, 959. (4) Royer, D. J. Bonding Theory; McGraw-Hill: New York, 1968. (5) Mulliken, R. S. J. Chem. Phys. 1934, 2, 782; 1935, 3, 573. (6) Allred, A. L.; Rochow, E. R. J. Inorg. Nucl. Chem. 1958, 5 , 264. (7) Allred, A. L. Ibid. 1961, 17, 215. (8) Gordy, W. Phys. Rev. 1951,69,604; J. Am. Chem. Soc. 1952,74,272. (9) Gordy, W.; Thomas, J. 0. J. Chem. Phys. 1956, 24, 439. ( I O ) Gao, X.-H. Acta Chim. Sin. 1961, 27, 190. ( I 1) Li, S. J. Ibid. 1957, 23, 234. (12) Sanderson, R. T. J . Chem. Educ. 1952, 29, 539; 1954, 31, 2, 238. ( 1 3) Sanderson, R. T. Chemical Bonds and Bond Energy, 2nd ed.; Academic Press: New York, 1976. (14) Phillips, J. C. Covalent Bonding in Crystals, Molecules, and Polymers; University of Chicago Press: Chicago, IL, 1969. (15) St. John, J.; Bloch, A. N. Phys. Rev. Lett. 1974, 33, 1095. (16) Simons, G.; Zandler, M. E.; Talaty, E. R. J. Am. Chem. SOC.1976, 98, 7869. ( 1 7) Wells, P. R. Prog. Phys. Org. Chem. 1968, 6, 111. (18) Mande, C.; Deshmukh, P.; Deshmukh, P. J. Phys. E 1977, 2293. (19) Inamoto, N.; Masuda, S. Tetrahedron Lett. 1977, 3287. (20) Inamoto, N.; Masuda, S.;Tori, K.; Yoshimura, Y. Ibid. 1978,4547. (21) Boyd. R. J.; Markus, G. E. J. Chem. Phys. 1981, 75, 5 3 8 5 . (22) Zhang, Y . H. Inorg. Chem. 1982, 21, 3886. (23) William, S. M.; Reynolds, W. F.; Taft, R. W.; Topsom, R. D. J. Org. Chem. 1984, 49,959. (24) Yuan, H. C. Acta. Chim. Sin. 1964, 30, 341.
3.5 Allred-Rochow
3 .O XA 2.5
3.5
-!O
-8
-6
-4
A A , Ho(C,H,X/CH,X)/kcul
-2
0 mol-'
Figure 1. Relations between AAfHo(C2H5/CH3X) and some electronegativity scales.
The Pauling scale is the most familiar and probably most widely used. However, the Mulliken and Allred-Rochow scales have been also widely accepted as alternatives to the Pauling scale. The other scales have not been widely used. Curiously, there are approximately linear relations, in fact, between all scales and the Pauling scale although they have different dimesions. Typical examples are xp, (energy)'/2; xM, energy; and xA, force. For the prediction of thermochemical data, in particular heats of formation, the method of group additivity has had great success and broad u t i l i ~ a t i o n . ~ However, ~-~~ it requires a rather large
The Journal of Physical Chemistry, Vol. 94, No. 2, I990
916
Luo and Benson
TABLE 11: Correlation Coefficients for the Five Best Electronegativity Scales for the Description of A A , W (RX/CH,X) rank order
AAiHO(C,H,X/CH,X)
example tested AAiHO(i-C,H,X/CH,X)
AAlHO(t-CdHoX/CHIX)
1
Vx (0.9806)
Vx (0.9674)
Vx (0.9910)
2 3
(0.9208) (0.9136) xs (0.9032) xp' (0.8892)
(0.9218) X B (0.9110) x p (0.9003) xp2(0.8707)
XS
4
5
XM
XB xp
(0.9569) (0.9459) x w (0.9357) x: (0.9189) xp
- 3.5
amount of data based on the values of ArHo for key molecules in homologous series. Much less demanding of data input is the method of bond additivity. However, it is also less precise, being only slightly better than the Pauling use of average bond energies. In consequence, there has been great interest in finding methods to derive group contributions from atomic proper tie^.^^ None of the various electronegativity scales have lent themselves to improving this situation. In recent years, Schleyer and c o - w ~ r k e r reported s ~ ~ ~ ~good linear relations between the differences of ab initio calculated heats of formation (ArHo) of compounds CH3X, C2H5X,and i-C3H7X and the Pauling scale or Allred-Rochow scale. However, when we use experimentally measured ArHo 33 instead of the calculated ArHo,the correlations turn out to have appreciable scatter with deviations of up to 2 and 3 kcal/mol. This is illustrated in Figure 1. The increased scatter must be attributed to the larger uncertainty of the MNDO estimated heats of formation. I n this paper, we will check and compare the correlations between AArHo(RX/CH3X&where R = CH>,,,(CH,),, m = 1-3, and various electronegativity scales. What may be labeled the "hydrogen anomaly" will be discussed.
- 3.0 - 2.5
XP
- 2.0
-17 -15 -13 -11 -9 AA,Ho( i-C,H,X/CH,X)/kcoi
-7 mol-'
Figure 2. Relations between AAfH"(i-PrX/CH3X) and some electronegativity scales.
Relations between AAfHo(RX/CH,X) and Various Electronegativity Scales
The values of AArHo(RX/CH3X),where R = C2Hs, i-C3H7, and t-C,H,, and the various electronegativity scales of X, where X represents the atoms and/or functional groups, are listed in Table la. Here AArHo(RX/CH3X)
E
ArHo(RX) - ArH'(CH3X)
(1)
is the difference in heats of formation of RX and CH3X. All heats of formation are from ref 33. There are 16 kinds of various electronegativity scales. These are listed in Table Ib along with the appropriate symbol and the assigned value of electronegativity for the atom or group X in that particular scale. Three measures of the goodness of fit (average absolute error, root-mean-square error, and correlation coefficient) for the linear correlations between AArHo(RX/CH3X) and various electronegativity scales have been determined for each of the three cases with Table 1a.b. We have actually explored 17 scales, the 16 scales in Table Ib plus,:x which is the square of xpbecause it has the dimensions of energy. We then chose for each of the three examples of AAfHo the correlation coefficient as a measure of goodness of fit. The electronegativity scales with the five highest Figure 3. Relations between AAfHo(r-BuX/CHx) and some electro(25) Benson, S. W. Thermochemical Kinerics, 2nd ed.; Wiley: New York,
negativity scales.
(26) Benson, S. W. Chem. Reu. 1978, 78, 23.
correlation coefficients are listed in Table 11. The values in the parentheses are the respective correlation coefficients. We have also made plots of AAfHo(RX/CH3X) and various electronegativity measures. Five scales (xe,Vx, xp,xM,and xA) are shown in Figure 1. As shown in Table I1 and Figures 1-3, the correlation with Vx is the highest for each example tested. The Boyd-Markus scale2' in which electronegativity xs is defined as the electrostatic force between the effective nuclear charge and an electron at a distance equal to the relative radius of the atom has the next highest correlation coefficients but significantly less than those for V x .
1976.
(27) Benson, S . W. In Thermochemistry and its Application to Chemical Biochemical Systems; Ribeiro da Silva, M. A. V., Ed.;Reidel: Dordrecht, The Netherlands, 1984, p 769. (28) Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. The Properties of Gases and Liquids, 3rd ed.; McGraw-Hill: New Ywok, 1977. (29) Luo, Y. R.; Benson, S. W. J. Phys. Chem., in press. (30) Clark, T.; Spitznagel, G. W.; Klose, R.; Schleyer, P. v. R. J . Am. Chem. Soc. 1984, 106,4412. (31) Schleyer, P. v. R. Pure Appl. Chem. 1987, 59, 1647. (32) Hehre,W. J.; Radom, L.; Schleyer, P. v. R.; Pople, J. A. Ab Initio Molecular Orbital Theory; Wiley: New York, 1986. (33) Pedley. J. B.; Naylor, R. D.; Kirby, S. P. Thermochemical Data of Organic Compounds, 2nd ed.; Chapman and Hall: London, 1986.
The Journal of Physical Chemistry, Vo1. 94, NO. 2, 1990 917
Heats of Formation vs Electronegativity Scales
TABLE 111: Pauling Stale and Y , of Main Group Elements'
xp Vx 8r
xp
0
2
4
"x/g '
Figure 4. Relation between Vx and
I
I
I
6
8
io
xp,
The two Pauling scales, xp and x:, are next highest in correlation. The correlation coefficients of all the other scales are less than 0.90 for all three sets of AAfHo.
Discussion As d e f i r ~ e d , Vx ~ ~ is, ~proportional ~ to the core potential of X at the covalent radius of X. The proportionality factor varies systematically with the group in the periodic table so that Vx is a plausible measure of the strength of the covalent bond in both diatomic and polyatomic molecules. We have found excellent correlations between Vx and AAfHo(RX/CH3X),34AAfHo(CH3X/HX),35the bond dissociation energies,36and the group parameter^.^^ These correlations are also quantitative for silicon-containing compound^.^^,^^-^^ Recently, we have found that V, plots linearly against the first ionization potentials of main group atoms.@ For all of the seven main groups, the average deviation frsm a linear relation is 0.16 eV with a maximum deviation of 0.4 eV. The average rootmean-square deviation is 0.18 eV. Yuan pointed out that Vx may be related to the Pauling scale by a linear equation: x p = av, + b (2) For main group atoms omitting H, we find that a = 0.337 and b = 0.724, as shown in Figure 4. We have compared the values of Vx, xp, and a revised Pauling scale derived from Vx using eq 2. The three items have been listed in Table 111. The values in the parentheses are reestimated. The averaged error between Pauling values and the reestimated ones is only 0.08 Pauling unit. The largest deviation is shown by H, and if we look at Figures 1-3, we note that H shows the largest deviations from the best linear plot. This has been commented on earlier40 and may be referred to as the hydrogen anomaly. We believe that the reason for the anomaly has to do with the electronegativity assigned to H in the various scales. In our Vx scale, we have used as the covalent radius for H atom the value obtained from the bond length in H2, namely 0.7414 8,. This is consistent with our and the generally accepted choice of values for the other univalent and polyvalent elements. In its compounds with other elements such as the halogens or oxygen, the bond lengths, X-H, show a considerable decrease from simple additivity. Thus, the C-H in CH, is 1.09 8, while the sum of the covalent radii rc + rH = 1 .I4 8,. If we choose a fixed value Luo, Y. R.; Benson, S. W. J . Phys. Chem. 1988, 92, 5255. Luo, Y. R.; Benson, S. W. J . Am. Chem. SOC.1989, 1 1 1 , 2480. Luo, Y. R.; Benson, S.W. J . Phys. Chem. 1989, 93, 3304. Luo, Y. R.; Benson, S.W. J . Phys. Chem. 1989, 93, 3306. (38) Luo, Y.R.; Benson, S. W. J . Phys. Chem. 1989, 93, 3791. (39) Luo, Y. R.; Benson, S.W. J . Phys. Chem. 1989, 93, 1674. (40) Luo, Y. R.; Benson, S.W. J . Phys. Chem. 1989, 93, 4643.
(34) (35) (36) (37)
B
C
N
O
F
2.0 (1.94) 3.66
2.5 (2.45) 5.19
3.0 (2.95) 6.67
3.5 (3.43) 8.11
4.0 (4.04) 9.915
Na
Mg 1.2 (1.22) 1.54
AI
Si
P
S
CI
1.5 (1.51) 2.40
1.8 (1.85) 3.41
2.1 (2.23) 4.55
2.5 (2.65) 5.77
3.0 (3.07) 7.04
Ca 1.0 (1.09) 1.15
Ga
Ge
Br
1.8 (1.79) 3.24
As 2.0 (2.11) 4.20
Se
1.6 (1.50) 2.38
2.4 (2.43) 5.13
2.8 (2.77) 6.13
Vx
xp
K 0.8
Vx
(0.87) 0.51
Vx I
Be 1.5 (1.40) 2.08
0.9 (0.92) 0.65
xp
IFr Cs
Li 1.0 (0.95) 0.75
xp Vx xp
Vx
Rb 0.8 (0.86) 0.48
Sr
In
Sn
Sb
Te
I
1.0 (1.06) .1.05
1.7 (1.37) 2.00
1.8 (1.65) . . 2.83
1.9 (1.92) . 3.62'
2.1 (2.18) '4.38'
2.5 12.47) '5.25'
Cs
Ba
TI
Pb
Bi
Po
At
0.7 (0.85) 0.43
0.9 (1.04) 1.01
1.8
1.8 (1.58) 2.60
1.9 (1.81) 3.29
2.0 (2.06) 4.03
2.4 (2.28) 4.67
Fr
Ra
0.7 (0.84) 0.42
0.8 (1.04) 1.00
'Note: From eq 2, we calculate a revised value of xp(H) = 1.61 (see Table Ib).
for C as rc = 0.77 8, (based on diamond and alkanes) then the C-H in CH4 yields rH = 0.32 8,. In HF, a similar procedure leads to rH = 0.21 8,;in H 2 0 , rH = 0.22 8,;and in HCI, rH = 0.28 8,. Clearly the rule of bond-radii additivity does not work well for H. It was suggested by Schomaker and Stevenson some time ago41 that r x y will always be less than the sum of rxo ryo derived from the elements by an amount proportional to their difference in electronegativity (absolute value). There is no obvious way to partition the observed bond shortening between the two elements in a covalent bond. This in turn means that one cannot arbitrarily assign a constant bond radius to one of the two elements rx or r y on the basis of an observation or r x y . If we use the Schomaker-Stevenson equation, then we can obtain from experimental data for r x y a corrected value that is a marked improvement on simple additivity. The large apparent variability of rH with a bonded element may then simply be a reflection of the need for a smaller value for VH than is assigned in Pauling's scale and other scales. From eq 2, we calculate a revised xp(H) = 1.63. The decrease in C-H bond length from 1.09 in alkanes to 1.06 in C2Hzwith an intermediate value for olefins can be seen as an increase in electronegativity of the C in the less saturated compounds, a view that agrees with a number of other properties of these species. We find a similar decrease in C-F bond lengths in the series CF,,H,_, (n I 1) going from 1.39 8, in CH3F to 1.32 8, in CF,, presumably representing a decrease in electronegativity of C with increasing fluorination. However, there is no corresponding decrease in the C-H distances, so the picture may not be so simple. If we make a plot of the differences between observed bond lengths in H-X compounds and the sum of covalent radii, rxo + r H o against , IV, - VHl in the spirit of the Schomaker-Stevenson plot,I4 we find a strong correlation with maximum deviations of 0.06 A and an average deviation of 0.026 8,. The line is given by (units in angstroms) rHx(0bS) = T H O + rxo - 0.023lVx - V,l (3)
+
It is a better correlation than was found with xp and has the virtue of including H,. Acknowledgment. This work has been supported by a grant from the National Science Foundation (CHE-87 14647). (41) Schomaker, V.; Stevenson, D. P. J . Am. Chem. SOC.1941, 63, 37.
