The concordance between the different values in each row of Table I11 and the close agreement between most of the experimental and calculated values of Table IV show the general applicability of the assumed relationships and constants. The calculated bond energies rarely differ more than 2 or 3 kcal. mole from the experimental values. The larger differences for H&, HzSe and HzTe are probably due to poor enthalpy data, based on
/COMMtJXICATIOX h-0. 1553 FROM
THE
experimental results published in 1885 and earlier. The discrepancies for the simple organic derivatives of sulfur and iodine may be due to error in the assumption that the C-H bond energy is strictly constant. Likewise, the calculation of the N-N bond energy in NzH4on the assumption that the N-H bond energy is the same as in NHS is probably unjustified (see Skinnerfi). ROCHESTER, S.Y.
KODAK RESEARCH LARORATORIES]
Atomic Radii. IV. Dependence of Interatomic Distance on Bond Energy’ BY MAURICE
L. HUGGINS
RECEIVED MARCH 27. 1953 I t is shown that most of the departures from strict additivity of radii in normal valence compounds can be attributed t o The simple relationship, variations in bond energy, DAB. A set of “constant energy radii,” d , has been computed. r.48 = r2 r s - ‘/z log DAB, yields interatomic distances which average within 0.02 A. of the best experimental values. If experimental bond energies are not available, values computed from electronegativities and non-polar bond-energy contributions (see the preceding paper) can be used, with but little loss of accuracy.
+
Introduction The hypothesis of constant additive atomic radii was introduced in 1920 by Bragg.2 Shortly thereafter the writer showed3 that the Lewis theory of valence was as applicable to crystals as to molecules and ions and pointed out4 that the valence electron distributions so obtained enable one to predict, in many cases, whether or not constancy and additivity of radii should exist. Various causes of variability were considered in a qualitative manner. It was shown, nevertheless, that a reasonable degree of constancy and additivity exists in certain classes of structures, the essential requirement being that each atom being considered has a sufficiently similar environment in the different substances being compared. Sets of radii for several such classes were computed, the most extensive being a set of tetrahedral radii,j computed from and for bonds joining atoms, each having a kernel charge +n, tetrahedrally to four others, each with a kernel charge of S - n. In 1934, Pauling and Huggins6 revised and extended these sets of radii and added a set of normal valence radii, differing for only six elements from the tetrahedral radii (see Table I). The writer has consistently pointed out that one should not expect close correspondence between the experimental interatomic distance and the sum of the appropriate radii from the standard set, if the environment of either or both of the atoms differs much from that of the corresponding atom in the class of substances from which the standard radii were derived. Such departures (1) This is a revision of portions of papers presented on Sept. 22, 1949. a t the 116th Meeting of the American Chemical Society in Atlantic City and on Sept. 13, 1951, a t the X I I t h International Congress o f Pure and Applied Chemistry in New York, N. Y. For the three previous papers in this series, see references 4, 5 and 6. (2) W. L. Bragg, Phil. Mw., [e] 40, 169 (1920). (3) M , L. Huggins, THISJ O U R N A L , 44, 1841 (1922). (41S I . L . Huggins, Phys. Rev., 19, 346 (1922). ( 5 ) M. I,. Huggins, ibid., 21, 205 (1923); 28, 1086 (1926). (6) L. Pauling and 11. I, Huggins, %. K i i s t . , 8 8 7 , 205 (19341: quoted in ref. : 7 ) .
from additivity are, in fact, often found. They are useful in supplying evidence regarding the dependence of the bond length on various factors. Many of the more marked departures from additivity have been interpreted’ as resulting from differences in the “degree of double-bond character.” On the other hand, Schomaker and Stevensons attribute these departures to varying degrees of bond polarity. They have published a set of non-polar radii ( Y , , ~ , A ) and proposed the following empirical equation for the calculation of singlebond distances from these radii and Pauling’s electronegati~ities~,~ (XA) rm
- rmp..4+ r n p , ~ O.OQjxr - X B I
(11
Although both of these explanations of departures from additivity seem reasonable and can be used to account for the observed distances in a considerable number of instances, Wellslo has concluded that the sum total of available evidence is against the general applicability of either. The present paper reports the results of an attempt to correlate interatomic distances with bond energies in a simple manner for single bonds in normal valence elements and compounds. Theoretical Background It is theoretically reasonable and experimentally
well established that, other things being equal, the greater the bond energy the shorter is the interatomic distance. The bond energy is minus the sum of the repulsion energy and the (negative) attraction energy, when the molecule is in its lowest energy state, ;.e., when the interatomic distance is YAB
DAB = - E when r = E = Ere, Eatt
+
r.4~
(2) (3)
(7) L. Pauling, “The Nature of the Chemical Bond,” Cornel1 University Press, Ithaca, N. Y., 1939. (8) V. Schomaker and D. P. Stevenson, THIS JOURA-AL, 63, 37 (1941). (9) L. Pauling, ibid., 64, 3570 (1932). , (1949). ‘10) A. F. Wells, J . Chem. S O C 55
Sept. 5, 1953
~ E P E N D E N C EOF
INTERATOMIC DISTANCE ON BONDENERGY
412 i
TABLE I ATOMICRADII Ha
Hb 1926 Tetrahedral P & H6 1934 Tetrahedral P & H6 1934 Normal valence S & S81941 Non-polar H 1953 Non-polar H 1953 Constant energy ( r i )
Hb
HC
0.375 0.28 0.28 0.28 0.28 .37 .37 .37 .37 .37 .38 .36 .34 .33 .32 .88 .86 .84 $83 .82 As
Ge
H5 1926 Tetrahedral P & H6 1934 Tetrahedral P&H61934Normalvalence S & S8 1941 Yon-polar H 1953 Non-polar (rnp,A) H 1953 Constant energy,(rT)
In Hz. In bonds with first row elements. In bonds with fourth row elements. (1
e
1.21 1.22 1.22 1.22 1.21 1.61
Elep= E& exp[a(rA*~- r)l
(5)
and r i arise from the These designations for fact that the former is the distance between atomic centers when the repulsion energy has the constant value DO,,,. A similar treatmentl4,l5 has been found satisfactory, a t interatomic distances not far from the equilibrium values, for diatomic molecules. Following Morse, l6 an exponential expression was also used for the attraction energy Eztt
exp[a’(r, - r)l
(6)
The constants E& and a’ do not have the same values for different molecules or for the same molecule in different energy states. For molecules containing only electronegative atoms or hydrogen, a uniform value of a could again be used, and again the YXB values computed were found to obey fairly well the adcitivity relationship (equation ( 5 ) ) . On assuming equations (3), (4) and (6) and deducing the constants from band spectral data for diatomic molecules, equation (2) is found to be far from accurate. Nevertheless, we shall tentatively (11) M. L. Huggins and J. E. Mayer, J . Chem. P h y s . , 1, 643 (1933) (12) M. L. Huggins, ibid., 6, 143 (1937). (13) M. Born and J. E. Mayer, Z . Physik, 7 6 , 1 (1932). (14) M. L. Huggins, J . Chem. P h y s . , 8, 473 (1935). (15) M . L. Huggins, ibid., 4, 308 (1936). (16) P. M. Morse, Phys. Rev., 84, 57 (1929).
Si
F
0
Br
Sn
S
P
C1
Sb
Te
I
1.29 1.36 1.41 1.41 1.45 1.83
1 . 2 3 1.19 1.32 1.28 1.37 1.33 1.37 1.33 1.41 1.35 1.79 1.73 In bonds with third row elements.
assume, as an approximation, that all four of these equations (and also equation ( 5 ) ) hold for single bonds in normal valence molecules containing only electronegative atoms and hydrogen, with the constants determined by experimental bond-energy and interatomic distance data. We thus have
- E&
DAB = -E:tt
exp[a(rA*a-
?,)I
(7)
From the equilibrium condition dE/dr = 0 when r = re
(4)
in the exponential varies but little from compound to compound and also that, assuming it to be accurately the same in all cases, the constant energy distances, r x B , can be additively computed from constant energy radii, r i , characteristic of the atoms concerned
E,tt =
N
1.12 1.09 1.36 1.14 1.11 1.40 1.17 1.14 1.40 1 . 1 7 1.14 1.40 1.17 1.14 1.42 1.58 1.56 1.80 In bonds with second row elements.
It was found that (with E$, assigned an arbitrary value, the same for all compounds) the constant a
+ ri
Se
C
0.77 0.70 0.65 0.61 1.14 1 . 0 8 1.02 0.97 .77 70 .66 .64 1.17 1.10 1.04 .99 .77 .70 .66 .64 1.17 1.10 1.04 .99 .77 .74 .74 .72 1.17 1.10 1.04 .99 .77 .75 .74 .72 1.15 1.11 1.04 1.00 1.22 1.12 1.12 1.11 1.57 1.53 1.46 1.44
1.16 1.18 1.21 1.21 1.24 1.63
The repulsion energy, due primarily to overlapping of the electron clouds of the two atoms concerned, would be expected to vary with the interatomic distance in a manner depending only slightly on the nature and magnitude of the attractions between the atoms. This expectation has been tested and verified for alkali halide crystals by Huggins and Mayer, 11,12 using the exponential repulsion law of Born and Mayer13
r& = r.:
He
Hd
(8)
one can deduce E& exp i a ( r X ~-
=
-Ett
re)
I
(9)
Combining equations (7) and (9), we have DAB=
(5 - l)EEp exp[a(rXa - re)]
(10)
We shall assume, for our present purpose, that the first factor in parentheses in this equation is essentially constant. This is equivalent to assuming that the repulsion energy a t equilibrium is proportional to the bond energy. The bandspectrum data for diatomic molecules give some justification for this assumption, as a rough approximation. It is less drastic than Morse’s hypothesis,16 that a = 2a’. Although essentially an ad hoc assumption, the data to be presented will testify as to its general validity. It is possible, nevertheless, that variability of this factor may, in some cases, not be negligible. Since the value of E&, is entirely arbitrary, we shall assume it to have such a value that
(3 Then, replacing I A R , we have
re
DAB ?AB
1) EZP = 1 kcal./mole
(11)
by its approximate equivalent, = eXp[U(rXB =
?‘A*B
-
In
rAR)1
DAB
(12) (13)
This equation, combined with the additivity relation (equation ( 5 ) ) ,has been tested by application to all available pertinent data. Complete sets of calculations have been made, in which a is taken equal to 4.00, 4.605 and 6.00 X 10; cm.-l, with less extensive sets for some other values.
MAURTCE L. HUGGINS
4125
VOl. 75
TABLEI1 INTERATOMIC DISTANCES Bond
TAB calcd. TAB calcd. from normal from Dnp,.4 valence radii* and x ~ l 7
YAB calcd. from exp.
?AB
DAB"
(em.)
0.749 1.094 1.102 1.014 0.957 0.926 1.456 1 424 1.419 1.334 1.284 1.478 1.52 f 0.01 1.523 1.6 1.423 1.700 f 0.015 1.711 1.712
H-H H-C
0.75 1.07
0.751 1.118
0.751 1.111
H-N H-0 H-F H-Si H-P
1.00 0.96 0.94 1.47 1.40
1.016 0.963 0.902 1.461 1.426
1.017 0.958 0.906 1.456 1.428
H-S H-Cl H-Ge
1.34 1.29 1.52
1.347 1.276 1.505
1.335 1.273
H-AS HSe H-Br H-Sn H-Sb
1.51 1.47 1.44 1.70 1.71
1.534 1.470 1.418 1.694 1.729
H-Te H-I
1.67 1.63 1 54
1 ,690 1.616 1.557
c-c
1.576 1,500
1.419
1.73 1.623 1.56 1.54
C--X
1.17
1.489
1.48 1.48 1.49
c-0
1.43
1.435 1.42
C-F
1.41
1.355
1.365
CClFa CClzFz CCLF CBrF8
C-Si
1 94
1.883
CBraF CIFa CHClFz CHClzF CHzClF SiC Si(C€L)(
1.89
Source
1.617 1.55 f 0.03 1.543 1.541 1.54 f 0.02 1.48 1.47 f 0.01 1.47 f 0.02 1.47 f 0.02 1.46 f 0.03 1.47 f 0.01 1.48 f 0.01 1.44 f 0.01 1.43 f 0.03 1.43 f 0.05 1.44 3= 0.03 1.44 f 0.02 1.36 3= 0.02 1.35 f 0.03 1.34 f 0.02 1.329 1.332 1.32 f 0.01 1.358 I: 0.001 1.357 f 0.017 1.398 1.42 4~ 0.02 1.39 5k 0.02 1.385 1.323 1.35 i.0.03 1.40 rk 0.04 1.44 i.0.04 1.321 1 . 3 3 f 0.015 1.44 f 0.06 1.33 i.0.015 1.36 f 0.03 1.41 f 0.03 1.40 f 0.03
or YAB
(em.)"
Sb s" Sd s" SC
S" s"
Sf Ma SC
Sh S" M< S', Mg S' Sh Mk
Sf MU
Sb E' Sd XC" E'
SC E' E' E' E' E' E' EL E' E' El El
E' E' E' S" M" SP MQ E' s" E' EL
M"
M'
1.89
E' E' E MU hi" E' M" E' E' E' XC"
1.888 f 0.02
EW
'.
Sept. 5, 1953
4129
DEPENDENCE OF INTERATOMIC DISTANCE ON BONDENERGY TABLE I1 (Continued) IAB calcd.
Bond
from normal from D n p , ~ valence radii6 and X A ' ~
c-s
1.87 1.81
1.865 1.812
c-CI
1.76
1.754
c-P
C-Ge C-AS C-Br
Molecule Or crystal
IAB calcd.
1.99 1.98 1.91
1.939 1.983 1.909
CH8SiFs (CHa)2SiCln P(CHs)t (CHslzS (CHa)iSz CCl,
*AB calcd. from exp. DAB"
1.83 1.749
CHCla
1.75
CH2Clz
1.75
CHIC1
1.75
CClFs CClzFi CC13F CHCliF CHClFz CHiClF CBrzClz CBrCla C( CHa)aCHzCI Si(CHs)3CHzCI Ge(CHI), AS(CH3)3 CBr,
1.908
CHBr3
1.91
CHzBrz CHaBr
1.91 1.91
CBrF3
C-Sn
2.17
2.140
c-I
2.10
2.114
CBrSF CBrC18 CBnClz Sn(CHd4 CHaSnHa (CH&SnCl CHaSnBrr (CH&SnBr CII CHI8 CHzG CHaI
N-N
N-0 N-F
1.40
1.36 1.34
1.487
1.458 1.377
CFaI NZH4 (CHa)2NzHz NS4 NHiOH NFs
2.14 2.15 2.14
1.49
1.356
Source ?AB
Of I A B
(exp.)
(expJa
1.88 1.83 f 0.06 1.87 f 0.02 1.82 f 0.01 1.78 f 0.03 1.755 i 0.005 1.765 f 0.015 1.761 1.77 f 0.02 1.75 f 0.01 1.767 1.77 f 0.02 1.81 1.772 1.77 f 0.02 1.77 f 0.01 1.780 f 0.002 1.779 1.7810 1.765 1.74 f 0.03 1.76 i 0.02 1.73 f 0.04 1.73 f 0.03 1.76 f 0.02 1.75 1.76 1.74 f 0.03 1.73 f 0.03 1.98 f 0.03 1.98 i 0.02 1.91 f 0.02 1.93 f 0.02 1.94 f 0.02 1.942 f 0.003 1.91 1.930 i 0.003 1.91 f 0.02 1.91 f 0.06 1.9391 1.936 1.91 f 0.02 1.91 f 0.04 2.01 1.93 2.18 f 0.03 2.143 i 0.002 2.19 f 0.03 ~2.17 2.17 i 0.05 2.12 i 0.02 2.15 f 0.02 2.12 f 0.03 2.12 f 0.04 2.132 2.1392 2.14 f 0.02 1.47 f 0.02 1.46 f 0.02 1.45 i 0.03 ' 1.47 1.46 1.37 f 0.02
M* E' E' E' E' E' E' E' E' XGY M0 E' XG" M' E' E' M"" Ma* M"" Mt E' E' E' E' E' E' E' Ead E' E' E' E' E' E'
M"' E' E' Mac MU
MU E' E'
E' E' Mk E' E' E' E' E' E' E' M"'
Mac MU E' XC"' E' E' Sah
E'
VOl. 75
MAURICE L. HUGGINS
4130
TABLE I1 (Conlinued) YAB calcd.
YAB calcd
from normal valence radii0
from Drip, 1 and XA"
?I-S
1.74
1.750
hT-C1
1.69
1 732
Bond
hIolecule or crystal
from exp. D..iBL7
N,S4
J 32
1.474
I ,479
0 -F
I 30
1.43H
1 ,403
1.83
1.688
NO& Si02 Si(OCH3), (CH3)eSi30a ( CH3)3SiOSi(CH~)3 Cl3Si0SiCl3
1.68
0-P
1.76
1.688
Pa06
0-c1
1.65
1.716
c120
0-AS
1.87
1,795
1.816
0-Sb
2.07
I.992
1.996
F-F F-Si
1.28 1.81
1.442 1.614
1.438 1.614
F-P
1.74
1.612
F-CI
1.63
1.661
C1F
F-Ge F-AS
1.86 1.85
1.663 1.716
GeF4 AsF~
F-Br F-Sb Si-Si
1.76 2.05 2.34
1.774 1.912 2.290
BrF SbF, Si Si?Hs
1.778 1.923 2.29
Si-S Si-CI
2.21 2.16
2.136 2.033
Si%
2.13 2.032
Sic14 SiHC17.
1.713
1.661 or 1.657 1.717
SiH&l? SiH3Cl
SiCIFJ
Si-Br
Si-I
2.31
2.50
TAB
(=e.)
2.197
SiBrCl3 SiBr4
2.430
SiHIBr SiBrFo SiBr2Fz Sir4
2.199
2.436
Of * A t
(exp.)
1.371 1.74 1.62 f 0.02
Ma E' E'
1.76 1.74 f 0.02 1.77 f 0.02 1.17f 0.02 1 .49 i- 0 . 0 2 1.18 I .41 f 0.05 1.38 f 0.03 1.42 f 0.05 1.61 1.64 f 0.003 1.66 f 0.04 1.63 f 0.03 1.64 f 0.05 1.67 i 0.03 1.65 f 0.02 1.63 1.68 f 0.03 1.701 i 0.020 1.80 f 0.02 1.78 f 0.02 2.0 f 0.1 2.00 1.435 f 0.01 1.54 f 0.02 1.561 f 0.005 1.593 f 0.002 1.560 f 0.005 1.560 i 0.005 1.52 f 0.04 1.535 1.6281
9'
I .77
0-0
0--Si
Source
YAB calcd.
1.67 f 0.03 1.72 f 0.02 1.712 f 0.006 1.759 2,352 2.32 f 0.03 2.14 2.02 f 0.02 1.98 f 0.02 2.00 f 0.03 2.01 f 0.03 2.05 i 0 . 0 3 2.02 f 0.03 2.06 f 0.05 2.035 2.048 f 0.004 1.989 i: 0.018 2 . 0 3 i: 0.03 2.05 f 0.05 2.14 f 0.02 2.15 i 0.02 2.209 2.153 i 0.018 2.16 f 0.02 2.43 f 0.02
El El E'
s' E'
S"' E' XC" EUm EO"
E"" E"" E' El Sa" E' E"P E' El XC"9 XC"' E' E' MUS M"t M"' MUS E' M""
M"" EUW
E' S"" M"V XC"" El
XC" E' XGY El
E' E' E' EL M bo Mbb
MUS
E' E'
El E' MbC
Mas El
E'
Sept. 5 , 1953
DEPENDENCE OF INTERATOMIC DISTANCE ON BONDENERGY TABLE I1 (Continued) YAB calcd.
Bond
YAB calcd.
from normal from D n p , ~ valence radii6 and 2 ~ 1 '
P-P
2.20
2.210
P-S P-c1
2.14 2.09
2.121 2.027
P-Br
2.24
P-I
s -s
Molecule or crystal
Normal P4 P (black)
YAB calcd. YAB
DAB''
(exd
2.205
P4S3
Pc13
2.023
2.191
PBr3
2.188
2.43
2.420
PI:,
2.415
2.08
2,070
Sa Sa
2.058 2.053
HzS? s2c12
s-Cl
SAs
2.03
2.25
2.007
2.241
SBr c1-Cl CI-Ge
2.18 1.98 2.21
2.167 1.998 2,085
CI-AS
2.20
2.137
Cl-se C1-Br CI-Sn
2.16 2.13 2.39
2.137 2.138 2,283
C1-Sb Cl-Te c1-I
2.40 2.36 2.32
2.336 2.331 2.313
Ge-Ge
2.44
2.419
Ge-Br
2.36
2.252
SCl? s2c12
2.015
AS&
2.230
AsrSe s2Br2 Cl? GeCI, GeH3C1 AsC13
2.230 2.156 1.998 2.097
(CH3)zAsCl SeC12 BrCl SnC14 CH3SnCla ( CH3)2SnC12 ( CH3)3SnCl SbC13 TeCl? I c1 Ge G2H6 GeBr4
2.138
2.138 2.141 2.296
2.334 2.319 2,408
Ge-I
2.55
2.490
GeI4
2.470
AS-AS AS -Rr
2.42 2.35
2,470 2.306
Normal As4 AsBra
2.470 2 308
AS-I
2.54
2.543
(CH3)~AsBr AS13
2.547
(CHalAsI Se-Se
2.34
2.348
Sea
Source
from exp.
Of YAH
(exp 1
2.21 f 0.02 2.17-2.20 2.15 2.00 f 0.02 2.03 f 0.02 2.043 f 0.003 2 . 2 3 f 0.01 2.18 f 0.03 2.52 f 0.01 2.46 2.43 f 0.04
E'
2.07 f 0.02 2.08 f 0.02 2.05 f 0.02 2.04 f 0.05 2.05 f 0.03 2.07 f 0.10 1.99 f 0 . 0 3 2.00 3t 0.02 1.98 f 0.05 1.99 f 0.03 2.01 f 0.07 2.23 i 0.02 2.19-2.27 2.25 f 0.02
E' E' E' E' E' E' E' E' E' E' E' E'
1.988 2.08 f 0.03 2.147 f 0.005 2.16 f 0.03 2.17 f 0.02 2.161 f 0.004 2 . 1 8 f 0.04
Sbf E' Mi E' E'
2.138 2.30 f 0.03 2.32 i 0.03 2.34 f 0.03 2.37 f 0.03 2.37 i 0.02 2.36 f 0.03 2.30 f 0.03 2.324 2.450 2.41 f 0.02 2.32 2.34 2.29 f 0.02 2.47 2.48 2.50 f 0 03 2.44 i 0.03 2.36 I!= 0.02 2.31 *2.33f 0.02 2.34 f 0.04 2.58 f 0.01 2.51 2.541 2.55 zk 0.03 2.52 zk 0 . 0 3 2.32
M*Q
SC"'
E' E' E' M bd El
E' E' E' E'
xcbe E'
Mbd
El E' E' E' E' E' E' E'
SA XC"' El
E' El
E' E' IS' E'
R' I
