7356
Journal of the American Chemical Society
(Polymers Program, Division of Materials Research). Thanks are also due the Dow Corning Corp., for the P D M S samples and for the sabbatical leave granted to J.R.F.
References and Notes (1) (a) University of Michigan. (b) University of Michigan. (c) University of Cincinnati. (2) P. J. Flory, "Principles of Polymer Chemistry", Cornell University Press, Ithaca. N.Y.. 1953. (3) L. R. G. Treloar, "The Physics of Rubber Elasticity", 3rd ed.,Clarendon Press, Oxford, 1975. (4) B. E. Eichinger, Macromolecules, 5, 496 (1972). (5) W. W. Graessley, Macromolecules, 8, 186, 865 (1975). (6) G. Ronca and G. Allegra. J. Chem. Phys., 63, 4990 (1975). (7) K. J. Smith, Jr., and R. J. Gaylord, J. Polym. Sci., Po/ym. Phys. Ed., 13,2069 (1975). (8)P. J. Flory, Proc. R. SOC.London, Ser. A, 351 351 (1976). (9) R. T. Deam and S.F. Edwards, Philos. Trans. R. SOC.London, Ser. A, 280, 317 (1976). (10) P. J. Flory, J. Chem. Phys., 66, 5720 (1977). (11) P. J. Flory, Macromolecules, 12, 119 (1979). (12) D. J. Walsh, G. Allend, and G. Ballard, Polymer, 15,368 (1974); G. Allen, P. A. Holmes, andD. J. Walsh, FaradayDiscuss. Chem. Soc.,57, 19(1974); G. Allen, P. L. Egerton, and D. J. Walsh, Polymer, 17, 65 (1976): J. R. Falender, G. S. Y. Yeh, and J. E. Mark, J. Chem. Phys., 70, 5324 (1979). (13) J. E. Mark and J. L. Sullivan, J. Chem. Phys., 66, 1006 (1977). (14) E. M. Valles and C. W. Macosko in "Chemistry and Properties of Crosslinked Polymers", S.S.Labana, Ed., Academic Press, New York, 1977. (15) J. E. Herz, P. Rempp, and W.Borchard. Adv. Polym. Sci., 28, 105 (1978), and pertinent references cited therein.
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November 21, 1979
(16) L. M. Dossin and W. W. Graessley, Macromolecules, 12, 123 (1979). (17) J. E. Mark, R. R. Rahalkar, and J. L. Sullivan, J. Chem. Phys., 70, 1794 (1979). (18) J. E. Mark, Macromol. Rev., 11, 135 (1976). (19) R. C. Smith, N. C. Angelotti, and C. L. Hanson in "Analysis of Silicones", A. L. Smith, Ed., Wiley-lnterscience, New York, 1974. (20) W. T. Grubb and R. C. Osthoff, J. Am. Chem. SOC., 77, 1405 (1955). (21) J. C. Saam and F. W. G. Fearon, U S . Patent 3 678 125 (1972). (22) A. Chapiro, Radiation Chemistry of Polymeric Systems", Wiley-lnterscience, New York, 1962. (23) M. L. Dunham, D. L. Bailey, and R. Y. Mixer, lnd. Eng. Chem., 49, 1373 (1957). (24) P. G. Bork and C. W. Roush in "Vulcanization of Elastomers", G. Alliger and I.J. Sjothun, Eds., Reinhold, New York, 1964. (25) A. J. Chalk and J. F. Harrod, J. Am. Chem. Soc., 87, 16 (1965). (26) J. E. Mark, J. Am. Chem. Soc., 92, 7252 (1970). (27) T.-K. Su and J. E. Mark, Macromolecules, 10, 120 (1977). (28) J. E. Mark and P. J. Flory, J. Appl. Phys., 37, 4635 (1966). (29) J. E. Mark, Rubber Chem. Techno/., 48, 495 (1975). (30) P. F. Flory and Y. Tatara, J. Polym. Sci., Polym. Phys. Ed., 13, 683 (1975). (31) M. Mooney, J. Appl. Pbys., 19, 434 (1948); R. S.Rivlin, Pbilos. Trans. R. SOC.London, Ser. A, 241, 379 (1948). (32) Very similar results were obtained when the segregated groups were placed at approximately every sixth unit at the ends of the chains (J. R. Falender, manuscript in preparation). This indicates that the vinyI-Si[OSi(CH,),H], cross-linking reaction is not very significantly affected by vinyl-vinyl proximity effects. (33) C. U. Yu and J. E. Mark, Polym. J., 7, 101 (1975). (34) M. A. Llorente and J. E. Mark, J. Chem. Pb s , 7 1 682 (1979). (35) The preliminary results published previously {PolymlPrepr.. 19,743 (1978)] were calculated using swelling equation parameters now thought to be less appropriate than those used in the present analysis.
Dependence of Bond Angles upon the Steric Effect. 1. XMX Bond Angles Marvin Charton Contribution from the Chemistry Department, Pratt Institute, Brooklyn, New York 11205. Received June 4, I979
Abstract: Bond angles X-M-X in tetrahedral and trigonal planar species are generally dependent on steric effects. This is shown by the success of correlations with the equation l/sin (Ox/2) = bo(l/rvx) b l , where Ox = LXMX and rvx is the van der Waals radius of X , and 1/sin (Ox/2) = dlrVM do, where r V M is the van der Waals radius of M. The correlation equations are useful for the prediction of bond angles. H atoms bonded to 0, N, or C do not fit the steric effect model. Groups X capable of d r - p a bonding also do not fit the model.
+
+
The valence shell electron pair repulsion (VSEPR) theory has been very widely used in the prediction of the shape of a chemical species on the basis of repulsions between electron pairs.' The theory does not successfully account for the deviation of the observed bond angles in many compounds from the predicted geometry. These deviations, although small, are experimentally significant. According to VSEPR theory, they are due to the electronegativity of the substituents attached to the central atom. Searcy has attempted to predict bond angles from an electrostatic modeL2 An alternative viewpoint has been presented3 which suggests that in many cases the observed bond angles can be accounted for by interactions between nonbonded atoms. This proposal has been criticized by Wilson4 on the grounds that equally good predictions in the case of trigonal planar carbon compounds could be made by simply assuming average values for the bond angles. Nonbonded interactions have been used recently to account for the geometry of a number of species. In this work we present evidence based on a very simple model which suggests that generally bond angles are determined by steric effects (nonbonded atom repulsions). Consider the structural fragment XI-M-X2 shown in Figure 1, where XI = X2, X is some atom or group of atoms, and M 0002-7863/79/1.501-7356$01.00/0
is a central atom to which the two X groups are bonded. If the bond angle is determined by the size of the X_gl_oup,the two X groups will be in contact, and the distance XA from the X nucleus to the point A at which the X groups are in contact will be equal to the van der Waals r a d N o f X if X is monatomic. If X is polyatomic, the distance XA will be from the group center to the point in contact and will be&en by the group van der Waals radius, r v x . The distance X M is simply the X M bond length. As triangles XI A M and X 2 A M are congruent, angle X 2 M A is equal to angle X ' M A and designating angle X 1 M X 2 as 8 X ' M A = 8x12
(1)
rvX = I M X sin (8x/2)
(2)
I/sin (8x12) = lMX/rVX
(3)
Then
We may write
I M X = rcM
+ rcx
(4)
where rCM and rex are the covalent radii of M and X, respectively. We have shown elsewhere5 that as suggested by
0 1979 American Chemical Society
Charton
1357
Dependence of Bond Angles upon the Steric Effect
Table 1. Results of Correlations with Equations 8,9, and 15
set Olf 1
2 3 13 14 15 16 21 22 31 32 33 34 35 36 151 152 153 154 155 156 157 161
boor C1
bl or G
ra
1 .oo
-0.349 0.872 1.169 1.211
0.9814 0.9680 0.9727 0.93 18 0.9159 0.9781 0.9909 0.9978 0.9644 0.9164 0.8509 0.7947 0.9 184 0.9633 0.9572 0.9401 0.9540 0.9970 0.9958 0.9910 0.9744 0.9972 0.9890 0.9972
0.590 0.268 0.230 0.377 0.267 0.255 0.254 0.328 0.285 0.336 0.4 I O 0.423 0.3 17 0.444 0.149 0.150 0.376 0.207 0.291 0.440 0.320 0.270 0.229
1.011
1.151 1.176 1.186 1.114 1.184 1.069 1.033 0.949 1.022 0.933 1.128 1.113 0.636 0.960 0.781 0.486 0.716 0.802 0.8 19
Fb
157.1 89.40 87.86 19.789 20.83g 110.3 217.0 916.6 39.921 10.49" 5.249k 6.855" 10.78" 90.15 21.86h 30.431' 30.41g 496.4 357.5 164.6 56.34 177.6h 44.55" 534.8
$est
0.0488 0.01 39 0.01 18 0.01 54 0.0289 0.006 98 0.005 18 0.002 51 0.0151 0.009 24 0.0154 0.0178 0.0188 0.006 20 0.0138 0.004 32 0.005 86 0.005 69 0.003 87 0.008 02 0.0225 0.001 70 0.002 86 0.004 58
SbOC
SblC
1OOrZ d
ne
0.0798 0.0624 0.0286 0.0518h 0.0826' 0.0254 0.0173 0.008 38 0.0519J 0.088 1 0.147' 0.157" 0.129" 0.0334 0.0949 0.02691' 0.0272" 0.0 169
0.131 0.0326 0.0153 0.0307 0.0455 0.0158 0.0106 0.005 14 0.0326 0.05241' 0.08721' 0.0935 0.0749j 0.0189 0.05521' 0.0151 0.05 12 0.0290 0.1090 0.0393 0.103" 0.0444h 0.0748 " 0.0176
96.32 93.71 94.62 86.83 83.89 95.66 98.19 99.57 93.01 83.98 72.41 63.15 84.35 92.79 9 1.62 88.38 91.02 99.40 99.17 98.21 94.94 99.44 97.80 99.44
8 8 7 5 6 7 6 6 5 4 4 6 4 9 4 6 5 5 5 5 5 3 3 5
0.01 I O
0.02271' 0.0586j 0.0240h 0.0404" 0.009 92
a Correlation coefficient. F test for significance of correlation. The superscript indicates the confidence level (CL). No superscript indicates a CL of 99.9%. Standard errors of the estimate, bo, and b l . The superscript indicates CL of the Student's t test for the significance of the coefficient. No superscript signifies CL of 99.9%. The percent of the variance of the data accounted for by the correlation equation. e The number of points in the set. f For set 01, bo is C1, bl is CO.g 97.5%CL. 95.0%CL. 98.0% CL. j 99.0% CL.
