Edward A. Robinsoni and Ronald J. Gillespie McMaster University Hamilton. Ontario L8S 4 ~ 1 Canada
I I
Bent Bonds and Multiple Bonds
This paper considers carbon-carbon multiple honds in terms of the hent hond model first proposed by Pauling in 1931 ( I ) , which allows douhle and triple honds in molecules such as ethvlene and acetvlene to he calculated directlv from the length i f a CC single h k d , and it is shown that the &ngths of these multiple honds may also he estimated with surprising accuracy from direct measurements on simple "bent-bond" models constructed from Prentice-Hall Molecular Framework Models. Cyclopropane and other molecules containing single honds that may he regarded as "hent" may he treated similary and their interatomic distances estimated from models or from suitable Pauling-type calculations. The douhle hond in cyclo~roneneis an unnsual case in which the two hent honds that m&e ;p the douhle hond are additionally hent in the plane of the ring, accounting for the unusually short C=C douhle hond distance and the exceptionally large H-C=C angle in this particular molecule. The hent hond model can he extended readily also to include multiple honds between other pairs of atoms and it accounts rather simply for the ohserved geometry of the recently prepared molecule FdS=CH2. Finally, Pauling models can he refined by taking into account the actual m&nitude of interorhital angles atcarhon calcnlated from ohserved XCX hond angles to give improved agreement hetween calculated and ohserved C=C hond lengths in suhstituted ethylenes of the type X2C=CX2 and in other molecnles with hent honds. Van't Hoff and T,eRel were the first to propose that the four valencies of carhon are directed toward the corners of a tetrahedron with the carhon atom a t its center, and they represented CC single, douhle, and triple honds, res~ectivelv,a s tetrahedra sharing a common apex, a common edge or a common face. Baeyer and other early chemists adopted this model, not only for ethylene and acetylene hut also for small ring molecules such as cyclopropane in which they snggested that the honds were hent or strained (2).and in 1931 Panline used a similar model to account for the douhle and triple hond lengths in ethvlene and acetvlene in terms of two and three hent single Iwnds, rrywcrively ( 1 I. Although it i.; the only simplr m d e l that allows m u l t i ~ ~hsnd l r ltwtlrhs to lw cnlculated with any accuracy from 'the length ;;f a single hond Pauling's approach seems to have been neglected in recent years. It is the purpose of this paper to show that the hasic Pauling model in fact works well for a variety of multiple honds. and that its refinement leads to rather accurate oredictionsof CC douhle hond lengths in a varietyof suhstit&ed ethylenes and other molecules containing hent hnnds. Paullng Models In calculating CC douhle and triple hond lengths Pauling assumed that each participating hent hond in a multiple hond can he represented as a constant arc equal in length to a CC sinele hond a n d h a r t i n e a t each carhon atom in one of t h ~ tet;ahedral direction.;, i s is shown in Figure I. Thus, t;~king twiw the P,lulinr sinrlt. hnnd radi~is mrhnn #f77 Dm, i.e. 154 pm, as the cc single hond length, Pauling's modkls give 131.6 pm and 118.0 pm, respectively, for the lengths of a CC douhle hond and triple hond. Experimentally, as is shown in Tahle 1.the ohserved hond lengths in a varietv of suhstituted ethylenes are in the range 131 to 138 pm, and for acetylenes triple CC hond lengths cover the range 118 to 122 pm. Specifically, if we take ethylene and acetylene as standard cases, their hond lengths [C2H4,133.7 f 0.2 ppm; C2H2,121.2 f 0.2
~ m (3.4) ] are within 2 to 3% of the lenzths - -predicted hv the iespective simple Pauling models. Double and triple hond lengths for other pairs of atoms may he calculated in an analogous way from appropriate single hond lengths and calculated douhle and triple hond lengths for a variety of pairs of atoms are compared&th experimentally observed values in Tahle 2. T h e agreement is seen to be auite satisfactorv. Molecules such as cyclopropane, where the CC single honds are commonly descrihed as hent (S),are also successfully treated using a Pauling-type model and some calculated and observed CC distances are shown in Tahle 3. The calculated CC distance of 150 pm in cyclopropane, for example, is to he compared to the observed distance (6)of 151.2 f 0.2 pm. Framework Models Instead of calculating the lengths of hent bonds, Pauling models may he constructed readily using Prentice-Hall Framework Molecular Models2, as is illustrated in Figure 2. Such models are readily cnnstructed from the tetrahedral forms and pieces of plastic tuhing from a standard kit and hond lengths may he measured directly. Bond lengths so ohtained are in surprisingly good agreement with the observed hond distances, as is shown in Tahle 4. The model of c.v c l .o ~ r.o ~ e nise oarticularlv informative (Figure 2e) hecause it clearly shows not only two bent single honds similar to those in cvclopro~anehut also an additional hending in the plane of the ring of the two bent honds that form the douhle hond. This additional bending not onlv acle counts for the exceptionally shc,rt oh*erved CC d n ~ ~ hi)(,nd distance of 130 pm in this molecule hut also tiw the very large H--C=C hond ancle " of-l:,Oo. Similar models may he constructed for many other molecules where the hondine mav .he represented hv hent honds. either single or multiple, and the results of measurements on models of some illustrative examples are also given in Tahle 4. It is supgested that construction of some of these modelsand measurement of douhle, triple, and hent single bond lengths in them would make a suitable and informative experiment for high school or university freshman chemistrv laboratories.
..
The Structure of F4S=CH2 The structure of the molecule FaSCH7. which has been determined recently (7), provides a'good &tration of the use of the hent hond model for the understanding of molecular geometry. The ohserved structure (I) is shown in Figure 3. It has the plane of the CH2 Group in the plane of the axial fluorine atoms. The alternative Peometrv ~-~~ ~. , .(11) , would have- the plane of the CH2 group coincident with the plane of the equatorial fluorine atoms. The hent hond model (III),which can he descrihed as an octahedron sharing an edge with a ~
~
' On leave from the Department of Chemistry, University of Tornnto 1477-7R
nient hut then plastic tubes &uld have to be obtained from some source. Volume 57, Number 5. May 1980 1 329
Flgure 1. Psuling bsnl bond dsacr.pllon of I.ethvlene. 11. awtylene,and mnhod of calculat ng douole b w d lengh re, and wlpls oond lengh ra, horn single bwut length rf
I (a) axes of tetrahedral orbitals in ethylene I (b) bent bond description of ewlene I (c) geomeby of the model: 0 ISme center of a clrcle of radius a of which hm s t a n t arc AB of lenolh r, forms a oart.
Table 1.
Observed and Calculated C=C Bond Lengths i n Ethylenes
X\ @,C=C X"'
x
X'
H H H H F CI
H H H H H H H H H H H
F CHI
H H H H H H H H F CI CHI H H H H F H H
F
F
CI
CI Br CHs
Cy, F CI Br CHI
Br
CHs H H H H H H F F
Br
CHs 01-
=n
a 0 lbmsav-
x"
F
CI
c% F
CI Cl
xX
&
H
11 6 2 113.2 115.4 111.2 109.3 123.6 124.1 115.8 111.4 116.5 112.4 114.7 118.5 109.2 105.6 114.0 106.0 110.5 112.4 115.6 115.2 112.6
F
CI CH3 F
CI Br
Cy, F CI CH, F CI CHs F F F
CHI F
CI
Sr CHr
+dl2
I1(a) axes of tetrahedral abitais In acetylene Il (b) bent band description of acetylene I1(c) geometv of lhe model 0 is the center of a clrcle of radius a of which the constant arc ABof length r, forms a part.
a*
2' 017
\x,, &"a
PLb
ratc
116.2 114.3 119.2 113.7 114.3 120.0 120.2 116.0 111.4 116.5 112.4 114.7 116.5 109.2 110.7 115.2 113.9 109.9 112.4 115.6 115.2 112.6
103.9 105.3 102.0 105.7 105.3 101.6 101.4 104.0 107.6 102.4 106.6 105.0 102.4 109.8 108.2 104.6 105.5 109.0 106.8 104.3 104.6 106.6
133.7 133.2 134.5 133.1 133.2 134.6 134.7 133.7 132.3 134.3 132.7 133.3 134.4 131.5 132.1 133.5 133.1 131.6 132.7 133.6 133.5 132.7
116.2 117.2 123.0 (116.2)' 119.3 (116.2)' (116.2)' (116.2)' 111.4 118.5 112.4 114.7 116.5 109.2 115.8 116.5 121.6 109.2 112.4 115.6 115.2 112.6
lmwbtal aqle lahslrro eleobon pain M i n g !hedoublebond (see tertl =r, = 151@IT 180 sin 012 pm. 1.e. calculated C = x bond length lpm) observed C=C boM lengm (pm); ~Explmantaliy 'A = loo (r, ,*)Ire. I.B. % dlneren~ebBtWm re and rdfAssumed.
-
330 / Journal of Chemical Education
,
'
rho
Ase
133.7 i 0.2 1 3 3 . 3 i 0.1 132.7 f 1.5 133.6 f 0.2 131.6 0.6 135.4 i 0.5 136.2 i 0.4 134.2 i 0.1 132.9 f 0.4 134.3 f 1.5 134.6 f 0.1 133.1 f 0.4 134.3 i 1.5 134.7 i 0.1 130.9 i 0.6 132 f 1 133 f 2 134.6 f 0.2 131.1 i 0.7 135.5 i 0.3 136.3 f 0.9 135.6 f 0 . 2
0.0 0.1 1.4 0.4 1.2 0.6 1.1 0.4 0.5 0.0 1.6 0.2 0.0 2.4 1.1 1.1 0.1 2.1 1.2 1.4 2.1 2.1
Table 2.
Comparlson of Observed Double and Triple Bond Lengths wlth Those Calculated on the Bask of Paullng's Tetrahedral Bant Bond Model Single Bond
Bond
Lengtha
Calc.
CC
154 147 143 140 136 182 178 181 174 170 214
132 126 122 120 116 156 152 155 149 145 183
CN CO NN NO SIN SiO SC SN SO PS
a sinale bond lengms calcvlatedfrom sum of
D w b l e Bond Length Observed"
131-138 (128) 121-123 121-125 115 157 151 156 154 141-143 187-189
Triple Bond Length Observedb
Cale. (xP=%)
~Rf+a) (XWNX) (NOg+) (Si=N) (Si=O) ( X e s ) S=N+=S) (x2S02) (XabS)
Paulingsingle bond cwalom radll.
a Most of the data taken Iran A. F. Wells. ''Suunwal inagank Chemistry." 4m Ed.. Clarendon Press. Oxford. 1975.
Table 3. Paullng Models and Reflned Models for Cyclopropranm, I, Perfluorocyclopropane,II, and Trlmethylenecyclopropane, Ill Y
1' 11.
Ill*
H
2
p
To'
r
(CCLIC
109.5 115.1 109.5 112.2 109.5 116.2 123.2
109.5 104.6 109.5 106.9 109.5 116.2 123.2
24.8 22.3 24.8 23.5 24.8 28.1 31.6
154 154 154 154 154 154 154
149.2 150.2 149.2 149.7 149.2 148.1 146.3
('Xb 151.2 150.5
145.6
*Teushedral o,bita1.n ss,h". 'Intemrbital angle. at carbon calculated horn xcx a Ethylene-llke arbltals at carbon. 'lnterwbital angles at carbon calwlated hm observed doubls bond lsngm (134.1 Dm).
(h.
Table 4. Comparlson of O h r y e d Bond Lenplhs wHh Bond Lengths Measured from Prentlce-Hall Framework Models F gure 2 Prentic%rlallMoleeularMOdelr: (alethylane. (blacalylene. (clcydopropane. (d) trimelhy enecyclopropane. (el cyclopropens. (1, ethylene oxide.
Molecule
HsC=CH*
0%
F G ~ F I
HA=--H
C 4 C-C
a
C-C
F X F F F
Bond Length (pm) Model * Observed
Bond
c-c 151
C-C
0
C
a
C-C
e N
fi S
fA
A
C-C C-0
153 149 129 148 140
C-C C-N
148 144
C-C C S
150 180
C=C
~
I. observed ~twcture,11. alternative geometry. Ill.bent band model.
-
Meawed from Prentic%HallFramework mxbls. 3 ~ v e r rae ~ wll~ lrom refs. loand 71.
Volume 57, Number 5. May 1980 1 331
tetrahedron, leads immediately to the observed geometry. In contrast, consideration of orbital overlap in a conventional treatment, which would describe the S=C hond as a dn-pn hond, does not lead to an obvious prediction of the correct structure. Refinement of the Pauling Models
In comparing the hent bond and a-n descriptions of, for examnle. the CC douhle bond i t is often assumed that in the bent Loid model use is made of sp3 hyhrid orhitals at each carhon atom (as in the simple Pauling model), while in the a-n description sp2 orbitals are used. Thus the bent bond description would imply a hond angle of 109.5' between the single honds in ethylenes while the sp2 orhitals used to form the a-bonded framework in the adescription require hond angles of 120". The observed bond angles in ethylenes in fact cover a range from close to the tetrahedral angle of 109.5' to about 125", while in ethylene itself the HCH bond angle is 116.2' (Table 1). Clearly both simple models are too crude to eive a precise descrintion and. in eeneral. the state of hvibridizaiion of the orbitals used to form the C-x single honds in ethvlenes depends on the nature of the substituents and is usualiy intermediate between sp2 and sp3. In fact it is onlv in its simplest form that the bent hond theory predicts an XCX hond angle equal to the tetrahedral angle of 109.5°. Slightly more detailed consideration following the ideas of VSEPR theory (8)leads to the prediction that the XCX hond angle in an ethylene will in general he greater than 109.5° and will devend on the electroneeativities of the lieands X in such a way t i a t the smallest hond angles will he observed for ligands of the greatest electronegativity (for example, fluorine). The two electron pairs forming a douhle hond are attracted hv the same twonuclei and are. therefore... pulled into closer relative proximity than if they were forming two single honds. This positioning allows the CX bonding pairs to move farther apart, thus tending to increase the XCX hond angle to greater than the tetrahedral angle of 109.5°, as is indeed almost always observed. The magnitude of this effect will depend on the electronegativities of the ligands X. For example, a highly electronegative ligand such as fluorine will leave the carhon with a residual pusitive charge causing it to rather strongly attract the two electron pairs forming the double hond, so that they are pulled into the carbon atom, thus preventing the angle between them from being as small as it would be in, for example, ethylene with less-electronegative hydrogen substituents. In consequence the electron pairs t hexample, ~ l ~ ~ ~ ra;niot ~~, in the i'-E'lu,nds in t e t r n t l t t ~ ~ ~ ~ efor niuve as far apart as can the elect run pairs in the C-H bundr; in cthvlrne. i r . the FCF nnrlr in F,C CF. is exoerred tube smaller than the HCH angie in H Z ~ = C H Moreover, ~. since
- .
the repulsion between CX bonding electron pairs themselves diminishes with increasing electronegativity of X, the FCF bond angle in C2F4 (9) is only 112.4' compared to 116.Z0 for the HCH bond angle in C2H4 (3). The fact that the angle between the two electron pairs in the double hond varies according to the substituents in an ethylene, and is, in general, less than the tetrahedral angle assumed in the simple Pauling model, affects the calculated CC douhle hond length. In fact, the CC douhle hond length would he expected in general to he longer than the value of 131.6 pm predicted by thesimple model. This feature is indeed almost always the case as can he seen from examination of the data in Tahle 1. On the basis of simple hyhrid orbital theory (lo), values for the interorhital angles between two electron pairs forming a double hond are readily calculated from observed XCX hond angles. Hence, a value for the CC double hond length in a substituted ethylene that is more precise than that given by the simple Pauling model (which assumes an interorbital angle of 109.5" in all ethylenes) may he calculated. The state of hybridization of orbitals involved in C-X bonding is calculated from the observed XCX hond angles and the state of hybridization of the orbitals used to form the double bond is readily calculated assuming that only the 2s and three 2p orbitals of carbon are utilized in overall bond formation (see Appendix). This improved value for the iuterorbital angle between the two hent bonds that make up the douhle hond, rather than the tetrahedral angle of the simple Pauling model, is then used to calculate an improved value for the CC double hond length, and other bent hond lengths, from the length of a standard single hond. For example, ethylene has an observed HCH bond angle of 116.2' (3)which implies 30.6% s character for each C-H bonding orbital and, hence, 19.4%s character for each hent hond orbital. The angle between the hent hond orbitals is then calculated to he 103.9" (rather than 109.5') which imples a C=C distance of 133.7 pm (rather than 131.6 pm). This distance is, in fact, in excellent agreement with the experimentally observed bond length of 133.7 f 0.3 pm. Thus, it may be claimed, at least in as far as the prediction of hond length is concerned, that the bent bond description provides a rather good mudel for the double bond. Application of a similar approach to the hent honds in cycloprupane (Table 3) leads to a predicted CC single hond distance of 150 pm, and for perfluorocyclopropane 149.5 pm, which are to be compared to the observed bond lengths of 151.2 f 0.3 pm (CaH6). ( 6 ) .and 150.5 f 0.2 pm (C3F6). (91. Analogous calculations on cyclopropene are not quite so straightforward because interorhital angles cannot he ohviously calculated from the geometry (Tahle 51, and there is some disagreement between the two reported structure de-
Table 5. Pauling Model and Refined Models for Cycloorooene - -
s=,. '133.0 149.9 '133.0 149.9
do -
109.5 109.5 116.2 116.2
8'" ~
125.3 125.3 121.9 121.9 ~~
i."" ~. 7.7 24.6 11.1 28.0
~
Tetrahedral orbital. at carbon. a Emylene-like orbitals at double bond. 'Double bond lensh in ethylene.
332 / Journal of Chemical Education
Substituted Ethylenes Calculated douhle bond lengths derived from XCX hond angles in substituted ethylenes are shown in Table 1.In fifteen of the twenty-two molecules listed the direction of the deviation of the C=C hond length from that in ethylene is correctly predicted and the calculated bond lengths agree with those observed within about 190,i.e. to better than 2 pm. In the case of CHCICH2, where the disagreement is 1.4%, the calculated bond length is barely outside of the rather large error in the reported experimental value. Among the remaining six cases. where agreement with exoeriment is rather poorerThan I%, the observed C=C hond iengths are longer than in ethylene, whereas they are predicted to he shorter. In all of these cases, however, there are two or more bulky methyl substituents or four bromine or chlorine substituents. It would appear that in these molecules there is some additional effect, such as interactions between the suhstituents, that makes the CC double hond length rather longer than expected (see for example (14,15)). Appendix The familiar examples of spS,spZ and sp hybridization correspond, respectively, to interorbital angles of 109.5', 120' and 180°,and intermediate states of hvbridization result in intermediate aneles, as has been discussed, fo; example, by Coulson and by Mislow i10): In general, if we represent the ith orbital to a carbon atom as having sp A; hybridization, where A j is the hybridization parameter, then itss character is given by 1/(1+Xi2),SO that for all of the orbitals to carbon, because the totals character must add up to one 2s orbital, XiZ)] = 1, and for an angle 8, between two hybrid we have x;[ll(l+ orbitals i and j
terminations for this molecule. Wiherg and cwworkers (12) and Chiang 1 1 1 ) give the ('C single bond rlistanre in the ring a i l3l.5 and 152.1 r 0.3 Dm. rpspertiveI\~,andtheC=Cdouble hond distance as 130 a i d 130.5 f 0.3 pm, respectively. However, they report rather different values for the H-C=C bond angle, of 149.g0 and 133 f 3', respectively. On the basis of these observed H-C=C hond angles and by assuming ethylene-like orbitals a t the double hond the C=C distance may be calculated (assuming that the C=C bond forms an arc of constant curvature of length 133.7 om equal to the douhle hond length in ethylene). 1Cis then a Bimplematter to calculate also the sinele C-C distances in the ring from the requisite interorbital"ang1es derived from the observed geometry. A value of 133O for the H-C=C angle gives 133 pm for the douhle bond distance and 144 pm for the single bond distances in the ring, while an angle of 149.9' leads to double bond and single bond ring distances of 129 pm and 148 pm, respectively. These are to be compared to the observed distances of 130 and -~ 151 nm. Wibere's structure is ao~arently more consistent with .. the calculations than is that of ~ h i a n g , i n dit is interesting to nnte ..... thnt - direct ~ ~measurement ~ ~ on a Prentice-Hall Molecular Framework model of cyclopropene (Figure 2e) gives an H-C=C hone,.anele ~.. close to 150' and bond lengths of 149 pm ( C - C ring) aud 129 pm (C=C ring). In l r i m ~ ~ r l ~ \ l ~ n u r ) c l ~ , p r oupparticularly onv short CC ring distance of 145.6 f i p& bas been reported (13) together with an exocyclic double bond length of 134.7 f 1pm. On the basis of a CC single bond length of 154 pm and the assumption that the orbitals a t each carbon atom are similar to those in ethylene (i.e. the angles between the carbon orbitals forming the ring are each 116.2") the calculated CC distances are 148 pm (ring) and 134 pm (exocyclic C=C). However, the calculation may he further refined by using the observed exocyclic C=C distance of 134.7 pm to calculate a more precise value for the magnitude of the interorbital angles a t the carbon atoms forming the ring. This calculation gives 101.3' for the angles between the orbitals forming the exocyclic douhle bonds (rather than 103.9" as in ethylene) and interorbital angles of 120.4O (rather than 116.2O) between the carhon orbitals forming the ring. The calculated ring CC distance then becomes 147 pm, which is in rather close agreement with that observed (145.6 f 1 pm). These calculations are shown in Table 3. ~
~
~
~
~~
~
1+X;Ajeos.9ij=0 so that for identical orbitals, eos Rii = -1/Ai2. Thus, for example, the characters of the orbitals involved in CH bonding in ethylene are readily calculated from the observed HCH hond angle of 116.Z0 (assuming that the interorbital angle is indeed the same as the hond angle), i.e. for Rii = 116.2', cos 8.. = -0.4415, whence the state of hybridization is sp2.2" corresponding to 30.6% s character. Since the s character of all four bonds to carbon must be 100%and there are two C-H bonds, it fallows that each bent bond that contributes to the double bond must have 19.4%s character, corresponding to sp4-'6hybridization, and the interorbital angle between the bent bonds, Rjj, is then given by
Literature Cited (1) Pauling,L., J Amer Chem Soc., 53,1367 (1981i:Slater. J.C.,Phys. R m , 37,481 11931): Pauling, L., in "Nature oftho Chemical Bond: 3rd Ed., Cornell Univ P r m , lthsea. 1960. pp. 136142. (2) See. for example. Ramray, 0. B., in '"van'tHofl~LeBe1 Centennial," Val. 12. A.C.S. Symponium Series, Amsr. Chem S n r . Washington, 1975, pp. 74-76, (3; Kuchitru. K..L Chem.Phys.. 44,906 (19661; Bsrtell.L.S..Roth.E.A..Hollowell,C. D.,Kuchifsu,K. K.,and Young,I.R., J. Chpm Phys.. 42,2683 (1965). (4) Tanimoto. M.. Kuchitsu. K., and Morino. Y.. Rull Chzm So
