P
W
4844 Pacific Southwest Association of Chemistry Teachers
Elihu Goldish University of Southern California LOS Angeles 7
I
I
Strain and interatomic Distances in s.1111-Ring Molecules
In connection with his research interests of recent years, the author has compiled a table of available molecular structures (configuration, bond lengths, and bond angles) of three- and four-membered organic ring compounds and their derivatives. This compilation is presented here (Table 1) with an accompanying discussion of the effects of strain on molecular dimensions. It is hoped that this table and the discussion, based on relatively simple mechanistic concepts, will prove of interest, since few texthoohs give appreciable space to this topic, and much of the puhlished literature is concerned with more complex quantum-mechanical aspects (la, b, c). Methods and Reliabilities of the Structure Determinations Table 1 contains the bulk of the structures of smallring molecules which have been determined. Sec-
ondary references, not directly related to the structure determination hut of structural interest, are also given. Techniques of structure determination have advanced greatly in the last decade, and sometimes caution should be used in considering results of work prior to about 1945. References to readable descriptions of the principal methods are given below. Infrared and Microwave Spectroscopy. The rotational energy of a gaseous molecule is quantized, and from the discrete absorption spectrum of energies in the microwave radio region or infrared light region it is possible to derive very accurately the moments of inertia of the molecule. Relationships between the atomic masses and interatomic separations are derived from these moments, which give, in the very simplest cases, the structural parameters directly; in general, however, for a complete determination, it is necessary to study isotopically substituted molecules also.
Table 1. Structure Data for Small-Ring Compounds
-
IR infrared spectrorrcopy; MW = microwave spectroscopy; ED = electron diffraction; X e = X-ray diffraction; assumed value; I = perpendicular to ring plane. Parameters (all distances in d)
Molecule HL-CH, Hs Cyclopropane HL-CH1
\c/ Cyelaprapyl chloride
H2C-CHs
c' /
C-C C C C-C C-C C-C C-C
+
1.526 0.03; C-H = 1 .08*. LHCH = 109' 28' 1.525; C-H = 1.07; LHCH'= 118.2" = 1.524 + 0.014 = 1.524; C-H = 1.07:; LHCH = 120°* = 1.518; C-H = 1.10 & 0.03; LHCH = 116 & 5" = 1.511; C-H = 1.086'; LHCH = 116"* =
=
=
based on
Reference ( 6 ) ED
(7) ED (8) I R (9)IR (10) ED (11) IR, ED
(1s) ED
= 1.52 i 0.02 C-C C-CI = 1.76 + 0.02 C-H = 1.09* LCI-C-I = 34 & 2" = 1.47 LHCH = 109" C-C LH-C-Cl = 109" C-C1 = 1.755 C-H = 1.09 (Unrefined values)
(13) MW
C-C = 1.5131 C-CI = I . 7780 C-H = 1.105
(83) MW
LHCH = 114-36' LH-C-C1 = 120'52.9'
C-C(ring) = 1.5131 C-Ccn = 1.4679 C=N = 1.1574
LHCH = 115'35' LH-C-CON = 119'35'
C-C=1.52&0.02 C C I = 1.76,& 0.02 C-H = 1.09
LCI-C-I=34*2" LCI-C-C1 = 112"
H/\cN Cyclopropyl cyanide
408
*
/ Journal of Chemical Education
(88) MW
Molecule
Parameters ( d l distances in
A)
Reference
H/'H Cyclopropaneearhohydrazide HL-CH, \N/ H Ethylenimine
C C = 1.480 C-N = 1.488 C-H = 1.083 N-H = 1.000*
LHCH = 116" 41' L H N I = 22" LHGC = 15g0 25' LCNC = 59" 39' LCCN = 60' 11' 'I2LHpCC-'I2 LHICN = 9' 30'
CC = 1.472 C-O = 1.436 C-H = 1.082
LHCH = 116" 41' LH,CC = 159" 25' LCOC = 61' 24' LCCO = 59O 18' LHsCC-'I? LHICO = 7- 40'
'/2
H,C--CH,
\o/
C-C C-0 C-H
= = =
Ethylene oxide
Propylene oxide
Epichlorohydrin
LHCH = 116' 15' + 21' LH&C = 158' 6' f 39' LCOC = 61" 17: LCCO = 59" 11 1 ' 1 LH1CC-1/2 LHGO = 7- 42' 1.470 f 0.001 1.435 f 0.001 1.084 i 0.002
LHCH(ring) = 116- 15" C C ( r i n g ) = 1 .471* LHCH(Me) = 109' 28'. C-CM. = 1.513 zk 0.020 C 4 = 1.436. LCCC = 120° 57' lo C H ( r i n g ) = 1.082* LCa.CI = 33' 47' f l o C H ( M e ) = 1.09: C-Ca. = 1.52f0.03 L C d 2 I =32+5' All other parameters assumed from (18) C-Cu.=1.52+0.03 LCx.CI=32+5' C C I = l76* CI almost trans against mid point of C-0 bond H, All other parameters assumed from (18)
(17) MW see a180 (10)
($0) ED ($0) EL)
-.
C-C = 1.492 C-S = 1.819 C H = 1.078
HL-CH?
\s/ Ethylene sulfide
I/,
LHCH = 116" 0' LH&C = 151'43' LCSC = 48" 24' LCCS = 65" 48' LH&C - '1. LHdX = 4' 37'
LHCH = 118"* 1.52s zk 0.02 LC=C-H = 152' 12' 1.288 f 0.04 1.OU1 f 0.04 (a") = 49' 52 LC=C-C = 65" 4' 1.515 LHCH = 114" 42' + 10' 1.300 LC=C-H = 149" 55' 1.087 i 0.004 (methylene) C-H = 1.070 (vinyl) LC=C--C = 64' 36' L C - C C = 50" 48'
C-C = C=C = C-H = LC-C-C C-C = C=C =
j
C-C(eentra1) = 1.48 f 0.03 C-C(periphers.1) = 1.51 + 0.04 C H = 1 .08*0 a = 61.5 2 LHCH = 120 i 8"
+
Spiropentane
&H* Methyleneoyclopropane HOOC
H (86)XR
&H, Feist's acid Volume 36, Number 8, August 1959
/
409
Molecule
Nor-tricyclene HL-CH2
I
I
HL-CH2 Cyclohutane HL-CH
/CHA
I
I
HL-CH, Methylcyclohutsne F2CCFl
I
I
FL-CF, Octafluorocyclohutane CIL-CCl*
I
1
CI2C-CCI? Oobehlorocyclobubne Q
/"
\
HC-CH
H U H
/
\
Q
Q
Parameters ( d l distances in
A)
Reference
CC(cyclopropme ring) = 1.50 10.02 C-C(other) = 1.54 10.02 L C C L = 19.7 12' See original paper for other parameters C4=1.5fi81:0.02 LHCH=11418" C-H = 1.OSs 1: 0.04 Ring non-planar(D2sor D4nsymmetry) with dihedrd angle -ma) 2oo(+lo", C-C(ring) = 1.56 i 0 .O3 LCCCuD= 118' C-Cm = 1.54 10.06 Angle between C-CM. and Iadjaoent C-C-C plane = 40 18' Non-planar ring; dihedral angle ahout 2W30° (SO) ED see also (31, 32)
C-C=1.60+0.04 LFCF=109513' CF=1.33+0.02 LCCC=89' Non-planar ring; dihedral angle 20 1: 4'
(53) XR
LCCC = 87.7' (a") C-C = 1.59 (av) = 109.4' (sv) LCI-C-CI C C I = 1.74 (av) C . . . C = 2.20 (av) Non-planar ring; dihedral angle ahout 22'
{i 2
:::;)I ,570 (av) C-C(ring) = ; 1; D C ( m ) = 1.388 (a") 10.021 (max dev) C-Cm = 1.504 (av) 192' L C & L = 45" (av) L L I + = ,390
1,2,3,4Tetraphenylcyclohutane C, ring planar; 6 very nearly reguler planar hexagons C-C(cyolobutane ring) > 1.54 Not completed Dinaphthylene cyclohutane; acemphthylene dimer C C = 1.54 C 4 = 1.46
HB-b Trimethylene oxide
H,CCH,
I I
HA-S Trimethylene sulfide
LCOC = 94.5' LCCC = 88.5" L C ~ O= 88,5a}~sumede w d LCOC = 94.2 f 2.5' C-C = 1.55a 10.03 C4=1.457+0.02 LCCC=86.81:2.5' = -113O 3R'* fav) LCCO = 89.5 f 2.5' LHCH ~LCCC = 84' C C = 1.55 C 4 = 1.44 LC. . . C-HZ = 4' C-H = 1.09 LHCH = 110' (adjacent to 0) LHCH = 111' (oooosite 0) C-H = 1.08 . .. Planar ~
LCSC(mrtx) = 78.0 11" C-C = 1.549 10.03 LCCC(msx) = 97 =t5' C-S = 1.85, 0.02 Large out-of-plane v~hrations
+
+
C-C = 1.56 0.03(av); C=C = 1.34 1: 0.02; a = 90' C-C = 1.55 10.02(av); C=C = 1.34 f 0.03; a = 92.5 Coplanarity assumed, hut investigation ineonolusive
a = 92.5 zt 1' C-C = 1.526 1 0 . 0 1 2 C=C = 1.328 f 0.012 Ring non-planar; dihedral angle = 5 zt 5'
Dimethylketene dimer
410
/
(22) MW dso (38)
see
Journal of Chemical Education
+ 2'
Molecule
Parameters (all distances in C-C = 1.53e zt 0.02 C-0 = 1.45a zt 0.03 C=C = 1.3'3. i 0.02 Coplanarity assumed
d)
Reference
a = 86.7 i 3' LCOC = 92.8"
3-Methylenetrimethylene oxide
+
C-C = 1.53 0.03 (a~sumedcqoal) C-0 = 1.45 zt 0 0 3 (ditto) LCOC = 89 i 3' LC-C=O = 143 zt 3" C=O = 1.19 + 0.03 u = 94.
Ketene dimer: diketene
C-C(+) = 1.37 (av) i 0.02 ( m a dev) C N ( r i n g ) = {1:42} 49 zt 0.010 C+-N=141fO.O10 C=0=1.15&0.010 131' 1' a = 93 1LCNC = 87 LCNC+ = {1340 L O = C l r = 87.5" L C + N l r = 79' Planar ring
+
Phenyl isocyanate dimw
.
.
I'otasnium benzylpenivillin HC=CH
I 1
H,C-CH, Cyclobotene
/CH, HC=C .
I
I
HL-CH, 1-Methylcyclohutene
+
I n the ring: LCNC = 95" C C = 1.52 in C-C=O C C = 1.51 in C-C-S a = 90.5" LNCC = 86.5' C N = 1.34 in N-C=O C-N = 1.47 in N-C-S LCCC = 87' Attached to ring: N4-C = 1.47 C=0 = 1.17 C,-N = 1.46 C-S = 1.84 CC=1.537f0.010(av) C=C = 1.32. i 0.04 C-H = 1.093 + 0.0lay(av)
C-C = 1.54 + 0.03 (av) C=C = 1.34 + 0.03
LC=C-C=94.0i1.4" A(C-C) S 0.08
LC-C-CM. LHC=C-CH,
125 zt 4" 93' 40' zt 3'
=
=
- ~ . .~ . (Refer to original psper far other simplifying assumptions)
\
C=CH
I
"
C-C
I
+y,
0 Phenyleyolohotenedione
C=C C-C C-C C-C C-Cm
= = = = =
1.340 L C = C C + = 134.3" 1.505; C=O = 1.196 (in 6-CC=O) 1.463; C=O = 1.230 (in H-CC=O) 1.536 (in O = C C = O ) 1.469 C-C(m) = 1.390 (a")
C-H a =
=
121
1.10' 3O
Biphenylene Volume 36, Number
8, August 1959
/
41 1
Table 2.
Bond
Standard distance (in A)
Reference Distances
Characteristic Features of Small Rings
Sources and references Based on values for ethane [I.536 (55,56),1.543(57)l and diamond
[I.544 (68)l
Based on velues for propene [I.498 (34)l. and cis-buten& -21.504 (9?4)1 Cf. iiaohutene [1.505(84)1 and ethaldehyde [l.501 (59)l Based on values for propene [1.338 ($411,isobutene [1.336(84)1,and ns-butene-2 [1.343 ($411. Cf. ethylene [1.334(601, 1.337 (611, 1.339 (69?)] Based on values for methylamine
[1.474(63,64), 1.48(65)1.
methylamine [I.465 (66)I
Cf.
hydrochloride
Based on values far methyl alcohol [l.42l (67), 1.434 (68),1.437 (69), 1.428 (70), 1.426 (71)] Based on values for methyl mercaptan [1.815 (73), 1.8170 (7.91,
1.8177 (74)l
(Otherwise, some of the less important distances and angles are assumed, and such assumptions may limit the accuracy of the final result.) Complete and highly accurate determinations of small molecules are now frequently found in the literature. Limits of error of 0.005 1and less are common in this work @a, 3). Electron Diffraction. The scattering from a beam of monoenergetic electrons passing through a gas jet in a vacuum chamber is recorded on a photographic plate perpendicular to the beam as a series of diffuse concentric rings. Analysis of this scattering pattern leads to a determination of interatomic distances in the gaseous molecule. For molecules of moderate complexity, limits of error of from 0.01 to 0.03 A on principal interatomic distances have been common in the past; however, improved methods of taking the photographs and refining the structural parameters have given results in recent years comparable in accuracy to those based on microwave measurements (3b, ha). X-ray Diffraction. Not only the structure of the individual molecules but also how they pack in the crystal is determined by this method, in which a single crystal is placed in a beam of monochromatic X-rays which are diffracted by the regular spacing of the atoms in the crystal; these diffracted X-rays are recorded as spots on photographic film or, for greater accuracy, measured individually by a counter tube (Geiger or other type). From the intensities and positions of these spots the shapes and relative locations of the molecules in the crystal are deduced. Refinement frequently gives standard errors of 0.01 to 0.04 A; in certain studies in recent years the accuracy has been even better. (Hydrogen atoms, however, are usually difficult to locate accurately by X-ray methods) ( 3 4~h 5 ) . 412
/
Journal of Chemical Education
All three- and four-membered organic ring molecules are highly strained. Strain energy may be defined for our purposes as the extra potential energy stored (in various ways; see below) in a molecule having at least one bond angle constrained to differ from its normal value. Thus, for example, the 60' and 90" ring angles in cyclopropane and cyclobutane differ significantly from the normal tetrahedral value of 109' 28' for bond angles at carbon; these molecules have strain energies per mole of 9.2 and 6.5 kcal/CH,, respectively (37, 54). (Cyclopentane has 1.3 kcal/CH2 strain energy, cyclohexane none.) Chemically, this results in such wellrecognized behavior as the great reactivity of cyclopropane (and, to a lesser extent, of cyclobutane) relative to other saturated hydrocarbons; less well known are the structural consequences, resulting partly from use of some of the strain energy to bend, and stretch or compress, bonds in the molecule. By perusal of the data in Table 1, two generalizations about small-ring structures can be made immediately. One is that bond distances in three-membered rings tend to be shorter than normal (as, for example, in cyclopropane, with C-C = 1.525 A); another is that distances in four-membered rings tend to be rather long (as in cyclobutane with C-C = 1.568 A). I n addition it appears in many cases that a the three-membered series). Cuuses of Anomalous Distances in Small Rings
The anomalous bond lengths mentioned above have been the subject of much discussion, most of it on a more advanced level than employed here. For the sake of visualization on an intuitive physical level, however, it is convenient to describe the causes of these abnormal distances by the concepts of "bent-bond shore ening" and "cross-ring repulsion"; in many molecules the effect of a third factor, "differential bond-angle strain," is superimposed on changes from the other two. I t is also convenient with regard to these concepts to think of small-ring molecules in terms of a mechanical analogue, something like the traditional ball-and-stick model. The "bonds" are made of rather stiff springs fixed a t normal, unstrained angles in rubber ball ',atoms;" two weaker diagonal springs, in the fourmembered rings, simulate the cross-ring repulsions discussed below. The potential force corresponding to a change from normal in a bond angle is provided in our model by deformation of the rubber-ball atoms as well as some bending of the spring bonds themse1ves.l The reader may find it helpful to bear this model in mind in the ensuing discussion. Table 2 is a list of reference interatomic distances. These are taken from reliably determined structures of relatively simple unstrained compounds, and will be used below as comparison standards; Table 3 gives some bond-stretching force constant ratios. (A Such a model would also show the characteristic widening of the H-C-H angle when the skeletal hononds of a ring carbon are forced closer together, as noted in the previous section.
Figure 1. Bent-bond shortening: skeletal orbitals in cyclopropone molecule, rhowlng bonds (long dasher) bent away from internuclew directions (short dashes). Cf. refs. ( I b , 761; see also Mmhima, M., J. Chem. Phys., 22, 1 7 8 5 (19541.
stretching force constant is defined for a small change in bond length, A~*-B,produced by a force t ~ - Bby , the equation: t * - ~ = k A - ~ AT*-^). See, for example, ref. (S), pp. 14, 26, 195, 331.) Bent-Bond Shortening. If we consider, with reference to the model described, the deformation of a C-C-C angle from its normal value of 109" 28' to, say, 60°, a shortening of the C-C internuclear distance would occur because of the bending of the C-C bonds; this "bent-bond" or "banana-bond" conceptjs useful for explaining qualitatively the short (1.525 A) C-C bonds in cyclopropane (76, 77). I n valence theory language, we would say that the normal tetrahedral orbitals on carbon are not able to bend toward each other as close as 60°, so that the maximum overlap of orbitals on adjacent carbons (and hence the line of maximum electron density defining the true bond) curves away slightly from the straight internuclear line. In three-membered rings, this shortening can be empirically estimated (from cyclopropane) to be 1% of the normal bond length;2 in four-membered rings the shortening is probably much less significant, perhaps '/a%. Cross-Ring Repulsions. A repulsive force between diagonally opposed (non-bonded) carbon atoms has been proposed (27) to explain the long C-C bond
length (1.568 A) in cyclobutane; allowing for a bentbond shortening of 0.005 A (see above) the cross-ring repulsiveforce lengthens each C-C bond in cyclobutane by 0.033 A, or more than 2%. In other fourmembered ring molecules the increase would depend on the magnitude of the repulsive energy and the stretching force constants of the individual bonds. There is also a small strain energy of adjacent CHI groups, and in CnClsand C4Fathe repulsion between halogen atoms on adjacent carbons may also contribute to the even lower C-C bonds of 1.59 A and 1.60 A., remec. tively (80). Diffwential Bmd-Angle Strain. Referring again to the rubber-ball-and-spring model, it will be seen that if the anele deformation enernv of one of the rubberhall atoms correspondi in^ to omp press ion of the bond angle from normal) differs from the others in the ring,
Eiomple~ of differentid bond-angle strain for various idealized molecule^ (see text], showing only ~ k e l e t a latoms and omitting bent-bond
Figure 3. effect.
unequal strains will be set up in the ring bonds; this is the general case of a heterocyclic ring, or one in which t.he skeletal atoms are not exactly equivalent. The angle deformation energy of an atom, transmitted by an attached bond, becomes a strain force on a neighboring bond, giving rise a t the same time to an equal but opposite "reaction" force.a The resultants of these forces, "differential bond-angle strain," compress and stretch the various bonds. As the reader can verify for himself, these angle strain forces and their reactions exactly cancel out for the highly symmetrical cases of cyclopropane and cyclob ~ t a n e . ~On the other hand, if each carbon in, for example, the less symmetricrtl ethylene oxide, H,C--CH,
i Figure 2. Schematic drawing of portion of rubber-boll-atom and spring-bond model of strained molecule, showing how m g i e deformation energy rtretche. neighboring bond! the forces a t A and B are of equal magnitude.
.
.
2 I n a more elaborate treatment the shortenine should oerhaos be n n d r inversely proportior~nlto thr bond-rtrrtehing forcr COIIatsnt; this refinenrent i ignored hrrr ereelt for thp extreme caw of the double bond, where thc ~ h o n m i n ei.i, srsumcJ to l x n l , u ~ 1
',q.
"0' This pair of forces is a t right angles to the "tmsmitting" bond; components of these forces are taken along the directions of the other bonds. The effect of the bending of the bonds is neglected in this approximation. Since the four oarbon atoms in either the olefinic or resonatine made. Volume 36, Number 8, August
1959, / 413
Table 3.
Stretching force constant ratio
Vslee
kc-clko-a kc-lko-o
1.48 0.78
kc-c/kcrr kc-clkc-
0.93 0.47
reaction would tend to shorten the C-C bond. By similar reasoning it is predicted that in trimethylene oxide,
Force Constant Ratios
Reference
'
Based on stretching force constant values quoted in (75)
has a greater angle deformation energy than the oxygen atom does (as seems reasonable) the angle strain forces do not cancel out, giving rise to a compressive force in the C-C bond and a stretching force about half as large in each C - 4 bond.5 A similar result is obtained for ethylene sulfide, etbylenimine, and cyclopropeue, i.e., the unique bond is compressed while the other two bonds are stretched by a force half as large. But in methylene cyclopropane the angle-stiffening property of the double-bond reverses the situation, and the unique ring bond is stretched by a force twice that compressing the other two ring bonds; this latter case is also that found in Feist's acid and ~piropentane.~ An alternative view of this important effect is obtained by considering what would happen in ethylene oxide, say, if the three ring atoms were suddenly free to move an infinitesimal distance. The more highly strained carbon atoms would tend to relieve their strain at the expense of the less strained oxygen by stretching the opposed C-0 bonds; as this occurred the resulting -
Wonsidered as a problem in plane statics, the more exact and general result is obtsjned that, for three-membered rings with one = -cas L C C 4 , where t is the tenatom unique, LC-oltc-c sion in the bond; this result does not depend on the particular bonds involved or the relative angle strain energies. For an equilateral triangle, cos 60' = I/,, which is the approximation used here. See ref. (78) for an analytic treatment of this topic. 6 Spiropentane is an exception to the simple theory presented here, since the large strain at the central carbon would be expected to lengthen the peripheral bonds and shorten the central ones, whereas it is found (see Table 1) that, although the relative lengths show the expected difference, all the bonds are shorter than in eyolopropsne. An explmation has been suggested ($31, but the molecule is of sufficient interest to warrant, perhaps, a confirmatory structure investigation.
Table 4.
Given bond (obs.)
Molecule HG-CHs
'/n(1.525-1.470)X0.78= +0.021
C-C
= 1.492
'/,(I ,525 - 1.492) + O 024
C-C
= 1.480
C-C
= 1.515
S
Hd-CH?
'3' HC=CH
\/
Calculated change in other bond
=-1.470
\/
P
Same Calculations
At least two major obstacles stand in the way of simple a priori calculations of small-ring structures by methods of plane statics: bond angles have changed so far from normal, especially in the three-membered rings, that the simple harmonic potential function cannot be assumed, and in four-membered rings it is difficult to assess the magnitude of the cross-ring repulsion potential. For three-membered rings, however, some check on the ideas presented here can be made by finding the ratio of the predicted lengthenings and shortenings of bonds, equivalent to solving the following problem: given one experimentally det,ermined bond length of a strncture, calculate the other. Consider, then, as an example of the results given in Table 4, ethylene oxide. The observed C-C bond is
Calculations for Three-Membered Rings
C-C
'5' H?CCH,
the C-C bonds are being compressed and the C-0 bonds stretched by approximately equal forces (exactly equal if the molecule were a perfect square); this situation also occurs in trimethylene sulfide. (Since the C-C bonds in these two four-membered rings are being stretched by cross-ring repulsions and simultaneously compressed by differential bond-angle strain, relatively little net change in the length of these bonds from their standard value (see Table 2 ) is expected compared to the C-0 and CS bonds, which are being stretched by both these forces. This is in accord with the experimental results listed for these molecules in Table 1.) In methylenecyclobutane the ring bonds adjacent to the double bond are compressed, the other two stretched (just the reverse of the trimethylene oxide case), and in cyclobutene the double bond is compressed and the opposed bond stretched by approximately equal forces, while the two side bonds are under (almost) no net strain from this source.
Standard length of other bond (corrected)
Expected length of other bond
Other bond length (obs.)
1.427X0.99=1.413
1.434
1.435
1.817 X 0 . 9 9 = 1.799
1.823
1.819
'/2(1.525-1.480)X0.93= t0.021
1.474X0.99=1.459
1.480
1.488
-2(1.515-1.502X0.99)X 0.47 = -0.026
1.338X0.995
1.331 1.305
1.300
1.531
1.535
x
1.48 =
=
r,h.
- re.,. 0.001
-0.004
0.008
-0.005
H*
HzCCH?
\c/
414
/
C-C(side)
= 1.484
2(1.502 X 0.99 +0.006
Journol o f Chemical Education
-
1.484) =
1.525
(uni ue c-8)
0.004
shorter (compressed because of differential bond-angle strain) than in cyclopropane by 1.525-1.470 = 0.055 A; a stretching of each C-0 bond is predicted, as noted before, corresponding to half the C-C compressive force (in the equilateral approximation). Taking 0.78 for the ratio of the bond-stretching force constants, kc-c/ko-o,7 a C-0 bond lengthening of (I/*) X (0.055) X (0.78) = 0.021 Ais calculated This is then added to the standard C-0 bond length corrected for a 1% bent-bond shortening: 1.427-0.014 = 1.413 A. 0.021 = The calculated C-0 bond length, 1.413 1.434 is in good agreement with the observed value, 1.435 A; results for other molecules are also in reasonably good agreement with the experimental values.
4,
+
Conclusion
Using the structure data collected in Table 1 an attempt has been made to show that strain energy of small-ring compounds is not stored solely in distorted bond-angles, but may also bend bonds, and stretch or compress them, with observable effects on bond distances in these molecules. Through the use of concepts based on an artificial model of a ring molecule, some degree of correlation and a simple basis for qualitative discussion have been provided. Although only organic small-ring molecules have been discussed here, the same concepts would also bear on small-ring inorganic molecules as well (diborane derivatives, for example (79)). An-example of strain in an inorganic molecule is provided by P4, with a strain energy per mole of 5.7 kcal/P atom (80, St). Examples of alternative points of view and further reading on the matters discussed in this paper may be found in (la, b, c, d, 64,81). Acknowledgment
This investigation was supported by the Office of Ordnance Research, U. S. Army. The author also thanks Prof. V. Schomaker for many helpful discussions, and Profs. N. Kharasch and J. Donohue for reading the manuscript and offering much constructive criticism. Literature Cited ( 1 ) ( a ) ConlsoN, C. A,, AND M o ~ m W. , E., J . C h a . Phys., 15, 151 (1947); ( b ) WALSH,A. D., Trans. Faraday Sac., 45, 179 (1949); ( c ) HANDLER,G. 8.) AND ANDERSON, J. H., Tetrahedron, 2 , 345 (1958); ( d ) KILPATRICK, J . E., AND SPITZER, R., J . Chem. Phys., 14,463 (1946). ( 2 ) ( a ) WILSON,E. B., JR., AND LIDE, D. R., JR.,in BRAWE, E. A,, AND NAcnoD, F. C., "Determination of Organic Structures by Physical Methods," Academic Press, Inc., New York, 1955, p. 503; ( b ) KARLE,J., AND KARLE,I. L., op. eit., p. 427; ( c ) ROBERTSON, J . M., op. cit., p. 463. ( 3 ) WEST, W., editor, "Chemical Applications of Spectroscopy," Interscience Publishers, Inc., New York, 1956. L. O., in WEISSBERGER,A,, "Phy~ical ( 4 ) (a) BROCKWAY, Methods of Organic Chemistry," Vol. I, 2nd cd., Interscience Publishers, Inc., New York, 1949, p. 1109; ( b ) FANKUCHEN, I., op. eit., p. 1073. J. M., "Organic Crystals and Molecules," ( 5 ) ROBERTSON, Cornell University Press, Ithaca, N. Y., 1953. 7 Althoueh a bond-strctchine force constant ~rohablv becomes smaller as b e bond is bent, because of decrertsfng atomic orbital overlap, it is assumed here that the ratio of the constants remains unchanged.
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(51) WONG,C., AND MARSH,R. E., private communicstion. (52) WASER, J., A N D S C H O X.AV.. ~.R J ., Am. Chem. Soc... 65.1451 . (1943). (53) WASER,J., AND Lu, C., J. Am. Chen~.Soc., 66,2035 (1944). (54) WESTHEIMER, F. H., in NEWMAN, M. S., editor, "Steric
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