Dwight F. Mowery, Jr.
Southeastern Massachusetts Technological Institute Dortmouth, Mossochusetts 02747
II
Criteria for Optical Activity in Organic Molecules
There has been and still is (1, 8) considerable lack of clarity regarding the reason for optical activity and criteria for prediction of the possible existence of optical isomers. The basis of this difficulty appears to be the tendency to favor a time-average (3) of the effect of a molecule or solution on a ray of plane polarized light passing through rather than a moleculeavevage although the latter must also be assumed in order to explain the facts. Actually the moleculeaverage is the most necessary of the two assumptions and, in view of the exceedingly short time required for light to pass through a molecule or a solution in a polarimeter tube, the timeaverage may well be neglected entirely. This is the basis for the present article, and it is hoped that the following presentation will lead to a more fundamental understanding of the criteria for predicting the possible existence of optical activity in organic molecules. It is not my purpose to consider the physical characteristics or methods of generation of plane polarized light. These can be found in any good physics text. Also I am not concerned with the mcchanism of the interaction between the nuclei and electrons of a molecule and a ray of plane polarized light passing through with a resulting rotation of the plane of polarization. I will assume the plane of polarization of a ray of light passing through a single molecule will always be twisted in one direction or the other except in one specific instance, a ray passing through a molecule possessing a plane of symmetry and passing in a direction exactly perpendicular to tbat plane. I n this case the plane of polarization is twisted in one direction upon traversing half the molecnle and to the same extent but in the opposite direction upon traversing the rest of the molecule (4) since the first half of the molecule is the mirror image of the second half (definition of plane of symmetry). This is an example of intevnal compensation erroneously used in some texts to explain the optical inactivity of meso tartaric acid. It should be noted that only an exceedingly small fraction, approximately (r/360)=, of the meso tartaric acid molecules exposed to plane polarized light in a polarimeter tube will on the average be oriented so that the perpendiculars to their planes of symmetry are within 1" of the path of the light so as to produce this type of compensation. Obviously a different explanation of the observed optical inactivity of meso tartaric acid and other compounds possessing a plane of symmetry is necessary. Since any solution with sufficient solute for optical measurements contains exceedingly large numbers of molccules, i.e., 6 X 10'8 in a 0.001 M solution contained i n a 1 X 10-cm polarimeter tube, it becomes evident
that optical inactivity must result principally from external compensation. For external compensation it is necessary that a second molecule of the solute be oriented exactly as the mirror image of a previous molecule through which the ray of polarized light has passed. The net twisting of the plane of polarization of the ray upon passage through a polarimeter tube will then be the resultant of small twists back and forth by each molecule through which it passes. Since exceedingly large numbers of molecules are involved it can be assumed tbat if these are randomly oriented in the solution there will be an equal probability for onentation of a molecule in one direction as in the exactly opposite (mirror image) direction. This will result, statistically, in half the molecules in a given polarimeter tube being oriented as the mirror images of the other half so as to produce a net rotation of zero. Criterion for Optical Activity
It is apparent that the fundamental criterion for the possible existence of optical activity is the criterion for ruling out external compensation, namely that a molecule is not identical with and therefore cannot be oriented as its own mirror image. This makes it impossible for a second molecule to exactly cancel the effect of a first in rotating the plane of polarization of a ray of plane polarized light. Since this is the one and only criterion for external compensation it is the only fundamental criterion for optical activity. Since in the case of complex molecules it is difficult to determine whether or not a molecule is identical with its mirror image without actually constructing threedimensional models, several rules of thumb have been introduced as so-called criteria for predicting optical activity. I n actuality these are all rules which may be applied to determine the identity or non-identity of a molecule and its mirror image. The simplest and most commonly occurring of these is the plane of symmetry. A molecule which is identical with its mirror image will in most cases possess a plane of symmetry. The reason for this will become apparent from the diagrams in Figure 1 of two identical molecules oriented as mirror images of each other. All bond angles are assumed to be the tetrahedral angle, the atoms (or groups) AI, c>A2 A%, B, and C are assumed to be symmetrical t and the bond lengths A2 j *I fixed (no vibration) and nlrmr done identical for identical groups. It is evident Figure 1. Two identical molecules that these identical moleoriented or mirror imager of each other. cules can be mirror
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iniagcs of each other only if AI is the same as As. If this is truc thc molecule must have a plane of symmetry passing through D and C and bisecting the angle bet~~~-cen A1 and A?. Although the principle is illust,rated here wit,h a simple five atom molecule it is apparent that it will apply to a molecule of any complexity. Suppose, for example, a highly complex molecule has a plane of symmetry represented by the vertical lines in the identical molecules pictured digramatically in Figure 2. A' must then be the mirror image of A in each molccule and the second molecule must be the mirror image of the first since A of the first molecule must he the mirror image of A' of the second and A' of the first the mirror image of A of the second. Although the molecules have been oriented with plancs of symmetry parallel for this comparisorr it is evident that if one molecule is twisted slightly in one direction and the other in the opposite direction they will still be mirror images of each other. The General Rule of Thumb
A more complex situation arises if the atoms (or groups) A,, A2, B, and C in the diagram of Figure 1 are not symmetrical. Suppose, for example, a molecule (8) of the type shown in Figure 3 in which R(+) and R(-) represent opposite forms of asymmetric groups, R(+) being the mirror image of R(-). Assuming all angles arc tetrahedral the second molecule can be oriented as the mirror image of the first by merely rotating it 90' about a horizontal line through its central carbon atom. This molecule possesses no plane of, symmetry as such a plane would have to bisect bothjangles between R(+) and R(-) groups. I t does, however, possess a rotation-reflection or alternating axis of symmetry (5). An axis of this type is present if a molecule is left unchanged after rotat,ion followed by reflection in a plane perpendicular to the axis of rotation. This is the same as saying that a molecule may be oriented as its mirror image by rotation about an axis perpendicular to the mirror plane and then displacement along that axis. If the angle of rotation is 90", as in this case, the molecule is said to possess a 3G0/90 or four-fold alternating axis of symmetry. The procedure for determining the presence or absence of an alternating axis of symmetry is the same as that of determining whether or not a molecule can be orient,ed as its own mirror image with suhstitut,ion of reflcction in a plane perpendicular to the axis of rotation for displacement along that axis. Another molecule having no plane of symmetry but nevertheless being capable of orientation as its own mirror image and therefore exhibiting no optical activity is shown in Figure 4. The ring is essentially planar due to overlapping p orbitals in the resonance
0 / form ( +NH=C / I
) and the approximately 120" bond
\,
angles of the nitrogen and carhonyl carbon atoms. I n t,his ease it is mcrcly necessary to rotate the molecule a t the left lSOo about the axis "a" and displace it along this axis to produce its mirror image shown a t the right,. This manipulation is equivalent to that necessary for dctcrmining t.he presence of the 360/180 or 270
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Figure 2. Diogrom of complex molecule with a plone of symmetry. n l r r d plane
Rl+R,-l("'
Rl-i--Rl-i-e:l*: "I*' 1
Figure 3. Diogrom of o complex molecuie with asymmetric groups orranged ro that it has on .Iternoting orir of symmetry.
R1.i
R*i I
RI.1
RI-l
.I.,.! P.I
k-;Aa1 xxr-;4n Figure 4. A molecule with no of rymmetry but still optically
H I 1 1
.,rro.
Pi.".
inactive.
two-fold alternating axis of symmetry. This molecule has a point of sym netry in the center of the ring. This will always be true for a molecule having a twofold alternating axis of symmetry and can be used as a quick test for determining the possible presence or absence of optical activity. A plane of symmetry can be seen t o be merely a 360/360 or one-fold alternating axis of symmetry. Time Versus Molecule Averaging
For the two molecules compared in Figure 1 in which A1 and As are identical atoms the plane of symmetry is lost if the bond lengths to A1 and Ap are not the same, a condition which would be true most of the time because of stretching vibration. This brings up the old question of whet,her the measurement of optical activity requires sufficient time so a time average of a molecule is involved. Admit,tedly the actual measurement. may take several minutes but the time t,he polarized light is passing through a molecule with consequent twisting of its plane of polarization is exceedingly short,-i.e., about 30 X 10-8/3 X 101° or lo-'? see for a 30 A diamet.cr molecule. The carbon-hydrogen stretching vibration, one of the fastest in a molecule, produces absorption a t about 3 fi wavelength and 3 X 101"/3 X or loL4 vihrations/see. The number of vibrations of a hydrogen atom during a single passage of polarized light through the molecule would therefore he 1014 X lo-" for i.e., O.lyoof a vibration per passage. This calculation shows that the light passes through a molecule so fast that even the highest frequency stretching vibrations are essentially stopped and therefore presumably also lower frequency motions such as bending, rotation about a single bond, change from one ring conformation to another, etc. Since this is the case optical inactivity must be due not to an averaging out of motions d h i n a symmetrical molecule but rather to the presence of a second molecule in the ray of light which is the mirror image of the first molecule. For example, the molecule on the left in Figure 1 (assuming A, the same as A*) would havc no plane of symmetry if the C-A1 bond were stretched longer than the C-A? bond. The molecule on the right would, however, still be its mirror image and counteract the effect upon a ray of polarized light if the C-A? bond were stretched longer than the C-A, bond.
Since A1 and AI arc similarly (mirror relationship) oriented with respect to the rest of the molecule, vibrations of A1 and Ap in one molecule should be exactly duplicated by vibrations of A2 and Al, respectively, in a second molecule and a t any given instant for a given number of molecules present in tho solution in a specific vibration st,ate there should be present an equal number in the mirror image vibration state, producing an optically inactive solution. This may be thought of as a special case of a complex racemic mixture in which an infinite number of pairs of enantiomers are present. Thc energy barrier to interconversion of the enantiomers is so low that they cannot be isolated and the rate of interconversion so fast that only an almost instantaneous measurement such as the passage of a ray of light would not involve a time-average of the atomic positions. Rotational Conformers
Tartaric acid exists in a D, an L, and a mcso form. They are shown in Newman convention in Figure 5. I n comparing these formulas for identity or lack thereof the carboxyl groups, most conveniently, may be kept in the trans position. Thi.; also is the most favored conformation in the actual molecule. It is apparent that the D and L forms are non-superimposable mirror images and therefore separately would show optical activity. They possess no planes or alternating axcs of symmetry. The two formulas shown for the meso form are mirror images but one, by rotation 180' about a horizontal axis so as to interchange carboxyl groups, can be seen to be identical with the other. This is just the manipulatio~i required to show the presence of a two-fold alternating axis. The optical inactivity of meso tartaric acid is therefore due to orientation of half the molecules as mirror images of the other half (external compensation) rather than passage of a ray of plane polarized light through the molecule with no net change of the plane of polarization (internal compensation) (3). This effect would occur only for the eclipsed form having carboxyl groups, hydroxyl groups, and hydrogen atoms opposite each other and therefore possessing a plane of symmetry. A ray of polarized light perpendicular to this plane of symmetry would pass through the molecule with no change of rotation but all other rays of light through the molecule would experience a rotation of the plane of polarization. Expressed in another way, a molecule oriented with its plane of symmetry exactly perpendicular to the ray of polarized light would produce no net change in its plane of polarization, hut if oriented in any other way would twist the plane. I n a solution molecules must be oriented a t random in all possible directions. A question that arises in the case of rotational conformers is how molecules in transition between stable conformations will affect the plane of polarization. Let us use meso tartaric acid as an example. Suppose a slight rotation about the center C-C bond has taken place in the meso tartaric acid formula on the left in Figure 5 to produce formula A in Figure 6. The energy increase and probability of this occurrence will be exactly the same as that of the opposite rotation in its mirror image (right hand meso formula in Figure 5) to produce formula B in Figure 6. The molecule represented by A is not the same as B
as the two formulas cannot be superimposed. There is no plane or alternating axis of symmctry iu either molecule. If thesc conformations were stahlc the molecules could be separated into a D and an L form. Here again we have an example of a racemic mixturc in which t,he enantiomers are rapidly interconverting. The optical inactivity of a solution containing apprcciahle amounts of these cooformations is due to the fact that each one would he formed to exactly the same extent and therefore there would be cqual numhers of both forms producing a racemic mixture with net rotation of zero. The reason for the optical inactivity of a substituted biphenyl (G) in which one ring has a plane of symmetry other thari the plane of the ring is vcry similar to that for the optical i11activit.y of meso tartaric acid. As an example, 2,G-dintro-2'-hydroxy biphcnyl is shown diagramatically in Figure 7 in an end view using the Kewman convent,ion. Optical inact,ivity is due to a racemate consisting of equal numbers of the lower energy hydrogen bonded conformers R and C , which are asymmetric hut mirror images of each other, rather than to the single higher energy symmetric transition conformer A. This case differs from that in which neither ring has a plane of symmetry (other than the plane of the ring) in that ortho groups on the two rings do not have to pass each other for racemization. Energy of racemization would therefore he expected to he low and independent of the size of thc ortho substituents. This is in contrast to substituted biphenyls in which neither ring has a plane of symmetry. Here the interconversion of enantiomers involves passage of ortho groups and therefore the energy of racemization depends upon their size and the rate of racemization may vary all the way from very rapid and optical inactivity, through moderate and fleeting activity, to slow and relatively stable optical isomers. The inactivity, however, is due in all cases to the existcncc of a racemate rather than high energy time-averaged symmetric molecules (both rings in the same plane, the plane of symmetry). Ring Conformers
The disubstituted cyclohexane derivatives offer a slightly more complex situation thari the essentially planar rings of Figure 4. The 1,2-cis disubstituted isomer is shown in Vigure S along with its mirror image. It possesses no plane of symmetry, simple or alternab ing, and is not identical with its mirror imagc. I t can, howcvcr, he converted to its mirror image by
I ~,p;
coon
COO"
OW COOW rnlll., A
plm.
0"COOR B
Figure 6. The mero form of tartaric acid after o sliaht rotation
Figure 5.
The D, 1, and meso
form, of tartanc
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Figure 7. Three forms d dinitro-2-hydroiy biphenyl.
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have planes of symmetry and are idcntical with their mirror images and therefore optically inactive. Conclusion
Figure 8. The 1.2-cis dirvbrtitvted cycloheione and its mirror image.
Figure 9. The 1.2-Irons diwbrtitvted cycloherone and its mirror image.
altering the ring conformation as shown by the arrows. This t,ransposes the ring to its mirror image, shown on the right, and a t the same time interchanges axial and equatorial positions of the substituents so that the molecule is converted to its mirror image. Since there is an equal probability for a molecule to he in one form as the other, optical inactivity results in the same way as for a n, L, or racemic mixture. I n this case there would be no ray of plane polarized light passing through a molecule in any direction which would not have its plane of polarization rotated. This explanation of the optical inactivity of cis cyclohexane derivatives (7) is prefcrahle to the improbable assumption of a time-average for the change of a molecule from one ring conformation to the other producing planes of symmetry in all the molecules. A questionable requirement of this assumption is the short time allowable for the shift of the ring conformation relative t,o the time of passage of a ray of light through the molecule. The 1,2-trans disubst,itutcd cyclohexane is shown in Figure 9 along with its mirror image and the alternate ring conformation forms. The molecule has no plane of symmetry, is not identical with its mirror image and cannot be convcrted to its mirror image by alternating the ring conformation. Two enantiomers exist. The 1,3-lvam disuhstituted cyclohexane has one group axial and the other cquat,orial and therefore has no plane of symmet,ry. It is not identical with and cannot be convcrted into its mirror image by alternating the ring conformation. Two enantiomers exist. The l,&cis and 1,4-trans disubstituted cyclohexanes all
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It is evident that because of the short time required for passage of a ray of plane polarized light through a molecule the assumption for a time-average effect for a given molecule is both invalid and unnecessary. The molecule-average, which is also necessary in the explanation of optical activity or inactivity, can by itself explain the facts and it appears accordingly, that optically inactive substances ordinarily thought to be composed of achiral molecules are in reality racemic assemblies of chiral molecules (be they chiral due to rotational conformational isomerism, molecular vibrations, or simply orientation in an incident ray of polarized light). The difference between this racemic mixture and the usual racemic mixture is merely the low energy of interconversion of the enantiomers, a condition which also makes their separation impossible. The fundamental criterion for optical activity is then simply that a molecule cannot exist in the solution as its own mirror image either by (a) simple reorientation or (b) bond vibration or rotation or ring conformation shift in conjunction with reorientation. The most general rule of thumb for predicting optical activity is the absence of a rotation-reflection or alternating axis of svmmetrv in the molecule or anu" o" f its eonformalions produced by bond vibration or rotation including ring alternation. Literature Cited (1) Mow~mu,D. F., JR.,J. Chem. E ~ u c .29, , 138 (1952). (2) THOMPSON, H. B., 3. CHEM. EDUC., 37, 530 (1960). (3) WHKLAND, G. W., "Advanced Organic Chemistry," John Wiley 61 Sons, Inc., New York, 1949, pp. 139-151, 191-2. K. F., Ada Aead. Aboensis Math. et Phtis.. 4. (4) LINDMAN,
.
.,
.
.
(5) MISLOW,K., 'Tnt&duck,im to Stereochernistry," W. A. Benjamin, New York, 1966, p. 24. (6) ELIKL,E.. L., "Stereochemistry of Carbon Compounds," McGraw-Hill Book Co., New York, 1962, p. 1.57. (7) ELIEX, ALLINGI.:~, ANDMI.,and Mo~msoN,"Coniormationsl Analysis," John Wiley & Sons, Inc., New York, 1965, p. 51.
