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JOURNAL of SCIENCE Published on the First Day of October, January, April, and July IOWA STATE COLLEGE JOURNAL OF SCIENCE Published on the first day of October, January, April, and July EDITORIAL BOARD EDITOR-IN-CHIEF, Joseph c. Gilman. AssISTANT EDITOR, H. E. Ingle. CONSULTING EDITORS: R. E. Buchanan, C. J. Drake, I. E. Melhus, E. A. Benbrook, P. Mabel Nelson, V. E. Nelson, C. H. Brown, Jay W. Woodrow. From Sigma Xi: E. W. Lindstrom, D. L. Holl, B. W. Hammer. All manuscripts submitted should be addressed to J. C. Gilman, Botany Hall, Iowa State College, Ames, Iowa. All remittances should be addressed to Collegiate Press, Inc., Iowa State College, Ames, Iowa. Single Copies: One Dollar. Annual Subscription: Three Dollars; in Can­ ada, Three Dollars and Twenty-Five Cents; Foreign, Three Dollars and Fifty Cents. Entered as second-class .:me«Eir '~~~ 1~;. :t935.- Qt. the postoffice at Ames, Iowa, under the,..... ae,t . .df. ?,JEO:'cn 3, 11119.: :. .',: . .:. .: ..: ... .. .... .... .. ... .. .. : .. ·... ···.. ·.. · .. ::·::··.. .. ·. .. .. ·: . .· .: .. .. .. :. ·.... · .. .. :. : : .: : . .. : .. ·: .: ·. ..·... ··::.:: .. : ·:·.·.: ....... THE NUMBER OF STEREOISOMERIC ALCOHOLS EDWARD s. ALLEN AND HARVEY DIEHL From the Departments of Mathematics and Chemistry, Iowa State College Received February 10, 1941 The problem of establishing the number of possible isomers of the alcohols of increasing carbon content has attracted the attention of var­ ious workers during the past sixty-five years. No general solution of the problem appears to be possible, but the brilliant attack by Henze and Blair1 makes the calculation of the number of isomeric alcohols of any given carbon content from the number of those of fewer carbon atoms a not too arduous task. Going further and with a similar approach, Henze and Blair2 devised formulas for calculating the number of stereoisomeric and non-stereoisomeric alcohols of a given carbon content, again, of course, presupposing such a knowledge for the alcohols of one less carbon content. By stereoisomeric was meant any compound capable of resolu­ tion into optically active forms, irrespective of the number of centers of asymmetry or the number of such isomers possible for a given structural isomer; non-stereoisomers are those simply lacking a center of asym­ metry and for which, of course, only one isomer is known. The method thus counts all of the isomers, and distinguishes those which have asym­ metric carbon atoms from those that do not, but does not state how many have two, three, etc. asymmetric carbon atoms. We have now adapted the method of approach of Henze and Blair to a more general treatment of the problem which does yield this informa­ tion. It has been necessary to introduce a factor, which for convenience is designated as the "asymmetry number" and to recast the notation of Henze and Blair in a somewhat more symbolic and condensed form. The asymmetry number, a, is the number of possible stereoisomeric forms of a given structural isomer. For all compounds having no center of asymmetry, it has the value l; these are the non-stereoisomeric forms of Henze and Blair. For a structural isomer containing one asymmetric carbon atom, a equals 2. For a structural isomer having two asymmetric carbon atoms, a may be 3 or 4 depending on whether the groups sur­ rounding the asymmetric atoms are the same in the two cases or not; where the groups are not the same, a has the value 4, or in general, if n be the number of different asymmetric carbon atoms, the number of stereoisomers is 2n. Tartaric acid is, of course, the most common example of the other case, the asymmetric carbon atoms being surrounded by the same groups; a has the value 3, corresponding to the d-, l-, and meso­ forms. This situation cannot arise among the alcohols but does among the hydrocarbons and glycols. 'Henze, H. R., and C. M. Blair. J. Amer. Chem. Soc., 53, 3042, (1931). See this paper also for a bibliography of earlier work. • J. Amer. Chem. Soc., 54, 1098 (1932). [161] 162 EDWARD S. ALLEN AND HARVEY DIEHL For a given number n of carbon atoms, the number of structural isomers of any given asymmetry number is represented by T n,a and the number of stereoisomers of any given asymmetry number is then T'n, a, which is equal to aTn, a· The total number of stereoisomers of all types for a given number of carbon atoms, T"n , is then the sum of these T'n .a values for values of a which occur for this n, T"n =~a T'n. a The general process of counting the number of isomers for a given n consists in considering the groups attached to the carbon atom carrying the hydroxyl group and coupling them in all possible manners. In the method of development the primary, secondary, and tertiary alcohols are considered separately. The primary alcohols may be considered as being derived by the attachment of an alkyl group having one less carbon atom to the carbon atom carrying the hydroxyl group. The number of primary alcohols is then the same as the total number of alcohols of all types having one less carbon atom, for the hydroxyl group may simply be replaced in each of these by-CH2-0H. Moreover, no change in asymmetry number occurs since no new asymmetric carbon atom is introduced. Sym­ bolically, then, Type 1 Pn,a = T n-1, a The secondary alcohols may be considered as being derived by the attachment of two alkyl groups, R 1 and R2, to the carbon atom carrying R1 H the hydroxyl group "'C / and the isomerism of the alcohol R2/ "'OH thus represented will depend on the similarities and differences of the attached groups. If the two attached groups have different numbers of carbon atoms, they are necessarily totally dissimilar, and this is repre­ sented symbolically by R 1 - 0 R2 • The two groups may have the same number of carbon atoms, but if their asymmetry numbers are different they are alike in only this one respect, and this is represented by R 1 - 1 R 2 ; if the asymmetry numbers are also the same, the groups may or may not be alike structurally; thus the propyl groups H I HaC-C-CHa and H 3C- CH2-CH2- I have the same asymmetry number, 1, but are structurally different. This is represented by R 1 - 2 R2. If, on the other hand, they are also structur­ ally alike, and the asymmetry number is two or greater, they may still differ from each other enantiomorphically, that is, d from l; similarity to this extent is represented by R 1 - 3 R2. The symbol R 1 - 4 R2 to indicate absolute identity in all respects-number of carbon atoms, asymmetry NUMBER OF STEREOISOMERIC ALCOHOLS 163 number, structural arrangement, and stereochemical configuration­ would seem natural but turns out to be unnecessary. Where three groups are attached to the carbon atom bearing the hydroxyl group, R 1 - 0 R 2 - 0 Ra indicates that the three attached groups are all of unequal carbon content; R 1 - 1 R2 - 0 R 3 that two of the groups are of equal carbon content but of different asymmetry number; R 1 - 3 R2 - 1 Ra that all have the same carbon content and that two groups also have the same asymmetry number and are structurally alike but that the third group has a different asymmetry number; and so on. The first of these statements as given would make a greater similarity between R 1 and Ra possible. We shall, however, adopt the convention that radicals connected by such similarity chains have the lowest degree of similarity mentioned in any intermediate link. In the case of the secondary alcohols, the sum of the carbon atoms in the two attached groups is one less than the total number of carbon atoms. Two cases arise: (a) if n is an even number, then the groups cannot be of equal carbon content, and (b) if n is odd, they may. For either case a contribution to the number of secondary alcohols, s,.,a, is given by Type2 ~i,j ~fl,'Y T ;,,13T1,-y Rl-0R2 i + j = n - 1 a = 2fl'Y i> j where i is the number of carbon atoms in one of the attached groups, j the number of carbon atoms in the other, and~ and y are the asymmetry numbers of the groups. A new center of asymmetry has been created in this process, since all of the groups attached to the central carbon atoms are different, and the asymmetry number is two times the product of the asymmetry numbers of the two groups. In case (b) , n odd, there are in addition the further combinations in which the two groups have the same number of carbon atoms. This gives rise to three types Type 3 T,._1 T,._1 ~ fJ,'Y - .{3 - .'Y a = 2p-y 2 ~ Type 4 1hTn - 1 (T,._1 -1) - .13 - ,{3 2 2 a = 2p' Type 5 T,._1 2 - ,{3 a = 2fl 2 s,.,a is then the sum of the numbers found for these four types. The formula for type 3, the first of these three formulas, is immedi­ ately obvious, since for each of the groups T ,._1 there can be attached to -,{3 2 the carbon atom carrying the -OH group T ,._1 groups, and since the -,"/ 2 164 EDWARD S. ALLEN AND HARVEY DIEHL groups are of different asymmetry number and different structurally a new asymmetric carbon atom has been created, and therefore a = 2~y. Type 4 deals with the case where the two groups have the same asym­ metry number but may still be structurally different; the total number is given by3 C2 T,,_1 -,(3 2 This is the number of distinct pairs of the T ,,_1 isomers.
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