L. J. OOSTERHOFF THE ISOMERS OF CYCLOHEXANE BY P. HAZEBROEK AND L. J. OOSTERHOFF Received 2nd February, I 95 I The equilibrium between the rigid and the flexible isomer of cyclohexane is calculated. The results indicate that a t ordinary temperatures the flexible isomer occurs a t most to a very slight extent. At higher temperatures it may occur in concentrations which are large enough to allow an experimental con- firmation. Cyclohexane resembles in many respects the saturated paraffinic hydrocarbons. This has led to the hypothesis that the valence state of the carbon atom in cyclohexane is similar to the valence state in saturated molecules where, according to Van't Hoff and Le Bel, one has to assume a tetrahedral arrangement of the four valencies. On the basis of this hypothesis, Sachse derived by geometrical considerations that two isomers of the cyclohexane molecule might be possible.The angles between two consecutive carbon-carbon bonds being less than IzoO-in our case 1 0 g O 28' 16"--it is possible to construct an infinite number of configurations which can be divided into two groups. In one group the six-membered ring can pass continuously through an infinite number of configurations. Sachse, Rer., 1890, 23, 1363 ; 2. physik. Chem., 1892, 10, 203.88 ISOMERS OF CYCLOHEXANE The other group consists of only one configuration which cannot change into the other configurations without a temporary alteration of the valence angles. The first group may be indicated as the flexible isomer, the second as the rigid isomer.Current names are boat and chair form, but these are less appropriate since the boat configuration is only one of all the possible configurations of the flexible isomer, whereas the pictures which are sup- posed to suggest a similarity with a chair do not emphasize the high sym- metry (D,d) of the rigid isomer. The ideas of Sachse, which were amplified by Mohr,2 were very fruitful in many chemical investigations of Boeseken and Hermans 4 into the properties of substituted cyclohexanes. A more definite proof of the structure of the cyclohexane molecule was based on the analysis of infra- red 4 and Raman spectra 6# and of the entropy and specific heat in the ideal gaseous state.'* These investigations proved the preponderant occurrence of the rigid isomer. The results of X-ray analysis of the crystal and electron diffraction of the gaseous molecules,D although less convincing, point in the same direction.The study of cyclohexane de- rivatives and other molecules containing a six-membered ring has also led to the conclusion that the ring preferably has the rigid s t r u c t ~ r e , ~ although exceptions are known (cf. Dallinga II). Cyclohexanedione-I : 4, for example, has a dipole moment of 1.3 D, which indicates that at least in part the flexible structure is present.lO, l1 At first sight, the higher free energy of the flexible isomer is some- what astonishing. The internal motion may be expected to add to the free energy the (negative) contribution of a fully excited degree of freedom, whereas the corresponding degree of freedom of the rigid form will be a partly excited vibration.Besides the rigid form has the higher sym- metry and therefore, also for this reason, would have a higher free energy. Thus one is led to the conclusion that there will be a difference in potential energy making the flexible isomer less stable. It is a plausible suggestion to hold the interactions between the C-H bonds responsible for this potential energy difference, as will be clear from a closer inspection of the geometry of both isomers. In Fig. I the rigid form and two configurations of the flexible form are pictured. The rigid form has a threefold axis together with the other symmetry ele- ments of the symmetry group D a d . The boat configuration has the sym- metry Czv, one twofold axis, and two planes of symmetry containing this axis.The stretched configuration of the flexible form has the symmetry V , three mutually perpendicular twofold axes. Tie other configurations of the flexible isomer have a twofold axis only. From the Fig. I it can be seen that in the rigid structure all pairs of consecutive CH, groups are in the staggered position, whereas in the boat configuration two pairs are in the eclipsed and four pairs are in the stag- gered position. I f , following Pitzer, we assume an energy difference of 2750 cal./mole between the staggered and the eclipsed position, the energy difference between the rigid and the boat configuration is 5500 cal./mole. The potential energy of the other configurations is more difficult to determine, but it will be shown that the potential energy is nearly a con- stant as far as the interactions of consecutive CH, groups are concerned. Mohr, J .prakt. Chem., 1.918 (z), 98, 322. Boeseken, Bull. SOC. Chzm., 1933, 53, I. Hermans, Rec. trav. chim., 1938, 57, 333. 6 Kohlrausch and Wittek, 2. physik. Chem. B, 1941, 48, 177. 6 Gerding, Smit and Westrik, Bec. trav. chim., 1942, 61, 561. 7 Aston, Schumann, Fink and Doty, J . Amer. Chem. Soc., 1941, 63, 2029. 8 Pitzer and Spitzer, J . Amer. Chem. Soc., 1947, 69, 2488. 9 Hassel and Vieroll, ArcJz. Math. Naturvidenskab., 1944, 47, no. 13. 10 Le Fkvre, Dipole Moments, (London, 1948)~ 2nd edn. *I Dallinga, Thesis (Leyden, 1951). Hassel and Viervoll, Acta Chem. Scand., 1947, I, 149.P. HAZEBROEK AND L. J. OOSTERHOFF 89 Pitzer,s in the interpretation of the experimentally determined entropy of cyclohexane reduced to the ideal gas state, only takes account of the boat configuration when calculating the entropy of the flexible isomer. Therefore the difference between the entropies of the two isomers is mainly due to the different symmetry.This difference, which is equal to R In 3, is large enough to conclude, on the basis of the experimental value of the entropy, that the rigid form occurs predominantly, at least at ordinary temperatures. If one takes account of the other configura- tions of the flexible isomer as well, the calculated value of the entropy of this isomer will be increased, which strengthens the argument of Pitzer. \/ 0 - 0 8 - y Sji-etched con fiy urabions. Boa1 cotrfigura~;on~ R/>l’d I5omer Flexi6ie homer FIG.I.-The isomers of cyclohexane. It would not be correct, however, to set the free energy difference equal to 55oo-RRT In 3 and to conclude that this value is in accord with the slight occurrence of the flexible isomer, since for a calculation of the equilibrium between both isomers one has to consider accurately the effect of the internal motion. In this paper we will indicate a method enabling the calculation of the free energy contribution of the internal motion of the flexible isomer, from which an estimate of the equilibrium ratio of the two isomers will z m , &Jq fQ9-5 be derived. Method Of CalCUlatiOn.-GEOMETRICAL 31 13 CONS ID ERAT I 0 N S .-h the description Of the possible coilfigurations of the cyclohexane molecule we will provisionally leave the hydrogen atoms out of consideration and fix our attention on the relative positions of the carbon atoms.Thus we have to treat FIG. 2. the problem of an appropriate description of the configurations of an equilateral and equiangular hexagon with angles of 1 0 g O 28’ 16”. The length of a side we put equal to I. The six sides are represented by six unit vectors, a,, u2, . . . a, which fulfil the equation 34 3 + Zai = 0. . * ( 1 ) + --+ Scalar multiplication with successively a, . . . a6 leads to the equations 6 1 Sij 7 0 J = I , * . . . 6, . * (2) i - 090 ISOMERS OF CYCLOHEXANE --++ with Sij = ( ~ i , aj). According to the suppositions Si; = I , Sii = 1/3, when i and j refer to consecutive vectors. In consequence of these equations and Sij = SF, there remain nine unknown Sij, which have to be determined by the six equations ( 2 ) and by additional equations which may be derived in the following way.Four vectors ai are always linearly dependent, so that we may write, e.g., --+ Sll Sl, Sl3 S2 1 ( 3 ) . . - - s44 - 1s41* ! I f and similar equations for all other four-rowed determinants contained in the matrix Sij. Of the equations ( 3 ) only three need be added to the six equations ( 2 ) as the other ones in the case of the flexible isomer do not restrict the consequences of the eight equations mentioned. We will now try to find a variable on which the Sij will depend in a symmetrical way. From equations ( 2 ) and the numerical values already mentioned it can be derived that s 1 3 = s46 ; S14 = - Q - S1, - S24 ; s ---6-s -s s24 = s51 ; s35 = s 2 S i and S25 =- Q - S24 - S35 ; a6 - 3 a5 46.With the substitutions 3S1, = 5 - I ; 3S,, = 7 - I ; 3S,, = 5 - I eqn. (3) can be written and two corresponding equations which can be obtained by cyclic per- mutations of [, 9 and <. Beside the trivial solution f = 9 = 5 = 0, which corresponds to the rigid structure, these equations possess a one-dimensional manifold of solutions, corresponding to the flexible isomer. Subtracting eqn. (4) in pairs leads, after some transformations, to the equations t 2 T 2 - 24(S2 + q2 - 677) - 32([ + 9) = 0, - * (4) and Eqn. (5) suggests the introduction of variables s and 0 by the substitutions 7 f q = ss cos 3e cos 8, or 5 = - s p - 2 cos2 8), 1 I I 5 = - s[J - 2 cos2 8 + - ( ;.,I*} - - = - s[: - 2 COSZ (8 - y ) 1.J The value of s follows from (6), which after substitution of (8) reads Sa - -cos2 38 + 36s - 32 = 0. . a (9) To a first approximation s = 8/9 and a more accurate value is 8 8 s =- 9 (I + T-COS2 39) . .P. HAZEBROEK AND L. J. OOSTERHOFF 91 The variable 6 is appropriate to define the possible configurations of the flexible isomer in a symmetrical way. The value 6 = o corresponds with a configuration of symmetry F' where the distance between carbon atoms I and 4 is as large as possible and which accordingly may be in- dicated as a stretched configuration (see Fig. I). 6 = 7~/2 corresponds w-ith a boat configuration, where the ca.rbon atoms I and 4 have the closest approach to one another, 8 = 7~ corresponds again with a stretched configuration which is the optical isomer of configuration 0 = 0, etc.POTENTIAL ENERGY.-For a calculation of the potential energy of the isomers of cyclohexane it is necessary to make an assumption about the interaction of the CH, groups. It looks plausible to assume an inter- action of the same kind as between the CH2 groups in %-butane and other straight chain paraffins for which Pitzer suggests a value of 3600 cal./mole for the potential barrier. On the other hand, Spitzer and Huffman,14 on account of heat of combustion data of cycloparaffins, consider even a value of 2750 cal./mole too high. This latter value has been used by Pitzer in considering cyclic molecules. Provisionally we will also assume a value of 2750 cal./mole. Besides these interactions between neighbouring CH, groups, there may also occur important repulsions between H atoms linked to opposite carbon atoms, since in a boat configuration the distance between two of these hydrogen atoms (1.84 A) is much lower than the sum of the van der Waals radii (2.4 A) according to Pauling.ls To begin with, however, we will restrict ourselves to the first type of interactions for which we will assume the formula v = I375 (cos 3 41 + cos 348 + cos 343).- * (11) + - - t 4 - P Defining the angle 43 as the angle between the vectors [a,, a,] and [%, as] for iqstance we find and similar formulae for the other angles. The total energy according to formulae (11) is The rigid isomer is characterized by [ = q = t = o . In this case we get the result For the flexible isomer we have to make use of (8) and (10) leading to Y (rigid) =- 3 x 2750.- (14) 95 7 T;.' (flexible) = - 81 x 2750 + x 2750 (r - cos 68). . (15) The energy of the flexible isomer is seen to be nearly constant, the term proportional to cos 6 8 not exceeding RT at room temperature. This picture may change if we include the repulsive interactions between the hydrogen atoms linked to opposite carbon atoms. These repulsions, which are most pronounced in the boat configurations and, to a less extent, in the neighbouring configurations, will increase the l2 Spitzer and Huffmann, J . Amer. Chem. SOL, 1947, 69, 211. 13 Pauling, The Nature of the Chemical Bond (1945), 2nd edn.92 ISOMERS OF CYCLOHEXANE potential energy of the flexible isomer at values of 8 equal to 30°, goo, 150°, etc.The magnitude of V , is difficult to assess. According to calculations of Barton l4 it may be as high as several kcal./mole. In the free energy calculations we will consider also the effect of a term like (16). MOMENT OF INERTIA.-For a calculation of the moment of inertia of the internal motion it is advantageous to introduce cylindrical co-ordinates to mark the position of the atoms. The twofold axis is taken as z-axis, Y measures the distance to this axis, and x the azimuth. The centre of gravity of the molecule is taken as origin. In order to study the internal motion separately, unperturbed by the overall rotation, the moment of momentum of the molecule has to be zero. This condition is expressed by So we may add to (15) a term of the type V,(I - cos 68) * (16) the summation extending over all atoms.calculated from The moment of inertia can be I = z m { ( - ) dz 2 + ( $ ) 2 + Y 2 ( g ) 2 } . d9 Using these formulae we find PARTITION FUNCTION.-In consequence of the nearly constant value of the moment of inertia the calculation of the partition function of the internal motion of the flexible isomer is particularly simple. In the case of a constant value of the potential energy the partition function of the completely free rotation is given by I g 39-6 x I O - ~ O + 0.4 x I O - ~ O cos 68 g.cm.2 . * (19) 277 (2rIkT)+ Qj=; 7 ( 2 0 ) where U; is the symmetry number of the internal motion. As all the distinguishable configurations are already represented by the &values in the range o < 9 Q r / 3 , ui = 6.Now if we add to the constant potential energy a term V,(I - cos 68) the value of the partition function Q can easily be deduced from the tables of Pitzer and Gwinn.lS In these tables R In (QlQj) is given as a function of Qj and of the ratio of the height of the potential barrier to RT. Results Equilibrium Concentrations of the Isomers of Cyclohexane .-For a calculation of the equilibrium concentrations of the isomers in gaseous cyclo- hexane we have to estimate the contribution to the partition function of the rigid isomer of the vibration, which we will consider as the counterpart of the internal motion of the flexible isomer. From a table of calculated vibration frequencies of the rigid isomer (Ramsay and Sutherlandls) we will select the vibration with the lowest frequency (w = 206 cm.-]) which will give an upper estimate for the partition function Qw and hence for the concentration of the rigid isomer as far as this vibrational contribution is concerned.Since the symmetry number of the rigid isomer is q = 6 and that of the overall rotation of the flexible isomer is U, = 2 the ratio of the number of mole- cules of the flexible isomer N , to the umber of molecules of the rigid isomer N is given by l4 Barton, J . Chem SOC., 1948, 340. 15 Pitzer and Gwinn, J . Chem. Physics, 1942, 10, 428. 16 Ramsay and Sutherland, Proc. Roy. SOC. A , 1947, 190, 235.P. HAZEBROEK AND L. J, OOSTERHOFF 93 where U is the potential energy of the flexible isomer, without the (I - cos 68) term, minus the potential energy of the rigid isomer.The percentage of flexible isomer of cyclohexane present in equilibrium with the rigid isomer is plotted in Fig. 3 against the absolute temperature for different values of v6. The dotted curve refers to a lower value of U (1900 X 148/81) which is taken into consideration following a suggestion of Spitzer and Huff mann.12 Calculations along similar lines have been performed for cyclohexanedione- I : 4. The results indicate that the stability of the flexible isomer and the magnitude of the dipole moment can be explained, assuming that the interaction between the 6 0 group and the CH, group is the same as that encountered in acetone. FIG. 3.-% flexible isomer in cyclohexane as a function of temperature for two U = 1900 x 148/81 cal./mole v 6 = 3600 cal./mole. values a t V , : and for Discussion In view of the uncertainties in the potential energy difference between the rigid and the flexible isomer of cyclohexane, the results, plotted in Fig. 3, should be considered as giving only a rough indication of the con- centrations of the flexible isomer which might occur. These results suggest the possibility of an experimental study of the flexible isomer of cyclohexane at temperatures above room temperature. A closer study of the isomers of cyclohexane and of cyclohexane derivatives will contribute not only to a better understanding of the behaviour of these molecules, but also to the solution of the problem of the forces hindering free rotation. The authors wish to express their thanks to Prof. H. A. Kramers for his interest and advice during the course of this research, to Mr. J. H. Kruizinga for his assistance in several mathematical derivations and for his indispensable help in carrying out many of the numerical calculations, and to Mr. J. A. van der Heiden for his help in preparing the figures. The paper is published by permission of the Management of the N.V. De Bataafsche Petroleum Maatschappij, The Hague. KoninklijkelShell-Laboratorium, N . V . De Bataafsche Petroleum A msterdam. Maatschappij, The Hague .