KENNETH S. PITZER 73 THE PRESENT STATUS OF THE PROBLEM OF HINDERED ROTATION IN HYDROCARBONS AND RELATED COMPOUNDS BY JOHN G. ASTON Received 13th February, 1951 A review of the facts and theories related to barriers hindering internal The thermodynamic functions for hindered rotation in hydrocarbons is given. rotations are discussed. It is proposed to review briefly the history and present status of our knowledge about the potential barriers hindering rotation about a single bond with particular reference to hydrocarbons. Historical.-In 1935 Teller and Topleyl made a calculation of the equilibrium constant for the hydrogenation of ethylene using the heat of hydrogenation and the spectroscopic and molecular data. They found a discrepancy which would disappear if the barrier hindering interna 1 rotation were actually 3000 cal.mole-' instead of a value of 360 cal. mole-1 previously calculated by Eyring. In the meantime measurements of the third law entropy of tetramethylmethane by Messerly and Aston 3 and of ethane by Witt and Kemp l4 had given values which did not agree with values calculated from spectroscopic and molecular data. These considerations led Kemp and Pitzer l4 to suggest that a barrier of about 3000 cal. mole-1 hindering internal rotation was responsible in all cases and that this was the rule rather than the exception. Following this, the brilliant paper of Pitzer * appeared in which a number of hydrocarbons were treated and typical barriers evaluated. A general approximate treatment of the problem of the thermodynamic functions for a hindered rotor was given, accompanied by tables for the thermodynamic functions.For the potential function hindering internal rotation the expression is used, where V,, is the barrier height and n is the number of equivalent minima. v = *v,(I - cos we), . - (1) I Tellyr and Topley, J. Chem. SOC., 1935, 876. Eyring, J . Amer. Chem. Soc., 1932, $4, 3191. Aston and Messerly, J. Chem. Physzcs, 1936, 4, 391 ; J. prner. Chem. SOC., 1936, 58, 2354. Pitzer, J . Chem. Physzcs, 1937, 5, 473.74 HINDERED ROTATION The energy and heat capacity are tabulated directly. Instead of recording the entropy value the quantity tabulated is the difference between the entropy S,, and that, S f , calculated using the free rotational partition function. The tabulated values are thus the quantities Srr- S f , with Sj = R(-0.775 +Q In T + Q In I r e d x 1040 - In n) .(2) Instead of the free energy, (Ff - F J / T is tabulated, where F,, is the value for restricted rotation and - F’/T = R(-1-275 + 8 In T + Q In I r e d x 1040 - In n) . ( 3 ) All values are tabulated as a function of Vo/RT and n 2 / I r e d V , with V , expressed in cal. mole-l and I r e d the reduced moment of inertia in c.g.s. units per molecule. The thermodynamic functions for the restricted rotation are added to the usual ones €or translation and for the three degrees of freedom of external rotation. Pitzer computed barrier heights to get the best fit with the experi- mental data. Some of the potential barriers assigned to internal rotations selected by Oosterhoff from a paper of French and Rasmussen6 are given in Table I.In most instances these have been computed by the methods discussed below. TABLE I.--METHYL GROUP BARRIERS FOR TYPICAL COMPOUNDS Compound Potential Barrier (cal./mole) Ethane . Propane isoButane . neoPentane . Me thylamine . Dimethylamine Trime thy lamine Me thylalcohol Dimethylether . Acetone . Me th ylme rcap t an Dime thylsulphide Tetrame thylsiIane 2750 3300 3870 4800 1520 3460 4270 I350 2700 5 60- I 240 1460 1100-1500 2000 Methods of Calculation.-In Pitzer’s paper, discussed in the last section, it was assumed that the rotating groups were sufficiently light so that cross product terms in the rotational kinetic energy could be neglected.7 More recently Pitzer and Gwinn 8 have extended their treat- ment for the case where the rotating groups are symmetrical, so that such cross product terms are taken into account, and have given tables for making the computation, In addition to tabulating the quantities in the last section they have tabulated the restricted rotational entropy and the corresponding FIT values.In this case the tabulation is made in terms of V,/RT and I/QT, where QT is related to the reduced moment of inertia I by the relation QT = 2-815 (1088 IT)llz/n - (4) For a symmetrical group such as methyl attached to a heavy rigid frame I , is approximately equal to the moment of a methyl group. If the group has a small moment of inertia and is not symmetrical (e.g. 5 Oosterhoff, Diss. (University of Leiden, 1949). 8 French and Rasmussen, J .Chew. Physics, 1946, 14, 389. 7 Eidinoff and Aston, J . Chew. Physics, 1935, 3, 379. 8 Pitzer and Gwinn, J . Chem. Physics, 1942, 10, 428.JOHN G. ASTON 75 OH and NH,) and the rigid frame is sufficiently heavy, I is approximately the moment of inertia about the bond; otherwise I can be obtained by the method given by Pitzer.* At this point it is appropriate to discuss the entry of n into eqn. ( 2 ) and ( 3 ) . For a symmetrical group it enters essentially as a symmetry number which is equal to the number of equivalent minima. It now remains to discuss the case where the minima are not equivalent. I f the minima are not equivalent, the simplest procedure is to consider that isomers exist, differentiated by the fact that their low energy states are oscillations about each of the positions of different minimum energy.The energy levels are calculated from a potential function of the form of eqn. (I), with n usually the fraction of a revolution existing between the two adjacent maxima. The thermodynamic functions for each isomer are calculated by the usual methods, but since the partition function of each is restricted to ~ / n of a revolution, n in eqn. (2) and (3) is retained to allow for this fact, even though the symmetry number be unity. To calculate the entropy or free energy, one must add the entropy of mixing of these isomers and for this, one must know their mole fractions. The mole fraction can be calculated by combining the total free energy functions of each isomer with the value of AEoo, the difference between the energy zeros of the two isomers, to obtain the free energy difference between the isomers and hence the equilibrium constant for their con- version.Thus for the isomerization of isomer i into isomer j , where Xi and X j are the mole fractions of the isomers i and j and the symbols represent differences in the molal properties between the isomers. It must be emphasized that Fjo and Fi0 include translational, external rotational and vibrational contributions as well as that due to restricted rotation, and have been brought to a common energy zero. The total heat content is given by ( H O - Eoo) = EXi(Ho - Eoo)i, . - (6) where the sum is to be taken over all isomers and (HO - Eoo)i is the total heat content of each isomer (i.e. 4RT has been added for translation and external rotation and the vibrational contributions also included as well as that for the restricted rotation).Thus the total heat capacity is given by or dXi Cpo = XiCii + ( H O - Eoo)i-;i7: . The first term on the left-hand side is the sum of the contributions of the individual isomers to the heat capacity, while the second can be obtained from the individual heat contents and the change in mole fraction of the isomers with temperature.IO An example of such a calculation is that for butadiene-1 : 3 which consists of a mixture of trans and cis forms The potential energy function of the trans form which best fitted the third law entropy and gaseous heat capacities was while that for the cis was Vets = 2,425 + 1,288 [I - cos ( 6 - n)/0.594] cal. mole-1 .(10) As can be seen from these equations the minima of the cis and trans are lo hston, Szasz, Wooley and Brickwedde, J . Chem. Physics, 1946, 14, 67. Vt,,,, = 2,500 [I - cos 6/0.406] cal. mole-' . ' (9) Pitzer, J . Chem. Physics, 1946, 14, 239.76 HINDERED ROTAT ION separated by 180' but the maxima are so located that the cis form occupies only 0-594 of a revolution the rest being occupied by the trans. The constant term in the expression for the cis form indicates that its energy zero is 2-425 kcal. higher than that of the trans. The solid curve in Fig. I represents these equations while the dotted curve is for the case where each isomer occupied half a revolution. In the first case n in eqn. (2) and (3) is taken as 1/0*5g4 €or the cis form and 1/0.406 for the trans while in the second case it is z for both isomers.Both curves fit the data equally well. Parr and Mulliken l1 have recently concluded from a wave mechanical calculation including n levels and repulsive terms that such a difference in energy should exist between the isomers. By making a slight ap- proximation this method of computation reduces to the method originally given by Pitzer.12 Calculation of Barriers.-It is rare that the energy levels of the hindered rotor give rise to lines in the Raman or infra-red. When they do, these lines may be used to calculate the barrier hindering internal r ~ t a t i o n . ~ ~ l3 In the case where there are no rotationa.1 isomers possible it is obvious from the foregoing that the third law entropy, compared with that cal- culated from spectroscopic data, is a precise method of calculating the barrier.The chief objection to this method is that it assumes that there is no random orientation at the absolute zero. The likelihood of such a situation will be discussed presently. This was the first method used to determine the barrier in ethane by Kemp and Pitzer l4 but it was soon followed by a determination of the barrier from a careful measurement of the gaseous heat capacities of ethane over a wide temperature range beginning at 143' K.161 16 Where there are rotational isomers, in order to get the barriers with respect to the bond in question as well as the energy differences between the isomers, both the third law entropy' data and gaseous heat capacities over a wide temperature range are necessary for comparison with values from the spectroscopic and molecular data.In the example previously cited this was the way by which Fig. I was obtained. Before leaving the subject of barrier determination it is appropriate to discuss the effect of zero point entropy in the crystal. Any undeter- mined entropy in the crystal at the absolute zero makes the barrier appear higher. For example, the barrier hindering internal rotation in methyl alcohol, obtained from the rotational energy levels determined from microwave data,l' is only 932 cal. mole-l, whereas the value cal- culated by comparing the third law entropy with that calculated from spectroscopic and molecular data l* is 1600 f 700 cal. mole-I. Among other explanations, Halford has attributed the difference to possible zero point entropy at the absolute zero due to hydrogen bonding but has pointed out that uncertainty in the gas imperfection correction and, less likely, in heat of vaporization and vapour pressures, could be re- sponsible.The possible occurrence of zero point entropy in crystalline hydrocarbons will be discussed later. At the present there is no reason to believe that the crystalline hydrocarbons used in key barrier deter- minations have entropies at the absolute zero. l1 Parr and Mulliken, J . Chem. Physics, 1950, 18, 1345. p2 Pitzer, J. Chew. Physics, 1941, 8, 711. l3 Aston and Sagenkahn, J . Amer. Chem. Soc., 1944, 66, 1171. l4 Kemp and Pitzer, J . Chem, Physics, 1936, 4, 749 ; J. Amer. Chem. SOL, l5 Kistiakowsky and Nazmi, J .Chem. Physics, 6, 1938, 18. 16 Euken and Parts, 2. physih. Chem. B, 1933, 20, 184 ; Euken and Betram, 1' Koehler and Dennison (private communication). 18Halford, J . Chem. Physics, 1950, 18, 363. 1937, 59, 277 ; see also Witt and Kemp, J . Amer. Chem. SOC., 1937, 59, 273. 2. 9hysik. Chem., B, 1936, 31, 3.63.JOHN G. ASTON 77 Naturally the barrier height, calculated by comparing measured thermodynamic properties with those calculated from spectroscopic and molecular data, depends on the form of the potential function. This dependence has recently been discussed by Blade and Kimball.lD Rotational Isomerism.-The rotational isomers discussed previously, as exemplified by the cis and trans forms of butadiene-I : 3, often have sufficiently different vibrational spectra so that their presence may be detected by examination of the Raman spectrum.The stable form of n-butane is the trans form with symmetry C2&. By rotation of one-third of a revolution on either side of this are obtained two isomers which have the symmetry C, and are optical isomers. Szasz, Sheppard and Rank 2o have taken the ratio of the intensities of the members of a pair of lines in 5 Fig. I. the Raman spectrum over a wide temperature range. The line at 325 cm.-l is due to the trans while the corresponding line for both of the cis isomers is at 432 cm.-l. This latter line disappears in the solid. The change of the ratio of the intensities of these lines gives a value of the AH" of isomerization from which the corresponding value of AEoo was found to be 770 & go cal.mole-1 in excellent agreement with the value found by Pitzer.12 On the other hand an attempt to carry out a similar investigation in the case of 2-methylbutane and z : 3-dimethyl- butane 2 1 did not give results which were as clear cut. To start with all the main lines persisted in the solid phase. The change of intensity with temperature of a line pair for 2-methylbutane was investigated. The change was very small and indicated either a very small energy difference or that each of the chosen pair of lines was due to one isomer which was the only one in appreciable concentration. This alternative interpretation would yield a large energy difference. Recently Scott, McCullough, Williamson and Waddington 22 have determined the third law entropies and vapour heat capacities of both compounds and compared them with values calculated from spectroscopic 22 Scott, McCullough, Williamson and Waddington, J .Amer. Chem. SOC. (in press). 19 Blade and Kimball, J. Chem. Physics, 1950. 18, 1030. 2 O Szasz, Sheppard and Rank, J. Chem. Physics, 1950, 16, 704. 21 Szasz and Sheppard, J. Chem. Physic$, 1949, 17, 93.78 HINDERED ROTATION and molecular data. From these results it is evident that for a-methyl- butane the rotational isomer with C, symmetry is several kilocalories less stable than that with C, sym-metry. For z : 3-dimethylbutane the energy difference between the rotational isomers was very small. According t o their third law results it is not impossible that some residual entropy exists in crystalline a t the absolute zero.The fact that all lines persist in the solid indicates that the isomers form solid solutions with each other. Below the melting point a : 3-dimethylbutane rotates in the solid state and forms solid solution with other branched hydrocarbons that rotate in the solid state. In the case of mixtures of 2 : a-dimethylbutane and cyclopentane a complex is formed in which there is evidence that rotation (or complete random orientation) persists down to very low temperature~.~~ It is thus possible that the solid solution of rotational isomers of z : 3-dimethylbutane per- sists down to sufficiently low temperatures that the transformation would be so slow that it would not occur, thus leaving zero point entropy. Theory of Origin of Barriers.-As has already been pointed out simple theory does not predict the high barriers hindering the rotation of methyl groups and this is true for other simple groups.The energy difference between the rotational isomers discussed in the last section is undoubtedly steric in origin while other rotational isomers, e.g. those of ethylene dichloride owe their origin t o dipole interaction. The barrier hindering internal rotation of simple groups has no such explanation. An em- pirical treatment of the problem z 4 used a repulsive potential in terms of the distance vij between the hydrogen atoms only, k v=- . Yi,6 Because I : I : I-trifluoroethane,26 I : I : I-trichloroethane 86 and per- fluoroethanc 27 show barriers of the same order as ethane, it is obvious that other atoms act in the same way as hydrogen and the empirical treatment just mentioned needs to be revised.The fact that the same forces cause the cyclopentane ring to be non-planar leaves little doubt that the forces are repulsive. French and Rasmussen have made a satis- factory empirical treatment in terms of the distance of nearest approach of the atoms, with all atoms included but not electron pairs. Lassettre and Dean 3 0 n 31 have carried out a theoretical treatment which makes certain simplifying assumptions about the distribution of molecular electrons. An expansion of the electrostatic potential in inverse powers of the distance was carried as far as quadrupole terms. Most of the barriers can then be accounted for. Oosterhoff has also carried out a thcoretical treatment of the problem and does not agree entirely with the conclusion of Lassettre and Dean that interaction between bonds with fixed charge distributions will explain the barrier.At present neither theoretical treatment is sufficiently unique to make a priori conclusions. Values of Methyl Group Barriers in Unsaturated Hydrocarbons .- In view of the uncertainty of the part played by unshared electrons, the values of the barriers in certain unsaturated compounds are listed in Table 11. In all of these, the methyl or ethyl group is attached to a doubly or triply bound carbon atom or one to which a vinyl or acetyleiiic group is attached. 23 Fink, Cines, Frey and Aston, J . Amer. G e m . SOC., 1947, 69, 1501. 24 Aston, Isserow, Szasz and Kennedy, J . Chem. Physics, 1944, 12, 36. 26 Russell, Golding and Yost, J . Amer. Chem. SOC., 1944, 66, 16. 26 Rubin, Levidahl and Yost, J . Amer. Chem. SOC., 1944, 66, 279. 27 Pace and Aston, J . Amer. Ch.em. Soc., 1948, 70, 566. 28 Aston, Schumann, Fink and Doty, J . Amer. Chem. SOC., 1941, 63, 2029. 29 Aston, Fink and Schumann, J . Amev. Chem. SOC., 1943, 65, 341. 3 0 Lassettre and Dean, J . Chem. Physics, 1948, 16, 157. 553. 31 Lassettre and Dean. J . Chew. Physics, 1949, 17, 317.JOHN G. ASTON TABLE 11.-METHYL AND ETHYL GROUP BARRIERS IN UNSATURATED Compound Propylene . cis-Butene . trans-Butene . Butadiene-1 : 2 . Butene-1 . Butyne-2 . Butyne-1 . COMPOUNDS VO (cal . mole-1) Me, 2,000 Me, 700 Me, 1,900 Me, 1,650 Me, 2,700 ; Et, 2,000 Me, 500 Me, 3,000 79 References 33 and 34 35 36 and 10 37 10 and 38 38 40 The School of Chemistry and Physics, The Pennsylvania State College, Pennsylvania, U.S.A. 33 Kistiakowsky and Rice, J . Chem. Physics, 1940, 8, 610. 34 Telfair, J . Chem. Physics, 1942, 10, 167. 35 Scott, Ferguson and Brickwedde, J . Res. Nat. Bur. Stand., 1944, 33, I. 36 Guttman and Pitzei, J . Amer. Chem. SOC., 1945, 67, 324. 37 Aston and Szasz, J . Amer. Chem. Soc., 1947, 69, 3108. Aston, Fink, Bestul, Pace and Szasz, J . Amer. Chem. SOC., 1946, 68, 52. 39 Osborne, Garner, Doescher and Yost, J Amer. Chem. SOC., 1941, 63, 3496. Aston, Mastrangelo and Moessen, J. Amer. Chem. SOC., 1950, 72, 5291.