Herbert 1. Slrauss
University of Colifornio Berkeley, 94720
Interconversion of the Cyclohexane Conformers
The understanding of conformations of cyclohexane has played a major role in the understanding of the conformation of saturated carbon compounds in general (1). This role has been emphasized by the award of the 1970 Nobel Prize in Chemistry to Hassel and Barton (2). Now the lowest energy conformations of cyclohexane are well understood. The molecule exists almost entirely in the chair conformation (Fig. 1) a t mom temperature. At higher temperature a number
, Figure 1.
0
Boot
Choir
i
Twist
The chair, boot, and h i r t conformdionr of cyclohexane.
of molecules also exist in the higher energy boat and twist forms (Fig. 1). The high energy of the boat and twist forms makes it very difficult to establish their properties experimentally, and one must rely on calculations based on the meager available experimental data. About ten years ago it was discovered that the rate of conversion of one chair form to its equivalent inverted chair form could he measured by nuclear magnetic resonance spectroscopy (5). This measurement and those that followed have stimulated enormous interest in the mechanism of inversion of cyclohexane and of other ring compounds. In the following we describe a way of picturing the inversion process and the geometrical relationships between the conformers. We start by seeking a set of coordinates iu which to write the conformations of cyclohexane. We only need deal with the coordmates of the molecules that change significantly. The methylene groups may be considered as rigid units, and the carbon-carbon distances may be considered fixed. We are not, of course, interested in the rotations or translations of the molecule as a whole. This leaves us with six independent coordinates which turn out to he combinations of the C-C-C bending motions and the methylene torsional motions. It will turn out that it will be relatively easy to picture the inversion in these coordinates, but we pay the price of having to define the coordinates in a complicated way. Let us start with an imaginary flattened cyclohexane. molecule in which the six carbon atoms are arranged in a regular hexagon. We use the plane of the hexagon as a reference plane from which to me* sure the positions of the carbon atoms in the real conformations of cyclohexane. We then define the three coordinates $, 7, and which we will use to describe the inversion, ~h~~ are drawn in j?igure and are defined so that 5 takes the hypothetical planar molecule into a
r,
boat-like conformation, 7 takes it into a twist-like conformation, and f takes it into a chair-like conformation. The other three of our six independent coordinates are in-plane coordinates and are also drawn in Figure 2. What do conformations defined by a combination of the three out-of-plane coordinates look like? They are intermediate between the boat, twist, and chair conformations and are just what we are looking for. We may formalize our description if we designate the diitance of each carbon atom from our reference plane by z~ and write
=
1,2,3.4,5,6
Our description can he made even simpler if we do one more trausformation of the coordinates. We transform €,7, and into spherical polar coordinates. That is rz = SP 'q rg
r
+ +
0 =
I
--I
(2)
r
which yields (with eqn. (1)) =
1.
pjt ax (&)I cos a
+ [ax (F- +)]ain
(3)
Thii looks complex hut the significant changes in conformation involve changes in only two coordinates, 6 and 4. The two chair conformations are defined by 6 = 0 or r a t any value of 6 The boat conformations are defined by 6 = r / 2 and q5 = 0, u / 3 , 2 n / 3 , r, 4 r / 3 , 5 r / 3
Out of plane displocements
f
* In plane displacements
Figure 2. The displacementr fmm a regular hexagon whish doflno tho three out-of-plane ring coordinate* 6 q, and and the three in-done coordin.ter.
r
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(note there are 6 of them!) and the twist conformations are defined by 8 = n / 2 and + = n/6, a / 2 , 5 n / 6 , 7 a / 6 , 3a/2, l l a / 6 . The remaining coordinate, r, measures the average deviation of the conformation from the hypothetical planar structure. The r, 8, and coordinates and the three in-plane coordinates of Figure 2 may all change appreciably with a change in conformation. We can determine their values by specifying values for 0 and and then requiring that the energy of the conformation be a minimum with respect to the four other changing coordinates. In order to do this we must first know the energy as a function of the internal coordinates of the molecule. We have derived such an energy function from structural and vibrational data (4) and have calculated the conformations for given 8 and (6). The calculations show that r for minimum energy remains relatively constant (within about 20%) over the whole range of 8 and 6, and thus specifying the value of 8 and is enough to define the conformation. The coordinate r measures the average "ouhf-planeness" of the cyclohexane ring and so it is not surprising that the ring remains a t about the same value of r for different values of e and +. The map of the conformational energy which results from the calculations is shown in Figures 3 and 4 as contour lines of constant energy plotted as a function of 0 and +. A cut through the contours a t = a is shown in Figure 5. Since we are working in spherical polar coordinates, the contour lines should he plotted on the surface of a sphere. Figure 3 is a cylindrical projection of that surface and Figure 4, a polar projection.' The figures can be used to
+
+
+
twist Figure 4. Conformationol energy map of cydohexme; polar projection. The pole is a t the center of the diagram and tho equator at the outer edge. Compare to Figure 3.
+
+
Figure 3. Conformational energy map of cydohexone: cylindrical proiection. Tho values of tho energy contour. are listed on the right in kiiocalorie* The dashed line* are ot maximum volues of the energy and the elored contours near B = 90' ore local minima in the energy. The b o d and twist conformsr. are represented b y poinb on the B = 90' line. The b o d conformer is a t 8.5 kcal/mols and tho twist at 7.9 kcal/mele. Tho chair conformer. are represented b y the lines a t 8 = 0' and B = 180'.
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Journal of Chemical Education
Figure 5. 1809
Conformotional energy of cyclohexane as a function of 8 a t 4 =
describe the process of inversion of cyclohexane. A molecule normally has the conformation represented by the pole of the sphere. When it acquires enough energy to react, it moves away from the pole to a region of higher potential energy until it gets to the maximum or transition state represented by the dashed line.% It then travels down the potential energy hill to the boattwist conformation represented by the equator. From the boat-twist conformation the molecule either retraces its steps back to the original chair conformer or continues on to the other chair conformer a t the opposite pole. The contour lines nm a t least approximately parallel to the equator, and this means that the molecules are relatively free to move in the motion represented by the + coordinate. For the boatAwist conformations this coordinate just takes the molecules between the various hoat and twist forms. For the chair conformation the + coordinate does nothing a t all and the chair form is rigid. Indeed in a recent article in THIS JOURNAL, Dunitz (7) pointed out that the hoattwist form of cyclohexane is less rigid than the chair 1 These d i i a m are idealized versions of the original diagrams of reference (6). The calculations of reference (6) required one more transformation of coordinstes which is complex but not relevant to the qualitative argument. The calculations were done using a particular form of the potentid functions on which there is not universal agreement. This leads to certain energy values (for example, the energy of the boaetwist conformer relative to the chair conformer) at variance with those derived farm other calculations. Again this does not change the qualitativeargument. a The cdculation of the kinetic energy parameters is discwed in reference (6). Also see a text which discusses transition state theory such as reference (6).
w Figure 6. Diagram of the model. The bare of the model and the motor which turns the cam shift ore not shown. Note the shape of the com.
form because of its lower symmetry. We can extend this argument to the various conformations that lie between the boat-twist and the chair form-the most interesting of these is the transition-state conformation. The contour lines for this conformation are almost exactly parallel to the equator, and as a consequence, a molecule in the transition state should pseudorotate freely. The directly observable consequence of this non-rigidity is a large positive entropy of activation. In order to understand the motion without using more mathematics we describe a mechanical model which pseudorotates. A diagram of the model is shown in Figure 6. The ring atoms are symbolized by balls attached to one end of each rod. Each rod is uivoted a short distance from the other end so that each ball can move in a vertical direction. The short end rides on a cam wbich determines the conformation of the halls (our model has a small Teflon wheel on the end of each rod wbich rides on the cam, but this is not necessary). The cam which generates the more interesting conformations has two maxima and two minima around its edge. It can be easily machined from a cylindrical piece of metal by cutting a circle a t right angles to the axis of the cylinder. The piece cut from the cylinder will have ends of about the correct shape. The various conformers can be made as follows: the chair conformer is made by removing the cam (or substituting a flat cam) and bending alternate ring atoms above and below the plane of the pivots. The boat form is made by using the cam and adjusting it so that a maximum lines up with one of the rods. If the cam is now rotated (we use a small clock motor), the various twist and boat conformations will be generated in order. Conformations intermediate
between the boat-twist and the chair are obtained by using the cam and a t the same time bending the rods as for the chair conformation. The different conformations are obtained by different amounts of bending or by adjusting the depth of the cam. Rotation of the cam for any conformation (except the chair!) gives the pseudorotation motion. Let us return to a consideration of the geometry and energy of the various conformers. The boat and twist conformers differ by only a small amount, 600 cal/mole from our calculation. The twist is lower in energy than the boat and all other conformers which are intermediate between the boat and the twist conformers are also intermediate in energy. The difference in energy between the boat and twist is rather small and the boat and twist conformers must be considered as a set of interconverting fdrms rather than as distinct chemical entities. It is appropriate to call the resulting conformation a twist-boat conformation. The calculation also provides more information on the transition state. If the cam of the model is adjusted so that a maximum of the cam lines up with one of the rods, the rods can be bent so that the result is a half-chair. This is a conformer in which one carbon atom is above the plane formed by the other five and is approximately the shape of the transition conformation a t fixed 4. Rotation of the cam will then give the geometry of the transition state as a function of the pseudorotation coordinate. Acknowledgment
It is a pleasure to acknowledge the invaluable help provided by Professors J. Cason, R. P. Frosch, and R. E. Powell and especially by Professor N. C. Craig. Literature Cited
222, No. 1.58 (1970); HANACK. M.. "Canformstion Theory," Academic Press.New Yprk. 1965: ELIEL. E. L.. ALZIN~ER. N. L..ANOTAL8. J.. AND MORRISON. G . A,. "Conformitiond ~ n a h i s , " interscience pub.. New York. 1965. (2) BARTON. D. H. R., Scimca. 169, 539 (1970); HABBEL, 0.. SC~PWS, 169, (1) L h n e m ~J. , B.. Scientilio Amcricon.
-".
407,107") ~
(3) JENBEN.F. R., NOTCE, D. 8., SEDERROLU. C. H.. AND BBBLIN.A. J.. J . A m ? . C b m . Soc.. 82, 1256 (1960): ANET,F. A. L.,AND BOUXN. A. J. R.. J . A m ? . Cham. Soc., 89,760 (1967). ( 4 ) PIOBETT. H. M.. A N D ~ T R A Y B B . H. L., J . Chen. P h y ~ . 53, , 376 (1970): BARTELL, L. S., J. CSEX.EDUO.. 45. 754 (1968) disusses the nature of theiorcesvhioh determinemoleeular geometry. (5) PIOKETT. H. M.. A N D S T R A ( TH. ~ ~L.,. J . Amei. Chsm. SOC..92. 7281 I,".,"\
,LW,",.
(6) J o a ~ s ~ o w H., 8.. "Gas Phase Reaotion Rate Theory." Ronald Prees.
New York. 1966. (7) Dnwme. I.D..J. Cnex.Eouo., 47.488 (1970).
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