[CONTRIBUTION FROM NOYES CHEMICAL LABORATORY, UNIVERSITY OF ILLINOIS]
A Quantitative Relationship between Dissociation Constants and Conformational Equilibria. Cyclohexanecarboxylic Acids1 BY ROBERT D. STOLOW~ RECEIVED JANUARY 30, 1959 The preparations of cis- and trans-4-t-butylcyclohexanecarboxylicacid are described. Their apparent dissociation constants are reported and compared to t h a t of cyclohexanecarboxylic acid. A quantitative relationship between dissociation constants and conformational equilibria has been derived and applied to t h e prediction of the dissociation constants of some 3- and 4-alkylcyclohexanecarboxylic acids. Published dissociation constants were employed to calculate t h e conformational equilibrium constants IK = ( u ) / ( e ) , where in chair conformation a, the carboxyl group is axial; and in e, equatorial) of cisand trans-4-methylcyclohexanecarboxylicacid and cyclohexanecarboxylic acid. The average calculated values of K a t 25' for these acids are 1, 0.003 and 0.06, respectively.
Different conformations of a single mobile mole- with one another. Similar equations may be decule are distinct chemical species, each having dif- rived involving equilibrium among more than two ferent chemical and physical properties. The conformations (such as equation 12, below). A t properties of the mobile molecule are an average first, only the two chair conformations of cyclohexof the individual properties of its several conforma- ane derivatives are to be ~onsidered.~The equitions, weighted according to the population of each libria involved are illustrated in Fig. 1,where A is an c ~ n f o r m a t i o n . The ~ purpose of this paper is to acid, B is its conjugate base, "a" designates the chair demonstrate the application of these principles to conformation in which the functional group is axial, equilibria such as those involved in the ionization of KA carboxylic acids. A , + S _r A , + S A discussion of the connection between the rate of a reaction and the distribution of the reacting species among various conformations has been preKB sented by Winstein and Holness. 3 , 4 Exclusive B,-kSH+ B, +SH' attention was directed toward the two chair conFig. 1. formations of simple cyclohexane derivatives. X method of quantitative conformational analysis "e" designates the chair conformation in which the based on rate measurements was derived with the functional group is equatorial, Ka and Ke are the speaid of transition state theory and the assumption cificdissociation constants for the pure "a" and "e" that conformational equilibration is rapid relative Conformations, respectively, and K is the conformato the rate of reaction. A similar derivation has tional equilibrium constant. The equilibria inbeen reported independently by Eliel and L ~ k a c h . ~volved are defined by equations 1-4. The experiAnalogous equations based on equilibrium con- mental thermodynamic dissociation constant is exstants rather than rate constants are derived in pressed by equation 5.* Substitution of relations this paper.6 from equations 3 and 4 into equation 5 gives equaDerivation.-The equations derived below arc tion 6. Substitution of the relation (Aa) = KA. applicable to all suitable reversible dissociation ( A e ) from equation 1, and solution for the conreactions. However, since the examples to be dis- formational equilibrium constant of the acid, concussed involve proton transfer to solvent (S), the verts equation 6 to 7. A similar derivation which derivation has been expressed in terms of the disso- combines equations 2-5 leads to equation 8, the reciation of a proton. The derived equations are lationship between the conformational equilibria generally applicable to dissociation reactions, a t of the acid and its conjugate base.g Equations 7 equilibrium, of molecules which can be assumed to and 8 together express the simple quantitative reexist in two conformations which are in equilibrium lationship between dissociation constants and conformational equilibria. (1) Taken in p a r t from t h e authur's 1'h.D. Thesis, University of Illinois, 1956: Disserlation Abslr., 17, 751 (1057). Presented in par1 a t t h e 135th bleeting of t h e American Chemical Society, Boston. M a s ? April S , 1959. ( 2 ) Xational Science Foundation Pre-doctoral Fellow, 19::kI University Fellow, 1955-1956. Present address: Llepartmciit of Chemistry, T u f t s University, Medford 5 5 , Mass. (3) S.Winstein and N. J. Holness, THISJ O U R N A L . 7 7 , Odii3 (lH.jT,). (4) A concise discussion of t h e conformational concept as aydic(l i o reactivity has been presented by W. G. Dauben and K. S. Pitzer, in 11. S. A-ewman, e d . . ''Steric Effects in Organic Chemistry," John Wiley and Sons, Inc., New York, N. Y . , 1956, pp, 44-47; sce also 1%' Hiickel and 31. Hanack, Ann., 616, 18 (1958); R. C. Cuuksiin, Ann. Reports (London). S4, 172 (1957). ( 5 ) E. L. Eliel and C . A. Lukach, THISJ O U R N A L , 79, 5980 (1957). (6) Background references concerning cyclohexane stereochemistry a n d conformational analysis are cited in refs. 3-5. For a n excellerit review of the relationship betmeen dissociation constants and the structure of organic compounds, see H . C . Brown, D. H. 1IcDaniel and 0. Hifliger, in E. A . Braude and F. C. S : x h o d , eds.. "Determination of Organic Structures by Physical Rletliods," .\cn(ler.ic Press, In',., New Y o r k , N. I-.,lg.55, 111) X 7 4 . 7 L
KA = (&)/(AB) K5 (Ba)/(Be) (Be)(SH+)/(Ae)(Sj IC, = ( & ) ( S H + j / ( A d ( S ) [(H,,) 1- ( 1 3 c , ) ] ( S I l i ) / [ ( A ~i) (A,)I(S) k-e
k' = ii
.~ .~
.
=
=
; K 8 ( A d $- K,(A,)I/KX,) i(-L)l Kh ( K - K,)/(KB- K ) Kf, (K,/K,)KA
(118
(21 (3) (41 (5) (ti)
(71 (81
.
( 7 ) T h e relative importance of other conformalions will be discussed in connection with equation 12, below. ( 8 ) T h e symbols in parentheses represent activities. (9) Replacing the terms (A,) and (A,) of equation 5 by (SH'). (Be)./Ke(S) and (SH +) (B,)/KB(S) from equations 3 and 4, dividing numerator and denominator b y Be, replacing ( B d / ( B e ) by K T ~ (equation 2 ) . sr,lvirlp f,x K 5 , and replacing the tern1 ( I < -- t i e ) / f i l h > K,t f c < l i i a i i , ? n7 ) xivrs eqiiation X. (ICR
Nov. 5, 1959
CYCLOHEXANECARBOXYLIC ACIDS
Substitution of the mole fractions of each conformation, defined by equations 9 and 10, converts equation 6 to 11. The general equation 12, for the case where n different conformations are considered to exist in equilibrium, may be derived analogously. N A= ~ (AS)/[(&) 4- (&)I N A= ~ (&)/[(A,) (-%)I K = K ~ N A -k , K~NA, K = K,LVA, K ~ N A .~. . K,NA,
+
+
+
+
(9) (10) (11) (12)
5807
RwcooH - Rwco skewed
boat
ti
(n)
It
I
(a)
R
I
COOH
Fig. 2.
Discussion and Results Equations 11 and 12 state that the experimental dissociation constant, K, is a weighted average of the specific dissociation constants. The weighting factor is the mole fraction (or relative population) of each conformation.1° Consideration of conformations other than the two chair conformations of cyclohexane derivatives is necessary a t this point.ll The skewed (or stretched) conformation (s) is of greatest interest, because it is predicted to be the most stable flexible conformation of cyclohexane.12,1a
Although the possibility is recognized that flexible conformations might contribute up to 10% to the value of K for cis-4-alkylcyclohexanecarboxylic acids (if Fc - F, > 4.0 kcal./mole), refinement of the calculations to include small contributions from flexible conformations would seem unwarranted until they are shown by experiment to be non-negligible. Therefore, in the calculations t o follow, the contributions of flexible conformations are assumed to be negligible (as would be expected if F, - F, > 5.8 kcal./mole). The quantities needed for the calculation of the conformational equilibrium constants (KAand KB) by the use of equations 7 and 8 are the particular values of K, Ke and Ka for the compound in question; K can be measured experimentally; K e and (S) K,can be evaluated by introducing the assumption By a method of computation “based upon a crude that they are constant for a suitable series of comapproximation,” Howlett has calculated that pounds. An example of such a series would be the This aswhereas the boat conformation (b) is 8 kcal./mole 4-alkylcyclohexanecarboxylic acids. sumption opens the way toward two approaches for less stable than the chair conformation (c), the skewed12 or “ h a l f - r ~ t a t e d ”conformation ~~ (s) is the evaluation of Ke and Ka: (1) the solution of only 4 kcal./mole less stable than the chair.I3?I4 simultaneous equations, and ( 2 ) the use of “pure” Assuming the value ca. 4 kcal./mole, the corre- axial and equatorial conformations.lg The use of sponding population ratio, (s)/(c), is 0.001 a t 25’ simultaneous equations in the evaluation of K e and for cyclohexane. Although negligible for cyclo- Ka is illustrated by the 4-alkylcyclohexanecarboxylic hexane, the population of a flexible conformation acids. For this series of compounds, the necessary might be much larger, for example, for certain cis- data for the calculation of K , and Ka a t 25” have l,-l-disubstituted cyclohexanes. For cis-4-alkylcy- been taken from the literature. Three equations clohexanecarboxylic acids (Fig. 2), the correspond- (7a, 7b and 7c) can be written in the form of equaing population ratio of the most stable flexible (n) tion 7, one for each of the compounds: trans-4and chair (generally a) conformations is estimated (n)/(a) = ca. 0.02 at 25’. (The energy difference between axial (a) to be (n)/(a) = ca. 0.02, a t 25”, and “n” could con- and equatorial (e) cyclohexanecarboxylic acid ( F a - F . ) C ~ H ~ ~ C O O H was calculated t o he ca. 1.7 kcal./mole ( d e infva). Assuming (1) tribute 2-10% to the value of K.16 the environments of the substituents of ‘ x ” are equivalent to equa-
w -
(10) T h e form of t h e relationship between dissociation constants and conformational equilibria is identical t o t h a t reported between rate constants and conformational equilibria in refs. 3-5. However, with dissociation constants rather t h a n rate constants, one has the advantage t h a t no assumptions are necessary concerning the p a t h of the reaction, the transition s t a t e or t h e r a t e of equilibration of the conformations involved. (11) It follows from equation 12 t h a t t h e contribution of a given conformation, n , to the experimental dissociation constant, K , will be negligible if K.N*, K . I n general, this condition can be con. sidered satisfied when Ai,, 0.001. ~ With this criterion, t h e importance of other conformations may be estimated. (12) P. Hazebroek a n d L. J. Oosterhoff, Disc. Faraday SOL.,10, 87 (19.51); R. E . Reeves, A n n . Rev. Biochem., 27, 15 (1958). (13) K .E . Howlett, J . Chem. Soc., 4353 (1957). (14) See Dauben and Pitzer, ref. 4, pp. 14-15, for a review of other calculations of the energy difference between t h e boat a n d chair conformations of cyclohexane, which give values ranging from 1-11 kcal./mole. (15) For the cis isomer, the chair conformations (a and e) must have one group axial, while for t h e flexible conformations, (b and n), both substituents may have a nearly equatorial environment. T h e resulting free energy difference between the most stable chair and flexible conformations would he ahcriit 2.3 kcal./mole, equivalent to
Kg for cyclohexanecarboxylic acid) are the same factors which are responsible for the greater acidity of the equatorial carboxyl group relative to the axial carboxyl group (K, > Ka). Equation 8 states this concisely. The major factor involved is probably the greater steric hindrance to solvation of the functional group when it is in the more crowded axial conformation. This is a far more important destabilizing factor for the carboxylate anion than for the uncharged carboxyl The calculated value of K, leads to the prediction that all trans-4-alkylcyclohexanecarboxylic acids have the experimental thermodynamic dissociation constant K = 1.30 X The small population of the axial conformation (K < 0.004) (23) T h e acid-weakening steric effect found with few exceptions when bulky alkyl groups are accumulated about a carboxyl function has been attributed mainly t o steric hindrance t o solvation of the carboxylate anion; G . S. Hammond and D. H. Hogle, {bid.. 77, 338 (1D.i.5).
KOV.
5, 1959
CYCLOHEXANECARBOXYLIC XCIDS
5so9
should make a negligible contribution to the dis- acid would be calculable from their measured valsociation constant in each case. Based on the cal- ues of K by the use of equation 7. culated value of Ka, cis-4-alkylcyclohexanecarDissociation Constants.-The apparent dissociaboxylic acids are predicted to have dissociation tion constants of cis- and trans-4-t-butylcyclohexconstants in the range K = 0.26 X 10-5 to 0.93 X anecarboxylic acid and cyclohexanecarboxylic acid The experimental thermodynamic dissocia- in 66% dimethylformamide are recorded in Table tion constant of the cis-4-t-butyl acid is predicted to 11. The “pure” equatorial carboxyl group of the be ca. 0.46 0.2 x In order to test these trans acid is more strongly acidic than the “pure” predictions, and a t the same time to determine K e axial carboxyl group of the cis acid. The apparent and K a by the use of “pure” conformations, the dissociation constants are consistent with the insyntheses of cis- and trans-4-t-butylcyclohexane- terpretation that the unsubstituted cyclohexanecarboxylic acids were undertaken b i d e infra). carboxylic acid exists as amixture of the “a” and “e” The assumptions made (concerning the absence conformations, in which the population of the “e” of polar substituent effects for alkyl groups) lead to conformation is greater by roughly a factor of 10. the prediction that all cis-3-alkylcyclohexanecar- Therefore, the apparent dissociation constants for boxylic acids, as well as the trans-4-acids, should the cis- and trans-4-t-butyl acids are in agreement have essentially the same experimental K, ca. 1.30 with the thermodynamic dissociation constants X IO-j. The value reported for cis-3-methylcyclo- predicted by the solution of simultaneous equahexanecarboxylic acid (K = 1.31 X 10-j) is in tions. 25 agreement.I7 Also, each trans-3-alkyl acid should TABLE I1 have essentially the same K as the corresponding cis-4-alkyl acid. For trans-3- and cis-4-methylcy- DISSOCIATION CONSTANTSOF CYCLOHEXASECARBOXYLIC ACIDS I N 6 6 7 , DIME’IHYLFORMAMIDE-34% LvATER“ clohexanecarboxylic acids, the values reported for Acid PKb lo6 K are 9 and 9.2, respectively.17 The success of cis-4-t-Butylcyclohexanecarboxylicc 8 . 2 3 =k 0 . 0 1 the predictions for the case of the methyl group Cyclohexanecarboxylicd 7.82f .01 leads one to suggest that one could determine the trans-4-t-Butylcyclohexanecarbox~licC i ,i‘9 i .01 cis or tram configuration of any 3- or 4-alkylcycloa The author is indebted t o Dr. H. Boaz of Eli Lilly and hexanecarboxylic acid simply by measuring its disCo., Indianapolis, Ind., for the determinations r i f the dissociation constant, even when the epimer is not sociation constants reported here. The pK is the negative available for comparison. logarithm of the measured dissociation constant of the acid. In the case of the 2-alkylcyclohexane deriva- The values reported are “apparent” +I; values, not the tives, the conformational equilibria will be altered thermodynamic pK values; the reported relative error is recorded. Portions of the anal!-tical sample described in by steric interactions between the substituent and this paper. X sample submitted by Prof. E. L . Eliel of the dissociating group. This direct interaction the Univ. of Kotre Dame. plus steric inhibition of solvation would alter the Calculation of the apparent conformational dissociation equilibria which would depend on the size and shape of each alkyl group. Because of equilibrium constant for cyclohexanecarboxylic these and other factors, the dissociation constants acid by substituting the values (from Table 11) of would not be expected to be simple functions of the the apparent dissociation constants of cis-&-butyl trans-4-t-butyl (K,) and cyclohexanecarboxconformational equilibria. The reported17 values (Ka), for cis- and tvans-2-methylcyclohexanecarboxylic ylic acid ( K ) into equation 7, gives the value K.4 acid ( K = 1.84 X 10-5and 0.92 X lo+, respec- = 0.12 =t 0.08. Thus, the carboxyl group of cyclohexanecarboxylic acid prefers the equatorial tively) augur no simple correlation. The second approach for the evaluation of Ka orientation over the axial orientation in a chair conand K e requires “pure” ~onformations.l9~~~ In formation by 1.0-2.0 kcal./mole. The data in Tathis case, determination of the experimental dis- ble I1 do not permit a more accurate calculation of sociation constants of cis- and trans-4-t-butylcyclo- this free energy difference (31)because the dissociahexanecarboxylic acid would give Ka and Ke, re- tion constants of the irans-4-t-butyl and the unsubstituted cyclohexanecarboxylic acids differ by so spectively. Then the conformational equilibria small an amount. However, the free energy value of other 4-alkylcyclohexanecarboxylic acids (such calculated by the use of “pure” conformations as 4-A1e, Et, i-Pr, etc.) and cyclohexanecarboxylic (1.O-2.0 kcal./mole) brackets the value calculated by the solution of simultaneous equations (1.7 f (21) For a cyclohexane derivative which m a y be considered to exist entirely as t h e “a” conformation, the experimental dissociation constant 0.2 kcal./mole). ( K ) is equal t o K, for t h a t compound. Similarly. for a “pure” equaThe results confirm the hypothesis that polar torial cyclohexane derivatiye, K = K e . If both of these cyclohexane effects are negligible as compared to the conformaderivatives belong t o a series of similar compounds for which K. and tional eRect in controlling the relative acidities of K , are assumed t o be constant, then t h e conformational equilibrium coustant of any other compuund in the series can he calculated from these compounds. If polar effects were significant, its measured value of K plus t h e measured K values of t h e two conone would expect that the electron-releasing propformationally “pure” compounds. I n the series of 4-alkylcyclohexyl derivatives, it is assumed t h a t for the tra~is-4-t-butylcyclohexylderivative, X = K O , and for the cis-i-l-butylcyclohexyl derivative, K = K,. These assumptions must be applied with caution, since they are not valid when any other conformation makes a significant contribution t o K . For example, when t h e functional group is bulky, t h e equilibrium concentrztion of t h e “e” conformation of t h e cis-4-I-butyl isomer can no longer be considered neglible, and its K # Ka. Even when t h e equilibrium concentration of a minor conformation, n , is very small, i t could be significant if K , >> K .
( 2 5 ) There is gnod reason t o expect t h a t t h e chanfie of solvent from water t o 6G% dimethylformamide-34% water would not alter the relative acidities of these acids because for each t h e substituent is f a r from the site of dissociation, and the substituent has no free electron pairs, no hydrogens which are likely t o enter into hydrogen bondinp and no other features which would result in important specific differences in solvation of the subsLituetit.?e (21:) C. K . Inpold. “Structure and llechaniam i n OrA.inic Chetnistry,” Curucll University Press, I t h a w , X . Y., 1922, y. 727.
erties of the t-butyl group would make the trans4-t-butyl acid weaker than the unsubstituted acid. Such is not the case. The importance of the conformational effect upon the dissociation constants is demonstrated by the difference between the dissociation constants of the cis- and trans-4-t-butyl acids, the inductive effects of which would be expected to be identical. Preparation and Configuration.-ck-.l-t-Butylcyclohexanecarboxylic acid was prepared by the catalytic hydrogenation of p-t-butylbenzoic acid. Hydrogenation with platinum oxide in acetic acid proceeded readily a t room temperature and 2 to 4 atmospheres pressure to produce a mixture which contained about of the cis- and 30% of the trans acid. The cis acid was isolated easily from dilute hexane solutions of the mixture by slow crystallization (Experimental procedure B) . Several recrystallizations from hexane yielded cis-4-t-butylcyclohexanecarboxylic acid, m.p. 117.5-118.5". Hydrogenation of sodium P-t-butylbenzoate in 107' aqueous sodium hydroxide (with Raney nickel catalyst a t 200" and 3000 lb./sq. in.) was employed ~ in the preparation of the t r ~ n s - a c i d . ~-Although adequate for this work, the procedure was repeatedly unsatisfactory because of incomplete hydrogenation and experimental difficulties.23 The resulting mixture of reduced acids and p-t-butylbenzoic acid was dissolved in dilute ammonium hydroxide and the solution was boiled to expel excess ammonia. The trans isomer crystallized from the solution. Recrystallization from hexane gave trans-4-t-butylcyclohexanecarboxylic acid, m.p. 176-177', The assignment of steric configuration to the isomers of 4-t-butylcyclohexanecarboxylic acid is consistent with the data available a t present and with accepted generalizations based on analogy. -Although no rigorous proof of configuration has been carried out, the evidence supports the assignment beyond any reasonable doubt. Xccording to conformational considerations, the tvans isomer should be thermodynamically more stable than the cis isomer. Equilibration has been employed to demonstrate the greater thermodynamic stability of the higher melting isomer of I - t butylcyclohexanecarboxylic acid.27 Therefore, the higher melting isomer should possess the trans configuration. The energy difference between cis- and frtzns-4-tbutylcyclohexanecarboxylate anions can be approximated by the difference in energy ( 2 ) between the axial and equatorial chair conformations of cydohexanecarboxylate anion, calculated above to be ole (corresponding to a mixture frnns and
