ACIDDISSOCIATIOS OF METHYL SUBSTITUTED PHESOLY
l h y , 196 L
81f
THERMODYKAMIC THEORY OF ACID DISSOCIATION OF METHYL SUBSTITUTED PHENOLS I N AQUEOUS SOLUTION BY I,. G. HEPLER' AXD W. F. O'HARA Dppnrtment of Chemistry, University of Virginza, Charlottesville, Virginia Recezued A-ovember 18, 1960
Rccentll- reported free energies, enthalpies and entropies of dissociation of aqueous phenol and nine methyl substituted phenols are interpreted theoretically in a way that can be extended to other acids. The enthalpy and entropy changes for the reaction HA.(aq) AJaq) = A,-(aq) HA,(aq), where the subscripts s and u refer to a substituted phenol and to unsubstituted phenol, are considered in terms of 0-H bond dissociation energies and solute-solvent interactions. This treatbasanin which a and b are constants, vS and vu represent the 0-H stretchment leads to the equation AHa' = a( ye2 - vnz) ing frequencies of a substituted phenol and of unsubstituted phenol and AH20 and ASSO are the enthalpy and entropy changes for the reaction above. This equation is in excellent quantilative agreement with the experimental data. From this equation other thermodynamic equations are derived and used for calculation of dissociation constants over a range of temperatures.
+
+
+
Introduction Papee, Canada#y, Zamidzki and Laidler2 have calorimetrically determined AH0298 of dissociation of phenol and of nine methyl substituted phenols (three cresols and six xylenols) in aqueous solution. Some of their calorimetric results have been confirmed in this L a b ~ r a t o r y . ~Papee, et U Z . , ~ have also tabulated AI;"0298of dissociation and have calculated ALSO^,^ of dissocia,tion in aqueous solution for these ten compounds. Our paper is concerned Jvith theoretical analysis of these dat'a by a method which can be extended to other acids. Canaday4 recently has observed that a plot of AF0298of dissociation against the 0-H st'ret'ching frequency (hereaft'er denoted by v) gives a fair straight line. Similar empirical correlations of AF'!9sor p K , hare been observed by several others. In spite of the occasional ut,ility of such correlations, they arts fundamentally incomplete and inadequate, especially when intended for use as aids in theoretical interpretation of substituent effects on acid ihrengths. Correlations of acid st'rengths wit,h a molecular property such as v for the 0-H stretch are of very limited significance in general and especially so unless the relative strengths of the a,cids under consideration are independent of temperature. This last condit,ion is usually not satisfied by real acids as has been pointed out^ by Bell5 and by several others. For example, among the niet'hyl subst'ituted phenols we see from the data of Papee, et CAZ.,~ that phenol is a stronger acid than is 3,5-xylenol a t 25" in acjueous solution. But above 30°, 3,s-xylenol is n stronger acid than is phenol. Such reversals of acid strengths at different temperat'ures are common. Anot'her objection to attaching much fundamental significance to such empirical correlations of acid st'rengths is that they attribute all of the differences in acid strengths of a series of acids to properties of the acid and possibly its anion to the cxcliision of solute--solvent interactions. Hammett (1) -4lfred
and others have criticized this neglect of solutesolvent interactions and it has been shown recently3 that differences in solute-solvent interactions are a t least as important as all other effects in comparing the thermodynamics of dissociation of some nitrophenols and chlorophenols in aqueous solution. Bell,5 Ingold6 and a few others have pointed out that differences in acid dissociation constants can have no simple significance, in part because the magnitude and even the sign of the difference depends on the temperature. But we are especially concerned with interpretation of these acid strength differences in such acids as the methyl substituted phenols so it has been our purpose to develop a thermodynamic theory which will account quantitatively for the dissociation constants of these acids as a function of temperature. It is alFo hoped that the method developed in this paper will be a start toward a theoretical treatment applicable to the effects of substituents in general on strengths of acids in aqueous solutions a t all accessible temperatures. In order to do this it has been and will be neressary to consider solute-solvent interactions as well as the properties of the acid molecules. Theoretical We begin by writing the equations
(1)
(2 )
for the acid dissociation of unsubstituted phenol (1) and a methyl substituted phenol ( 2 ) . Since our concern is with the effect of substitution on AFO, AHo and ASo of dissociation, we combine (1) and (2) to obtain HA,(aq)
+ A,-(aq)
=
A,-(aq)
+ HA,iaq)
(3)
The standard free energy, enthalpy and entropy changes for this reaction are A&'
AHj'
APSO - AF,' AHa0 - AHun
(4) (5 )
and ASao = ASS0 - AS"'
P. Sloan Foundation Research Fellow.
(2) H. M. Papsse. TT. J . Canaday. T. W. Zairidaki and K. J. Laidler, Tr?ns. Faradag Soc.. 55, 1734 (1959). (3) L. P. Feriiimdez and L. G . Hepler, J . Am. Chem. Soc.. Si, 1783 (19591, and unpublished #lata. (4) 'X. J. Cane.dag. Can. J. Chem., 38. 1018 (1960). ( 5 ) R. P. Bell, "The Proton in Chemistry," Cornell University I'rpss. Ithnca, S e w T o r k , 1959.
+ Adaq) HA4,(aq) = H+(aq) + A,-(aq)
HA4,(aq) = H+(aq)
and
(6)
where the subscripts 3, s and u refer to thermodynamic quantities associated with reactions 3, 2 and 1, respectively. (0) C. K Ingold, "Structure and Mechanism in Organic Clirinis" G Bell and Sons. Ltd , London, 1963.
try
L. G. HEPLERAND W. F. O'HARA
812
Our attention is first confined to AHSO and AS3O because we can calculate AF2 and also the temperature dependence of AF2 from these quantities. Both AHS0and AS$' are taken to be the sum of internal and external contributions as represented by
+
AH3' = AHo,*$ AHOext
( 7)
and AX3O =
ASOint
ASOext
( 8)
External contributions to the enthalpy and entropy are associated with solute-solvent interactions and internal contributions arise from differences in enthalpy and entropy within the acid molecule and its anion. Pitzer' already has investigated the entropy of dissociation of weak acids and by applying his methods to our problem we deduce that ASoint for reaction 3 is very nearly zero. Hence we take Asso = ASoert
(9)
The external enthalpy change represented by AHo,,$ in (7) is the sum of the solute-solvent interaction enthalpies for A,-(aq) and HA,(aq) minus the sum of these same interaction enthalpies for A,-(aq) and HA,(aq). ASoext is another measure of these same interactions so we are led to suggest that AHoextis directly proportional to ASoext. It has been suggested previously by Latimer, Pitzer and Slanskys and by others that enthalpies of hydration are directly proportional to entropies of hydration. These suggestions lead to AHOext
bA8'e.t = bA.Sio
F = z2/2rD
(11)
where z, r and D represent charge, radius and dielectric constant, respectively. Differentiation with respect to temperature and combination with (bFlbT)p = -Sand F = H - TSgives 1
H = [(d In D / d T )
+T] S
(12)
showing that for this model solute-solvent interaction enthalpy is indeed proportional to interaction entropy. It is well known that the Born equation does not account quantitatively for the thermodynamic properties of real solutes in real solvents but the Born equation and its derivatives do correctly predict the relative solvation free energies, enthalpies and entropies for series of similar ions. Because of the similarity of the species on the left and right sides of (3), it is to be expected that many of the quantitative inadequacies of the Born equation and its derivatives will be cancelled when applied to this reaction. We therefore have confidence in the applicability of (10) to the problem at hand. Now we are left with the problem of evaluating AHoint, which represents that part of the total enthalpy change for (3) not due to solute-solvent (7) K. 9. Pitzer, J . Am. Chem. Soc., 59,2365 (1937).
interactions. Thus we are concerned with the internal energy changes involved in transferring a proton from H4, to A,- to form 4,- and HA,. We approach evaluation of this energy by way of consideration of the dissociation energies of diatomic molecules. The dissociation energy of a diatomic molecule, the potential energy of which is given by the Morse function, is proportional to the square of the fundamental stretching frequency times the reduced mass.g Even though this relation strictly applies to dissociation into atoms, we have calculated, using data from NBS Circular 500,1° A H 0 for the gas phase dissociation of HC1, HBr and H I into hydrogen and halide ions and obtained a straight line when these AHovalues were plotted against the squares of the corresponding stretching frequencies." Thus, neglecting the 2% difference in reduced mass between HC1 and HI, we observed that the heat of dissociation into ions is directly proportional to y2. If we consider phenol and the various methyl substituted phenols as diatomic molecules in which the phenolic hydrogen is one atom and the rest of the molecule is the other atom, the reduced masses differ by less t,han 0.05%. On the basis of our above considerations of real diatomic molecules, we therefore take AH (leaving out solvent effect) of dissociation of phenol and the methyl substituted phenols into ions to be directly proportional to the squares of their 0-H stretching frequencies. Thus we have
(10)
Equation 10 is in accord with and actually was partly suggested by the Born equation for the interaction of a spherical solute particle with a continuous dielectric solvent medium. The Born equation for the free energy is written
T'ol. 65
AHOint
a(vs2
-
vu2)
(13)
where v, and vu represent the 0-H stretching frequencies of a methyl substituted phenol and unsubstituted phenol and a is the proportionality constant. Since we are concerned in (3) and (13) with differences in dissociation energies of similar molecules, we expect that any small deviations from direct proportionality will largely cancel. We now combine (7), (10) and (13) to obtain AH30 = a(vs2 - vu2)
+ bASs"
(14)
This equation is rearranged to
--AH3' v,2
vu2
-a+(-]
A&'
(15)
Data given by Papee, et al.,? have been used in (5) and (6) for calculation of AHS0 and ASSo for nine methyl substituted phenols and the spectral data of Bavin and Canaday12 have been used for calculating vs2 - vu2 for these same methyl substituted phenols. All the relevant data are given in Table I. It is important to note that the spectral data were all obtained by t h c same investigators using the same solvent and instrument. Figure 1 is a graph of AH3°/(vs2 - vuz) against AS3O/(vS2 - vu2) and it is seen that a good straight (8) W. M. Latimer, K. S. Pitzer and C. M . Slaneky, J . Chem. Phys., 7 , 108 (1939). (9) L. Pauling and E. B. Wilson, Jr., "Introduction to Quantum Mechanics," MrGraw-Hill Book Co., Inc., New York, N. Y.. 1935. (10) "Selected Values of Chemical Thermodynamic Properties," Circular 500, National Bureau of Standards. 1952. (11) G. Heraberg, "Spectra of Diatomio Molecules," D. Van Nostrand Co.. Inc.. New York, N. Y.,1950. (12) P. M. G. Bavin and W. J. Canaday, Can. J . Chern., S I , 1555 (1957).
ACIDDISSOCIATION OF METHYL SUBSTITUTED PHENOLS
May, 1961
TABLEI SUMMARY OF TBERMODYNAMIC AND SPECTRALDATA Compound
cal./mole
cal./mole
ASO, cal. mole Leg.
(1) Phenol
13,630 14,030 13,760 14,000 14,140 14,290 13,900 14,460 13,860 13,670
5600 7130 4900 4290 6610 7680 6220 4950 8240 7510
-27.0 -23.2 -30.7 -32.8 -25.3 -22.1 -25.7 -31.9 -18.9 -20.7
(2) (3) (4) (5) (6) (7) (8) (9) (10)
AFO,
o-Cresol m-Cresol p-Cresol 2,3-Xyienol 2,4-Xyl.enol 2,5-Xylenol 2,6-Xylenol 3,CXylenol 3,B-Xylenol
&Yo,
0.14
0.12
r(OH)
(am. -1)
3600 3606 3605 3604 3608 3610 3606 3618 3605 3602
line is obtained as predicted by (15). Least squares treatment of the data gives a = 0.00865 and b = 284.0 from the intercept and slope of the straight line. The average magnitude of the difference between the experimental values of AH30 and the values calculated from (14) with the above values of a and b and experimental AS30 values is 86 cal./ mole, which is slightly greater than the experimental uncertaini:y in the A H 3 0 values. As mentioned by l'apee, et the p K a of 2,6-xylenol was determined by different investigators from those who determined the other pK, values used in calculating AF:? and thence AS30. If we choose to leave the$ data for 2,6-xylenol out of the least squares treatment, we obtain a = 0.00910 and b = 282.9 and find that the average difference between calculated and experimental values of AHso is decreased to only 22 cal./mole, which is less than the experimental uncertainty in the AH3O values. The empirical value of b should be compared with [l/(d ln Dl'd?') TI, in (12). Malmberg and Maryott13 give (d In D/dt) = -4.543 X at 25" so we caiciilate that [l/(d In D/dT) TJ = 77.4 a t 298°K. The empirical value of either b given above is 3.7 times this value, which is about as expected from the results of earlier calculations with the Born equation by Latimer and others.
+
+
Therniodynamic Relations The values found above for a and b may be used with experimental or theoretical values for either AH30 or ASil for calculation of AFO , and thence the equilibrium constant for (3), all a t 298°K. This equilibrium constant, which is denoted by K3, may then be combined with the dissociation constant for HA,, K,, to give the dissociation constant for HA,, K,. Then the already known AS2 or AHg0, combined with ASUo or AHu0,permits calculation of AX,? or AH2 and thence K , a t other temperatures. Equations for these calculations, which yield values in excellent quantitative agreement with experimental data, are given later in this paper. All this suggests the desirability of considering methods of predicting AS3'' or AH,O, a t least semi-theoretioally. Theoretical prediction of AH3O values for various substituted phenols is impractical because there are presently no means of calculating AHo,,$ or even of empirically correlating A H o e , t with known substituent properties. The situation is more promis(13) C. G. hlalmberg rind A. A. Meryott, J . Research Natl. Bur. Standards, 66, 1 (1956).
0.10
0.08
5 0.06
-
1
0.04
-
"2
0.02
-
ir
v
813
\
0 -0.02 -0.04
-4
-
0 2 4 6 AS,~/Y: v: x 104. Fig. 1.-Graph based on equation 15. The numbers represent the corresponding compounds in Table I. -2
-
ing with respect to theoretically predicting AS30 because we have shown that Ass0 = ASo,,,and quite a lot is known about entropies and solute-solvent interactions. First, we consider the standard partial molal entropies of the anions of phenol and the methyl substituted phenols. The ortho substituted anions are all expected to have more positive entropies than the other anions because the solute-solvent interaction (ordering of the water molecules and hence loss of entropy) around the oxygen is less for ortho substituted anions than for the other anions. The entropy of the anion of 2,6-xylenol is therefore expected to be the most positive of all the anions under consideration. Xext most positive are the entropies of the anions of o-cresol, 2,3-xylenol, 2,4-xylenol and 2,5-xylenol. Present knowledge is insuEcient t o permit any confident assignment of even the relative order of these latter entropies. Similarly, present knowledge permits us to predict only that entropies of anions of phenol, m-cresol, p-cresol, 3,4-xylenol and 3,sxylenol are all less than any of the entropies of ortho substituted anions. Similar reasoning applied to the entropies of the undissociated phenols suggests that the entropy of 2,6-xylenol is the most positive and that the entropies of o-cresol, 2,3-xylenol, 2,4-xylenol and 2,5-xylenol are all less positive but fall in some as yet unknown order with respect to one another. The entropy of unsubstituted phenol is probably next most positive, followed in some unknown order by the remaining meta and para substituted phenols. It is shown easily that the relative orders of entropies given above are consistent with the experimental values of AS30 but they do not permit quantitative prediction or calculation of values of ASao for substituted phenols. Since we are presently unable to predict or calculate numerical values of AH30or AS30 from knowledge of molecular properties and solute-solvent interactions, we must derive our equations for the
8 14
L. G. HEPLER AND W. F. O'HARA
equilibrium constant of reaction 3 and the dissociation constants of the various acids in terms of experimentally determined values of AS3'' or AHso. Of these quantities, only AH30 can be evaluated experimentally without knowing the equilibrium constant we want to calculate. Therefore we derive thermodynamic equations for K 3 , K , and K , in terms of a, b, vs2, vu2, AH3O and T . Combination of (14) with
Vol. 65
thermodynamic equations in this paper. We have been forced to consider all enthalpies and entropies to be temperature independent, which is only true if ACpo = 0 for the reactions under consideration. KO experimental data are available that permit reliable evaluation of ACpoof dissociation of phenol or any of the methyl substituted phenols and it is certainly pushing the applicability of the Born equation too far to use it for calculation of ACPo. Since it has been observed many times that values AF3' = AH3' - TAS3' (16) of ACpo of dissociation are all about the same gives for similar acids (about -35 cal./deg. mole for uncharged acids), we expect that all the unknown values for AC," of dissociation of the phenols are far from zero but all of the same magnitude. There1J7ealso write (17) as fore calculations for reaction 3, or its generalized version, for which AC," is small, are reliable over wide ranges of temperature. Calculations for -RT In K I (18) reactions 1 and 2 are reliable only for temperatures -RT In K,/K, (19) close to 25'. Free energies and equilibrium constants, at several The treatment described in this paper, based on temperatures, as calculated from (17), (18) and Consideration of solute-solvent interactions and (19) are in good agreement with the experimental 0-H stretching frequencies, can be applied to the thermodynamics of dissociation of some carvalues.2 The temperature a t which AF3O = 0, K3 = 1.0 boxylic acids in aqueous solution as is described in a and K , = K , is denoted by T* and is obtained paper in preparation. It also seems likely that this treatment can be satisfactorily applied to the acid from (17), (18) or (19) as dissociation of anilinium ions and similar species. bAH3" T* = There are, however, restrictions on the applicaAHSa - a(v2 - vu2) bility of this method to interpretation of the Since it is often of interest to compare two sub- thermodynamics of dissociation of acids. For exstituted acids with each other rather than to com- ample, this method cannot be applied without pare a substituted acid with the unsubstituted modification to acids with intramolecular hydroacid, me may generalize our equations by substi- gen bonds. One such acid IS o-nitrophenol. tuting subscripts x and y for subscripts s Further, it is possible that this treatment, although and u and understanding that x and y refer to qualitatively satisfactory, may be less quantitaany two acids in the series under consideration. tively accurate when applied to acids with highly In this fashion we obtain polar substituents which make the magnitudes of the effects considered here much larger. This possibility cannot be settled at present because there are insufficient thermodynamic and spectral where AH3"' refers to the enthalpy change for the data for such acids to permit an adequate comgeneralized version of reaction 3. parison of theory with experiment. It already has been pointed out that phenol and XOTEADDEDIS PriooF.-Professor R.Bruce Martin has 3,3-xylenol have the same dissociation constant at reminded us that the Born model is not unique in leading t o 30'. Equation 20 gives T* = 303Ok'. = 30'C. solute-solvent interaction enthalpy proportional to entropy. I t may he calculated from the data of Papee, This led us to investigate the oriented dipole model of Powell et a1.,2 that the dissociation constants of o-cresol and Latimer, J . Chem. P h y s , 19, 1139 (1951). We have their equation for the electrostatic potential energy of and p-cresol are equal at 296'K. Equation 21 used interaction with their equation 8 in the same manner that gives T* = 297'K. Similar calculations for other led to equation 12 in this paper. The value deduced by pairs of acids chosen from the phenols under con- Powell and Latimer for the temperature coefficient terms Fideration give values of T* in agreement with the was then used in calculating a value of 800 for the proporconstant we have called b. temperatures at which dissociation constants art. tionality The actual solhte-solvent interactions might be expected equal ac. calculated directly from the experimental to be between the extremes of the oriented dipole model thermodynamic data.2 For some pairs of acids and the Born continuous dielectric model The value for b T* is out of the range of existence of aqueous solu- deduced in this paper (284) is betneen the extremes of the dielectric value (77) and the oriented dipole tions and ha< no physical significance. The dis- continuous value (800). sociation constants of such acids do not become Acknowledgment.-We are grateful to the Naequal to each other at any accessible temperature. tional Science Foundation and the hlfred P. Discussion Sloaii Foundation for financial support of this There is one fundamental limitation on the and related research.
