J. Phys. Chem. 1083, 87, 2411-2416
consumed was desorbed as water. Thus, most of the hydrogen consumed was retained in the salts, plausibly in the form of protons. Scheme of Acid-Site Generation. The catalytic activities and the infrared spectra of adsorbed pyridine in AgTP, CuTP, and AlTP demonstrate the role of water and hydrogen on the acid-site generation. In the case of A m , there is no catalytic activity for xylene isomerization and pyridinium ion is not formed by adsorption of pyridine. The catalytic activity develops and pyridinium ion is formed, however, after the treatment of AgTP with hydrogen. Thus, there is no Bronsted acidity in AgTP as prepared, but the Bronsted acid sites are generated probably according to reaction 2. The observation of 0-D bands, after treatment with deuterium supports the scheme. As for CuTP, the general features are the same as for AgTP, except that a small band due to pyridinium ion was observed without treatment with hydrogen. Thus, the mechanism for CuTP is similar to reaction 2 and may be expressed as follows: Cu2++ 1/2H2(or H) Cu+ + H+ (3) Cu+ + 1/2H2(or H) Cuo + H+ The high reducibility of Ag and Cu salts of heteropolyacids has been demonstrated previously with temperature-programmed reduction by Niiyama et al.22 It is also found that methanol or butanol can play a role similar to that of hydrogen in acid-site formation by donating hydrogen (molecular or atomic) during their conversion. This explains why silver and copper salts have high activity in methanol conversion.
--
2411
For AlTP, the situation is different. The activity for xylene isomerization exists and pyridinium ions are formed on AlTP, which was evacuated at 300 "C. There is no enhancement of the catalytic activity or of the intensity of the band due to pyridinium ion by treatment with hydrogen, but the activity and the intensity of the band are enhanced by the treatment with water. Thus, water has an essential role in creating Bronsted acid sites, and the scheme may be expressed by reaction 1, as proposed by Niiyama."J2 It is very plausible that the acidity formed in AlTP evacuated at 300 "C is also developed by the dissociation of residual crystalline water according to reaction (1). The scheme with reduction of metal cations like reaction 2 or 3 has no contribution to the acidity, for hydrogen has no effect on the acidity. This is expected since aluminum cations are difficult to reduce. The small acidity in CuTP as observed through the IR spectrum of adsorbed pyridine may be formed also according to reaction 2. Acknowledgment. We are deeply grateful to Professor Emeritus Tominaga Keii for his fruitful discussion and his continuous encourangement throughout the work. The present work is partially supported by a Grant-in-Aid for Scientific Research Nos. 56040039 and 57040012 from the Ministry of Education, Science, and Culture and by a Grant from the General Sekiyu Research and Development Encouragement and Assistance Foundation. Registry No. HTP, 1343-93-7; AlTP, 83714-04-9; CuTP, 77839-56-6; AgTP, 78897-19-5; o-xylene, 95-47-6; 1-butanol,7136-3; pyridine, 110-86-1; methanol, 67-56-1; hydrogen, 1333-74-0.
Intramolecular Unsymmetrical OH0 Bonds. Thermochemistry K. I. Lazaar and S.
H. Bauer'
Department of Chemistry, Cornel1 University, Ithaca, New York 14853 (Received: June 28, 1982; In Final Form: December 14, 1982)
Most 0-- -H-0 bonds are unsymmetrical; this report deals with nearly symmetric potential wells, slightly perturbed by the next to nearest neighbor environment. The ratios of residence times of the hydrogen atom in the two-well potentials in unsymmetrically substituted acetylacetones were derived from NMR chemical shifts at the carbonyl carbons (13Cnatural abundance),recorded for dilute solutions of the enol isomers in nonpolar solvents. Cross-checksbetween several equilibrium constants indicate that the motion of the bridging hydrogen atom is strongly coupled to (indeed, is largely determined by) the dynamics of the molecular skeleton.
Introduction The OH0 bond has been treated theoretically and experimentally as the prototype of a hydrogen atom moving in a one-dimensional, double-minimum potential with a low barrier. The literature is extensive.' However, it has been recognized for many decades that the model is highly idealized because this potential is not uncoupled from the multidimensional potential surface due to the medium in which O H 0 is embedded. In the real world completely symmetric cases are relatively rare, as are "strong" H bonds (1) (a) P. Schuster, G. Zundel, and C. Sandorfy, Eds., "The Hydrogen Bond",Vol. 1-111, North-Holland Publishing Co., 1976;(b) Reports from the Hydrogen Bond Project at the Institute of Chemistry, University of Uppsala.
0022-365418312087-2411$01.50/0
characterized by single-minimum potentials. Recently there appeared theoretical treatments2 in which two-well potentials were considered to be weakly coupled to the dynamics of the molecular framework. There are numerous theoretical discussions, but few experiments3 have been described which permit exploration of the shape of the potential function.4a Practically, little information is (2)(a) C. W. Bock, M. Trachtman, and P. George, Chem. Phys., 62, 303-18 (1981);(b) K.M.Christoffel and J. M. Bowman, J.Chem. Phys., 74,5057(1981);(c) Y.Bouteiller, A. Allavena, and J. M. Leclercq, Chem. Phys. Lett., 84, 361-4 (1981). (3)(a) R. Rosseti, R. Rayford, R. C. Haddon, and L. E. Brus, J . Am. Chem. SOC.,103,4303-7 (1981);(b) G. J. Woolfe and P. J. Thistlewaite, ibid., 103,6916-23 (1981). (4)(a) M.M.Kreevoy and B. A. Ridl, J. Phys. Chem., 85, 914-7 (1981);(b) J. R. de la Vega et al., J.Am. Chem. SOC.,104,3295 (1982).
0 1983 American Chemical Society
2412
Lazaar and Bauer
The Journal of Physical Chemistry, Vol. 87, No. 13, 1983
-
-ENOLS
EST IMA T ED RELATIVE ENTHALPIES
= 20 (X=CF, ; Y z C H 3 )
=7
-
=5
E9
DIKETONES
-
Figure 1. Significant molecular configurations assumed by the acetylacetones, wlth postulated transition structures.
available on differences in well depths, heights of barriers, the location of vibrational states within the wells, and estimates of the residence times of the hydrogen in each of the wells, that is, the kinetics of isomerization. Our current explorations of the thermodynamics and kinetics of intramolecular transformations over low barriers consist of several objectives: (i) to determine the equilibrium constants for partition within slightly asymmetric double or triple potential wells; (ii) to measure the relaxation times for such systems (or, at least, to place upper bounds on their magnitudes); and (iii) to develop a model which incorporates the motion of the hydrogen atom into the structural isomerization of the rest of the molecule. The following are several prototypes of symmetrical O H 0 bonds, listed in sequence of the ring size of the diketones, which incorporate alternate singledouble-bonded atom pairs:
( 5 - or 7 - atom ring) I1 0-H-0
2
(or, 9-hydroxyphenalenone, or naphthazarin)
IiI
0---H-0
I
2
IV
Because of their availability and ease of manipulation we chose to investigate the acetylacetones (AcAc, 1111, the classical enol-keto tautomers; specifically, we studied Z = H; X, Y = CH,, CF3,phenyl, OCH,, OC2H5,NH(C6H5). Asymmetry is introduced when different substitutents are present at the left- and right-hand sides. In AcAc a sequence of equilibria coexist, as illustrated in Figure 1. In this communication we are concerned with the equilibrium at the left end of this sequence. Demonstration that the well depths are unequal requires measurement of the equilibrium constant, K L I R = [R]/[L]. Even a slight departure from unity is significant. Effects of small departures from symmetry in the double-minimum potential for O H 0 have been explored.4b A convenient model which focuses on the energetics of the L i=t R transformations (Figure 1) is to consider the relative
stabilities of four subunits (there are corresponding small entropy increments, as discussed below): O-H-----o O-----H-O I
(7x1
II
(PY1
/I
(Ox1
I
(YY1
In terms of these parameters (assuming that they are approximately additive) asymmetry in well depths is determined by the magnitude of mL/Ro
[(PX -k YY) - ( P Y
+ 7 X ) l or [(Px- 7x1 - (PI - YY)I (1)
Thus, the difference in well depth is set by the "relative preference" of Y and X for the p vs. y structures; this determines the ratio of residence times of the H atom in the two wells. The duration of the residence times is determined not only by the mass of the migrating atom (e.g., D vs. H) but also by the time required for the heavier skeletal atoms to adjust from one set of interatomic distances to another. (Note the microwave structure of the simplest species, mal~naldehyde.~)This intramolecular conformational change is strictly analogous to a valence isomerization. With respect to the barrier height, the one in AcAc is close to that in semibullvalene (5.5 f 0.1 kcal/mol),6 but it is considerably lower than in bullvalene (13.3 f 0.1 kcal/mol).' Whether the kinetics of these transformations is collisionally controlled or has a large component of "tunneling" is sensitively dependent on the height and shape of the barrier, and on the temperature.8 The equilibrium constants in several nonpolar solvents for various combinations of substituents in I11 were estimated from NMR chemical shifts at the 2- and 4-carbon atoms; the temperature dependence of KLIRfor (trifluoromethy1)acetylacetone was also measured. The experimental procedure is described below.
Experimental Section The chemicals employed were obtained from commercial sources. All liquid materials were degassed and dried by passing them, under vacuum, through a column containing (5)S. L. Baughcum et al., J. Am. Chem. SOC.,103, 6296 (1981). (6)A. K. Cheng, F. A. L. Anet, J. Mioduski, and J. Meinwald, J. Am. Chem. SOC.,96,2887 (1974). (7)H. Gunther and J. Ulmen, Tetrahedron, 30, 3781 (1974). (8)(a) K.-H. Grellmann, U. Schmitt, and H. Weller, Chem. Phys. Lett., 88,40 (1982);(b) E.F. Caldin, Chem. Reu., 69, 140 (1969);(c) P. D.Pacey, J. Chem. Phys., 71,2966(1979);(d) P.H. Cribb, S. Nordholm, and N. S. Hush, Chem. Phys., 69, 259 (1982).
The Journal of Physical Chemistry, Vol. 87, No. 13, 1983
Intramolecular Unsymmetrical OH0 Bonds
a molecular sieve ( m e 4A, 1/16 in.). The original samples of ethyl acetoacetate and diethyl malonate were not further treated since their relatively high boiling points (181 and 199 OC, respectively) made it difficult to handle them in the vacuum line. The purity of these materials was checked by 13C NMR and occasionally lH NMR. With liquid specimens, 5% solutions in prepurified solvents were prepared volumetrically under vacuum manipulation. Solutions 5% by weight were prepared of the solid reagents. Since acetoacetanilide is not very soluble in benzene, saturated solution of the compound (less than 3%) was used. The NMR spectra were run in 10-mm flat-bottom tubes (not sealed). When tetramethylsilane (Me4Si)was not used as a solvent, a small amount (less than 2% by volume) was added to the solutions to provide an internal reference. The I3C resonances were measured in natural abundance on a Bruker-WM300 spectrometer, pulsed Fourier transform mode, operating a t 75.47 MHz, using a proton broad-band decoupler. The experimental parameters were kept constant for each series of measurements with the various solvents; generally 400-8000 scans were accumulated. Chemical shifts are reported in the tables in parts per million relative to Me4Si. (A positive shift is in the downfield direction.) The shifts listed in this report are estimated to be precise to within A0.02 ppm. The Bruker variable-temperature (B-VT 1000) unit controlled the probe temperature during the NMR scans. The actual temperature was measured before and after each run by placing a thermocouple into a nonspinning sample tube filled with acetone. The decoupler was turned off momentarily before the temperature was read.
Determination of the Equilibrium Constants For a general system such as
2413
TABLE I: Effect of H Bonding on I3C Chemical Shifts ( m m ) compd no.
compd
C'
C2
c3
12 16
H3C'-C20-C3H=CH~a'b
25.93
196.81
137.51
H3C-CO-CH=CH2"'b
25.93
198.07
137.42
17
H~C'--C~O-CH~~
30.07
203.65
18
H~C--C?--CH~'
30.01
205.51
HO-CHI
Cl-H
I
-
a Concentrations of methyl vinyl ketone, acetone, methanol, and hydrogen chloride were 5% (volumetric) in benzene. Chemical shift of atom of methylene group cannot be measured exactly because it overlaps with the solvent resonance line(s).
benzene) and then a 5% solution of 12 to which 5 % methanol was added (in benzene) were examined. In the latter, the resonance line appeared 1.26 ppm downfield compared to its position in the former. The addition of hydrogen chloride to acetone (in benzene) shifted the carbonyl line downfield (by 1.86 ppm). (Acetone was used instead of methyl vinyl ketone, since the latter might have undergone extensive polymerization in the presence of HC1.) We accepted 1.86 ppm as the H-bonding correction (as induced by the stronger acid) in computing the equilibrium constants, because more complete H bonding is achieved in this mixture. From an experimental value for 84xy and K L I R one can estimate the magnitudes of the intrinsic chemical shifts, Furthermore, if the conjugate vinyl compound and
I
H H
H
KLIR E [RlI[Ll = ~ R / T L rRand rLare the residence times which the enol isomers spend in the right and left conformations, respectively. The observed chemical shifts at C2and C4 (which are easily distinguished) are the averages of the intrinsic shifts weighted by the corresponding residence times. Thus 6ZLTL 62XY
=
?L
+ 62R7R - 62L + h R K L / R + rR
1 + KLJR
(2)
For the symmetric species X-X
62xx =
(62L
+ 62R)/2
(3)
Analogous expressions (for aUy and 6,m) apply to C4. The additional datum required to evaluate K L is provided (approximately) by a model structure, an d-bonded vinyl ketone (HQ is a strong acid): !?----Ha
I
H
Thus, 62R can be estimated from spectra obtained under conditions identical with those used for the diketone. Then, eq 3 permits evaluation of bZL, and eq 2 leads to K L I p For (trifluoromethyl)acetylacetone,for example, was estimated from the chemical shift of the carbonyl C in compound 12 (Table I). First, a 5% solution of 12 (in
is available, 64L may be measured directly and a second value for K L obtained, to check on how reliably these chemical shiits can be transferred from the models to the diketones.
Results Chemical shifts for the various 13C's for both the enol and keto tautomers of P-diketones in Me4&, hexane, and benzene solutions are listed in Table 11. As expected, values for are less than 62R since the ketonic structures provide less shielding of the carbon atom than do the hydroxy structures. In Table 111, the chemical shifts are given for the relevant vinyl ketones. With the exceptions listed below no impurity lines were recorded. In methyl acetoacetate a weak line appeared at 176.18 ppm which must be assigned to a trace of impurity, rather than to either of the carbonyl 13C's in the enol tautomer. Similarly, the very weak lines at 176.03 and 173.04 ppm found in the spectrum of ethyl acetoacetate are probably due to an impurity. The contrary assumptions lead to negative values for the KLIR's. Physically acceptable ICs place these resonances between 183.37 and 198.67 ppm. Although the ratio of integrated proton signals for CH3(en)/CH3(ke)of 8 indicated that the enol form was present at 16% in a 0.1 mole fraction of solute in b e n ~ e n e and , ~ our own rough estimates, based on 13Cintegrated resonance lines, are in the range 14.5-18.2% (for a 5% solution), neither we nor any previous investigators1° could detect resonance lines (9) M.
T.Rogers and J. L.Burdett, Can. J . Chen., 43, 1516 (1965).
The Journal of Physical Chemistty, Vol. 87,
2414
No. 13,
Lazaar and Bauer
1983
TABLE 11: Chemical Shifts ( p p m ) of 13C Atoms of P-Diketones in Different Solvents a t 27.5 "C compd no. 1
2 3
4 5 6 7 8
comud
solvent
CZ
C4
acetylacetone (H,C-C(OH)=CH-CO-CH,)
benzene 191.02 en 201.07ke Me,Si 190.17e n hexane 190.49en hexafluoroacetylacetone benzene 176.14 en' (F,C...CF,) Me,Si 177.40en' hexane 177.65 en' (trifluoromethy1)acetylacetone benzene 175.70 en' 177.05en' (F,C...CH,) Me,Si hexane 177.23en' benzene 186.03en dibenzoylmethane (Ph...Ph) benzene 183.76 en 1-benzoylacetone (Ph...CH,) 4,4,4-trifluoro-l-phenyl-l,3- benzene 186.52en butanedione (Ph...CF,) methyl acetoacetate benzene unobsd en 167.35 ke (H,CO.*.CH,) ethyl acetoacetate benzene unobsd en (H,CCH,O...CH,) 166.87ke
acetvl
unobsd e n 199.03 ke unobsd en
20.81 en 29.38 ke 20.82en
198.89 ke
29.36 ke
benzene 166.59ke
166.59ke
10
benzene 166.21 ke
166.21 ke
11
acetoacetanilide (Ph-NH...CH,)
benzene unobsd e n 163.54 ke
unobsd e n 204.45ke
' Mean value for quartet.
-C
24.37 e n 100.27en 30.05 ke 58.07 ke 24.44 en 99.69 en 24.37 en 99.81 e n 93.66 e n 93.66e n 93.71 en 23.99 en 96.32en 24.28 en 95.92 en 24.18 en 96.00en 93.31 en 25.22 en 96.71en 92.55 en
dimethyl malonate (H,CO...OCH,) diethyl malonate (EtO...OEt)
9
OL
191.02 e n 201.07ke 190.17 en 190.49 en 176.14e n 177.40e n 177.65en 194.16en 192.70en 193.05e n 186.03e n 193.44e n 177.02ena
other
116.83 (F,C) ena 117.03(F,C) en" 117.21 (F,C) en' 117.73(F,C) en' 117.52(F,C) en' 117.71 (F,C) ena Phb Phb
117.88(F,C) en: Phb
89.61 en 50.71 (H,CO) en 51.61 ke 49.47 (H,CO) ke 89.92en 14.23 en (H,C of Et), 59.91 en (H,C of E t ) 49.81 ke 14.02 ke (H,C of Et), 60.90ke (H,C of E t ) 41.02ke 51.81 (H,CO) ke 41.64 ke 13.98 ke (H,C of Et), 61.07ke (H,C of E t )
21.33 e n unobsd en 30.12ke 49.61 ke Phb
' 13Cresonance lines of phenyl group were not assigned.
TABLE 111: Chemical Shifts ( p p m ) of 13C Atoms of Vinyl Compounds compd no.
12 13 14 15
compd methyl vinyl ketone (H,C= C H - C O - C H , ) phenyl vinyl ketone (H,C= CH-CO-Ph) methyl acrylate (H,C= CH-CO-OCH,) ethyl acrylate (H,C= CH-CO-OCH,-CH,)
solvent
H,C=
Me,Si benzene benzene
b
= CH
126.38
co
other
25.99 (CH,) 25.93 (CH,)
132.63
138.22 137.51 b
194.66' 196.81' 196.31a
PhC
benzene
130.10
b
166.02
51.03 (CH,)
benzene
129.80
129.07 or 128.86'
165.62
60.24 (OCH,), 14.19 (CH,)
'
Cannot be measured because of overlapping with solvent resonance Values prior t o correction for lack of H bonding. line(s). ' 13C resonance lines of phenyl group were not assigned. __
for the carbonyl C's in the enol tautomer for 7 and 8. The presumption is that these carbonyl resonances are too broad to be detected because of the weakness of the
bridge, so that lifetimes in the open and closed structures are in the millisecond range at room temperature. Finally, the sample of phenyl vinyl ketone available to use had an impurity, which appeared to be methyl vinyl ketone. We s on therefore place somewhat less reliance on K L , ~ 'based NMR spectra of phenyl vinyl ketone. Values of K L I R calculated from these data are listed in Table IV. All the magnitudes depart from unity in the direction expected. For (trifluoromethy1)acetylacetone in benzene, Me,Si, and hexane, respectively, the solvent effects are negligible. The two equilibrium constanta derived for the phenylacetylacetone agree within 13% of their mean value, which is larger than the precision error in reading the resonance frequencies. We are inclined to believe that the lower value is more reliable because (i) the model compound (phenyl vinyl ketone) was contaminated, (10)J. B. Stothers and P. C. Lauterbur, Can. J. Chem., 42, 1563 (1964).
TABLE IV: K L , R Values in Different Solvents at 27.5"C compd no.
compd
benzene
Me,Si
hexane
3 5
F,C.,.CH, Ph...CH,
0.43
0.40'
6
Ph...CF,
0.42 0.52,' 0.68' 1.08
a The chemical shift of the carbonyl C of 12 in Me,Si was used t o calculate this value. Based o n 1 and 12. Based o n 4 and 13. If a larger correction is assumed for the effect of H bonding (2 x 1.86),the corresponding KL,R values are 0.59 and 0.72. We consider these t o be upper limits. (Compare acetone dissolved in benzene with that dissolved in methyl alcohol: G. F . Maciel and J. J. Natterstad, J. Chem. Phys., 42,2752 (1965).)
'
and (ii) it is possible that benzene as a solvent perturbed the model somewhat more than it did the diketone. K L i R for the Ph ...CF3diketone (1.08) is 13% less than the expected 1.24, based on the ratio 05210.42. Thus, while the experimental value for 6 is qualitatively consistent with those for 3 and 5, quantitatively it indicates that the assumption that a diketone may be treated e.s a superposition of separable subunits is at best approximate. This underscores again the artificiality of the abstraction that the OH0 potential is uncoupled from the rest of the molecule;
The Journal of Physlcal Chemistry, Vol. 87, No. 13, 1983
Intramolecular Unsymmetrical OH0 Bonds
2415
TABLE V : Temperature Dependence o f Chemical Shifts ( p p m ) o f lPC4for Compounds 1, 3, and 12 and the Deduced KLIR's' temp, K
compd no.
compd
229.7
252.7
274.7
300.6
1 3 12
H,C-C4(OH)=CH-C40-CH, F,C-C4( O H ) = C H - C 4 0 - C H , H,C=CH-C40-CH,
190.14 192.37 195.14b
190.13 192.44 194.92b
190.15 192.54 194.7gb
190.17 192.68 194.64b
0.51
0.48
0.46
0.43
KLIR
These values were corrected for lack o f H bonding ( i . e . , t 1 . 8 6 ) before the K L I R ' Swere calculated. N o In benzene. additional corrections need be made for the temperature dependence of the magnetic susceptibility. All the NMR scans were obtained for dilute solutions o f the reference compounds and the acetylacetones, under identical conditions.
T
the entropy increment is a significant factor in determining the magnitude of the equilibrium constant. The ratios of the integrated I3C resonances indicated that, in the solutions we used, 1 was present entirely as the enol in MelSi and hexane, and predominantly as the enol in benzene. Compounds 2 and 3 exist entirely as enols in all three solvents. Enolization is complete for 4-6 in benzene solutions. For 7 and 8 both tautomers are present with Kenjkegreater than unity, whereas 9 and 10 exist entirely in the keto form. The approximate partition of molecular energies into subunits may be extended to include the enol F? keto conversion by introducing a third subunit:
( O K )
300.0
280.0
260.0
240.0
I
I
I
I
0
II
i
"X
I
3.20
I
I
I
3.60
I
4.00
I
I
4.40
W T ) x io3 Figure 2. Dependence of K,,, for (trifluoromethyl)acetylacetoneon
We estimate that ASen/keo = +7 eu, due to increase in molecular flexibility upon ring opening. In these terms one can specify the parameters which account for a high level of enol (=go% in benzene) for 1 while 8 has an equilibrium constant Kenke = 5.39 in benzene. For simplicity assume that, for 8, is also equal to unity. Then
k,,,
temperature.
aen/keo(l)
deductions based on such a model bear but a modest resemblance to reality. The temperature dependence of KLIR for (trifluoromethy1)acetylacetone was evaluated from the data summarized in Table V, and illustrated in Figure 2. The deduced magnitude for A H L / R o = -323 f 23 cal/mol is presumed to be 1 order of magnitude greater than the estimated energy difference between the s and a states in acetylacetone. (The reported6 split between s and a states in malonaldehyde is 26 cm-1 (74 cal).) The value for ASL/RO (-2.74 f 0.09 eu) is rationalized on the basis of incomplete cancellation of the changes in the rotatoryoscillational frequencies around the C1-C2vs. C W 6bonds upon conversion from yma+ &H3 to @ma + yCHs.I1 Clearly,
-
(11) The effect of the transformation L R on the barrier for rotation of the CH3 group can be estimated by comparing the barrier in methyl vinyl ketone (1250 cal/mol, P. D. Foster et al., J. Chem. Phys., 43,1064 (1965)) with that in isobutene (2210 cal/mol, V. W.Laurie, J. Chen. Phys., 39,1732 (1963)). This leads to a lowering of the molecular entropy. Corresponding but smaller changes (from a higher to a lower barrier) take place for rotation of the CF3group. The net effect is a decrease by 2.7 units.
aen/keo(8) (aOEt
= 2aCH3- iYCH3 + PCH3)
'0
=
+ %Ha) - %((POEt + PCH3) + (YOEt + YCH3))
0 (5)
The usual rules of thumb regarding relative acidities of the protons on the C3 atom, and conjugative contributions involving the X or Y moieties with C=O and C=C .rr orbitals, are of little help in predicting the relative magnitudes for Kenke, since small differences between large terms are invoived, and entropy factors cannot be neglected. Finally, all the observed resonance line widths were no greater than the instrumental values. This indicates that in these solutions the enol e keto lifetimes are considerably longer than milliseconds and that the lifetimes in the double-minimum wells are considerably shorter than milliseconds.
Acknowledgment. This investigation is supported by the Directorate of Chemical Sciences, Air Force Office of Scientific Research, under Grant No. AFOSR-80-0046. Acknowledgment is made to the National Science Foun-
J. Phys. Chem. 1983, 87, 2416-2419
2416
dation Instrumentation Program (CHE-7904825) for support of the Cornel1 Nuclear Magnetic Resonance Facility. We sincerely thank Dr. S. Huang for his instruction on the use of this spectrometer, and Professor Charles F. Wilcox, Jr., for interesting discussions regarding the energetics of
the tautomer equilibria. Registry No. 1, 123-54-6;2,1522-22-1; 3,52902-93-9;4, 12046-7; 5,g3-g1-4; 6,326-06-7; 7, 105-45-3; 8, 141-97-9;9, 108-59-8; 10, 105-53-3;11, 102-01-2; 12, 78-94-4; 13, 768-03-6; 14,96-33-3; 15, 140-88-5; methanol, 67-56-1; hydrochloric acid, 7647-01-0.
Thermodynamic and Klnetic Descriptions of Equlllbrlum P. L. Corlo Department of Chemistry, University of Kentucky, Lexington, Kentucky 40506-0055 (Received: November 5, 1982)
A method is given for determining a necessary and sufficient condition for consistency of the kinetic and thermodynamic descriptions of the equilibrium state. It is assumed that the kinetics satisfies Wei’s axioms, but no special assumptions are introduced regarding the form of the reaction rate near equilibrium. The computational procedure is illustrated in several examples.
Introduction will be denoted K when the coefficients have been determined so that, in addition to satisfying the mass conserInvestigations of the consistency of the thermodynamic vation conditions, they are all positive integers whose only and kinetic descriptions of the equilibrium state have been common factors are f l . These conditions determine the principally concerned with rate laws expressible as the si uniquely and fix the scale of the reaction. When eq 1 difference of forward and reverse reactions.’-’ In paris scaled by multiplication with a positive constant a, the ticular, a necessary and sufficient condition for consistency associated equilibrium constant is K“. The constant Q is in the case of a reversible reaction whose forward and determined by the initial conditions, that is, the proporreverse rates are each proportional to a product of powers tions in which the reactants are mixed, in the laboratory, of concentrations is that the ratio of forward and reverse rate constants be a power of the equilibrium ~ o n s t a n t . ~ , ~ or the steps of a theoretical mechanism. The condition for thermodynamic equilibrium may be This investigation presents a theory of consistency that written u = 0, where makes no assumptions as to the form of the rate expression, does not impose stringent conditions of differentiability,2 and is applicable to systems in which several independent reactions occur in the neighborhood of the equilibrium state.8 We consider a closed, homogeneous, and x i is the concentration of Xi.Although u is defined isothermal system, with a stable, unique state of thermofor nonequilibrium states, virtually all subsequent equadynamic equilibrium, and adopt Wei’s axiomsg for the tions containing the x i refer to relations subsisting at kinetics: (1)mass conservation; (2) nonnegative concenequilibrium, so we shall not introduce special symbols for trations; (3) rates of change that are continuous functions the equilibrium state. I t will be convenient to introduce of the concentrations; (4) microscopic reversibility; ( 5 ) the algebraic stoichiometric coefficients vi, defined by vi = -si, existence of an appropriate Liapounov function. These fori = 1,2, ...,p; vi = si, for i = p + 1, p + 2, ..., p + a. Thus, axioms imply convergence to a unique, stable equilibrium the equilibrium expression can be written state. K = nxp (3) The initial discussion is limited to uncataiyzed reactions i in nonionic, ideal systems; the relaxation of these restrictions is indicated in the concluding section. Here we introduce the convention that, unless otherwise indicated, the range of a lower case Latin index in a sum Consistency Condition or product extends from 1 to p + a. Let the equilibrium system include p reactants X1,X2, The rate of the reaction associated with eq 1may be an ..., X,,and a products Xp+l, Xp+2, ..., X,+*,with p + a assumed expression designed to fit experimental data, or v = 1, where v is the number of mass conservation cona theoretical expression inferred from a proposed mechaditions. The equilibrium constant for the stoichiometric nism. In any case, since p + a - v = 1,there is at most one relation independent reaction in the neighborhood of equilibrium, SIX1 + szx2 + ... + spxp = and its rate r will be a function of xl,x 2 , ..., xp+*. At equilibrium r = 0. Sp+lXp+1 + sp+zxp+2 + + Sp+*Xp+r(1) The equilibrium conditions u = 0, r = 0 may be interpreted as surfaces in Euclidean space of p + a dimensions. (1) M. Manes, L. J. E. Hofer, and S. Weller, J. Chem. Phys., 10, 1355 (1950). Consistency of these two modes of description requires that (2) C. A. Hollingsworth, J. Chem. Phys., 20,921, 1649, (1952). these surfaces touch, but only at a single point, as otherwise (3) J. Horiui, 2.Phys. Chem., 11, 358 (1957). the equilibrium state would not be unique. Consequently, (4) K.G. Denbigh, T h e Principles of Chemical Equilibrium”,4th ed., Cambridge University Press, London, 1981, Chapter 15. the surfaces have a common tangent plane, so that cor( 5 ) E. H. Blum and R. Luus, Chem. Eng. Sci., 19,322 (1964). responding derivatives at the point of tangency are pro(6) R. K.Boyd, Chem. Reu., 77,93 (1977). portional: (7)L. P.Hammett, “Physical Organic Chemistry“, 2nd ed., McGraw.e*
Hill, New York, 1970, pp 54-7. (8) By a “neighborhood of the equilibrium state”,we always mean an infinitesimal neighborhood. (9) J. Wei, J. Chem. Phys., 36, 1578 (1962).
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