NORMAN N. LICHTINAND HARRYP. LEFTIN
164
Vol. 60
IONIZATION AND DISSOCIATION EQUILIBRIA IN SULFUR DIOXIDE. IV. ALKYL AND ARYL DERIVATIVES OF TRIPHENYLCHLOROMETHANE AT 0 AND -8.9' BYNORMAN N. LICHTINAND HARRYP. LEFTIN' Contribution from the Department o j Chemistry, Boston University, Boston, Mass. Received August 17, 1966
Conductivity data in liquid sulfur dioxide solution have been secured for triphenylchloromethane, its mono-, di- and tri-mphenyl, mono-, di- and tri-p-phenyl, mono-m- and mono- tbutyl derivatives at -8.9' and, where previously lacking, at 0'. Shedlovsky's procedure has been applied to the data in dilution range 2000 to 80,000 liters per mole to calculate values and Ao. Values of AHO,,, have been approximated by assuming constancy of this quantity over the temperature of K e x p range employed. The relationship between AHO,,, and AFO.., values for this series of compounds is quite different from that observed with KCl, KI, KBr and (CHa),NBr. This difference is presumed to be associated with the occurrence of covalent association in the present cases. Ion-pair dissociation constants have been calculated on the basis of an assumed model and some of the consequences of this procedure have been explored.
tg
Introduction
K,,, = (R+)(Cl-)/[(RCl)
There has been described already2 a considerable amount of evidence that solutions in sulfur dioxide of derivatives of triphenylchloromethane belong to a class of electrolytes the equilibria of which have been investigated relatively little, namely, those in which ions associate covalently as well as electrostatically. Perhaps most pertinent is the observationzbthat at 0" Koxp, the dissociation constant obtained from conductance data, is about five times greater for the mono-p-phenyl derivative than for the parent compound, whereas for the mono-m-phenyl derivative it is only three-fourths as large as for the parent compound. Since the mono-m- and p-substituted triphenylcarbonium ions are of similar size and shape this cannot be a consequence of structural influence on electrostatic ionic association which is now known3 to adhere remarkably closely to Bjerrum's equation in this medium. Conductance data for most of these solutes were secured in the earlier work a t only one temperature, 0") two compounds having been investigated at - 17" as we1L2* Measurements reported herein have been carried out largely a t -8.9" and in part a t 0")so that useful information on the temperature dependence of K e x pfor eight solutes of this type is now available. The observed relationship between Kexpand its temperature dependence is quite different from that found with solutes consisting of spherical ions which associate only electrostatically. It has been assumed2 that the equilibria represented by eq. 1 maintain in these systems. Here RCl represents the covalently associated species Ki --c, Ka
+ (R+Cl-)]
=
+ 1/Ki)
(2)
Equation 2 is identical in form with the relationship (their eq. 10) which Sadek and Fuoss4 have recently presented. It differs, however, in that K 1 measures the equilibrium of ionization of a covalent bond and its dependence on structure iu therefore subject to correlation with established group effects. The observed temperature dependence is consistent with this analysis. An attempt also has been made to apply these equations to the quantitative correlation of the measured equilibria with structure.
Experimental
and R+Cl- represents tho electrostatically associated species (ion pair). K, and K2, the.constants for ionization and dissociation, respectively, are related to K e x p as shown in eq. 2, where the quantities in parentheses are activities.
Apparatus, procedures and solvent were those referred to in the preceding paper.* The samples of triphenylchloromethane, its mono-m-t-butyl, mono-p-t-butyl, mono-mphenyl and mono-p-phenyl derivatives were drawn from batches of thcse compounds which have been described previouslyza~b and which, havin been stored in desiccators in the dark, gave no evidence of decomposition. Syntheses.-Solutes were prepared with emphasis on purity and little regard for yield. All-glass equipment was used throughout and moisture and oxygen were excluded6 wherever necesgary. Solvents were of C.P. grade or higher and thoroughly dried before use. Grignard solutions were subjected to acidimetric analysis of aliquots of hydrolysate before use. The di- and tri-m-phenyl derivatives of triphenylcarbinol were pre ared by the reaction of Eastman Kodak Co. "White Label'' ethyl benzoate or ethyl carbonate' (dried over anh. CaS04), respectively, with m-biphenylylmagnesium bromide. The Grignard reagent was prepared by treating freshly prepared turnings of Dow Chemical Company "Super Pure" magnesium with one equivalent of m-bromobiphenyl in dry ether containing 5-10% dry benzene.7 The carbinols were obtained after steam distillation and attempted purification by crystallization as impure yellow powders of wide melting range. Di-m-biphenylylphenylchlorornethane, previously unreported, was prepared from the impure carbinol (5 g.) by refluxing ( 4 hr.) with freshly distilled acetyl chloride (15 ml.). Removal of excess acetyl chloride by suction, trituration of the resulting oil and standing for 3 days with petroleum ether followed by washing with this solvent produced a yellow powder. This material was recrystallized three times from 1: 1 acetyI chloride-petroleum ether, washed with petroleum ether and dried at room temperature under
(1) Based in part on a dissertation submitted by H. P. Leftin in fulfillment of a requirement for the Ph.D. degree, Boston University, May, 1955 (2) (a) N. N. Lichtin and P. D. Bartlett, J . A m . Chem. Soc., 78, 5530 (1951); (b) N. N. Lichtin and H. Glaaer, ibid., 78, 5537 (1951). (3) N. N. Lichtin and H. P. Leftin, THISJOURNAL, 60, 160 (1956).
(4) H. Sadek and R. M. Fuoss, J . Am. Chem. Sac., 76,5905 (1954). (5) For instance, cf. L. F. Fieser, "Experiments in Organio Chemistry," Part 11, D. C. Heath and Co., New York, N. Y.,1941, p. 404. (6) C. 8.Marvel, E. Ginsberg and M. B. Mueller, J . A m . Chem. Soc.. 61, 77 (1939). (7) N. N. Lichtin and H. P . Leftin, ibid., 14, 4207 (1952).
RCl J_ RfC1-
zR + + Cl-
(1)
--hh
Feb., 1956
ALKYLAND ARYLDERIVATIVES OF TRIPHENYLCHLOROMETHANE 165
high vacuum to yield a white powder, m.r. 113.5-114.5'.) AnaL8 Calcd. for CalH&l: C, 86.4; H, 5.4; Cl, 8.2. Found: C, 86.4; H, 5.5; C1,8.3. Another sample was obtained from partially purified carbinol by reaction with acetyl chloride as before. Solid product was obtained from the reaction mixture on standing in the freezing compartment of a refrigerator for ten days and was purified by recrystallizing twice from acetyl chloride, washing with petroleum ether and pumping under high vacuum for eight The analdays to yield white crystals, m.r. 117.6-118.5'. ysiss for CI was 8.2%. Conductivity data obtained with the two samples at -8.9' were in excellent agreement. Tri-rn-biphenylylchloromethanewas obtained by refluxing the crude carbinol (5 g.) with freshly distilled acetyl chloride (25 ml.) for 3 hr., adding enough benzene (15 ml.) to bring the product into solution and chilling for 12 hr. The product was recrystallized once from benzene containing a few drom of acetvl chloride and dried bv DumDing give minite white "needles, m.r. 201-202% lcoi-:). AnaZ.8 Calcd. for CIrH&l: C, 87.6; H, 5.4; C1, 7.0. Found: C, 86.7,88.2; H, 5.3,5.2; C1, 6.7. A portion of this material was recrystallized twice more from- dry benzene containing a few drops of acetyl chloride, then once from dry benzene and dried by pumping to yield crystals with m.r. 200-201 ' (cor.). The analysiss for C was 87.1%, H, 5.8y0; C1, 7.1,7.0%. Conductivity data obtained with the two samples a t both 0.1 and -8.9' were in excellent agreement. Di-p-biphenylylphenylcarbimol*was prepared by the reaction of phenylmagnesium bromide with di-p-bi henylyl ketone.I0 Di-p-biphenylyl ketone was synthesizeaby distilling phosgene (1 mole) into a mixture of biphenyl (0.88 mole),. AlgCle (0.62 mole) and carbon disulfide (450 ml.) and stirring for 14 hours. After hydrolysis, the crude product was extracted in a Soxhlet apparatus with 1 :1 acetonebenzene. The extract deposited crystals which were recrystallized from 1 :1 benzene-pyridine and then from tolu; ene to yield small yellow-green crystals, m.r. 235-237 (cor.). Phenyl Grignard (0.22mole) was prepared in ether solution and the ether replaced by anisole (200 ml.) before adding a solution of the ketone (0.073 mole) in anisole (70 ml.). After completion of the reaction (steam-bath for 4 hours, room temperature for 12 hours), hydrolysis and steam distillation, the product was crystallized from 1 :1 benzeneether (90 ml.) to yield white di-p-biphenylylphenylcarbinol (0.043 mole), m.r. 153.5-154.5' (cor.). Di-p-biphenylylphenylchloromethanewas prepared from n sample of carbinol (3.0g.) by refluxing (0.5 hour) with 25 ml. of 1 :4 benzene-ether containing acetyl chloride; the solvents were removed on the aspirator and petroleum ether added. The precipitate was recrystallized from 25 ml. of 4:1 petroleum ether-methylene chloride containing a few drops of acetyl chloride and dried at 35' by pumping a t 0.1 mmd for 12 hours. The crystals were pale purple, m.r. 136-137 (cor .). The purity, estimated by acidimetric analysis of the hydrolysate, was 100.3 f 0.3%. The analysis* for C was 86.1%, H 5.4%. Tri-p-biphenylylcarbin~l~J~ was prepared by the reaction of p-bromohiphenyl with ethyl carbonat8e and sodium. A solution of 0.32 mole of p-bromobiphenyl (Eastman Kodak Co. "White Label" recrystallized twice from 95% ethanol) and 0.10 mole of ethyl carbonate (Eastman Kodak Co. "White Label" freshly distilled, @D 1.3847)in 100 ml. of dry benzene was added with stirring over a period of 1 hour to 0.83 mole of sodium dispersion (National Distillers Products Corp. 50% dispersion in toluene) in 100 ml. of boiling dry benzene. After the reaction was complete (2hr. in all), ethanolysis, steam distillation and recrystallization from toluene-ligroin gave 0.042 mole of almost white product, m.r. 205-208' (uncor.). Tri-p-biphenylylchloromethane was prepared from the carbinol (5.0g.) by refluxing with 5 ml. of freshly distilled acetyl chloride in 60 ml. of 5:l benzene-petroleum ether for one hour. The resulting crystals were recrystallized from 2:1 methylene chloride-petroleum ether (10 ml.) and pumped at 0.1 mm. for 12 hours to yield pale violet crystals, m.r. 197-198' (cor.). The purity, estimated by acidimetric
to
(8) Compounds were analyzcd by Dr. Carol H. Fitz. (9) W. Schlenk, Ann., 868, 301 (1909). (IO) P. Adam, Ann. chim. phys.. [a], 16, 200 (1888). (11) W. E. Bschmann and F. Y. Wiaelogle, J . Org. Chem., 1, 371 (1936).
analysis of the hydrolysate, was 99.9 ='c 0.2%. ysiss for C was 87.5%, H 5.5%.
The anal-
Data Conductance data from typical individual runs are recorded in Table I. Values of Ao and Kexp were calculated from the data of a t least three runs for each compound a t each temperature except for the mono-m-t-butyl and mono-m-phenyl derivatives a t -8.9', for which the data of two runs were employed. The average scatter of experimental points about the smoothed plot of A us. V was found to be of the order of j = l % (of A) or less for each compound in the dilution range employed in the Shedlovsky calculations. Values of K,,, and A0 were calculated by applying Shedlovsky's p r o c e d ~ r e ' ~to~ individual '~ data points in the dilution range 2000-80,000 liters per mole (cf. the preceding paper3 for the basis for choice of this range). Values of the dielectric constant and viscosity were calculated by means of eqs. 1 and 2 of the preceding paper.a The leastsquare procedure was employed to determine the slope [1/K(A0)2]and intercept (l/Ao) of the Shedlovsky line. New values of K e x pand A. were also calculated in exactly the same way from earlier datazaOb so that more accurate comparisons of structural and temperature effects could be made. Table I1 summarizes these quantities. Data, not reported here, also were collected a t -8.9' for the di-p-t-butyl and tri-p-phenyl derivatives of trityl chloride. In these cases the percentage change in the Shedlovsky ordinate [l/AS(z)] over the range of abscissa (As(z)f;/V) corresponding to the selected dilution limits was too small (relative to the precision of the data) to permit accurate determination of the slope of the Shedlovsky line. Thus, in addition to the fundamental ~ n c e r t a i n t y 'in~ volved in application of the Shedlovsky procedure to solutes that are almost completely dissociated, an upper limit to the values of Kexpwhich can be determined with some degree of accuracy is imposed by the precision of the data. This difficulty must also be recognized in the case of the above two compounds at 0" where the data yield an approximately 5% change in Shedlovsky ordinate over the selected range. Also assembled in Table I1 are values of AFoexp calculated from the relationship AF'exp = -RT In Kexpand of AHoexpcalculated by means of eq. 3 on the assumption that AH'exp is temperature independent over the range employed. The data for trityl chloride are, however, contrary to this AHoeXp= R In (K/Kf)/[(l/Tf) - (1/T)I
(3)
assumption. Although the data are too few to permit accurate analysis, it is possible to state AH'exp as a function of temperature for trityl chloride by establishing an empirical expression relating R In K to temperature and differentiating this expression with respect to T-l. This process yields A H o = 1.178 X lo5 - 3.372 X lo7 T-1. Substitution of appropriate values of T suggests that over the range +O.l to -17" AH" varies linearly with T. It is thus possible to use the rela(12) T. Shedlovsky, J . Frank. Inst., IB6, 739 (1938). (19) H.M.Daggett, J . Am. Chcm. SOC.,18, 4977 (1951). 68, , 673 (1954). (14) C. A. Kraus, T H ~JSO V ~ N A L
166
NORMAN N. LICHTINAND HARRY P.LEFTIN
Vol. 60
to all runs with the indicated solute and temperature. All TABLE I conductances are corrected for solvent conductivity which CONDUCTANCE DATA FOR TYPICAL SINOIX R T T N S " ~ ~ fell in the range 0.7 - 2.0 X 10-7 mhos cm.-'. b Solutes V A I' il are indkated in terms of the substituents introduced int,o triphenylchloromethane. Thus, di-m-phenyl indicat,es Unsubstituted Mono-p-&Butyl c1 -8.93 f 0.03' -8.85 f 0.07" Q / l 260.2 90.8 585.8 40.5 600.2 109 C '& 1333 55.5 1385 127 3042 74.9 3194 140 6942 97.8 7370 156 CeHs' 15860 122 17020 164 tionship of Douglas and Crockford16 (eq. 4) to 36200 144 39300 168 determine the temperature for which the value 83750 161 90928 173 arrived at by means of eq. 3 is the correct value Mono-m-t-but yl Mono-p-phen yl of AH'. For the interval + O . l to -8.9" this is -8.77 f 0.02" -8.93 f 0.03" -5.2". If it is assumed that for all compounds of 526.9 68.1 631.9 78.6 TO= ln(T2/Td/Kl/Td - (1/T2)l (4) 1223 88.7 1448 99.4 this series AH" varies linearly with temperature 2841 110 3303 120 over the range +O.l to -8.9, then it follows that 6600 132 7557 138 the tabulated values all correspond to this tem15340 148.5 17300 151 perature and are comparable. Additional data are, 35650 157 39600 156 of course, needed to test this assumption. Values 82860 164 90760 161 of AS" have not been tabulated since the values of Mono-m-penyl Di-p-p henyl AH" and AFO correspond to different temperatures. -8.93 f 0.03' -8.90 i 0.01" Discussion 433.5 27.1 994.7 38.4 873 * 7 114 If the significance ascribed to values of A H O e x p 2288 53.5 1991 131 is justified, these quantities can be compared with 5265 72.5 4543 145 the predictions of theory. Denison and Ramsey's 12110 94.7 10363 154 semi-empiricalS~16 equation (eq. 5, L = -8 In D/ 27850 118 23671 160 dT = 6.676 X lo+), which yields a trend in AHo 64070 138 54061 161.5 AH'DR = AFoexP(l- LT) (5) Di-m-p henyl Tri-m-phenyl similar to that obtained experimentally for equilib-8.93 f 0.03' -8.93 f 0.03' ria involving electrostatic association of spherical ions in sulfur dioxide obviously fails for the present 194.9 14.9 1032 23.2 series of compounds since it cannot accommodate 444.0 21.2 2355 32.8 1012 30.0 the increase in exothermicity of AHoexp associated 5374 45.7 2298 41.8 with decrease in endothermicity of A F O e x p observed 12270 62.4 5223 57.1 for compounds 1 through 6. It is equally clear 28000 81.7 that there is no relationship between AHoerpand 11910 76.4 63870 102 ionic dimensions as would be expected from a Bjer27200 98.4 145800 119 rum type of expression for AHo (eq. 8). 62230 119 141700 141 The observed variation of A H O e x p with A F O e x p is consistent, however, with eq. 2 and the assumpDi-p-pheny 1 Di-m-phenyl tions upon which it is founded. Equation 6 is 0.12 f 0.03" 0.10 i 0.01" readily derived from eq. 2. At the limit of large 434.2 96.6 309.5 15.2 K1 the second term of eq. 6 vanishes and AHoexp 711.2 21.7 978.4 118.5 becomes identical with AHoz. 1631 30.9 2206 138 AH"exp= AH'z f IR/(1 1/K1)1[b(l/Kl)/b(l/T)I (6) 3742 43.4 4974 153 11210 164 8588 59.7 At the limit of small K 1the second term of eq. 6 19680 79.2 25300 171 becomes equal to [R/(l/Kl) ][~3(l/K1)/d(l/T>I 45110 100 57060 176 and AHoexpequals AHoz AHol. Meta- and 103300 120 para-substituents which influence K1 through electronic effects acting on the ease of ionization of the Tri-p-phenyl Trim-phenyl carbon to chlorine bond would be expected t o 0.ld i 0.01' 0.12 f 0.03" affect AHo, and AFol similarly." A qualitative 823.3 132.5 690.0 16.3 parallelism of this sort between A H O e x p and AFOexp 1871 148 1600 23.7 is apparent in the data of Table I1 for compounds 3706 33.9 4254 159 1 through 6. This suggests that, for these solutes, 9668 166 8583 47.7 K , is sufficiently small that AHoexp reflects changes 21980 169 19880 65.2 (15) T.B. Douglaa and H. D.Crookford, J . Am. Chem. SOC.,67, 97 50013 170 46040 86.1 (1935). 106700 106 (16) J. T. Denison and J. B. Ramsey, ibid., 77, 2615 (1955). a T7 is in liters/mole, A in mhos cm.*/mole. Tempera(17) L. P. Hammett, "Physical Organic Chemistry." McGraw-Hill Book CO., New York, N. Y;, 1940,p i . 70-78, 123, -184-194. tures ("c.) and their average mean deviations correspond
L A - 0
0
+
+
ALKYLAND ARYLDERIVATIVES OF TRIPHENYLCHLOROMETHANE 167
Feb., 1956
TABLE I1 QUANTITIES DERIVEDFROM SHEDLOVSKY TREATMENT Solutea
T,"C.
'io, mhos cm.e/mole
106 K ~ F D , moles/liter
AF'expb
Scatter about Shedlovsky line,c %
AH'expb
kcal./mole
1 Tri-m-C&
-8.93 157 +5*70 -5.62 11.40 1.94 1.08 1.36 f6.09 Tri-m-CaHI +0.12 159 2 Di-m-C& -8.93 162 +5'44 -5.80 1.14 3.10 0.77 163 2.22 $5.82 Di-m-C& $0, 12d 3 Monom-C&6 -8.93 177 4.63 +5'24 -6.54 1.22 Mono-m-CBH6 $0. 12 184 3.07 +5.64 1.30 4 Unsubstd. 17# 168 13.0 +4.56 -11.2 2.14 Unsubstd. -8.93 188 6.69 +5.05 -7.53 1.24 Unsubstd. 0.00" 207 4.15 f5.48 1.41 5 Mono-m-methyl 0.00" 20 1 9.53 f5.02 0.85 6 Mono-m-t-butyl -8.77 173 32.3 +4.23 -10.2 0.94 Mono-m-t-butyl 0 * 00" 200 17.3 +4.73 0.78 7 Mono-p-C& -8.93 172 42.4 +4.08 - 9.02 0.55 Mono-p-C& $0. 12d 189 24.1 $4.52 0.74 8 Mono-p-t-butyl -8.85 177 101 f3'63 -5.58 0.64 0.43 Mono-p-t-butyl 0. ooe 192 72 +3.93 1.05 9 Mono-p-methyl 0. ooe 196 77 +3.89 .50 10 Di-p-C&, -8.90 168 147 +3.42 181 105 +3,72 - ' 40 .74 $0.10 Di-p-CsHb 177 310 +3.13 .66 +o. 10 11 Tri-p-C& 191 330 +3.10 1.33 12 Di-pt-butyl 0. ooe b Calculated from the relationship, 0 Solute8 are indicated as derivatives of triphenylchloromethane a8 in Table I. I n terms of the ordinate, l/AS(z). Cf. ref. 2b for conductance data. * Cf.ref. 2a A P , , , = -RT In K,,,, and eq. 3. for conductance data.
-
in both AHol and AHoz. For the remainder of the series (compounds 6-12), insofar as they are known, AHoexpand AFoexp vary in opposite directions. Here, the contribution of AHol is presumequals ably diminishing and the limit where AHoexp AHozis approached. Confirmation of these conclusions requires a knowledge of values of Kl and K z and the corresponding thermodynamic quantities. Spectroscopic measurements with solutions of triphenylchloromethane in sulfur dioxide solution have indicatedl* that, for this solute a t 1 f 0.3",K L 3 3 X Lacking further experimental information, we are limited to estimation of these quantities on the basis of theory.lQ The results3 of the investigation of KCl, KBr, K I and (CH3)4NBr enable a small amount of progress to be made in this direction. Since Bjerrum's equation, with a taken equal to the sum of ionic radii has been shown to yield unusually good agreement with values of Kexpfor these spherical ions in sulfur dioxide sohtion, it is possible to predict ion pair dissociation (18) P. D. Bartlett and R. E. weston, Jr., ONR Technical Report No. 6 under Project No. NR-056-095, Contract No. N5 ori-76, Task XX, April 10, 1952. (19) Although the analysis represented by eq. 2 is capable of accommodating available information, i t is recognized that it does not constitute a unique interpretation. The data are, for instance, also consistent with the notion t h a t the ions of each solute a t a given temperature associate in only one way. The species SO formed would, for the weakest electrolytes, closely approximate a normal covalent molecule, for the strongest a Bjerrum ion-pair. For intermediate cases the carbon-chlorine bond would have varying degrees of ionic character in excess of t h a t characteristic of the normal amount for this bond. This hypothesis is, in principle, experimentally distinguishable from that of eq. 2, but not via conductance data. Equation 2, however, permits quantitative, theoretical speculation which is not possible with the alternative model. This, to be sure, constitutes a hazard as well as an advantage for eq. 2.
constants for such solutes with considerable confidence. A triphenylcarbonium ion, however, does not have spherical symmetry and information on the relationship between Bjerrum's equation and the behavior of analogous non-spherical ions in this medium is not yet available. A speculative approach to this question can be made by postulating a basis for selecting equivalent Bjerrum radii for these cations. For this purpose, it is proposed that these cations occupy the spherical spatial region defined by the largest van der Waals radius about the center of gravity of the ion.20 Approximate estimates of these radii can be made conveniently with the aid of Fisher-Hirschfelder-Taylor models. These are listed in Table I11 along with corresponding values of K B j e r r u m a t 0" and AHOBjerrum a t -5.2" calculated by means of eqs. 7 and 8.a.z1 Values of K1 calculated from eq. 2 by identifying Kz with these KBjerrum values are also tabulated. These quantities prove to be consistent with equation 2 and the experimental data. Thus, K B j e r r u m values are all greater than the corresponding values of K e x p , the difference becoming small for the K - ' B ~ ~=, ~(4,N/loOO)(~~/okT)"(b) ,~ = R(1 - LT)[3 eb/Q(b)b31
AH'Bjerrum
+
(7)
(8)
stronger electrolytes for which it has been suggested that Kexp approaches Kz. I n addition, AHo~jerrum varies but little for this series and takes values less exothermic than AHo,,, for the strongest electrolyte. The fact that values of Kexpfor trityl chloride (20) This suggestion differs from t h a t offered some time ago.28 The earlier proposal is not yet subject to quantitative elaboration. (21) Cf.H. 8. Harned and B. B. Owen, "Pli~sicalChemistry of Eloctrolytic Solutions," Reinhold Publ. Corp., New York, N. Y., 1950, pp. XIII-XXVII, for meaning of symbols
NORMAN N. LICHTINAND HARRYP. LEFTIN
168
TABLE I11 THEORETICAL EQUILIBRIUM QUANTITIES 74
3
I O * X L I ~10aXlbJ . ~ ~ ~ ~ AH ~ o~isrrum at Oo at -5.2a moles/l. kcal./mo
