Calculations on Ionic Solvation Part 2.-Entropies of Solvation of Gaseous Univalent Ions using a One-layer Continuum Model BY MICHAEL H. ABRAHAM" Department of Chemistry, University of Surrey, Guildford, Surrey JANOS LISZI Department of Physical Chemistry, University of VeszprCm, 8201 Veszprkm, Hungary AND Received 17th February, 1978 The electrostatic entropy of solvation of gaseous ions has been calculated using our previous model in which the ion is surrounded by a local solvent layer, immersed in the bulk solvent. The calculate6 electrostatic entropy is combined with the nonelectrostatic entropy of solvation, obtained from experimental data on entropies of solution of gaseous nonpolar solutes, to yield the total entropy of solvation of a gaseous ion. The only new parameters involved in the calculations are the variations with temperature of the solvent bulk dielectric constant (a known property) and the dielectric constant in the local solvent layer.We found that if the latter parameter is taken as - 0.001 60 (a reasonable value for a region of low dielectric constant), there is excellent agreement with experiment for entropies of solvation of univalent cations and anions in a wide variety of aprotic solvents, and for entropies of transfer of these ions between aprotic solvents. Since no adjustable parameters are used in the calculations, the method can be used to predict entropies of solvation or of transfer in aprotic solvents. Agreement with experiment is no+ 'xnd for solvation entropies of ions in hydrogen bonded solvents.We have reported previously results of calculations on the free energy of solvation of gaseous univalent ions, using a continuum model in which an ion of radius a and dielectric constant ci is surrounded by a local solvent layer of thickness (b-a) and dielectric constant For all ions in all solvents we took a as the ionic crystal radius, (b -a) as the solvent radius, E~ = 1, c1 = 2 (since for many solvents the value of E: is approximately 2) and c0 as the bulk dielectric constant. We then calculated the electrostatic free energy of solvation by the method of Beveridge and Schnuelle and combined this with the nonelectrostatic free energy, obtained from experimental data on nonpolar solutes, to obtain the total free energy of solvation via eqn (1) Agreement between calculated and observed free energies of solvation was so good as to encourage us to carry out similar calculations on ionic entropies of solvation.Our aim was first to see if the simple one-layer model could be used to reproduce known entropies of solvation of cations and anions in a variety of solvents; several workers have suggested that for ions in water and highly structured solvents a second, disorganised, solvent layer will make an additional contribution to the total entropy. Secondly, we hoped to be able to predict entropies of solvation and immersed in the bulk solvent of dielectric constant c0. AG," = AG,"+AG,". (1) 2858M. H . ABRAHAM AND J. LISZI 2859 entropies of transfer of ions, especially for the less polar solvents where there are very great experimental difficulties in obtaining such data.Few calculations on entropies of solvation of ions in nonaqueous solvents have been reported. Although Eley and Evans successfully calculated entropies of solvation of ions in water using a discontinuous method, a similar calculation failed completely to account for ionic entropies of solvation in methanol. More recently, de Ligny et aL7 have carried out calculations on the division of entropies of transfer of electrolytes into single ion contributions, but to date calculations on actual entropies of solvation in nonaqueous solvents have met with but little success. THEORY OF THE METHOD In terms of the one-layer model, the electrostatic part of the ionic solvation equation with entropy is obtained by differentiating Beveridge and Schnuelle's respect to temperature.Taking only the relevant first term, we derive eqn (2) in which 2 is the charge on the ion (taken as unity in all the present calculations), 6c1/6T is the variation of local dielectric constant with temperature, and 6co/6T is the like variation of the bulk dielectric constant. We retain exactly the same values of a, b, c1 = 2 and c0 as used in our free energy calculations,' so that the only para- meters left are 6c1/6Tand 6c0/6T. The latter is a property of the bulk solvent and the values we have used are given in table 1. Thus the only variable parameter that remains is 6cl/6T, and even this is constrained because for all solvents we have fixed cl = 2. For many solvents with c0 N 2, values of 6c0/6T are about -0.001 60; furthermore, although data are rather limited, it seems as though for most aprotic solvents values of 6&2/ST also approach the same value.We, therefore, set for all solvents Sc,/6T = -0.801 60, and thus retain no variable parameter at all. This procedure has the advantage of rendering the method entirely predictive; the dis- advantage is that the method may well not apply to hydrogen bonded solvents where ~EJBT is usually numerically smaller than 0.001 60. We now combine the calculated AS," values with a nonelectrostatic contribution, AS,", obtained from experimental data *-12 on the entropy of solution of gaseous nonpolar solutes (table 2), via eqn (3) AS: = AS,O+AS,O. (3) As with the corresponding free energies of solvation, this procedure removes all difficulties over standard states.There is some uncertainty in the experimental values given in table 2, and within this uncertainty, values for solution of nonpolar solutes in many nonaqueous solvents can be fitted to the linear eqn (4)" AS: = mr+c (4) where nz = 6.96, Y = solute radius in A, and c = 1.2 cal K-l mol-l, (see table 3). For solvents other than water, the AS," values are quite small (e.g., -8.5 cal K-I mol-1 for Naf and - 16.5 cal K-l mol-' for I- in many solvents)? and to a large * Data we have used to calculate m and c in eqn (4) cover only the range r = 1.29 (He) to 2.03 (Xe). Recent work l4 on the entropy of solution of tetramethyltin, with r = 3.07, indicates that for many nonaqueous solvents eqn (4) does in fact hold up to at least r = 3.07 A.t 1 cal = 4.184 J.2860 CALCULATIONS ON IONIC SOLVATION extent reflect the change in standard state from 1 atm (gas) to unit mol fraction (soln) ; for most nonaqueous solvents such a change corresponds to an entropic contribution of about - 12,cal K-l mol-l. For water,12 the AS," values are much more negative. TABLE 1.-vALUES OF -6&,/6T USED IN THE CALCULATIONS solvent - 68016T water methanol ethanol 1-propanol formamide N-methylformamide N,N-dime thy1 formamide dimethylsulphoxide acetoni trile nitromethane acetone nitro benzene ammonia 1 ,Zdichloroethane 1 , 1 -dichloroethane tetrahydro furan 1 ,Zdimethoxyethane ethyl acetate chlorobenzene bromo benzene pentyl acetate diethyl ether di-isopropyl ether butyl stearate di-isopen t yl ether benzene cyclohexane 0.3595 * 0.197 b* C 0.147 0.142 0.72 (0.40) 1.62 0.178 d 0.106 (0.126)f 0.160 C 0.161 0.0967 (0.0977) C 0.180 0.0780 g 0.0560 C 0.0480 0.0299 (0.0298) 1 0.0410 0.015 0.0168 C 0.0143 0.012 0.020 0.018 0.0053 C 0.0050 0.001 99 0.001 5 5 ' a R.L. Kay, G. A. Vidulich and K. S. Pribadi, J. Phys. Chem., 1969,73,445 ; B. B. Owen, R. C. Miller, C. E. Milner and H. L. Cogan, J. Phys. Chem., 1961, 65, 2065. b From data quoted in Landolt-Bornstein, Zaklenwerte und Funktionen, Sechste Auflage, Band 11, Teil6, and by J. Timmer- mans, Physico-chemical Constants of Pure Organic Compounds (Elsevier, Amsterdam, 1950 and 1965). C A. A. Maryott and E. R. Smith, Table of Dielectric Constants ofPure Liquids, NBS Circular 514, Washington D.C., 1951. d G. R. Leader and J.F. Gormley, J. Amer. Chem. Soc., 1951, 73, 5731. e R. Garney and J. E. Prue, Trans. Faradar Soc., 1968, 64, 1206. f J. F. Casteel and P. G. Sears, J. Chem. Eng. Data, 1974,19,196. 0 H. M. Grubb, J. F. Chittum and H. Hunt, J. Amer. Chem. Soc., 1936,58,776. h J. T. Denison and J. B. Ramsey, J. Amer. Chem. Soc., 1955,77,2615. i C. Carvajal, K. J. Tolle, J. Smid and M. Szwarc, J. Arner. Chem. Soc., 1965,87,5548. f D. J. Metz and A. Glines, J. Phys. Chem., 1967, 71, 1158. k V. Viti and P. Zampetti, Chem. Phys., 1973, 2, 233. * R. H . Stokes, J. Chem. Thermodynamics, 1973,5, 379. Experimental ionic solvation entropies can be obtained from known entropies of hydration,13 together with ionic entropies of transfer from water to the given solvents. In order that the same standard states apply to both nonpolar solutes and ions, we recalculated the entropies of hydration tabulated by Noyes l3 to standard states of 1 atm (gas) and unit mol fraction (as).There is still the problem that the assignment of cationic and anionic contributions carried out by Noyes l3 may not be appropriate or consistent with the constants we use in the present calculations. We find, however,M. H. ABRAHAM AND J . LISZI 2861 that it comparatively small adjustment (by 5 cal K-l mol-l)* to Noyes’ cationic and anionic values yields single ion entropies of hydration that are perfectly compatible with the present calculations for all solvents, provided that single ion entropies of transfer are assigned by the correspondence plot method. We used ionic entropies of transfer, given by Abraham and c o - ~ o r k e r s , ~ ~ - ~ ~ that are based on the mol fraction TABLE 2.-sTANDARD ENTROPIES OF SOLUTION OF GASES (1 atm GAS AND UNIT MOL FRACTION SOLUTION) IN cal K-l mol-1 AT 298 K a gas : solvent hexane cyclohexane benzene toluene iodobenzene bromobenzene chloro benzene nit robenzene acetone average value calc.from eqn (4) b N-met hylacetamide isobutanol ethanol methanol water -AS,” He Ne Ar Kr Xe 10.0 9.9 10.6 10.0 9.4 10.4 12.3 9.4 8.6 10.1 10.2 11.2 12.3 8.3 11.1 11.1 12.1 10.6 8.5 9.9 10.6 10.9 14.1 13.7 13.0 12.7 12.9 13.7 13.7 13.4 13.3 13.4 13.1 13.6 13.4 13.3 14.3 14.1 14.6 14.2 13.3 13.9 13.8 13.7 12.4 15.0 14.3 14.1 15.4 16.8 13.0 13.4 15.5 16.3 14.6 14.9 17.3 17.4 24.1 26.3 30.9 32.3 a From data in ref.(8)-(12) ; b -AS,” = 6.96r+ 1.2. 14.5 15.0 15.0 15.1 15.7 14.3 14.9 15.3 34.2 TABLE 3.-cONSTANTS IN THE EQUATION - AS,”(cal K-l mol-l) = mr+ c solvent m C aprotic solvents in table 2 6.96 1.2 N-met h ylace t amide 6.96 3.2 isobutanol 6.96 4.0 ethanol 6.96 4.0 methanol 6.96 5.3 water b 10.51 12.84 a With AS: in cal K-I mol-l and Y in A. b Using data for Ar, Kr, Xe, Ch, C&, C3Ha, n-C4H1,, and neo-CsH12. standard state, separated into cationic and anionic contributions by the correspon- dence plot method, and smoothed out by the method developed by Abraham.15 Details of the final solvation entropies are in table 4. Absolute solvation entropies of ions have also been tabulated by Criss and Salomon.l * Taking into account the change in standard state, there is quite good agreement with the values we give in * This adjustment leads to ASg(H+) = -42.3 cal K-I mol-1 with our standard states.Given that F(H+, gas, 1 atm) = 26.0 this yields for $(H+, aq) a value of - 16.3 on the mol fraction scale or - 8.3 cal K-l mol-’ on the usual molal scale. The latter value is quite close to suggested ‘‘ abso- lute ” values on the molal scale ( - 5 to -6 cal K-l mol-1).18 Noyes original single ion division corresponds to a value of -3.3 cal K-l mol-l for the “absolute ” value of*(€€+, aq, I mol kg-l).2862 CALCULATIONS ON IONIC SOLVATION table 4. We also give in table 4 ionic solvation entropies obtained by Strong and Tuttle for the solvents tetrahydrofuran (THF) and 1,2-dimethoxyethane (1'2- DME). Unfortunately there seem to be rather large uncertainties in the data that preclude any exact application of the correspondence plot method.Na+ K+ Rb+ cs+ Me4Nf Et4N+ c1- Br- I- c l o y TABLE 4.-ENTROPIES OF SOLVATION OF IONS (1 atm GAS + UNIT MOL FRACTION SOLUTION) IN Cal K-' M01-l AT 298 K waterb MeOHc EtOHC h"OHC F NMF DMSO DMF MeCN -36.8 -47 -50 -54 -42 -45 -50 -57 -55 -28.7 -42 -45 -49 -37 -40 -45 -52 -50 -25.9 -38 -41 -45 -33 -36 -41 -48 -46 -25.2 -36 -39 -43 -31 -334 -39 -46 -44 -39.8 -35 -38 -42 -30 -33 -338 -45 -43 -59.1 -43 -42 -46 -34 -37 -42 -49 -47 -23.3 -33 -36 -4-0 -28 -31 -36 -43 -41 -19.3 -31 -34 -38 -26 -29 -34 -41 -39 -14.3 -27 -30 -34 -22 -25 -30 -37 -35 -13.0 -26 -29 -33 -21 -24 -29 -36 -34 acetone - 60 - 55 - 51 - 49 - 48 - 52 -46 -44 - 40 - 39 ammonia 1,2-DCE 1,l-DCE d THF * 1,2-DME Na+ - 60 - 69 - 70 - 82 - 58 K+ - 55 -64 - 65 - 67 - 74 Rb+ - 51 -60 - 61 -70 -104 cs+ - 49 - 58 - 59 - 89 - 88 Me4N+ - 48 - 57 - 58 Et,N+ - 52 - 61 - 62 c1- - 46 - 55 - 56 - 63 - 73 BI- -44 - 53 - 54 I- -40 - 49 - 50 ClO; - 39 - 48 - 49 Obtained from the values for water, together with the smoothed entropies of transfer from the correspondence plot method in ref.(15). b From ref. (13) adjusted as stated in the text. C Ref. (17). d Ref. (16). e From ref. (19) ; the cationic and anoinic contributions have been roughly assigned by the correspondence plot method. There is also a difficulty over AS," for Me4N+ and Et4N+. Quite discordant values have been calculated for S0(Me4N+, gas) : 73.7 cal K-l rnol-1 by Boyd 2o and 58.2 by Ladd.21 Combination with Johnson and Martin's value 22 for S"(Me,N+, aq, 1 mol kg-l) yields on our scale values for ASi(Me4N+) of -39.8 (Boyd) or -24.1 (Ladd).We chose the value based on Boyd's calculation on the grounds that Ladd's value for So(Me4N+, gas) seems too low by comparison with that for neo- pentane (73.2 cal K-l r n ~ l - l ) . ~ ~ For Et4N+ we used data by Boyd 2o and Johnson and Martin,22 but there is no independent value to confirm these calculations. RESULTS AND DISCUSSION We give in table 5 details of the calculations for 1,l-dichloroethane (1'1-DCE). Agreement between calculated and observed AS," values is in general excellent, except for Me,N+ and Et4N+ where the calculated values are too positive by 10 and 15 cal K-l mol-1 respectively. These differences are not special to 1,l -dichloroethane, and we find for a11 the solvents studied that on average the calculated AS," values areM. H.ABRAHAM AND J . LISZI 2863 too positive by 8 and 12 cal K-I mol-l. This cannot be due to incorrect values of the solute radii, a, because wide variations in a still do not yield agreement. We think it possible (see discussion above) that the observed values of AS,” are in error ; this would then result in exactly the same error in AS,” for all the solvents studied. In the event we have incorporated in all the calculated AS,” values, extra entropic contributions of - 8 and - 12 cal K-l mol-1 for Me4N+ and Et4N+, respectively. TABLE 5.-RESULTS FOR SOLVENT. 1,I-DICHLOROETHANE IN Cal K-’ mOl-’ AT 298 K ion ASZ(ca1c) AS: a ASi(ca1c) ASi(obs) Na+ - 67.3 - 8.5 - 75.8 - 70 K+ - 53.7 - 10.5 - 64.2 - 65 Rb+ -50.1 -11.1 - 61.2 - 61 a+ -43.5 - 12.7 - 56.2 - 59 Me4N+ -28.6 - 19.1 - 47.7 (- 55.7) - 58 Et4N+ - 24.0 - 22.8 -46.8 (-58.8) - 62 c1- - 40.1 - 13.8 - 53.9 Br- - 37.3 - 14.8 - 52.1 I- - 33.3 - 16.5 -49.8 c104 - 30.1 - 18.2 -48.3 - 56 - 54 - 50 - 49 Q -AS: = 6.96r-1- 1.2.b Extra entropic contributions included (see text). TABLE 6.xALCULATED AND OBSERVED VALUES OF IONIC SOLVATION ENTROPES, As,” IN i on Na+ Kf Rb+ cs+ Me4N+ EtdN+ c1- Br- I- Clog ion Na+ K’ Rbf Csf Me4N+ Et,N’ c1- Br- I- ClO, d K-’ rno1-l AT 298 K 1, 1-DCE 1,2-DCE THF 1,2-DME ammonia calc. obs. calc. obs. calc. obs. calc. obs. calc. obs. -75.8 -70 -78.0 -69 -78.4 -82 -87.7 -58 -64.3 -60 -64.2 -65 -66.2 -64 -66.6 -67 -75.3 -74 -53.7 -55 -61.2 -61 -63.2 -60 -63.6 -70 -72.1 -104 -51.0 -51 -56.2 -59 -58.0 -58 -58.4 -89 -66.5 -88 -46.6 -49 -55.7 -58 -57.2 -57 -57.5 - 64.3 -48.0 -48 -58.8 -62 -60.2 -61 -60.5 - 66.6 -51.9 -52 -53.9 -56 -55.6 -55 -56.0 -63 -63.8 -73 - 4 .6 -46 -52.1 -54 -53.8 -53 -54.2 - 61.8 -43.2 -44 -49.8 -50 -51.4 -49 -51.8 - 59.0 -41.4 -40 -48.3 -49 -49.8 -48 -50.1 - 57.0 -40.3 -39 acetone acetonitrile DMF 2MSO calc. obs. calc. obs. calc. obs. calc. obs. -63.8 -60 -57.7 -55 -59.3 -57 -55.1 -50 -53.0 -55 -47.5 -50 -48.9 -52 -45.0 -45 -50.3 -51 -45.0 -46 -46.3 -48 -42.5 -41 -45.9 -49 -40.9 -44 -42.1 -46 -38.5 -39 -47.3 -48 -43.3 -43 -44.1 -45 -41.3 -38 -51.2 -52 -47.6 -47 -48.3 -49 -45.7 -42 -43.9 -46 -39.1 -41 -40.2 -43 -36.8 -36 -42.5 -44 -37.9 -39 -38.9 -41 -35.6 -34 -40.7 -40 -36.3 -35 -37.3 -37 -34.2 -30 -39.6 -39 -35.5 -34 -36.4 -36 -33.4 -29 a Calculated values using -AS: = 6.96rf 1.2.All values for Me4N+ and Et4N+ include the extra entropic contributions (see text). Observed values from table 4. b If - 8eo/8T is taken as 0.126 (see table l), the calculated values would be more negative by only about 0.3 cal K-’ mol-‘.2864 CALCULATIONS O N IONIC SOLVATION In table 6 are summarised results of our calculations on the non-hydrogen bonded solvents for which data are available. As before,l we define two parameters as measures of agreement between calculated and observed AS," values : the standard deviation 0 = { lAS,"(calc) - AS:(obs)J2/(n - 1)}&, and a measure of systematic devia- tion = AS,"(calc, average) -AS,"(obs, average).For the solvents l,l-DCE, 172-DCE, ammonia, acetone, acetonitrile, dimethylformamide (DMF) and dimethyl- sulphoxide (DMSO) there is good agreement between calculated and observed values, with CJ averaging about 2 cal K-l mol-1 and 5 averaging numerically only about 1 cal K-1 mo1-I for the 7 solvents (see table 7). These deviations are of the order of experimental errors in AS," values, and it seems therefore that we have achieved one of our main aims, namely to reproduce observed entropies of solvation, especially in the less polar solvents, by a method that is predictive in nature. The two aprotic solvents for which there is only poor agreement are THF and DME (table 7), but it TABLE VA VALUES OF 5 AND Q JN cal K-' rnoP solvent no. of ions 5 d 1, 1-dichloroethane 1 , 2-dichloroethane tetrahydrofuran 1,2-dirnethoxyethane ammonia acetone acetoni trile dimet hyl formamide dimethylsulphoxide 10 10 5 5 10 30 10 10 10 0.8 9.6 6.3 -0.1 0.6 0.3 1.3 - 1.9 - 2.4 2.7 3.4 16.1 24.7 1.9 2.0 1.9 2.2 3.2 N-methylformamide 10 - 8.9 9.6 formamide 10 - 13.5 14.4 1 -propano1 C 10 - 12.4 13.2 ethanol 10 - 12.5 13.3 methanol d 10 - 13.4 14.3 water 10 - 26.8 29.6 0 -AS: = 6.96~+3.2.b -AS: = 6.96r+5.3 ; if - Sq,/ST was taken as 0.40 (see table l), 5 would be about - 12.5 cal K-' mol-'. C -AS: = 6.96~+4.0. d -AS: = 6.96rf5.3. AS: ob- tained from a plot of AS: against Y (inert gas) for Na+, K+, Rbf and Cs+ and from -AS: = 10.51r + 12.84 for the larger ions. must be borne in mind that for these rather nonpolar solvents there is considerable technical difficulty in obtaining experimental values and that this is reflected in possible errors.Attempts to apply the correspondence plot method to experimental values for THF and DME suggests that they are self-consistent to only about 10 cal K-I mol-l (THF) and 20 cal K-I mol-1 (DME). Strong and Tuttle I9 themselves estimate that errors in enthalpies of solution could be " several kcal mol-I ", so that it is not surprising that there is only poor agreement between calculated and observed entropy values. Indeed, we might claim that for solvents as nonpolar as THF and DME, with dielectric constants 7.4 and 7.3, our method of calculation yields entropies of solvation that are more reliable than those that can be obtained by present experi- mental methods. Since values of 5 and CT (table 7) are so small for the solvents that are not hydrogen bonded, it seemed that we might also achieve the aim of calculation of reliable ionic entropies of transfer.Details are in table 8 for the typical case of (Rbf and Br-), with acetonitrile taken as an arbitrary reference solvent. Agreement with observedM, H. ABRAHAM AND J . LISZI 2865 values is quite remarkable, and the present method clearly can be used for the calcula- tion of entropies of transfer between non-hydrogen bonded solvents to within present experimental error (usually about 1-2 cal K-l mol-l). For hydrogen bonded solvents, we found substantial differences between calcu- lated and observed AS," values. We do not give details for these solvents, but list in table 7 the [ and 0 values found.Since the arithmetical value of [ is approximately the same as 6, the differences are systematic and not merely random. One cause TABLE CALCULATION OF ENTROPIES OF TRANSFER ON THE MOL FRACTION SCALE, M cal K-l mol-1 AT 298 K solvent Rb+(calc) Br-(cab) Rb+(obs) Br-(obs) a dimethylsulphoxide + 3 +2 +5 + 5 dimethylformamide - 1 - 1 -2 -2 acet onitrile 0 0 0 0 acetone - 5 - 5 - 5 - 5 ammonia -6 -5 - 5 -5 1,2-dichloroethane - 18 - 16 - 14 - 14 1, 1-dichloroethane - 16 - 14 - 15 - 15 a Table 4. of the discrepancies is that for this type of solvent -6~,/6T is probably less than 0.001 60, the value used in all calculations. This cannot be the only cause because choice of other values for 6~,/6T still does not yield agreement. We hope to investi- gate the hydrogen bonded solvents in more detail later.We also thought it useful to investigate for the aprotic solvents the effect of altering the thickness of the local layer, (b -a), whilst keeping a constant value of Q. It is not obvious how changes in b will influence the calculated AS," (and AS,") values, because both terms in eqn (2) include the b parameter. We find that an increase in b TABLE CA EFFECT OF THE THICKNESS OF THE LOCAL LAYER (b-a) ON CALCULATED ENTROPIES AND FREE ENERGIES OF SOLVATION OF Rb+ AT 298 K a solvent (b-a)lA AS,"(calc) AS,"(obs) AGXcalc) AG:(obs) 2.0 4.444 MeCN { 2,222 4.970 C 5.200 -44.1 - 45.0 - 49.7 - 49.3 - 50.3 - 53.1 -61.9 - 61.2 - 59.8 - 73.0 - 71.9 - 75.3 - 46 - 73.9 - 64.3 - 74.9 - 51 - 72.2 - 64.7 -72.1 - 61 - 69.3 - 62.8 - 68.4 I a Free energies (in kcal mol-') calculated as in ref.(1) ; entropies in cal K-l mol-I. In all calculations a = 1.43 A. b Solvent radius. C Solvent diameter. results in a more negative entropy of solvation in the case of polar solvents, but in a more positive calculated entropy of solvation for the less polar solvents ; results for acetonitrile, acetone and 1, 1-DCE are in table 9 where Q has been taken as 1.43 A (Rb+). Small variations in (b-a) produce only minor changes in AS," (calc), so that it requires a quite detailed analysis to show that choice of (b-a) as the solvent radius leads to better agreement with observation than choice of (b-a) = 2 A for all solvents. It does seem, however, that with the values of and ael/aTused in our2866 CALCULATIONS ON IONIC SOLVATION calculation, the choice of solvent radius does lead to slightly better agreement in terms of entropies of solvation.The free energy term is affected more by the change in (b -a) than is the entropy term (table 9), and in this case also the choice of solvent radius leads to better agreement than does that of taking (b -a) = 2 A. We also give in table 9 results of calculation with (b-a) taken as the solvent diameter, but there is now considerable discrepancy between calculated and observed values, especially in the free energy term. Many previous workers have modified the simple Born equation for the eiectro- static free energy or enthalpy of solvation of an ion, by means of empirical adjust- ments, A, to the ionic radii. For cations, these adjustments are often ~ 0 .8 A not only in water 24 but also in propylene carbonate 2 5 and DMF,26 but for anions the adjustments range from 0.1 A in water 24 to 1.00 A in DMF.26 These empirical, non-predictive, modifications to the simple Born equation are in no way equivalent to the present one-layer model. In the modified Born equation a charged region of radius (a+A) and E = 1 is surrounded by the bulk solvent with E = E ~ , whereas in the present model a charged region of radius a and E = 1 is surrounded by the solvent layer of thickness (b -a) and E = 2, followed by the bulk solvent with E = co. Our calculations indicate that this solvent layer must be of about a solvent radius thick; for most solvents this corresponds to 2-3 A. CONCLUSION Ionic entropies of solvation and ionic entropies of transfer can be calculated accurately by use of a simple one-layer model in which the dielectric constant of the solvent layer is taken as = 2 and in which &/8T is taken as -0.001 60, provided that the solvents investigated are non-hydrogen bonded.The method uses no adjustable constants and is predictive in nature. As an example, entropies of solva- TABLE 1o.-PREDICTION OF ENTROPIES OF SOLVATION IN Cal K-' m01-l AT 298 K " solvent Rb+(calc) Br-(calc) Rb+(obs) Br-(obs) dimethylsulphoxide ni tromethane dimet hylformamide acetonitrile nitro benzene acetone ammonia 1 ,2-dichloroethane 1,l-dichloroethane tetrahydro fur an 1 ,Zdimet hoxyet hane ethyl acetate chlorobenzene bromobenzene pentyl acetate diethyl ether di-isopropyl ether butyl stearate di-isopentyl ether benzene cyclohexane - 42 - 45 - 46 - 45 - 48 - 50 - 51 - 63 - 61 - 64 - 72 - 58 - 63 - 61 - 63 - 86 - 87 - 61 - 66 - 57 - 57 - 36 - 38 - 39 - 38 -40 - 43 - 43 - 54 - 52 - 54 - 62 - 49 - 53 - 52 - 54 - 74 - 75 - 52 - 56 - 48 - 48 =Values of AS: taken as -11.1 for Rb+ and -14.8 for Br- -AS: = 6.96r+ 1.2.b Table 4. - 41 - 34 - 48 - 41 - 46 - 39 - 51 -44 - 51 -44 - 60 - 53 - 61 - 54 - - - - in all cases, as calculated fromM . H . ABRAHAM AND J . LISZI 2867 tion for Rbf and Br- are predicted in table 10 for a very wide range of aprotic solvents? The only solvent properties required for such predictions are the solvent density (used to obtain the thickness of the solvent layer) and the bulk properties co and SEO/~T. Even for the aprotic solvents of some degree of solvent structure, such as DMSO and DMF, a one-layer model is sufficient to account for the ionic entropies of solvation.This result is slightly at variance with the suggestions of Parker and co-workers that entropic contributions from a second (disorganised) solvent layer are negligible in the case of acetonitrile (as we would also suggest) but are not negligible for DMSO and DMF. Parker and his coworkers stress, however, that any contributions from this second layer in the case of DMSO and DMF are much less than with highly structured hydrogen bonded solvents. M. H. Abraham and J. Liszi, J.C.S. Furuduy I, 1978,74,1604. D. L. Beveridge and G. W. Schnuelle, J. Phys. Chem., 1975,79,2562. H. S . Frank and M. W. Evans, J. Chem.Phys., 1945,13,507. B. G. Cox, G. R. Hedwig, A. J. Parker and D. W. Watts, Austral. J. Chem., 1974, 27, 477. D. D. Eley and M. G. Evans, Trans. Furaduy SOC., 1938,34,1093. D. D. Eley and D. C. Pepper, Trans. Furuduy SOC., 1941,37,581. ' D. Bax, C. L. de Ligny and M. Alfenaar, Rec. Trav. chim., 1972, 91,452; D. Bax, C. L. de Ligny and A. G. Remijnse, Rec. Truv. chim., 1972,91,965. * H. L. Clever, R. Battino, J. H. Saylor and P. M. Gross, J. Phys. Chem., 1957, 61, 1078; R. Battino, F. D. Evans, W. F. Danforth and E. Wilhelm, J. Chem. Thermodynamics, 1971, 3, 743; J. H. Saylor and R. Battino, J. Phys. Chem., 1958,62,1334. R. H. Wood and D. E. DeLaney, J. Phys. Chem., 1968,72,4651. (University of New Zealand, 1949). mentuls,) 1976, 15, 336. lo A. J. Beckwith, MSc. Thesis (University of New Zealand, 1949) ; J. T. Law, PhD. Thesis l1 C. L. de Ligny, N. G. van der Veen and J. C. van Houwelingen, Ind. and Eng. Chem. (Fundu- l2 E. Wilhelm, R. Battino and R. J. Wilcock, Chem. Rev., 1977,77,219. l3 R. M. Noyes, J. Amer. Chem. SOC., 1962,84, 513. l4 M. H. Abraham and A. Nasehzadeh, unpublished observations. l5 M. H. Abraham, J.C.S. Furuduy I, 1973,69, 1375. l 6 M. H. Abraham, A. F. Danil de Namor and R. A. Schulz, J. Solution Chem., 1976,5, 529. l7 M. H. Abraham and A. F. Danil de Namor, J.C.S. Faraday I, 1978, 74, 2101. l 8 C. M. Criss and M. Salomon, in Physical Chemistry of Organic Solvent Systems, ed. A. K. l9 J. Strong and T. R. Tuttle, J. Phys. Chem., 1973, 77, 533. 2o R. H. Boyd, J. Chem. Phys., 1969,51,1470. '' M . F. C. Ladd, 2. phys. Chem. (N.F.), 1970,72,91. 22 D. A. Johnson and J. F. Martin, J.C.S. DuZton, 1973, 1583. 23 D. R. Stull, E. F. Westrum and G. C. Sinke, Chemical Thermodyriumics oforgatzic Compounds 24 W. M. Latimer, K. S. Pitzer and C. M. Slansky, J. Chem. Phys., 1939,7, 108. 25 Y.-C. Wu and H. L. Friedman, J. Phys. Chem., 1966,70,501. 26 C . M. Criss and E. Luksha, J. Phys. Chem., 1968, 72,2966. * An interesting prediction (see table 10) is that entropies of solvation pass through minimum Covington and T. Dickinson (Plenum Press, London, 1973). (John Wiley, New York, 1969). values ( i z , more negative values) in solvents with E~ N 4. (PAPER 8/280)