J . Chern. Soc., Faraday Trans. 1 , 1987, 83 (lo), 3129-3138 The Ethane- 1,2-dio1-2-Methoxyethanol Solvent System The Dependence of the Dissociation Constant of Picric Acid on the Temperature and Composition of the Solvent Mixture Gian Carlo Franchini, Lorenzo Tassi and Giuseppe Tosi* University of Modena, Department of Chemistry, Via G. Campi 183, 41100 Modena, Italy Dissociation constants of 2,4,6-trinitrophenol (picric acid) in a series of ethane- 1,2-diol-2-methoxyethanol mixtures have been determined by the conductance method in the temperature range from -10 to 80°C. The dissociation constants exhibit different variations with temperature in different solvent systems, but they are well fitted by identical equations of the type In K = a, + a, T+ a,/ T+ a3 In T, from which thermodynamic functions data were also evaluated and discussed.With regard to solvent systems in which a K,,, value is observed [i.e. for the solvent mixtures having X(ethane-1,2-diol) > 0.26771, the correlations between K and T obtained from fitting the equation above by Harned's theory and by a pK us. I/& graph, gave results consistent with each other. As part of systematic studies on the influence of solvents on the dissociation constant of weak electrolyte^,^-^ in a recent study5 we reported the dissociation constant of picric acid in ethane-1,2-diol and 2-methoxyethanol at 19 temperatures between - 10 and 80 O C , evaluated from conductance data. The dependence of K on temperature was different for the two solvents, showing a maximum in the case of ethane-1,2-diol and a continuous decrease in the case of 2- methoxyethanol.Thermodynamic data have also been evaluated, and an explanation of their variations with temperature, which were very different for the two solvents, was suggested in terms of the ability of the ion pairs and/or other species to orient the sol- vent molecules in their immediate neighbourhood and the dependence of the dielectric constant and viscosity on temperature. In order to investigate the influence of structural changes in the solvent systems on the dissociation of weak electrolytes in amphiprotic media, we have planned to extend the study to binary mixtures of ethane- 1,2-diol (E = 41.06 at 25 "C) and 2-methoxyethanol (E = 16.94 at 25 "C), using picric acid as solute. One of the most difficult problems in studies involving non-aqueous solvents is the nature of the ion-solvent interaction ; the inapplicability of the models used to define the phenomena in aqueous solution suggests the use of mixed solvent systems since, for example, these systems make a continuous variation of the dielectric constant possible, and it can be assumed that the behaviour of ions reveal more clearly their unusual properties.Experimental Materials 2,4,6-Trinitrophenol (picric acid), supplied by Fluka (reagent grade) was used without further purification. The solvents ethane- 1,2-diol and 2-methoxyethanol (containing < 0.10 and 0.05 YO water, respectively, found by Karl Fischer titration) were Carlo Erba high-purity grade. 31293130 The Ethane- 1,2-diol-2-Methoxyethanol System Apparatus Conductances of the solutions were measured with an Amel model 123 conductivity bridge operating in the 0.1 x 10-'-0.3 S (scale-end) range, with a sensitivity of f 1 .O %, and using platinized platinum electrodes (cell constant 0.98 cm).The temperature control was provided by a Lauda K2R thermostatic bath maintained to f0.02 "C. Viscosity measurements were performed using a Schott-Gerate AVS 400 viscosity-measuring system, equipped with a series of Ubbelhode viscometers. Densities at the different temperatures were determined with calibrated Hg densimeters (sensibility 0.0005 g ~ m - ~ ) . The dielectric constants were measured using a WTW-DMO 1 dipolometer equipped with two stainless-steel cylindrical cells: MFL2 (7 < E < 21) was calibrated with dichloromethane ( E = 9.08 at 20 "C), pyridine ( E = 12.30 at 25 "C), butan- 1-01 ( E = 17.80 at 20 "C) and acetone (E = 20.70 at 25 "C);' MFL3 (21 < E < 90) was calibrated with ethanol (E = 24.30 at 25 "C), methanol ( E = 32.63 at 25 "C), glycerol ( E = 42.50 at 25 "C) and water ( E = 80.37 at 20 oC).s A frequency of 2.0 MHz was used.Karl Fischer titrations were performed with an automatic titration system (Crison model KF43 1) equipped with a digital burette (Crison model 738). Procedure The solvent mixtures were prepared by weight and all weights were corrected to vacuum. Solutions of picric acid of different concentrations were obtained by successive dilutions of stock solutions, prepared as previously reported. Conductance readings were recorded when they became invariant with time; this took ca.30min for each measurement. Solvent conductance corrections were applied to all the data at different temperatures. Results and Discussion In order to determine the dissociation constants and the A, values for 2,4,6- trinitrophenol in mixed solvents, conductance measurements were performed at temperatures in the range from - 10 to 80 "C for at least six different concentrations for each mixture; the experimental data, corrected at each temperature with the specific conductances of the solvent systems, are reported in six tables available as supplementary publication no. SUP 56682.f. The corrected values were analysed and processed by the method of Fuoss and Shedlowsky,' as was done in our previous paper for the picric acid in the pure ethane- 1,2-diol and 2-methoxyethanol solvent^,^ using the relation 1 1 cASf2 -- - -+- AS A, KA; where the symbols have the usual significance.By plotting l/AS us. cASf 2, A, and K were evaluated from intercept and slope, respectively; an initial value of A, is estimated from a plot of A us. c1I2 and then only a few iterations are necessary to obtain A, and K. In order to calculate the thermodynamic parameters AGO, AW and AS", all the experimental K values for each solvent mixture were fitted to an equation of the form where T is the absolute temperature. Table 1 summarizes the composition of the solvent systems ( X = mole fraction) and the values obtained for a,, a,, a2 and a3. Tables 2-7 contain the values of A, and K obtained at different temperatures for all the systems investigated.An interesting variation of K with temperature for picric acid in the six mixtures and f' See Notices to Authors, J. Chem. SOC., Faraday Trans. 1, 1987, 83, January issue. lnK= a,+a,T+a2T1+a31nT ( 1 )G. C. Franchini, L. Tassi and G. Tosi 3131 Table 1. Composition of solvent systems and fitted a,, u,, a, and a3 coefficients of eqn (1) solvent X(ethane- system 1,2-diol) a0 a1 a2 a3 A" B C D E F G H a 1.000 0 0.927 2 0.849 9 0.679 9 0.485 6 0.261 4 0.135 9 0.000 0 317.253 1 38.538 3 533.212 3 1841.847 0 1003.626 3 - 18.428 9 - 1402.572 7 -2148.102 6 0.046 48 0.102 04 0.472 65 0.238 32 -0.042 21 -0.033 30 -0.463 92 -0.690 28 - 11 683.34 -4 820.59 - 17 711.32 -52 593.54 - 29 655.13 - 789.85 35 320.46 54 518.42 - 52.290 8 -2.779 1 -89.587 4 -318.250 0 - 172.454 6 3.939 0 247.967 4 379.184 5 2.5 - 2.0 - 1.5 - a Ref.(5). A B C qq 0 1 I I I I c F -10 10 30 50 70 t/" c Fig. 1. Dissociation constant of picric acid in ethane- 1,2-dio1-2-methoxyethanol solvent systems as a function of the temperature (for the meaning of letters A-H see table 1). in the two pure solvents5 is shown in the fig. 1; the letters A and H refer to the pure ethane- 1,2-diol and 2-methoxyethanol solvents, respectively, and B-G to the above- mentioned mixtures (see table 1). The maximum in the K value shifts to lower temperatures on passing from A to E, i.e. as X(ethane-1,2-diol) decreases (fig. 1 and table 8), whereas curves F-H show a continuous decrease as the temperature increases.Table 8 reports a comparison of the KmaX values obtained by fitting eqn (1) by3132 The Ethane- 1,2-diol-2-Methoxyethanol System Table 2. Dissociation constants and limiting molar conductances of picric acid in solvent system B K/ 103 A0 t/OC E , mol dm-3 /S mol-' cm2 A,q A0q0.7 - 10 -5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 46.04 44.88 43.74 42.64 41.56 40.51 39.48 38.48 37.51 36.56 35.63 34.73 33.85 33.00 32.16 3 1.35 30.55 29.78 29.03 1.70 f 0.01 1.86 f 0.01 1.97 f 0.01 2.08f0.01 2.18 fO.01 2.27 f 0.0 1 2.32 f 0.01 2.37f0.01 2.37 f 0.01 2.38 f 0.01 2.36 f 0.02 2.34 f 0.02 2.30 f 0.02 2.25 f 0.02 2.17 fO.01 2.09 f 0.01 1.99 k 0.02 1.88 f 0.02 1.80 f 0.01 7.1 fO.l 8.3fO.l 10.3 f 0.1 12.1 f 0.2 13.8 f0.2 16.0 f 0.2 18.5 f0.3 21.6 f 0.3 24.9 f 0.9 29.3 f 0.4 32.6 f 0.4 36.7 f 0.5 41.2f 0.5 46.0 f 0.6 51.1 f0.6 56.6 f 0.7 62.3 k0.8 68.4 f 0.8 74.4 f 0.9 5.04 5.59 4.42 5.34 4.18 5.47 3.80 5.39 3.38 5.16 3.12 5.10 2.89 5.04 2.76 5.12 2.65 5.19 2.61 5.39 2.47 5.36 2.38 5.42 2.31 5.49 2.25 5.56 2.19 5.63 2.13 5.70 2.07 5.76 2.02 5.80 1.95 5.80 Table 3.Dissociation constants and limiting molar conductances of picric acid in solvent system C K / 103 A0 t/OC E , mol dmP3 /S mol-' cm2 Aoq A0q0.7 - 10 -5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 43.19 42.10 41 -03 39.99 38.98 37.99 37.03 36.09 35.18 34.28 33.42 32.57 31.74 30.94 30.16 29.39 28.65 27.92 27.22 1.50f0.01 7.0fO.l 3.97 4.71 1.66 f 0.01 8.4f0.1 3.58 4.62 1.77 f 0.02 9.9f0.1 3.23 4.52 1.87 f 0.01 1 1.8 f 0.2 2.98 4.50 1.94 f 0.02 13.8 f 0.2 2.77 4.48 1.99f0.02 16.0f0.2 2.56 4.44 2.00f0.02 18.7f0.2 2.43 4.48 2.00 f 0.02 21.4 f 0.3 2.29 4.48 2.01 f0.02 24.3f0.3 2.17 4.48 2.00 f 0.02 27.5 f 0.4 2.06 4.48 1.99 & 0.02 30.8 f 0.4 1.98 4.50 1.94 f 0.02 34.4 f 0.5 1.90 4.52 1.92f0.02 38.1 f0.5 1.83 4.55 1.86 f 0.02 41.9 f 0.6 1.76 4.56 1.79 k0.02 45.8 f 0.6 1.70 4.57 1.71 f 0.02 50.2 f 0.7 1.65 4.60 1.65 f 0.02 55.4 f 0.7 1.62 4.68 1.57 f 0.02 60.3 f 0.8 1.58 4.71 1 S O f 0.02 65.1 f 0.9 1.53 4.71G. C.Franchini, L. Tassi and G . Tosi 3133 Table 4. Dissociation constants and limiting molar conductances of picric acid in solvent system D K / 103 A0 t/OC E, mol dm-3 /S mol-' cm2 A. q A. qO.' ~ - 10 -5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 37.60 36.61 35.65 34.72 33.81 32.92 32.06 31.22 30.40 29.61 28.83 28.08 27.34 26.63 25.93 25.25 24.59 23.95 23.32 0.775 & 0.006 0.865 f 0.007 0.932 f 0.008 0.972 f 0.009 1.02 f 0.01 1.03 fO.O1 1.04f 0.01 1.02 f 0.01 0.998 f 0.009 0.952 f 0.009 0.920 f 0.009 0.889 f 0.009 0.861 &0.010 0.837 & 0.009 0.804 f 0.009 0.768 f 0.009 0.743 f 0.009 0.702 f 0.009 0.664 f 0.009 7.7 f 0.1 9.0 f 0.1 10.4k0.2 12.2k0.2 14.1 k0.2 16.2 k 0.2 18.5k0.2 21.1 k0.3 24.0 & 0.3 27.2 k0.3 30.5 f0.4 34.1 f 0.5 37.7 f 0.5 41.4f0.6 45.3 f0.7 49.5 f0.7 51.9 f 2.7 56.1 f3.1 60.2 f 3.8 2.17 3.17 2.03 3.17 1.89 3.15 1.79 3.18 1.69 3.19 1.61 3.22 1.54 3.25 1.49 3.29 1.44 3.35 1.40 3.41 1.36 3.45 1.31 3.49 1.26 3.50 1.22 3.50 1.17 3.50 1.14 3.54 1.08 3.44 1.06 3.50 1.06 3.57 Table 5.Dissociation constants and limiting molar conductances of picric acid in solvent system E ~ 1 1 0 4 All tl°C El3 mol dm-3 /S mol-1 cm2 Aoq AOqO.' - 10 -5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 31.76 30.91 30.08 29.27 28.49 27.73 26.99 26.26 25.56 24.88 24.21 23.56 22.93 22.32 21.72 21.14 20.57 20.02 19.49 6.16 f 0.06 6.44 f 0.06 6.58 f 0.07 6.76 & 0.07 6.68 f 0.07 6.68 & 0.07 6.67 f 0.07 6.45 f 0.07 6.14f0.07 5.80 f 0.07 5.55 f0.07 5.30 f 0.07 5.04 f q.07 4.75 f 0,07 4.50 f 0.06 4.25 rfr 0.06 4.02 f 0.09 3.73 f 0.06 3.47 f 0.06 6.9f0.1 8.6k0.2 10.3 f0.2 12.0f0.2 14.0 & 0.2 16.0 f 0.3 18.2f0.3 20.5 f 0.3 23.1 f 0.3 25.8 f0.4 28.7 f 0.4 31.7 f 0.5 34.8 f 0.5 38.1 & 0.6 41.5 f 0.7 44.9 & 0.7 48.4 k 0.8 51.9 f0.9 55.7 f 0.9 0.95 1.73 1.00 1.90 1.01 2.02 0.99 2.10 0.97 2.16 0.93 2.18 0.90 2.21 0.86 2.24 0.84 2.27 0.82 2.30 0.80 2.34 0.79 2.38 0.78 2.43 0.77 2.49 0.76 2.53 0.75 2.57 0.74 2.60 0.72 2.61 0.70 2.613134 The Ethane- 1,2-diol-2-Methoxyethanol System Table 6.Dissociation and limiting molar conductances of picric acid in solvent system F - 10 -5 0 5 10 15 20 25 26.17 25.39 24.63 23.90 23.17 22.48 21.81 21.16 2.63 f 0.06 9.6f0.2 0.68 1.50 2.54 f 0.06 11.2 f 0.3 0.67 1.56 2.45 k 0.05 12.9 f 0.3 0.66 1.61 2.35 f 0.05 14.7 f 0.3 0.64 1.64 2.26f0.05 16.6f0.3 0.62 1.67 2.16k0.05 18.6f0.4 0.61 1.70 2.02 k 0.05 20.8 f 0.4 0.60 1.89f0.05 23.2f0.5 0.59 30 20.52 1.79i0.05 25.5s0.5 0.57 35 19.91 1.71f0.05 27.9k0.6 0.56 40 19.31 1.60f0.05 30.5f0.7 0.55 45 18.73 1.50f0.05 33.2k0.8 0.54 50 18.17 1.41f0.04 35.9f0.9 0.53 .73 .76 .79 .80 .83 .86 .88 55 17.63 1.31k0.05 38.8f1.0 0.52 1.90 60 17.10 1.22f0.04 41.8k1.1 0.51 1.92 65 16.59 1.15k0.05 44.4f1.2 0.50 1.92 70 16.09 1.06f0.04 47.7f1.3 0.50 1.95 75 15.61 0.971f0.040 50.9f1.5 0.49 1.97 80 15.14 0.887f0.036 54.2f 1.4 0.48 2.00 Table 7.Dissociation constants and limiting molar conductances of picric acid in solvent system G K/ 104 A0 t/OC E, mol dm-3 / S mol-l cm2 Aoq A0q0.7 - 10 -5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 22.79 22.21 21.66 21.1 1 20.59 20.06 19.56 19.07 18.59 18.12 17.67 17.22 16.79 16.37 15.96 15.56 15.16 14.78 14.4 1 1.82 f 0.03 10.3 f 0.2 0.50 1.23 1.64f0.03 12.1 f0.2 0.49 1.31 1.45f0.03 14.2f0.3 0.51 1.39 1.23 f 0.03 16.7 f 0.3 0.53 1.48 1.04f0.02 19.6f0.4 0.54 1.59 0.880 f0.022 22.8 f 0.5 0.56 1.70 0.73 1 k 0.016 26.6 k 0.6 0.58 1.82 0.673 f0.014 29.7 f 0.6 0.57 1.87 0.578 f 0.016 33.7 f 0.8 0.58 1.98 0.506 f 0.014 37.9 f 0.9 0.59 2.06 0.431 f0.013 42.9f 1.1 0.61 2.18 0.365 f 0.013 48.3 f 1.3 0.62 2.30 0.304f0.010 54.7k 1.4 0.65 2.45 0.254f0.010 61.7f 1.9 0.67 2.60 0.213 f 0.009 69.2 f 2.1 0.69 2.75 0.171 IfrO.009 78.8f2.6 0.73 2.97 0.138 f 0.007 89.4 f 3.2 0.77 3.20 0.114+0.008 99.8f4.4 0.80 3.40 0.0907 f.0.0057 1 13 f 5 0.85 3.68G. C. Franchini, L. Tassi and G. Tosi 5.0 4 . 5 % 4.0 3.5 3.0 3135 . ' 5.5 1 2 . 5 - jH 2 3 I I I I I I 2 3 4 5 6 7 lo2 € - I Fig. 2. pK us. 1 / E plot for picric acid in ethane- 1,2-di01-2-methoxyethanol solvent systems. Curves A and B at the bottom right of the figure have been vertically shifted in order to present the results more clearly. Table 8.K,,, and t for the solvent systems eqn (1) Harned pK us. 1/~ solvent system lo3 K,,, t/OC lo3 K,,, t/OC lo3 K,,, t/OC A 2.49 34.4 2.47 38.4 2.61 34.3 B 2.39 33.5 2.32 36.0 2.56 32.5 C 2.03 27.5 1.98 31.3 2.18 25.5 D 1.03 18.0 1.10 21.3 1.08 16.4 E 0.675 8.2 0.659 3.9 0.711 10.2 Harned's methods and by a pK us. 1 / ~ correlation; good agreement is observed for the three methods. As regards the pK us. 1 / E correlation, the dielectric-constant values have been directly measured for pure ethane- 1 J-diol and 2-methoxyethanol and for all mixtures at all investigated temperatures, and the data obtained have been optimized using a relation of the type log& = at+/? (where t is in "C) and reported in tables 2-7. Fig. 2 shows that for the solvent systems A-E the plots are smooth curves with a minimum whose ordinate should provide the pK,,, value and whose abscissa should provide corresponding 1 / ~ value, and as a consequence the temperature of K,,,.In order to simplify the problem the two branches of the curves have been approximated as two straight lines (r = 0.923 and 0.980 for A; r = 0.954 and 0.983 for B; r = 0.965 and 0.985 for C; r = 0.954 and 0.998 for D; r = 0.987 and 0.994 for E), whose intersection leads to the data reported in table 8. For systems F, G and H pK values increase linearly (r = 0.999, 0.999 and 0.998 for F, G and H, respectively). The Walden products9 for picric acid in these mixtures (tables 2-7) indicate that, increasing the temperature, the deviation from the equation Aoq = constant is more pronounced as the percentage of ethane- 172-diol in the mixture increases ; this behaviour may be related to the dramatic variation with temperature of the viscosity of ethane-1,2-diol (from 97.12 CP at - 10 "C to 3.04 CP at 80 "C) and consequently of the3136 The Ethane- 1,2-diol-2-Methoxyethanol System 35 0 u Q 2 25 m I 15 -10 10 30 50 70 tl OC 0 I 'H I 1 I I -10 10 30 50 70 tl "c -2001 -..-.-. ..\. -.-.-. --...,;. \. \. \. \.\.\.\. \. \. \ \. -200 - .\ \. -\ \. .\. \ \ H I I I I I -10 10 30 50 70 tl OC Fig. 3. Dependence on the temperature of AGO (in J mol-l), AH" (in J mol-l) and AS" (in J mol-1 K-l) of the dissociation of picric acid in ethane- 1,2-diol-2-rnethoxyethanol solvent systems. mixtures containing more ethane- 1,2-diol. Another feature of these A.q values is the fact that they decrease as the temperature increases for the solvent systems A-F, while they increase for G and H. For mixtures containing higher percentages of ethane- 1,2-diol, i.e. solutions that are more viscous, a better correlation is obtained by using a modified law, A. q0.7 = constant, as suggested by Stokeslo for high-density aqueous solutions. Starting from the best-fitting eqn (1) the thermodynamic data AGO, AH" and AS' for the dissociation reaction of picric acid have been evaluated, although many difficultiesG. C. Franchini, L. Tassi and G. Tosi 3137 2 3 4 5 6 7 1 0 2 E-’ Fig. 4. A, 1’s. 1 / E isotherms for picric acid in ethane- 1,2-di01-2-methoxyethanol solvent systems. The curves have been vertically shifted in order to present the results more clearly.(a) -10, (b) 0, (c) 10, ( d ) 20, (e) 30, (f) 40, (g) 50 and (h) 60 “C. and uncertainties are associated with these calculations. However, AW and ASO should be especially useful in obtaining information on solute species and solute-solvent and solvent-solvent interactions. In fact, considering the entropy of dissociation in detail, we are concerned with entropy changes due to differences ‘within’ the acid and its anion and with changes due to differences in solute-solvent interactions and also in solvent- solvent interactions if we are in the presence of mixed solvents. Thus, since in the present work we have the same solute in different mixed-solvent systems, ASo could be useful in studying the properties of the media. Data for thermodynamic functions have been plotted against t for each solvent system and are shown in fig.3. ASo us. t and A F us. t curves indicate the existence of a particular composition of the solvent mixture, very close to that of F, which denotes a sudden change in the properties of the system under study. In order to determine the composition of this supposed mixture, we have examined in detail the A. values which are reported in tables 2-7. A plot of A,, us. 1 / ~ at each temperature results in two straight lines intersecting in a point whose abscissa is then used to obtain for each temperature the E value of the so-called ‘limiting’ mixture; fig.3138 The Ethane- 1,2-diol-2-Methoxyethanol System 4 shows examples for some temperatures. Now, since over a large temperature range (- 10 to 60 “C) the relationship between the dielectric constant of the solvent system and its mole fraction is linear (r ranges from 0.994 to 0.996), the E value calculated above allows us to obtain at each temperature the composition of the ‘limiting’ mixture; these compositions are practically constant at all temperatures and average values of X(ethane- 1,2-diol) = 0.2677 and X(2-methoxyethanol) = 0.7323 [X(2-methoxyethanol)/ X(ethane- 1,2-diol) = 2.751 were obtained.A possible explanation of this behaviour of the solvent system under study may be found by considering the ‘selective solvation’ which occurs when the composition of the solvent components in the neighbourhood of the charged species is different from the composition of the bulk solution; normally, assuming a mole fraction of 0.5, the concentration of the solvent component with higher dielectric constant increases around the charged species. In the system under study it is possible that solvation occurs preferably for ethane- 1,2-diol up to the so-called ‘ limiting’ mixture, where the solvation shell probably changes, passing from a prevalence of ethane- 1,2-diol molecules to a prevalence of 2-methoxyethanol ones.By considering the entropies of dissociation (fig. 3), from the data for the pure solvents we have suggested in a previous paper5 that the formation of ion pairs should be more likely for 2-methoxyethanol (AS‘ ranging from - 38.0 to - 63.2 cal mol-‘ K-l)? than for ethane-1,2-diol (from -4.0 to - 17.9 cal mol-1 K-l) and that the solute species are more effective in orienting 2-methoxyethanol than ethane- 1,2-diol solvent molecules.Also, the AS‘ data for mixtures B-E are in accord with the above suggestions; the variation of AS‘ with the composition of the solvent system is slow, and this fact cannot be justified simply by the decrease of E of the bulk solution, but rather also by the above- mentioned selective solvation. For mixtures F and G an abrupt increase in AS‘ occurs, and this behaviour is in accord with the suggested change in preferential solvation corresponding to the ‘limiting’ mixture. The effect of temperature on the A P values is a minimum for mixture F, which is very close to the ‘limiting’ case. We thank Prof. C . Preti for stimulating discussions and the helpful suggestions, the Centro di Calcolo Elettronico of Modena University for the computing facilities and the Minister0 della Pubblica Istruzione, Italy for financial support. References 1 C. Preti and G. Tosi, Anal. Chem., 1981, 53, 48. 2 C. Preti, L. Tassi and G. Tosi, Anal. Chem., 1982, 54, 796. 3 E. Neviani Giliberti, C. Preti, L. Tassi and G. Tosi, Ann. Chim. (Rome), 1983, 73, 527. 4 C. Preti, L. Tassi and G. Tosi, Ann. Chim. (Rome), 1985, 75, 201. 5 G. C. Franchini, E. Ori, C. Preti, L. Tassi and G. Tosi, Can. J . Chem., 1987, 65, 722, and references 6 Handbook of Chemistry and Physics, ed. R. C. Weast (Chemical Rubber Co., Boca Raton, Fla, 7 R. M. Fuoss and T. Shedlovsky, J. Am. Chem. SOC., 1949, 71, 1496. 8 H. S. Harned and I. Kazanijian, J. Am. Chem. SOC., 1936, 58, 1912. 9 P. Walden and E. Birr, Z . Phys. Chem., Teil A , 1931, 153, 1. 10 J. M. Stokes and K. H. Stokes, J. Phys. Chem., 1958, 62, 497. therein. 1982-1983), p. E-50. Paper 6/ 1920; Received 29th September, 1986 t 1 cal = 4.18 J.