J . Chem. SOC., Faraday Trans. I, 1982, 78, 225-236 Osmotic Coefficients of the Ternary System Water + Potassium Chloride + Sucrose at 0 O C BY THELMA M. HERRINGTON* AND CHRISTOPHER P. MEUNIER Department of Chemistry, University of Reading, Reading RG6 2AD Received 10th February, 198 1 The freezing points of aqueous solutions of potassium chloride have been measured in the concentration range 0.2-2.8 mol kg-'. Osmotic coefficients have been calculated and compared with the results of previous workers. The freezing points of the ternary system water + potassium chloride + sucrose have been measured for I : I mole ratio sucrose : KCl from 0.15- 1.5 mol kg-' KCl, and for a 1 : 3 mole ratio sucrose : KCl from 0.2-2.0 mol kg-' KCl. Osmotic coefficients were calculated at 0 OC and fitted to an analytic formula.The experimental results are discussed in terms of the McMillan-Mayer theory of solutions. During the last decade there has been considerable interest in the properties of ternary aqueous solutions containing non-electrolytes and electrolytes. Salting-in and salting-out effects of alkali-metal and alkaline-earth halides on alcohols, amino acids and peptides have been determined by various worker~.l-~ The objective, of gaining some understanding of solute-solute interactions in these systems, has been interpreted using several different theories - electrostatic, classical therm~dynamic,~ scaled- particle,6 and surface effects' - with little pattern emerging in the final picture. A break- through in the understanding of solutions containing a single non-electrolyte was the work of Stigters on aqueous solutions of sucrose and glucose, which he interpreted using the Hill9 and McMillan-Mayer'O theories.We have extended this approach to ternary systems using an extension of the Debye-McAulay theory for the electrostatic interactions.ll> l2 It is found in crystallizing sucrose from solutions containing other electrolytes and non-electrolytes, e.g. molasses, that the solubility of the sucrose is considerably increased. It has been suggestedl3 that 'association' of sucrose and KCl occurs on a one-to-one basis. It was decided to extend our earlier work14 on the ternary system sucrose + KCl +water to lower temperatures and investigate the effect of solute-solute interactions on the freezing point of the system.EXPERIMENTAL The equilibrium method of determining freezing-point depressions of solutions was used. At equilibrium it is necessary to determine both the temperature and the concentration of solute in the solution in equilibrium with ice. Full experimental details are given in ref. (12). APPARATUS Two identical silvered Dewars (one for solution and one for water) were each suspended from the top of a cylindrical copper can (the internal heat shield) by six springs. The two copper cans were suspended from the iid of an oval copper submarine water-tight jacket (the external heat shield) itself completely immersed in the thermostat. Each Dewar had a close-fitting polythene lid, through which passed stirrers, sampling tubes and closed glass tubes for a 225226 FREEZING POINTS OF AQUEOUS SOLUTIONS thermopile and platinum resistance thermometer. The internal arrangement in each Dewar was identical to remove any differential errors from the effect of hydrostatic pressure on the freezing points or on the gas solubility.Two electronic controllers kept the temperature of each copper heat shield the same as the contents of its Dewar; thus each Dewar was in an adiabatic environment and heat leakage was negligible. A mixture of methanol and water was used for the thermostatted bath in which the whole apparatus was immersed; its temperature was ca. 0.2 "C below the freezing point of the solution. Efficient stirring of the solutions was essential as the viscosity of concentrated solutions of sucrose increases rapidly with decreasing temperature. Both gas and mechanical stirrers were used.The mechanical stirrers were made of nylon-coated stainless steel. The gas stirrers were based on the design of Brown and Prue;15 the nitrogen gas was pre-cooled and saturated with water vapour. Initially KCl or KCl and sucrose in the required mole ratio were dissolved in cooled conductivity water in the Dewar flask and seeded with small ice particles. A glass sample tube, containing a sintered glass filter to prevent ice particles entering, enabled samples to be withdrawn for conductivity measurements, and also allowed further dilutions to be made; a small amount of solution was first sucked into a dummy cell, before the sampling cell was connected, to remove any solution in the sampling tube.After sampling, an equal volume of conductivity water was added under nitrogen pressure. A sample was taken only when the temperature drift was less than two thousandths of a degree per hour. The solutions were analysed conductometrically. The conductance cell was designed according to the recommen- dations of Jones and Bollinger.16 The cell was calibrated with solutions of known molality saturated with nitrogen. The thermopile was made of chromekonstantan and had 25 junctions. The e.m.f. was determined by a null method using a Keithley microvoltmeter (model 150B) and a Tinsley Vernier potentiometer (model 4363-E Auto). It was calibrated using the freezing-point data of Scatchard and Prentiss17 and of Harkins and Roberts.la The calibration was checked periodically.The precision obtained was & 2 x lo-* K, giving at 0.1 mol kg-l an error of 0.1 % in the freezing point. The most concentrated solution for which data are given is 1.3 mol kg-l,17 so that, for solutions more concentrated than 1 mol kg-l, a platinum resistance thermometer, calibrated to 2 x K, was used. There is a negligible error introduced between the difference in hydrostatic head between a solution and pure water. The most concentrated solution used had a density of 1.2 g ~ m - ~ , so that for a 10 cm head the difference in freezing point would be 3 x K. The effect of change in atmospheric pressure was usually negligible; an increase of 10 cm of mercury decreases the freezing point of water by lop3 K. The nitrogen gas used for stirring is not as soluble in a solution as it is in pure water at the same temperature.The freezing-point depression for nitrogen in water at 273 K is 2 x K at saturation; nitrogen is less soluble in potassium chloride solutions than in water so that the measured difference in freezing point will be too small. However, the solubility of nitrogen increases with decreasing temperature and this acts in the opposite direction to the 'salting-out' effect. It is estimated that the net effect gives an error of 3 x K for 3 mol kg-' ; if the salting-out is negligible then the temperature difference would be too large by K at 1 mol kg-l and 3 x K for 1 mol kg-' potassium chloride and 4 x K at 3 rnol kg-'. MATERIALS All solutions were prepared using once distilled, previously deionized water.The water was not used unless the conductivity was < 1 x lop6 W1 cm-l. The sucrose was supplied by the research department of Tate and Lyle Ltd; it was of 99.993% purity containing 0.002% invert sugar, 0.002% ash and 0.003% organic matter. It was stored in a desiccator before use. The potassium chloride was a B.D.H. AR reagent, recrystallized three times from conductivity water, dried at 150 "C and then stored as the sucrose. Solutions were made up by mass and buoyancy corrections applied to give a precision of & 0.1 mg. The molar mass of KC1 was taken to be 74.555 g mol-l, sucrose 342.3019 g mol-l and that of water 18.0153 g mol-l.T. M. HERRINGTON AND C. P. MEUNIER 227 RESULTS POTASSIUM CHLORIDE The osmotic coefficient at 0 "C was calculated from the formulaelg where a, is the activity of water, m3 is the molality of KCl, M , is the molar mass of water in kg mol-l, T, is the freezing point of pure water in K, 8 is the depression of freezing point, T = - 8, AH: is the heat of fusion of pure water at T,, L, is the relative partial molar enthalpy of water in the solution at T,, J , is the relative partial molar heat capacity of water in the solution at T,, ACi, is the change in heat capacity of water on fusion at T, and p is the temperature coefficient in the relation ACpl = a +pT, where ACpl is the difference between the partial molar heat capacity of water in the solution and the molar heat capacity of ice.Values for L, and J1 were taken from Randall and Rossini,20 and for AH: Giauque and Stout's value of 6007 J m o t 1 was used.21 The data of Osborne et a1.22 and O ~ b o r n e ~ ~ give AC;l as 38.1 J K-l mol-l.is a function of the composition of the solutions but in the absence of any data for our systems it was estimated from the temperature variation of the heat capacities of ice and water near 0 0C;24 the value was -0.196 J K-2 mol-l. (For 8 values < 5 "C a simpler formula could have been used to calculate 4 at 0 "C but, as the calculations were computer programmed, it was more convenient to use the exact formula throughout.) Our results are plotted in fig. 1 ; the estimated error in the osmotic coefficient at different values of the molality is shown in the figure. I 0.96 0.94 0.92 4 3 0.90 0.88 0861 1 1 I 1 1 1 0 1 .o 2 .o 3.0 m 3/m0l kg-' FIG. 1 .-Osmotic coefficient of potassium chloride plotted as a function of molality at 0 "C.The smooth curves is the least-squares fit to eqn (4); 0, our data; 0, Scatchard and Prentiss;" 0, Damkohler and Wein~ierl.~'228 FREEZING P O I N T S OF A Q U E O U S S O L U T I O N S POTASSIUM CHLORIDE+SUCROSE Two series of mixtures of fixed mole ratios were used for the measurements; the mole ratios chosen were 1 : 3 and 1 : 1 sucrose: KCl. These ratios were exact to the accuracy of weighing. TABLE 1 .-FREEZING-POINT DEPRESSIONS AND OSMOTIC COEFFICIENTS AT 0 O C FOR AQUEOUS SOLUTIONS OF 1 : 1 SUCROSE : POTASSIUM CHLORIDE AND 1 3 SUCROSE POTASSIUM CHLORIDE sucrose: KCl = 1 : 1 sucrose: KCl = 1 : 3 m,/mol 4 1 1 4 1 1 m3lmol m,/mol 4 1 3 4 1 3 kg-' 8/K (obs) (calc) kg-l kg-l 8/K (obs) (calc) 0.1505 0.1553 0.2743 0.3028 0.3776 0.3838 0.39 15 0.3919 0.4550 0.4744 0.5193 0.5304 0.5453 0.5642 0.5958 0.6012 0.6859 0.7380 0.81 17 0.8157 0.8852 0.8939 0.91 15 1.0108 1.0652 1.0756 1.0912 1.1075 1.1222 1.2188 1.3677 1.4982 0.7965 0.8194 1.4399 1.5870 1.9760 2.008 1 2.05 12 2.0520 2.3815 2.48 13 2.7137 2.7697 2.8506 2.9467 3.1089 3.1371 3.5803 3.8500 4.2360 4.2559 4.6188 4.6665 4.7573 5.2773 5.561 1 5.6166 5.6964 5.78 14 5.8605 6.3692 7.1505 7.8392 0.9489 0.9460 0.941 6 0.9402 0.939 1 0.9391 0.9403 0.9397 0.9396 0.9390 0.9384 0.9377 0.9389 0.938 1 0.9374 0.9375 0.9383 0.9380 0.9387 0.9387 0.9390 0.9393 0.9394 0.9404 0.9407 0.9410 0.9408 0.9410 0.941 5 0.9427 0.9444 0.946 1 0.948 1 0.9478 0.9421 0.941 3 0.9398 0.9397 0.9396 0.9396 0.9389 0.9388 0.9385 0.9385 0.9384 0.9383 0.9383 0.9384 0.9385 0.9387 0.9390 0.9390 0.9395 0.9396 0.9397 0.9406 0.941 1 0.9412 0.9414 0.9415 0.941 7 0.9428 0.9447 0.9463 0.2336 0.3590 0.3721 0.3837 0.3910 0.4773 0.5027 0.5185 0.5553 0.6035 0.6365 0.65 12 0.7790 0.7904 0.8728 0.9705 0.9904 1.0710 1.0839 1.1246 1.2330 1.2975 1 .3940 1.4108 1.4137 1.4358 1.4682 1.5080 1.5458 1.5668 1.6371 1.6713 1.6887 1.7024 1.7844 1.8038 1.8270 1.9608 2.0163 0.07787 0.1197 0.1240 0.1279 0.1303 0.1591 0.1676 0.1728 0.1851 0.2012 0.2122 0.2171 0.2597 0.2635 0.2909 0.3235 0.3301 0.3570 0.3613 0.3749 0.41 10 0.4325 0.4647 0.4703 0.4712 0.4786 0.4894 0.5027 0.5153 0.5223 0.5457 0.5571 0.5629 0.5675 0.5948 0.601 3 0.6090 0.6536 0.672 1 0.9323 1.4192 1.4678 1.5136 1.5407 1.8772 1.9738 2.0345 2.1724 2.3576 2.4860 2.5410 3.0261 3.0716 3.3826 3.7567 3.8284 4.1386 4.1866 4.3410 4.7484 4.9928 5.3593 5.4202 5.4309 5.5169 5.6387 5.7896 5.9295 6.0098 6.278 1 6.4009 6.4729 6.5236 6.8307 6.907 1 6.99 17 7.4920 7.7023 0.9202 0.9 120 0.9 100 0.9101 0.909 1 0.9077 0.9064 0.9059 0.9034 0.9023 0.9023 0.9016 0.8983 0.8988 0.8968 0.8964 0.8954 0.8955 0.8952 0.8950 0.8938 0.8936 0.8935 0.8932 0.8932 0.8935 0.8934 0.8935 0.8929 0.893 1 0.8935 0.8927 0.8936 0.8935 0.8932 0.8939 0.8936 0.8937 0.8941 0.9 198 0.91 17 0.91 10 0.9105 0.9101 0.9066 0.9057 0.9052 0.9040 0.9027 0.9018 0.901 5 0.8989 0.8987 0.8973 0.8960 0.8958 0.8950 0.8949 0.8945 0.8939 0.8935 0.8932 0.8932 0.8932 0.8932 0.893 1 0.8930 0.8930 0.8930 0.8930 0.893 1 0.893 1 0.893 1 0.8933 0.8934 0.8935 0.8940 0.8943T.M.HERRINGTON AND C. P. MEUNIER 229 The osmotic coefficients were calculated from the equation bij = -In a1/(m2 + 2m3) M , (3) where ij denotes the mole ratio of sucrose: KCl, and eqn (2). Values of L, and J1 for the sucrose+ KCl solution were calculated from the data for aqueous solutions of KC120 and for aqueous sucrose s01utions~~~ 26 assuming that the interactions between the two solutes were negligible. (It was estimated that this introduced an error of < 0.0001 in bij in the molality range investigated.) For sucrose solutions data for L, were taken from Gucker et al.25 and the values of Gucker and Ayres26 were used for J,. The observed osmotic coefficients are denoted by 4ij(obs) and are given in table 1. The results are plotted in fig. 2; the estimated errors in the osmotic coefficient are also shown in the figure.DISCUSSION Our results for potassium chloride are plotted in fig. 1 together with those of Scatchard and Prentiss17 and Damkohler and W e i n ~ i e r l . ~ ~ These values (except those of Damkohler and Weinzierl below 1 mol kg-l) were fitted to the Pitzer equation:28 (4) 43 - 1 = - Ad m3-f/( 1 + bm3-f) +$m3 +D1 m3 c a m s i + A , = (2n Np,); (e2/4n E kT)f/3 where e is the charge of a proton, N is Avogadro's constant, E is the rationalized permittivity and pw is the density of water (expressed in kg per unit volume); values for the relative permittivity of water were those of Malmberg and M a r y ~ t t , ~ ~ and for the density of water the values of Tilton and Taylor30 were used. At 0 O C A , = 0.3773 kgi mol-i.For b and a, Pitzer's optimized values of 1.2 and 2.0 kg-f mol-4 were used. The values obtained by a least-squares fit for the other coefficients were P/kg mol-l = 0.02661 k 0.001 05 pl/kg mol-1 = 0.17806&0.00500 C/kg mol-1 = 0.00248 0.000 37. The smooth curve for values of 4,(calc) calculated according to eqn (4) is plotted in Literature data were used to calculate the osmotic coefficients of sucrose at 0 "C. The freezing-point data of Aschaffenburg and King3, at 0.2714 mol kg-I and of Morse et a/., quoted by Timmerman~,~~ from 0.5 to 1.0 mol kg-l were used; from 1 to 7 mol kg-l the vapour-pressure data of Berkeley et al.33 gave accurate values of the osmotic coefficient. The osmotic coefficient of sucrose, b2, was fitted to the formula fig.1. where b2-1 = Pm2+Qmi Plmo1-I kg = 0.124 18 k 0.001 69 Q/mol-2 kg2 = - 0.005 13 & 0.000 29. The coefficients of the power series in eqn ( 5 ) can be used to give information on solute-solute interactions in aqueous sucrose solutions. From eqn (7 1) of Garrod and Herringtonll (6) NB;; = Vy+ V P - ~ R T K where l?:: = -b:2 (b:2 is the cluster integral for two molecules of solute in pure230 0.92 @ij 0.91 0.90 FREEZING POINTS OF AQUEOUS SOLUTIONS - - - 0'93 t 0.89 I I I ' I I I I 0 0.5 1 .o 1.5 2.0 rn ,/mol kg- ' FIG. 2.-Osmotic coefficients of aqueous solutions of (a) 1 : 1 sucrose:potassium chloride and (b) 1 : 3 sucrose: potassium chloride plotted as a function of the molality of potassium chloride at 0 O C . The smooth curves are the least-squares fit to our data, eqn (18); 0, experimental values. solvent), A,, is the linear coefficient in the expansion of In y , as a power series in the mole ratio m,, i.e.In y , = A2,m2 -t Btm; -k . . . , ( 7 ) V(: is the molar volume of pure solvent, P'P is the partial molar volume of solute at infinite dilution, K is the solvent compresibility, and m, = N , / N , ( N , and N , are the number of molecules of solute and solvent, respectively). The coefficient A,, is related to P of eqn ( 5 ) by A,, = 2P/M,, where M, is the molar mass in kg mol-l; thus, using our value for P, A,, = 13.786. For sucrose at 0 OC, V p is 206.31 cm3 mol-l l4 and, for water, RTK is 1.14 cm3 mol-1 and Vy is 18.02 cm3 m01-l.~~ These values give a value for NBZ*," of 330 cm3 mol-1 at 0 O C compared with Stigter'ss value of 305 cm3 mo1-1 at 25 OC.We can consider B:: as being composed of an attractive and a respulsive contribution from the intermolecular forces.8 Now B,*,O = 2n [I -exp ( -wo2/kT)]r2dr (8) r where coo, is the potential of average force between two molecules of solute in the pure solvent, including averaging of the force over all rotational coordinates. Let R,, be the distance of closest approach between the centres of two molecules, thenT. M. HERRINGTON AND C. P. MEUNIER 23 1 4 . 2 a3 [I - exp ( - mo2/k T)]r2 dr + 271 Bz: = 271 [ 1 - exp ( - mo2/k T)]r2 dr (9) I. I R 2 2 = S + # A where S is the repulsive and 4A the attractive contribution. I ~ i h a r a ~ ~ has evaluated B,*,O for non-interacting rigid ellipsoids: his formula is NS =f(4v2) where u, is the particle volume andfis equal to unity for a sphere, but increases when the particles become more asymmetrical.From crystallographic data35 the sucrose molecule may be approximated by a prolate ellipsoid of semiaxes 5.9 and 3.5 A. From eqn (28) of ref. (34)fis 1.07. The volume of a prolate ellipsoid is given by 4n1:1,/3 for I, > I,. These figures give a value for NS of 783 cm3 mol-1 and hence at 0 OC, the attractive contribution to Bt: is given by N4A = NB:: - NS = - 453 cm3 mol-l indicating considerable attraction (presumably via hydrogen bonding) between sucrose molecules in solution. The activity coefficients y: and y: of the non-electrolyte and electrolyte, respectively, in the ternary system may be written as In y z = In y2+ln 7 2 3 (10) and In y; = In y3+1n 732 (1 1 ) where y , and y 3 are the activity coefficients in the appropriate binary systems of molalities m, and m3, respectively, y d 3 represents the contribution of the electrolyte to the activity coefficient of the non-electrolyte in the ternary system and ~ 3 2 that of the non-electrolyte to the electrolyte activity coefficient.The theory of Debye and M ~ A u l a y ~ ~ has been extended to more concentrated so1utions,12 and the activity coefficients In 723 and In 7 3 2 can be written as power series in the molalities kT In y23 = vlm3+q2m3%+ . . . vkTln y 3 2 = a,m,fa,m,m,~+. . . where the coefficients ri, oi are functions of the ionic strength and the dielectric constant of the solution. A quantity Aij is defined in terms of experimental quantities by (14) where 4, and 43 are the osmotic coefficients of aqueous solutions of component 2 only and of 3 only, respectively, and ij is the mole ratio of m, to m3.If Aij is represented as a polynomial in the form Aij = 4ij(m2 + vm3) -m2 4 2 - vm3 4 3 Aij/m,m3= Z X,,mfmg (1 5 ) p-o,1,2. . . q=o, &l' * * then it can be shown1, by application of the Gibbs-Duhem equation that232 FREEZING POINTS OF AQUEOUS SOLUTIONS If we consider the first three terms in the above expansions, then Aii/m2m3 = Xoo+Xo~m,f+Xo,m3+ . . . In y2, = ~~~m~ + XOpz,f + ;xol m3, + . . . lny,, = . .. It was found12 that only two X,, coefficients were required to represent our data; an equally adequate fit for eqn (1 8) was obtained with a constant term plus a term in m34 or a constant term and a term in m3.The values found for the coefficients were Xo0/mol-l kg = -4.416 x (a = 8.33 x lo-,) Xo,t/mol-i kgg = - 1.716 x lo-, (a = 7.00 x lo-,) (21) and XoO/mol-l kg = -5.141 x (a = 8.33 x Xo,/mo1-2 kg2 = -9.316 x lo-, (a = 6.95 x (22) Our data extended over a relatively limited molality range, so osmotic coefficient data on other aqueous electrolyte plus non-electrolyte systems extending over a wider molality range were analysed to see whether in general the inclusion of terms in non-integral powers of the molality gave an improved fit over an integral power series. The effect of the order of adding terms, i.e. m2 or m3 first, was also investigated. In general adding m, first gave better results corresponding to the greater effect of m3 on the value of Aij/m2m3.It was found that for the systems water + sucrose + NaC13’ and water + urea+ NaC13* the best fit was obtained by using half powers of m3 and fitting the terms in m3 before those of similar order in m,. In the system of water + urea + NaCl, the seven data points for the highest concentrations, which Bower and Robinson excluded from their polynomial fit, were given a zero weighting. The polynomial of Bower and Robinson using whole powers of m3 cannot fit these data. However, our polynomial using half-powers of m, continues to give a good fit even for these points. [The reader is referred to ref. (12) for further details.] This tends to confirm the experimenters’ assertion that the extrapolations used to calculate these points should not be greatly in error, and underlines the usefulness of the Debye- McAulay theory even for these very high concentrations.The activity coefficients of sucrose and of potassium chloride in the mixture were compared with those in their single-solute solutions. For sucrose In yz = 2Pm2+3Qmi/2 and for potassium chloride The values of y2, y3, y; and y$ calculated from eqn (lo), (1 I), (21), (23) and (24) are given in table 2. The effect of sucrose on potassium chloride is to decrease the activity coefficient of the latter and similarly the addition of potassium chloride decreases the activity coefficient of sucrose. Let us consider further the coefficient Xoo. The solute-solute interactions of the electrolyte may be notionally subdivided into the purely electrostatic interactions and ‘ non-electrolyte ’ interactions.Then for the ternary system, considering non-electrolyte contributions only, eqn (25) of ref. (1 1) givesT. M. HERRINGTON A N D C. P. MEUNIER 233 TABLE 2.-ACTIVITY COEFFICIENTS OF POTASSIUM CHLORIDE AND OF SUCROSE IN THE TERNARY SYSTEM AT 0 O C . THE NUMBERS ARE yj/yf. rn,/mol kg-l m2 /mol kg-' 0 0.1 0.5 1 .o 1.5 2.0 0 1 .O/ 1 .O 0.769/ 0.641 / 0.587/ 0.559/ 0.543/ 0.1 /1.025 0.768/1.021 0.639/1.000 0.585/0.971 0.557/0.941 0.541/0.910 0.5 /1.130 0.760/1.125 0.632/1.102 0.578/1.070 0.550/1.037 0.534/1.003 1 .o /1.272 0.752p.266 0.624p.241 0.569/1.205 0.541 /1.167 0.525p.129 1.5 /1.427 0.743/1.420 0.615/1.391 0.560/ 1.351 0.532/1.309 0.516p.266 2.0 / 1.594 0.734/ 1.586 0.606/ 1.554 0.552/ 1.5 10 0.523/1.462 0.507/ 1.414 1 m, + vm, # .. - I = (iA2,m;+ A23Ei2iTi3+&A33Ei:)+. . . where m, and m, are mole ratios (m, = N , / N , and m3 = N J N , ) . From eqn ( 5 ) and (7) 4,- 1 = ;A2,m2+. . . and ~ ( 4 ~ - 1) = $A3,m3 + . . . then from eqn (14) and (18) and Aij = A,,~i,tii,+ . . . A,, = XOO/Ml. A,, gives information on the non-electrolyte-solute 2 plus non-electrolyte-solute 3 interactions in solution, as it is directly related to the McMillan-Mayer cluster integral Bz;.lo From eqn (72) of ref. (1 1) NB:: = A,, Vy+ V g + VP-RTK, (30) where B;: = -bill (b&l is the cluster integral for one molecule of solute 2 and one molecille of solute 3 in pure solvent), Vy is the molar volume of pure solvent, V g and V p are the partial molar volumes of solutes 2 and 3, respectively, at infinite dilution and K is the solvent compressibility.Using eqn (21) for Xoo and taking V p as 206.31 cm3 mol-1 at 0 OC14 and V p as 23.63 cm3 mol-l at 0 0C,39 then NB:: is 185 cm3 mol-I at 0 O C . Again, let us consider B:: as being composed of an attractive and a repulsive contribution from the intermolecular forces. Let R,, be the distance of closest approach between the centres of two molecules, then B:: = 471 [l -exp (-c;o0l1/kT)]r2dr+4n [l -exp (-c;o0l1/kT)]r2dr r3 6 = S+@ (31) where S is the repulsive and 4* the attractive contribution. The signs of the integrals can be obtained from considerations of the magnitudes of cool1. In the region 0 < r < R,,, cool1 is mostly positive and very much greater than kT, so that S is positive. For R23 < r < 00, moll is negative and comparable with kT so that q5* is negative.Thus the sign and magnitude of B:: depends on the relative magnitudes of S and #*. For a hard sphere of radius a, and a hard prolate ellipsoid with long axis234 FREEZING POINTS O F AQUEOUS SOLUTIONS 2b, and short axis 2u, 34 S = ~ n a ~ + ~ n a ~ b , + 2 n a l b , a,[(l -c2)~+sin-1s/~]+a, From the crystallogra hic data35 for sucrose b, is 5.9 A and a, is 3.5 A. An average and hence at 0 O C the attractive contribution to Bz: is N#A = - 304 cm3 mol-l. This figure can be compared with that for sucrose + water interactions in a binary system. Now according to Garrod and Herringtonll eqn (69) ionic radiusg0 of 1.57 8: was used for potassium chloride. Then NS is 489 cm3 mol-l, NB:O= VP-RTK (33) where B,*P = -byl (byl is the cluster integral for one molecule of solvent and one of solute in pure solvent).Using the above values for sucrose and assuming the water molecule to be a hard sphere of radius 1.52 A, NdA is found to be - 271 cm3 mol-1 for the attactive contribution. In table 3 these values for the interactions of different TABLE 3 .-ATTRACTIVE CONTRIBUTION TO THE PARTICLE-PARTICLE INTERACTION COEFFICIENT NB,*,O NB:; NB,*,o NS -N#A /cm3 /cm3 /cm3 /cm3 /cm3 T/K mol-' mo1-I mol-l mol-1 mo1-I hexamethylenetetraminea ureaa sucrosea sucroseb sucrose + waterb hexamethylenetetramine + sucrose + potassium chlorideb glycine + uread or-alanine + uread a-aminobutyric acid + uread glycylglycine + uread glycine +calcium chloridee 8-alanine +calcium chloride" y-aminobutyric acid +calcium E-aminocaproic acid +calcium glycylglycine + calcium waterC chloride" chloridee chloride" 298 298 29 8 273 273 29 8 273 298 298 298 298 298 298 298 298 298 - - 302 1 - - - - 3220 396 179 783 783 476 300 489 352 416 469 469 196 235 277 342 277 58 178 498 453 27 I 191 304 348 3 70 378 445 ' 521 2375 2816 3363 3497 a Ref.(41); this work; ref. (42); ref. (43); " ref. (2). molecules are compared with those for two molecules of the same species (urea, hexamethylenetetramine and s u c r o ~ e ) . ~ ~ ~ 42 The implications are that sucrose + sucrose interactions are stronger than sucrose_+ water, presumably indicating a greater degree of hydrogen bonding. The interparticle interactions in urea and hexamethylene- tetramine are weak indicating perhaps surprisingly little hydrogen bonding in the former case.Hydrogen bonding between two molecules of hexamethylenetetramine is unlikely, but hydrogen bonding between hexamethylenetetramine and water is possible and is indicated by our results. The surprising result is that the interactionT. M. HERRINGTON AND C. P. MEUNIER 235 between sucrose +potassium chloride is of comparable magnitude with that of sucrose + water; if hydration radii were used for potassium chloride in the calculations then the attraction would be even greater. The results for sucrose + KCl interactions in the ternary system can be compared with those of other workers on aqueous solutions of two non-electrolytes or electrolyte + non-electrolyte. Lilley and Scott4, have used a similar approach for aqueous solutions of amino acids and peptides.Frompsmotic coefficient data at 25 OC they calculated B& for the interaction of amino acids with urea: the values are given in table 3. The hard-sphere contributionll is 71 s = - ( R 2 + R )3 (34) 6 where R, and R, are the hard-sphere diameters. R, and R, were calculated from the partial molar volumes at infinite dilution.44 The attractive contributions, NdA, are presented in table 3. The attractive contributions of the three amino acids are very similar, but the interaction of the dipeptide glycylglycine with urea is considerably greater. However, all of these attractive values are greater than those for sucrose + KC1. Briggs et a/., used a calcium ion-exchange electrode to determine the activity coefficients of calcium chloride in aqueous solutions of various amino acids at 25 "C.They tabulate values of a parameter A , where our parameter A',, = 3 In 1 OA . We have used their data to calculate values of 82. and these are also given in table 3. The hard-sphere contribution was again calculated from an R, value based on V'p 44 and an R, value from V'? = 17.78 cm3 m01-l.~~ The striking feature is the large negative values of both NB:; and N#A. The system sucrose+KCl exhibits salting-in of the non-electrolyte by the electrolyte, whereas most non-electrolytes are salted-out by the addition of electrolyte. The data on the amino acids correspond to very extensive salting-in of the amino acids by calcium chloride. Salting-in and salting-out effects are usually expressed in terms of the Setchenov coefficient.46 Following Friedmann47 we define the Setchenov coefficient by If the solubility of the non-electrolyte is measured in a series of solutions of the electrolyte and In m, plotted against m,, then the slope of the graph at any point is - k , for the solution.Experimentally it is found that for m3 < 1 mol kg--l and m, < 0.1 mol kg-l, k , is a constant. Then if we define k , by where p: is the chemical potential of the non-electrolyte in the ternary system. From eqn (7), (10) and (19) k , = Xoo. (37) In other words the leading term in the expansion for Aij/mzm3 is the Setchenov coefficient. 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