J. Chem. Soc., Faraday Trans. I , 1982, 78, 213-223 Apparent Molar Volumes, Temperatures of Maximum Density and Osmotic Coefficients of Dilute Aqueous Hexame t hylene te t ramine Solutions BY THELMA M. HERRINGTON* AND ELSPETH L. MOLE Department of Chemistry, The University of Reading, Whiteknights, Reading RG6 2AD Receiued 10th February, 1981 Apparent molar volumes for solutions of hexamethylenetetramine are determined over a wide range of concentrations, using both pyknometric and dilatometric techniques, at 5, 15 and 25 OC. The temperatures of maximum density of dilute aqueous solutions of hexamethylenetetramine are found. The osmotic coefficient and solute activity coefficient are determined at 25 O C for hexamethylenetetramine using the isopiestic technique. Theories of dilute solutions are applied to solute-solute and solute-solvent interactions, assuming a rigid particle model for the repulsive potential.A rigorous statistical-mechanical theory of concentrated aqueous solutions of non-electrolytes is still a distant goal, but considerable advances have been made in establishing formal connexions between the thermodynamic properties of dilute solutions and molecular behaviour.l> Determination of the partial molar volume of the solute at infinite dilution gives information on solute-solvent interaction and together with data for the osmotic coefficient of the solvent an understanding of solute-solute interaction may be attained. In order to test the theories a solute + solvent system was chosen where the solute-solvent interaction would be anticipated to be considerably larger than the solute-solute interaction. The system chosen was hexamethylenetetramine + water.Hexamethylenetetramine has four sites available for hydrogen bonding with water but no hydrogen electron donors for hydrogen bonding with another molecule of hexamethylenetetramine. There is considerable interest in the effect of hydrogen-bonding solutes on the temperature of maximum density of water. It is primarily the rate of change of the solute-solvent intermolecular forces with temperature that determines the magnitude of the depression of the temperature of maximum density. It was decided to investigate whether the behaviour of hexamethylenetetramine was consistent with the formation of four hydrogen bonds with water.This work is an extension of our earlier investigations of aqueous solutions of non-ionic solutes. EXPERIMENTAL SECTION MATERIALS The hexamethylenetetramine was recrystallized three times from an ethanol + water mixture; the purified material gave the correct percentages on C, H, N analysis. A.R. grade sodium chloride was purified three times by precipitation from a saturated solution with hydrogen chloride gas. Deionized water from a mixed-bed ion-exchange resin was used in the preparation of all solutions; any non-ionic impurities were present to less than one part in los. The conductivity water was outgassed on a water pump and then further degassed by keeping 213214 DENSITIES OF AQUEOUS SOLUTIONS at 10 O C above thermostat temperature for the volume measurements. Solutions were made up by weight to give a precision of kO.1 mg.The molar mass of N,(CH,), was taken to be 140.1876 g mol-l and water 18.0153 g mol-l. DETERMINATION OF PARTIAL MOLAR VOLUMES The densities of the concentrated solutions were determined using Ostwald-Sprengel pyknometers; from these the apparent molar volumes of dilute solutions were determined with a dilatometer, the theory and operation of which has been described previ~usly.~ Further experimental details are given in ref. (4). TEMPERATURE OF MAXIMUM DENSITY The temperature of maximum density of hexamethylenetetramine solutions was found after each dilatometric measurement at 5 O C . A six-degree Beckmann thermometer was placed in the thermostat. The thermostat temperature was lowered 0.2 O C , the dilatometer and contents allowed to come to thermal equilibrium and the level of the liquid in the capillary stem noted.This was repeated until the level of liquid had fallen and risen again by approximately the same height. The method of analysis of the measurements to determine the temperature of maximum density, Om, is described in ref. (3). ISOPIESTIC MEASUREMENTS A standard type of isopiestic apparatus was Silver dishes with close-fitting lids were fitted into copper blocks mounted in a vacuum dessicator fitted with rubber sealing rings. The evacuated dessicator was rocked gently in a water thermostat set at 25.00f0.01 O C for up to 6 days to ensure that isopiestic equilibrium had been attained. RESULTS PARTIAL MOLAR VOLUMES Densities and apparent molar volumes of hexamethylenetetramine solutions were determined at 5 , 15 and 25 OC.Density values for the concentrated solutions are given in table 1. Data for the density of water were taken from Bigg.6 Values for the apparent molar volumes 4 V of the concentrated solutions were smoothed to give the values recorded in table 1, and these smoothed values were used to calculate the apparent molar volumes of the dilute solutions, also recorded in table 1. The apparent molar volumes can be fitted to a polynomial in the molality using the equation (see later) 4V = Vp+RT (iA’rn+!B’f’ m2+. . .). (1) The coefficients are given in table 2. The results at 5, 15 and 25 OC are represented to within 0.01, 0.03 and 0.02 cm3 mol-1 in 4 V, respectively. Our results are compared with those of White’ and Creszenzi et aZ.* in fig.1. Both these sets of workers determined the densities by pyknometric methods only. The greater self-consistency of our results is undoubtedly helped by the dilatometric technique enabling the apparent molar volume to be determined in very dilute solution so that little extrapolation is required for Vp. TEMPERATURE OF MAXIMUM DENSITY The change in the temperature of maximum density of water produced by a solute, AO,, is defined by AO,/K = O,/”C - 3.980. (2) The observed values of A&, are given in table 3.TABLE 1 .-DENSITIES AND APPARENT MOLAR VOLUMES OF AQUEOUS HEXAMETHYLENETETRAMINE SOLUTIONS AT 5 , 1 5 AND 25 O C T = 278.15 K T = 288.15 K T = 298.15 K molality molality molality /mol kg-' p/g cmP3 4V/cm3 mol-l /mol kg-l p/g cm-3 4V/cm3 mol-l /mol kg-l p/g cm-3 4V/cm3 mol-' '-1 __ water 0.008 13" 0.0 15 26" 0.020 84" 0.025 38" 0.028 86" 0.069 90" 0.089 33" 0.106 42" 0.503 4 0.969 9 1.493 2 2.003 0 2.558 0 2.969 6 3.450 8 3.886 9 4.425 1 0.999 964 - - - - - - - - 1.014 98 1.027 66 1.040 70 1.052 35 1.064 04 1.072 01 1.080 67 1.087 94 1.096 42 - 108.87 108.88 108.86 108.85 108.86 108.86 108.84 108.85 108.72 108.62 108.49 108.36 108.21 108.14 108.08 108.05 107.98 water 0.004 35" 0.015 27" 0.020 70" 0.025 88" 0.031 81" 0.069 52" 0.089 36" 0.494 6 0.992 9 1.501 2 2.009 0 2.488 6 2.954 4 3.466 0 4.015 1 5.029 2 5.474 0 0.999 101 - - - - - - - 1.013 46 1.026 63 1.038 9 1 1.050 15 1.056 00 1.068 73 1.077 69 1.086 5 1 1.100 81 1.106 34 ~~ - 109.72 109.76 109.75 109.74 109.77 109.73 109.71 109.67 109.52 109.39 109.28 109.15 109.10 109.02 108.97 108.96 108.99 water 0.025 70" 0.029 85" 0.047 04" 0.088 18" 0.108 36" 0.494 5 1.005 9 1.501 6 2.022 6 2.493 0 2.940 3 3.276 8 3.286 1 3.996 6 4.078 5 4.400 1 5.551 6 0.997 047 - - - - - 1.011 01 1.024 34 1.030 60 1.047 27 1.056 56 1.064 73 1.070 60 1.070 79 1.082 03 1.083 28 1.087 82 1.102 64 - 110.58 110.59 110.57 110.52 110.51 110.46 110.29 110.20 110.08 110.02 109.98 109.92 109.90 109.85 109.84 109.85 109.84 a Dilatometric determinations.216 DENSITIES OF AQUEOUS SOLUTIONS TABLE 2.-cOEFFICIENTS OF THE POLYNOMIAL v = v$+ + RT($A’m + !jBt’mz + .. . ) A’/ 10-4 kg B+’/ 10p5 kg2 T/K V$+/cm3 mo1-l mol-l atm-l molP2 atm-l 278.15 108.87 - 0.273 288.15 109.76 - 0.263 298.15 110.58 - 0.245 0.32 0.38 0.37 TABLE 3 .-TEMPERATURE OF MAXIMUM DENSITY OF HEXAMETHYLENETETRAMINE AQUEOUS SOLUTIONS molality / 1 0-2 mol kg-l - AB,/K water 0.8 1 1.52 2.08 2.54 2.89 6.99 8.93 10.64 0.000 0.085 0.125 0.185 0.210 0.260 0.530 0.610 0.700 1 I 1 73.15 283 15 293.15 303.15 313 I! T/K FIG.1 .-Temperature dependence of the apparent molar volume at infinite dilution of hexamethylene- tetramine aqueous solutions. Values at 5, 15 and 25 O C are from e, this work; 0, ref. (6) and 0, ref. (7).T. M. HERRINGTON A N D E. L. MOLE 217 ACTIVITY AND OSMOTIC COEFFICIENTS The osmotic coefficients of aqueous solutions were determined over the range 0.5-4.5 mol kg-l at 25 *C against aqueous sodium chloride solutions as the isopiestic standard. Table 4 gives molalities of isopiestic pairs of sodium chloride and hexa- methylenetetramine solutions.The values for the osmotic coefficients of sodium chloride were taken from the compilation of Robinson and stoke^.^ The osmotic coefficients of hexamethylenetetramine are well represented by the equation 4 = 1 +0.2277 m- 1.60 x lop3 m2 (3) In y, = 0.4554 m - 2.40 x m2. (4) ( 5 ) (the mean deviation in 4 is 0.005), and hence the activity coefficient is given by Creszenzi et a1.* found that 4 is linear in molality up to 4.7 mol kg-l and given by 4 = 1 +0.220 m. TABLE 4.-ISOPIESTIC SOLUTIONS OF SODIUM CHLORIDE AND HEXAMETHYLENETETRAMINE molality molality sodium hexamethylenetetramine chloride/mol kg-l /mol kg-l 0.402 90 1.182 32 1.528 61 1.712 17 2.016 04 2.454 51 2.566 13 3.025 31 3.321 98 3.619 16 4.254 58 0.643 54 1.628 14 2.015 60 2.224 70 2.537 44 2.990 72 3.087 99 3.536 33 3.819 04 4.105 30 4.747 66 DISCUSSION Let us write for the Gibbs energy of a solution of mole ratio of solute to solvent m10 G-'/N,k T = p:/k T+ m&/k T - fii+ mlnm+ iA,,iiP + ~B2,,W' + Then for the partial molecular volume of the solvent V, = ~ ~ - k T [ ~ ( c ? A ~ ~ / t ? p ) ~ f i i ~ + ~ ( t ? B ~ ~ ~ / Q ~ ) , f i i ~ + .. .] and for the solute u, = v ~ + k T [ ( a A , 2 / i 3 p ) T ~ + ( a B 2 2 2 / i 3 p ) T l ~ 2 + . . .I. Thus the total volume, V, of the solution is given by V/ N , = V: + mvp + k T[i A,,'@, + +B2,,'m3 + . . . ] 8 . . .218 DENSITIES OF AQUEOUS SOLUTIONS where the dash represents differentiation with respect to pressure, and the apparent molar volume is given by (10) 4 % = Vp+RT(iA’m+$Bm2+.. .) where A’ = A2,’M1, B’ = B,,,/M;, etc. solvent the osmotic pressure, n, is given by According to the theory of McMillan and Mayerl for a solution of a solute in a n/kT = n+ B,*n2+ B,*,,n3+. . . . ( 1 1) (12) Hillll has shown that the coefficients A,, etc. may be related to the coefficients B,*, etc. For example where B:: = -b:,. [In relating coefficients higher than A,, (i.e. B,,, etc.) to the coefficients B**, it must be remembered that these equations are only applicable in dilute solutions when y - 1 21 In y ; in more concentrated solutions A,, V? = 2B;: - V? + byl In y = A,,m+ B t m 2 + . . . (13) B t = B,,,-4A222 etc.] (14) (15) Thus instead of eqn (10) in general we write #& = Vp+RT(+A’m+$Bt’rn2+.. .). Now byl, the solute-solvent cluster integral, is related to the partial molecular volume of solute at infinite dilution bylo byl = -up+kTu. (16) Thus values for the solute-solvent interaction can be calculated from eqn (1 6). From the experimental values for the activity coefficient values for the solute-solute interaction can be calculated from eqn (1 2), (1 3) and (1 6). SOLUTE-SOLVENT ATTRACTION Values for the solute-solvent interaction, NB:: (where B;f;O = -byl) are given in table 5. Compressibility data for water were taken from the compilation of Bradley and Pitzer.12 Now byl is given by byl = -471 [ I -exp (-co11/kT)r2dr] (17) sb; where coil is the potential of mean force between one molecule of solute and one of solvent in the pure solvent (including averaging of the force over all rotational coordinates) and Y is the distance apart of the centres of the molecules.The cluster integral byl consists of an attractive and a repulsive contribution; it can be arbitrarily split into those components in the following way. Let R be the distance of closest approach of the two molecules, then the repulsion will occur for Y < R, and attraction for Y > R, and the integral can be split into repulsive and attractive parts as follows: B;E,O = 4n [ -exp ( - d l / k T ) ] r2dr+4n [I -exp (-coll/kT)] r2dr (18) (19) s:‘ s:: = S+@* where S is the repulsive and @* the attractive contribution.T. M. HERRINGTON AND E. L. MOLE 219 If the form of the potential function mlr is known, then the integration could be performed to yield BrF. The simplest potential function regards the molecules as rigid spheres.For two hard spheres of diameters R, and R, (20) n S = - (R, + R,)3. 6 X-ray studies13 have shown the hexamethylenetetramine molecule to be almost spherical with slight protrusion of the>CH, groups; from this data and Corey-Pauling models the diameter was taken as 6.8 A. The water molecule can be considered to be a sphere of diameter 3.04 A. Then NS = 300 cm3 mol-1 and the attractive contri- bution at 25 O C is given by NOA = NB,*,o- NS = - 191 cm3 mol-l. Values at other temperatures are given in table 5; on this model the attractive contribution increases with decreasing temperature. TABLE 5.-ATTRACTIVE CONTRIBUTIONS TO THE SOLUTE-SOLVENT INTERACTION COEFFICIENT VF R TIC NB,*P NS - N<DA T/K /cm3 mol-' /cm3 mol-l /cm3 mol-l /cm3 mol-l /cm3 mol-l hexame t h ylenete tra- mine + water 278.15 108.87 1.14 107.73 300 193 hexamethylenetetra- mine + water 288.15 109.76 1.13 108.63 300 192 hexamethylenetetra- mine + water 298.15 110.58 1.11 109.47 300 191 sucrose + water 298.15 211.49 1.11 210.38 476 266 urea + water 298.15 44.2 1.11 43.10 176 143 glucose +water 298.15 112.2 1.11 11 1.10 358 246 It is interesting to compare the values of NOA for hexamethylenetetramine with those for solute-solvent interaction in aqueous solutions of urea, glucose and sucrose.In order to calculate the rigid sphere contribution to B;T,O these three solute molecules are assumed to approximate to prolate ellipsoids. For a hard sphere of radius a, and a hard prolate ellipsoid with short axis 2a, and long axis 2b2l4 +a, 1+- 1 +&)I (21) ( 1 - c 2 2& I - & 4 4 3 3 S = -nai +-naib, + 2na,b, where c2 = (b;--@)/b;.The sucrose molecule has semiaxes of 5.9 and 3.5 A using Corey- Pauling models and crystallographic data,15 the urea molecule is assumed to be an ellipsoid with semiaxes 3 and 2.4 A and glucose to have semiaxes 4.8 and 3.2 A. At 25 O C V p for sucrose is 21 1.49 cm3 m ~ l - l , ~ for urea 44.2 cm3 mol-1 l6 and for glucose 1 12.2 cm3 m ~ l - ~ . ~ ' The attractive contributions to the Solute-solvent interaction at 25 O C are given in table 5. The attraction between hexamethylenetetramine and water molecules is intermediate between that for sucrose +water and urea + water. This is in agreement with hexamethylenetetramine having four hydrogen-bond accepting sites.The interaction for glucose with probably five hydrogen-bond sites is only slightly less than that for sucrose with more sites available. The solute-water attraction is weaker for urea and rather less than might be anticipated; it is, however, in agreement with spectroscopic evidencela which shows that urea-water hydrogen-bond interactions exist but are very short-lived. 8-2220 DENSITIES OF AQUEOUS SOLUTIONS SOLUTE-SOLUTE ATTRACTION From eqn (12) and (16) 2B:: = A,,$ + 2143 - kTK (22) so that a value for B,*,O can be calculated from the partial molar volume at infinite dilution and the activity-coefficient data. Hexamethylenetetramine shows large deviations from ideal behaviour. At 4.7 mol kg-l # is 2.04 compared with 1.45 for sucrose.1g The molar volume of water at 25 O C is 18.07 cm" mo1-1.20 From eqn ( 5 ) A,, = 0.4554/M1 and hence NB:: is 338 cm3 mol-l.Now BZ: can be considered, like B::, to be composed of an attractive and a repulsive contribution from the intermolecular forces, thus B:: = S+(DA (23) where S is the repulsive and @A the attractive contribution. Isiharal* has calculated the repulsive contribution for hard ellipsoids and finds that S =f(4u2) where u, is the volume of a solute molecule and f is unit for a sphere. Taking the hard-sphere diameter of hexamethylenetetramine as 6.8 i 1 3 gives an attractive contribution N(DA of - 58 cm3 mol-l. This value of (DA is a measure of the pairwise interaction between two hexamethylenetetramine molecules in water.In table 6 it is compared with values for urea, sucrose and glucose calculated by the same method. The calculations for urea and sucrose are given in ref. (21). For glucose the osmotic-pressure data of Morse2, were used for B,*,O. The glucose molecule approximates to a prolate ellipsoid (v, = 4d:1,/3 for I, > I,) with semiaxes 3.2 and 4.8 A andffactor 1.05. TABLE 6.-ATTRACTIVE CONTRIBUTION TO THE SOLUTE-SOLUTE INTERACTION COEFFICIENT AT 25 O C NB;; NS -N@AA -N@AB /cm3 mol-l /cm3 mol-l /cm3 mol-l /cm3 mol-' - hexamethylenetetramine 338 396 58 urea 1 179 178 174 glucose 117 520 403 229 sucrose 285 783 498 557 a Our values. Kauzmann and As can be seen from table 6, the attraction between two solute molecules decreases in the series sucrose > glucose > urea > hexamethylenetetramine.This is consistent with no suitable hydrogen electron donors for hydrogen bonding in the molecule of hexamet h ylene tet ramine. have calculated values of the pairwise attraction for many non-electrolytes in aqueous solution. They used the following equation to calculate B,*,O Kauzmann and NB:; = ( V p - V,") + V,"(+ - B) where B is the coefficient in the equation for the expansion of the logarithm of the solvent activity coefficient,f,, in a power series of the mole fraction of solute, x,, thusT. M. H E R R I N G T O N A N D E. L. MOLE 22 1 Eqn (25) is shown to be equivalent to our eqn (22) in ref. (4). They then considered the repulsive contribution to be given by S = 4vF Their results are compared with ours in table 6.It can be seen from both methods that the magnitude of the attractive contribution increases with the number of groups capable of hydrogen bonding with another solute molecule. TEMPER A T U R E OF MA X I MUM D ENS1 TY-SO LU TE-SO L U TE A N D SOLUTE-SOLVENT INTERACTION There has been a great deal of interest in the effect of solutes on the temperature of maximum density of water. Whether a solute raises or lowers the temperature of maximum density depends on the nature of the solute. Despretzz4 found that many electrolytes lowered the temperature of maximum density, the effect being proportional to the solute concentration. However, it has been foundz5 that alcohols raise the temperature of maximum density. Let us write V = V,O/M,+m@V and V,O = v:" [ 1 + a,(8- 8,)2] for the temperature dependence of the molar volume of water in the neighbourhood of its temperature of maximum density.Then substituting eqn (27) into (26) and differentiating with respect to temperature gives for the solution at its temperature of maximum density The first term on the right-hand side of eqn (28) is the 'ideal dilute' contribution to the change in the temperature of maximum density caused by the addition of a solute, and the second and higher terms are the non-ideal contribution. From eqn (16) the ' ideal-dilute ' contribution involves the rate of change of solute-solvent interactions only with temperature, whereas since3 (29) the higher terms also involve the rate of change of solute-solute interactions with temperature.If we consider [a Vp/llTIom and [a( TA')/i3T],n, to be constant for small values of A@,, then (30) In table 7 the values obtained for [ and < for hexamethylenetetramine are compared with those obtained for other substances3 The value of (?Vp/3T)Qm is 9.25 x lop2 cm3 mol-l K-l. Using values of 8.0 x lop6 Kp2 and 18.02 cm3 mol-l for a, and K*", respectively,20 gives a calculated value for [ of - 5.8 K mol-1 kg, compared with the experimental value of -6.3 K mol-l kg. The sign and magnitude of [ is determined by the sign of c? Vp/c?T. For butan-2-01 and ethanol this is negative, which results in the temperature of maximum density of water being raised by these solutes. For the others it is positive and 8, is lowered. Hexamethylenetetramine has an effect intermediate between that of glycerol and sucrose as would be expected for a molecule with four sites capable of hydrogen bonding with water.A 2 , 4 = - 2bi2 + 2by1 - k TIC A$, = [m+<m2+xm3+. . . .222 DENSITIES OF AQUEOUS SOLUTIONS TABLE 7.-ANALYSIS OF THE EFFECT OF SOLUTE-SOLVENT INTERACTION ON THE TEMPERATURE OF MAXIMUM DENSITY but an-2-01 4.2 - 10.3 ethanol 1.5 - 2.0 ethylene glycol - 3.2 - 0.4 glycerol -4.1 - 1.5 hexamethylenetetramine - 6.3 25 sucrose - 18.3 20 CONCLUSION The analysis of the McMillan-Mayer interaction coefficients for the solute-solute and solute-solvent cluster integrals of aqueous solutions of hexamethylenetetramine supports our predictions. The solute-solute attractive contribution to B,*,O is consid- erably less than the solute-solvent contribution to and yet each lies in the correct sequence with respect to other hydrogen-bonded molecules. Thus, in spite of the nake model assumed for the repulsive interactions, the calculated quantities are of the anticipated order of magnitude.This gives us confidence in applying the McMillan- Mayer and Hill theoretical equations to aqueous solutions of non-electrolytes at a molecular level. It is instructive to obtain an approximate estimate of the effect of hydrogen bonding on the attractive part of BrF. It is assumed that at the surface of the hexamethylenetetramine molecule is a square-well of depth d1 and width d. Then @* = [I -exp( -coll/kT)] dx dy dz. d l is taken as 16 kJ mo1-1,26 the known order of magnitude for the enthalpy of formation of this hydrogen bond.A value of - 191 cm3 mol-1 for the integral gives d = 0.80 A which is a reasonable figure for atomic vibrations in the 0-H * - N bond. We thank the S.R.C. for the award of a Research Studentship to E.L.M. GLOSSARY OF SYMBOLS cluster integral for two molecules of solute in pure solvent cluster integral for one molecule of solute and one of solvent in pure solvent Gibbs energy Boltzmann's cons tan t molar ratio of solute to solvent (N2/N1) molality of solute molar mass of solvent in kg mol-l number density of solute Avogadro's constant number of molecules of solvent number of molecules of solute gas constantT. M. HERRINGTON AND E. L. MOLE repulsive contribution to the cluster integral absolute temperature partial molecular volume of solvent molecular volume of pure solvent partial molecular volume of solute partial molecular volume of solute at infinite dilution partial molar volume of solute at infinite dilution apparent molar volume of solute activity coefficient of solute on the molality scale osmotic pressure osmotic coefficient attractive contribution to the configuration integral density of solution isothermal compressibility of solvent temperature of maximum density of the solution 223 W.G. McMillan and J. E. Mayer, J . Chem. Phys., 1945, 13, 276. T. L. Hill, J. Am. Chem. SOC., 1957, 79, 4885. J. E. Garrod and T. M. Herrington, J. Phys. Chem., 1970, 74, 363. E. L. Mole, Thesis (Reading University, 1975). R. A. Robinson and R. H. Stokes, J . Phys. Chem., 1961, 65, 1954. P. H. Bigg, Br. J. Appl. Phys., 1967, 18, 521. 'I E. T. White, J. Chem. Eng. Data, 1967, 12, 285. V. Creszenzi, F. Quadrifoglio and V. 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Washington, 1914, 198. 23 J. J . Kozak, W. S. Knight and W. Kauzmann, J . Chem. Phys., 1968, 48, 675. 24 M. C. Despretz, Ann. Chim. Phys., 1839, 70, 49. 25 F. Franks and B. Watson, Trans. Faraday Soc., 1967, 63, 329. 26 G . C. Pimentel and A. L. McClellan, The Hydrogen Bond (W. H. Freeman, New York, 1950). (PAPER 1 /2 19)