460 Acta Chim. Slov. 2007, 54, 460–468 Scientific paper Apparent Molar Volume and Apparent Molar Expansibility of Tetraethyl-, Tetra-n-propyl-, Tetra-n-butyl-, and Tetra-n-pentylammonium Cyclohexylsulfamates in Aqueous Solution† Cveto Klofutar, Jaka Horvat and Darja Rudan-Tasi~* Biotechnical Faculty, University of Ljubljana, Jamnikarjeva 101, 1000 Ljubljana, Slovenia * Corresponding author: E-mail: darja.rudan.tasic@bf.uni-lj.si Received: 25-05-2007 †Dedicated to Prof. Dr. Jo`e [kerjanc on the occasion of his 70th birthday Abstract The apparent molar volume of tetraethyl-, tetra-n-propyl-, tetra-n-butyl, and tetra-n-pentylammonium cyclohexylsulfa-mates was determined from the density data of their aqueous solutions at 293.15, 298.15, 303.15, 313.15, 323.15 and 333.15 K. From the apparent molar volume the partial molar volume of the salts and water was determined. The limiting apparent molar volume and apparent molar expansibility were apportioned into their ionic components. The limiting apparent molar volume of the salts was found to be a linear function of the cation molecular weight. The partial molar ionic volumes at infinite dilution were treated by a model proposed by Marcus. From the second derivative of the limiting apparent molar volume with temperature it was found that the investigated solutes behave as structure-making ones in water. The packing density of the ions in solution were evaluated and compared with the corresponding data for tetra-n-alkylammonium halides. The density of aqueous solutions of the investigated salts can be adequately represented by an equation derived by Redlich. Keywords: Density data, partial molar volume, expansibility, cyclohexylsulfamates, aqueous solutions 1. Introduction Aqueous solutions of tetra-n-alkylammonium salts are of great interest because of their anomalous properties. Their physicochemical properties such as activity coefficients, partial molar volumes, viscosity, and electrical conductivity are quite different from those of most simple electrolytes.1 It is well known that tetra-n-alkylam-monium salts dissolved in water undergo hydrophobic hydration with the formation of a more ordered and rigid structure of water surrounding the ions. Therefore, tetra-n-alkylammonium salts have been and still are extensively used as model systems for the characterization of hydrop-hobic hydration and various types of interactions. From the valuable literature data, e. g.,2–4 it follows that tetra-n-alkylammonium ions do not form strong hydrogen bonds with water molecules in solution, nor do they exhibit strong electrostatic interactions. Volumetric studies of such electrolyte solutions can provide useful information concerning ion-solvent, ion-ion, and solvent-solvent inter-actions.3 From a theoretical point of view, the most useful thermodynamic quantities are the limiting values of the apparent molar volume and the apparent molar expansibility since these values depend on the intrinsic size of the ion and on ion-solvent interactions. So, the ion-solvent interactions manifest themselves in all molar functions obtained by extrapolation to infinite dilution.5 The present work deals with the apparent molar volumes and expansibilities of dilute aqueous solutions of some lower numbers of the tetra-n-alkylammonium cyclo-hexylsulfamates, i. e. with salts where both of the ions, cation and anion, undergo hydrophobic hydration. Through this study we extended our knowledge of the volumetric properties of cyclohexylsulfamates as potential artificial sweeteners.6,7 The partial molar and apparent molar volumes of various solutes have been used in the past in accessing drug potency and in sweet taste chemoreception, e. g.8–10 With this in mind, we determined some volumetric properties of tetra-n-alkylammonium cyclohexylsulfa- Klofutar et al.: Apparent Molar Volume and Apparent Molar Expansibility of Tetraethyl-, Tetra-n-propyl-, ... Acta Chim. Slov. 2007, 54, 460–468 461 mates which can provide an important insight into the interactions of sweet solutes with water and perturbation effects on the structure of water. 2. Results and Discussion The values of density, d (kg dm–3) of all investigated salts are given in Table 1 as a function of the temperature-independent molalities, m (mol kg1). The apparent molar volume, V€,(dm3 mol–1), of a solute with molar mass, M2 (kg mol–1), was calculated from the relation11 (1) where d0 is the density of pure water (kg dm–3) and c is the concentration of the salt (mol dm–3) calculated from c = md/(1 + mM2) . In calculation of the uncertainty of the apparent molar volume, SV^, only the uncertainty in density measurement was taken into account since Vo is not seriously influenced by errors in molarity: (2) The relative error of the apparent molar volume is about 1 per cent at the lowest concentration and 0.1 per cent at the highest concentration. In Fig. 1 the values of V0 of Et4NCy are plotted against c1/2 at 293.15, 313.15 and 333.15 K; the values of V0 of the other salts give similar plots. From Fig. 1 it can be seen that deviations in V0 are more pronounced at lower temperature. In dilute solutions Vj, values of the salts investigated decrease at all temperatures with increase of the square root of the molar concentration. It has been established that, except for the tetramethylammonium salts, plots of V^ against c1/2 show negative slopes, indicating a type of behaviour not normally associated with 1:1 electrolytes.12 The dependence of V^ on the square root of molarity at a definite temperature was fitted with an equation of the (3) type where V| represents the apparent molar volume of the salt at infinite dilution, equal to the limiting partial molar volume of the salt, V2 (dm3 mol-1), A1, A2 and A3 are empirical constants which depend on the salt, solvent and temperature. In calculation of the empirical constants we considered also the V^ values calculated from the sum of the limiting conventional partial molar ionic volumes of the tetra-n-alkylammonium ions at the relevant temperature. The V o£ values for temperatures other than 273.15 K, 298.15 K and 323.15 K were obtained by analytical interpolation to Millero’s data3 and the cyclohexylsulfamate ion from ref.7 For n-Pe4N+ ion we used the average value (0.3996 dm3 mol-1) at 298.15 K given in refs.3, 12 The conventional limiting partial molar ionic volume of the n-Pe 4 N+ ion at other temperatures, calculated from the difference between V^ and VC y– , is given in Table 2. To represent the experimental data adequately for Et4NCy we used three and for the other salts only two empirical constants in rel. (3). The regression values of V£ and the empirical constants of rel. (3), together with the standard error of the estimate, sv are given in Table 2. As can be seen from this Table, the sign of the limiting slope, A1, is negative for all investigated salts. It is interesting to note that aqueous solutions of Me4NCy studied previously6 follow the limiting Debye-Hückel law. Furthermore, the limiting slope increases with increasing size of the R4N+ ion. However, the effect of temperature on the limiting slope is different. So, dAJdTat 298.15 K is negative for Et4NCy, n-Pr4NCy and n-Bu4NCy, while for n-Pe4NCy it is positive. In recent years many investigators have studied the V^ values of tetra-n-alkylammonium salts in aqueous solution and they have shown that these salts have large negative deviations from the limiting slope, e. g. 12, 13, 14 On the contrary, Franks and Smith15 determined V^ values of some tetra-n-alkylammonium salts in dilute aqueous solutions and showed that V^ values approach limiting behaviour (a positive limiting slope) in extremely dilute solutions in accordance with the Debye-Hückel law. From these it may be concluded that for such systems in extremely dilute range, c < 0.01 mol dm-3, the initial slope being positive. The values of V| and the empirical constants of rel. (3) obtained were tested by a Redlich type of equation16 which follows from the combination of rel.(1) and 3) d = dv +\M1 - V£d0 )c - AldBci'2 - A2dc2 - A^d0c5n (4) In calculation of the density via rel. (4), the last term was used only for Et4NCy solutions. The calculated densities are, within experimental uncertainty, equal to those given in Table 1. From the standard deviation, sd presented (5) in Table 2, it follows that the parameters (rel. 3) obtained correctly represent the experimental density data. The partial molar volume of solute, V 2 (dm3 mol–1) and solvent, Vx (dm3 mol–1) were computed from V^ by the following relations:11 (6) (7) Klofutar et al.: Apparent Molar Volume and Apparent Molar Expansibility of Tetraethyl-, Tetra-n-propyl-, ... Acta Chim. Slov. 462 Table 1. Density of aqueous solutions of tetraethyl- ( Et4NCy), tetra-n-propyl- (n-Pr4NCy), tetra-n-butyl- (n-Bu4NCy), and tetra-n-pentylammonium (n-Pe4NCy ) cyclohexylsulfamates at the indicated molalities and temperatures. m d (kg dm–3) at T(K) (mol –1 293.15 298.15 303.15 313.15 323.15 333.15 Et4NCy 0.00499 0.99840 0.99724 0.99584 0.99240 0.98822 0.98337 0,01036 0.99862 0.99746 0.99605 0.99261 0.98842 0.98357 0,01498 0.99881 0.99764 0.99624 0.99278 0.98859 0.98375 0,02026 0.99902 0.99786 0.99645 0.99299 0.98879 0.98392 0.02506 0.99923 0.99805 0.99664 0.99317 0.98897 0.98410 0.03037 0.99945 0.99827 0.99685 0.99338 0.98917 0.98429 0.05074 1.00028 0.99909 0.99766 0.99416 0.98992 0.98503 0.06100 1.00072 0.99950 0.99806 0.99454 0.99029 0.98539 0.07128 1.00114 0.99992 0.99847 0.99493 0.99067 0.98575 0.07514 1.00130 1.00007 0.99862 0.99508 0.99081 0.98589 0.08124 1.00156 1.00031 0.99885 0.99532 0.99104 0.98609 0.10049 1.00235 1.00109 0.99961 0.99604 0.99174 0.98679 0.12458 1.00332 1.00205 1.00055 0.99694 0.99261 0.98763 n-Pr4NCy 0.02635 0.99897 0.99778 0.99636 0.99289 0.98867 0.98379 0.05031 0.99967 0.99847 0.99702 0.99352 0.98926 0.98434 0.07657 1.00046 0.99923 0.99776 0.99420 0.98990 0.98495 0.10035 1.00117 0.99991 0.99842 0.99481 0.99048 0.98549 0.12618 1.00194 1.00065 0.99914 0.99549 0.99110 0.98607 0.15169 1.00271 1.00139 0.99985 0.99614 0.99172 0.98664 0.17753 1.00348 1.00214 1.00056 0.99681 0.99233 0.98722 0.19408 1.00398 1.00260 1.00101 0.99721 0.99271 0.98757 0.25226 1.00568 1.00428 1.00262 0.99870 0.99405 0.98884 n-Bu4NCy 0.02701 0.99885 0.99765 0.99621 0.99271 0.98846 0.98357 0.05166 0.99943 0.99820 0.99673 0.99316 0.98885 0.98391 0.07686 1.00003 0.99876 0.99726 0.99363 0.98926 0.98425 0.10195 1.00063 0.99932 0.99778 0.99409 0.98966 0.98459 0.12777 1.00126 0.99992 0.99833 0.99456 0.99006 0.98492 0.14773 1.00175 1.00037 0.99876 0.99493 0.99038 0.98519 0.17718 1.00246 1.00103 0.99937 0.99545 0.99081 0.98555 0.20069 1.00304 1.00156 0.99986 0.99587 0.99117 0.98585 0.22877 1.00374 1.00220 1.00046 0.99637 0.99159 0.98620 0.25358 1.00434 1.00276 1.00097 0.99680 0.99195 0.98649 0.28523 1.00512 1.00348 1.00162 0.99733 0.99238 0.98685 0.34975 1.00672 1.00493 1.00294 0.99842 0.99327 0.98757 n-Pe4NCy 0.00450 0.99828 0.99711 0.99571 0.99227 0.98807 0.98322 0.01128 0.99840 0.99722 0.99581 0.99234 0.98813 0.98325 0.01567 0.99847 0.99729 0.99587 0.99239 0.98817 0.98328 0.02033 0.99855 0.99737 0.99594 0.99245 0.98820 0.98330 0.02709 0.99868 0.99748 0.99604 0.99252 0.98826 0.98333 0.03022 0.99874 0.99753 0.99609 0.99256 0.98829 0.98335 0.03494 0.99882 0.99761 0.99616 0.99261 0.98832 0.98338 0.04131 0.99895 0.99772 0.99625 0.99269 0.98838 0.98341 0.04795 0.99907 0.99783 0.99635 0.99277 0.98843 0.98344 0.05259 0.99917 0.99792 0.99643 0.99282 0.98847 0.98347 0.06193 0.99936 0.99808 0.99657 0.99293 0.98855 0.98351 0.06576 0.99943 0.99815 0.99663 0.99298 0.98858 0.98353 2007, 54, 460–468 Figure 1. Dependence of apparent and partial molar volume of Et4NCy on square root of molar concentration; apparent molar volume: 293.15 K -, 313.15 K 8, 333.15 KV. The full lines are calculated using the rel. (3) and respective parameters given in Table 2; partial molar volume: 293.15 K --------, 313.15 K . . . . and 333.15 K ----- . -; the lines were calculated according to rel. (9) using the parameters given in Table 2. where Vo is the molar volume of water and dVJdc1/2, according to rel. (3), is given as where the last term of rel. (8) was used only for Et4NCy. The partial molar volumes of the salts are compiled in Table 3 as a function of temperature-independent molalities, m and presented in Fig. 1 for Et4NCy at three temperatures. At low solute concentration the error caused by neglecting the concentration terms within the square brackets of rels. (6) and (7) is less than 1 × 10–4 dm3 mol–1 for V 2 and 5 × 10–5 dm3 mol–1 for Vx and the values of square brackets are close to 0.5 (rel. 6) and 1.0 (rel. 7). Taking into account rels. (3), (6) and (8), the partial molar volume of solute can be given at low solute concentration as where the last term is used only for Et4NCy. For the solutions investigated rel. (9) is valid over the entire concentration range used for Et4NCy and n-Pe4NCy, while for n-Pr4NCy and n-Bu4NCy the upper concentration at which rel. (9) can be used gradually increases with increasing temperature; at 293.15, 298.15 and 303.15 K the upper concentration is about 0.150 mol dm–3, at 313.15 K it is 0.200 mol dm–3 and at 323.15 and 333.15 K rel. (9) is valid up to the maximal concentration used. For simple 1:1 electrolytes the upper limit of validity of rel. (9) is about 1.0 mol dm–3.11 The partial molar volume of the solvent in the whole concentration and temperature range studied was found to be close to the molar volume of water at a definite temper-– ature within ±5 × 10–5 dm3 mol–1. The V values of some Klofutar et al.: Apparent Molar Volume and Apparent Molar Expansibility of Tetraethyl-, Tetra-n-propyl-, ... Acta Chim. Slov. 2007, 54, 460–468 463 Table 2. Limiting apparent molar volume, VFo, empirical constants of rel. (3), standard error of estimate, sv, and standard deviation of density, sd, of tetraethyl- (Et4NCy), tetra-n-propyl-(n-Pr4NCy), tetra-n-butyl- (n-Bu4NCy), and tetra-n-pentylammonium (n-Pe4NCy ) cyclohexylsulfamates in aqueous solution, and the conventional partial molar volume of tetra-n-pentylammonium ion, Vno-Pe4N+ as a function of temperature. Solute T (K) 293.15 298.15 303.15 313.15 323.15 333.15 Et4NCy V× 103 (dm3 mol–1) 271.96 ± 0.07 272.67 ± 0.11 273.38 ± 0.23 275.61 ± 0.17 277.86 ± 0.15 280.43 ± 0.06 A1 × 103 (dm9/2 mol–3/2) -38.9 ± 1.8 -49.3 ± 2.7 -52.7 ± 5.8 -51.0 ± 4.2 -60.9 ± 3.9 -52.4 ± 1.6 A2 × 103 (dm6 mol–2) 99.7 ± 12 192.3 ± 18 232.3 ± 39 214.6 ± 29 279.5 ± 26 209.0 ± 11 A3 × 103 (dm15/2 mol–5/2) -96.6 ± 23 -267.4 ± 35 -346.2 ± 75 -307.1 ± 55 -412.1 ± 50 -282.0 ± 21 sv × 103 (dm3 mol–1) 0.09 0.13 0.29 0.21 0.19 0.08 sd × 106 (kg dm-3) 4.4 5.0 8.5 6.6 8.8 7.2 n-Pr4NCy V× 103 (dm3 mol–1) 337.05 ± 0.03 338.25 ± 0.03 339.56 ± 0.03 342.25 ± 0.04 345.41 ± 0.04 348.44 ± 0.05 A1 × 103 (dm9/2 mol–3/2) -6.6 ± 0.03 -8.0 ± 0.2 -5.9 ± 0.3 -7.0 ± 0.4 -9.3 ± 0.4 -8.1 ± 0.5 A2 × 103 (dm6 mol–2) –4.1 ± 0.7 –4.2 ± 0.5 -3.3 ± 0.6 1.3 ± 0.7 7.0 ± 0.8 5.1 ± 0.9 sv× 103 (dm3 mol–1) 0.04 0.03 0.04 0.05 0.05 0.06 sd × 106 (kg dm-3) 7.3 3.7 4.7 7.4 5.2 5.6 n-Bu4NCy V£× 103 (dm3 mol–1) 397.85 ± 0.07 399.57 ± 0.06 401.81 ± 0.02 405.86 ± 0.02 410.95 ± 0.02 415.42 ± 0.04 A1 × 103 (dm9/2 mol–3/2) -0.35 ± 0.54 -0.39 ± 0.5 -2.1 ± 0.1 -3.9 ± 0.2 -9.5 ± 0.2 -11.6 ± 0.4 A2 × 103 (dm6 mol–2) -12,8 ± 0.9 -10.4 ± 0.8 -6.9 ± 0.2 -1.0 ± 0.3 8.5 ± 0.3 14.0 ± 0.8 sv× 103 (dm3 mol–1) 0.09 0.08 0.02 0.03 0.03 0.09 sd × 106 (kg dm–3) 7.7 6.4 4.8 5.1 6.7 16.1 n-Pe4NCy V× 103 (dm3 mol–1) 460.85 ± 0.06 462.99 ± 0.04 464.60 ± 0.03 469.03 ± 0.02 474.20 ± 0.01 480.11 ± 0.01 A1 × 103 (dm9/2 mol–3/2) -1.4 ± 1.0 -1.5 ± 0.6 -0.3 ± 0.4 -0.1 ± 0.4 -1.7 ± 0.2 -3.1 ± 0.3 A2 × 103 (dm6 mol–2) -33.8 ± 4 -29.5 ± 2 -19.3 ± 2 -8.2 ± 2 1.8 ± 0.8 2.4 ± 1.0 sv× 103 (dm3mol–1) 0.08 0.05 0.03 0.03 0.02 0.02 sd × 106(kg dm-3) 2.8 1.3 0.9 1.5 0.5 0.6 × 103(dm3 mol 1) 337.7 ± 0.4 339.6 ± 0.4 340.2 ± 0.5 343.5 ± 0.6 347.6 ± 0.6 352.3 ± 0.9 tetra-n-alkylammonium bromides in aqueous solution have been determined previously by Schiavo et al.17 They found that for a relatively high concentration of the salts – (2.5 mol dm–3), V1 increased with increasing concentration of solute and temperature. Conway et al.13 found that in aqueous solution VFo of tetra-n-alkylammonium halides is a linear function of the –o cation molecular weight, MR4N+ where Vx– is the limiting (10) partial molar ionic volume of the anion and w is the slope of the Conway plot.13 The dependence of the limiting partial molar volume of the tetra-n-alkylammonium cyclo-hexylsulfamates, together with Me4NCy5 on the molecular weight of R4N+ cation is presented in Fig. 2 at 298.15 K, from which it is evident that a linear correlation exists. This linear relation (10) was also obtained at other temperatures studied. The intercepts and slopes of rel. (10) are given in Table 4 together with the linear correlation coefficient and the standard error of the estimate. The val-–o ues of the intercept, i. e. VCy–, are within experimental uncertainties equal to those given in ref.7 though with a higher error of about 2.5 × 10–3 dm3 mol–1. A relatively – high scatter of Vxo– was also observed earlier on studying the partial molar volumes of tetra-n-alkylammonium hydroxides in aqueous solution8 or even of tetra-n-alky- lammonium perchlorates in non-aqueous solutions.19 The slope w of rel. (10) slightly increases with increasing temperature (see Table 4), from (1.13 ± 0.1) dm3 kg–1 at 293.15 K to (1.194 ± 0.005) dm3 kg–1 at 333.15 K, with a positive value of ?w/?t = (7.4 ± 1.3)10–4 dm3 kg–1 K–1. This increase of the slope w with temperature was also observed by Krumgalz earlier.5 Figure 2. Dependence of the limiting apparent molar volume VFo f some tetra-n-alkylammonium cyclohexylsulfamates in aqueous solution on the molecular weight of the R4N+ ion at 298.15 K; the VFo value of Me4NCy is given in ref.5 n-Pe N Klofutar et al.: Apparent Molar Volume and Apparent Molar Expansibility of Tetraethyl-, Tetra-n-propyl-, ... 464 Acta Chim. Slov. 2007, 54, 460–468 V Table 3. Partial molar volume, 2 of tetraethyl-, tetra-n-propyl-, tetra-n-butyl-, and tetra-n-pentylammonium cyclohexylsulfamates in aqueous solutions at indicated molalities and temperatures. m (mol kg–1) 293.15 V x IO3 (dm3 mol x) at T (K) 298.15 303.15 313.15 323.15 333.15 Et4NCy 0.00499 0,01036 0,01498 0,02026 0.02506 0.03037 0.05074 0.06100 0.07128 0.07514 0.08124 0.10049 0.12458 268.8 267.8 267.4 267.0 266.8 266.6 266.2 266.1 266.0 266.0 265.9 265.8 265.7 269.1 268.4 268.2 268.0 267.9 267.9 267.9 267.8 267.7 267.6 267.4 266.8 265.5 269.8 269.2 269.1 269.0 269.1 269.1 269.3 269.2 268.9 268.9 268.7 267.7 265.9 272.1 271.5 271.3 271.2 271.2 271.2 271.4 271.3 271.2 271.1 270.9 270.2 268.8 273.8 273.3 273.2 273.2 273.3 273.4 273.9 273.9 273.8 273.7 273.5 272.6 270.7 276.7 276.0 275.8 275.7 275.7 275.7 275.9 275.9 275.9 275.8 275.7 275.4 274.2 n-Pr.NCy 0.02635 0.05031 0.07657 0.10035 0.12618 0.15169 0.17753 0.19408 0.25226 335.3 334.5 333.8 333.2 332.7 332.2 331.7 331.5 330.6 336.6 335.9 335.2 334.7 334.2 333.7 333.3 333.0 332.2 338.0 337.3 336.7 336.2 335.8 335.3 334.9 334.7 333.9 340.6 340.1 339.6 339.3 339.2 338.7 338.5 338.3 337.9 343.5 343.0 342.7 342.5 342.3 342.2 342.1 342.1 341.9 346.8 346.3 345.9 345.7 345.5 345.4 345.2 345.2 345.0 m (mol kg–1) 293.15 V x IO3 (dm3 mol x) at T (K) 298.15 303.15 313.15 323.15 333.15 n-Bu4NCy 0.02701 0.05166 0.07686 0.10195 0.12777 0.14773 0.17718 0.20069 0.22877 0.25358 0.28523 0.34975 397.1 396.5 395.8 395.2 394.6 394.2 393.5 393.1 392.5 391.9 391.4 390.2 398.9 398.4 397.9 397.4 396.9 396.5 396.0 395.6 395.1 394.7 394.2 393.3 400.9 400.4 400.0 399.5 399.1 398.8 398.4 398.0 397.6 397.3 396.9 396.2 404.9 404.5 404.1 403.9 403.6 403.5 403.2 403.1 402.9 402.7 402.5 402.2 409.1 408.6 408.4 408.2 408.1 408.0 408.0 408.0 408.0 408.1 408.1 408.3 413.3 412.9 412.8 412.7 412.8 412.8 413.0 413.1 413.3 413.4 413.6 414.2 n-Pe.NCy 0.00450 0.01128 0.01567 0.02033 0.02709 0.03022 0.03494 0.04131 0.04795 0.05259 0.06193 0.06576 460.4 459.9 459.5 459.2 458.7 458.5 458.2 457.7 457.3 456.9 456.3 456.1 462.6 462.1 461.8 461.5 461.1 460.9 460.6 460.2 459.8 459.5 459.0 458.7 464.4 464.1 463.9 463.8 463.5 463.4 463.2 463.0 462.7 462.5 462.2 462.1 468.9 468.8 468.8 468.7 468.6 468.5 468.4 468.3 468.2 468.2 468.0 468.0 474.0 473.9 473.9 473.9 473.9 473.9 473.9 473.9 473.8 473.8 473.8 473.8 479.8 479.7 479.6 479.6 479.5 479.5 479.4 479.4 479.3 479.3 479.3 479.3 The average value of the slope in different organic solvents was established to be 1.235 dm3 kg–1 at 298.15 K.5 This value is higher than values obtained earlier from the partial molar volumes of tetra-n-alkylammonium hydroxides in aqueous solution, where w = (1.107 ± 0.012) dm3 kg–1,18 and from aqueous solutions of different ions or tetra-n-alkylammonium perchlorates in non-aqueous solu-tions,5 which varies from w = 1.162 dm3 kg–1 to w = 1.199 dm3 kg–1.19 The Conway method was criticized by Hefter and Marcus20 since extrapolation (rel. 10) neglects the solvent exclusion volume which depends on the size of the solvent and on the electrostriction caused by charge. The limiting apparent molar volume can be separated into the individual volumes for the cation and anion:3 (11) The conventional values of VRo4N+ and VCo y– take the standard partial molar ionic values of hydrogen ion to be zero at all temperatures. The absolute partial molar ionic – value, Vio,abs of any ion i of charge zi can be obtained from5 vm + z no (12) where Vo(H+bs) was suggested by Conway21 to be -5.5 × 10–3 dm3 mol–1, -(4.9 × 10–3 ± 7 × 10–4) dm3 mol–1 by King22 and -5.4 × 10–3 dm3 mol–1 by Zana and Yeager23 at 298.15 K. The V o abs values of the investigated ions, –o Table 4. Dependence of intercept, Vx–, and slope, w, of rel. (10) on temperature, linear correlation coefficient, r, and standard error of estimate, s. Ion 293.15 T(K) 298.15 303.15 313.15 323.15 333.15 V × 103 (dm3 mol 1) w (dm3 kg-1) s × 103 (dm3 mol 1) 125.0 ± 2.7 125.5 ± 2.6 126.0 ± 2.6 127.0 ± 2.5 127.0 ± 2.5 125.0 ± 1.1 1.13 ± 0.01 1.13 ± 0.01 1.14 ± 0.01 1.15 ± 0.01 1.17 ± 0.01 1.194 ± 0.005 0.9997 0.9997 0.9997 0.9997 0.9997 0.9999 3.4 3.3 3.3 3.3 3.1 1.1 r Klofutar et al.: Apparent Molar Volume and Apparent Molar Expansibility of Tetraethyl-, Tetra-n-propyl-, ... Acta Chim. Slov. 2007, 54, 460–468 465 together with Me4N+ and NH4+ ions, are given in Table 5 – where for Vo(H+abs) a value of –5.5 × 10–3 dm3 mol–1 was –o used. The Vi,abs values of cations are within experimental error equal to those given in ref.4 The standard partial molar ionic volumes at 298.15 K were treated by a model4 which involves a hydration shell of specified thickness, where Vi,hydr is the volume of a hydrated ion when not electrostricted, Vi,el 1 is the electrostriction caused by the charge in the hydration shell, Vi,el 2 is the electrostriction caused in the water surrounding this shell and Vi,str is the volume increment which is a consequence of the structuring of water molecules around hydrophobic ions: (13) In the adopted model it was supposed that the hydration shell of the ion contains ni molecules of water, given by (14) where ri is the ionic radius (pm) and A = 360 pm a constant. The values of ionic radii, given in Table 5 were calculated from King’s van der Waals volumes22 while the ionic radius of the cyclohexylsulfamate ion was determined from X-ray structural data previously24. The number of water molecules bound in the hydration shell of ions (see Table 5) is approximately one molecule of water per ion. Previously we found that ni for the cyclohexyl-sulfamate ion is about 0.54.24 The molar volume of the hydrated ion was calculated from (15) where NA is Avogadro’s constant and ?r is the (15) thickness of the unelectrostricted hydration shell, obtained from (16) where d = 276 pm is the diameter of a water n.lxd:s/6 = (4K/3)Ur[+dr.f-r?~\ (16) molecule. From the results collected in Table 5 it can be seen that Vi,hydr contributes a fraction of about 0.75 of the partial molar ionic volume of Me4N+, 0.72 of the Et4N+ ion and then a constant value of 0.68 for higher members of the series of R4N+ ions. A relatively high value, of 2.1 was obtained for the NH4+ ion and 1.13 part of cyclo-hexylsulfamate ion. The electrostriction contribution to the ionic volume, i. e. diminution of the volume within the hydration shell was calculated by (17) and given in Table 5. Except for the NH4+ ion, Vi,el 1 contributes a minor and negative value to the absolute partial molar ionic volume. The electrostriction contribution in the surroundings of the hydrated ion was calculated by and Vuli =-417.5|Z||[J/(rç +^)J (18) given in Table 5. Like Viel 1, the values of Vi el 2 are small and negative and increase from the Me4N+ ion to the n-Pe4N+ ion. The rest of the partial molar ionic volume was ascribed to the volume increment caused by the structuring of water molecules around hydrophobic ions with an open, clathrate-like structure, yielding a positive value of Vi,str. It was found4 that if Vi str was assigned a value of 6.1 × 10–3 dm3 mol–1 per CH3-group and 5.4 × 10–3 dm3 mol–1 per -CH2-group, the value of Vi str for hydrophobic R4N+ ions can be determined. The calculated values of Vi,str of R4N+ ions are presented in Table 5. The sum of the various contributions to the partial molar ionic volumes given by rel. (13), i. e. V o cal, are presented in Table 5, from which it follows that the adopted model satisfactorily describes the behaviour of highly hydrophobic R4N+ ions. The greatest difference between V o obs and V o cal was observed for the Et4N+ ion (~ 2.5 %), while for the other ions this difference was smaller than one per cent. The Vostr values of NH4+ and Cy– ions were obtained as the difference between V o obs and the other ionic volume contributions. Surprisingly Vistr of the cyclohexylsulfa-mate ion is negative in spite of the fact that it contains a cyclohexyl entity which causes structural effects on water molecules like the phenyl group, to which a positive value of 23.3 × 10–3 dm3 mol–1 was assigned.4 It seems that a theory which can satisfactorily describe the behaviour of large hydrophobic ions in an aqueous environment is at present beyond reach. Some difficulties arise from the inadequacy of the values of radii assigned to the ions. Thus, for example, the ionic radii of R4N+ ions, given by Robinson and Stokes25 are substantially larger than those given in Table 5. As comented on by Marcus4 the inadequacy of the ionic radii is possibly the main reason for the observed discrepancies. The second reason arises from the charge of the ion. Recent quantum mechanical ab initio calculations show that the positive charge of the R4N+ ion is distributed on the a and ß methylene carbon atoms rather than on the nitrogen atom.26, 27 Similarly, in the case of the cyclohexyl-sulfamate ion, the charge is distributed along the three oxygen atoms and the hydrocarbon portion of the ion.7, 24 As a consequence the charge density of such ions may be low because electron delocalization and its polarizing effect on water molecules could be unusually small. In the last column in Table 5, the packing density of the investigated ions in solution, defined by28 are given (19) where VvdW is van der Waals volume of the ion. This represents the fraction of the partial molar ionic volume occupied by the ions of the solute in solution at infinite dilution. The Klofutar et al.: Apparent Molar Volume and Apparent Molar Expansibility of Tetraethyl-, Tetra-n-propyl-, ... 466 Acta Chim. Slov. 2007, 54, 460–468 Table 5. Partial molar ionic volumes, ionic radii, hydration numbers, various volume increments of partial molar ionic volumes (rel. 13), calculated partial molar ionic volumes and packing densities of some R4N+ ions, NH4+ and Cy– ions at 298.15 K. Ion V,°obS X 103 (dm3 mol x) (pm) V i,hydrXl°3 -V i,ellXl°3 -V i,el 2Xl°3 V^Xltf VU x lo3 (dm3 mol x) (dm3 mol x) (dm3 mol x) (dm3 mol x) (dm3 mol x) V /Vo i,vdW i,obs NH + 12.6 ± 0.4a 168d 2.1 25.8 13.7 1.9 2.4 12.6 0.944 Me4N+ 85.1 ± 0.5b 280d 1.3 64.1 2.8 1.4 24.4 84.3 0.650 Et4N+ 143.4 ± 0.4 337d 1.0 103.6 1.4 1.2 46.0 147.0 0.671 n-Pr4N+ 209.0 ± 0.4 379d 0.9 142.9 0.8 1.1 67.6 208.6 0.656 n-Bu4N+ 270.3 ± 0.4 413d 0.9 184.3 0.7 1.0 89.2 271.8 0.659 n-Pe4N+ 333.7 ± 0.4a 443d 0.8 225.3 0.5 0.9 110.8 334.7 0.656 Cy- 118.3 ± 0.4c 370e 1.0 134.1 0.9 1.1 -13.8 118.3 1.084 a) Ref.4 b) Ref.6 c) Ref.7 d) Ref.22 e) Ref.24 VvdW values of the NH4+ ion and R4N+ ions were taken from22 and for the cyclohexylsulfamate ion from.24 The average value of the packing density, considering all the investigated R4N+ ions, is 0.658 ± 0.008, which is in good agreement with the value 0.655 ± 0.003 given by King,22 with a plateau density of 0.660.28 The packing density of the cyclohexylsulfamate ion is greater than unity. It seems that molecules of water are packed more tightly around R4N+ ions than around the cyclohexylsulfamate or NH4+ ion. The temperature derivative of the limiting apparent molar volume, 0^ = (dV^ ldt)P, was calculated from the temperatre dependence of V^ given as where a0, a1 and a2 (20) are empirical constants depending on salt and solvent, and T = 298.15 K. Rel. (20) is a quadratic function and consistent with the volumes going through a maximum; the latter behaviour is characteristic of most salts that have been studied in aqueous solution, as shown by Helgeson and Kirkham.29 The values of the constants of relation (20) are given in Table 6 together with the standard error of the estimate, s. From relation (20) it follows that 0^ is a linear function of temperature and that O^ = a1 at 298.15 K. From the results collected in Table 6 it follows that ®°E at other temperatures studied increases with temperature. The fact that (&% is a linear function of temperature with a positive coefficient for the temperature term means that the second derivative of V – o with respect to temperature is also positive. As Hepler30 has pointed out that (21) where A C P is the partial molar isobaric heat capacity of aqueous electrolyte at infinite dilution and P is a pressure, a negative value of (dA CpdP)T is evidence that the investigated ionic solutes behave as structure-making solutes in water over the experimental temperature range studied. On the other hand, NH4Cy7 and Me4Ncy,6 having a negative coefficient for the temperature term with a negative second derivative of V^ with respect to temperature, may be classified as structure-breaking solutes in water. This difference with respect to other ionic R4NCy solutes can be explained by the different solvation behaviour of NH4+ andMe4N+ ions, e. g.4, 6, 13 In Fig. 3 the dependence of O^ at 298.15 K on the molecular weight of the salts is given and where the corresponding data on NH4Cy7 and Me4NCy6 are included. From the plot it can be seen that the limiting partial molar expansibility at 298.15 K increases with increasing molecular weight of the salts. From the limiting partial molar ionic expansibility of the investigated ions, ER4 N+ and ECy– , the limiting partial molar expansibility of the salts can be calculated from31 (22) the E o N+ values were obtained from the temperature 01 = Er^- + Ecf (22) dependence of VRo4N+ values using Millero’s data3 at 273.15, 298.15 and 323.15 K for Et4N+, n-Pr4N+ and n-Bu4N+ ions, while for the n-Pe4N+ ion we used an average value of VRo4N+ ion at 298.15 K obtained from,312 while for other temperatures the V o N+ values given in Table 2 were used. The temperature dependence of VRN+ was expressed by rel. (20). The values of ER4 N+ at various temperatures are given in Table 7. Table 6. Values of the constants ai of relation (20) and the standard error of the estimate, s. Solute a0 X 103 (dm3 mol"1) a2 x 103 s x IO3 (dm3 mol"1 K"1) (dm3 mol"1 K"2) (dm3 mol"1) aj x IO3 Et4NCy 272.67 ± 0.06 0.165 ± 0.009 0.0017 ± 0.0003 0.12 n-Pr4NCy 338.26 ± 0.05 0.254 ± 0.007 0.0011 ± 0.0002 0.08 n-Bu4NCy 399.71 ± 0.13 0.402 ± 0.020 0.0014 ± 0.0006 0.24 n-Pe4NCy 462.73 ± 0.10 0.368 ± 0.015 0.0037 ± 0.0005 0.17 r n Klofutar et al.: Apparent Molar Volume and Apparent Molar Expansibility of Tetraethyl-, Tetra-n-propyl-, ... Acta Chim. Slov. 2007, 54, 460–468 467 Table 7. Limiting partial molar ionic expansibility of some R4N+, NH4+ and Cy– ions at the indicated temperatures. Ion Eioon × 103 (dm3 mol 1K 1) at T(K) 293.15 298.15 303.15 313.15 323.15 333.15 NH + 0.0344 0.0325 0.0307 0.0269 0.0233 0.0196 Me4N+ 0.0469 0.0522 0.0574 0.0678 0.0782 0.0886 Et4N+ 0.0760 0.0826 0.0892 0.1025 0.1158 0.1291 n-Pr4N+ 0.1070 0.1274 0.1478 0.1886 0.2294 0.2702 n-Bu4N+ 0.2398 0.2780 0.3162 0.3927 0.4692 0.5457 n-Pe4N+ 0.2116 0.2487 0.2857 0.3598 0.4338 0.5079 Cy– 0.1150 In Table 7 we included the corresponding data for NH4+ and Cy– ions obtained in an analoguous way previ-ously.6,7 From the data collected in Table 7, it can be seen –o that ENH4+ linearly decreases with increasing temperature, –o –o ECy– is constant at the indicated temperatures, while ER4N+ –o values linearly increase with tem– perature. In Fig. 4 ER4N+ the values are plotted versus MR4N+ of the ions at 298.15 K. From this plot it follows that the limiting partial molar ionic expansibility of R4N+ and NH4+ ions gradually increases with increasing ionic molecular weight, with the exception of the n-Bu4N+ ion whose different solvation behaviour from other ions was pointed out by Wen and Saito.12 They proposed a clathrate-like structure for – n-Bu4NBr with ?Vno-Bu4N+/?? between (0.2 – 0.5) × 10–4 dm3 mol–1 K–1 at room temperature. According to Hepler30 (see rel. 21) the investigated cations are structure-making in water, while the cyclohexylsulfamate anion according to this view is neither a structure-making nor structure-breaking ion. The large cyclohexylsulfamate ion causes structural effects which are different from those of simple Figure 3. Dependence of the limiting partial molar expansibility of some tetra-n-alkylammonium and ammonium cyclohexylsulfa-mates in aqueous solution on molecular weight at 298.15 K; the dotted line was calculated using rel. (22) and corresponding data given in Table 7; the data for NH4Cy and Me4NCy are given in refs.7,6 monovalent ions.7 A similar plot to those given in Fig. 4 – was obtained earlier by Krumgalz.5 ERo4N+ values on his plots fall on two straight lines with a point of intersection, showing that the specific hydrophobic hydration of R4N+ ions begins to play a dominant role only starting from the n-Pr4N+ ion. The calculated sum of rel. (22) at 298.15 K within experimental uncertainty for n-Pr4NCy, n-Bu4NCy and n-Pe4NCy is equal to those given in Table 6, while for Et4NCy it is higher by about three standard deviations (see Fig. 3). –o –o Figure 4. Dependence of ENH4+ and ER4N+ on molecular weight of the cation at 298.15 K; the data are from ref.3 3. Experimental 3. 1. Materials All tetraalkylammonium salts investigated, i. e. tetraethyl- (Et4NCy), tetra-n-propyl- (Pr4NCy), tetra-n-butyl- (Bu4NCy), and tetra-n-pentylammonium cyclo-hexylsulfamates (Pe4NCy) were obtained by careful neutralization of cyclohexylsulfamic acid (purchased from Sigma) with the corresponding base (Fluka). The purity of the salts were checked after recrystallization from various organic solvents (Et4NCy from ethyl methyl ketone, Pr4NCy from ethyl acetate, Bu4NCy and Pe4NCy from cyclohexane) by analysis of the elements C, H and N (Perkin Elmer, 2400 Series II CHNS/O Analyzer) and also by ion exchange of the cation of the salt by the hydrogen ion (DOWEX, Type 50 WX8); a purity of 99.7 % at least (Et4NCy) or better, e. g. 99.9% for Bu4NCy was determined. The salts were kept in a vacuum desiccator over P2O5. The solutions investigated were prepared on a molal concentration scale by precise weighing, using a digital balance (Mettler Toledo, model AT201, Switzerland) accurate to within ±1 × 10–8 kg. Before use the solutions were degassed by ultrasound (ultrasonic bath, Bandelin Sonorex, type TK 52, Berlin, Germany). Klofutar et al.: Apparent Molar Volume and Apparent Molar Expansibility of Tetraethyl-, Tetra-n-propyl-, ... 468 Acta Chim. Slov. 2007, 54, 460–468 3. 2. Density measurements The density, d, of aqueous solutions of all sweeteners was measured by a vibrating-tube density meter (Anton Paar, model DMA 60, Graz, Austria) equipped with a measuring cell (Anton Paar, type 602) and a digital thermometer (Anton Paar, DT 100–20) with a precision of ± 0.01 K. The apparatus was calibrated with doubly distilled water32 and dry air33 at each investigated temperature at atmospheric pressure. The temperature in the measuring cell was regulated to better than ± 0.01 K, using an external bath circulator (Haake, DC3-B3, Karlsruhe, Germany). The uncertainty of the density measurements was ±2 × 10–5 kg dm–3. 4. References 1. H. S. Frank, W. Y. Wen, Discuss. Faraday Soc. 1957, 24, 133– 140. 2. F. Franks, Water; The Royal Socity of Chemistry, London, 1984, 57–68. 3. F. J. Millero in: Water and Aqueous Solutions, Structure, Thermodynamics and Transport Properties (R. A. Horne, Ed.); Wiley-Interscience, New York, 1972, 519–595. 4. Y. Marcus, J. Chem. Soc. Faraday Trans. 1993, 89, 713–718. 5. B. S. Krumgalz, J. Chem. Soc. Faraday Trans. 1980, 76, 1887–1904. 6. C. Klofutar, D. Rudan-Tasic, Monatsh. Chem. 2005, 136, 1727–1736. 7. C. Klofutar, J. Horvat, D. Rudan-Tasic, Monatsh. Chem. 2006, 137, 1151–1162. 8. S. Shamil, G. G. Birch, M. Mathlouthi, M. N. Clifford, Chem. Senses, 1987, 12, 397–409. 9. M. Mathlouthi, C. Bressan, M. O. Portmann, S. Serghat in: Sweet–Taste Chemoreception (M. Mathlouthi, J. A. Kanters, G. G. Birch, Eds.); Elsevier Applied Science, London, 1993, 141–174. 10. G. G. Birch, K. A. Haywood, G. G. Hannifly, C. M. Coyle, W. J. Spillane, Food Chem. 2004, 84, 429–435. 11. H. S. Harned , B. B. Owen, The Physical Chemistry of Electrolytic Solutions (3rd ed.); Reinhold Publishing Corp., New York, 1958, 358–370. 12. W.Y. Wen, S. Saito, J. Phys. Chem. 1964, 68, 2639–2644. 13. B. E. Conway, R. E. Verall, J. E. Desnoyers, Trans. Faraday Soc. 1966, 62, 2738–2749. 14. P. S. Nikam, A. B. Sawant, J. Chem. Eng. Data, 1997, 42, 585–589. 15. F. Franks, H. T. Smith, Trans. Faraday Soc. 1967, 63, 2586–2598. 16. O. Redlich, J. Phys. Chem. 1940, 44, 619–629. 17. S. Schiavo, B. Scrosati, A. Tommasini, Sci. Chim. 1967, 37, 211–218. 18. C. Klofutar, D. Rudan-Tasic, V. Mancic-Klofutar, J. Solution Chem.. 1997, 26, 1037–1047. 19. J. Krakowiak, D. Bobicz, W. Grzybkowski, J. Chem. Thermodynamics, 2001, 33, 121–133. 20. G. Hefter, Y. Marcus, J. Solution Chem.. 1997, 26, 249–266. 21. B. E. Conway, J. Solution Chem.. 1978, 7, 721–770. 22. E. J. King, J. Phys. Chem. 1970, 74, 4590–4592. 23. R. Zana, E. Yeager, J. Phys. Chem. 1967, 71, 521–536. 24. D. Rudan-Tasic, C. Klofutar, Food Chem. 2004, 84, 351– 357. 25. R. A. Robinson, R. H. Stokes, Electrolyte Solutions (2nd ed.); Dover Publications, New York, 2002, 123–126. 26. J. Nagano, H. Mizumo, M. Sakiyama, J. Phys. Chem. 1991, 95, 2536–2540. 27. A. V. Goldberg, A. A. Varnek, C. J. Stepanov, A. M. Chekma-rev, R. P. Ozerov, J. Phys. Chem. (Russian), 1991, 65, 2397– 2404. 28. E. J. King, J. Phys. Chem. 1969, 73, 1220–1232. 29. H. C. Helgeson, D. H. Kirkham, Am. J. Sci. 1976, 276, 97– 240. 30. L. G. Hepler, Can. J. Chem. 1969, 47, 4613–4617. 31. F. J. Millero, J. Phys. Chem. 1968, 72, 4589–4593. 32. G. S. Kell, J. Chem. Eng. Data, 1975, 20, 97–105. 33. F. Kohlrausch, Prakt. Phys. 1968, 3, 40. Povzetek Na osnovi merjenja gostote vodnih raztopin nekaterih tetraalkilamonijevih soli cikloheksilsulfaminske kisline pri ra-zli~nih temperaturah smo dolo~ili njihove volumenske lastnosti. Razlike v navideznem molskem volumnu ter navidezni molski ekspanzibilnosti preiskovanih topljencev, zlasti njihove limitne vrednosti, ka`ejo na vpliv velikosti tetraalkilam-onijevega kationa oz. nara{~ajo~ega hidrofobnega karakterja v izbrani homologni vrsti sinteti~nih sladil. Klofutar et al.: Apparent Molar Volume and Apparent Molar Expansibility of Tetraethyl-, Tetra-n-propyl-, ...