Volumetric manifestation of van der Waals interactions between cholesterol and organic solvents of linear structure

Fluid Phase Equilibria 167 Ž2000. 207–221 www.elsevier.nlrlocaterfluid

Volumetric manifestation of van der Waals interactions between cholesterol and organic solvents of linear structure Paweł Goralski ´

)

Department of Physical Chemistry, UniÕersity of Łodz, ´ ´ Pomorska 165, 90-236 Łodz, ´ ´ Poland Received 7 July 1999; accepted 21 October 1999

Abstract The densities of solution of cholesterol in several selected n-alkanes, alkan-1-ols and tertiary amines have been measured at 298.15 K. The values of standard partial molar volume of cholesterol have been calculated and the effect of solute–solvent and solvent–solvent interactions on the partial molar volume of cholesterol have been discussed. The increase in the alkyl chain length Ž R i . of solvents such as alkanes and amines enhances the dispersion solvent–solvent interactions Žper –CH 2 – . and weakens the contribution of dispersion forces in the cholesterol-solvent interactions. Both these effects bring about an increase in the partial molar volume of cholesterol with increasing the alkyl chain length of solvent molecules. In alcohols, the cholesterol–solvent dispersion interactions increase with R i . This effect may be accounted for by the self-association of the solvent. In consequence, there is no distinct dependence of partial molar volume of cholesterol on the alkyl chain length of alcohol. q 2000 Elsevier Science B.V. All rights reserved. Keywords: Density; Partial molar volume; Cholesterol; Alkanes; Alkan-1-ols; Tertiary amines; Van der Waals interactions; Dispersion forces; Solvation

1. Introduction Cholesterol plays a significant part in the processes that control the permeability and fluidity of cell membranes, with the specific structure of hydrophobic fragment of this molecule being a decisive factor. The non-polar part of cholesterol is held between the hydrocarbon chains of other lipids that form membranes by van der Waals interactions, and, as a matter of fact, by the dispersion forces of these interactions. The cholesterol hydroxyl group interacts through hydrogen bonds in the polar layer of membranes. Although van der Waals forces occur wherever a direct contact between molecules appears Ž condensed phases. , most physico-chemical methods fail to detect directly their presence. The )

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0378-3812r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 Ž 9 9 . 0 0 3 0 9 - X

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P. Goralskir Fluid Phase Equilibria 167 (2000) 207–221 ´

difficulties in the observation of van der Waals interactions concern also cholesterol. Therefore, so far cholesterol has been examined mainly in terms of specific interactions — hydrogen bonds formed through its hydroxyl group. Many experimental techniques have been used. The ability of cholesterol to self-associate in non-polar solvents has been studied by means of IR w1–3x, NMR w4,5x, VPO w5,6x, relaxation methods w7,8x and dielectric permittivity measurements w9x. The thermodynamics of hydrogen bond formation with molecular and ionic proton acceptors has been studied by means of IR w1,10–14x, calorimetry w15–19x or IRrcalorimetry w20–23x. Densimetric measurements as the basis for the determination of the volumic properties seem to be of particular importance. From our previous paper w24x, it follows that the formation of hydrogen bonds between cholesterol and solvents has no decisive effect on the partial molar volume of cholesterol Ž V20 . or its partial molar expansibility. This concerns also strong proton acceptors such as amides or HMPA. For most already examined solvents w24–26x with different donor–acceptor ability and different polarities, V20 of cholesterol has higher values than for pure cholesterol. A distinct decrease of cholesterol volume Ž contraction. was observed w24x only in the case of solvents with linear structures Ž n-heptane, di-n-butylamine, di-n-butylether, heptan-1-ol. independently of the possible appearance of solute–solvent specific interactions. The observed contraction has been regarded as a result of van der Waals interactions between the hydrocarbon fragment of the cholesterol molecule and alkyl solvent chains. This allows one to believe that volume measurements present one of the few physico-chemical methods capable of providing direct information about non-specific interactions Žparticularly about the dispersion forces of these interactions. taking place in solutions. These interactions are of particular importance in the formation of micellar or biologic systems, e.g., those between the hydrocarbon ‘‘tails’’ in the non-polar part of lipid bilayers. Therefore, the aim of the present study is to examine the volumic properties of cholesterol in solvents consisting of linear alkyl chains and showing different donor–acceptor properties: n-alkanes, alkan-1-ols and tertiary amines, i.e., solvents showing differences in the solute-solute and solute– solvent interactions.

2. Experimental Cholesterol Ž Sigma, standard for chromatography. was dried for a few days at about 608C over P2 O5 in vacuum. Reagent-grade solvents were dried by standard methods, distilled and degassified immediately before measurements. The preparation of solutions was carried out in a dry box. The densities of cholesterol solutions relative to the densities of pure solvents were measured using a digital flow densimeter Ž Sodev, model 03, Sherbrook, Quebec. . The densimeter was calibrated with reference to pure water and nitrogen gas Ž 1 atm. . The density of water was taken from Kell w27x and that of nitrogen were calculated by means of the van der Waals equation of state. The uncertainty in the measured densities was lower than 5 = 10y6 g cmy3.

3. Results The density measurements of cholesterol solutions in solvents in which cholesterol is slightly soluble were carried out within the concentration range from about 0.03 mol kgy1 to that of near

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Table 1 . of cholesterol solutions in alkanes, densities of pure solvents and partial molar volumes of Relative densities Ž dy dU 1 cholesterol at 298.15 K n-Hexane

n-Heptane y1 .

m Žmol kg

y3 .

D d Žg cm

y1 .

m Žmol kg

n-Octane y3 .

D d Žg cm

m Žmol kgy1 .

D d Žg cmy3 .

0.03167 0.002908 0.03240 0.002968 0.03534 0.003237 0.03799 0.003482 0.03803 0.003492 0.03970 0.003645 0.04059 0.003721 0.04079 0.003727 0.04112 0.003771 0.04178 0.003833 V20 s 374.78"0.34 cm3 moly1 dU1 s 0.65443 g cmy3 dU1 s 0.65484 a g cmy3

0.04026 0.003489 0.04042 0.003498 0.04026 0.003487 0.04042 0.003513 0.04026 0.003497 0.04042 0.003503 0.03886 0.003363 0.04032 0.003493 0.04214 0.003651 0.04443 0.003838 V20 s 379.49"0.29 cm3 moly1 dU1 s 0.67918 g cmy3 dU1 s 0.67946 a g cmy3

0.03601 0.002972 0.03682 0.003020 0.03721 0.003062 0.03817 0.003139 0.03824 0.003136 0.03850 0.003162 0.03881 0.003191 0.03890 0.003202 0.03924 0.003238 0.03955 0.003256 V20 s 383.22"0.35 cm3 moly1 dU1 s 0.69855 g cmy3 dU1 s 0.69862 a g cmy3

n-Decane m Žmol kgy1 .

n-Dedecane m Žmol kgy1 .

n-Hexadecane m Žmol kgy1 .

D d Žg cmy3 .

0.03640 0.002724 0.04052 0.003033 0.04283 0.003184 0.04429 0.003311 0.04581 0.003409 0.04622 0.003447 0.04737 0.003525 0.04957 0.003688 0.05033 0.003734 0.05260 0.003927 V20 s 389.19"0.34 cm3 moly1 dU1 s 0.72646 g cmy3 dU1 s 0.72635a g cmy3 a b

D d Žg cmy3 .

0.03087 0.002170 0.03118 0.002183 0.03176 0.002231 0.03214 0.002259 0.03491 0.002447 0.03676 0.002559 0.03755 0.002627 0.03916 0.002733 0.04026 0.002814 0.04427 0.003101 V20 s 391.12"0.33 cm3 moly1 dU1 s 0.74614 g cmy3 dU1 s 0.74587 b g cmy3

D d Žg cmy3 .

0.03074 0.001981 0.03318 0.002146 0.03354 0.002169 0.03658 0.002347 0.03829 0.002480 0.03875 0.002497 0.03981 0.002558 0.03985 0.002575 0.04144 0.002668 0.04152 0.002680 V20 s 392.13"0.30 cm3 moly1 dU1 s 0.77020 g cmy3 dU1 s 0.77030 b g cmy3

Ref. w28x. Ref. w29x.

saturation. In solvents in which cholesterol shows a good solubility Ž some amines and alcohols. , the maximum cholesterol concentration in the solutions under investigation did not exceed 0.14 mol kgy1. Experimentally measured differences in solution densities in relation to that of pure solvent are given in Tables 1–3, which contain also the experimentally found densities of the pure solvents at 298.15 K and those available in the literature. The apparent molar volume of cholesterol in solution, VF ,2 , was calculated from the following relationship: M2 1000 Ž d y dU 1 . VF ,2 s y Ž1. U d mdd1

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Table 2 . Relative densities Ž dy dU 1 of cholesterol solutions in alkan-1-ols, densities of pure solvents and partial molar volumes of cholesterol at 298.15 K Ethanol

Propan-1-ol

Butan-1-ol

Hexan-1-ol

Octan-1-ol

m Dd m Dd m Dd m Dd m Dd Žmol kgy1 . Žg cmy3 . Žmol kgy1 . Žg cmy3 . Žmol kgy1 . Žg cmy3 . Žmol kgy1 . Žg cmy3 . Žmol kgy1 . Žg cmy3 . 0.04609 0.002951 0.05333 0.003412 0.05449 0.003495 0.05732 0.003652 0.05899 0.003759 0.05908 0.003775 0.06252 0.003977 0.06286 0.003995 0.06286 0.004010 0.06649 0.004231 V20 s 387.17" 0.22 cm3 moly1 y3 dU 1 s 0.78501 g cm U a d1 s 0.78493 g cmy3 a b

0.04600 0.002822 0.05195 0.003186 0.05858 0.003581 0.06603 0.004034 0.07022 0.004264 0.07541 0.004594 0.08267 0.004989 0.08470 0.005118 0.08761 0.005318 0.08859 0.005401 V20 s 386.24" 0.32 cm3 moly1 y3 dU 1 s 0.79975 g cm U a d1 s 0.79965 g cmy3

0.05471 0.003274 0.06930 0.004164 0.07129 0.004242 0.08164 0.004886 0.09547 0.005678 0.09812 0.005829 0.10489 0.006205 0.11927 0.006994 0.12022 0.007036 0.12903 0.007537 V20 s 385.79" 0.38 cm3 moly1 y3 dU 1 s 0.80598 g cm U a d1 s 0.80574 g cmy3

0.04851 0.002804 0.05033 0.002893 0.05486 0.003182 0.06743 0.003893 0.06905 0.003962 0.07177 0.004098 0.07894 0.004531 0.08411 0.004797 0.09520 0.005458 0.10591 0.006032 V20 s 385.94" 0.35 cm3 moly1 y3 dU 1 s 0.81518 g cm U b d1 s 0.81534 g cmy3

0.05816 0.003237 0.06021 0.003364 0.06581 0.003640 0.08267 0.004566 0.09003 0.005003 0.09694 0.005359 0.11053 0.006091 0.12332 0.006749 0.13066 0.007134 0.14103 0.007708 V20 s 386.42" 0.30 cm3 moly1 y3 dU 1 s 0.82142 g cm U a d1 s 0.82155 g cmy3

Ref. w30x. Ref. w28x.

Table 3 . Relative densities Ž dy dU 1 of cholesterol solutions in tri-n-alkylamines, densities of pure solvents and partial molar volumes of cholesterol at 298.15 K Triethylamine

Tri-n-propylamine

Tri-n-butylamine

Tri-n-hexylamine

Tri-n-octylamine

m Dd m Dd m Dd m Dd m Dd Žmol kgy1 . Žg cmy3 . Žmol kgy1 . Žg cmy3 . Žmol kgy1 . Žg cmy3 . Žmol kgy1 . Žg cmy3 . Žg cmy3 . Žg cmy3 . 0.06482 0.005527 0.07055 0.006004 0.08397 0.007134 0.09147 0.007724 0.09882 0.008369 0.10341 0.008691 0.11820 0.009907 0.12035 0.010104 0.14110 0.011803 0.14756 0.012286 V20 s 369.11" 0.33 cm3 moly1 y3 dU 1 s 0.72276 g cm U a d1 s 0.72305 g cmy3 a

Ref. Ref. c Ref. d Ref. b

w28x. w31x. w32x. w33x.

0.05719 0.004352 0.06232 0.004734 0.07740 0.005888 0.08046 0.006118 0.09134 0.006904 0.09780 0.007389 0.10356 0.007803 0.11412 0.008552 0.12023 0.008983 0.12708 0.009544 V20 s 377.06" 0.34 cm3 moly1 y3 dU 1 s 0.75203 g cm U b d1 s 0.75234 g cmy3

0.05904 0.004164 0.07293 0.005116 0.08114 0.005680 0.08658 0.006049 0.09577 0.006683 0.10395 0.007238 0.10950 0.007567 0.11528 0.008006 0.12461 0.008638 0.13221 0.009145 V20 s 379.92" 0.25 cm3 moly1 y3 dU 1 s 0.77384 g cm U b d1 s 0.77378 g cmy3

0.05109 0.003285 0.05411 0.003473 0.05886 0.003786 0.06078 0.003879 0.06494 0.004159 0.06613 0.004254 0.06707 0.004292 0.07002 0.004495 0.07156 0.004556 0.07238 0.004627 V20 s 383.12" 0.30 cm3 moly1 y3 dU 1 s 0.79471g cm U c d1 s 0.7937 g cmy3

0.03879 0.002348 0.04222 0.002576 0.04740 0.002881 0.05458 0.003297 0.05745 0.003479 0.06241 0.003755 0.06803 0.004130 0.07022 0.004230 0.07376 0.004447 0.07683 0.004628 V20 s 384.12" 0.31 cm3 moly1 y3 dU 1 s 0.80817 g cm U d y3 d1 s 0.8068 g cm

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where: M2 is the molar weight of cholesterol, d is the density of solution, dU 1 is the density of pure solvent, m is the molality of cholesterol. As in the case of many previously examined systems w24,26x, no concentration dependence of the apparent molar volume of cholesterol was found in the concentration range considered in the present investigation. Thus, the partial molar volume of cholesterol, equal to its apparent molar volume in the infinite dilute solution, V20 s VF ,2 Žfor m 0., was calculated as the average value from VF ,2 values determined within the investigated concentration range. The calculated average values V20 for all systems are also given in Tables 1–3.

™

4. Discussion 4.1. Relationship between cohesion energy and solÕent Õolume One of the methods of expressing the intermolecular interactions forces in liquids is the use of cohesive energy that can be calculated from the approximate relation: DUm s D Hv y RT q pVm

Ž2.

where: DUm is molar cohesive energy, D Hv is molar enthalpy of vaporisation, p is vapour pressure and Vm is molar liquid volume at temperature T. The molar cohesive energy DUm , of the examined substance, in the case of homologous series, reflects mainly the effect of changes in the interaction energy resulting from the changes in molecule dimensions. Therefore, usually w34x the so-called cohesive energy density per unit volume, DU s DUmrVm is used, which describes the magnitude of cohesive forces acting between molecules in a volume unit, e.g., 1 cm3. However, the use of this value to describe the changes in interaction energy in a homologous series fails to be reliable since it comprises also an effect resulting from the changes in molecule packing in the liquid. Therefore, in further considerations, it is proposed to use the average cohesive energy per –CH 2 – group of solvent, DU CH 2 s DUmrn, where n is the number of equivalent –CH 2 – groups in the examined homologue. The number of equivalent –CH 2 – groups in the compounds under investigation may be calculated on the basis of van der Waals volume of particular groups forming the solvent molecule. For instance, the –CH 3 – group is equivalent to 1.336 –CH 2 – Ž VwCH 3rVwCH 2 s 1.336., the hydroxyl group involved in H-bonding is equivalent to 0.683 –CH 2 –, etc. w35x. The number of equivalent –CH 2 – groups in any compound may be expressed as follows:

Ý n i Vwi ns

i

VwCH 2

s

Vw VwCH 2

Ž3.

where n i is the number of i segments in the molecule, Vwi is the van der Waals volume of i segment, VwCH 2 is the van der Waals volume of –CH 2 – group, and Vw is the total van der Waals volume of the molecule. Thus, the cohesive energy per ‘‘average’’ –CH 2 – group described as above is expressed by the following relationship: DU

CH 2

s DUm

VwCH 2 Vw

Ž4.

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In a similar way one can express the average volume of the –CH 2 – group in the examined solvent: V

CH 2

s Vm

VwCH 2

Ž5.

Vw

and the ‘free’ volume Žempty space. per the –CH 2 – group, resulting from the difference between the liquid volume at a given temperature and van der Waals volume: CH 2 Vfree s Ž Vm y Vw .

VwCH 2 Vw

s ž V CH 2 y VwCH 2 /

Ž6.

CH 2 . The calculated cohesive energy values Ž DU CH 2 . and volumes Ž V CH 2 and Vfree per the ‘‘average’’ –CH 2 – group for several normal hydrocarbons, alcohols and primary amines at 298.15 K are given in Table 4. Ž The lack of vaporisation enthalpy data makes it impossible to calculate the cohesive energy for the tertiary amines under discussion in this work.. As is seen from Table 4, in the case of aliphatic hydrocarbons, the increase in the chain length is accompanied by a slight increase in the cohesive energy per the –CH 2 – group and a drop in the volume of –CH 2 – group. It seems that the only forces which combine hydrocarbon molecules in the liquid state are the dispersion forces. In the series from hexane to hexadecane, the energy of solvent–solvent dispersion interactions increases linearly with the increase in matter packing expressed by the drop in the volume of the –CH 2 – group Ž V CH 2 . . The molecules of the discussed polar solvents, alkan-1-ols as well as primary amines, in addition to dispersion interactions, are subjected also to specific and dipole interactions. To describe the self-association of the mentioned compounds in a pure state, a continuous linear association model is often used w39–42x. It is assumed in this model that solvent molecules form linear associates with various chain lengths and the formation energy of each H-bond does not depend on the length of the resultant self-associate. From the data given in Table 4 it is seen that the formation of H-bonds brings about a very strong increase in the cohesive energy per the ‘‘average’’ –CH 2 – group of alcohols and amines. Fig. 1 shows the relationship DU CH 2 s f Ž V CH 2 . , which is ascending in the case of alcohols Žline a. and amines Ž line b. and descending in the case of hydrocarbons Ž line c. . The value of DU CH 2 is the highest for alcohols and amines with short chains and decreases with the increase in the chain length, contrary to hydrocarbons. 0 . Knowing the association enthalpy of solvent molecules Ž D Hass , one may try to correct the molar cohesive energy Ž DUm , in Eq. Ž4.., taking into account the contribution of specific interactions, and to convert it to the contribution of dispersion interactions concerning the ‘‘average’’ –CH 2 – group.

CH 2 DUdisp s

Ž DUm y < D Hass0 < .

VwCH 2 Vw

Ž7.

Some values of the association enthalpy of alcohols and amines are available in the literature w39–47x. The cited values are different depending on the used method and calculation model or the selected homomorph Ž if its use is necessary. . From the data given by Treszczanowicz et al. w42x, it 0 calculated from Mecke–Kempter’s and Flory–Ginell’s follows that the average value of D Hass y1 models is y26.7 kJ mol for alkan-1-ols and -8.7 kJ moly1 for primary amines. The values of

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Table 4 The values of cohesive energy and volumes of several liquid alkanes, alkan-1-ols and primary amines Žaverage shares for –CH 2 – groups. CH 2 DUdisp ŽkJ moly1 .

V CH 2 Žcm3 moly1 .

CH 2 Vfree Žcm3 moly1 .

4.36 4.44 4.50 4.55 4.58 4.62 4.64 4.66 4.68 4.73

4.36 4.44 4.50 4.55 4.58 4.62 4.64 4.66 4.68 4.73

19.74 19.23 18.85 18.58 18.35 18.19 18.01 17.91 17.80 17.63

9.51 9.00 8.62 8.35 8.12 7.96 7.78 7.68 7.57 7.40

42.309 a 47.321a 52.340 a 56.940 a 61.850 c 66.810 c 70.980 c 76.860 c 81.500 c 91.960 c

13.19 11.16 9.93 9.05 8.46 8.02 7.59 7.42 7.17 6.87

4.35 4.51 4.61 4.61 4.65 4.69 4.63 4.76 4.75 4.82

19.44 18.70 18.32 18.06 17.86 17.71 17.58 17.46 17.38 17.25

9.21 8.47 8.09 7.83 7.63 7.48 7.35 7.23 7.15 7.02

31.260 a 35.740 a 40.080 a 45.100 d 49.960 d

6.59 6.20 5.91 5.79 5.68

4.60 4.58 4.54 4.60 4.64

19.01 18.50 18.31 18.04 17.86

8.78 8.27 8.08 7.81 7.63

Vm Žcm3 moly1 .

D Hv ŽkJ moly1 .

131.69 147.53 163.52 179.69 195.85 212.28 228.29 244.93 261.13 294.00

31.552 a 36.550 a 41.490 a 46.442 a 51.367 a 56.430 b 61.287 a 66.230 a 71.090 b 81.380 b

Alkan-1-ols C2 58.69 C3 75.15 C4 91.96 C5 108.72 C6 125.34 C7 142.04 C8 158.54 C9 174.98 C 10 191.54 C 12 224.59 Primary amines C3 83.01 C4 99.26 C5 116.56 C6 132.90 C7 149.40

Ri Alkanes C6 C7 C8 C9 C 10 C 11 C 12 C 13 C 14 C 16

a

Ref. Ref. c Ref. d Ref. b

DU CH 2 ŽkJ moly1 .

w28x. w36x. w37x. w38x.

CH 2 DUdisp calculated on the basis of the above mentioned data are given in Table 4. These values for CH 2 primary amines and alkan-1-ols presented as a function of V CH 2 or Vfree form a common straight CH CH line ŽFig. 1, line c. and coincide with the straight line DU 2 s f Ž Vfree 2 . formed by hydrocarbons. CH 2 The good consistency of the values DUdisp calculated for self-associating solvents and aliphatic 0 hydrocarbons allows one to believe that it would be possible to estimate D Hass values of associating liquids on the basis of converted Eq. Ž 7. . In this case a hypothetical hydrocarbon with the same average volume of –CH 2 – group Ž V CH 2 . would be the homomorph for the examined associating 0 compound. Obtained results of D Hass would be comparable with those obtainable for Mecke–

214

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CH 2 . Ž Fig. 1. Cohesive energy per –CH 2 – group vs. its volume Ž V CH 2 . or ‘free’ volume Ž Vfree ; B: alcohols; v: primary . amines; ': hydrocarbons ; solid symbols: total cohesive energy per –CH 2 – group; empty symbols: the cohesive energy of CH 2 . dispersion interaction per –CH 2 – group Ž Vdisp .

Kempter’s or Flory–Ginell’s models for compounds for which mass of specific group –xŽ –OH, –NH 2 , etc.. is hardly different from mass of –CH 3 group. CH 2 CH 2 . The common linear relationship DUdisp vs. V CH 2 Žor Vfree observed for the series of hydrocarbons, alkan-1-ols and primary amines allows one to assume the following. Ži. The dispersion interactions Žper –CH 2 – group. acting between liquid solvent molecules increase with increasing length of the alkyl chain Ž R i .. This is true also for strongly associated liquids for which the cohesive energy density per unit volume decreases, e.g., for alcohols: DUmrVm s 679 J cmy3 Žethanol. and DUmrVm s 398 J cmy3 Ždodecanol.. Žii. The increases in short range forces is accompanied by a decrease in the average volume of CH 2 . –CH 2 – group of the given solvent or a decrease in the ‘‘free’’ volume around –CH 2 – group Ž Vfree . Žiii. It may be assumed that for other solvents containing also linear chains, e.g., tertiary amines where the lack of data makes it impossible to calculate the cohesive energy, the relationship CH 2 CH 2 . DUdisp s f Ž Vfree will show the same character. CH 2 Thus, the values of Vfree , easy to determine on the basis of density data, could be, in the given homologous series, a relative measure of dispersion interactions between the alkyl solvent chains. The

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CH 2 higher the values of Vfree the weaker the dispersion solvent-solvent interaction concerning the CH 2. Ž –CH 2 – group DUdisp .

4.2. Partial molar Õolumes of cholesterol — effect of solÕent–solÕent and solute–solÕent interactions The partial molar volumes Ž V20 . of cholesterol as a function of the carbon atom number of the alkyl chain Ž R i . in n-alkanes, alkan-1-ols and tertiary amines, are shown in Fig. 2. As is seen, in the case of solvents which are not subject to self-association, i.e., alkanes and tertiary amines, V20 increases with increasing R i in a molecule. This increase is the highest for homologues with short chains and rapidly decreases with increasing R i . In the case of alcohols, the changes in V20 with increasing solvent chain length are small. The partial molar volume of solute at infinite dilution depends on three basic factors: the intrinsic volume of the non-solvated solute, the term that takes into account the volume changes resulting from solute–solvent interaction and the term resulting from solvent–solvent interaction. As was shown CH 2 earlier, the solvent–solvent interaction forces expressed by DUdisp of dispersion interactions for the solvents that form a homologous series are linearly correlated with the ‘free’ volume around –CH 2 –

Fig. 2. Dependence of partial molar volume of cholesterol on the solvent alkyl chain length R i ; Ž': hydrocarbons; v: tertiary amines; B: alcohols..

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CH 2 . groups of solvent. This volume Ž Vfree , when solute is present, may Ž but not necessarily. be partially contracted due to the solute–solvent interactions. This will result in decreasing partial molar volume CH 2 of the solute Ž V20 . with increasing Vfree values. Indeed, for the non-associated solvents with CH 2 . predominant non-specific interactions the function V20 s f Ž Vfree is descending. This is shown in Fig. 3, which illustrates the dependence of the standard partial molar volume of cholesterol on the ‘free’ CH 2 volume Vfree calculated from Eq. Ž6. for the solvents under investigation. In the case of hydrocarbons and tertiary amines, there is a quasi linear drop in the partial molar volume of cholesterol as the ‘free’ volume around –CH 2 – group increases. The functions V20 s CH 2 . f Ž Vfree for the alkane and tertiary amine series are parallel. This means that the ‘free’ volume around –CH 2 – group, appearing when R i is changed, is contracted to similar extent for both series of the compounds. Lower volumes of cholesterol in tertiary amines in relation to hydrocarbons with the CH 2 same Vfree may result from additional specific and dipole cholesterol–amine interactions. In the case of alkan-1-ols, the course of the discussed relationship is different. Despite the increase CH 2 . in the ‘free’ volume Ž Vfree with decreasing of alcohol chain R i Žfrom 7.35 cm3 moly1 for octan-1-ol to 9.21 cm3 moly1 for ethanol. the partial molar volume of cholesterol is hardly changed. One may assume that the appearing ‘free’ volume is not contracted due to the solute–solvent dispersion interactions. One should think that this is due to hydrogen bonds which combine alcohol molecule into associates whose ‘‘stiffened structure’’ inhibits the dispersion interactions between alkyl solvent chains and the hydrocarbon core of cholesterol. As the alcohol alkyl chain length increases, there is a

Fig. 3. Dependence of partial molar volume of cholesterol on the ‘free’ volume per –CH 2 – group of solvent. Ž': hydrocarbons; v: tertiary amines; B: alcohols..

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growing possibility of bending and fitting at least its part to the surface of cholesterol molecule. Therefore, the dispersion solute–solvent interactions should gradually acquire greater importance despite the self-association of the solvent. They should bring about a decrease in V20 of cholesterol, with its extent growing with the chain length R i . The effect of growing of cholesterol–alcohol dispersion interactions with increasing R i is however compensated by the decrease in the ‘free’ CH 2 . Ž volume of solvent Ž Vfree i.e., the volume which may be contracted. . In consequence, the 0 dependence of V2 on the alcohol hydrocarbon chain length assumes a flat shape, different from that CH 2 Ž of alkanes. Comparing alkanes and alcohols with large R i and comparable Vfree Fig. 3., it is seen that the contraction of cholesterol in alcohols becomes more distinct than that in alkanes. Another way to relate V20 of solute to solute–solvent interactions results from the possible use of the scaled particle theory Ž SPT. , according to which the partial molar volume of solute in a solution at infinite dilution may be expressed as follows w48,49x: V20 s Vcav q Vint q k 10 RT

Ž8.

where: Vcav is the volume contribution to the partial molar volume of solute associated with cavity formation in a liquid solvent, Vm is the volume contribution resulting from the solute–solvent interactions, k 10 is the isothermal compressibility of a solvent and R is the gas constant. The last term Ž k 10 RT . is a correction for the standard state change from the gaseous to liquid state. The cavity volume Ž Vcav . formed by the solute in a liquid solvent constitutes by definition a positive contribution to the partial molar volume Ž V20 . expressed by Eq. Ž8. . The contribution of Vint is negative, which results from the existence of intermolecular solute–solvent attractive forces which bring about the contraction of the formed cavity. To calculate the contributions appearing in Eq. Ž 8. , we have used the procedure described by Klofutar et al. w26x and Klofutar and Nemec w50x. The cavity volume, V˙cav was calculated from the relationship given by Sakurao w51x: Vcav s k 10 RT

ž

y q 1yy

3 yz Ž 1 q z .

Ž1 y y .

2

9 y2z2 q

Ž1 y y .

3

/

q

Ps 23N 6

Ž9.

where: z s s 2rs 1, s 2 and s 1 are hard sphere diameters of solute and solvent, respectively. Parameter y is the ratio of hard sphere volume occupied by 1 mol of solvent to its molar volume V10 : ys

Ps 13NA

Ž 10.

6V10

In order to obtain a set of coherent data, this value was calculated on the basis of the following relationship w52x:

k

0 1s

V10 Ž 1 y y .

4

RT Ž 1 q 2 y .

2

Ž 11.

The hard sphere diameter of cholesterol Ž s 2 s 0.961 nm. calculated on the basis of De Ligny’s empirical equation w53x was taken from the paper by Klofutar et al. w26x. The effect of the solvent–solvent interactions on the volume Vcav calculated by the SPT methods is, to some extent, taken into account through the isothermal compressibility of solvent. This value Ž k 10 ., as follows from the data given in Table 5, within the homologous series is linearly correlated with the

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CH 2 . define above ‘free’ volume of solvent Ž Vfree . It seems that the drop in liquid compressibility with increasing R i is connected with the decreases in the ‘free’ volume around solvent molecules. The obtained values of Vcav and Vint and the hard sphere diameters calculated via Eqs. Ž 10. and Ž11. for the solvent under investigation are given in Table 5. As is seen from the given data, Vint clearly has the most negative value for alkanes. This confirms the opinion that this function is connected mainly with dispersion interactions w52x. The solute–solvent interactions which bring about the contraction of the cavity formed by the solute seems to be the strongest for alkanes with short chains. This is indicated by high values of ŽyVi nt .. Short alkyl chains of alkanes seem to be better fitted to the flat non-polar cholesterol core. At the same time, compounds CH 2 with shorter chains show higher Vfree capable of contracting and higher cavity volumes Ž Vcav . created by cholesterol. It may be concluded that for the homologous series of non-associating solvents, the observed changes in the partial molar volume of cholesterol are the resultant of two cumulative effects: – the decreases in the solute–solvent dispersion interactions with increasing the length of solvent alkyl chain R i , Žshowed by Vi nt .

Table 5 Partial molar volumes Ž V20 ., cavity volumes Ž Vcav ., interaction volumes Ž Vint . of cholesterol in solution and hard sphere diameters Ž s 1 . of the solvent under investigation Ri Alkanes C6 C7 C8 C 10 C 12 C 16

CH 2 Vfree Žcm3 moly1 .

k 10 Ž10 10 Pay1 .

V20 Žcm3 moly1 .

Vcav Žcm3 moly1 .

yVint Žcm3 moly1 .

s l Žnm.

9.51 9.00 8.62 8.12 7.78 7.40

17.06 a 14.27 a 12.73 a 10.74 a 9.90 a 8.30 a

374.78"0.34 379.49"0.29 383.22"0.35 389.19"0.34 391.12"0.33 392.13"0.30

463.4 457.5 454.3 450.1 449.3 445.8

92.8 81.6 74.3 63.6 60.6 55.8

0.561 0.597 0.628 0.682 0.730 0.812

11.5 b

369.11"0.33 377.06"0.34 379.92"0.25 383.12"0.30 384.12"0.31

446.4

80.2

0.594

11.53 c 10.06 c 9.42 c 8.36 c 8.00 c 7.77 c

387.17"0.22 386.24"0.32 385.79"0.38 385.94"0.35 386.83"0.33 d 386.42"0.30

428.7 428.6 429.9 430.7 431.2 431.9

44.4 44.8 46.4 46.0 47.2 47.4

0.412 0.466 0.510 0.582 0.613 0.641

Tertiary amines C2 8.61 C3 8.03 C4 7.60 C6 7.22 C8 6.97 Alkan-1-ols C2 9.21 C3 8.47 C4 8.09 C6 7.63 C7 7.48 C8 7.35 a

Ref. Ref. c Ref. d Ref. b

w54x. w55x. w56x. w24x.

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– the increase in the solvent–solvent dispersion interactions with increasing R i Žobserved as a drop CH 2 . in the ‘free’ volumes Vfree . Both phenomena facilitate the increase in the partial molar volume of cholesterol Ž drop in contraction. with increasing alkyl chain length in the non-associating solvent molecule. For the tertiary amines under investigation, the calculation of the volume contribution due to the solute–solvent interactions Ž Vint . by the SPT method was possible only for triethylamine Ž TEA. . The high value ŽyVint . seems to result from relatively strong non-specific interaction between cholesterol and this solvent. In this respect, TEA becomes similar to a hydrocarbon with a hypothetical chain CH 2 length containing about 7.5 carbon atoms for which both Vint and Vfree would be close to those calculated for TEA ŽTable 5.. In the case of alkan-1-ols series, the volumes V20 , Vcav and Vint are not changed to such a clear extent as in alkanes. The volume contribution due to the solute–solvent interaction ŽyVint . increases gradually with increasing alkyl chain length of alcohol R i , contrary to that in alkanes. This seems to confirm the above presented model of the cholesterol–alcohol dispersion interactions consisting in the specific fitting of alkyl chain fragments of alcohols to the surface of the flat hydrocarbon core of cholesterol molecule and its side chain. In the case of molecules with long R i , such a fitting, and consequently the cholesterol volume contraction, may take place without disturbing H-bonds of the alcohol self-associates. One may conclude that the partial molar volumes of cholesterol in alcohols result from the following three effects: – the increase in the solute–solvent dispersion interactions with increasing alkyl chain length of alcohol Ž R i . — contrary to alkanes; CH 2 . – the increase in the solvent–solvent dispersion interactions with increasing R i Ždecrease in Vfree ;

– the self-association of alcohol via H-bonds, with its influence on V20 of cholesterol in solution being decreased with increasing R i . Analogously as in non-associating solvents, the second of the mentioned effects facilitate the increase in the partial molar volume of cholesterol Ž decrease in cholesterol contraction. with increasing alkyl chain length in alcohol molecule. The first and third effects bring about the decrease in partial molar volume of cholesterol with increasing of alcohol R i . The common compensation of the above mentioned effects results in the fact that the partial molar volume of cholesterol in the homologous series of alcohols is changed only to a slight extent. The dispersion interactions which affect cholesterol in lipide bilayers seem to be similar to those existing in alcohols and hydrocarbons. The observed contraction of cholesterol in these solvents is connected with the contribution of analogous forces Ž van der Waals. which bring about the well known condensing effect that cholesterol has on the liquid–crystalline phase of the bilayer w57x. List of symbols d Density of solution Density of pure solvent dU 1 m Molality of cholesterol M2 Molar weight of cholesterol n Number of equivalent –CH 2 – groups in solvent molecule

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p R Ri DUm D Hv 0 D Hass DU CH 2 CH 2 DUdisp Vm VF ,2 V20 Vw VwCH 2 CH 2 Vfree Vcav Vint

k 10 s 1, s 2

Vapour pressure of solvent Universal gas constant Alkyl chain length of solvents molecule Molar cohesive energy of solvent Molar enthalpy of solvent vaporisation Association enthalpy of solvent molecules Average cohesive energy per –CH 2 – group of solvent Average cohesive energy of dispersion interaction per –CH 2 – group Molar volume of solvent Apparent molar volume of cholesterol in solution Partial molar volume of cholesterol in solution van der Waals volume of the solvent molecule van der Waals volume of –CH 2 – group Average ‘free’ volume Ž empty space. per one –CH 2 – group of solvent Volume contribution to the partial molar volume of solute associated with cavity formation in a liquid solvent, Volume contribution to the partial molar volume of solute resulting from the solute– solvent interactions Isothermal compressibility of a solvent Hard sphere diameters of solvent and solute, respectively

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