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Thermodynamics of transfer of naphthaleneand 2-naphthoic acid from water to
(water + ethanol) mixtures atT 298:15 K
Baoxue Zhou a,b,1, Weimin Cai a,b,*, Lizhuang Zou c
a Shool of Environmental Science and Engineering, Shanghai Jiao Tong University,
Shanghai 200240, PR Chinab Harbin Institute of Technology, Harbin 15000, PR China
c School of Chemical and Environmental Engineering, China University of Mining and Technology,
Beijing 100083, PR China
Received 14 January 2003; accepted 28 April 2003
Abstract
Standard thermodynamic functions of transfer of naphthalene and 2-naphthoic acid from
water to (water + ethanol) mixtures at T 298:15 K have been determined from solubilitymeasurements at different temperatures. Standard free energies of transfer of both naphthalene
and 2-naphthoic acid showed decreasing tendency with the increasing x(EtOH), and the stan-
dard entropy and enthalpy of transfer exhibited a change of double peaks with x(EtOH). The
DtrG0 of 2-naphthoic acid decreased more rapidly than that of naphthalene when x(EtOH)
< 0.746 and lower than that of naphthalene when x(EtOH)>0.746 atT 298:15 K. The dou-ble peaks in the curves of standard entropy and enthalpy of transfer illustrated that the micro-
structure of the series of mixed solvents of (water + ethanol) underwent a variable process fromordered to disordered and then from disordered to ordered. The results mean that there is a
relatively ordered structure near x(EtOH) 0.13 in the (water + ethanol) solutions besidesthe existence of a clathrate structure in the water-rich region.
2003 Elsevier Science Ltd. All rights reserved.
Keywords: Naphthalene; 2-Naphthoic acid; Ethanol; Thermodynamic Functions of transfer; Intermolec-
ular interactions
J. Chem. Thermodynamics 35 (2003) 14131424
www.elsevier.com/locate/jct
* Corresponding author. Tel.: +86-21-5474-7461; fax: +86-21-5474-0825.
E-mail addresses: [email protected](B. Zhou), [email protected] (W. Cai).1 Also corresponding author.
0021-9614/$ - see front matter 2003 Elsevier Science Ltd. All rights reserved.
doi:10.1016/S0021-9614(03)00112-5
http://mail%20to:%[email protected]/http://mail%20to:%[email protected]/ -
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puriss. P0.990 mass fraction) were recrystallized three times from absolute ethyl al-
cohol, respectively, before use. Ethanol (Shanghai Chemical Reagent Corporation,
China Medicine (Group), GR) was purified by standard methods of fractional dis-
tillation. The experimental values of the refractive index at T 293:15 K (20 C)(n20D ) was 1.3614, in good agreement with literature data [12]. The solvents were pre-
pared by mixing weighed quantities of water and ethanol, and the mole fraction of
ethanol x(EtOH) was in the range of 0 to 0.8. Double-distilled water, treated with
ion exchange resin before distillation, was used and its conductivity was 1.18 104
S m1.Saturated solutions of naphthalene and 2-naphthoic acid were obtained by fol-
lowing procedures: 30 mL of mixed solvents of different composition and a slight
excess amount of naphthalene and 2-naphthoic acid were added to different glass
tubes each with a ground mouth. Then the tubes were sealed with a tight-fitting
ground stopper and parafilm, followed by equilibration in a shaking thermostatat T (298.15, 303.15, 308.15 and 313.15) K, respectively. The equilibrium timewas 10 to 14 d depending on the temperature. The selection of equilibrium time
was based on the condition test by following procedures: when maintained at con-
stant temperature for up to 3 to 4 days, the supernatant solution was analyzed by
spectrophotometry. This operation was repeated at intervals of 3 to 4 days until
there was no change of absorbance beyond the experimental error. After the condi-
tion test, the sealed glass tubes were not opened before the determination of solubil-
ities in order to avoid a tiny effect of volume change of solvent on experimental
results. The concentration of the saturated solutions of naphthalene in different
mixed solvents was measured by using ultraviolet visible spectrophotometry after
proper dilution with its respective solvents at 275 nm wavelength. The saturated sol-
ubilities of 2-naphthoic acid in series of mixtures were also analyzed by using ultra-
violet visible spectrophotometry just like the measurement of naphthalene at 280 nm
wavelength after an appropriate correction using a series of respective standard
curves. The method was also corrected by a different method of titrating with stan-
dard NaOH solution and phenolphthalein indicator. The two methods agree to
within 1%.
3. Results and discussion
Based on thermodynamic principles [1,3,6], the standard Gibbs free energies
change (DG0s ) of naphthalene or 2-naphthoic acid dissolving in a solvent at a certain
temperature can be calculated by using equation (1)
DG0s RT lns; 1
where s is the saturated solubility of solute in mol dm3. The standard Gibbs freeenergies of transfer (DtrG
0) for a solute from medium 1 to medium 2 can be calcu-
lated by using equation (2)
DtrG0 RTlns2=s1; 2
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wheres1 ands2 in equation (2) are the saturated solubilities of solute in medium 1 to
medium 2. The functional relationship ofDtrG0 with the temperatureTcan be fitted
to a nonlinear equation. The selected function as follows [1,3]:
DtrG0 a bT cTln T; 3
where a, b, c are constants independent of temperature, their values vary with the
composition and species etc., of mixed solvents.
Adding equations (2) and (3) produces equation (4)
RTlns2=s1 a bT cT ln T; 4
then
lns2=s1 a=RT b=R c=R ln T: 5
Equation (5) can be regressed by a method of least squares based on the experi-
mental points of ln (s2/s1)versus T. Designing a programme for equation (5) by using
Origin 6.0, the a, b, c and nonlinear correlation coefficient c can be obtained.
The standard free energy of transfer of solute from water to (water + ethanol)
mixtures is then calculated using equation (6)
DtrG0 DG0s DG
0w as aw bs bwT cs cwTln T; 6
where subscripts s and w refer to the solvent mixtures and water, respectively. Ac-
cording to the functional relationship of GibbsHelmholtz, the standard entropy of
transfer (DS
0
tr) can be derived with equation (7)DtrS
0 oDtrG0=oTp: 7
Combination of equation (6) with (7) gives (8)
DtrS0 bw bs cw cs1 ln T: 8
The standard enthalpy of transfer (DtrH0) can be calculated using equation (9) based
on the thermodynamic relationship DH0 DG0 TDS0
DtrH0 DtrG
0 TDtrS0: 9
The solubilities of naphthalene and 2-naphthoic acid in various mixtures of(water + ethanol) at T (298.15, 303.15, 308.15, and 313.15) K were presented intable 1, figures 1 and 2, respectively. As shown in table 1, figures 1 and 2, on the
whole, the solubilities of both naphthalene and 2-naphthoic acid in (water + ethanol)
solution increase with increasing x(EtOH) very slowly at low x(EtOH) and sharply
at x(EtOH) P0.08. This implies that the effect of ethanol concentrations on the
solubilities (s) of naphthalene and 2-naphthoic acid is not remarkable at low con-
centration. However, when the concentration of ethanol exceeds a certain level of
0.08 of x(EtOH), the effect becomes much stronger. A similar result has been ob-
served in the (water + t-butyl alcohol) mixtures for benzene, toluene, trimethylben-
zene in our previous papers [4,5]. It also can be seen from table 1, figures 1 and 2 thatthe replacement of the H atom of naphthalene in the 2 position by the COOH group
changes the solubilities considerably. The solubilities of 2-naphthoic acid are always
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higher than those of naphthalene in the ranges ofxEtOH0 to 0.746 for T 298:15K, 0 to 0.680 for T 313:15 K, 0 to 0.616 for T 318:15 K, 0 to 0.526 forT 323:15 K, respectively. However, over the ranges, lower solubilities for2-naphthoic acid than those of naphthalene are observed. In addition, it also can be
seen from the above results that the maximum value of xEtOH for the solubility of
2-naphthoic acid is higher than that of naphthalene.
Coefficientsa,b,c and nonlinear correlation coefficient cin equation (5) are given
in table 2. The data of nonlinear correlation coefficients c show that the nonlinear
regressions of ln (s2/s1)versus T in various (water + ethanol) solvents using equation
(5) are reliable as reflected by the values of c> 0:99. Table 3 and figure 3 show thevariations ofDtrG
0 of naphthalene and 2-naphthoic acid withx(EtOH) from water to
(water + ethanol) mixtures. From figure 3, it can be observed that, on the whole,
TABLE 1
Solubilities of naphthalene and 2-naphthoic acid in water and (water + ethanol) solutions
xEtOH Naphthalene s/(104 mol dm3) 2-Naphthoic acid s/(104 moldm3)
T/K: 298.15 303.15 308.15 313.15 298.15 303.15 308.15 313.15
0 2.4198 2.9245 3.5028 4.1466 2.4345 2.9430 3.7366 4.8383
0.01 2.7325 3.3100 3.9607 4.6089 2.8133 3.5198 4.4948 5.8243
0.02 3.1019 3.8117 4.5675 5.4585 3.2203 4.0629 5.2146 6.7517
0.03 3.6152 4.4059 5.3799 6.2445 3.7222 4.7307 6.0896 7.8738
0.04 4.0213 5.1014 6.3422 7.7668 4.3644 5.5374 7.4315 9.6766
0.045 4.3537 5.7335 7.2100 8.8775 4.6365 6.1006 8.1203 10.474
0.05 4.5978 6.2369 7.8428 9.7683 4.9585 6.5699 8.8166 11.434
0.06 5.4583 7.4933 9.5581 11.679 5.9946 8.1059 10.630 13.798
0.08 7.5317 11.107 14.495 17.604 9.4992 13.373 18.248 23.875
0.10 11.277 16.092 20.608 25.694 17.396 22.253 30.065 40.478
0.20 99.241 142.52 176.39 214.03 194.18 238.55 297.20 348.620.40 691.99 964.22 1279.1 1626.6 1251.3 1400.5 1671.0 1873.6
0.60 1601.94 2090.0 2833.5 3572.8 2294.3 2534.9 2917.8 3200.0
0.80 3290.7 4331.2 5229.9 6344.9 2942.3 3312.7 3745.8 4056.4
FIGURE 1. Plot of solubility of naphthalene in (water + ethanol) solutions against mole fraction ethanol.
() T 298:15 K; (s)T 303:15 K; (M) T 308:15 K; (O) T 313:15 K.
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DtrG0 of naphthalene and 2-naphthoic acid decrease with the increase ofx(EtOH).
According to thermodynamic principles, the value ofDtrG0 indicates the ease of
spontaneous transfer of solutes from water to the mixed solvent. The larger the value
ofDtrG0, the easier is the transfer of solutes from water to the mixed solvent. Hence
the larger the value ofDtrG0, the more favourable is the distribution for solutes in
the mixed solvent. It can be concluded that the distribution of naphthalene and
2-naphthoic acid are more and more favourable, in general, with the increase of
x(EtOH) in the mixed solvent. From figure 3, it also can be seen that the curves of
DtrG0 for both naphthalene and 2-naphthoic acid exhibit several different downward
trends. At the initial stages, DtrG0 decreases slowly. Whenx(EtOH) rises above 0.08,
the DtrG0 decreases more readily. When x(EtOH) rises to about 0.27, the decrease of
DtrG0 slows again. This indicates that the intermolecular interactions among the sol-
ute, water and ethanol in the series of mixed solvent are not similar to each other.
As a comparison of naphthalene with 2-naphthoic acid, the changing trend of
DtrG0 is slightly different. The DtrG
0 of 2-naphthoic acid decreases more rapidly than
that of naphthalene when x(EtOH) < 0.746, but more slowly forx(EtOH)> 0.746 at
T 298:15 K. This implies that the substitution product of naphthalene by theCOOH group is easier to transfer than is naphthalene when x(EtOH)< 0.746 andmore difficult to transfer than is naphthalene for x(EtOH)> 0.746 for changes fromlow concentrations to high concentrations of ethanol at T 298:15 K. This phenom-enon should be attributed to the result of direct formation of H-bonds of the hydro-
philic COOH group with water molecules in 2-naphthoic acid in low concentrations
of ethanol following Kundu [1], a formation of hydrogen-bonded solvent cage
referred as skin phase, the formed H-bonds cause a greater stabilization of the
2-naphthoic acid molecule than that of naphthalene and more negative DtrG0 values
than those of naphthalene in water and (water + EtOH) mixtures. However, along
with the decrease of mole fraction of water and the increase of x(EtOH), theH-bonds between the carboxylic group and water become weak and the hydrophobic
interaction between ethanol molecules increase gradually. The existence of the polar
FIGURE 2. Plot of solubility of 2-naphthoic acid in (water + ethanol) solutions against mole fractionethanol. ()T 298:15 K; (s) T 303:15 K; (M) T 308:15 K; (O)T 313:15 K.
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TABLE 2
Regression coefficients a, b, c of regression equation (5) and correlation coefficient c
xEtOH Naphthalene 2-Naphthoic acid
a b c c a b
(J mol1) (J mol1 K1) (J mol1 K1) (J mol1) (J mol1 K1 )
0 70368.9 )958.924 139.03 1.000 )362195.8 8703.97
0.01 162464.1 )3000.19 442.91 0.9999 )203943.7 5258.01
0.02 103747.6 )1671.98 244.17 0.9999 )162082.0 4348.32
0.03 201406.6 )3832.63 565.68 0.9991 )121030.3 3452.58
0.04 133288.6 )
2232.88 324.85 1.0000 )
166713.0 4509.26
0.045 281963.8 )5454.12 802.58 0.9999 51313.3 )274.42
0.05 368730.2 )7328.00 1080.31 0.9994 56674.4 )372.79
0.06 514560.8 )10527.60 1555.78 0.9999 153213.7 )2506.83
0.08 878895.7 )18462.90 2733.60 0.9998 450697.1 )8939.67
0.10 600562.0 )12370.10 1827.47 0.9992 )287276.7 7194.30
0.20 724061.8 )15165.40 2242.21 0.9965 185543.2 )3477.39
0.40 390846.0 )7752.60 1134.49 1.0000 )34203.3 1171.66
0.60 116153.2 )1755.25 242.37 0.9983 15599.6 )1.36
0.80 356067.0 )7200.00 1055.70 0.9984 166315.6 )3334.66
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carboxylic group can enervate the hydrophobic interaction between the naphthalene
and ethanol molecules. Thus, the solubility of 2-naphthoic acid increase slowly and
DtrG0 decreases gently at high x(EtOH). Particularly when x(EtOH) > 0.746 at
T 298:15 K, the solubility of 2-naphthoic acid is lower and DtrG0 is higher than
that of naphthalene.
The standard entropy (DtrS0) and enthalpy (DtrH
0) of transfer of naphthalene
and 2-naphthoic acid from water to (water + ethanol) mixtures are given in table 3and figure 4. As illustrated in figure 4, the TDtrS
0 curves for both naphthalene and
2-naphthoic acid show a change in the double peak curves with the increase
FIGURE 3. Plot ofDtrG0 from water to (water + EtOH) against mole fraction ethanol. () naphtha-
lene; (s) 2-naphthoic acid.
TABLE 3
Thermodynamic functions for transfer of naphthalene and 2-naphthoic acid from water to (water +
ethanol) solutions at T 298:15 K
xEtOH Naphthalene 2-Naphthoic acid
DtrG0
DtrS0
DtrH0
DtrG0
DtrS0
DtrH0
(kJ mol1) (J K1 mol1) (kJ mol1) (kJ mol1) (J K1 mol1) (kJ mol1)
0 0 0 0 0 0 0
0.01 )0.3013 6.0169 1.4926 )0.4442 20.5527 5.6836
0.02 )0.6157 8.8885 2.0344 )0.6935 27.2590 7.4338
0.03 )0.9953 16.1678 3.8251 )1.0526 33.3140 8.8800
0.04 )1.2593 29.4463 7.5201 )1.4472 40.1811 10.5328
0.045 )1.4563 51.0429 13.7622 )1.5008 60.8065 16.6287
0.05 )1.5887 64.7990 17.7311 )1.5926 65.3331 17.8865
0.06 )
2.0168 79.8521 21.7911 )
2.2341 73.2931 19.61820.08 )2.8150 126.7244 34.9679 )3.3754 121.3191 32.7959
0.10 )3.8158 102.6638 26.7934 )4.8755 49.6245 9.9200
0.20 )9.2077 120.2604 26.6480 )10.8568 64.3645 8.3335
0.40 )14.0194 126.4570 23.6839 )15.4760 32.1277 )5.8971
0.60 )16.1052 104.1860 14.9578 )16.9791 28.5490 )8.4672
0.80 )17.8883 101.6013 12.4041 )17.5959 40.1888 )5.6136
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of x(EtOH). The first peaks appear at x(EtOH) 0.0838 for naphthalene andx(EtOH) 0.0801 for 2-naphthoic acid. The second peaks appeared atx(EtOH) 0.285 for naphthalene and x(EtOH) 0.267 for 2-naphthoic acid.
The change of thermodynamic functions of transfer depends on the change of
structure of mixed solvents. When a solute is introduced into solvent systems, a per-
turbation of the solvent structure and the solvents physical and chemical properties
would occur. However, this perturbation and change determines the variation of
macroscopic thermodynamic properties in the systems. In general, the transfer of
a solute from water to mixed solvent may be conceived to occur in four successive
steps. Firstly, the solute is removed from the water molecules. Secondly, the water
molecules reissue. (The cavity in the water is closed.) Thirdly, a new cavity is formed
in the co-solvent solution. Finally, the solute is solvated again by mixed solvents. It is
well known that the (water + monohydroxy alcohol) mixture has a high degree of
structure in the water-rich region [2,1315], which makes it easy to form the clathrate
structure of the alcohol molecule surrounded by water molecules. For example, the
clathrate structure of ethanol and t-butyl alcohol surrounded by several water mol-
ecules in water-rich region is suggested in the literature [4,5,16]. However, as the sizeof the alcohol molecule is different, the ratio of aqua alcohol in clathrate is different.
The smaller the size of alcohol molecule, the more difficult it is to determine the ex-
istence directly and to characterize the structure of clathrate. This perhaps is the
main reason why there are no reports in the literature where the structure of clath-
rate has been determined directly. Nevertheless, from the information of this exper-
iment, it is concluded that there is a clathrate structure in the (ethanol + water)
mixture in the water-rich region. Das [1] and Hu et al. [2] also suggested that there
is a clathrate structure (or solvent cage) in the (ethanol + water) mixture in the water-
rich region based on the studies of thermodynamics of the transfer of benzoic acid in
(water + ethanol) systems.Ethanol is characterized by the presence of the hydrophilic and hydrophobic
groups leading to two different types of hydration. Thus both hydrophilic and hydro-
FIGURE 4. Plot ofTDtrS0 andD trH
0 from water to (water + EtOH) against mole fraction ethanol. ()
TDtrS0, naphthalene; (s)TDtrS0, 2-naphthoic acid; (M) DtrH0, naphthalene; (O) DtrH0, 2-naphthoicacid.
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phobic interactions compete for organization of the water structure. In the low con-
centration region of the alcohol (water-rich region), free ethanol is scarce due to
the existence of clathrate. Naphthalene can not enter the clathrate easily, which re-
sults in its solubilities increasing slowly with x(EtOH). In this process, the standardfree energy of transfer exhibits a slow change. Although 2-naphthoic acid has polar
carboxylic group leading the hydrophilic interaction (skin phase), the large hydro-
phobic group in 2-naphthoic acid makes it difficult to enter the clathrate. Subse-
quently, the clathrate structure broke down gradually with increasing x(EtOH),
and the ethanol molecules are removed from the clathrate. The probability of hydro-
phobic interactions occurrence between hydrophobic groups of ethanol and naph-
thalene or hydrophobic groups of 2-naphthoic acid would increase. This made the
solubilities increase sharply and continuously. The corresponding free energy for
transfer decreases quickly when x(EtOH) is above 0.08. When x(EtOH) rises to
about 0.27, the decreasing tendency of DtrG0 became slow once again. The abovephenomenon indicated that the microstructure and intermolecular interactions
change in a different way in the series of mixed solvents. From the mathematical
point of view, the changing characteristics ofDtrG0 can be ultimately expressed by
the change ofDtrS0 since the DtrS
0 is dependent on the slop ofDtrG0.
It is well known that DtrS0 can be used as a probe for elucidating the structure of
mixed solvents. Therefore, the introduction of alcohol would enhance the stabiliza-
tion of the clusters corresponding to a gradual organization into a water clatharate
surrounding an ethanol molecule. Initially, the water structure is little affected but
becomes more organized until it reaches a maximum structure. Beyond that point,
further addition of ethanol initiates a rapid structural collapse. A peak in the
TDtrS0 curves at x(EtOH) 0.0838 for naphthalene and at x(EtOH) 0.0801 for
2-naphthoic acid accompanies this process. Along with increasing x(EtOH), for
x(EtOH) up to 0.134 for naphthalene and 0.139 for 2-naphthoic acid, corresponding
a low valley ofTDtrS0, a relatively ordered structure appears in the mixed solvents.
This relatively ordered structure should be attributed to the maximum formation of
H-bonds between ethanol and water after the rupture of H-bonds between water and
water. After that, H2OEtOH H-bonds are replaced gradually by EtOHEtOH
H-bonds and a maximum ofTDtrS0 is observed again at x(EtOH) 0.285 for naph-
thalene and at x(EtOH) 0.267 for 2-naphthoic acid. These correspond to the for-mation of a new relatively disordered structure (accompanied by inflection in thecurves ofDtrG
0 near 0.27 forx(EtOH) in mixed solvents). This phenomenon, to some
degree, should be attributed to the maximum rupture of H-bonds between ethanol
and water though DtrG0 decreased continuously. Henceforth, the mixed solvent sys-
tem enters the region of rich ethanol, and then the degree of EtOHEtOH H-bond-
ing increases. The ordered degree of the solvent system increases gradually
accompanied by the increasing hydrophobic interaction of ethanol molecules. It is
noted that both enthalpy and entropy of 2-naphthoic acid decrease up to about
x(EtOH) 0.483, corresponding to the second low valley ofTDtrS0 for 2-naphthoic
acid. Beyond that mole fraction, an increasing trend of TDtrS0 is observed in thecurve of 2-naphthoic acid. This result can also be attributed to the effect of the
substituted carboxylic group in molecule evidently. A similar result has been
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observed in the (water + EtOH) mixtures of benzoic acid [17]. It should be pointed
out that various studies, including those for some thermodynamic properties of
(water + ethanol) mixed solvents, indicate the maximum peak ofDtrS0 in water-rich
region. However, only one peak of DtrS0 was found and the reported values ofx(EtOH) were much different. Kundu et al. [1,3] suggested a peak ofDtrS
0 at about
0.061 ofx(EtOH) forp-nitroaniline and at about 0.144 ofx(EtOH) for benzoic acid
in (water + ethanol) mixed solvents. Huet al. [13] suggested a maximum DtrS0 region
ofx(EtOH) about 0.1 to 0.15 for amino acid and benzoic acid [2],x(EtOH) > 0.2 foraminobenzoic acid andx(EtOH) > 0.25 for glycine [14,15]. In comparison our work,it is found that the points of experimental data with described in the literature for the
water-rich region are fewer than that of our work. Some small change in the series of
solvent system may be compensated by using the deficient data. Therefore, it is nec-
essary to obtain a numbers of data to confirm the region ofx(EtOH) for this change.
It can be seen from our studies that there were at least two peaks of TDtrS0 in(water + ethanol) for both naphthalene at x(EtOH) 0.0838, x(EtOH) 0.285, re-spectively, and for 2-naphthoic acid at x(EtOH) 0.0801, x(EtOH) 0.267, corre-sponding to two maximum disordered structures in the series solvent system. The
low values ofTDtrS0 at x(EtOH) 0.134 for naphthalene and at x(EtOH) 0.139,
0.483 for 2-naphthoic acid, correspond to the maximum formation of newly ordered
structures in our solvent system and are observed in our studies. All of these results
have not yet been reported in literature.
It can be seen from these studies that both naphthalene and 2-naphthoic acid are
easily transferred from water to (water + ethanol) mixtures. However, the trends of
transfer for both naphthalene and 2-naphthoic acid are not significant in the
water-rich region. Analysis from the microstructure of mixed solvents in the
water-rich region shows that this phenomenon may be attributed to the effect of
clathrate formation. However, the solubilities increase dramatically and free energies
of transfer decrease with x(EtOH) when x(EtOH)> 0.08. This is obvious due to theincrease of hydrophobic interactions between the hydrophobic group of ethanol and
naphthalene or 2-naphthoic acid after a rapid structural collapse of the clathrate.
Subsequently, the solvent system undergoes a series of changes from ordered to dis-
ordered, and ordered again, though the change in trend of solubility is not obvious.
The replacement by polar a carboxylic group in the naphthalene molecule can in-crease its solubility at low x(EtOH) and decrease its solubility at high x(EtOH).
Acknowledgement
This project was supported by China Postdoctoral Science Foundation ([2002]
No. 11).
References
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