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Surface tension regularity of non-polar, polar, and weak electrolyte liquid hydrocarbons
Fluid Phase Equilibria 234 (2005) 101–107
Surface tension regularity of non-polar, polar, and weak electrolyte liquid hydrocarbons Mohammad Hadi Ghatee ∗ , Leila Pakdel Department of Chemistry, College of Science, Shiraz University, Shiraz 71454, Iran Received 2 April 2004; received in revised form 12 May 2005; accepted 13 May 2005
Abstract The correlation of reduced surface tension σsc∗ versus reduced temperature Tsc∗ which produces family of linear curves is investigated ∗ for universality. Each curve is characterized by the reduced temperature index Tindex which is included in Tsc∗ . The correlation is based on phenomenological scaling law and consideration of law of corresponding states. The reduced temperature Tsc∗ , scaled distance from the boiling temperature Tb , has the practical advantage of being independent of critical temperature. The correlation is highly accurate when applied to 10 classes of non-polar, slightly polar, polar, and weak electrolyte liquid hydrocarbons. The slopes of linear curves of each class of liquids ∗ at given Tindex ’s are rather constant and a universal regularity among all liquid is very likely, however small differences identify three classes of non-polar, slightly polar, and polar plus weak electrolytes. Consideration of the pervious application to molten alkali halides, molten salts, and molten metals, and considering that the reduced surface entropy S s∗ = (∂σsc∗ /∂Tsc∗ ) is rather constant for all liquid tested, strengthen the idea that σsc∗ –Tsc∗ correlation is likely universal. © 2005 Elsevier B.V. All rights reserved. Keywords: Law of corresponding states; Liquid hydrocarbons; Molten metals; Molten salts; Surface tension; Correlation for surface tension; Weak electrolytes
1. Introduction Surface tension is an important property where wetting, penetration, foaming, or droplet formation of a liquid is considered, and where environmental profile is encountered [1,2]. Values of gas–liquid interfacial tension are used in studying of liquid–liquid and liquid–solid interfaces, as well as the design of fractionators, absorbers, separators, two-phase pipeline, and in reservoir design calculations [1]. It is also a parameter required for drug formulation and drug product design [3,4]. Surface tension, among other physical and thermodynamic properties of liquids, is a basis for practical and theoretical modeling of the liquid state of matters [5]. It varies with temperature linearly and vanishes non-linearly close to critical point Tc , where the interface fades out. Although surface tension can be measured accurately by experimental methods, it has been ∗
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correlated with many physical properties, for both prediction and validation of theories of interface. Relationship involving the melting point, molar volume, and heat of vaporization are the most meaningful [6]. For best application correlations based on the law of corresponding states have been established and thus, the applicability becomes universal [7]. The correlation of surface tension with temperature, given by van der Waals and then developed by Guggenheim [7], as σ = σ o (1 − Tr )m , requires knowledge of the substancedependent constant σ o , and the critical temperature Tc , where σ is the surface tension, T the absolute temperature, and the reduced temperature Tr = T/Tc . The constant m has universality and its reported values are in the range 1.16 [8], which are different from the theoretical values of 1.26 [9]. Further works have progressed to express σ o in terms of Tc , the critical pressure Pc , and a substance-dependent constant in terms of boiling temperature [10]. Based on phenomenological scaling, Lielmezs and Herrick [8] have presented a correlation as a single linear
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curve in the form of corresponding states, applicable to a wide range of atomic, spherical, alkanes, and slightly polar organic liquids. This correlation and its modified form [5] have been applied to molten metals [5,11], while requiring Tc and predicts surface tension within 15% for molten metals [11], and within 14% of experimental values of atomic and molecular normal fluids [8]. This and similar correlations result in the corresponding states of surface tension in terms of temperature scaled distance from Tc . One of the present author has presented a reasonably accurate modified formed of the above correlation in which surface tension is given as function of temperature scaled distance from boiling temperature [5], and thus, it has found practical application for the fact that it is independent of critical temperature. It is applicable to molten alkali metals over a wide range of temperature with maximum average deviation of 2.43% [5], and to molten alkali halides with maximum average deviation of 7.5% [6]. ∗ –T ∗ In the present study, we investigate the features of σsc sc correlation for universality by considering different classes of liquid hydrocarbons from non-polar to slightly polar and weak electrolytes. These hydrocarbons are of importance as solvent and as the building block of industrial products, biochemical, and biological systems [12,13]. The results of this study and the previous results of application to important materials of engineering type are used to qualify the universality.
2. Surface tension regularity Based on phenomenological scaling law [8], and consideration of law of corresponding states, relations have been presented for surface tension, which are mainly aimed to correlate surface tension in terms of physical properties of fluid bulk. The accuracy of these correlations has been noted by a number of applications to normal liquids [14], molten salts [15], and molten metals [16]. A reasonably accurate modified form of the above correlation in which the reduced surface tension σ * = (σ/T)(σ f /Tf ) is given as function of reduced temperature T* = [(Tb − T)/(Tb − Tf )](Tf / T) (scaled distance from boiling point) has been presented, where subscripts b and f stand for boiling and freezing points, respectively. It has found practical applications for the fact that it is independent of critical temperature. The correlation of σ * with T* results in a single curve from freezing point to boiling point. Following successful applications, it has been argued that since surface tension is a measure of intermolecular interaction potential should not have necessarily the same weighting as T which is a measure of thermal energy. Consistence with these theories, then we have found empirically a new regularity for the surface tension in which a family of linear functionals were produced by [6], n+1 Tf σ ∗ σsc = (1) σf T
as a function of n Tf Tb − T σ Tsc∗ = Tb − T f T σf
(2)
where σ is the surface tension, n a constant value, Tf and σ f the absolute temperature and surface tension at the freezing point, respectively, and Tb is the absolute temperature at the ∗ with T ∗ (is referred boiling point. The fact that correlation σsc sc ∗ ∗ to as σsc –Tsc , hereafter) is independent of the critical temperature makes it of special interest for practical application. The first application to molten alkali halides indicates that ∗ –T ∗ is highly accurate, with linear correlation coefficient σsc sc (LCC) between 0.9992 and 0.9998. Each curve of the family of curves is indexed by Tb − T ∗ Tindex (3) = Tb − T f involved in Tsc∗ . In Eqs. (1) and (2), n is a constant equal ∗ –T ∗ is reduced to the previous σ * –T* to 4.0. With n = 0, σsc sc correlation [5], in which no indexing was applicable and only a single curve is produced. When molten alkali halides are treated with σ * –T* correlation, a single linear curve with LCC = 0.9906 is produced, below the limit for considering ∗ –T ∗ , the high a curve satisfactorily linear. Therefore, for σsc sc ∗ accuracy is gained by including Tindex in the definition of Tsc∗ and thus, it is the key point for producing a quite accurate universal relationship. Since the form of Tsc∗ is rather com∗ plex, it worth mentioning that Tsc∗ = Tsc∗ (Tindex , σ; σf , n = 4.0) equally clarify the physical structure of Tsc∗ as Tsc∗ = Tsc∗ (T, σ; Tf , σf , Tb , n = 4.0) does. However, the former has a special feature presented in particular, leading to simple ∗ –T ∗ correlation. procedure for the construction of σsc sc While the accuracy is excellent, the critical temperature is ∗ –T ∗ preferential over the counexcluded and this makes σsc sc terpart correlations. This is of practical interest for both prediction and validation of theory of interface when the organic liquids with high Tc , susceptible to decomposition at high T’s, are involved. Empirically, we have found that the value 4.0 is an exact lower limit of n for molten alkali halides [6]. This has been the case for the subsequent reported applications [14–16], and a higher limit cannot be specified [6]. For n > 4.0, the linear behavior persists but the accuracy will be decreased, as indicated by decreasing LCC.
3. Results and discussion ∗ –T ∗ has been applied to molten alkali The correlation σsc sc halides [6], alkanes [14], molten salts [15], and molten metals [16]. (See the notes in Refs. [14] and [16].) In this study, we investigate to demonstrate its application as a universal correlation. To achieve this, we have selected 10 sets of organic liquid of different polarity. The liquids and their physical properties are listed in Table 1 . The liquids involved are mainly long chain hydrocarbons and their
M.H. Ghatee, L. Pakdel / Fluid Phase Equilibria 234 (2005) 101–107 Table 1 The physical properties of liquid hydrocarbons
103
Table 1 (Continued )
Substances
Tf (K)
Tb (K)
σ f (mN m−1 )
Acids Methanoic acid Ethanoic acid Propionic acid Butanoic acid 2-Methylpropionic acid Pentanoic acid 2-Methylbutanoic acid Hexanoic acid Heptanoic acid
281.4 290 250 268 226 255 244 299 264.5
374 391 414 436.7 427 458 450 519 387
38.96 27.91 30.98 28.82 31.22 30.51 29.86 39.72 30.61
Alcohols Methanol Ethanol 1-Propanol 2-Propanol 1-Butanol 2-Butanol 1-Pentanol 3-Methyl-1-butanol 2-Pentanol 2-Methyl-2-butanol 2-Propen-1-ol 2-Methoxyethanol 2-Ethoxyethanol 2-Butoxyethanol 1-Octanol 2-Octanol 1-Nonanol 1-Decanol
175 143 146 184 184 165 194 156 223 265 144 188 173 203 257 235 266 280
337.7 351 370 355 391 381 412 404 392 375 370 397 410 444 468 451 488 503
31.59 34.88 35.14 29.93 35.19 33.13 34.46 35.37 31.00 24.79 39.18 41.68 39.57 33.90 30.37 31.09 30.33 29.86
Aldehydes Ethanal Propanal Butanal Pentanal Heptanal Benzaldehyde o-Hydroxybenzaldehyde
148 285 177 181 230 247 275
294 397 348 376 426 452 470
40.92 27.02 35.56 37.27 32.61 43.57 45.15
Alkyl halides 1-Bromopropane 1-Bromobutane 1-Bromopentane 1-Bromohexane 1-Bromoheptane 1-Chlorobutane 1-Chloropentane 1-Chlorohexane 1-Chloroheptane 1-Iodoethane 1-Iodopropane 1-Iodobutane 1-Iodoheptane
165 161 178 188 215 150 174 179 204 165 172 170 225
344 377 403 428 453 352 381 409 434 346 376 403 478
41.47 41.34 39.49 38.04 36.45 39.73 37.76 38.09 35.59 45.58 43.13 41.45 36.45
Esters Ethyl methanoate Methyl propionate Ethyl propionate Propyl propionate Ethyl butanoate Ethyl-2-methylbutanoate Methyl hexanoate Ethyl heptanoate Ethyl octanoate
193 185 200 197 180 174 202 207 226
327 352 372 397 393 406 424 462 481
37.01 38.67 35.26 34.92 36.28 35.77 35.91 34.53 33.96
Tf (K)
Tb (K)
σ f (mN m−1 )
293 307 175 189 195 175 174 193 173
596 437 330.5 350 399 390 385 443 307
29.41 28.88 40.60 36.06 35.90 35.53 36.20 36.25 44.03
Amines Methylamine Ethylamine Propylamine Butylamine Isobutylamine sec-Butylamine tert-Butylamine Allylamine Hexylamine Dimethylamine Trimethylamine Diethylamine Dipropylamine Diisopropylamine Tripropylamine Dibutylamine Diisobutylamine Di-sec-butylamine Tributylamine Ethylenediamine Cyclohexylamine Benzylamine Aniline N-methylaniline
180 192 190 223 187 169 201 185 250 236 156 223 210 177 180 222 200 203 203 283 255 283 267 216
267 290 322 351 339 336 318 328 404 327 275 329 378 357 429 434 412 407 489 389 407 458 457 469
36.70 33.76 35.20 31.87 33.89 34.76 26.86 38.84 30.64 34.20 29.51 28.44 31.31 32.19 32.76 31.37 30.67 31.71 32.30 43.40 36.35 41.14 45.50 44.86
Alkanes 1-Butane 1-Pentane 1-Hexane 1-Heptane 1-Octane 1-Nonane 1-Decane 1-Undecane 1-Dodecane 1-Tetradecane 1-Pentadecane
134.82 143.5 178 183 216 220 244 247.56 263.6 285 283.10
272.5 309.07 341.74 371.43 398.66 427.79 447.12 468.98 489.28 526.59 543.74
31.55 32.54 30.16 30.94 28.95 29.69 28.35 28.77 27.97 27.27 27.93
Ethers Ethyl methyl ether Ethyl propyl ether Diethyl ether Dipropyl ether Diisopropyl ether Dibutyl ether Butyl ethyl ether Butyl methyl ether Dihexyl ether
160 145.60 156.80 147 186.30 177.90 170 157.60 230
280.60 336.36 307.60 363.23 341.66 413.43 365.40 343.31 499.60
33.46 35.36 29.49 35.81 28.99 33.68 33.57 34.38 31.20
Paraffins 3-Ethylpentane 2-Methylhexane 3-Methylhexane 2-Methylheptane 3-Methylheptane
154.50 154.90 153.70 164.16 152.60
366.60 363.19 364.99 390.81 392.09
34.77 32.61 33.32 33.19 34.94
Substances Methyl tetradecanoate Methyl hexadecanoate Methyl ethanoate Ethyl ethanoate Butyl ethanoate 2-Butyl ethanoate sec-Butyl ethanoate Hexyl ethanoate Methyl methanoate
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M.H. Ghatee, L. Pakdel / Fluid Phase Equilibria 234 (2005) 101–107
Table 1 (Continued ) Substances 4-Methylheptane 3-Ethylheptane 4-Ethylheptane 2-Methyloctane 3-Methyloctane 2,2-Dimethylbutane 2,2-Dimethylpentane 2,4-Dimethylpentane 2,2,3-Trimethylbutane 2,2-Dimethylhexane 2,5-Dimethylhexane 2,2,5-Trimethylhexane 2,2,3-Trimethylhexane 4-Ethyl-2-methylhexane 3-Ethyl-3-methylhexane 4-Ethyl-3-methylhexane 3,3-Diethylpentane 2,2,3,3-Tetramethylpentane 2,2,4,4-Tetramethylpentane 2,3,3,4-Tetramethylpentane 2,3-Dimethylheptane 2,4-Dimethylheptane 2,5-Dimethylheptane 3-Ethyl-2-methyl pentane 2,2,3-Trimethyl pentane Olefins 1-Decene 1-Undecene 1-Dodecene 1-Tridecene 1-Tetradecene 1-Pentadecene 1-Hexadecene 1-Heptadecene 1-Octadecene 1-Nonadecene 1-Eicosene
Tf (K)
Tb (K)
σ f (mN m−1 )
152 160 160 192.78 166 174 149.30 153.20 248 151.97 182 167.39 153 160 160 160 240.10 263.30 206.61 171 160 160 160 158.20 160.89
390.87 416.20 414.40 416.43 417.40 322.88 352.30 353.64 354.01 380.01 382.27 397.24 406.80 407 413.80 413.60 419.30 413.44 395.44 414.72 413.70 406.10 409.20 388.81 383
35.01 35.30 35.30 31.32 34.29 28.11 31.79 31.74 23.15 32.09 29.99 31.08 34.50 34.40 35.84 35.71 28.61 26.05 28.24 33.27 35.02 33.72 33.72 34.07 32.51
206.9 224 237.9 250.1 260 269.4 277.3 284 290.8 297.0 302.0
443.6 465.8 486.9 505.9 524.3 541.5 558.0 573.2 588.0 601.7 614.9
31.93 31.12 30.52 30.05 29.72 29.37 29.11 28.92 28.69 28.48 28.36
isomers with particular functional groups, and are used in organic synthesis as solvent as well as organic reagent. On the other hand, they are the building block of molecular structure of living organism for which molecular interfacial behavior is responsible for some living activity [12,13]. The vapor–liquid interfacial tension of these liquids are also important when drug formulation is concerned. Only acids and alkyl halides (and to some extent amines) are weak electrolytes and the rest can be considered as molecular compounds. In this way, a variety of liquids of different structures and molecular shapes are involved. The widest range of surface tension at freezing point is due to the class of amines (26.86–45.50 mN m−1 ), aldehydes (27.02–45.15 mN m−1 ), and esters (29.41–44.03 mN m−1 ). The class of alkanes (27.27–32.53 mN m−1 ), ethers (28.99–35.81 mN m−1 ), and olefins (28.36–34.82 mN m−1 ) have the least ranges of surface tension. All the surface tension data were taken from tabulation given by Jasper [17]. Extrapolations or interpolations were made when required. A check on the values of slope and intercept (of surface tension that have) given in the
∗ vs. T ∗ for alcohols in the range T ∗ Fig. 1. Plots of σsc sc index = 0.9–0.5.
tabulations did not lead to serious discrepancy. The values of physical properties were taken from CRC Handbook [18]. ∗ versus T ∗ for the class of alcohols are shown Plots of σsc sc ∗ = 0.9–0.5 are shown only. In the in Fig. 1. The plots for Tindex ∗ first step, the temperature corresponding to the selected Tindex is calculated, and then the value of σ is determined by either interpolation or extrapolation. In the case of those liquids for which σ f values were reported in literature, no discrepancy was seen as compared to Jasper [17]. The slopes and intercepts of the plots for alcohols in Fig. 1 are shown in Table 2. It can be seen that there are excellent accuracies as indicated by the LCC well above the limit (e.g., 0.995) for the satisfactorily linear behavior. We have considered the same plot for nine other classes: acids, aldehydes, alkyl halides, esters, amines, alkanes, paraffins, olefins, and ethers. Typically, we have shown in Figs. 2–5 plots for classes of alkanes, paraffins, acids, and alkyl halides, respectively. Although the general form ∗ of the plots are the same, but diversity of ranges of σsc ∗ versus Tsc among the classes, which is essentially originated from the diversity of values of surface tension and the physical properties characteristics of each class of liquid, are noticeable. For instance acids, alcohols, esters, and amines have the largest ranges; paraffins, olefins, and ethers have
∗ vs. T ∗ for alkanes in the range T ∗ Fig. 2. Plots of σsc sc index = 0.9–0.5.
M.H. Ghatee, L. Pakdel / Fluid Phase Equilibria 234 (2005) 101–107
105
Table 2 ∗ –T ∗ for different The intercept α and the slope β of linear correlation σsc sc classes of liquids ∗ Tindex
β
0.9 0.8 0.7 0.6 0.5
1.26 1.36 1.49 1.66 1.90
α
LCC
β
−0.1392 −0.1066 −0.0807 −0.0604 −0.0446
0.9996 0.9996 0.9995 0.9994 0.9991
1.19 1.29 1.42 1.58 1.82
Acids
1.18 1.27 1.38 1.54 1.75
1.23 1.31 1.42 1.56 1.76
−0.0824 −0.0643 −0.0490 −0.0370 −0.0273
0.9942 0.9946 0.9949 0.9957 0.9961
1.32 1.38 1.46 1.56 1.71
−0.1188 −0.0839 −0.0584 −0.0402 −0.0272
0.9993 0.9996 0.9996 0.9992 0.9985
1.29 1.32 1.37 1.43 1.52
1.24 1.33 1.43 1.57 1.75
−0.1247 −0.0875 −0.0599 −0.0400 −0.0260
0.9945 0.9958 0.9968 0.9978 0.9984
1.22 1.31 1.42 1.58 1.79
−0.1123 −0.0830 −0.0607 −0.0440 −0.0315
0.9994 0.9993 0.9991 0.9988 0.9983
−0.1096 −0.0768 −0.0525 −0.0350 −0.0228
0.9980 0.9984 0.9988 0.9989 0.9988
−0.0756 −0.0547 −0.0389 −0.0270 −0.0182
0.9994 0.9993 0.9993 0.9991 0.9989
Ethers −0.1773 −0.1124 −0.0698 −0.0420 −0.0245
0.9975 0.9981 0.9983 0.9983 0.9981
Olefins 0.9 0.8 0.7 0.6 0.5
0.9948 0.9952 0.9958 0.9964 0.9970
Amines
Alkanes 0.9 0.8 0.7 0.6 0.5
−0.0866 −0.0705 −0.0554 −0.0427 −0.0320
Alklyl halides
Esters 0.9 0.8 0.7 0.6 0.5
LCC
Alcohols
Aldehyde 0.9 0.8 0.7 0.6 0.5
α
1.22 1.30 1.40 1.53 1.71 Paraffins
−0.1577 −0.0902 −0.0499 −0.0264 −0.0130
0.9995 0.9995 0.9994 0.9991 0.9987
1.16 1.24 1.34 1.47 1.65
the middle ranges; aldehydes, alkyl halides, and alkanes ∗ versus T ∗ . have the smallest ranges of σsc sc ∗ –T ∗ correlation correThe slope and the intercept of σsc sc sponding to other (total 10 classes) are also shown in Table 2. ∗ –T ∗ correlation form a The LCC values indicate that σsc sc
∗ vs. T ∗ for paraffins in the range T ∗ Fig. 3. Plots of σsc sc index = 0.9–0.5.
∗ vs. T ∗ for acids in the range T ∗ Fig. 4. Plots of σsc sc index = 0.9–0.5.
promising functional for all classes of the liquids considered. ∗ In general, as Tindex decreases, the deviation of the data from a linear behavior increases slightly but, still stay well above the limit for the satisfactorily linear behavior. ∗ For a given Tindex , the slopes of all the classes of liquids are ∗ close to one another. For instance, at Tindex = 0.9 the slopes are in the range 1.16 (for paraffins) to 1.32 (for alkanes). The small differences may be attributed to the fact that intercept ∗ is not only determined by Tindex , but also determined by the set Tf , Tb , and σ f characteristics of class of liquids under consideration. Since the slopes of different classes of liquids are almost close to one another, it is expected that all the liquids of dif∗ –T ∗ correlation in a universal way ferent classes follow σsc sc ∗ at any given Tindex .The differences between the slopes of dif∗ ferent classes at Tindex = 0.9 rise to 12.1% (due to paraffins ∗ and alkanes) and at Tindex = 0.5 rise to 20% (due to acids and olefins). Thus, based on the values of the slopes, we have identified three categories: (I) alcohols and aldehydes; (II) alkanes, olefins, and paraffins; (III) alkyl halides, acids, esters, amines, and ethers. It is not surprising that polar and weak electrolytes form the same group and non-polar long chain hydrocarbons without functional group form a different group. Thus, the above
∗ vs. T ∗ for alkyl halides in the range T ∗ Fig. 5. Plots of σsc sc index = 0.9–0.5.
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M.H. Ghatee, L. Pakdel / Fluid Phase Equilibria 234 (2005) 101–107 Table 4 The values of surface entropy for 10 classes of liquids in Table 1 Ss
Liquid class
Acids Alcohols Aldehydes Alkyl halides Esters Amines Alkanes Ethers Olefins Paraffins ∗ vs. T ∗ for all liquids of the 10 classes in the range Fig. 6. Plots of σsc sc ∗ Tindex = 0.9–0.5.
universality is followed by all the liquids except for some differences due to the difference in the group properties of the each class of liquid, e.g., cationic and anionic partial charges and the extent of polarity. The same plot for 150 liquids of the 10 classes considered in this study is shown in Fig. 6. It can be seen that all (experimental) data points are very close to the corresponding fitted linear curve. The related slopes and intercepts are shown in Table 3. Considering the excellent accuracy of the correlation for these plots indicated by LCC’s, it can be concluded that ∗ there exists a promising universality at any Tindex , in the form of the linear relation (see Table 3 for values of parameters α and β) ∗ = α + βTsc∗ σsc
(4)
The negative value of temperature dependency of surface tension (∂σ/∂T)p is the surface entropy Ss . It represents an important property of the surface especially of the vapor–liquid interface. By comparing the values of Ss listed in Table 4, it can be seen that for the liquids in this study it varies at most by a factor of two. (Compare columns 2 and 3 in Table 4.) On the other hand, the value of surface energy Es for different classes of liquids varies by three orders of magnitude. Verification of these properties is crucial because the surface free energy Gs is determined by both surface entropy and the surface energy. Surface free energy essentially may be used to quantify the tendency for wetting, spreading, and penetration in many real systems such as biological membranes, and thus is responsible for some biological activity. Table 3 ∗ –T ∗ involving all The intercept α and the slope β of linear correlation σsc sc liquids of 10 classes in Table 1 ∗ Tindex
β
α
LCC
0.9 0.8 0.7 0.6 0.5
1.22 1.30 1.41 1.56 1.77
−0.1108 −0.0797 −0.0561 −0.0393 −0.0271
0.9978 0.9983 0.9985 0.9982 0.9974
Maximum
Minimum
Average
0.1098 0.1004 0.1360 0.1286 0.1572 0.1488 0.1206 0.10271 0.10271 0.10320
0.0763 0.0703 0.0920 0.0887 0.0775 0.0831 0.0858 0.0841 0.0841 0.0865
0.0923 0.0833 0.1087 0.1067 0.1084 0.11595 0.0966 0.09006 0.0901 0.09381
The fact that surface entropy is essentially a constant has been attributed to the fact that it is determined at the vapor side of interfacial region (more appropriately, at the vapor side of the Gibbs dividing surface), and therefore, it is independent of the nature of particular liquid. It turns out that the surface entropy is a generic property. On the other hand, the surface energy is a specific property, attributing that it is determined from liquid side of the dividing surface [1,19]. ∗ /∂T ∗ ) is shown In this work, the slope of Eq. (4) β (= ∂σsc sc to be rather constant for all the liquids, and in a sense it may be treated as the reduced surface entropy S s∗ , indexed by ∗ Tindex . Having considered the surface entropy of all liquids with diverse values (within a factor of 2), S s∗ may be treated as a generic property. This is the case for other liquid reported in literature, for which the parameters of their corresponding ∗ –T ∗ correlation are given in Table 5 [6,14,15]. Thus, we σsc sc ∗ –T ∗ and shown that it have implemented the functional σsc sc is a constant characteristics of all liquids considered in this study (and in other works). This resembles the conventional law of corresponding states given by van der Waals for flu∗ as a universal function of T ∗ ids. The resemblance is like σsc sc ∗ comprising Tindex . It has rather a more complex form as compared to reduced temperature Tr (of the van der Waals law of corresponding states), in the sense that it is constrained by two physical properties Tb and Tf characteristics of the liquid–vapor and the solid–liquid phase transitions, respec∗ tively. Indexing the temperature by Tindex is indeed to select the temperature T (and thus the corresponding surface tension σ) as a given fraction of (Tb − Tf ) less Tb : ∗ (Tb − Tf ) T = Tb − Tindex
(5)
Table 5 ∗ –T ∗ for systems The intercept α and the slope β of linear correlation σsc sc reported in literature [6,15] ∗ Tindex
0.9 0.8 0.7 0.6 0.5
Alkali halides
Molten salts
β
α
LCC
β
α
LCC
1.31 1.39 1.51 1.67 1.89
−0.1717 −0.1241 −0.0885 −0.0623 −0.0432
0.9997 0.9996 0.9995 0.9994 0.9992
1.19 1.28 1.54 1.72 1.98
−0.0795 −0.0512 −0.0800 −0.0567 −0.0401
0.996 0.997 0.994 0.996 0.996
M.H. Ghatee, L. Pakdel / Fluid Phase Equilibria 234 (2005) 101–107
In other words, if one divides the surface tension of classes of liquid by let say 10 segments on the temperature scale, then there exists certain regularity between the liquids properties at the temperature of each segment. One may pursue to conclude that physical and thermodynamic properties shown by ∗ liquids of a class are universal subjected to Tindex , e.g., at T1 liquid 1 shows the same properties shown by liquid 2 at T2 , ∗ etc., corresponding to the same Tindex . The importance of this statement becomes more clear by verifying that T is specified by Eq. (5) in terms of Tb and Tf , which are both accurate measures of intermolecular interaction potential energy and the molecular structure, and also by the fact that the surface properties are not simply determined by the effect of feature functional group in a molecule but also by the structure in general and by the molecular symmetry in particular. The theoretical study on the dissolving process is benefited by the information on the interfacial properties of solute and solvent, because the dissolving process is not only governed by intermolecular interaction but also is determined by the interfacial free energy of admixing partners. The correlation with the nature of this study would allow the determination of physical states at which the extent of solubility of different solvents for a given solute is the same. It may have theoretical application in drug design and formulation, though it should be verified in detail. The value of n which has been determined empirically, having the exact lower limit 4.0 for all liquids studied in this ∗ –T ∗ correlawork. Whether the mathematical structure of σsc sc tion and its feature, in particular the lower limit of n, described and verified in this study correspond to fundamental physical or thermodynamic properties remains the topics of further investigation. 4. Conclusion ∗ –T ∗ has shown to be universally appliThe correlation σsc sc cable to non-polar, polar, slightly polar, and weak electrolytes of liquid hydrocarbons. The slope of the correlation has been termed as reduced surface entropy and shown to be rather ∗ constants at each Tindex for all liquids of the 10 classes. It may be stated that if one divides the surface tension of a liquid by let say 10 segments on the temperature scale in the range Tf to Tb , then there exists certain regularity between the liquids properties at the temperature of each segment. Considering the result of application to 10 classes of liquids in the present study and pervious application to molten alkali halides, molten salts, and molten metal reported in literature, it has been verified and shown that a universal regularity is very likely. This correlation which has been given by phenomenological scaling law contains the empirical constant n with the exact lower limit 4.0, and has been shown to be applicable to the all liquids considered in this study.
List of symbols E internal energy G Gibbs free energy
P S T
107
pressure entropy absolute temperature
Greek symbols α intercept of linear equation β slope of linear equation proportionality constant σo σ surface tension Superscripts m constant exponent n constant exponent s surface * reduced quantity Subscripts b boiling c critical f freezing r reduced sc scaled
Acknowledgement The authors are indebted to research councils of Shiraz University for supporting this study. References [1] J. Lyklema, Surf. Colloid: Phys. Eng. Asp. 156 (1999) 413–421. [2] P. Lam, K.J. Waynne, G.E. Winek, Langmuir 18 (2002) 948–951. [3] A. Martin, Physical Pharmacy, fourth ed., LEA & Febiger, Philadelphia, 1993. [4] S. Schreier, S.V.P. Malheiros, E. de Paula, Biochim. Biophys. Acta 1508 (2000) 210–234. [5] M.H. Ghatee, A. Boushehri, High Temp. High Press. 26 (1994) 507–513. [6] M.H. Ghatee, M.H. Mousazadeh, A. Boushehri, High Temp. High Press. 29 (1997) 717–722. [7] E.A. Guggenheim, J. Chem. Phys. 13 (1945) 256. [8] J. Lielmezs, T.A. Herrick, Chem. Eng. J. 32 (1986) 165–169. [9] C.D. Holcomb, J.A. Zillweg, J. Phys. Chem. 97 (1993) 4797–4807. [10] R.C. Reid, J.M. Prausnitz, B.E. Poling, The Properties of Gases and Liquids, McGraw-Hill, New York, 1987. [11] R. Chhabra, High Temp. High Press. 23 (1991) 569–573. [12] A. Felmeister, J. Pharm. Sci. 61 (1972) 151–164. [13] P. Seeman, Pharmacol. Rev. 24 (1972) 583–589. [14] Note: This paper includes incorrect results for slope and intercept. An erratum would be effective. See Table 2 (this work) for correct values. S. Sheikh, A. Boushehri, High Temp. High Press. 31 (1999) 197–202. [15] H. Eslami, High Temp. High Press. 33 (2000) 245–251. [16] Note: This paper includes incorrect results too. S. Sheikh, A. Boushehri, High Temp. High Press. 32 (2000) 233–238. [17] J.J. Jasper, J. Phys. Chem. Ref. Data 1 (1972) 841–1009. [18] D.R. Lide (Ed.), CRC Handbook of Chemistry and Physics, 80th ed., Boca Raton, London, 1999–2000. [19] M.H. Ghatee, A. Maleki, H. Ghaed-Sharaf, Langmuir 19 (2003) 211–213.
Surface tension regularity of non-polar, polar, and weak electrolyte liquid hydrocarbons Mohammad Hadi Ghatee ∗ , Leila Pakdel Department of Chemistry, College of Science, Shiraz University, Shiraz 71454, Iran Received 2 April 2004; received in revised form 12 May 2005; accepted 13 May 2005
Abstract The correlation of reduced surface tension σsc∗ versus reduced temperature Tsc∗ which produces family of linear curves is investigated ∗ for universality. Each curve is characterized by the reduced temperature index Tindex which is included in Tsc∗ . The correlation is based on phenomenological scaling law and consideration of law of corresponding states. The reduced temperature Tsc∗ , scaled distance from the boiling temperature Tb , has the practical advantage of being independent of critical temperature. The correlation is highly accurate when applied to 10 classes of non-polar, slightly polar, polar, and weak electrolyte liquid hydrocarbons. The slopes of linear curves of each class of liquids ∗ at given Tindex ’s are rather constant and a universal regularity among all liquid is very likely, however small differences identify three classes of non-polar, slightly polar, and polar plus weak electrolytes. Consideration of the pervious application to molten alkali halides, molten salts, and molten metals, and considering that the reduced surface entropy S s∗ = (∂σsc∗ /∂Tsc∗ ) is rather constant for all liquid tested, strengthen the idea that σsc∗ –Tsc∗ correlation is likely universal. © 2005 Elsevier B.V. All rights reserved. Keywords: Law of corresponding states; Liquid hydrocarbons; Molten metals; Molten salts; Surface tension; Correlation for surface tension; Weak electrolytes
1. Introduction Surface tension is an important property where wetting, penetration, foaming, or droplet formation of a liquid is considered, and where environmental profile is encountered [1,2]. Values of gas–liquid interfacial tension are used in studying of liquid–liquid and liquid–solid interfaces, as well as the design of fractionators, absorbers, separators, two-phase pipeline, and in reservoir design calculations [1]. It is also a parameter required for drug formulation and drug product design [3,4]. Surface tension, among other physical and thermodynamic properties of liquids, is a basis for practical and theoretical modeling of the liquid state of matters [5]. It varies with temperature linearly and vanishes non-linearly close to critical point Tc , where the interface fades out. Although surface tension can be measured accurately by experimental methods, it has been ∗
Corresponding author. Fax: +98 711 228 6008. E-mail address: [email protected] (M.H. Ghatee).
0378-3812/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2005.05.011
correlated with many physical properties, for both prediction and validation of theories of interface. Relationship involving the melting point, molar volume, and heat of vaporization are the most meaningful [6]. For best application correlations based on the law of corresponding states have been established and thus, the applicability becomes universal [7]. The correlation of surface tension with temperature, given by van der Waals and then developed by Guggenheim [7], as σ = σ o (1 − Tr )m , requires knowledge of the substancedependent constant σ o , and the critical temperature Tc , where σ is the surface tension, T the absolute temperature, and the reduced temperature Tr = T/Tc . The constant m has universality and its reported values are in the range 1.16 [8], which are different from the theoretical values of 1.26 [9]. Further works have progressed to express σ o in terms of Tc , the critical pressure Pc , and a substance-dependent constant in terms of boiling temperature [10]. Based on phenomenological scaling, Lielmezs and Herrick [8] have presented a correlation as a single linear
102
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curve in the form of corresponding states, applicable to a wide range of atomic, spherical, alkanes, and slightly polar organic liquids. This correlation and its modified form [5] have been applied to molten metals [5,11], while requiring Tc and predicts surface tension within 15% for molten metals [11], and within 14% of experimental values of atomic and molecular normal fluids [8]. This and similar correlations result in the corresponding states of surface tension in terms of temperature scaled distance from Tc . One of the present author has presented a reasonably accurate modified formed of the above correlation in which surface tension is given as function of temperature scaled distance from boiling temperature [5], and thus, it has found practical application for the fact that it is independent of critical temperature. It is applicable to molten alkali metals over a wide range of temperature with maximum average deviation of 2.43% [5], and to molten alkali halides with maximum average deviation of 7.5% [6]. ∗ –T ∗ In the present study, we investigate the features of σsc sc correlation for universality by considering different classes of liquid hydrocarbons from non-polar to slightly polar and weak electrolytes. These hydrocarbons are of importance as solvent and as the building block of industrial products, biochemical, and biological systems [12,13]. The results of this study and the previous results of application to important materials of engineering type are used to qualify the universality.
2. Surface tension regularity Based on phenomenological scaling law [8], and consideration of law of corresponding states, relations have been presented for surface tension, which are mainly aimed to correlate surface tension in terms of physical properties of fluid bulk. The accuracy of these correlations has been noted by a number of applications to normal liquids [14], molten salts [15], and molten metals [16]. A reasonably accurate modified form of the above correlation in which the reduced surface tension σ * = (σ/T)(σ f /Tf ) is given as function of reduced temperature T* = [(Tb − T)/(Tb − Tf )](Tf / T) (scaled distance from boiling point) has been presented, where subscripts b and f stand for boiling and freezing points, respectively. It has found practical applications for the fact that it is independent of critical temperature. The correlation of σ * with T* results in a single curve from freezing point to boiling point. Following successful applications, it has been argued that since surface tension is a measure of intermolecular interaction potential should not have necessarily the same weighting as T which is a measure of thermal energy. Consistence with these theories, then we have found empirically a new regularity for the surface tension in which a family of linear functionals were produced by [6], n+1 Tf σ ∗ σsc = (1) σf T
as a function of n Tf Tb − T σ Tsc∗ = Tb − T f T σf
(2)
where σ is the surface tension, n a constant value, Tf and σ f the absolute temperature and surface tension at the freezing point, respectively, and Tb is the absolute temperature at the ∗ with T ∗ (is referred boiling point. The fact that correlation σsc sc ∗ ∗ to as σsc –Tsc , hereafter) is independent of the critical temperature makes it of special interest for practical application. The first application to molten alkali halides indicates that ∗ –T ∗ is highly accurate, with linear correlation coefficient σsc sc (LCC) between 0.9992 and 0.9998. Each curve of the family of curves is indexed by Tb − T ∗ Tindex (3) = Tb − T f involved in Tsc∗ . In Eqs. (1) and (2), n is a constant equal ∗ –T ∗ is reduced to the previous σ * –T* to 4.0. With n = 0, σsc sc correlation [5], in which no indexing was applicable and only a single curve is produced. When molten alkali halides are treated with σ * –T* correlation, a single linear curve with LCC = 0.9906 is produced, below the limit for considering ∗ –T ∗ , the high a curve satisfactorily linear. Therefore, for σsc sc ∗ accuracy is gained by including Tindex in the definition of Tsc∗ and thus, it is the key point for producing a quite accurate universal relationship. Since the form of Tsc∗ is rather com∗ plex, it worth mentioning that Tsc∗ = Tsc∗ (Tindex , σ; σf , n = 4.0) equally clarify the physical structure of Tsc∗ as Tsc∗ = Tsc∗ (T, σ; Tf , σf , Tb , n = 4.0) does. However, the former has a special feature presented in particular, leading to simple ∗ –T ∗ correlation. procedure for the construction of σsc sc While the accuracy is excellent, the critical temperature is ∗ –T ∗ preferential over the counexcluded and this makes σsc sc terpart correlations. This is of practical interest for both prediction and validation of theory of interface when the organic liquids with high Tc , susceptible to decomposition at high T’s, are involved. Empirically, we have found that the value 4.0 is an exact lower limit of n for molten alkali halides [6]. This has been the case for the subsequent reported applications [14–16], and a higher limit cannot be specified [6]. For n > 4.0, the linear behavior persists but the accuracy will be decreased, as indicated by decreasing LCC.
3. Results and discussion ∗ –T ∗ has been applied to molten alkali The correlation σsc sc halides [6], alkanes [14], molten salts [15], and molten metals [16]. (See the notes in Refs. [14] and [16].) In this study, we investigate to demonstrate its application as a universal correlation. To achieve this, we have selected 10 sets of organic liquid of different polarity. The liquids and their physical properties are listed in Table 1 . The liquids involved are mainly long chain hydrocarbons and their
M.H. Ghatee, L. Pakdel / Fluid Phase Equilibria 234 (2005) 101–107 Table 1 The physical properties of liquid hydrocarbons
103
Table 1 (Continued )
Substances
Tf (K)
Tb (K)
σ f (mN m−1 )
Acids Methanoic acid Ethanoic acid Propionic acid Butanoic acid 2-Methylpropionic acid Pentanoic acid 2-Methylbutanoic acid Hexanoic acid Heptanoic acid
281.4 290 250 268 226 255 244 299 264.5
374 391 414 436.7 427 458 450 519 387
38.96 27.91 30.98 28.82 31.22 30.51 29.86 39.72 30.61
Alcohols Methanol Ethanol 1-Propanol 2-Propanol 1-Butanol 2-Butanol 1-Pentanol 3-Methyl-1-butanol 2-Pentanol 2-Methyl-2-butanol 2-Propen-1-ol 2-Methoxyethanol 2-Ethoxyethanol 2-Butoxyethanol 1-Octanol 2-Octanol 1-Nonanol 1-Decanol
175 143 146 184 184 165 194 156 223 265 144 188 173 203 257 235 266 280
337.7 351 370 355 391 381 412 404 392 375 370 397 410 444 468 451 488 503
31.59 34.88 35.14 29.93 35.19 33.13 34.46 35.37 31.00 24.79 39.18 41.68 39.57 33.90 30.37 31.09 30.33 29.86
Aldehydes Ethanal Propanal Butanal Pentanal Heptanal Benzaldehyde o-Hydroxybenzaldehyde
148 285 177 181 230 247 275
294 397 348 376 426 452 470
40.92 27.02 35.56 37.27 32.61 43.57 45.15
Alkyl halides 1-Bromopropane 1-Bromobutane 1-Bromopentane 1-Bromohexane 1-Bromoheptane 1-Chlorobutane 1-Chloropentane 1-Chlorohexane 1-Chloroheptane 1-Iodoethane 1-Iodopropane 1-Iodobutane 1-Iodoheptane
165 161 178 188 215 150 174 179 204 165 172 170 225
344 377 403 428 453 352 381 409 434 346 376 403 478
41.47 41.34 39.49 38.04 36.45 39.73 37.76 38.09 35.59 45.58 43.13 41.45 36.45
Esters Ethyl methanoate Methyl propionate Ethyl propionate Propyl propionate Ethyl butanoate Ethyl-2-methylbutanoate Methyl hexanoate Ethyl heptanoate Ethyl octanoate
193 185 200 197 180 174 202 207 226
327 352 372 397 393 406 424 462 481
37.01 38.67 35.26 34.92 36.28 35.77 35.91 34.53 33.96
Tf (K)
Tb (K)
σ f (mN m−1 )
293 307 175 189 195 175 174 193 173
596 437 330.5 350 399 390 385 443 307
29.41 28.88 40.60 36.06 35.90 35.53 36.20 36.25 44.03
Amines Methylamine Ethylamine Propylamine Butylamine Isobutylamine sec-Butylamine tert-Butylamine Allylamine Hexylamine Dimethylamine Trimethylamine Diethylamine Dipropylamine Diisopropylamine Tripropylamine Dibutylamine Diisobutylamine Di-sec-butylamine Tributylamine Ethylenediamine Cyclohexylamine Benzylamine Aniline N-methylaniline
180 192 190 223 187 169 201 185 250 236 156 223 210 177 180 222 200 203 203 283 255 283 267 216
267 290 322 351 339 336 318 328 404 327 275 329 378 357 429 434 412 407 489 389 407 458 457 469
36.70 33.76 35.20 31.87 33.89 34.76 26.86 38.84 30.64 34.20 29.51 28.44 31.31 32.19 32.76 31.37 30.67 31.71 32.30 43.40 36.35 41.14 45.50 44.86
Alkanes 1-Butane 1-Pentane 1-Hexane 1-Heptane 1-Octane 1-Nonane 1-Decane 1-Undecane 1-Dodecane 1-Tetradecane 1-Pentadecane
134.82 143.5 178 183 216 220 244 247.56 263.6 285 283.10
272.5 309.07 341.74 371.43 398.66 427.79 447.12 468.98 489.28 526.59 543.74
31.55 32.54 30.16 30.94 28.95 29.69 28.35 28.77 27.97 27.27 27.93
Ethers Ethyl methyl ether Ethyl propyl ether Diethyl ether Dipropyl ether Diisopropyl ether Dibutyl ether Butyl ethyl ether Butyl methyl ether Dihexyl ether
160 145.60 156.80 147 186.30 177.90 170 157.60 230
280.60 336.36 307.60 363.23 341.66 413.43 365.40 343.31 499.60
33.46 35.36 29.49 35.81 28.99 33.68 33.57 34.38 31.20
Paraffins 3-Ethylpentane 2-Methylhexane 3-Methylhexane 2-Methylheptane 3-Methylheptane
154.50 154.90 153.70 164.16 152.60
366.60 363.19 364.99 390.81 392.09
34.77 32.61 33.32 33.19 34.94
Substances Methyl tetradecanoate Methyl hexadecanoate Methyl ethanoate Ethyl ethanoate Butyl ethanoate 2-Butyl ethanoate sec-Butyl ethanoate Hexyl ethanoate Methyl methanoate
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M.H. Ghatee, L. Pakdel / Fluid Phase Equilibria 234 (2005) 101–107
Table 1 (Continued ) Substances 4-Methylheptane 3-Ethylheptane 4-Ethylheptane 2-Methyloctane 3-Methyloctane 2,2-Dimethylbutane 2,2-Dimethylpentane 2,4-Dimethylpentane 2,2,3-Trimethylbutane 2,2-Dimethylhexane 2,5-Dimethylhexane 2,2,5-Trimethylhexane 2,2,3-Trimethylhexane 4-Ethyl-2-methylhexane 3-Ethyl-3-methylhexane 4-Ethyl-3-methylhexane 3,3-Diethylpentane 2,2,3,3-Tetramethylpentane 2,2,4,4-Tetramethylpentane 2,3,3,4-Tetramethylpentane 2,3-Dimethylheptane 2,4-Dimethylheptane 2,5-Dimethylheptane 3-Ethyl-2-methyl pentane 2,2,3-Trimethyl pentane Olefins 1-Decene 1-Undecene 1-Dodecene 1-Tridecene 1-Tetradecene 1-Pentadecene 1-Hexadecene 1-Heptadecene 1-Octadecene 1-Nonadecene 1-Eicosene
Tf (K)
Tb (K)
σ f (mN m−1 )
152 160 160 192.78 166 174 149.30 153.20 248 151.97 182 167.39 153 160 160 160 240.10 263.30 206.61 171 160 160 160 158.20 160.89
390.87 416.20 414.40 416.43 417.40 322.88 352.30 353.64 354.01 380.01 382.27 397.24 406.80 407 413.80 413.60 419.30 413.44 395.44 414.72 413.70 406.10 409.20 388.81 383
35.01 35.30 35.30 31.32 34.29 28.11 31.79 31.74 23.15 32.09 29.99 31.08 34.50 34.40 35.84 35.71 28.61 26.05 28.24 33.27 35.02 33.72 33.72 34.07 32.51
206.9 224 237.9 250.1 260 269.4 277.3 284 290.8 297.0 302.0
443.6 465.8 486.9 505.9 524.3 541.5 558.0 573.2 588.0 601.7 614.9
31.93 31.12 30.52 30.05 29.72 29.37 29.11 28.92 28.69 28.48 28.36
isomers with particular functional groups, and are used in organic synthesis as solvent as well as organic reagent. On the other hand, they are the building block of molecular structure of living organism for which molecular interfacial behavior is responsible for some living activity [12,13]. The vapor–liquid interfacial tension of these liquids are also important when drug formulation is concerned. Only acids and alkyl halides (and to some extent amines) are weak electrolytes and the rest can be considered as molecular compounds. In this way, a variety of liquids of different structures and molecular shapes are involved. The widest range of surface tension at freezing point is due to the class of amines (26.86–45.50 mN m−1 ), aldehydes (27.02–45.15 mN m−1 ), and esters (29.41–44.03 mN m−1 ). The class of alkanes (27.27–32.53 mN m−1 ), ethers (28.99–35.81 mN m−1 ), and olefins (28.36–34.82 mN m−1 ) have the least ranges of surface tension. All the surface tension data were taken from tabulation given by Jasper [17]. Extrapolations or interpolations were made when required. A check on the values of slope and intercept (of surface tension that have) given in the
∗ vs. T ∗ for alcohols in the range T ∗ Fig. 1. Plots of σsc sc index = 0.9–0.5.
tabulations did not lead to serious discrepancy. The values of physical properties were taken from CRC Handbook [18]. ∗ versus T ∗ for the class of alcohols are shown Plots of σsc sc ∗ = 0.9–0.5 are shown only. In the in Fig. 1. The plots for Tindex ∗ first step, the temperature corresponding to the selected Tindex is calculated, and then the value of σ is determined by either interpolation or extrapolation. In the case of those liquids for which σ f values were reported in literature, no discrepancy was seen as compared to Jasper [17]. The slopes and intercepts of the plots for alcohols in Fig. 1 are shown in Table 2. It can be seen that there are excellent accuracies as indicated by the LCC well above the limit (e.g., 0.995) for the satisfactorily linear behavior. We have considered the same plot for nine other classes: acids, aldehydes, alkyl halides, esters, amines, alkanes, paraffins, olefins, and ethers. Typically, we have shown in Figs. 2–5 plots for classes of alkanes, paraffins, acids, and alkyl halides, respectively. Although the general form ∗ of the plots are the same, but diversity of ranges of σsc ∗ versus Tsc among the classes, which is essentially originated from the diversity of values of surface tension and the physical properties characteristics of each class of liquid, are noticeable. For instance acids, alcohols, esters, and amines have the largest ranges; paraffins, olefins, and ethers have
∗ vs. T ∗ for alkanes in the range T ∗ Fig. 2. Plots of σsc sc index = 0.9–0.5.
M.H. Ghatee, L. Pakdel / Fluid Phase Equilibria 234 (2005) 101–107
105
Table 2 ∗ –T ∗ for different The intercept α and the slope β of linear correlation σsc sc classes of liquids ∗ Tindex
β
0.9 0.8 0.7 0.6 0.5
1.26 1.36 1.49 1.66 1.90
α
LCC
β
−0.1392 −0.1066 −0.0807 −0.0604 −0.0446
0.9996 0.9996 0.9995 0.9994 0.9991
1.19 1.29 1.42 1.58 1.82
Acids
1.18 1.27 1.38 1.54 1.75
1.23 1.31 1.42 1.56 1.76
−0.0824 −0.0643 −0.0490 −0.0370 −0.0273
0.9942 0.9946 0.9949 0.9957 0.9961
1.32 1.38 1.46 1.56 1.71
−0.1188 −0.0839 −0.0584 −0.0402 −0.0272
0.9993 0.9996 0.9996 0.9992 0.9985
1.29 1.32 1.37 1.43 1.52
1.24 1.33 1.43 1.57 1.75
−0.1247 −0.0875 −0.0599 −0.0400 −0.0260
0.9945 0.9958 0.9968 0.9978 0.9984
1.22 1.31 1.42 1.58 1.79
−0.1123 −0.0830 −0.0607 −0.0440 −0.0315
0.9994 0.9993 0.9991 0.9988 0.9983
−0.1096 −0.0768 −0.0525 −0.0350 −0.0228
0.9980 0.9984 0.9988 0.9989 0.9988
−0.0756 −0.0547 −0.0389 −0.0270 −0.0182
0.9994 0.9993 0.9993 0.9991 0.9989
Ethers −0.1773 −0.1124 −0.0698 −0.0420 −0.0245
0.9975 0.9981 0.9983 0.9983 0.9981
Olefins 0.9 0.8 0.7 0.6 0.5
0.9948 0.9952 0.9958 0.9964 0.9970
Amines
Alkanes 0.9 0.8 0.7 0.6 0.5
−0.0866 −0.0705 −0.0554 −0.0427 −0.0320
Alklyl halides
Esters 0.9 0.8 0.7 0.6 0.5
LCC
Alcohols
Aldehyde 0.9 0.8 0.7 0.6 0.5
α
1.22 1.30 1.40 1.53 1.71 Paraffins
−0.1577 −0.0902 −0.0499 −0.0264 −0.0130
0.9995 0.9995 0.9994 0.9991 0.9987
1.16 1.24 1.34 1.47 1.65
the middle ranges; aldehydes, alkyl halides, and alkanes ∗ versus T ∗ . have the smallest ranges of σsc sc ∗ –T ∗ correlation correThe slope and the intercept of σsc sc sponding to other (total 10 classes) are also shown in Table 2. ∗ –T ∗ correlation form a The LCC values indicate that σsc sc
∗ vs. T ∗ for paraffins in the range T ∗ Fig. 3. Plots of σsc sc index = 0.9–0.5.
∗ vs. T ∗ for acids in the range T ∗ Fig. 4. Plots of σsc sc index = 0.9–0.5.
promising functional for all classes of the liquids considered. ∗ In general, as Tindex decreases, the deviation of the data from a linear behavior increases slightly but, still stay well above the limit for the satisfactorily linear behavior. ∗ For a given Tindex , the slopes of all the classes of liquids are ∗ close to one another. For instance, at Tindex = 0.9 the slopes are in the range 1.16 (for paraffins) to 1.32 (for alkanes). The small differences may be attributed to the fact that intercept ∗ is not only determined by Tindex , but also determined by the set Tf , Tb , and σ f characteristics of class of liquids under consideration. Since the slopes of different classes of liquids are almost close to one another, it is expected that all the liquids of dif∗ –T ∗ correlation in a universal way ferent classes follow σsc sc ∗ at any given Tindex .The differences between the slopes of dif∗ ferent classes at Tindex = 0.9 rise to 12.1% (due to paraffins ∗ and alkanes) and at Tindex = 0.5 rise to 20% (due to acids and olefins). Thus, based on the values of the slopes, we have identified three categories: (I) alcohols and aldehydes; (II) alkanes, olefins, and paraffins; (III) alkyl halides, acids, esters, amines, and ethers. It is not surprising that polar and weak electrolytes form the same group and non-polar long chain hydrocarbons without functional group form a different group. Thus, the above
∗ vs. T ∗ for alkyl halides in the range T ∗ Fig. 5. Plots of σsc sc index = 0.9–0.5.
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M.H. Ghatee, L. Pakdel / Fluid Phase Equilibria 234 (2005) 101–107 Table 4 The values of surface entropy for 10 classes of liquids in Table 1 Ss
Liquid class
Acids Alcohols Aldehydes Alkyl halides Esters Amines Alkanes Ethers Olefins Paraffins ∗ vs. T ∗ for all liquids of the 10 classes in the range Fig. 6. Plots of σsc sc ∗ Tindex = 0.9–0.5.
universality is followed by all the liquids except for some differences due to the difference in the group properties of the each class of liquid, e.g., cationic and anionic partial charges and the extent of polarity. The same plot for 150 liquids of the 10 classes considered in this study is shown in Fig. 6. It can be seen that all (experimental) data points are very close to the corresponding fitted linear curve. The related slopes and intercepts are shown in Table 3. Considering the excellent accuracy of the correlation for these plots indicated by LCC’s, it can be concluded that ∗ there exists a promising universality at any Tindex , in the form of the linear relation (see Table 3 for values of parameters α and β) ∗ = α + βTsc∗ σsc
(4)
The negative value of temperature dependency of surface tension (∂σ/∂T)p is the surface entropy Ss . It represents an important property of the surface especially of the vapor–liquid interface. By comparing the values of Ss listed in Table 4, it can be seen that for the liquids in this study it varies at most by a factor of two. (Compare columns 2 and 3 in Table 4.) On the other hand, the value of surface energy Es for different classes of liquids varies by three orders of magnitude. Verification of these properties is crucial because the surface free energy Gs is determined by both surface entropy and the surface energy. Surface free energy essentially may be used to quantify the tendency for wetting, spreading, and penetration in many real systems such as biological membranes, and thus is responsible for some biological activity. Table 3 ∗ –T ∗ involving all The intercept α and the slope β of linear correlation σsc sc liquids of 10 classes in Table 1 ∗ Tindex
β
α
LCC
0.9 0.8 0.7 0.6 0.5
1.22 1.30 1.41 1.56 1.77
−0.1108 −0.0797 −0.0561 −0.0393 −0.0271
0.9978 0.9983 0.9985 0.9982 0.9974
Maximum
Minimum
Average
0.1098 0.1004 0.1360 0.1286 0.1572 0.1488 0.1206 0.10271 0.10271 0.10320
0.0763 0.0703 0.0920 0.0887 0.0775 0.0831 0.0858 0.0841 0.0841 0.0865
0.0923 0.0833 0.1087 0.1067 0.1084 0.11595 0.0966 0.09006 0.0901 0.09381
The fact that surface entropy is essentially a constant has been attributed to the fact that it is determined at the vapor side of interfacial region (more appropriately, at the vapor side of the Gibbs dividing surface), and therefore, it is independent of the nature of particular liquid. It turns out that the surface entropy is a generic property. On the other hand, the surface energy is a specific property, attributing that it is determined from liquid side of the dividing surface [1,19]. ∗ /∂T ∗ ) is shown In this work, the slope of Eq. (4) β (= ∂σsc sc to be rather constant for all the liquids, and in a sense it may be treated as the reduced surface entropy S s∗ , indexed by ∗ Tindex . Having considered the surface entropy of all liquids with diverse values (within a factor of 2), S s∗ may be treated as a generic property. This is the case for other liquid reported in literature, for which the parameters of their corresponding ∗ –T ∗ correlation are given in Table 5 [6,14,15]. Thus, we σsc sc ∗ –T ∗ and shown that it have implemented the functional σsc sc is a constant characteristics of all liquids considered in this study (and in other works). This resembles the conventional law of corresponding states given by van der Waals for flu∗ as a universal function of T ∗ ids. The resemblance is like σsc sc ∗ comprising Tindex . It has rather a more complex form as compared to reduced temperature Tr (of the van der Waals law of corresponding states), in the sense that it is constrained by two physical properties Tb and Tf characteristics of the liquid–vapor and the solid–liquid phase transitions, respec∗ tively. Indexing the temperature by Tindex is indeed to select the temperature T (and thus the corresponding surface tension σ) as a given fraction of (Tb − Tf ) less Tb : ∗ (Tb − Tf ) T = Tb − Tindex
(5)
Table 5 ∗ –T ∗ for systems The intercept α and the slope β of linear correlation σsc sc reported in literature [6,15] ∗ Tindex
0.9 0.8 0.7 0.6 0.5
Alkali halides
Molten salts
β
α
LCC
β
α
LCC
1.31 1.39 1.51 1.67 1.89
−0.1717 −0.1241 −0.0885 −0.0623 −0.0432
0.9997 0.9996 0.9995 0.9994 0.9992
1.19 1.28 1.54 1.72 1.98
−0.0795 −0.0512 −0.0800 −0.0567 −0.0401
0.996 0.997 0.994 0.996 0.996
M.H. Ghatee, L. Pakdel / Fluid Phase Equilibria 234 (2005) 101–107
In other words, if one divides the surface tension of classes of liquid by let say 10 segments on the temperature scale, then there exists certain regularity between the liquids properties at the temperature of each segment. One may pursue to conclude that physical and thermodynamic properties shown by ∗ liquids of a class are universal subjected to Tindex , e.g., at T1 liquid 1 shows the same properties shown by liquid 2 at T2 , ∗ etc., corresponding to the same Tindex . The importance of this statement becomes more clear by verifying that T is specified by Eq. (5) in terms of Tb and Tf , which are both accurate measures of intermolecular interaction potential energy and the molecular structure, and also by the fact that the surface properties are not simply determined by the effect of feature functional group in a molecule but also by the structure in general and by the molecular symmetry in particular. The theoretical study on the dissolving process is benefited by the information on the interfacial properties of solute and solvent, because the dissolving process is not only governed by intermolecular interaction but also is determined by the interfacial free energy of admixing partners. The correlation with the nature of this study would allow the determination of physical states at which the extent of solubility of different solvents for a given solute is the same. It may have theoretical application in drug design and formulation, though it should be verified in detail. The value of n which has been determined empirically, having the exact lower limit 4.0 for all liquids studied in this ∗ –T ∗ correlawork. Whether the mathematical structure of σsc sc tion and its feature, in particular the lower limit of n, described and verified in this study correspond to fundamental physical or thermodynamic properties remains the topics of further investigation. 4. Conclusion ∗ –T ∗ has shown to be universally appliThe correlation σsc sc cable to non-polar, polar, slightly polar, and weak electrolytes of liquid hydrocarbons. The slope of the correlation has been termed as reduced surface entropy and shown to be rather ∗ constants at each Tindex for all liquids of the 10 classes. It may be stated that if one divides the surface tension of a liquid by let say 10 segments on the temperature scale in the range Tf to Tb , then there exists certain regularity between the liquids properties at the temperature of each segment. Considering the result of application to 10 classes of liquids in the present study and pervious application to molten alkali halides, molten salts, and molten metal reported in literature, it has been verified and shown that a universal regularity is very likely. This correlation which has been given by phenomenological scaling law contains the empirical constant n with the exact lower limit 4.0, and has been shown to be applicable to the all liquids considered in this study.
List of symbols E internal energy G Gibbs free energy
P S T
107
pressure entropy absolute temperature
Greek symbols α intercept of linear equation β slope of linear equation proportionality constant σo σ surface tension Superscripts m constant exponent n constant exponent s surface * reduced quantity Subscripts b boiling c critical f freezing r reduced sc scaled
Acknowledgement The authors are indebted to research councils of Shiraz University for supporting this study. References [1] J. Lyklema, Surf. Colloid: Phys. Eng. Asp. 156 (1999) 413–421. [2] P. Lam, K.J. Waynne, G.E. Winek, Langmuir 18 (2002) 948–951. [3] A. Martin, Physical Pharmacy, fourth ed., LEA & Febiger, Philadelphia, 1993. [4] S. Schreier, S.V.P. Malheiros, E. de Paula, Biochim. Biophys. Acta 1508 (2000) 210–234. [5] M.H. Ghatee, A. Boushehri, High Temp. High Press. 26 (1994) 507–513. [6] M.H. Ghatee, M.H. Mousazadeh, A. Boushehri, High Temp. High Press. 29 (1997) 717–722. [7] E.A. Guggenheim, J. Chem. Phys. 13 (1945) 256. [8] J. Lielmezs, T.A. Herrick, Chem. Eng. J. 32 (1986) 165–169. [9] C.D. Holcomb, J.A. Zillweg, J. Phys. Chem. 97 (1993) 4797–4807. [10] R.C. Reid, J.M. Prausnitz, B.E. Poling, The Properties of Gases and Liquids, McGraw-Hill, New York, 1987. [11] R. Chhabra, High Temp. High Press. 23 (1991) 569–573. [12] A. Felmeister, J. Pharm. Sci. 61 (1972) 151–164. [13] P. Seeman, Pharmacol. Rev. 24 (1972) 583–589. [14] Note: This paper includes incorrect results for slope and intercept. An erratum would be effective. See Table 2 (this work) for correct values. S. Sheikh, A. Boushehri, High Temp. High Press. 31 (1999) 197–202. [15] H. Eslami, High Temp. High Press. 33 (2000) 245–251. [16] Note: This paper includes incorrect results too. S. Sheikh, A. Boushehri, High Temp. High Press. 32 (2000) 233–238. [17] J.J. Jasper, J. Phys. Chem. Ref. Data 1 (1972) 841–1009. [18] D.R. Lide (Ed.), CRC Handbook of Chemistry and Physics, 80th ed., Boca Raton, London, 1999–2000. [19] M.H. Ghatee, A. Maleki, H. Ghaed-Sharaf, Langmuir 19 (2003) 211–213.