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On the relation between surface energy, internal pressure and molar volume in pure fluids
Colloids and Surfaces, 30 (1988) 405-411 Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands
405
Brief Note
On the R e l a t i o n b e t w e e n S u r f a c e E n e r g y , I n t e r n a l P r e s s u r e and Molar Volume in P u r e Fluids I. VAVRUCH Route du Centre 6, CH-1723 Marly (Switzerland)
(Received27 July 1987; accepted in final form 16 November 1987)
In a recent paper [ 1 ], I derived the following expression relating the total molar surface energy E~m of a pure liquid to its internal pressure Pi and its molar volume Vm: Pi V m / E ~ = k
(1)
in which k is a dimensionless geometric factor. Equation (1) is based on the Onsager sphere in a continuum model, with an internal surface [2,3 ] and although based on a molecular model, it contains exclusively macroscopic parameters of the system. This relation can be interpreted as follows: the work Pi Vm done against the attractive intermolecular forces in bringing one mole of liquid molecules from the bulk to the interface is proportional to the total molar surface energy of the system. The numerical value of factor k in Eqn (1) depends on the shape of liquid molecules which may exhibit considerable departure from spherical symmetry. The geometry of the liquid cavity in the surface can then be better approximated by using a polyhedron, a spheroid or a rod [ 4 ]. If Pi is expressed in J c m - 3, Vm in cm 3 m o l - 1 and E s in J m o l - 1, it can be shown that the numerical values of k are theoretically equal to 3.22 for a sphere, 3.81 for a octahedron, 4.00 for a cube and 4.81 for a tetrahedron. Since the sphere has the m i n i m u m ratio of surface to volume, its geometric factor k is always the lowest. The way of calculating factor k has been described elsewhere [ 1,5 ]. We shall later apply Eqn (1) exclusively to systems at constant temperature T and constant pressure P. The use of the total molar surface enthalpy H ~ instead of the surface energy E ~ would therefore be appropriate. However, since the product P d V is usually negligible in liquids at not too elevated pressures, H~m= E ~ is a good approximation [6]. The total molar surface energy is given by [ 7 ] E~ =/~ - T(Op~/OT)
0166-6622/88/$03.50
(2)
© 1988 Elsevier Science Publishers B.V.
406
where/Am is the molar surface free energy which can be evaluated from the equation
p~ = N~/31/¢/3/~
(3)
Here, NA is the Avogadro number a n d / l is the surface free energy (surface tension). It follows from Eqn ( 2 ) that for the calculation of E ~ not only the molar surface energy Pro, but also its temperature coefficient dp~/dT has to be determined. A direct calculation of this coefficient f r o m / ~ values at different temperatures may charge the values of E ~ with a considerable error, however, especially in liquids for which the dependence/1 (T) is known with less accuracy. Furthermore, it is very tedious. Therefore, an alternative procedure is usually preferred. It leads in general to more reliable results, since especially the values of densities and of the temperature coefficient of density are known for many liquids with a high degree of accuracy and they may be easily found in the literature. To derive the alternative equation for the determination o f E ~ , we first substitute Eqn (3) into Eqn (2) and, after rearranging, obtain the following relation [ 8 ]
E ~ = N~/3 V2/3/~- T O( N 1/3 Vc/3/~) / OT =N1/3[ I/¢/3/~- TV2/3( Op/ OT) - T~ ( OV2/3/ OT) ]
(4)
and further
0 V ~ 3/0 T = M 2/3 ( Op- 2/3/0 T) = - ( 2/3 ) MC/3p- 5/3 ( Op/OT) -- - ( 2 / 3 ) ( V c / ~ / p )
(Op/OT)
(5)
where M is the molecular weight and p is the density. Now, we take from Eqn (5) the expression for 0 V~ 3/0 T and substitute it into Eqn (4) to obtain the final relation
E~ = NIA/3V 2/3 [ (/~- TO/~/OT) + (2/3) T(#/p ) (Op/OT) ]
(6)
From Eqn (6), the surface energy E ~ can be easily calculated. Both temperature coefficients, dp/dT as well as dp/dT, are negative. The first term in the square brackets defines the total surface energy E s and Eqn (6) may therefore be simplified to
E~ =N~/3 V¢/3 [E ~+ (2/3) T(/~/p ) ( Op/OT) ]
(7)
Equation (1) forms a suitable basis for further theoretical treatment of liquid interfaces. It seemed therefore worthwhile to investigate its behaviour in different classes of fluids in more detail.
407 RESULTS AND DISCUSSION
Equation (1) has been applied to 76 fluids, both organic and inorganic. All data for the calculation of the molar volume Vm as well as the values of the density p and of its temperature coefficient dp/dT for organic systems were taken from Ref. [ 9 ]. Most experimental values of the surface tension tt and of its temperature coefficient d,u/dT were taken from Ref. [ 10 ] and the rest from Refs [9] and [11]. Data on the internal pressure Pi were taken from original publications (see Table 1), or they were evaluated from the thermodynamic relation [ 12 ] Pi = ( OZ/KT) T - P
(S)
where a is the coefficient of thermal expansion and K~T is the coefficient of isothermal compressibility. The values of these coefficients were taken mostly from Ref. [ 9 ], but in some cases it was necessary to calculate the value of a from the relation [ 9 ]
a= [(Pl/P2)-l]/(t2-tl)
(9)
where Pl and P2 are the density values at temperatures tl and t2 (in oC), and t2 > tl. In this last case, the data for densities were taken from Ref. [9 ]. A few h~T values were taken from original papers (see Table 1 ). An analysis of the probable errors in numerical data used for the calculation of the geometric factor k showed that the values are charged with an absolute error of about 0.06-0.25 units. Numerical values of k calculated from Eqn (1) for 28 selected liquids are listed in Table 1. The values of the surface energy E ~ were calculated from Eqn (6). For all these liquids, the value of factor k at 25 ° C lies in the range of 3.22 + 0.25, i.e. within the reproducibility of the results. It thus corresponds to the numerical value for a spherical cavity in a liquid surface. The agreement is excellent especially for such simple liquids, both orgnic and inorganic, in which the intermolecular forces do not depart at all from spherical symmetry, or depart from it only slightly (liquids 5, 7, 14, 25 and 28). It is also very good for some liquids in which this agreement between theory and experimental data is rather unexpected, e.g. owing to the complex structure of the liquid molecules, as in liquids 19, 20 and 22 in Table 1, because of their very dipolar character (especially liquids 11 and 23), or because of the hydrogen bonds occurring in the system, as in liquids 8 and 9, and especially in 21 and 24. The explanation for this finding lies probably in the enhanced symmetry of the molecular holes at the surface of pure liquids, as compared with the bulk [ 13 ]. On the other hand, the individual deviations from the theoretical value for a sphere are somewhat greater in liquids composed of molecules which are cylindrical ( liquids 1 and 3 ) or fiat (liquids 2, 4 and 6). In some liquids composed of complex molecules with great polarity and/or
408 TABLE 1 Numerical values of factor k in Eqn (1) for selected liquids at 25 ° C No
Liquids
Vm (cm3 mo1-1 )
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Hexane Cyclohexane 1-Hexene Cyclohexene Benzene Toluene Diethyl ether Acetone 2-Butanone Ethyl acetate Propylene carbonate Dichloromethane Chloroform Carbon tetrachloride Trichloroethylene Tetrachloroethylene Chlorobenzene 1-Bromobutane 1,2-Dibromoethane 1,1,2-Trichloro-1,2,2trifluoroethane 1,2-Ethanediol Nitromethane Acetonitrile Formamide Carbon disulfide Phosphorus trichloride Silicon tetrachloride Tin tetrachloride
131.62 108.75 125.90 101.91 89.42 106.87 104.75 74.05 90.17 98.49 85.42 64.50 80.68 97.09 90.02 102.73 102.24 108.00 86.62 119.84
21 22 23 24 25 26 27 28
55.92 53.96 52.86 39.88 60.64 87.68g 114.6g 116.8g
Pi (J cm -~)
E~ (kJ mo1-1 )
k
243.0 319.0 244.4 346.0 374.3 348.9 250.2 329.7a 298.6 345.4b 424.4 374.9c 378.6 338.9¢ 406.9d 402.2d 382.8 325.1 432.5 258.8
9.51 10.32 9.22 10.60 10.22 10.97 8.32 7.53 8.80 9.80 10.96 7.89 9.24 10.08 10.95 13.23 11.46 10.34 11.80 9.78
3.36 3.36 3.34 3.33 3.27 3.40 3.15 3.24 3.06 3.47 3.31 3.06 3.31 3.26 3.35 3.12 3.42 3.40 3.18 3.17
501.6e 606.0 409.0 581.3 453.0 f ~ 348g ~257 g ~ 311g
8.46 9.42 6.95 7.25 8.79 9.86 8.57 11.06
3.32 3.47 3.11 3.20 3.13 3.1 3.4 3.3
aRef. [141. bRefs [9,15]. CRef. [16]. dRefs [9,17]. eRefs [9,18]. rRef. [19]. gRef. [ 20 ].
e l e c t r i c a l a s y m m e t r y , t h e d e v i a t i o n s o f f a c t o r k i n E q n (1) f r o m t h e t h e o r e t i c a l v a l u e for a s p h e r e b e c o m e m o r e e v i d e n t . F o r e x a m p l e , for p y r i d i n e k e q u a l s 3.6 a t 25 ° C, for e t h y l e n e d i a m i n e a n d d i m e t h y l s u l f o x i d e 3.9, for i o d o b e n z e n e 4.0, a n d for n i t r o b e n z e n e a n d a n i l i n e 4.2. F o r h i g h e r a l k a n e s a n d p r o b a b l y a l s o h i g h e r a l k e n e s , for m o n o h y d r i c a l c o h o l s a n d for c a r b o x y l i c a c i d s , k b e c o m e s
409 TABLE 2 Extreme values of factor k in selected fluids at 25 °C
Fluids
k
Fluids
k
Octane Decane Dodecane Hexadecane Methanol Ethanol 1-Butanol 1-Hexanol
3.9 4.2 4.5 4.9 2.7 3.1 3.5 4.0
1-Octanol Formic acid Acetic acid Propionic acid Isobutyric acid Water Mercury~ Argonb
4.2 3.0 3.2 3.5 3.6 0.46 0.76 ~ 2.0
"Experimentaldata taken from Refs [20-22]. bAt 87 K; experimental data taken from Refs [20,23]. systematically higher with increasing chain length (see Table 2). The agreem e n t with theory is therefore in these systems only semiquantitative. Equation (1) breaks down for very strongly hydrogen-bonded water, mercury and other liquid metals. The numerical value of k in cryogenic fluids is low ( close to 2.0 ). In addition to the data given in Table 2, the following results are of interest: for liquid sodium at 370 K, k equals ~ 0.8 and for liquid nitrogen, oxygen and chlorine at 74, 80 and 233 K, respectively, it is equal to about 1.9-2.2. The total molar surface energy is independent of temperature for normal liquids which are not near the critical point [ 7,8]. On the other hand, both the internal pressure of liquids and their molar volume change with temperature. However, since Pi decreases with increasing temperature and Vm increases at the same time, the numerical value of the geometric factor k in Eqn (1) depends on temperature only slightly. For example, for carbon tetrachloride, k equals 3.31 at 15°C and 3.20 at 35°C (for experimental data for Pi, see Ref. [24]), for acetone 3.18 at 35°C and 3.08 at 55°C [14] and for methanol k equals at the same temperatures 2.60 and 2.59 [ 14], respectively. For water, however, the opposite is true, as in this special case the numerical value of k increases substantially with increasing temperature: k equals 0.14 at 10 oC, 0.56 at 30°C, 1.04 at 60°C and ~ 1.6 at 100°C [ 12]. This may be explained by the cleavage of hydrogen bonds in the system, due to enhanced thermal motion with increasing temperature. The results of my investigations show at the same time the limits of the liquid model on which Eqn (1) is based. SUMMARY The behaviour of Eqn (1) which relates the surface energy of a pure liquid to its internal pressure and molar volume has been investigated for different
410
classes of fluids. Equation (1) is based on a molecular model and it contains a dimensionless geometric factor h. The numerical value of this factor depends on the shape of liquid molecules and for a sphere, it theoretically equals 3.22. Equation (1) has been applied to 76 fluids, both organic and inorganic. Table 1 lists the numerical values of factor k for 28 selected liquids at 25 ° C. The agreement with theory is very good not only for simple liquids in which the intermolecular forces do not depart greatly from spherical symmetry, but also in some interacting systems (liquids 11 and 23, as well as 8, 9, 21 and 24). On the other hand, the individual deviations from the theoretical value for a sphere are somewhat greater for liquids composed of cylindrical or flat molecules (liquids 1, 3 and 2, 4, 6). In some other systems composed of asymmetric molecules, the value of k lies approximately between 3.6 and 4.2; e.g. for pyridine, dimethylsulfoxide, iodobenzene and nitrobenzene. Systems in which Eqn (1) is invalid are listed in Table 2. They include higher alkanes, monohydric alcohols, carboxylic acids, water, mercury and cryogenic fluids. The numerical value of k decreases with increasing temperature only slightly. Water is an exception to this rule, however. It may be concluded that the principle of the calculations discussed in this work is of rather limited validity. In my opinion, there is therefore little to be gained in pursuing this line of research further.
REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
I. Vavruch, Colloids Surfaces, 15 (1985) 57. L. Onsager, J. Am. Chem. Soc., 58 (1936) 1486. D.R. Rosseinsky, J. Phys. Chem., 81 {1977) 1578. I. Vavruch, J. Phys. Chem., 82 (1978) 2752. P. Becher, J. Colloid Interface Sci., 38 (1972) 291. A.W. Adamson, Physical Chemistry of Surfaces, 3rd edn, Wiley-Interscience, New York, NY, 1976, p. 47. W.A.P. Luck, Angew. Chem., 91 {1979) 408. W.A.P. Luck and W. Ditter, Tetrahedron, 27 (1971) 201. J.A. Riddick, W.B. Bunger and T.K. Sakano, Techniques of Chemistry, Vol. II, Organic Solvents, 4th edn, Wiley-Interscience, New York, NY, 1986. J.J. Jasper, J. Phys. Chem. Ref. Data, 1 (1972) 841. G. Kiiri~si and E.sz. Kov~its, J. Chem. Eng. Data, 26 (1981) 323. J.V. Leyendekkers, J. Phys. Chem., 87 (1983) 3327. J.S. Rowlinson, Chem. Soc. Rev., 7 (1978) 329. E.B. Bagley, T.P. Nelson and J.M. Scigliano, J. Paint Technol., 43 (1971) 35. D.J. Tyrer, J. Chem. Soc., 105 (1914) 2534. J.H. Hildebrand, J.M. Prausnitz and R.L. Scott, Regular and Related Solutions, Van Nostrand-Reinhold, New York, NY, 1970, p. 216. D.M. Newitt and K.E. Weale, J. Chem. Soc., {1951) 3092. R.E. Gibson and O.H. Loeffler, J. Am. Chem. Soc., 63 (1941) 898.
411 19 J.A.R. Renuncio, G.J.F. Breedveld and J.M. Prausnitz, J. Phys. Chem., 81 (1977) 324. 20 Landolt-Biirnstein, Eigenschaften der Materie in ihren Aggregatzus~nden, II. Band, 1. Teil, Springer-Verlag, Berlin, 1971. 21 M.C. Wilkinson, Chem. Rev., 72 (1972) 575. 22 E.A. Moelwyn-Hughes, J. Phys. Colloid Chem., 55 (1951) 1246. 23 R.A. Pierotti, Chem. Rev., 76 (1976) 717. 24 H. Benninga and R.L. Scott, J. Chem. Phys., 23 {1955) 1911.
405
Brief Note
On the R e l a t i o n b e t w e e n S u r f a c e E n e r g y , I n t e r n a l P r e s s u r e and Molar Volume in P u r e Fluids I. VAVRUCH Route du Centre 6, CH-1723 Marly (Switzerland)
(Received27 July 1987; accepted in final form 16 November 1987)
In a recent paper [ 1 ], I derived the following expression relating the total molar surface energy E~m of a pure liquid to its internal pressure Pi and its molar volume Vm: Pi V m / E ~ = k
(1)
in which k is a dimensionless geometric factor. Equation (1) is based on the Onsager sphere in a continuum model, with an internal surface [2,3 ] and although based on a molecular model, it contains exclusively macroscopic parameters of the system. This relation can be interpreted as follows: the work Pi Vm done against the attractive intermolecular forces in bringing one mole of liquid molecules from the bulk to the interface is proportional to the total molar surface energy of the system. The numerical value of factor k in Eqn (1) depends on the shape of liquid molecules which may exhibit considerable departure from spherical symmetry. The geometry of the liquid cavity in the surface can then be better approximated by using a polyhedron, a spheroid or a rod [ 4 ]. If Pi is expressed in J c m - 3, Vm in cm 3 m o l - 1 and E s in J m o l - 1, it can be shown that the numerical values of k are theoretically equal to 3.22 for a sphere, 3.81 for a octahedron, 4.00 for a cube and 4.81 for a tetrahedron. Since the sphere has the m i n i m u m ratio of surface to volume, its geometric factor k is always the lowest. The way of calculating factor k has been described elsewhere [ 1,5 ]. We shall later apply Eqn (1) exclusively to systems at constant temperature T and constant pressure P. The use of the total molar surface enthalpy H ~ instead of the surface energy E ~ would therefore be appropriate. However, since the product P d V is usually negligible in liquids at not too elevated pressures, H~m= E ~ is a good approximation [6]. The total molar surface energy is given by [ 7 ] E~ =/~ - T(Op~/OT)
0166-6622/88/$03.50
(2)
© 1988 Elsevier Science Publishers B.V.
406
where/Am is the molar surface free energy which can be evaluated from the equation
p~ = N~/31/¢/3/~
(3)
Here, NA is the Avogadro number a n d / l is the surface free energy (surface tension). It follows from Eqn ( 2 ) that for the calculation of E ~ not only the molar surface energy Pro, but also its temperature coefficient dp~/dT has to be determined. A direct calculation of this coefficient f r o m / ~ values at different temperatures may charge the values of E ~ with a considerable error, however, especially in liquids for which the dependence/1 (T) is known with less accuracy. Furthermore, it is very tedious. Therefore, an alternative procedure is usually preferred. It leads in general to more reliable results, since especially the values of densities and of the temperature coefficient of density are known for many liquids with a high degree of accuracy and they may be easily found in the literature. To derive the alternative equation for the determination o f E ~ , we first substitute Eqn (3) into Eqn (2) and, after rearranging, obtain the following relation [ 8 ]
E ~ = N~/3 V2/3/~- T O( N 1/3 Vc/3/~) / OT =N1/3[ I/¢/3/~- TV2/3( Op/ OT) - T~ ( OV2/3/ OT) ]
(4)
and further
0 V ~ 3/0 T = M 2/3 ( Op- 2/3/0 T) = - ( 2/3 ) MC/3p- 5/3 ( Op/OT) -- - ( 2 / 3 ) ( V c / ~ / p )
(Op/OT)
(5)
where M is the molecular weight and p is the density. Now, we take from Eqn (5) the expression for 0 V~ 3/0 T and substitute it into Eqn (4) to obtain the final relation
E~ = NIA/3V 2/3 [ (/~- TO/~/OT) + (2/3) T(#/p ) (Op/OT) ]
(6)
From Eqn (6), the surface energy E ~ can be easily calculated. Both temperature coefficients, dp/dT as well as dp/dT, are negative. The first term in the square brackets defines the total surface energy E s and Eqn (6) may therefore be simplified to
E~ =N~/3 V¢/3 [E ~+ (2/3) T(/~/p ) ( Op/OT) ]
(7)
Equation (1) forms a suitable basis for further theoretical treatment of liquid interfaces. It seemed therefore worthwhile to investigate its behaviour in different classes of fluids in more detail.
407 RESULTS AND DISCUSSION
Equation (1) has been applied to 76 fluids, both organic and inorganic. All data for the calculation of the molar volume Vm as well as the values of the density p and of its temperature coefficient dp/dT for organic systems were taken from Ref. [ 9 ]. Most experimental values of the surface tension tt and of its temperature coefficient d,u/dT were taken from Ref. [ 10 ] and the rest from Refs [9] and [11]. Data on the internal pressure Pi were taken from original publications (see Table 1), or they were evaluated from the thermodynamic relation [ 12 ] Pi = ( OZ/KT) T - P
(S)
where a is the coefficient of thermal expansion and K~T is the coefficient of isothermal compressibility. The values of these coefficients were taken mostly from Ref. [ 9 ], but in some cases it was necessary to calculate the value of a from the relation [ 9 ]
a= [(Pl/P2)-l]/(t2-tl)
(9)
where Pl and P2 are the density values at temperatures tl and t2 (in oC), and t2 > tl. In this last case, the data for densities were taken from Ref. [9 ]. A few h~T values were taken from original papers (see Table 1 ). An analysis of the probable errors in numerical data used for the calculation of the geometric factor k showed that the values are charged with an absolute error of about 0.06-0.25 units. Numerical values of k calculated from Eqn (1) for 28 selected liquids are listed in Table 1. The values of the surface energy E ~ were calculated from Eqn (6). For all these liquids, the value of factor k at 25 ° C lies in the range of 3.22 + 0.25, i.e. within the reproducibility of the results. It thus corresponds to the numerical value for a spherical cavity in a liquid surface. The agreement is excellent especially for such simple liquids, both orgnic and inorganic, in which the intermolecular forces do not depart at all from spherical symmetry, or depart from it only slightly (liquids 5, 7, 14, 25 and 28). It is also very good for some liquids in which this agreement between theory and experimental data is rather unexpected, e.g. owing to the complex structure of the liquid molecules, as in liquids 19, 20 and 22 in Table 1, because of their very dipolar character (especially liquids 11 and 23), or because of the hydrogen bonds occurring in the system, as in liquids 8 and 9, and especially in 21 and 24. The explanation for this finding lies probably in the enhanced symmetry of the molecular holes at the surface of pure liquids, as compared with the bulk [ 13 ]. On the other hand, the individual deviations from the theoretical value for a sphere are somewhat greater in liquids composed of molecules which are cylindrical ( liquids 1 and 3 ) or fiat (liquids 2, 4 and 6). In some liquids composed of complex molecules with great polarity and/or
408 TABLE 1 Numerical values of factor k in Eqn (1) for selected liquids at 25 ° C No
Liquids
Vm (cm3 mo1-1 )
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Hexane Cyclohexane 1-Hexene Cyclohexene Benzene Toluene Diethyl ether Acetone 2-Butanone Ethyl acetate Propylene carbonate Dichloromethane Chloroform Carbon tetrachloride Trichloroethylene Tetrachloroethylene Chlorobenzene 1-Bromobutane 1,2-Dibromoethane 1,1,2-Trichloro-1,2,2trifluoroethane 1,2-Ethanediol Nitromethane Acetonitrile Formamide Carbon disulfide Phosphorus trichloride Silicon tetrachloride Tin tetrachloride
131.62 108.75 125.90 101.91 89.42 106.87 104.75 74.05 90.17 98.49 85.42 64.50 80.68 97.09 90.02 102.73 102.24 108.00 86.62 119.84
21 22 23 24 25 26 27 28
55.92 53.96 52.86 39.88 60.64 87.68g 114.6g 116.8g
Pi (J cm -~)
E~ (kJ mo1-1 )
k
243.0 319.0 244.4 346.0 374.3 348.9 250.2 329.7a 298.6 345.4b 424.4 374.9c 378.6 338.9¢ 406.9d 402.2d 382.8 325.1 432.5 258.8
9.51 10.32 9.22 10.60 10.22 10.97 8.32 7.53 8.80 9.80 10.96 7.89 9.24 10.08 10.95 13.23 11.46 10.34 11.80 9.78
3.36 3.36 3.34 3.33 3.27 3.40 3.15 3.24 3.06 3.47 3.31 3.06 3.31 3.26 3.35 3.12 3.42 3.40 3.18 3.17
501.6e 606.0 409.0 581.3 453.0 f ~ 348g ~257 g ~ 311g
8.46 9.42 6.95 7.25 8.79 9.86 8.57 11.06
3.32 3.47 3.11 3.20 3.13 3.1 3.4 3.3
aRef. [141. bRefs [9,15]. CRef. [16]. dRefs [9,17]. eRefs [9,18]. rRef. [19]. gRef. [ 20 ].
e l e c t r i c a l a s y m m e t r y , t h e d e v i a t i o n s o f f a c t o r k i n E q n (1) f r o m t h e t h e o r e t i c a l v a l u e for a s p h e r e b e c o m e m o r e e v i d e n t . F o r e x a m p l e , for p y r i d i n e k e q u a l s 3.6 a t 25 ° C, for e t h y l e n e d i a m i n e a n d d i m e t h y l s u l f o x i d e 3.9, for i o d o b e n z e n e 4.0, a n d for n i t r o b e n z e n e a n d a n i l i n e 4.2. F o r h i g h e r a l k a n e s a n d p r o b a b l y a l s o h i g h e r a l k e n e s , for m o n o h y d r i c a l c o h o l s a n d for c a r b o x y l i c a c i d s , k b e c o m e s
409 TABLE 2 Extreme values of factor k in selected fluids at 25 °C
Fluids
k
Fluids
k
Octane Decane Dodecane Hexadecane Methanol Ethanol 1-Butanol 1-Hexanol
3.9 4.2 4.5 4.9 2.7 3.1 3.5 4.0
1-Octanol Formic acid Acetic acid Propionic acid Isobutyric acid Water Mercury~ Argonb
4.2 3.0 3.2 3.5 3.6 0.46 0.76 ~ 2.0
"Experimentaldata taken from Refs [20-22]. bAt 87 K; experimental data taken from Refs [20,23]. systematically higher with increasing chain length (see Table 2). The agreem e n t with theory is therefore in these systems only semiquantitative. Equation (1) breaks down for very strongly hydrogen-bonded water, mercury and other liquid metals. The numerical value of k in cryogenic fluids is low ( close to 2.0 ). In addition to the data given in Table 2, the following results are of interest: for liquid sodium at 370 K, k equals ~ 0.8 and for liquid nitrogen, oxygen and chlorine at 74, 80 and 233 K, respectively, it is equal to about 1.9-2.2. The total molar surface energy is independent of temperature for normal liquids which are not near the critical point [ 7,8]. On the other hand, both the internal pressure of liquids and their molar volume change with temperature. However, since Pi decreases with increasing temperature and Vm increases at the same time, the numerical value of the geometric factor k in Eqn (1) depends on temperature only slightly. For example, for carbon tetrachloride, k equals 3.31 at 15°C and 3.20 at 35°C (for experimental data for Pi, see Ref. [24]), for acetone 3.18 at 35°C and 3.08 at 55°C [14] and for methanol k equals at the same temperatures 2.60 and 2.59 [ 14], respectively. For water, however, the opposite is true, as in this special case the numerical value of k increases substantially with increasing temperature: k equals 0.14 at 10 oC, 0.56 at 30°C, 1.04 at 60°C and ~ 1.6 at 100°C [ 12]. This may be explained by the cleavage of hydrogen bonds in the system, due to enhanced thermal motion with increasing temperature. The results of my investigations show at the same time the limits of the liquid model on which Eqn (1) is based. SUMMARY The behaviour of Eqn (1) which relates the surface energy of a pure liquid to its internal pressure and molar volume has been investigated for different
410
classes of fluids. Equation (1) is based on a molecular model and it contains a dimensionless geometric factor h. The numerical value of this factor depends on the shape of liquid molecules and for a sphere, it theoretically equals 3.22. Equation (1) has been applied to 76 fluids, both organic and inorganic. Table 1 lists the numerical values of factor k for 28 selected liquids at 25 ° C. The agreement with theory is very good not only for simple liquids in which the intermolecular forces do not depart greatly from spherical symmetry, but also in some interacting systems (liquids 11 and 23, as well as 8, 9, 21 and 24). On the other hand, the individual deviations from the theoretical value for a sphere are somewhat greater for liquids composed of cylindrical or flat molecules (liquids 1, 3 and 2, 4, 6). In some other systems composed of asymmetric molecules, the value of k lies approximately between 3.6 and 4.2; e.g. for pyridine, dimethylsulfoxide, iodobenzene and nitrobenzene. Systems in which Eqn (1) is invalid are listed in Table 2. They include higher alkanes, monohydric alcohols, carboxylic acids, water, mercury and cryogenic fluids. The numerical value of k decreases with increasing temperature only slightly. Water is an exception to this rule, however. It may be concluded that the principle of the calculations discussed in this work is of rather limited validity. In my opinion, there is therefore little to be gained in pursuing this line of research further.
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