An anisotropic surface model for the Parachor

An Anisotropic Surface Model for the Parachor LISE JEANNE PAQUETTE AND DOUGLAS E. GOLDSACK Department of Chemistry, Laurentian University, Sudbury, Ontario P3E 2C6, Canada Received February 17, 1982; accepted April 30, 1982 An anisotropic surface model for the surface layer of liquids has been utilized to yield an expression for the Parachor [P] defined as ~r~/4Vbwhere ~r is the surface tension of the liquid and Vb is the molar volume of the bulk liquid phase. The Parachor is shown to be related to the molecular properties of the liquid by:

[p] = [Z1/2NaI + 2.42× XYVe 10a#211/*V05/6(1_ T/Tc)-l/~ where N is Avogadro's number, a is the molecular polarizability, I is the first ionization potential, Z is a geometric factor, Vc is the molar critical volume, Vo is the volume of the liquid at 0*K, and Tc is the critical temperature of the liquid. Agreement between calculated and observed values of the Parachor is obtained for a large number of polar and nonpolar fluids to within an average error of 5%. INTRODUCTION

volumes when compounds were in corresponding states with respect to the intermolecular forces in the surface. This argument is fallacious since corresponding states occur at corresponding temperatures and pressures, and not when the surface tension for each compound is unity. However, since unit surface tension for most compounds occurs near their critical point, then to a first approximation, the parachor, calculated under those conditions, would be close to the value of the molar critical volume. Since the molar critical volume is roughly additive, the parachor would be then roughly additive. It was also pointed out (4) that since the surface tension appears in Eq. [1] to the exponent 1/4, its influence is dampened and hence it does not contribute to the additive character of the parachor as strongly as does the molar volume. Numerous correlations have been proposed to relate the parachor to other properties of the liquid such as the molar refraction (5, 6), the critical volume, Vc (2, 7), isothermal compressibilities, × (8), as well as the

The parachor is an additive and constitutive property of organic and inorganic compounds which has been of great utility in structural studies of closely related compounds since it is affected by unsaturation, branching, rings, and functional groups (1). It has also been shown (1, 2) that the parachor is independent of temperature although a slight dependence is observed over a large temperature range. The parachor relates the surface tension and the density of a liquid (3) and was expressed by Sugden (2) as: [P]

=

(TI/4V B

[1]

where [P] is the parachor, a is the surface tension in ergs/cm z, and VB is the molar volume of the bulk liquid in cm3/mole. Several attempts have been made to explain the additive property of the parachor. Sugden suggested that the determination of the parachor at a common surface tension value, e.g., unity, was a comparison of molar 154

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Journal of Colloid and Interface Science, Vol. 92, No. 1, March 1983

ANISOTROPIC SURFACE MODEL FOR THE PARACHOR

critical pressure, Pc and temperature, Tc (9). By making use of van der Waals' and Katayama's relations, Ferguson and Kennedy (I0) found [2] [P] - ATI/4V ¢ ¢7/8 where A is a constant particular to a compound. Similarly, Lennard-Jones and Comer (11) derived an expression for the parachor through the application of the Eotvos and Guggenheim relations /4 [P] = C V S~/ 6 T t ~

[P]

= KII4TI/4V~/6

the enthalpy, A//sv, required to move I mole of molecules from the surface phase into the vapor phase. The former quantity may be obtained from the temperature dependence of the molar surface free energy, AGs. In order to estimate the molar surface free energy, it may be assumed that the surface layer consists of molecules of a particular geometry. It can then be shown that (13) the molar surface free energy in units of Joules/mole is given by: A ~ S -~ X t r Z 213

[3]

where C is an empirical constant characteristic of the compound. McGowan (12) then developed the Lennard-Jones and Comer relation further and showed that: [4]

where K is a constant typical of a molecule and Vo is the volume of the liquid at 0°K. It should be noted that although the above expressions show good agreement with observed parachor values, the relations are purely empirical and yield no simple molecular explanation for the behavior of the parachor. Furthermore, the constants in the above equations are values derived from the experimental Eotvos and van der Waals relations which merely fit the observed data. The purpose of this paper is to describe an anisotropic surface model for the surface layer of a liquid which leads to a molecular theory for the parachor, gives an explanation for the relative insensitivity of this parameter to temperature, and shows its relation to the molar volume of the liquid. THEORY

In order to obtain a simple model for the gas-liquid interface, it will be assumed that the surface consists of a monolayer and can be treated as a separate phase. The energetics of vaporization can then be related to two distinct processes viz. the enthalpy, A/-/s, required to move 1 mole of molecules from the bulk liquid phase into the surface layer and

155

[5]

where a is the surface tension in ergs/cm2, Vs is the molar volume of the liquid in the surface layer, and X is a geometric factor of 10.21 for spherical geometry. From this expression, assuming a and Vs are functions of temperature, the molar surface enthalpy is given as: AISIs =

Xo.V2/3[1 TOaa dT-2/3asT]

[6]

where as is the coefficient of thermal expansion of the liquid in the surface and T is the absolute temperature. It has already been shown (13) that the classical model for the London dispersion forces between nonpolar molecules (14) can be modified to obtain an expression for A//sv. This new model, called the anisotropic surface model, assumes that one of the normal frequencies of vibration in the classical London model (15), namely the frequency of vibration in the Z direction, taken as the direction of movement of the molecule from the surface to the vapor phase, can be allowed to become zero. A/-/sv is then found to be given by:

Z NaI 4 YVs

ABsv = -- - -

[7]

where Z is the coordination number of the molecule in the surface (approximately equal to 4 for spherical geometry), a is the coefficient of polarizability, I is the first ionization potential, in Joules per mole, Vs is the molar surface volume, and Y is a geometric factor Journal of Colloidand InterfaceScience, Vol. 92, No. 1, March 1983

156

PAQUETTE AND GOLDSACK

which is 0r/6) for a spherical model of the space occupied by the molecule in the surface. Note that the dispersion enthalpy varies as the inverse of the volume for this model whereas for the ordinary isotropic London model, the dispersion enthalpy varies as the inverse square of the volume. The molar enthalpy of vaporization, A/lvap, is therefore given by

Z NaI

-Vs-

xgzO

/

Tc rc-

I

"

I13j

By rearranging Eq. [13] and incorporating Eq. [10], an expression for Vs can be obtained: ~---~Tc/

V~/3.

[ 14]

Applying the conditions at the critical point T = Tc and Vs = VB = Vc, the Eotvos constant may be determined as follows in terms It has been observed that if bulk molar propof molecular constants: erties, such as the bulk molar volume, VB, and the bulk liquid coefficient of thermal Z NaI expansion, aB, are used in the expression for K - 4 XYVcTc" [15] A/ts instead of surface molar properties, the value of A//s is approximately one-half of By replacing this expression into Eq. [14], a AH~p. This observation is known as Ste- simple relationship for the molar volume of phan's Law (16). This law implies that the a liquid in the surface layer is found. energy required to move a molecule from the Vs = V~SV~/5. [161 bulk liquid to the surface phase is equivalent to that when moving from the surface layer Equation [ 16] also leads to a simple expresto the vapor phase. sion for the coefficient of thermal expansion Applying Stephan's Law to Eqs. [6] and in the surface phase: [7], the following relation is found: as = 2/5aB. [17] Z NaI [TOur TI -4 ~ss = XaV2/3 1 gOT 2/3as • [91 Equation [15] was used to calculate the Eotvos constant for a number of compounds A simple relationship may be derived for 1/ showing excellent agreement with the oba(Oa/OT), by making use of the empirical served values and giving confidence in the Eotvos equation: anisotropic surface model as a predictive tool aV z/3 = K(Tc - T) [101 (13) for surface properties. Dipolar effects may be taken into account, where K is a constant characteristic of a comassuming no orientation takes place in the pound and is known as the Eotvos constant. surface layer of the liquid, by adding to Hence: A/-/sv, a dipolar interaction term derived 1 0g 1 from the energy of interaction between two ~ 0--~= -2/3a. Tc- T" [111 like molecules (13): m / l v a p ----- A / ~ S + A / ~ S V .

[8]

Equation [6] becomes with this substitution:

2

A fls = X~V~/3 × [2/3T(aB- Ots) + (TcT--C_T)] . [12] A further simplification is possible through the approximation 2/3T(aB - as) ~ [Tc/Tc - T)]. Consequently, Eq. [9] becomes: Journal of Colloid and Interface Science,

A/-Isv = Z2.42 × 104#2

Vol. 92, No. 1, March 1983

[18]

YVs

where t~ is the dipole moment in Debye units. Hence, by including dipolar and dispersion contributions in the energy of interaction between molecules in the surface phase and again applying Stephan's Law, the following relation results:

ANISOTROPIC SURFACE MODEL FOR THE PARACHOR

[Z 1/2NaI+ 2.42 X 104#2]

DISCUSSION

YVs

It has often been noted that the parachor = × f l Y 2/3

Tc

Tc- T"

[19] shows additive, constitutive, and almost tem-

Rearranging this expression leads to an explicit equation for the surface tension of polar and nonpolar liquids:

[Z1/2NaI+ =

104/.t21

2.42X

XYV2/3Vc × (l -

T/Tc).

[2O]

It has been shown (17) that for many compounds, the density of the liquid, o~, may be related to the reduced temperature T/To to a first approximation by the expression. oB = po(1 -

T/Tc) 1/3

[21]

where Oo is the density of the liquid at 0°K and may be related to a molar volume Vo. Consider Eq. [20] where the substitution of Eq. [21 ] leads to:

o=

[2 1/2Na1+2"42×104#2] XYVcM2/3 T[Tc) 11/9.

× Og/3(1 -

[22]

The parachor may then be defined as follows:

XYVc M × 04/6 M ( 1 OB

T/Tc) 11/36

[23]

which may be simplified further: [P] = o-1/4VB

=[ZI/2NaI+

157

2.42X

104/.t2]1/4

XYVc × Vo5/6(1 - T / T c )

-1/36.

[24]

Hence the parachor may be calculated in terms of molecular properties in an expression derived from the anisotropic surface model.

perature-independent properties. The anisotropic surface model has led to an expression for the parachor which provides a molecular explanation for these properties. Equation [24] contains three components which account for the behavior of the parachor. First, it can be observed from Table I, that the combination of molecular constants [Z/2(1/2NaI + 2.42 × 104#2/XYVc) 1/4 is practically constant for a wide range of compounds (5.70 +_0.19). Second, the parachor is proportional to a molar volume Vg/6 where Vo represents the molar volume of the liquid at 0°K. This molar volume VB displays both additive and constitutive characteristics. Finally, the slight temperature dependence of the parachor is exhibited in the term (1 - T/Tc) -~/36 since all other terms are constants, characteristic of the molecule and independent of temperature. Table I displays excellent agreement between the observed and the calculated values of the parachor, at the boiling point, using Eq. [24] and Fig. 1 illustrates this agreement for the spherical geometry approximation. The slope has a value of 1.132 with a correlation coefficient of 0.985. The average error for the calculation of the parachor is 5%. It is extremely significant that this theory accounts for the parachor values of not only nonpolar materials but also of dipolar and hydrogen bonding compounds such as water and the alcohol. The excellent agreement implies that at the boiling point, these compounds show no orientation effect. Table II compares the observed and calculated parachor values with respect to temperature which has only a minor effect. It should be noted that the term 2/3(aa - as)T, which is much smaller than Tc/Tc - T, could be included in the derivation, but the final expression would be far more complex with no real improvement in the understanding of the parachor. Journal of Colloidand InterfaceScience, Vol.92, No. 1, March 1983

158

PAQUETTE AND GOLDSACK TABLE I Comparison of Observed and Calculated Parachor Values, [P], at the Boiling Point _z0/2 tc~t + 5.78 × 103~2!]'/'

vo

Compound

T~, °K

#v¢~.~

(cm:~)

[P],~

[P],~,~

1. CC14 2. C2C14 3. CrH6 4. C6HsCH3 5. C3Hs 6. CsHto 7. C6HI2 8. C6H14 9. CsHI2 10. C12 I1. CS2 12. CH2C12 13. C6HsF 14. CrHsCI i 5. C6HsBr 16. C6H5I 17. C3H60 18. SO2 19. C4HloO 20. CO 21. CH3CN 22. CH3CH2CN 23. (CH3)3N 24. CH3CHO 25. CrHTN 26. CrH60 27. CH3NH2 28. CH3CH2NH2 29. HCN 30. C7HsN 31. CsHsN 32. CH3OH 33. C2HsOH 34. C3HvOH 35. HCOOH 36. CH3COOH 37. H20 38. C6H5CH2OH

350 394 353 384 231 326 354 342 309 239 320 313 358 405 429 462 329 263 308 82 355 371 276 294 457 455 267 290 299 464 389 338 352 370 374 381 373 478

0 0 0 0.4 0 0 0.3 0 0 0 0 1.8 1.4 1.6 1.5 1.4 2.9 1.6 1.3 0.1 3.5 3.7 0.6 2.5 1.6 1.6 1.3 1.3 3.0 3.2 2.3 1.7 1.7 1.7 1.5 1.3 1.8 1.7

76.44 81.59 69.02 83.72 54.66 70.25 83.20 96.76 82.72 35.93 46.62 48.00 72.51 81.52 85.23 92.61 57.01 33.10 74.23 25.36 40.66 55.08 64.47 39.84 75.05 71.64 31.06 46.91 27.76 84.32 63.76 29.91 42.72 55.29 27.49 44.83 13.52 81.29

225.2 251.3 206.5 243.7 150.1 202.5 240.9 268.8 230.4 103.8 143.6 147.8 213.9 243.3 253.7 276.7 164.2 101.4 211.9 61.2 131.2 175.1 180.6 119.7 238.7 266.4 98.7 139.5 81.1 255.0 197.6 89.7 127.9 167.6 89.9 131.8 52.3 217.1

226.5 234.9 201.9 233.3 163.9 204.5 232.7 260.2 203.5 120.9 156.6 151.1 208.6 232.3 242.4 261.5 168.7 111.4 205.9 82.6 130.2 168.0 172.2 125.3 214.4 214.8 95.7 138.9 89.7 242.9 188.5 97.3 132.7 164.1 91.2 136.3 52.0 231.3

The empirical expression between the parachor, Vo and Tc as observed by McGowan (12) and given Eq. [4] in the introduction may readily be explained by the anisotropic surface model for the parachor. The term [Z/2(1/2Nal + 2.42 × I04t~2/XYVc)] can be replaced by KTc from Eq. [ 15]. Therefore the relationship for the parachor may be rewritten as: Journal of Colloid and Interface Science, Vol. 92, No. 1, March 1983

[P] =

2

xYvc

vo/Vc

5.93 5.83 5.77 5.66 5.68 5.74 5.68 5.59 5.65 5.97 6.22 5.85 5.71 5.77 5.80 5.84 5.64 5.87 5.51 5.43 5.77 5.78 5.19 5.65 5.69 5.93 5.32 5.47 5.46 5.86 5.74 5.55 5.62 5.61 5.60 5.57 5.80 5.72

0.28 0.28 0.27 0.27 0.27 0.27 0.27 0.26 0.27 0.29 0.27 0.25 0.26 0.27 0.26 0.26 0.27 0.27 0.27 0.27 0.24 0.24 0.25 0.26 0.28 0.31 0.22 0.26 0.20 0.26 0.25 0.25 0.26 0.25 0.25 0.26 0.24 0.24

(KTc)I/4V~/6(I -

T / T c ) -1/36.

[25]

It may be noted that the term (1 T/Tc) -1/36 approaches the value of 1, hence Eq. [25] is in exact agreement with the empirical equation derived by McGowan. Table I also indicates that the ratio Vo/Vc is approximately constant for a wide variety of polar and nonpolar compounds (0.26 - -

159

ANISOTROPIC SURFACE MODEL FOR THE PARACHOR 3.0

®~ 2.0

N 0 x o

.1,0

@

II( 0

I 1.0

I 2.0

I 3.0

l'P~calc x 10 -2

FIG. 1. Observed Parachor values, [P]o~, versus calculated Parachor values, [P]~c, using spherical geometry for a variety of liquids: the numbers correspond to the compounds in Table I.

+ 0.02). Therefore Eq. (25) may be rewritten as:

[p] =

/ lz \5/6

[PI~-(KTc)I/4[~}

V5/6-c

[261

c:

~ - r + 2.42 × 104~

XYVc X V05/6(1 - T/Tc) -1/36. [29]

which in accordance with the empirical Eq. [3] given by Lennard-Jones (11). Further relationships (6) of the parachor with other properties may be predicted from this model. For example, the molar refraction, R, is given by: n2 - 1 R = VB nZ + 2 = 4/3~rNa

[27]

where n is the refractive index. It follows that 3R Na = 47r which may be substituted into Eq. [24]:

[28]

The relation between the parachor and molecular volumes has often been noted (1 6, 18). In an attempt to show that the parachor is directly proportional to the molecular volume, the anisotropic surface model shows that, as other empirical relations have noted, the parachor is in fact proportional to the molecular volume to the 5/6 power (V]/6) and not to the molecular volume (Vo). CONCLUSIONS

The anisotropic surface model for the parachor accounts for the additive, constitutive, Journal of Colloid and Interface Science. Vol. 92, No. 1, March 1983

160

PAQUETTE AND GOLDSACK TABLE II

Temperature Dependence of the Parachor, Observed and Calculated Values T OK

[P]~

[Pl*,~

T OK

CCLt 250 273 298 323 348 373 398 423 448

218.8 219.6 220.2 220.3 219.9 218.5 215.9 211.2 203.0

242.9 243.7 244.5 245.1 245.4 245.4 244.9 243.8 241.9 238.6 233.6

[P]oac

C6H6 224.4 224.8 225.4 226.1 226.8 227.6 228.5 229.6 230.9

280 298 323 348 373 398 423 448 473

C6HsC1 228 250 273 298 323 348 373 398 423 448 473

[P]o~

205.8 206.3 206.3 206.2 205.1 203.1 199.3 192.9 181.1

199.9 200.3 200.8 201.5 202.2 202.9 203.9 205.0 206.4

C3H60 227.2 228.4 229.6 230.9 232.2 233.5 234.7 235.9 237.0 238.0 238.9

180 198 223 248 273 298 323 348 373 398 423 448

156.5 157.6 159.1 160.5 161.8 163.9 163.9 164.7 164.9 163.8 162.2 157.2

165.9 166.2 166.5 166.9 167.4 167.9 168.6 169.2 170.0 171.0 172.2 173.9

and temperature-independent character of the parachor for a wide variety of polar and nonpolar compounds in terms of molecular constants. This model also explains the parachor in terms of dispersion and dipolar intermolecular forces as they occur in the surface of a liquid. Previous attempts at providing a rationale for the Parachor have been, of necessity, of an empirical nature (l), focusing on atomic, bonding, and branching effects to explain minor deviations from additive behaviors of the parachor. The anisotropic surface model also offers a sound explanation for the empirical rela-

Journalof Colloidand InterfaceScience.Vol.92, No. I, March 1983

tions between the parachor and other molecular properties such as the critical volume, critical temperature, and molar refraction. The explicit assumptions made in deriving the model are therefore justified and valid. The most important conclusion that this analysis has yielded is that there are no significant orientation effects occurring in the surface layer of most liquids. These ideas should also be extended to other types of liquids such as liquid metals and liquid salts. ACKNOWLEDGMENTS We would like to thank the National Science and Engineering Research Council for funds in aid of this project. REFERENCES 1. Quayle, O. R., Chem. Rev. 53, 439 (1953). 2. Sugden, S., "The Parachor and Valency." Routledge, London, 1930. 3. Macleod, D. B., Trans. FaradaySoc. 19, 38 (1923). 4. Exner, O., Nature (London) 196, 890 (1962). 5. Lagemann, R. I., and Dunbar, W. S., J. Phys. Chem. 49, 428 (1945). 6. Putnam, R. C., Canad. J. Chem. 44, 1343 (1966). 7. Costello, J. M., and Bowden, S. T., Chem. Ind. 1041 (1956). 8. McGowan, J. C., Recuei176, 155 (1957). 9. Lautir, R., Bull. Soc. Chim. 589 (1946). 10. Ferguson, A., and Kennedy, S. J., Trans. Faraday Soc. 32, 1474 (1936). 11. Lennard-Jones, J. E., and Corner, J., Trans. Faraday Soc. 36, 1156 (1940). 12. McGowan J. C., Recuei176, 147 (1957). 13. Goldsack, D. E., and Paquette, L. J., M.Sc. thesis, Laurentian University, 1982. 14. Hirshfelder, J. O., Curtiss, C. F., and Bird, R. B., "Molecular Theory of Gases and Liquids." Wiley, New York, 1954. 15. Ibid, p. 856. 16. Partington, J. R., "An Advanced Treatise on Physical Chemistry," Vol. 2. Longmans, London, 1951. 17. Rowlinson, J. S., "Liquids and Liquid Mixtures." Butterworths, London, 1959. 18. Edwards, J. T., Chem. Ind. 774 (1956).