Ind. Eng. Chern. Fundarn. 1980, 79, 309-311 Kletz, T. A., Chem. Eng. frog., 70(4), 80-84 (1974). Lawley, H. G., Chem. Eng. frog., 70(4), 45-56 (1974). McHugh, B., Chalmers Tekniska Hhskola, Sweden, Report R 73-48 (1973). Menzles. R. M., Strona, R.. The Chem. Ena.. 151-160 (1979). Nlelsen, D. S.. Ris0-M-1374 (1971). Nlelsen, D.,Platz, O., Kongs0, H. E., Ris0-M-1903 (1977). Powers, 0. J., Tompklns, F. C. Jr., AIChE J . , 20(2), 376-387 (1974).
309
Schneeweiss, W., “Zwerliissigkeitstheorie”,Springer-Vetlag, Berlln, Mldelberg, New York, 1973. Wolfe, W. A,, AECL-6172 (1978).
Received for review January 15, 1980 Accepted April 24, 1980
Gradient Theory of Surface Tension of Water M. I. Guerrero and H. T. Davis*
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Subdireccion de Investigacion Basica de Procesos, Instituto Mexican0 del Petroleo, Mexico 14, D.F., Mexico
The gradient theory of inhomogeneous fluids has been applied to the prediction of the surface tension of water. Over temperatures ranging from 300 K to the critical point, 647.3K, the difference between theory and experiment is no more than 20%. Introduction of a single scale factor for the influence parameter of inhomogeneous fluid reduces the difference between theory and experiment to 5% or less. The inputs of the theory include a molecular theoretical estimate of the influence parameter and the equation of state of homogeneous fluid modeled by Fuller.
1. Introduction
The gradient theory of inhomogeneous fluid has been applied successfully by Carey et al. (1978, 1980) to the prediction of surface tensions of normal alkanes and their mixtures. The equation of state requirements of the theory were met by the Peng-Robinson equation (1976) (a modification of van der Waals’ original equation) which is accurate for nonpolar fluids. The purpose of the present paper is to use gradient theory to predict the surface tension of water. Here we use a modification of the van der Waals equation introduced by Fuller (1976) instead of using the Peng-Robinson, which is not recommended for water. Fuller’s equation has the advantage over the Peng-Robinson (and similar, earlier equations of Redlich and Kwong (1949) and Soave (1972)) that it predicts accurately the vapor pressures and liquid and vapor coexistence densities of water. The gradient theory yields a single parameter, the influence parameter (Bongiono et al., 1976),which along with the homogeneous fluid equation of state characterizes inhomogeneous fluid structures such as interfaces. This parameter is estimated from a molecular model for the calculations presented herein. 2. Outline of Theory The surface tension of an interface between a liquid phase a t density nl and a vapor phase a t density ng is expressed in the gradient theory by
Y=
filrd-
dn
(2.1)
where WB = w ( n J = w(n,) is a constant and the thermodynamic potential w(n)is determined from the homogeneous fluid properties by =fob) -np (2.2) Here fo(n)is the Helmholtz free energy of homogeneous fluid computed at density n and p is the chemical potential
of the fluid. The influence parameter c carries the information on the molecular structure of the interface and is given rigorously (Bongiono et al., 1976; Yang et al., 1976) by c = krJs2Co(s;n) 6 d3s
or, to a good approximation (McCoy et al., 1979) by the mean field theoretical formula 1 c - - 1 s ’ U ( S ) go(s;n) d3s (2.4) 6 where u(s) is the molecular pair potential, go(s;n)the pair correlation function of homogeneous fluid, and Co(s;n)the direct correlation function of homogeneous fluid. go and Cocarry equivalent structural information in that they are related by the transformation g,(s;n) = 1 + Co(s;n) n l C o ( l s- s‘ln)[go(s’;n) - 11 d3s’ (2.5)
+
The liquid and vapor densities along the coexistence vapor pressure curve are determined by the usual conditions of thermodynamic equilibrium po(n1) = po(n,);
POhl)
= Po(n,)
(2.6)
where p o ( n ) and Po(n)are the chemical potential and the pressure of homogeneous fluid. Since the chemical potential p is constant it can be evaluated in bulk phase, so that p = po(n,) = po(nl). Also, from the definition of w ( n ) it follows that WB = -Po(n;,) = -Po(nJ. Thus, solution of eq 2.6 provides the quantities ng,nl, p , and OB needed to compute tension at a given temperature. The formula for f o ( n )can be determined from the equation of state. The equation of state of homogeneous fluid proposed by Fuller is
nkT n2a Po(n) = -1-nb 1 + n$b ~
* Department of Chemical Engineering and Materials Science University of Minnesota, Minneapolis, Minn. 55455
(2.3)
(2.7)
where a is the van der Waals energy parameter and b is a molecular excluded volume. $ is a temperature-de-
0196-4313/80/1019-0309$01.00/0 0 1980 American Chemical Society
310
Ind. Eng. Chem. Fundam., Vol. 19, No. 3, 1980
pendent parameter such that in the limit $ = 0 eq 2.7 becomes the van der Waals equation and in the limit $ = 1the Redlich-Kwong-Soave equation; Fuller allows a and $ to be universal functions of the fluid critical parameters and acentric factor and of the reduced temperature TIT,. However, Fuller adds a new fluid parameter in the equations for a and 4 and relates this new parameter to the fluid parachor. The formulas of Fuller used for the computations presented herein are given in the Appendix. By integration of the thermodynamic relationship dFo = -Po d V of homogeneous fluid (Fo= f o V )we obtain
0.65
0.60
K 0.55
0.50
I
na
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- In (1 + n$b) (2.8)
*b where p+( r ) is the contribution of the ideal gas chemical potential arising from the kinetic and internal energies of the particles. In our analysis this quantity can be set to zero without loss of generality since it cancels out all expressions used for computation. The chemical potential of homogeneous fluid can be computed from fo(n) as follows
= ~‘(2‘) - RT In
(:
-b)
-
For a 6-12 Lennard-Jones fluid, it has been demonstrated (McCoy et al., 1979) that the parameters a and c as defined by eq 3.1 and 2.4 are almost independent of density. Thus, the dimensionless group c/ab2i3 is also expected to be almost independent of density. Carey et al. (1978) in fact found that the assumption that the group c/ab2i3 is a universal constant gives pretty good estimates of the surface tensions of the homologous series of normal alkanes. On the basis of these findings, then, we shall estimate the ratio c/ab2i3from the low density limit (where go = e - u ( s ) / k r )
where the excluded volume has been expressed as b = 2 / 3 ~ u 3u, being the molecular diameter. For the pair potential of water we take the “angle averaged” Stockmeyer potential (Hirschfelder et al., 1954) ;)12
I
I
I
600
700
800
-
($1
0 2
- 3kTs6 -
(3.3)
For water, the Lennard-Jones parameters are e l k = 209.1 K, u = 3.329 A, and the dipole moment is D = 1.85 Debye
I
T(K)
Figure 1. K vs. T for water parameters.
- experimental
80
---
c= K O ~ ‘ ‘ ~
I
nu (2.9) 1 +nGb
3. Results To complete the input needed for computing tension we need to estimate the influence parameter c. To get this parameter we shall exploit the relationship between c and the mean field theoretical formula for the van der Waals energy parameter a, namely 1 a = - -2 l u ( s ) go(s;n)d3s (3.1)
(
I
500
a In (1 + n$b) + &b nbRT 1-nb
4 s ) = 4r[
I
400
400
I
500
’..
600
1
700
T(K)
Figure 2. Comparison of predicted and measured surface tension of water.
(Reed and Gubbins, 1973). K is plotted in Figure 1 as a function of temperature. It increases slowly with temperature, from a value of 0.490 at 373 K to 0.634 a t 646.5 K very near the critical point of water. To predict the surface tension of water we now set c = Kab2I3 (3.4) where a and b are determined from Fuller’s equation of state and K is estimated from eq 3.2. Predicted tensions vs. temperature are compared with experiment in Figure 2. The error between predicted and measured tension is largest at the lowest temperature, being about 20% at the boiling point of water. Considering we are dealing with water, one of the hardest substances to model theoretically or empirically, and that the parameter c has not been adjusted but has been estimated a priori from molecular theoretical and equation of state input, we believe the agreement between theory and experiment is quite satisfying. If, to adjust roughly for density and other model dependent errors inherent in eq 3.4, an empirical constant is introduced into eq 3.4, the agreement error between experiment and gradient theory is substantially reduced. The recommended relation between c and ab2i3,found by averaging about a dozen points of data over the temperature range included in Figure 2, is
c = 0.888Kab2I3 (3.5) with eq 3.5 computed and measured tensions shown in Figure 2 agree to within about 5%. We believe the results shown in Figure 2 demonstrate that with Fuller’s equation of state gradient theory gives pretty good estimates of surface tension even for hydrogen bonding fluids. The simplest empirical improvement, eq
Ind. Eng. Chem. Fundam., Vol. 19, No.
3.5, leads to quite good results for water. We speculate that large classes of fluids could be accurately treated by gradient theory if a universal function were determined for c/ab213 similarly to such determinations of a , b, and 1c/ in the various modifications of the van der Waals equations suggested by Redlich and Kwong, Soave, Peng, and Robinson, Fuller and others.
Acknowledgment We are indebted to M. A. Leiva for useful discussions on Fuller’s equation and for providing us with a copy of his computer program. One of us (H.T.D.) is also grateful to the U. S. Department of Energy for financial support of this research.
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Appendix In this appendix we summarize the formulas of Fuller’s equation of state
3, 1980 311
Nomenclature a = molecular iteraction parameter b = excluded volume parameter c = influence parameter Co = direct correlation function D = dipole moment fo = bulk phase Helmholtz free energy density Fo = bulk phase Helmholtz free energy go = radial correlation function K = dimensionless parameter = c / a b 2 i 3 n = density P = bulk pressure P = parachor q = parameter introduced in Fuller’s equation R = universal gas constant T = absolute temperature v = specific volume 2 = compressibility factor = PV/RT Greek Symbols cy = Fuller’s temperature dependence on the “a” parameter (3 = parameter introduced in Fuller’s equation y = surface tension c = intermolecular potential well depth
= chemical potential = ideal gas chemical potential $ = coefficient on “b” in Fuller’s equation p p’ D
= molecular diameter
w = thermodynamic potential w, = Pitzer acentric factor
0 = coefficients in the “a” and “b” parameters
Subscripts
114
[0.480
Pc
+ 1 . 5 7 4 ~ ~0.176~,2] -
= ncb
PO
- = 7.788 - 36.83162, + 50.70712,’ P C
0 = 10.9356
+ 0.0285P
For water Tc = 647.3 K, Pc = 220.5 bar, nc = 0.0179 mol/cm3, 2, = 0.229, and wa = 0.344.
a = refers to the “a” parameter b = refers to the “b” parameter c = refers to the critical point g = refers to gas phase 1 = refers to liquid phase r = reduced respect to critical value Literature Cited Bongiorno, V., Scriven, L. E., Davis, H. T., J . Colloid Interface Sci., 57, 462 (1976). Carey, B. S., Scriven, L. E., Davis, H. T., AIChE J., 24, 1076 (1978). Carey, B. S., Scriven, L. E., Davis, H. T., AIChE J., to appear 1960. Fuller, G. G., Ind. Eng. Chem. Fundam., 15, 254 (1976). Hirschfelder, J. O., Curtiss, C. F., Bird, R. B., “Molecular Theory of Gases and Liquids”, Wiley, New York, 1954, p 26. McCoy, B. F., Scriven, L. E., Davis, H. T., J . Chem. Phys.. submitted for publication. Peng, D. Y.. Robinson, D. B., Ind. Eng. Chem. Fundam., 15, 59 (1976). Redlich, O., Kwong, J. N. S., Chem. Rev., 44, 233 (1949). Reed, T. M., Gubbins, K. E., “Applied Statistical Mechanics”, McGraw-Hill, New York, 1973, p 163. Soave, G., Chern. Eng. Sci., 27, 1197 (1972). Yang, A. J. M., Fleming P. D. 111, Gbbs, Gbbs, J. H., J. Ghem. phys., 64, 3732 (1976).
Received for review September 19, 1979 Accepted April 21, 1980
