J. Phys. Chem. 1985,89, 2647-2648
2641
Effect of an Electric Field on the Vapor-Liquid Equilibrium of a Dielectric Fluid Jacques R. Quint, Laboratoire de Thermodynamique et Cinetique Chemique, University of Clermont-Ferrand-2, F. 631 70 Aubiere, France
Jeffrey A. Gates, and Robert H. Wood* Department of Chemistry, University of Delaware, Newark, Delaware 19716 (Received: May 14, 1984; In Final Form: August 17, 1984)
Equations are derived for calculating the effect of a change in the electric displacement on the liquid-vapor equilibrium pressure of a dielectric fluid. The change in equilibrium pressure is very small for electric displacements accessible in the laboratory. However, near an ion in the gas phase the electric displacement is very large and the effect is not negligible.
Introduction Previous articles from this laboratory have explored the thermodynamic properties of a charged hard sphere in a compressible dielectric fluid1i2in the hopes that this model would prove useful for ions in the gas phase near or above the critical temperature. Calculations for this model show that for ions just above the critical point there are large increases in density and dielectric constant of the solvent near the ion due to the compression of the dielectric fluid in the electric field of the ion.2 Because of this large compression one would expect that, for an ion in the gas phase, the electric field will condense the gas, creating a liquid drop surrounding the ion. In order to explore this effect quantitatively, it is necessary to know the effect of electric displacement on the liquid-vapor saturation pressure. The present paper derives the appropriate equations for calculating this effect and presents some calculations on the magnitude of the effect for water.
(del/ W
P
= (ar/m,/ (aD/ W P
(5)
together with Frank's eq 12 ( a ~ / a E ) p= -@xVm
(6)
where E is the electric field strength, eo is the vacuum permittivity, x is the electric susceptibility, and Vmis the molar volume of the fluid. We also have the standard electrostatic equations x = €/to- 1 =K - 1 (7) where K is the relative dielectric constant and E is the permittivity and D = COKE From eq 8 we have (dD/dE)p =
(8)
E&+ ~oE(aK/dE)p
(9)
Derivation of the Equations Consider a sample of liquid and vapor in equilibrium at constant temperature with an electric field applied so that the electric field vector is perpendicular to the interface between the liquid and the vapor. Under these circumstances the electric displacement, D, is perpendicular to the liquid-vapor interface and equal in both the liquid and gas phases. Under these circumstances the chemical potential is a function of temperature, pressure, and electric displacement ( p [ T,P,D]). At D = 0, the saturation pressure, Pat, is equal to the saturation pressure in the absence of an electric field, Psa:, and we have the criteria for equilibrium
Substituting eq 6, 7, and 9 into eq 5 we obtain (dp/dD)p = -E(K - l)Vm/IK E(aK/aE)p]
cLv( T,PS,,o,O) = P1(TJ,,O,O) (1) where subscript v denotes the vapor phase and 1 the liquid phase. If the liquid and vapor are to remain at equilibrium upon applying an electric field then
Substituting eq 10 and 12 in eq 1 1 and using Frank's eq 13,3
+
Next we evaluate the denominator of eq 4 starting with the change of variables: (+/ W
D
= -(aP /aD),(aD/aP),
and from eq 8 we find (aD/aP), = t&(dE/aP),
(11)
+ eoE(dK/dP),
= E O ( ~ E / ~ P )+, [E(BK/BE),] K
(BP/BE), = EOEX
(12)
(13)
we find
dPV = dlrl
(2)
( ~ c L / ~ P= )Vm[K D + E ( a K / a E ) , I / [ K + E(aK/aE)pI
dPv = ( a P v / a p ) D , T , " d P + ( a P v / m P , T , n dD
(3)
Finally we substitute eq 14 and 10 into eq 4 and converting to Dz as the variable we have
with a similar equation for pl. Substituting eq 3 and its liquid analogue into eq 2 we find dP/dD = (aPmt/aD)T = - A ~ - y ( a p / a D ) p / A ~ - v ( a p / d P ) ~ (4) where we have used AI-& to denote X, - Xv for any quantity X evaluated at Psa, and the subscripts T,n have been dropped for simplicity because these will be held constant in the remaining equations of this paper. In order to proceed further we will need the equations for the thermodynamic properties of a compressible dielectric fluid derived by Frank.3 In order to evaluate the numerator of eq 4 we use Wood,R. H.: Ouint. J. R.;Grolier, J.-P. E. J . Phvs. Chem. 1981, 85,
2944.. (2) Quint, J. R.; Wood,
R. H.J . Phys. Chem. 1985, 89, 380.
(14)
-apnilt - a02
-1
AI,[(K - l)VmKeo/(K3eoZ + 2DZ(aK/aE2),1]
2 A,-"[Vm(K3rO2 + 2DZ(aK/aE2),]/(K3eo2 + 2DZ(aK/aEz)p)] (15)
The value of (dK/aE2),can be calculated from Frank's eq 163 (aK/aEZ), = ~ e ~ p ~ ( a ~ / a p ) ~ ~ /(16) 2 ~~~~~
(1)
(10)
~~
(3) Frank, H. S. J . Chem. Phys. 1955, 23, 2023. (4) Haar, L.; Gallagher, J. S.;Kell, G. S. National Bureau of Standards Internal Report No. 81-2253. (5) Uematsu, M.; Franck, E. U. J . Phys. Chem. Ref.Dura 1980,9, 1291.
0 1985 American Chemical Society
J. Phys. Chem. 1985.89, 2648-2651
2648
where this equation assumes there is no dielectric saturation; Le., (aK/aEZ)&q= 0. We also need to calculate ( a K / a E 2 ) , using
TABLE I: Values of eo(dP,/dD*)/ Calculated by 24 for WateP and APMtfor an Electric Field of 106 V m-*in the Liquid
( T - 273.15)/K 25 100 150 200 250 300 350
+
( a K / a E 2 ) , = ( a K / a p ) E ( d p / a E 2 ) p ( a K / d E 2 ) , (17)
again neglecting dielectric saturation (the last term in eq 17) and using Frank's eq 11 we have ( d K / 8 E 2 ) P = ( e o a P / 2 ) [ P ( a K / a P ) E 2 - ( K - I)(dK/+)El
(18)
Introducing eq 16 and 18 into eq 15 we get our final equation:
+
aP,,,/aD2 = [A,-~(K(K - ~ ) V , / [ K % ~p ~ ~ ( p ~ ( a ~ /-a p ) , ~ ( K - ~)~(~K/~~)E)III/[AI-~[~V~(K~~O + + D 2 p l ~ Z ( a K / a ~ -) E( K2 ~ ~ f l p ~ ( a ~ // a p h ~ ) 1MaK/%)ElI)l (19)
In order to use this equation we need to know p as a function of D2 along the saturation line. Start with (ap/aD2)p=Psa, = ( a p / a D 2 ) p+ (ap/aP)daPsat/aD2)
(20)
C ~ W . . , / ~ D ~ ) ~AP...lmPa /~O~ 0.088 0.005 1.82 0.050 6.86 0.1 17 19.1 0.204 43.8 0.284 87.3 0.317 146.1 0.224
"The equation of state of Haar, Gallagher, and Kel14 was used together with the dielectric constant of water from Uematsu and Fran~k.~ can be solved by standard numerical techniques to yield p as a function of Dz at P = P,,. Finally eq 15 is integrated to give Pat as a function of Dz. In order to explore the magnitude of the change in P,,, for reasonable values of electric displacement we can approximate PSatby a Taylor series around D = 0 which gives, from eq 15, P,,, = P,?
Differentiating eq 8 we find ( a D 2 / a p ) , = eoZE22K(dK/ap)p+ t t J ? ( a E 2 / a p ) p
(21)
+ (dPsat/dDZ)0D2+ ...
(23)
with
Now substituting eq 21 into eq 20 and using Frank's eq 11 for (dE2/ap), we have
A
where we have used (aK/ap), = ( ~ ? K / d p(since ) ~ we are assuming no dielectric saturation). We have no measurements of (ap/aP)Dexcept at D = 0 where ( ~ 3 p / d P=) ~pp. In using eq 22 we will have to make some assumption about the effect of electric displacement on the compressibility. In addition to assuming no dielectric saturation, (aK/dE2),,, = 0, an analogous assumption, no compressibility saturation, (d/3/dEZ),,, = 0, must be made. Thus, the compressibility is assumed to be a function of the density and temperature but not of the electric field. In the absence of experimental data this is probably the best assumption. We can substitute eq 15 into eq 22 and the result is ( a p / C ~ D ~ ) ,as , , a~ function ~ of p and d.This differential equation
where we have used superscript zeros to indicate evaluation of the quantity at D = 0. Table I gives values of co(dP,,/dd)O for water at several temperatures. At low temperatures the vapor term in eq 24 dominates and (aP,,,/aD2)0is close to (Kv- 1)/ (2c&,2) where K , is the relative dielectric constant of the vapor phase at PSat. Table I also gives values of the change in saturation pressure on applying an electric field of lo6 V m-I to the liquid. The changes in pressure are very small at all temperatures so the experimental detection of the effect will be very difficult. However, the electric displacement around an ion is extremely large and the change in Pat near an ion can be quite large. We are currently investigating the calculation of the properties of hard sphere ions in a gaseous dielectric fluid including the effect of electric displacement on P,,,. Acknowledgment. This work was supported by the National Science Foundation (Grant No. CHE77-809672). We thank the Franco-American Exchange Commission for assistance with travel expenses in connection with the interuniversity exchange between the University of Delaware and the University of ClermontFerrand-2.
The Potential of Electron Spin Resonance Spin-Labellng In Determlnlng Micelle Shapes D. D. Lasic and H.Hauser* Laboratorium fur Biochemie, Eidgenossische Technische Hochschule, ETH-Zentrum. CH 8092 Zurich, Switzerland (Received: May 14. 1984: In Final Form: January 8. 1985) The potential of ESR spin-labeling in determining micelle shapes is assessed. Micelles that tumble over slowly on the ESR time scale give rise to anisotropic spectra containing at least in principle morphological information. Elongated, rodlike micelles can be readily identified by ESR if the motional averaging about the c, symmetry axis is rapid on the ESR time scale. If, however, this averaging is slow, then rodlike micelles give smectic type spectra indistinguishable' from those arising from disklike or spherical micelles. In this case the anisotropic ESR spectrum cannot be interpreted in terms of micelle shape. Introduction Isotropic micellar solutions have &en the subject of extensive studies.l.2 Information as to the shape and size of micelles has (1) Wennerstrom, H.; Lindman, B. Phys. Rep. 1979, 52, 1.
0022-36S4/85/2089-2648$01 .50/0
been deduced mainly from scattering experiments using different radiation sources and hydrodynamic measurements. Information concerning the micelle shape is, however, not directly obtainable (2) Lindman, B.; Wennerstrom, H. Top. Curr. Chem. 1980, 87, 1.
0 1985 American Chemical Society
