THE VAPOR PRESSURE OF CURVED SURFACES' E. F. HAMMEL University of California, Los Alamos Scientific Laboratory, Los Alamos, New Mexico
IN1871 Lord Kelvin (I) showed that in a one component system the vapor pressure in equilibrium with a curved surface of a liquid enclosed in a capillary would be "less a t a concave than at a plane surface of liquid and less a t a plane surface than-at a convex surface." These different vapor pressures are related by the wellknown formula (2): where p, and p m are the vapor pressures of surfaces the radius of curvature of which is indichted by the subscript, a is the surface tension, and U L is the liquid volume per molecule. It will be shown below that this equation is applicable only to those curved surfaces in which the surface tension changes the pressure in the confined liquid, namely, for small liquid drops and capillary condensate, but not for vapor bubbles within the liquid. Since these limitations are repeatedly ignored in almost all textbooks, we repeat and elaborate upon the statement of Doring (3) that "the [eqnilibrium] vapor pressure [of bubbles] is completely independent of the form of the surface." The thermodynamical basis for this discussion is the Work performed under the auspices of the Atomic Energy Commission.
variation, at constant temperature, of the fugacity with pressure (4) d i n = d~
O K
k~
where p* is the fugacity and P the pressure in the condensed phase.2 Integrating t o obtain the relationship between the fngacity of a liquid bounded by a plane surface and that with a radius of curvature r a t pressures P t and PF, respectively, we have neglecting the compressibility of the liquid If we use as a reference state the plane liquid surface under its saturated vapor pressure, and consider only pressures so low that deviations from ideal gas behavior may be neglected, we may rewrite the above equation as where p , is the saturated vapor pressure in equilibrium with a plane surface of the liquid. The pressure in a liquid bounded by a spherical interface of radius r is given by the Laplace equation P; =
PY
=t2o/r
(3)
a Note that for a plane surface the pressure on the phase and in the phase are identical.
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where the plus sign is to be taken when the liquid surface is convex and the minus sign when i t is concave. P: is the pressure in the phase external t o the liquid. These are the fundamental equations (2 and 3), which we now apply to the following special cases: (1) The vapor pressure of small drops. If P : =p,, we derive for drops equation (1) with the plus sign. P: may have any positive value, however, so that the most general relationship is As pointed out by Volmer (6), for drops in equilibrium with their own saturated vapor (PT equal to p,) the small difference p,-p, is usually neglected. (2) The vapor pressure of liquids in capillaries. I n this case we have from equations (2) and (3) (with the minus sign) Sett,ing P: equal to p,, and neglecting the differencebetween p, and p, in comparison with 2r/r we again obtain equation (I), this time with the minus sign. (3) The vapor pressure of bubbles in liquids. As is well known (6), vapor bubbles in liquids may exist in unstable equilibrium only when the external pressure on the liquid is less than the saturated vapor pressure. This external (together with hydrostatic) pressure also defines, as shown below, the radius of the equilibrium bubble. If the pressure in the liquid a t the bubble is PL,the vapor pressure in the bubble is given directly by equation (2) except that the subscripts r are unnecessary, since there is no dependence of vapor pressure upon radius. The radius of the equilibrium bubble is then obtained, as shown in the figure, from the simultaneous solution of equations (2) and (3) or r., = 2 u ( p , exp [ ( P L- p S ) u ~ / k T ] PL]-I
Although the thermodynamics discussed above seems quite straightforward, the distinction between the treatment of bubbles and capillary condensate has not always been made. The history of the subject is of interest especially because it illustrates the confusion which has surrounded this problem for many years. I n 1934 Kaischew and Stranski (6) derived an expression for the rate of formation of bubble nuclei in liquids. I n their derivation they assumed that the vapor pressure within a bubble was given by equation (1) with the minus sign. Simultaneous solution of equations (1) and (3) then yielded the equilibrium radius. Their derivation of the rate of bubble nuclei formation required for its further development specification of the vapor pressure of other than equilibrium radius bubbles. Although it would appear consistent to have used equation (1) to provide the desired radial dependence of the bubble vapor pressure, Kaischew and Stranski chose to assign to bubbles of all sizes the vapor pressure of the equilibrium hized bubble. I n 1937 Doring (7) "corrected" this "omission" on the part of Kaischew and Stranski, rederiving an expression for the rate of bubble nuclei formation in liquids but this time retaining the radial dependence of the bubble vapor pressure as given in equation (1). Several months later he published a correction (3) t o his original paper based on the approach reiterated above. I n a footnote he acknowledges that, although Professor Stranski had VOLUME 35, NO. 1, JANUARY 1958
reg
RADIUS
Radius of the equilibrium bubble in asuperhested liquid, thessturated or orthobario vapor pressure of whioh i. p,, the equilibrium n.por preraure of lahioh under a combined external and hydroatetic messurc PL < p , is p.
meanwhile pointed out his error, he himself had independently arrived a t the same conclusion. Subsequently Volmer treated the problem again in his book "Kinetik der Phasenbildung (6)." There, although Doring's corrected version of bubble formation is presented, Volmer's treatment is not completely consistent. On page 157 it is pointed out that, since the thermodynamic potentials are the same in both liquid - 1 provided one avoids large and vapor phases p , / ~ , ~ = negative pressures in the liquid. On page 164, however, the erroneous relation between p, and p m obtained from eliminating &/r from equations (1) and (3) is still utilized. Interest in this subject has recently arisen due to the development of bubble chambers (8). Although no quantitative verification of these views is as yet available, calculations and engineering design work have both been based on the "corrected" Doring version. It is hoped that eventually conditions of bubble formation will be sufficiently well defined to provide an independent experimental verification of Doring's treatment. ACKNOWLEDGMENT
The author is indebted to T. R. Roberts and other members of the cryogenic group of this laboratory for suggestions and comments. LITERATURE CITED W., Phil. Mag. [ 4 ] ,42, 448 (1871). ( 1 ) THOMSON, ( 2 ) PAUL,M. A,, "Principles of Chemical Thermodynamics," 1st ed., McGraw-Hill Book Co., Inc., New York, 1951, p. 275. W . ,Z . phvsik. Chem. ( B ) ,38,292 (1938). ( 3 ) DORING, ( 4 ) GUGGENHEIM, E . A., "Thermodynamics," 2nd ed., North Holland Publishing Co., 1950, p. 166. ( 5 ) VOLMER, M . , "Kinetik der Phasenbildung," Dresden and Leipzig, 1939, p. 88. ( 6 ) KAIBCHEW, R., AND I. N. STRANSKI, Z. physik C h a . ( B ) , 26,317 (1934). W., Z.physik. C h a . ( B ) ,36,371 (1937). ( 7 ) DORING, (8) MARTELLI, G., NUOVOCimato, 1 2 , 250 (1954); BERTANZA, NUOVOCimmto, 1, 324 (1955); L., AND G. MARTELLI, GLASSER, D. A,, NUOVO Cimento, Supplement 2 to Vol. 11, Series 9, 361 (1954).
