Vapour pressure over curved surfaces

Figure 5.9 Vapour pressure over curved surfaces (a) over a drop p > p. ...

Such a relation between surface tension and vapour pressure was first established by Lord Kelvin, who demonstrated that the vapour pressure over a curved surface must be different from that over a plane surface of the same liquid. His proof cannot be given here, but the following simple considera-... 

Vapour pressure over a curved surface of liquid. The vapour pressure over a convex surface is greater than that over a plane and over a concave surfaoe it is less. The difference depends on the fact that condensation of vapour on a small convex drop of a liquid increases its surface... 

Lord Kelvin (1870) showed that the vapour pressure of a liquid in the form of drops, or otherwise having a curved surface, is different from that oyer a plane liquid surface. In Fig. 12.VIII L the liquid is supposed to wet the glass (0=0) and rise to a height A in a capillary tube of radius r, with a curved meniscus, which is assumed to be part of a sphere of radius r, exposed to the vapour of the liquid in the vessel. Let q be the densities of the liquid and vapour, and p, p the vapour pressures over the large plane surface in the vessel and over the curved surface of the meniscus, respectively. Then (when the variation in the density of the vapour over a height h is neglected) ... 

Kelvin s equation for surface energy and temperature, 138 for vapour pressure over a curved surface, 366 Khaikin s viscosity equation, 107 Kistiakowsky s equation for specific cohesion, 152, 162... 

Van Wijk and Seeder s viscosity equation, 91 vapour, density of saturated, 324 specific heat of saturated, 336, 346-7, 359 vapour pressure 226 alignment chart, 271 of aliphatic esters, 286 of alkali halides, 237,243 of benzene, 267 boiling-point method for, 235 in capillary tubes, 367 of carbon, 246 centri fugal force, effect of, on 292 constant, 335, 341 over curved surface, 366 determination of, 227-47 dew-point method, 241 of dibasic acids, 243 dynamical method, 235 effusion method, 241. electrification, effect of on, 238, 375 of elements 257 of esters, 250 f., 286 of fusible metal, 230 in... 

The vapour pressure over a curved surface, p, is related to the vapour pressure over a flat surface, po, by the Kelvin equation ... 

Sintering does not usually involve chemical reactions, and the driving force is a reduction in the surface area and the associated reduction in surface energy. This driving force can be illustrated for a flat surface that contains a spherical promberance and a similar spherical depression. The vapour pressure over a curved surface is related to the vapour pressure over a flat surface by the Kelvin equation. This shows that the vapour pressure over a protuberance will be greater than the vapour pressure over the flat surface and will increase as the radius of the curved surface decreases. Similarly, the vapour pressure over a depression will be less than the vapour pressure over a flat surface. When a solid is heated, vapour transfer of matter will take place from a protuberance to a depression, and the surface will tend to become flat. 

Since surface tension is a manifestation of intermolecular forces, it is to be expected that the curvature of a liquid-vapour interface will affect the ease of evaporation and, therefore, the vapour pressure. A relationship between the vapour pressure over a curved surface and the curvature of the surface may be obtained as follows. 

The equilibrium vapour pressure, P, over a curved surface is defined by tlie Kelvin equation... 

A direct experimental verification of equation (8) has been carried out by Thoma,1 by comparing the vapour pressures of various liquids over plane surfaces and over the curved surfaces in capillary tubes of a few tenths of a millimetre diameter. The very small differences of vapour pressure were measured by recording, with an interferometer, the movements of a membrane separating the vapour spaces over the two liquid surfaces. Verification of equation (8) was achieved within an accuracy of... 

As the area is diminished below some thousands of sq. A., where the molecules cover only a small fraction of the surface, the surface pressure rapidly becomes much smaller than that of a perfect gas, and in the four acids with the longest chains becomes constant over a considerable region. The curves are indeed a very faithful reproduction of Andrews s curves for the relation between pressure and volume, for carbon dioxide, at temperatures near the critical. The horizontal regions in the curves correspond to the vapour pressure of liquids, and indicate the presence of an equilibrium between two surface phases, the vapour film, and islands of liquid, coherent film. 

Structure of adsorbed films on dilute solutions. Gibbs s equations (7.1 to 7.7) permit the calculations of the area per molecule, if the surface tension is determined over a range of concentration of the solutions. The area per molecule is proportional to the slope of the curve relating the lowering of surface tension to the activity /2c2 (the vapour pressure of component 2 if the vapour obeys the simple gas laws). This assumes... 

Eq.(32) gives the relationship between the saturated vapour pressure of liquid over the curved (inside the pores) and the flat liquid surface, p and po, respectively, rj is the radius of the liquid meniscus, 7 is the surface tension of liquid, Vj is the molar volume and T is absolute temperature. It follows from Eq.(32) that the sharp emptying of the pore... 

The Kelvin equation is derived from the Young-Laplace equation and the principles of phase equilibria. It gives the vapour pressure, P, of a droplet (curved surface) over the ordinary vapour pressure (P ) for a flat surface (see Appendix 4.2 for the derivation) ... 

The most common method is the method of capillary rise or depression. It is truly static and can be used over a wide temperature range (up to the vapour-liquid critical point of a pure liquid). There are numerous variants developed and applied in the literature, high precision is obtained only by rigorously conforming to certain stractural and dimensional specifications in designing the capillarimeter. The mathematical theoiy is quite simple. As it results directly from the well-known Laplace equation for the difference of pressures Ap on the outer and inner sides of a curved surface of a liquid, its surface tension (7 and the main radii of curvature r and 2 at a given point of the interface. 

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