Pressure of Small Drops

These thermodynamical considerations are due in the main to Lord Kelvin. He also was the first to recognise the fact that the vapour pressure of a liquid is dependent on the magnitude and shape of the sruface. Let us calculate the work dtv which can be obtained by allowing the mass dm of a liquid whose surface tension is y, and whose density is s, to evaporate from a sphere of radius r. to a plane siuface. The mass m of the sphere is -jTrr s and its surface a is 4 7rr. When the mass of the small sphere of liquid is diminished 

The same amoimt of work may be obtained reversibly in a different manner, viz. by the isothermal distillation of the amount dm from the small sphere to the plane surface. Let jp be the ordinary vapour pressure of the liquid at the plane surface and jf its vapour pressure at the small sphere. If the vapour obeys the gas laws, the work done in the transference is 

The vapour pressure at a convex surface is therefore greater than the vapour pressure at a plane surface. In a similar way it may be shown that the vapour pressure at a concave surface is less than at a plane surface. 

The alteration in the vapour pressure is, however, very small. For water at the ordinary temperature (T=300) the difference 

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Ostwald2 pointed out that, just as the vapour pressure of small drops of liquid is greater than that of large drops, so the vapour pressure and solubility of small solid particles is greater than that of large. The relation between the radius, surface tension, and vapour pressure or solubility of spherical particles is the same as that deduced in Chap. I, 15, for small drops. 

This is one of the most important results in the thermodynamics of aerosols. It shows the increase in vapor pressure of small drops compared with the planar surface of the bulk material. Although the Kelvin effect appears often in the held of aerosol formation and growth, direct experimental verification is difficult and few such tests have been made (LaMer and Gruen. 1952). 

Vapor Pressure of Small Drops A bulk liquid is subject to a saturation vapor pressure of pig,r=oo- As a result of capillary pressure, the chemical potential of a drop of liquid is higher by... 

Figure 10.1 (a) The increase in vapor pressure of small drops of different liquids, with decreasing... 

Nuclei, and often even colloidal particles, are so small that their thermodynamic properties may differ considerably from those of a corresponding macrophase, A wei knowm example cf this behaviour is the increased vapour pressure of small drops as given by the equation of W, Thomson... 

Increases in system pressure lead to more rapid vapor blanketing of small drops so, to obtain rapid phase transitions at elevated pressures, more extensive prefragmentation is necessary. 

The vapour pressure of water in small drops is greater than that of water in mass, and the solubility of a solid is greater when in a state of fine subdivision than when in large pieces cf. Hulett, Z, physzkal. Ckem, 1901, 37, 385). The vapour pressure of small crystals is also greater than that of large ones (Pawloff, Z. physikaL Ckem, 8, 3x6). 

The tendency of small drops of liquid to coalesce to large ones is manifested in an increase of vapour pressure with diminution of radius. The relation between vapour pressure and drop radius is easily calculated. When a small amount of liquid, volume dV, evaporates from a drop of radius r, the change in surface area is dA. We have... 

Heat transfer to a liquid leads to an increase of the temperature up to the boihng temperature at which evaporation starts. The vapor pressure becomes equal to the pressure of the system. In the case of the evaporation of a liquid mixture all components or only some of them or perhaps only one component can be present in the vapor. Dealing with the evaporation of an aqueous solution of an inorganic salt with a veiy low vapor pressme approximately pure steam is leaving the liquid. (Note that an entrainment of small drops can take place with the resrrlt of small salt contents in the steam.) In general, all components of the liquid rrrixtirre will be present in the vapor when there are no great differences of the vapor pressirre of the components. 

This is Kelvin s equation for the vapour pressure of a drop of radius R the second (approximate) form of whidi holds only if NjkT v Piy. It is inapplicable to drops containing only a few molecides since they have no uniform interior for whidi the calculation of p is meaningful. iW a drop of water of radius 1 mm, ln(p /p ) is lOA and for a drop of radios 1 pm, it is 10. Hence in a mist the very small drops evaporate and the larger drops grow. All are unstable with respect to the pool of liquid that ultimately forms. 

The vapor pressure of a drop is higher than that of a liquid with a planar surface. One consequence is that an aerosol of drops (fog) should be unstable. To see this, let us assume that we have a box filled with many drops in a gaseous environment. Some drops are larger than others. The small drops have a higher vapor pressure than the... 

A volatile oil contains a relatively large fraction of lighter and intermediate oomponents which vaporise easily. With a small drop in pressure below the bubble point, the relative amount of liquid to gas in the two-phase mixture drops rapidly, as shown in the phase diagram by the wide spacing of the iso-vol lines. At reservoir pressures below the bubble point, gas is released In the reservoir, and Is known as solution gas, since above the bubble point this gas was contained in solution. Some of this liberated gas will flow towards the producing wells, while some will remain in the reservoir and migrate towards the crest of the structure to form a secondary gas cap. 

Gas has a much higher compressibility than oil or water, and therefore expands by a relatively large amount for a given pressure drop. As underground fluids are withdrawn (i.e. production occurs), any free gas present expands readily to replace the voidage, with only a small drop in reservoir pressure. If only oil and water were present in the reservoir system, a much greater reduction in reservoir pressure would be experienced for the same amount of production. 

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