Vapor pressure curvature dependent

Capillary condensation is the condensation of vapor into capillaries or fine pores even at vapor pressures below Pq. Lord Kelvin was the one who realized that the vapor pressure ofaliquid depends on the curvature of its surface. In his words, this explains why moisture is retained by vegetable substances, such as cotton cloth or oatmeal, or wheat-flour biscuits, at temperatures far above the dew point of the surrounding atmosphere [505]. 

At first we tried to explain the phenomenon on the base of the existence of the difference between the saturated vapor pressures above two menisci in dead-end capillary [12]. It results in the evaporation of a liquid from the meniscus of smaller curvature ( classical capillary imbibition) and the condensation of its vapor upon the meniscus of larger curvature originally existed due to capillary condensation. We worked out the mathematical description of both gas-vapor diffusion and evaporation-condensation processes in cone s channel. Solving the system of differential equations for evaporation-condensation processes, we ve derived the formula for the dependence of top s (or inner) liquid column growth on time. But the calculated curves for the kinetics of inner column s length are 1-2 orders of magnitude smaller than the experimental ones [12]. 

On the meniscus surface the deviation of vapor pressure from the saturation pressure Psat depends on the surface tension a, liquid density p( gas constant R, temperature T, and radii of curvature r. When p( > -Psat(T) < (2[Pg.354]

In this chapter we get to know the second essential equation of surface science — the Kelvin5 equation. Like the Young-Laplace equation it is based on thermodynamic principles and does not refer to a special material or special conditions. The subject of the Kelvin equation is the vapor pressure of a liquid. Tables of vapor pressures for various liquids and different temperatures can be found in common textbooks or handbooks of physical chemistry. These vapor pressures are reported for vapors which are in thermodynamic equilibrium with liquids having planar surfaces. When the liquid surface is curved, the vapor pressure changes. The vapor pressure of a drop is higher than that of a flat, planar surface. In a bubble the vapor pressure is reduced. The Kelvin equation tells us how the vapor pressure depends on the curvature of the liquid. 

The force only depends on the radius of the particles and the surface tension of the liquid. It does not depend on the actual radius of curvature of the liquid surface nor on the vapor pressure This is at first sight a surprising result, and is due to the fact that, with decreasing vapor pressure the radius of curvature, and therefore also x, decreases. At the same time the Laplace pressure increases by the same amount. 

The vapor pressure of a liquid depends on the curvature of its surface. For drops it is increased compared to the vapor pressure of a planar surface under the same conditions. For bubbles it is reduced. Quantitatively this is described by the Kelvin equation. 

One consequence of the curvature dependence of the vapor pressure is capillary condensation, that is the spontaneous condensation of liquids into pores and capillaries. Capillary condensation plays an important role for the adsorption of liquids into porous materials and powders. It also causes the adhesion of particles. The condensing liquid forms a meniscus around the contact area of two particles which causes the meniscus force. ... 

The fluid phase that fills the voids between particles can be multiphase, such as oil-and-water or water-and-air. Molecules at the interface between the two fluids experience asymmetric time-average van der Waals forces. This results in a curved interface that tends to decrease in surface area of the interface. The pressure difference between the two fluids A/j = v, — 11,2 depends on the curvature of the interface characterized by radii r and r-2, and the surface tension, If (Table 2). In fluid-air interfaces, the vapor pressure is affected by the curvature of the air-water interface as expressed in Kelvin s equation. Curvature affects solubility in liquid-liquid interfaces. Unique force equilibrium conditions also develop near the tripartite point where the interface between the two fluids approaches the solid surface of a particle. The resulting contact angle 0 captures this interaction. 

The vapor pressure over a pure liquid droplet at equilibrium ps depends on its radius of curvature. The Kelvin equation gives this relationship as ... 

The gas-liquid permporometry combines the controlled stepwise blocking of membrane pores by capillary condensation of a vapor, present as a component of a gas mixture, with the simultaneous measurement of the free diffusive transport of the gas through the open pores of the membrane. The condensable gas can be any vapor provided it has a reasonable vapor pressure and does not react with the membrane. Methanol, ethanol, cyclohexane and carbon tetrachloride have been used as the condensable gas for inorganic membranes. The noncondensable gas can be any gas that is inert relative to the membrane. Helium and oxygen have been used. It has been established that the vapor pressure of a liquid depends on the radius of curvature of its surface. When a liquid is contained in a capillary tube, this dependence is described by the Kelvin equation, Eq. (4-4). This equation which governs the gas-liquid equilibrium of a capillary condensate applies here with the usual assumption of a=0 ... 

According to deduction the classical Kelvin equation [44], which describes the dependence of the saturated vapor pressure on the curvature of interface in two-phase system, is rigorous one, when it is employed for the free (not confined) liquid because this equation takes into consideration only the action of the surface tension forces at the fiquid-vapor interface. 

Equations (17.21), (17.24), (17.26), and (17.27) are different forms of the Kohler equations (Kohler 1921, 1926). These equations express the two effects that determine the vapor pressure over an aqueous solution droplet—the Kelvin effect that tends to increase vapor pressure and the solute effect that tends to decrease vapor pressure. For a pure water drop there is no solute effect and the Kelvin effect results in higher vapor pressures compared to a flat interface. By contrast, the vapor pressure of an aqueous solution drop can be larger or smaller than the vapor pressure over a pure water surface depending on the magnitude of the solute effect term B/Dp relative to the curvature term A/Dp. Note that both effects increase with decreasing droplet size but the solute effect increases much faster. One should also note that a droplet may be in equilibrium in a subsaturated environment if DpA < B. 

Let us now consider how the vapor pressure of a droplet depends on its radius of curvature r. In equilibrium, the pressure difference across the droplet is given by Eq. 3.61. If we transfer atoms from, say, the liquid to the surrounding gas phase, there is a small and equal equilibrium displacement on both sides of the interface ... 

Differences in vapor pressure or solubility that depend on particle size (radius of curvature) can only be observed for particles smaller than r < 10 nm (100 A). If we assume representative values for a water droplet [7 = 72.8 ergs/cm (7.28 x 10 J/m ), = 18 cmVmole (18 x 10 mVrnole)], Pr/Po approaches unity... 

Liquid systems with curved surfaces exhibit many unique properties (capillary rise, curvature-dependent vapor pressure, contact angle, difficulty of nuclea-tion) because of differences between internal and external pressures at the interface. 

Let us now consider how the vapor pressure of a droplet depends on its radius of curvature r. We obtain the following ... 

Note that the vapor pressure over curved surfaces is dependent on the radius of curvature. This involves modifications to the vapor pressure relationships for highly curved systems like rain drops or liquids confined in small capillaries (Topic 3.1.1). 

To determine the distribution of pores with diameters smaller than 20 nm, a nitrogen desorption technique is employed which utilizes the Kelvin equation to relate the pore radius to the ambient pressure. The porous material is exposed to high pressures of N2 such that P/Po 1 and the void space is assumed to be filled with condensed N2, then the pressure is lowered in increments to obtain a desorption isotherm. The vapor pressure of a liquid in a capillary depends on the radius of curvature, but in pores larger than 20 nm in diameter the radius of curvature has little effect on the vapor pressure however, this is of little importance because this region is overlapped by the Hg penetration method. 

This does not divorce these values from the measured vapor-pres sure curve of the individual species, because vapor pressure. (If the curves in Figure 5.4 were totally straight instead of being gently curved, then if we knew vapor-pressure curve for species i with complete accuracy. In Figure 5.4 none of the experimental data curves cross one another, and the curvature is more or less proportional to (a, so it makes sense that there could be a universal =f(Tr, (o) function. Chao and Seader proposed such a function for hydrocarbons the methods described in the next section also do this, but in a somewhat different way.)... 

As an example, Figure 5.8 shows the force versus distance and the adhesion force versus humidity for a sphere of = 3 pm radius and a plane. It is compared to the force between a cylinder of = 200 nm radius with a conical end (opening angle = 88°) and a plane. Both surfaces are assumed to be perfectly wetted (0i = 2 = 0). Force versus distance was calculated at a relative vapor pressure of water of P/Po = 0.9 leading to the radius of curvature of r = —0.52 nm/lnO.9 = 5.0 nm (Eq. (5.24)). The humidity dependence is plotted for contact (D = 0) and is thus equal to the adhesion force. 

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