Vapor pressure Kelvin relation

Numerous mathematical formulas relating the temperature and pressure of the gas phase in equilibrium with the condensed phase have been proposed. The Antoine equation (Eq. 1) gives good correlation with experimental values. Equation 2 is simpler and is often suitable over restricted temperature ranges. In these equations, and the derived differential coefficients for use in the Hag-genmacher and Clausius-Clapeyron equations, the p term is the vapor pressure of the compound in pounds per square inch (psi), the t term is the temperature in degrees Celsius, and the T term is the absolute temperature in kelvins (r°C -I- 273.15). 

Use of the condition of constant meniscus volume is most appropriate when growth and dissolntion of the meniscus is comparatively slow. An alternative is to consider the Kelvin eqnilibrium condition. The Kelvin eqnation relates the eqnilibrinm meniscus curvature (also known as the Kelvin radius) to the relative vapor pressure and if Kelvin eqnilibrinm is maintained during the separation process, then the adhesive force becomes [19] ... 

In porous media, liquid-gas phase equilibrium depends upon the nature of the adsorbate and adsorbent, gas pressure and temperature [24]. Overlapping attractive potentials of the pore walls readily overcome the translational energy of the adsorbate, leading to enhanced adsorption of gas molecules at low pressures. In addition, condensation of gas in very small pores may occur at a lower pressure than that normally required on a plane surface, as expressed by the Kelvin equation, which relates the radius of a curved surface to the equilibrium vapor pressure [25],... 

The consequence of Laplace pressure is very important in many different processes. One example is that, when a small drop comes into contact with a large drop, the former will merge into the latter. Another aspect is that vapor pressure over a curved liquid surface, pcur, will be larger than on a flat surface, pf,at. A relation between pressure over curved and flat liquid surfaces was derived (Kelvin equation) ... 

Adsorption studies leading to measurements of pore size and pore-size distributions generally make use of the Kelvin equation which relates the equilibrium vapor pressure of a curved surface, such as that of a liquid in a capillary or pore, to the equilibrium pressure of the same liquid on a plane surface. Equation (8.1) is a convenient form of the Kelvin equation ... 

We know from theoretical principles, however, (and your text may explain this) that the vapor pressure of a liquid is related to its heat of vaporization (AHv), which is a physical constant characteristic of the liquid, to the gas constant (R = 1.987 cai/mole deg), and to the Kelvin temperature (T), by the equation... 

STRATEGY We expect the vapor pressure of CC14 to be lower at 25°C than at 57.8°C. The Clausius-Clapeyron equation gives the quantitative relation between vapor pressure and temperature, so substitute the temperatures and the enthalpy of vaporization to find the ratio of vapor pressures. Then substitute the known vapor pressure to find the desired one. To use the equation, convert the enthalpy of vaporization into joules per mole and all temperatures into kelvins. 

The Claussius-Claperyon equation relates the vapor pressure of a liquid, P, to the Kelvin temperature, T. 

It is interesting that equation (4.88) is completely identical to the Kelvin Thomson relation for the saturated vapor pressure over a spherical par tide of the condense phase of the same substance. It is also important that expression (4.88) corresponds to an increase in the adsorption heat of the... 

Mesopore size analysis is usually based on the application of Kelvin s relation between vapor pressure of a capillary condensed phase and pore size. It is conventionally admitted that mesopores are pores whose width lies between 2 and 50 run. For narrower pores, the capillary condensation phenomenon does not take place and Kelvin s relation is irrelevant. These methods are thus strictly limited to pores in which the capillary condensation phenomenon occurs, as can be visualized by the hysteresis loop. 

It is well established that the pore space of a mesoporous solid fills with condensed adsorbate at pressures somewhat below the prevailing saturated vapor pressure of the adsorptive. When combined with a eorrelating function that relates pore size with a critical condensation pressure, this knowledge can be used to characterize the mesopore size distribution of an adsorbent from its adsorption isotherm. The correlating function most commonly used is the Kelvin equation [1], Refinements make allowance for the reduction of the physical pore size by the thickness of the adsorbed film existing at the critical condensation pressure [1-2]. Still further refinements adjust the film thickness for the curvature of the pore wall [3]. 

Clapeyron equation. The equation dpldr = AH/TAV. It states that the rate of change of vapor pressure of a liquid (expressed in ergs per cubic centimeter) with absolute temperature (in Kelvins) equals the heat of vaporization (in ergs per gram) divided by the product of the absolute temperature and the increase in volume (V, in cubic centimeters) when a gram of the liquid changes to vapor. Other consistent units may be used. The approximate Clapeyron equation, dlnp/dT = AH/RT2, expresses the same relation in a less-exact form because in its derivation, it is assumed that AV is equal to the volume of vapor. This assumption (that the volume of liquid is negligibly small) is usually true within a few percent under ordinary conditions of temperature and pressure. 

The interface between the droplet and the gas is not discontinuous the average molecular density decreases over a narrow region from the liquid side to the vapor. When the size of the droplet becomes sufhctently small compared with the thickness of the transition layer, the use of classical thermodynamics and the bulk surface tension become inaccurate the Kelvin relation and Laplace formula no longer apply. This effect has been studied by molecular dynamics calculations of the behavior of liquid droplets composed of 41 to 2(X)4 molecules that interact through a Lennard-Jones (LI) intermolecular potential (Thomp.son et al., 1984). The results of this analysis are shown in Fig. 9.5, in which the nondimensional pressure difference between the drop interior and the surrounding vapor (Pd — p)rr / ij is... 

Figure 9.6 Equilibriunn vapor pressure curves for droplets composed of solvent aJone (Kelvin relation) and of a solvent with a fixed mass of nonvolatile solute.

Panicle huental Pressure Laplace s Formula 257 Limit of Applicability of Kelvin Relation 258 Hygroscopic Particle-Vapor Equilibrium 259 Charged Particle-Vapor Equilibria 263 Solid-Particle-Vapor Equilibrium 265... 

The related supported liquid-phase catalysts consist of traditional support materials such as y-AljOj having micropores filled with solvent and a dissolved catalyst. In small pores, because of the Kelvin effect, the vapor pressure of the solvent is small so that it will remain in the pore as a liquid, even when the catalyst is used at a high temperature in flowing vapor-phase reactants" . These catalysts are active for alkene hydroformy-lation the soluble catalyst can thus be used without the complications of corrosion and difficult separation from products—provided that it is stable (cf. 14.2.4). 

The Kelvin equation can be combined with the relative humidity, RH, if water is involved as the fluid relative humidity indicates how moist the air is. The amount of water vapor in the air at any given time is usually less than that required to saturate the air. The relative humidity is the percentage of saturation humidity, generally calculated in relation to the saturated vapor density. Relative humidity may be defined as the ratio of the water vapor density (mass per unit volume) to the saturation water vapor density, usually expressed in percent. Relative humidity is also approximately equal (exactly equal when water is assumed as an ideal gas) to the ratio of the actual water vapor pressure to the saturation water vapor pressure, RH = PJP°. The P° values corresponding to each temperature are given in tables which can be found in handbooks. If RH is measured in an experiment, then Pv can be calculated by using the saturation water vapor pressure tables and can be inserted into the Kelvin equation. 

If it is assumed that an absorbed layer of water exists before capillary condensation takes place and that this layer consists of ordered or oriented water molecules, then the contact angle in Kelvin s equation should be very close to zero. With zero contact angle, vapor pressures in the capillaries are calculated from Kelvin s equation for capillaries from 10 %. to 200 X, Table I. The vapor pressure of water below 0 C (15) is compared with the vapor pressures in the capillaries to obtain the freezing points. Figure 1 shows the relation between the freezing point depression and the capillary radius. 

The difference in thermodynamic equilibrium between crystals of different size is defined by the Gibbs-Thomson or Ostwald-Freundlich equation (Mullin 1993). The original Kelvin equation was written to describe droplets of liquid in a vapor in terms of vapor pressures around droplets of different size. However, in crystallization studies, the vapor pressure may be related, through the solution activity, to the equilibrium concentration in solution for crystals of different size, faking the assumption that concentration can be used directly fpr vapor pressure gives... 

The vapor pressure of A just above the particle surface can be related to the particle mass through the Kelvin equation, (9.86), which after expressing the particle diameter as a function of its mass by... 

Porosity in a powder can come from both closed and open pores. Two primary methods for the determination of porosity are gas adsorption and mercury porosimetry. It is assumed that gas adsorption is favored in small capillaries because of the overlapping surface potentials, which result in capillary condensation. Pores from 1.5 to 100 nm in diameter are determined using the Kelvin equation, which relates capillary radius r to the ratio of the vapor pressure P and the equilibrium vapor pressure of the same liquid over a plane surface Pq, as follows ... 

As the size of a particle decreases, the contribution of the surface to the total flee energy becomes increasingly significant. This trend is evident in the Kelvin equation, a classic thermodynamic equation relating the vapor pressure of a droplet to its physical size. This equation is often applied in the determinalion of the pore size distribution through adsorption porosimetry. For this appHcalion, the equation can be written ... 

AH estimations of the pore size of a material from its gas adsorption isotherm measurements are based on the well-known Kelvin equation suggested more than 100 years ago. It considers an equilibrium between the vapor phase and the bulk liquid at a constant temperature and relates the relative vapor pressure p/p to the radius r of the convex (plus) or concave (minus) spherical meniscus of the Hquid placed in a capillary ... 

Both approaches are useful. The energy approach relates interfadal tension to thermodynamics and thus allows useful results to be derived (e.g., the Kelvin equation of Example 1.1, which gives the effect of drop size on vapor pressure). The force approach is needed to justify using interfacial tension in boimdary conditions involving forces and stresses at interfaces. Such boundary conditions are employed in solving the governing equations of fluid mechanics when fluid interfaces are present. 

In capillary condensation phenomena, multilayer adsorption from the vapor phase into a porous medium proceeds until the pores are filled with condensed liquid from the vapor phase. In this process, vapor condensation occurs below the bulk saturation pressure of the liquid because of the increased number of interactions between vapor phase molecules inside the porous media. Once the condensation process is completed, a vapor-liquid interface is formed in the pores with an equilibrium vapor pressure, P, which is below the bulk saturation pressure, P, of the fluid. For very large pores (i.e., macropores), the relation between P and P, can be well described with the macroscopic form of the Kelvin equation (Melrose 1966) ... 

Another aspect is that vapor pressure over a curved liquid surface, pc rve will be larger than on a flat surface, Pdaf A relation between pressure over curved mdftat liquid surface was derived (Kelvin equation) ... 

It can see from the above-mentioned discussion that capillary cohesion is closely related to the curved liquid surface. The pressme boimdary causes capillary cohesion — the critical vapor pressure relates to the cmvatme radius of liquid surface. Kelvin equation has been derived from thermodynamics, where the curvature radius (rjs) of the meniscus of hemispherical (concave) liquid and the equilibrium vapor pressure (p) has the following relationships ... 

The Kelvin equation relates the vapor pressure above a meniscus of a liquid to a diameter of a containing pore. This section discusses the use of nitrogen isotherms to descriptions of mesoporosity <20 nm diameter. 

Sorption isotherms are plotted as Hr vs. water saturation of pore space, S . The corresponding capillary pressure as given by the Kelvin equation (1), which relates the relative vapor pressure p/p to the capillary pressure Pc, and the equivalent pore size and/or crack thickness is shown on a separate axis in Fig. 2a. In the Kelvin equation. Pc is expressed in terms of surface tension, a, and mean radius of curvature, r v is the liquid molar volume, R the universal gas constant and T the temperature. 

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. 

The technique is based on the Kelvin equation, which relates the reduced vapor pressure of a liquid with a curved interface to the equilibrium vapor pressure of the same liquid in bulk with a plane liquid-vapor interface [21]. The BET adsorption theory is frequently applied to gas adsorption in order to obtain specific surface areas. 

The vapor pressure of the liquid, Pv. is related to the curvature (or, equivalently, to the capillary pressure) by the Gibbs-Thompson (or Kelvin) equation,... 

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