Vapor pressure Kelvin equation

Here, r is positive and there is thus an increased vapor pressure. In the case of water, P/ is about 1.001 if r is 10" cm, 1.011 if r is 10" cm, and 1.114 if r is 10 cm or 100 A. The effect has been verified experimentally for several liquids [20], down to radii of the order of 0.1 m, and indirect measurements have verified the Kelvin equation for R values down to about 30 A [19]. The phenomenon provides a ready explanation for the ability of vapors to supersaturate. The formation of a new liquid phase begins with small clusters that may grow or aggregate into droplets. In the absence of dust or other foreign surfaces, there will be an activation energy for the formation of these small clusters corresponding to the increased free energy due to the curvature of the surface (see Section IX-2). 

While Eq. III-18 has been verified for small droplets, attempts to do so for liquids in capillaries (where Rm is negative and there should be a pressure reduction) have led to startling discrepancies. Potential problems include the presence of impurities leached from the capillary walls and allowance for the film of adsorbed vapor that should be present (see Chapter X). There is room for another real effect arising from structural peiturbations in the liquid induced by the vicinity of the solid capillary wall (see Chapter VI). Fisher and Israelachvili [19] review much of the literature on the verification of the Kelvin equation and report confirmatory measurements for liquid bridges between crossed mica cylinders. The situation is similar to that of the meniscus in a capillary since Rm is negative some of their results are shown in Fig. III-3. Studies in capillaries have been reviewed by Melrose [20] who concludes that the Kelvin equation is obeyed for radii at least down to 1 fim. 

The Kelvin equation (Eq. HI-18), which gives the increase in vapor pressure for a curved surface and hence of small liquid drops, should also apply to crystals. Thus... 

Bikerman [179] has argued that the Kelvin equation should not apply to crystals, that is, in terms of increased vapor pressure or solubility of small crystals. The reasoning is that perfect crystals of whatever size will consist of plane facets whose radius of curvature is therefore infinite. On a molecular scale, it is argued that local condensation-evaporation equilibrium on a crystal plane should not be affected by the extent of the plane, that is, the crystal size, since molecular forces are short range. This conclusion is contrary to that in Section VII-2C. Discuss the situation. The derivation of the Kelvin equation in Ref. 180 is helpful. 

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). 

The vapor pressure of a surface composed of a large number of very small pores, P, is influenced by the radius of the pores, as described by the Gibbs-Kelvin equation ... 

Calculated vapor pressure relationships of T2, HT, and DT have been reported (10) (see Deuteriumand tritium,deuterium). An equation for the vapor pressure of soHd tritium in units of kPa, Tin Kelvin, has been given (11) ... 

STRATEGY We expect the vapor pressure of CC14 to be lower at 25.0°C than at 57.8°C. Substitute the temperatures and the enthalpy of vaporization into the Clausius-Clapeyron equation 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 express all temperatures in kelvins. 

As an example, the ratio of the equilibrium vapor pressures for water, Pi6 and water. Pig, depends on temperature and is expressed by the following equation, derived from Faure (1977) (temperature is in kelvins) ... 

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],... 

Vapor pressure in bar can be predicted for temperature in Kelvin from the Antoine equation using coefficients in Table 4.73. Data... 

Example 4.5 2-Propanol (isopropanol) and water form an azeotropic mixture at a particular liquid composition that results in the vapor and liquid compositions being equal. Vapor-liquid equilibrium for 2-propanol-water mixtures can be predicted by the Wilson equation. Vapor pressure coefficients in bar with temperature in Kelvin for the Antoine equation are given in Table 4.113. Data for the Wilson equation are given in Table 4.126. Assume the gas constant R = 8.3145 kJ-kmol 1-K 1. Determine the azeotropic composition at 1 atm. 

The system methanol-cyclohexane can be modeled using the NRTL equation. Vapor pressure coefficients for the Antoine equation for pressure in bar and temperature in Kelvin are given in Table 4.176. Data for the NRTL equation at 1 atm are given in Table 4.186. Assume the gas constant R = 8.3145 kIkmol 1-K 1. Set up a spreadsheet to calculate the bubble point of liquid mixtures and plot the x-y diagram. 

Vapor sorption onto porous solids differs from vapor uptake onto the surfaces of flat materials in that a vapor (in the case of interest, water) will condense to a liquid in a pore structure at a vapor pressure, Pt, below the vapor pressure, P°, where condensation occurs on flat surfaces. This is generally attributed to the increased attractive forces between adsorbate molecules that occur as surfaces become highly curved, such as in a pore or capillary. This phenomenon is referred to as capillary condensation and is described by the Kelvin equation [19] ... 

Kelvin s equation determines the equilibrium vapor pressure over a curved meniscus of liquid ... 

Though not a general adsorption equilibrium model the Kelvin equation does provide the relationship between the depression of the vapor pressure of a condensable sorbate and the radius (r) of the pores into which it is condensing. This equation is useful for characterization of pore size distribution by N2 adsorption at or near its dew point. The same equation can also describe the onset of capillary condensation the enhancement of sorption capacity in meso- and macro-pores of formed zeolite adsorbents. 

As progressively higher pressures are used during N2 adsorption, capillary condensation will occur in pores that are increasingly larger. The Kelvin equation illustrates that the equilibrium vapor pressure is lowered over a concave meniscus of liquid, which is why N2 is able to condense in catalyst pores at pressures lower than the saturahon pressure ... 

As discussed in Section 1.4.2.1, the critical condensation pressure in mesopores as a function of pore radius is described by the Kelvin equation. Capillary condensation always follows after multilayer adsorption, and is therefore responsible for the second upwards trend in the S-shaped Type II or IV isotherms (Fig. 1.14). If it can be completed, i.e. all pores are filled below a relative pressure of 1, the isotherm reaches a plateau as in Type IV (mesoporous polymer support). Incomplete filling occurs with macroporous materials containing even larger pores, resulting in a Type II isotherm (macroporous polymer support), usually accompanied by a H3 hysteresis loop. Thus, the upper limit of pore size where capillary condensation can occur is determined by the vapor pressure of the adsorptive. Above this pressure, complete bulk condensation would occur. Pores greater than about 50-100 nm in diameter (macropores) cannot be measured by nitrogen adsorption. 

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) ... 

Use the Kelvin equation (Chapter 14.C.2) to show that it is true that the vapor pressure of pure water at 25°C is only 0.1% greater over a l-/u,m radius particle than over a flat surface, but 11% greater over a 0.01-/zm radius particle. The surface tension of water is 72 dyn cm 1 at 25°C. 

This Kelvin equation says that the vapor pressure over a droplet depends exponentially on the inverse of the droplet radius. Thus, as the radius decreases, the vapor pressure over the droplet increases compared to that over the bulk liquid. This equation also holds for water coating an insoluble sphere (Twomey, 1977). 

This has important implications for nucleation in the atmosphere. Condensation of a vapor such as water to form a liquid starts when a small number of water molecules form a cluster upon which other gaseous molecules can condense. However, the size of this initial cluster is very small, and from the Kelvin equation, the vapor pressure over the cluster would be so large that it would essentially immediately evaporate at the relatively small supersaturations found in the atmosphere, up to 2% (Prup-pacher and Klett, 1997). As a result, clouds and fogs would not form unless there was a preexisting particle upon which the water could initially condense. Such particles are known as cloud condensation nuclei, or CCN. 

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 ... 

In order to derive the Kelvin equation on thermodynamic grounds, consider the transfer of d moles of vapor in equilibrium with the bulk liquid at pressure Pq into a pore where the equilibrium pressure is P. This process consists of three steps evaporation from the bulk liquid, expansion of the vapor from Pq to P and condensation into the pore. The first and third of these steps are equilibrium processes and are therefore accompanied by a zero free energy change, whereas the free energy change for the second step is described by... 

EXAMPLE 6.2 Use of the Kelvin Equation for Determining Surface Tension. Figure 6.5 shows a plot of experimental data that demonstrates the validity of the Kelvin effect. Necks of liquid cyclohexane were formed between mica surfaces at 20°C, and the radius of curvature was measured by interferometry. Vapor pressures were measured for surfaces with different curvature. Use these data to evaluate 7 for cyclohexane. Comment on the significance of the fact that the linearity of Figure 6.5 extends all the way to a p/p0 value of 0.77. 

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... 

As indicated by the plots in Figure 10.13a, the vapor pressure of a liquid rises with temperature in a nonlinear way. A linear relationship is found, however, when the logarithm of the vapor pressure, In Pvap, is plotted against the inverse of the Kelvin temperature, 1 /T. Table 10.8 gives the appropriate data for water, and Figure 10.13b shows the plot. As noted in Section 9.2, a linear graph is characteristic of mathematical equations of the form y = mx + b. In the present instance, y = lnPvap, x = 1/T, m is the slope of the line (- AHvap/R), and b is the y-intercept (a constant, C). Thus, the data fit an expression known as the Clausius-Clapeyron equation. ... 

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 cause for this change in vapor pressure is the Laplace pressure. The raised Laplace pressure in a drop causes the molecules to evaporate more easily. In the liquid, which surrounds a bubble, the pressure with respect to the inner part of the bubble is reduced. This makes it more difficult for molecules to evaporate. Quantitatively the change of vapor pressure for curved liquid surfaces is described by the Kelvin equation ... 

An important application of the Kelvin equation is the description of capillary condensation. This is the condensation of vapor into capillaries or fine pores even for vapor pressures below Pequilibrium vapor pressure of the liquid with a planar surface. Lord Kelvin was the one who realized that the vapor pressure of a liquid 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 [17]. 

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. 

The pore volume and the pore size distribution can be estimated from gas adsorption [83], while the hysteresis of the adsorption isotherms can give an idea as to the pore shape. In the pores, because of the confined space, a gas will condense to a liquid at pressures below its saturated vapor pressure. The Kelvin equation (Eq. (4.5)) gives this pressure ratio for cylindrical pores of radius r, where y is the liquid surface tension, V is the molar volume of the liquid, R is the gas constant ( 2 cal mol-1 K-1), and T is the temperature. This equation forms the basis of several methods for obtaining pore-size distributions [84,85]. 

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

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