Kelvin equation condensation

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. 

If the adsorbent contains mesopores, capillary condensation will occur in each pore when the relative pressure reaches a value which is related to the radius of the pore by the Kelvin equation, and a Type IV isotherm will... 

As already indicated in Section 3.1, the study of mesoporous solids is closely bound up with the concept of capillary condensation and its quantitative expression in the Kelvin equation. This equation is, indeed, the basis of virtually all the various procedures for the calculation of pore size... 

As with all thermodynamic relations, the Kelvin equation may be arrived at along several paths. Since the occurrence of capillary condensation is intimately, bound up with the curvature of a liquid meniscus, it is helpful to start out from the Young-Laplace equation, the relationship between the pressures on opposite sides of a liquid-vapour interface. 

From the Kelvin equation it follows that the vapour pressure p over a concave meniscus must be less than the saturation vapour pressure p°. Consequently capillary condensation of a vapour to a liquid should occur within a pore at some pressure p determined by the value of r for the pore, and less than the saturation vapour pressure—always provided that the meniscus is concave (i.e. angle of contact <90°). 

It must always be borne in mind that when capillary condensation takes place during the course of isotherm determination, the pore walls are already covered with an adsorbed him, having a thickness t determined by the value of the relative pressure (cf. Chapter 2). Thus capillary condensation occurs not directly in the pore itself but rather in the inner core (Fig. 3.7). Consequently the Kelvin equation leads in the first instance to values of the core size rather than the pore size. The conversion of an r value to a pore size involves recourse to a model of pore shape, and also a knowledge of the angle of contact 0 between the capillary condensate and the adsorbed film on the walls. The involvement of 0 may be appreciated by consideration... 

In calculations of pore size from the Type IV isotherm by use of the Kelvin equation, the region of the isotherm involved is the hysteresis loop, since it is here that capillary condensation is occurring. Consequently there are two values of relative pressure for a given uptake, and the question presents itself as to what is the significance of each of the two values of r which would result from insertion of the two different values of relative pressure into Equation (3.20). Any answer to this question calls for a discussion of the origin of hysteresis, and this must be based on actual models of pore shape, since a purely thermodynamic approach cannot account for two positions of apparent equilibrium. 

The formation of a liquid phase from the vapour at any pressure below saturation cannot occur in the absence of a solid surface which serves to nucleate the process. Within a pore, the adsorbed film acts as a nucleus upon which condensation can take place when the relative pressure reaches the figure given by the Kelvin equation. In the converse process of evaporation, the problem of nucleation does not arise the liquid phase is already present and evaporation can occur spontaneously from the meniscus as soon as the pressure is low enough. It is because the processes of condensation and evaporation do not necessarily take place as exact reverses of each other that hysteresis can arise. 

Thus, as pointed out by Cohan who first suggested this model, condensation and evaporation occur at difi erent relative pressures and there is hysteresis. The value of r calculated by the standard Kelvin equation (3.20) for a given uptake, will be equal to the core radius r,. if the desorption branch of the hysteresis loop is used, but equal to twice the core radius if the adsorption branch is used. The two values of should, of course, be the same in practice this is rarely found to be so. 

In general, therefore, there are three processes, prior to the kind of capillary condensation associated with the hysteresis loop of a Type IV isotherm, which may occur in a porous body containing micropores along with mesoporesia primary process taking place in very narrow micropores a secondary, cooperative process, taking place in wider micropores, succeeded by a tertiary process governed by a modified Kelvin equation. 

The principle underlying surface area measurements is simple physisorb an inert gas such as argon or nitrogen and determine how many molecules are needed to form a complete monolayer. As, for example, the N2 molecule occupies 0.162 nm at 77 K, the total surface area follows directly. Although this sounds straightforward, in practice molecules may adsorb beyond the monolayer to form multilayers. In addition, the molecules may condense in small pores. In fact, the narrower the pores, the easier N2 will condense in them. This phenomenon of capillary pore condensation, as described by the Kelvin equation, can be used to determine the types of pores and their size distribution inside a system. But first we need to know more about adsorption isotherms of physisorbed species. Thus, we will derive the isotherm of Brunauer Emmett and Teller, usually called BET isotherm. 

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],... 

Table 16-4 shows the IUPAC classification of pores by size. Micropores are small enough that a molecule is attracted to both of the opposing walls forming the pore. The potential energy functions for these walls superimpose to create a deep well, and strong adsorption results. Hysteresis is generally not observed. (However, water vapor adsorbed in the micropores of activated carbon shows a large hysteresis loop, and the desorption branch is sometimes used with the Kelvin equation to determine the pore size distribution.) Capillary condensation occurs in mesopores and a hysteresis loop is typically found. Macropores form important paths for molecules to diffuse into a par-... 

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] ... 

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. 

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. 

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... 

Rarely have hysteresis loops been found which have not closed, at the low end, usually by relative pressures of 0.3. According to the Kelvin equation, pore radii corresponding to relative pressures less than 0.3 would be smaller than 15 A. Since monolayer formation is usually complete when the relative pressure reaches 0.3, the radius available for condensation would be diminished by the thickness of the monolayer or by about two molecular diameters. The radius of this center core would then be approximately one or two molecular diameters. It would be difficult to establish the validity of the Kelvin equation for pores this small, and it is entirely possible that adsorption and desorption in micropores proceed by the same path, thereby precluding hysteresis. 

A vapor will condense into pores of radius r according to the Kelvin equation, namely... 

For simplicity, let us assume that the liquid condensed in a pore has a surface that is part of a sphere of radius Rs, with Rs > r. For a spherical surface we may use the Kelvin equation (Equation (6.40) ) to calculate p/p0... 

A minus sign has been introduced in the Kelvin equation because the radius is measured outside the liquid in this application, whereas it was inside the liquid in the derivation of Chapter 6. In hysteresis, adsorption occurs at relative pressures that are higher than those for desorption. According to Equation (92), it is as if adsorption-condensation took place in larger pores than desorption-evaporation. Since the pore dimensions are presumably constant, we must seek some mechanism consistent with this observation to explain hysteresis. 

Specific surface areas of the materials under study were calculated using the BET method [22, 23]. Their pore size distributions were evaluated from adsorption branches of nitrogen isotherms using the BJH method [24] with the corrected form of the Kelvin equation for capillary condensation in cylindrical pores [25, 26]. In addition, adsorption energy distributions (AED) were evaluated from submonolayer parts of nitrogen adsorption isotherms using the algorithm reported in Ref. [27],... 

Figure 1. (a) Experimental relations between the capillary condensation pressure and the pore diameter (hollow circles) and between the capillary evaporation pressure and the pore diameter (filled circles) for nitrogen adsorption at 77 K. The dashed line corresponds to the Kelvin equation with the statistical film thickness correction. The solid line corresponds to Eq. 2 derived using the KJS approach, (b) Relation between pore diameters calculated on the basis of Eq. 1 and the KJS-calibrated BJH algorithm using nitrogen adsorption data at 77 K. 

Since capillary condensation pressure was found to be a gradually increasing function of the pore size, it was possible to find an empirical equation for this relation (see Fig. la). This equation [17] was similar to the well-known Kelvin equation for cylindrical pores [25] ... 

According to the classical treatment of Cohan [8], which is the basis of the conventional BJH method [14], capillary condensation in an infinite cylindrical pore is described by the Kelvin equation using cylindrical meniscus, while desorption is associated with spherical meniscus. In large pores the following asymptotic equation is expected to be valid [8] Pd/Po = (PA/P0)2, where Pd/Po and Pa/Po are the relative pressures of the desorption and adsorption, respectively. An improved treatment [9-11, 13], originated from Deijaguin [9], takes into... 

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]. 

Capillary condensation can be illustrated by the model of a conical pore with a totally wetting surface (Fig. 2.12). Liquid will immediately condense in the tip of the pore. Condensation continues until the bending radius of the liquid has reached the value given by the Kelvin equation. The situation is analogous to that of a bubble and we can write... 

Use the Kelvin equation to calculate the pore radius which corresponds to capillary condensation of nitrogen at 77 K and a relative pressure of 0.5. Allow for multilayer adsorption on the pore wall by taking the thickness of the adsorbed layer on a non-porous solid as 0.65 nm at this relative pressure. List the assumptions upon which this calculation is based. For nitrogen at 77 K, the surface tension is 8.85 mN m-1 and the molar volume is 34.7 cm3 mol-1. 

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]. 

Most models to calculate the pore size distributions of mesoporous solids, are based on the Kelvin equation, based on Thomson s23 (later Lord Kelvin) thermodynamical statement that the equilibrium vapour pressure (p), over a concave meniscus of liquid, must be less than the saturation vapour pressure (p0) at the same temperature . This implies that a vapour will be able to condense to a liquid in the pore of a solid, even when the relative pressure is less than unity. This process is commonly called the capillary condensation. 

When mesopores are present, capillary condensation will occur according to the Kelvin equation. A t-plot will therefore show an upward deviation, starting at the relative pressure at which the finest pores are just being filled. 

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