Pressure Kelvin equation

In this model also the decrease of the pore radius due to the formation of an adsorbed layer is incorporated. Flow 1 in Fig. 9.9 is the case of combined Knudsen molecular diffusion in the gas phase and multilayer (surface) flow in the adsorbed phase. In case 2, capillary condensation takes place at the upstream end of the pore (high pressure Pi) but not at the downstream end (P2), and in case 3 the entire capillary is filled with condensate. The crucial point in cases 3 and 4 is that the liquid meniscus with a curved surface not only reduces the vapour pressure (Kelvin equation) but also causes a hydrostatic pressure difference across the meniscus and so causes a capillary suction pressure Pc equal to... 

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. 

The equilibrium vapour pressure, P, over a curved surface is defined by tlie Kelvin equation... 

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 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. 

Equation (3.20) is conventionally termed the Kelvin equation. The tacit assumption is made at the integration stage that K is independent of pressure, i.e. that the liquid is incompressible. 

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. 

At the upper end of the pore size range there is no theoretical limit to the applicability of the Kelvin equation to adsorption isotherms so long as 9 < 90°. There is however a practical limitation, the nature of which may be gathered from Table 3.8 which gives the relative pressures corresponding to... 

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 ... 

Pore volumes are determined by forcing N2 (for micro- and mesoporous materials) or Hg (macroporous materials) under pressure into the pores. The quantity of N2 or Hg entering the catalyst is directly related to the pressure and the radius of the pores. The Kelvin equation describes this ... 

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 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] ... 

Thus, either type I or type IV isotherms are obtained in sorption experiments on microporous or mesoporous materials. Of course, a material may contain both types of pores. In this case, a convolution of a type I and type IV isotherm is observed. From the amount of gas that is adsorbed in the micropores of a material, the micropore volume is directly accessible (e.g., from t plot of as plot [1]). The low-pressure part of the isotherm also contains information on the pore size distribution of a given material. Several methods have been proposed for this purpose (e.g., Horvath-Kawazoe method) but most of them give only rough estimates of the real pore sizes. Recently, nonlocal density functional theory (NLDFT) was employed to calculate model isotherms for specific materials with defined pore geometries. From such model isotherms, the calculation of more realistic pore size distributions seems to be feasible provided that appropriate model isotherms are available. The mesopore volume of a mesoporous material is also rather easy accessible. Barrett, Joyner, and Halenda (BJH) developed a method based on the Kelvin equation which allows the calculation of the mesopore size distribution and respective pore volume. Unfortunately, the BJH algorithm underestimates pore diameters, especially at... 

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... 

---