The Kelvin Equation

The vapor pressure (P) of a liquid over a meniscus that stretches across a cylindrical pore is given by (Adamson and Gast, 1997 Gregg and Sing, 1982 Rouquerol et al 1999)  

Here Po is the saturated vapor pressure at absolute temperature P, y is the surface tension, V is the molar volume of the liquid, r is the radius of the cylindrical pore, and R is the gas constant. 

Adsorbents Fundamentals and Applications, Edited By Ralph T. Yang ISBN 0-471-29741-0 Copyright 2003 John Wiley Sons, Inc. 

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 

Since an actual crystal will be polyhedral in shape and may well expose faces of different surface tension, the question is what value of y and of r should be used. As noted in connection with Fig. VII-2, the Wulff theorem states that 7,/r,- is invariant for all faces of an equilibrium crystal. In Fig. VII-2, rio is the 

The difference in chemical potential or dispersed phase solubility between different sized droplets was first elucidated by Lord Kelvin in 1871, viz  

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  

To derive the Kelvin equation we consider the Gibbs free energy of the liquid. The molar Gibbs free energy changes when the surface is being curved, because the pressure increases 

5 William Thomson, later Lord Kelvin, 1824-1907. Physics professor at the University of Glasgow. 

We have assumed that the molar volume remains constant, which is certainly a reasonable assumption because most liquids are practically incompressible for the pressures considered. For a spherical drop in its vapor, we simply have AGm = 2-yVm/r. The molar Gibbs free energy of the vapor depends on the vapor pressure / o according to 

The theory for adsorption of vapor on to a porous solid is derived, from thermodynamic considerations, and leads to the Kelvin equation which is exact in the limit for large pores. However it becomes progressively less accurate as the pore size decreases and breaks down when the pore size is so small that the molecular texture of the fluid becomes important. 

Although nitrogen adsorption isotherms are readily determined with high precision, the extraction of pore size distributions from the experimental data is problematical with small pores, due to the restricted range of validity of the Kelvin equation and the difficulty of assigning a correct value to the residual thickness, t, when the core of a pore empties. 

Below a critical size, pores do not undergo capillary condensation, but fill continuously as the pressure is increased without a discontinuity in the single pore adsorption isotherm [8]. 

For nitrogen adsorbing on porous carbon this critical pore size corresponds roughly to the conventional boundary between micropores and mesopores at 2nm [9]. The pore filling mechanism is not accounted for in the thermodynamic methods, which are therefore incapable of determining pore size distributions in the micropore range. 

The Kelvin equation may be derived as follows. Consider a liquid within a pore in equilibrium with its vapor. Let a small quantity, da moles, be distilled from the bulk of liquid outside the pore, where its equilibrium pressure is Pq, into the pore where its equilibrium pressure is P. The total increase in free energy SG is the sum of three parts evaporation of 5a moles of liquid at pressure Pq(5Gi), expansion of da moles of vapor from pressure Pq to pressure P (BG, condensation of Sa moles of vapor to liquid at pressure PiSG-j.  

It is often also important to consider the pressure of the vapour in eqnilibrinm with a liquid. It can be demonstrated that this pressure, at a given temperatnre, actually depends on the curvature of the liquid interface. This follows from the basic equations of thermodynamics, given in Chapter 3, which lead to the result that 

since both cases are at equilibrium, there must be an equivalent decrease in chemical potential of the liquid, that is, 

But from the Taplace equation the change in pressure of the liquid (assuming the meniscns is, for simplicity, spherical) is given by 

Another common method used to measure the surface tension of liquids is called the Wilhelmy plate . These methods use the force (or 

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 a pore the overlapping potentials of the walls more readily overcome the translational energy of an adsorbate molecule so that condensation will occur at a lower pressure in a pore than that normally required on an open or plane surface. Thus, as the relative pressure is increased, condensation will occur first in pores of smaller radii and will progress into the larger pores until, at a relative pressure of unity, condensation will occur on those surfaces where the radius of curvature is essentially infinite. Conversely, as the relative pressure is decreased, evaporation will occur progressively out of pores with decreasing radii. 

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 

When the adsorbate condenses in the pore it does so on a previously adsorbed film thereby decreasing the film-vapor interfacial area. The free energy change associated with the filling of the pore is given by 

Equations (8.2) and (8.3), when combined using the assumption of a zero wetting angle, yield 

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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. 

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. 

Derive Eq. XVII-136. Derive from it the Kelvin equation (Eq. Ill-18). 

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... 

Thomson s original equation is not suitable for direct application to adsorption data the form used by later workers, the Kelvin equation , is... 

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. 

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... 

Fig. 3.8 Relation between r of the Kelvin equation (Equation (3.20)) and the core radius r for a cylindrical pore with a hemispherical meniscus 6 is the angle of contact.

Now, in principle, the angle of contact between a liquid and a solid surface can have a value anywhere between 0° and 180°, the actual value depending on the particular system. In practice 6 is very difficult to determine with accuracy even for a macroscopic system such as a liquid droplet resting on a plate, and for a liquid present in a pore having dimensions in the mesopore range is virtually impossible of direct measurement. In applications of the Kelvin equation, therefore, it is almost invariably assumed, mainly on grounds of simplicity, that 0 = 0 (cos 6 = 1). In view of the arbitrary nature of this assumption it is not surprising that the subject has attracted attention from theoreticians. 

At the middle of the capillary where the eflect of the walls on chemical potential is negligible, the radius of curvature will be equal to r as calculated by the Kelvin equation (3.20) but it will become progressively larger as the wall is approached. 

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. 

These models, though necessarily idealized, are sufficiently close to the actual systems found in practice to enable useful conclusions to be drawn from a given Type IV isotherm as to the pore structure of a solid adsorbent. To facilitate the discussion, it is convenient to simplify the Kelvin equation by putting yVJRT = K, and on occasion to use the exponential form ... 

Use of the Kelvin equation for calculation of pore size distribution... 

Closer examination reveals however that the Brunauer method is not fundamentally distinct from methods based on the Kelvin equation. As pointed out by de Vleesschauwer, equations such as (3.52) are not really employed in the integral form, inasmuch as the aim is to evaluate the surface areas of successive groups of cores. In effect Equation (3.52) is used after adaptation to small rather than infinitesimal increments and becomes... 

Now the left-hand side of Equation (3.54) is equal to the hydraulic radius r of the group of cores (cf. Equation (3.49)) and the right-hand side by the Kelvin equation (cf. Equation (3.20)) is equal to rjl. Consequently,... 

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... 

Effect of an error of 0-05 K in temperature of the solid on the value of r calculated from the Kelvin equation... 

The evaluation of pore size distribution by application of the Kelvin equation to Type IV isotherms has hitherto been almost entirely restricted to nitrogen as adsorptive. This is largely a reflection of the widespread use of nitrogen for surface area determination, which has meant that both the pore size distribution and the specific surface can be derived from the same isotherm. 

In principle, the use of a suitable adsorptive should also make it possible, as Kamaukhov has pointed out, to reduce the magnitude of the f-correction, which is always a source of some uncertainty. From the Kelvin equation... 

The computation of mesopore size distribution is valid only if the isotherm is of Type IV. In view of the uncertainties inherent in the application of the Kelvin equation and the complexity of most pore systems, little is to be gained by recourse to an elaborate method of computation, and for most practical purposes the Roberts method (or an analogous procedure) is adequate—particularly in comparative studies. The decision as to which branch of the hysteresis loop to use in the calculation remains largely arbitrary. If the desorption branch is adopted (as appears to be favoured by most workers), it needs to be recognized that neither a Type B nor a Type E hysteresis loop is likely to yield a reliable estimate of pore size distribution, even for comparative purposes. 

A vast amount of research has been undertaken on adsorption phenomena and the nature of solid surfaces over the fifteen years since the first edition was published, but for the most part this work has resulted in the refinement of existing theoretical principles and experimental procedures rather than in the formulation of entirely new concepts. In spite of the acknowledged weakness of its theoretical foundations, the Brunauer-Emmett-Teller (BET) method still remains the most widely used procedure for the determination of surface area similarly, methods based on the Kelvin equation are still generally applied for the computation of mesopore size distribution from gas adsorption data. However, the more recent studies, especially those carried out on well defined surfaces, have led to a clearer understanding of the scope and limitations of these methods furthermore, the growing awareness of the importance of molecular sieve carbons and zeolites has generated considerable interest in the properties of microporous solids and the mechanism of micropore filling. 

In writing the present book our aim has been to give a critical exposition of the use of adsorption data for the evaluation of the surface area and the pore size distribution of finely divided and porous solids. The major part of the book is devoted to the Brunauer-Emmett-Teller (BET) method for the determination of specific surface, and the use of the Kelvin equation for the calculation of pore size distribution but due attention has also been given to other well known methods for the estimation of surface area from adsorption measurements, viz. those based on adsorption from solution, on heat of immersion, on chemisorption, and on the application of the Gibbs adsorption equation to gaseous adsorption. 

It would be difficult to over-estimate the extent to which the BET method has contributed to the development of those branches of physical chemistry such as heterogeneous catalysis, adsorption or particle size estimation, which involve finely divided or porous solids in all of these fields the BET surface area is a household phrase. But it is perhaps the very breadth of its scope which has led to a somewhat uncritical application of the method as a kind of infallible yardstick, and to a lack of appreciation of the nature of its basic assumptions or of the circumstances under which it may, or may not, be expected to yield a reliable result. This is particularly true of those solids which contain very fine pores and give rise to Langmuir-type isotherms, for the BET procedure may then give quite erroneous values for the surface area. If the pores are rather larger—tens to hundreds of Angstroms in width—the pore size distribution may be calculated from the adsorption isotherm of a vapour with the aid of the Kelvin equation, and within recent years a number of detailed procedures for carrying out the calculation have been put forward but all too often the limitations on the validity of the results, and the difficulty of interpretation in terms of the actual solid, tend to be insufficiently stressed or even entirely overlooked. And in the time-honoured method for the estimation of surface area from measurements of adsorption from solution, the complications introduced by... 

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