Applications of the Kelvin equation

The Kelvin equation has numerous applications, e.g. in the stability of colloids (Ostwald ripening, see below), supersaturation of vapours, atmospheric chemistry (fog and rain droplets in the atmosphere), condensation in capillaries, foam stability, enhanced oil recovery and in explaining nucleation phenomena (homo- and heterogeneous). The Kelvin (as well as the Gibbs equations, see Equation 4.7a) are also valid for solids/solid-liquid surfaces, and they can be used for estimating the surface tensions of solids. We discuss hereafter several applications of the Kelvin equation. 

An important application of the Kelvin equation, of great importance to oil applications and environmental engineering (processes in soils), is the presence of liquids in capillaries. As there is a vapour pressure decrease outside the concave surface (negative curvature), we have condensation of the liquids inside the cracks, a phenomenon called capillary condensation. In this case the vapour pressure is reduced relative to that of a flat surface. Liquids that wet the solid wiU therefore condense into the pores at pressures below the equilibrium vapour pressure corresponding to a plane surface. 

When the radius is above 1000 nm, the normal vapour pressure is not affected, whereas for radii below 100 nm, and especially below 10 nm, there is an appreciable increase of the vapour pressure compared to the vapour pressure of flat surface. For liquids of high molar volume and surface tension, e.g. mercury, the effect is even more significant. 

In conclusion, liquids in capillaries condense and evaporate harder than in the open space. 

The Kelvin equation points out the difBcullies associated with the formation of a new phase. Suppose we start with a vapour phase and compress it to the saturation pressure, Pq. Formation of the liquid phase requires molecules coming together and coalescing. It is unlikely that a very large number of molecules will come together at once to form a drop. Even the formation of a small drop with r = 100 nm involves the simultaneous aggregation of 1.4 x 10 molecules What is likely to happen is that a few molecules 

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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 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 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. 

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. 

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 aim of this chapter is to discuss in general terms the use of adsorption measurements for the characterization of mesoporous solids (i.e. adsorbents having effective pore widths in the approximate range of 2-50 nm). Our approach here is mainly along classical lines and is based on the concept of capillary condensation and the application of the Kelvin equation. However it is appropriate to include a brief discussion of the relevant aspects of network percolation and density functional theory. 

Over the period 1945-1970 many different mathematical procedures were proposed for the derivation of the pore size distribution from nitrogen adsorption isotherms. It is appropriate to refer to these computational methods as classical since they were all based on the application of the Kelvin equation for the estimation of pore size. Amongst the methods which remain in current use were those proposed by Barrett, Joyner and Halenda (1951), apparently still the most popular Cranston and Inkley... 

According to the classical interpretation, the phenomenon is explained by application of the Kelvin equation this gives the relationship between the relative pressure and the meniscus curvature of liquid condensed in a pore. When the Kelvin equation is used to evaluate the pore size, it is tacitly assumed that there is a direct relationship between the meniscus curvature and the dimensions of the mesopores. In principle, it is also necessary to define the pore shape. 

This equation can be used to explain that stable adsorption on the inner wall of a capillary tube is possible up to a certain critical thickness. Capillary condensation starts from this critical thickness. It appears that capillary condensation in the cylindrical pores with radii between 2 and 7 run is well described by the corrected Kelvin equation. The deviation from the ideal behavior can amount to up to +20%. Nevertheless, it is difficult to estimate the limits of the applicability of the Kelvin equation. It seems likely that distortions of the meniscus in small pores may occur and that the local pore geometry may have a marked influence [1,15]... 

A complication arises because isotherms display hysteresis as shown in Fig. 7.17. Much has been written on the origin of these hysteresis cur xs, which provide information about the shape of pores. There are two extremes. Cylindrical or slit-shaped pores (Fig. 7.18a) give moderate hysteresis, such as Fig. 7.17a. The adsorption branch of the isotherm results from adsorbed layers on the cylindrical walls, which thicken until a miniscus forms. Applicability of the Kelvin equation to such a microscopic system is doubtful. Upon desorption, however, evaporation occurs at the larger miniscus and the Kelvin equation is valid. In this case, pore size distributions from the desorption branch arc recommended as more reliable. The other extreme... 

The occurrence of hysteresis in adsorption phenomena, caused by capillary condensation, has led to the application of the Kelvin equation for the desorption branch of a complete adsorption isotherm and thus to a complete distribution curve of the widths of the various pores as a function of their volumes (49). The results are mostly expressed in the form of radii of cylindrical pores. The method may be applied for radii between 20 and 300 A. The figures obtained have to be corrected for the thickness of the multilayer adsorption on the surface of the nonfilled capillaries (50), and various calculation methods have recently been published (51) which need not be discussed here, since Wheeler (5 ) gave an excellent review recently. 

Fisher, L. R. and Israelachvili, J. N., Experimental studies on the applicability of the Kelvin equation to highly curved concave menisci, J. Colloid Interface ScL, 80, 528-541 (1981). 

Nitrogen Capillary Condensation. Application of the Kelvin equation generally assumes circular pores but in reality for non-circular shapes the Kelvin equation evaluates a volume-surface capillary ratio. In view of the uncertainty of the Kelvin equation in terms of the variation observed between the adsorbed and the bulk physical properties of nitrogen, together with the inconsistency of t-curves and BET coefficients excessive refinement of the pore size distribution model has little warranty (13,17) and thus the modelless treatment was chosen (18). 

Example 4.2. Application of the Kelvin equation to water droplets. 

Mesopores, also known as intermediate or transition pores, having widths between 2 and 50 nm these mark the limit of applicability of the Kelvin equation. 

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