Kelvin s equation

Equation (6.50) is often referred to as the Thomson s (or Kelvin s) equation. As an example of the effect of this equation, the vapour pressure of a spherical droplet of molten Zn at the melting temperature is shown as a function of the droplet radius in Figure 6.14. 

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

Capillary condensation. We start with Kelvin s equation ... 

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. 

Langmuir s equation indicates that a droplet will not evaporate when S = pip, a 1.0. But according to the Kelvin equation, curvature effects will cause small droplets to evaporate, even when S exceeds 1.0. How can this apparent contradiction be resolved One way is to replace the p, - pt term in Eq. 15.14 with an equivalent term from Kelvin s equation which takes curvature into account. Recalling Kelvin s equation... 

In order to correct these discrepancies, Broekhoff and de Boer [5] generalized Kelvin s equation by taking pore shape into account, as well as the influence of surface curvature on the thickness of the adsorbed layer. The BdB method can be applied to both adsorption and desorption isotherms using four different pore models defined by a shape factor. Unfortunately, owing to computational difficulties, this last method, although more general, has been far less applied than the first two. 

Kelvin s equation for surface energy and temperature, 138 for vapour pressure over a curved surface, 366 Khaikin s viscosity equation, 107 Kistiakowsky s equation for specific cohesion, 152, 162... 

The study of this type of solid is based on the capillary condensation phenomenon and its quantitative expression is given by Kelvin s equation relating the adsorbate condensation pressure P to the radius of the pore r. ... 

The calculation methods for pore distribution in the microporous domain are still the subject of numerous disputes with various opposing schools of thought , particularly with regard to the nature of the adsorbed phase in micropores. In fact, the adsorbate-adsorbent interactions in these types of solids are such that the adsorbate no longer has the properties of the liquid phase, particularly in terms of density, rendering the capillary condensation theory and Kelvin s equation inadequate. The micropore domain (0.1 to several nm) corresponds to molecular sizes and is thus especially important for current preoccupations (zeolites, new specialised aluminas). Unfortunately, current routine techniques are insufficient to cover this domain both in terms of the accuracy of measurement (very low pressure and temperature gas-solid isotherms) and their geometrical interpretation (insufficiency of semi-empirical models such as BET, BJH, Horvath-Kawazoe, Dubinin Radushkevich. etc.). 

In deriving this equation the vorticity could not be neglected, as it was in deriving Kelvin s equation [3.6.63], because then there would have been no damping at all. For infinitely high elasticity, K - < , Reynolds derived ... 

Table 1 presents the textural parameters of the different materials studied using adsorption/ desorption isotherms before and after modifications or catalytic testing, corresponding to BET surface area, the total pore volume and the proportion of the micropore volume. The adsorption isotherm was found to be in agreement with the ones reported for MCM-41 materials with similar pore sizes [5]. Pore condensation of N, signified by a steep increase in the adsorbed volume in the N2 adsorption isotherm, was observed at a relative pressure (P/Po) of 0.26. Using Kelvin s equation, compensating for the multilayer adsorption the pore size was determined to be 2.5 nm. 

This refinement qualitatively captures that the chemical potential for a drop configuration is not Ap, = p — Pcoex = 0 but rather Ap = dAFjdn. The shift of the chemical potential increases with the interface tension and decreases as the size becomes larger (Kelvin s equation). 

The increased pressure of vapor of fine particles can be calculated from the Kelvin s equation [24] ... 

Wulff s theorem [25] states that cr/r is invariant for all faces. Therefore, the result obtained from the Kelvin s equation must be independent of the choice of a face. 

The present-day refined form of Kelvin s equation may be expressed as [45]... 

As a consequence of surface tension, there Is a balancing pressure difference across any curved Interface. Thus, the vapor pressure over a concave liquid surface will be smaller than that over a corresponding flat surface. This vapor pressure difference can be calculated from the Kelvin s equation ... 

If it is assumed that an absorbed layer of water exists before capillary condensation takes place and that this layer consists of ordered or oriented water molecules, then the contact angle in Kelvin s equation should be very close to zero. With zero contact angle, vapor pressures in the capillaries are calculated from Kelvin s equation for capillaries from 10 %. to 200 X, Table I. The vapor pressure of water below 0 C (15) is compared with the vapor pressures in the capillaries to obtain the freezing points. Figure 1 shows the relation between the freezing point depression and the capillary radius. 

Table I Capillary radii, vapor pressures and freezing point of water from Kelvin s equation.

We also need data for water. Kelvin s equation relates the water pressure P to the relative humidity. The water vapor density as function of P and T is... 

There are still other types of transformers. A particular constraint equation could look like dXj + dX 2 = 0. In this case, the conversion of different energy forms occurs. For example, in the derivation of Kelvin s equation of surface tension such a constraint appears. The volume of a sphere V = is connected to the area... 

The situation is similar as in the case of the derivation of Kelvin s equation for the surface tension of soap bubbles. 

Equilibrium is established only when the intensive variables corrected by the transformation factors are equal. We may not conclude what will happen if the system is not in equilibrium, whether it will move to an equilibrium state or rather away. We may think that a system is more far from equilibrium as the difference of intensive variables is more large. In the example above, we may take a, the width of the lamella w, the mass m, and g as constants. In this case, equilibrium is at crw = gm. Under laboratory conditions, we may easily change the width of the lamella w and the hanging mass m, but the other parameters are difficult to change. In this case, equilibrium is established only in one singular point. If only one parameter does not fit, equilibrium is not established. On the other hand, in the case of the soap bubble, inside there is a compressible gas. We have the well-known Kelvin s equation (for a double surface)... 

Even though Kelvin s equation is of questionable validity for narrow mesopores it was employed and the results are unexpectedly in good agreement with nitrogen results. 

This work has been continued by Rayleigh, who modified Kelvin s equation and derived expression [32] ... 

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