Kelvin equation derivation

With the preceding introduction to the handling of surface excess quantities, we now proceed to the derivation of the third fundamental equation of surface chemistry (the Laplace and Kelvin equations, Eqs. II-7 and III-18, are the other two), known as the Gibbs equation. 

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). 

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. 

As an example, the ratio of the equilibrium vapor pressures for water, Pi6 and water. Pig, depends on temperature and is expressed by the following equation, derived from Faure (1977) (temperature is in kelvins) ... 

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. 

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) ... 

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... 

In our derivation of the Kelvin equation, the radius is measured in the liquid. For a gas bubble in a liquid, the same equilibrium is involved, but the bubble radius is measured on the opposite side of the surface. As a consequence, a minus sign enters Equation (40) when it is applied to bubbles. Since y is on the order of millijoules while RT is on the order of joules, Equation (40) predicts that In (p/p0) is very small. It is important to realize, however, that y/R is also divided by the radius of the spherical particle and therefore becomes more important as Rs decreases. For water at 20°C, the Kelvin equation predicts values of p/p0 equal to 1.0011, 1.0184, 1.1139, and 2.94 for drops of radius 10 6, 10"7, 10 8, andl0 9m, respectively. For bubbles of the same sizes in liquid water, p/p0 equals 0.9989, 0.9893, 0.8976, and 0.339. These calculations show that the effect of surface curvature, while relatively unimportant even for particles in the micrometer range, becomes appreciable for very small particles. 

What assumptions have been made in the derivation of the Kelvin equation in the text How restrictive are these assumptions ... 

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. 

Laplace equation A thermodynamic derivation Determining surface tension from the Kelvin equation Heat of immersion from surface tension and contact angle Surface tension and the height of a meniscus at a wall Interfacial tensions from the Girifalco-Good-Fowkes equation... 

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. 

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... 

A simple derivation of the Kelvin equation is presented by Broekhoff and van Dongen [16]. Imagine a gas B in physical adsorption equilibrium above a flat, a convex, and a concave surface, respectively (see Fig. 12.8). Considering a transfer of dN moles of vapour to the adsorbed phase at constant pressure and temperature, equilibrium requires that there will be no change in the free enthalpy of the system. 

Around 1967, Broekhoff and de Boer [16], following Derjaguin [17], pointed out that the supposition introduced in the derivation of the Kelvin equation, viz. the equality of the thermodynamic potential of the adsorbed multilayer to the thermodynamic potential of the liquified gas (see Eqn. 12.28), cannot be correct. This can be seen immediately from an inspection of the common t curve (Fig. 12.5) at each t value lower than, say, 2 run, the relative equilibrium pressure is lower than 1, the equilibrium pressure of the liquefied gas. 

Introduction of Eqn. 12.35 into the derivation of the Kelvin equation, along the same lines of reasoning as following in Section 12.10, gives the corrected Kelvin equation ... 

Here ps is the saturated vapor pressure at temperature T, y the surface tension, Vm the molar volume of the liquid, and the curvature radius r is conventionally taken as negative for concave interfaces. Kelvin equation for a non-ideal multicomponent mixture was derived by Shapiro and Stenby (1997). 

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... 

Evans and his co-workers (1986) have shown that a statistical mechanical treatment may be used to derive the Kelvin equation. This approach, which was designed to avoid the difficulties associated with the exact form of the meniscus, led to a new mathematical description of the effect of confining a fluid in pores of different size and shape on its liquid-gas coexistence curve. An equation of the same mathematical form as Equation (7.10) was obtained, provided that the undersaturation was not too great, i.e. that pjp° was not too low. It was shown that this simple equation becomes less accurate as rK is reduced and is no longer applicable beyond a capillary critical point . At a lower rK or higher T, the two-phase relation fails because of the existence of only one stable fluid configuration in the pore. 

As Ic = 2ji / A this is the first example of a dispersion equation, giving the wave length as a function of frequency. It is historically interesting that [3.6.63] was already derived by Lord Kelvin very long ago. Hence it is called the Kelvin equation (for damping). 

As with experimental sorption data, it is possible to obtain the ratio Ff as a function of pressure (and thus pore size via the Kelvin equation) for simulations of the nitrogen sorption experiment on the model grids derived from NMR images. In order to determine the effect, if any, of the macroscopic heterogeneities (non-randomness) in the spatial distribution of voidage and pore size on the nitrogen desorption isotherm it is... 

In the case of SiC12T sample and benzene as a wetting liquid the desorption curve is smooth without characteristic step. It may be explained by not satisfactory wetting of this silica by benzene and restricted penetration of narrow pores. Observed effect is confirmed by small pore volume for the same sample, derived from benzene desorption data. The localization of desorption steps on temperature axis corresponds to emptying of pores with dominant share in total pore volume. Converting the temperature into pore radius, by using the Kelvin equation, the dimensions of pores and pore size distributions PSD, AV Affp vs. R, may be calculated in the manner described earher [9]. 

In this paper, a modified HK method is presented which accounts for spatial variations in the density profile of a fluid (argon) adsorbed within a carbon slit pore. We compare the pore width/filling pressure correlations predicted by the original HK method, the modified HK method, and methods based upon statistical thermodynamics (density functional theory and Monte Carlo molecular simulation). The inclusion of the density profile weighting in the HK adsorption energy calculation improves the agreement between the HK model and the predictions of the statistical thermodynamics methods. Although the modified Horvath-Kawazoe adsorption model lacks the quantitative accuracy of the statistical thermodynamics approaches, it is numerically convenient for ease of application, and it has a sounder molecular basis than analytic adsorption models derived from the Kelvin equation. 

For deriving the Kelvin equation one uses the changes in the free energy AG upon changing the number of moles in the gas phase and the condensed phase. When dN moles of a substance change between the condensed phase and the gas phase at equilibrium conditions we can write... 

As a conclusion, one can say that the thermogravimetric technique is a usefull method in the investigations of the porosity of solids. Our analysis, based on the Kelvin equation, of the thermogravimetric curves for silica gel wetted with liquid n-butanol and carbon tetrachloride leads to core/pore size distributions curves which are similar, but not identical in shape to the pore size distribution curves derived by standard procedure from low temperature nitrogen adsorption/desorption isotherms. The linear heating mode is... 

Classical thermodynamic models of adsorption based upon the Kelvin equation [21] and its modihed forms These models are constructed from a balance of mechanical forces at the interface between the liquid and the vapor phases in a pore filled with condensate and, again, presume a specihc pore shape. Tlie Kelvin-derived analysis methods generate model isotherms from a continuum-level interpretation of the adsorbate surface tension, rather than from the atomistic-level calculations of molecular interaction energies that are predominantly utihzed in the other categories. 

The classical model for describing adsorption in simple geometric pores is based on the Kelvin equation [125], which is derived from the condition of mechanical equilibrium for a curved interface between coexisting vapor and liquid phases in a pore. If the adsorbed liquid completely wets the pore walls, as shown in Fig. 17a, and the vapor phase is assumed to be an ideal gas, then mechanical equilibrium requires that... 

Nitrogen adsorption isotherms, as shown in Fig. 7.9, when extended to p/po values approaching unity, include the region where nitrogen condenses in the pores. Figure 7.16 demonstrates the region for mesopore condensation. This phenomenon is governed by the Kelvin equation, first derived for capillary condensation ... 

Equation (1) is a useful route to calculating headspace concentrations above a pure substance from vapour pressures c is the gas phase concentration in g 1 1, p° is the saturated vapour pressure in mmHg, and T is the temperature in Kelvin. Equation (2) is the same equation restated in terms of concentration m in mol 1 1 at a temperature of 25 °C for cases where the molecular mass is unknown. These equations derive directly from the ideal gas equation. 

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