Porous solids Kelvin equation

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... 

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] ... 

Use the Kelvin equation to calculate the pore radius which corresponds to capillary condensation of nitrogen at 77 K and a relative pressure of 0.5. Allow for multilayer adsorption on the pore wall by taking the thickness of the adsorbed layer on a non-porous solid as 0.65 nm at this relative pressure. List the assumptions upon which this calculation is based. For nitrogen at 77 K, the surface tension is 8.85 mN m-1 and the molar volume is 34.7 cm3 mol-1. 

Over the years, vapour adsorption and condensation in porous materials continue to attract a great deal of attention because of (i) the fundamental physics of low-dimension systems due to confinement and (ii) the practical applications in the field of porous solids characterisation. Particularly, the specific surface area, as in the well-known BET model [I], is obtained from an adsorbed amount of fluid that is assumed to cover uniformly the pore wall of the porous material. From a more fundamental viewpoint, the interest in studying the thickness of the adsorbed film as a function of the pressure (i.e. t = f (P/Po) the so-called t-plot) is linked to the effort in describing the capillary condensation phenomenon i.e. the gas-Fadsorbed film to liquid transition of the confined fluid. Indeed, microscopic and mesoscopic approaches underline the importance of the stability of such a film on the thermodynamical equilibrium of the confined fluid [2-3], In simple pore geometry (slit or cylinder), numerous simulation works and theoretical studies (mainly Density Functional Theory) have shown that the (equilibrium) pressure for the gas/liquid phase transition in pores greater than 8 nm is correctly predicted by the Kelvin equation provided the pore radius Ro is replaced by the core radius of the gas phase i.e. (Ro -1) [4]. Thirty year ago, Saam and Cole [5] proposed that the capillary condensation transition is driven by the instability of the adsorbed film at the surface of an infinite... 

In the following discussion we will consider the application of percolation theory to describing desorption of condensate from porous solids. In Sections III,A-III,C we briefly recall types of adsorption isotherms, types of hysteresis loops, and the Kelvin equation. The matter presented in these sections is treated in more detail in any textbook on adsorption [see, e.g., the excellent monographs written by Gregg and Sing (6) and by Lowell and Shields (49) Sections III,D-III,H are directly connected with percolation theory. In particular, general equations interpreting the hysteresis loop are... 

Figure 8 PSDs for model porous silica glasses [10]. A, B, C, D are sample glasses prepared by Quench Molecular Dynamics, and differ in mean pore size and porosity. The solid curves are the exact geometric PSDs for the models the dashed lines are PSDs predicted by analyzing simulated nitrogen adsorption isotherms for these materials using the BJH method (a form of the modified Kelvin equation). The BJH method gives mean pore sizes that are too small by about I nm in each case.

Porous solids having a regular pore structure have gathered much attention in the fields of chemistry and physics[l-7]. Those solids are expected to elucidate the interaction of gas with pores from the microscopic level. lUPAC classified pores into micropores, mesopores, and macropores using pore width w ( micropores w< 2nm, mesopores 2 nm < w< 50 nm, and macropores w> 50 nm)[8]. Physical adsorption occurs by the mechanism inherent to the pore width. Vapor is adsorbed on the mesopore wall by multilayer adsorption in the low pressure range and then vapor is condensed in the mesopore space below the saturated vapor pressure P . This is so called capillary condensation. Capillary condensation has been explained by the Kelvin equation given by eq. (1). 

In the case of adsorption of a vapor by a porous material, a three phase system in terms of SAS is produced pore/adsorbed film or capillary condensed vapor/solid. Since the s.l.d. of H2O and D2O are known while the pore space s.l.d. equals to zero, contrast matching conditions are achieved if an appropriate mixture of H2O/D2O that has the same s.l.d. as the solid is used as the adsorbate. In this case the adsorbed film as well as the condensed cluster of pores will cease to act as scatterers, and only the remaining empty pores will produce measurable scattering. In terms of SANS, contrast matching reduces the solid/film/pore system to a binary one [1]. By determining a number of scattering curves corresponding to the same sample equilibrated at various relative pressures, for both the adsorption and desorption branches of the adsorption isotherm, a correlation of the two methods could be possible. If the predictions of the Kelvin equation are in accordance with the SAS analysis, a reconstruction of the adsorption isotherm can be obtained from the SAS data [2]. 

On the other hand, the Kelvin equation has been extensively used in research on gas adsorption onto porous solids (see Sections 8.4 and 8.5) and capillary condensation. 

Lord Kelvin realized that, instead of completely drying out, moisture is retained within porous materials such as plants and vegetables or biscuits at temperatures far above the dew point of the surrounding atmosphere, because of capillary forces. This process was later termed capillary condensation, which is the condensation of any vapor into capillaries or fine pores of solids, even at pressures below the equilibrium vapor pressure, Pv. Capillary condensation is said to occur when, in porous solids, multilayer adsorption from a vapor proceeds to the point at which pore spaces are filled with liquid separated from the gas phase by menisci. If a vapor or liquid wets a solid completely, that is the contact angle, 0= 0°, then this vapor will immediately condense in the tip of a conical pore, as seen in Figure 4.8 a. The formation of the liquid in the tip of the cone by condensation continues until the cone radius, r, reaches a critical value, rc, where the radius of curvature of the vapor bubble reaches the value given by the Kelvin equation (r = rc). Then, for a spherical vapor bubble, we can write... 

Determination of the meximanr pore size in a porous solid can be performed using the Kelvin equation 1... 

It has long been known that the freezing point of a liquid inside a porous solid is lower than that of the bulk liquid. A number of expressions may be derived to predict the actual freezing point based on the Kelvin equation. One such... 

Adsorption of nonionic surfactants on porous solids has been studied by Huinink et al. in a series of p ers [ 149,150]. They elaborated a thermodynamic approach that accounts for the major features of experimental adsorption isotherms. It is a very well known fact that during the adsorption of nonionic surfactants there is a sharp step in the isotherm. This step is interpreted as a change from monomer adsorption to a regime where micelle adsorption takes place. Different surfactants produce the step in a different concentration range. The step is more or less vertical depending on the adsorbate. The thermodynamic analysis made by Huinink et al. is based on the assumption that the step could be treated as a pseudo first order transition. Their final equation is a Kelvin-like one, which shows that the change in chemical potential of the phase transition is proportional to the curvature constant (Helmholtz curvature energy of the surface). 

The capillary condensation phenomenon was discovered by Zsigismody [139], who investigated the uptake of water vapour by silica materials. Zsigismody proved that the condensation of physicosorbed vapours can occur in narrow pores below the standard saturated vapour pressure. The main condition for the capillary condensation existence is the presence of liquid meniscus in the adsorbent capillaries. As it is known, the decrease of saturated vapour pressure takes place over the concave meniscus. For cylindrical pores, with the pore width in the range 2-50 nm, i.e., for the mesopores, this phenomenon is relatively well described by the Kelvin equation [14]. This equation is still widely applied for the pore size analysis, but its main limitations remain unresolved. Capillary condensation is always preceded by mono- and/or multilayer adsorption on the pore walls. It means that this phenomenon plays an important, but secondary role in comparison with the physical adsorption of gases by porous solids. Consequently, the true pore width can be assessed if the adsorbed layer thickness is known. 

Kelvin equation thus suggests that if liquid is present in a porous material, such as cement, then the difference in vapor pressure exists between two pores of different radii. Infact, in cement industry, one reduces the value of y, such that the vapor pressure difference between pores of different radii is reduced. Similar consequence of vapor pressure exists when two solid crystals of different size are concerned. The smaller sized crystal will exhibit higher vapor pressure and will also result in fester solubility rate. 

Halenda) or modem DFT (density fimctional theory) methods also allow to evaluate the pore size distribution from the same data. Sample preparation, highly defined experimental conditions, and very precise pressure measurements are the key factors for accurate surface analysis. While sorption experiments probe pores in the size range from approximately 0.3 to 100 nm, mercury porosim-etry is the method of choice to determine the total pore volume and the pore size distribution from 5 nm up to 500 p,m. The method pushes liquid mercury under high pressure into the porous material and the Hg volume accommodated in the solid is monitored as a function of pressure. Following the Kelvin equation, a higher pressure is necessary to push the mercury into smaller pores. Therefore, from the amount of mercury infiltrated into the solid as a function of pressure the pore size distribution can be obtained. 

Capillary condensation in a porous solid is a secondary process since it can only occur after an adsorbed layer has been formed on the pore walls. According to the Kelvin equation the relative pressure at which condensation occurs is ... 

Capillary cohesion phenomenon — Kelvin equation. The theory of capillary cohesion and Kelvin equation are the theoretical basis of physical vapor adsorption. When the steam of adsorbate contacts with porous solid surface, it will form liquid film of the adsorbate on the surface adsorption field. The films in the pore bend variously with the pore diameter, while the films in the outer surface of particles are relatively flat. The film thickness of liquid of adsorption increases with increase in vapor pressure. When it reaches a certain moment, the gravity between the curved liquid surfaces sufficiently liquidity the vapor from gaseous automatically, and completely fill the pores. This phenomenon is known as capillary cohesion. 

The physical adsorption isotherm in the p/po range for multilayer adsorption usually shows a hysteresis on porous solids, as indicated in Figure 1.17. Although several theories have been put forward to explain the hysteresis, it is not yet totally resolved. However, the theory advanced by Cohan (1944) is helpful in understanding why the desorption isotherm is used for the determination of pore size distributions. He argues that, upon adsorption, pores are not filled vertically but rather radially. On desorption, on the other hand, pores are emptied vertically, for which the Kelvin equation of Eq. 1.55 applies. 

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. 

When dealing with porous solids, it is common to classify the pores according to their size into three main categories. This classification lacks precision and universality but a tentative one, usually accepted, includes mieropores with sizes below 2 nm, mesopores ranging from 2 to 50 nm, and finally maeropores whose sizes are larger than 50 nm. The reason for this classification is the applieability of the Kelvin equation with N2 at 77 K as adsorbate. 

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