Bulk isothermal

A MC study of adsorption of living polymers [28] at hard walls has been carried out in a grand canonical ensemble for semiflexible o- 0 polymer chains and adsorbing interaction e < 0 at the walls of a box of size C. A number of thermodynamic quantities, such as internal energy (per lattice site) U, bulk density (f), surface coverage (the fraction of the wall that is directly covered with segments) 9, specific heat C = C /[k T ]) U ) — U) ), bulk isothermal compressibility... 

These are the 2D analogs of the bulk isothermal compressibility and bulk modulus, respectively. The scaling concepts introduced above clearly make sense in terms of the compressibility/elasticity as well. For polymers where the interface is a good solvent, the lateral modulus, sometimes called static di-lational elasticity, is small whereas it becomes larger as the interface becomes poorer. 

The above discussion leads to the consideration of another empirical rule involving the (bulk) isothermal compressibility. It is defined as... 

Isofliermal.— With bulk isothermal crystallization, the conversion-time inter-... 

From the mechanism it can be seen that material is added to or depleted from the gas phase by adsorption/desorption with the exception of hydrogen which is assumed to be consumed directly from the gas phase. In formulating a theoretical model for the system it was assumed that the adsorption/desorption kinetics played an important role in the dynamics of the periodic operation and these kinetics were incorporated into the dynamic equations. Furthermore, it was assumed that there was neither bulk nor pore diffusional heat and mass transfer resistances, that the reactor was isothermal (both in the bulk gas phase and locally) and that the flow pattern in the reactor could be approximated by plug flow. Most of the above assumptions (i.e. plug flow, bulk isothermal conditions, no pore diffusion limitations) could be... 

A comparison was made between Eq 8 and 6, the latter having been generated from the bulk, isothermal fatigue of a similar Sn-Ag-Cu alloy. The two equations conhrm that the fatigue be-... 

Figures shows the bulk isothermal crystallization curves for H84, which is typical for all the copolymers. The relative crystallinity Xt, is defined as, ...

The limiting compression (or maximum v value) is, theoretically, the one that places the film in equilibrium with the bulk material. Compression beyond this point should force film material into patches of bulk solid or liquid, but in practice one may sometimes compress past this point. Thus in the case of stearic acid, with slow compression collapse occurred at about 15 dyn/cm [81] that is, film material began to go over to a three-dimensional state. With faster rates of compression, the v-a isotherm could be followed up to 50 dyn/cm, or well into a metastable region. The mechanism of collapse may involve folding of the film into a bilayer (note Fig. IV-18). 

The adsorption isotherm corresponding to Eq. X-51 is of the shape shown in Fig. X-1, that is, it cannot explain contact angle phenomena. The ability of a liquid him to coexist with bulk liquid in a contact angle situation suggests that the him structure has been modihed by the solid and is different from that of the liquid, and in an enmirical way, this modihed structure corresponds to an effective vapor pressure F , F representing the vapor pressure that bulk liquid would have were its structure that of the... 

Polymers typically exhibit a high-affinity adsorption isotherm as shown in Fig. XI-5 here the adsorbed amount increases very rapidly with bulk concentration and then becomes practically independent of concentration. 

This description is traditional, and some further comment is in order. The flat region of the type I isotherm has never been observed up to pressures approaching this type typically is observed in chemisorption, at pressures far below P. Types II and III approach the line asymptotically experimentally, such behavior is observed for adsorption on powdered samples, and the approach toward infinite film thickness is actually due to interparticle condensation [36] (see Section X-6B), although such behavior is expected even for adsorption on a flat surface if bulk liquid adsorbate wets the adsorbent. Types FV and V specifically refer to porous solids. There is a need to recognize at least the two additional isotherm types shown in Fig. XVII-8. These are two simple types possible for adsorption on a flat surface for the case where bulk liquid adsorbate rests on the adsorbent with a finite contact angle [37, 38]. 

The first term on the right is the common inverse cube law, the second is taken to be the empirically more important form for moderate film thickness (and also conforms to the polarization model, Section XVII-7C), and the last term allows for structural perturbation in the adsorbed film relative to bulk liquid adsorbate. In effect, the vapor pressure of a thin multilayer film is taken to be P and to relax toward P as the film thickens. The equation has been useful in relating adsorption isotherms to contact angle behavior (see Section X-7). Roy and Halsey [73] have used a similar equation earlier, Halsey [74] allowed for surface heterogeneity by assuming a distribution of Uq values in Eq. XVII-79. Dubinin s equation (Eq. XVII-75) has been mentioned another variant has been used by Bonnetain and co-workers [7S]. 

A rather simpler situation arises when the bulk concentrations are sufficiently small that the adsorption isotherms approach linearity. Then (7,4), for example, shows that... 

The most direct test of the tensile strength hypothesis would be to compare the value of Tq calculated from the closure point of the isotherm by Equation (3.61) with the tensile strength of the bulk liquid determined directly. Unfortunately, experimental measurement of the tensile strength is extremely difficult because of the part played by adventitious factors such as the presence of solid particles and dissolved gases, so that the values in the literature vary widely (between 9 and 270 bar for water at 298 K, for example). 

However, it is not practical to set the gas temperature in steady state without equally setting the temperature of the surface and bulk phases hounding the gas. Consideration of the response of the system as a vacuum environment can then provide a sufftciendy precise prediction of the pressure P and the surface coverage 9 at temperature Tfor molecules of a known species in a known state on a known surface. For example, an isotherm is estabhshed between the surface of the condensed and the gaseous phases, depending, eg, on the heat of desorption. For submonolayer coverage on a... 

The Van t Hoff isotherm identifies the free energy relationship for bulk chemical reactions. 

When the two liquid phases are in relative motion, the mass transfer coefficients in eidrer phase must be related to die dynamical properties of the liquids. The boundary layer thicknesses are related to the Reynolds number, and the diffusive Uansfer to the Schmidt number. Another complication is that such a boundaty cannot in many circumstances be regarded as a simple planar interface, but eddies of material are U ansported to the interface from the bulk of each liquid which change the concenuation profile normal to the interface. In the simple isothermal model there is no need to take account of this fact, but in most indusuial chcumstances the two liquids are not in an isothermal system, but in one in which there is a temperature gradient. The simple stationary mass U ansfer model must therefore be replaced by an eddy mass U ansfer which takes account of this surface replenishment. 

From isotherm measurements, usually earried out on small quantities of adsorbent, the methane uptake per unit mass of adsorbent is obtained. Sinee storage in a fixed volnme is dependent on the uptake per unit volume of adsorbent and not on the uptake per unit mass of adsorbent, it is neeessary to eonvert the mass uptake to a volume uptake. In this way an estimate of the possible storage capacity of an adsorbent can be made. To do this, the mass uptake has to be multiplied by the density of the adsorbent. Ihis density, for a powdered or granular material, should be the packing (bulk) density of the adsorbent, or the piece density if the adsorbent is in the form of a monolith. Thus a carbon adsorbent which adsorbs 150 mg methane per gram at 3.5 MPa and has a packed density of 0.50 g/ml, would store 75 g methane per liter plus any methane which is in the gas phase in the void or macropore volume. This can be multiplied by 1.5 to convert to the more popular unit, V/V. 

The equlibrium between the bulk fluid and fluid adsorbed in disordered porous media must be discussed at fixed chemical potential. Evaluation of the chemical potential for adsorbed fluid is a key issue for the adsorption isotherms, in studying the phase diagram of adsorbed fluid, and for performing comparisons of the structure of a fluid in media of different microporosity. At present, one of the popular tools to obtain the chemical potentials is an approach proposed by Ford and Glandt [23]. From the detailed analysis of the cluster expansions, these authors have concluded that the derivative of the excess chemical potential with respect to the fluid density equals the connected part of the fluid-fluid direct correlation function (dcf). Then, it follows that the chemical potential of a fluid adsorbed in a disordered matrix, p ), is... 

However, before proceeding with the description of simulation data, we would like to comment the theoretical background. Similarly to the previous example, in order to obtain the pair correlation function of matrix spheres we solve the common Ornstein-Zernike equation complemented by the PY closure. Next, we would like to consider the adsorption of a hard sphere fluid in a microporous environment provided by a disordered matrix of permeable species. The fluid to be adsorbed is considered at density pj = pj-Of. The equilibrium between an adsorbed fluid and its bulk counterpart (i.e., in the absence of the matrix) occurs at constant chemical potential. However, in the theoretical procedure we need to choose the value for the fluid density first, and calculate the chemical potential afterwards. The ROZ equations, (22) and (23), are applied to decribe the fluid-matrix and fluid-fluid correlations. These correlations are considered by using the PY closure, such that the ROZ equations take the Madden-Glandt form as in the previous example. The structural properties in terms of the pair correlation functions (the fluid-matrix function is of special interest for models with permeabihty) cannot represent the only issue to investigate. Moreover, to perform comparisons of the structure under different conditions we need to calculate the adsorption isotherms pf jSpf). The chemical potential of a... 

During the materials selection procedure isothermal corrosion testing may indicate the suitability of a material for handling a corrosive process fluid. In many cases where heat transfer is involved the metal wall temperature experienced in service is higher than the bulk process fluid temperature. This, and the actual heat transfer through the material, must be taken into account since both factors can increase corrosion rates significantly. 

There are few systematic guidelines which can be used to predict the properties of AB2 metal hydride electrodes. Alloy formulation is primarily an empirical process where the composition is designed to provide a bulk hydride-forming phase (or phases) which form, in situ, a corrosion— resistance surface of semipassivating oxide (hydroxide) layers. Lattice expansion is usually reduced relative to the ABS hydrides because of a lower VH. Pressure-composition isotherms of complex AB2 electrode materials indicate nonideal behaviour. 

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