Isotherm, defined

Figure 11.15 Observed sorption of dodecylpyridinium on a soil (EPA-12) exhibiting an overall cation exchange capacity of 0.135 mol-kg"1. Two Langmuir isotherms (defined with particular values of C,s max and K/l, recall Eq. 9-5) are placed on the data to illustrate how different portions of the observed isotherm may reflect the influence of different materials in the complex soil sorbent or possibly different mechanisms (data from Brownawell et al., 1990).

Langmuir isotherm or model Simple mathematical representation of a favorable (type I) isotherm defined by Eq. (2) for a single component and Eq. (4) for a binary mixture. The separation factor for a Langmuir system is independent of concentration. This makes the expression particularly useful for modeling adsorption column dynamics in multicomponent systems. 

Consider again the heat capacity data of Fig. 38. The isotherm defines the phase diagram of a protein—water system. Below 0.38 h th re is a solution of water in protein—more accurately, water on the prote in. At 0.38 h there is phase separation, and at higher hydration levels the hydrated protein phase is constant in composition. This end point is well... 

In the initial studies carried out to date, poly-s tyrene-supported dimethylamine polymers crosslinked with 2 and 15% divinylbenzene (DVB) were complexed with m-chlorobenzoic acid and the adsorption isotherms defined. The corresponding binding constants were calculated to be 126.19 and 67.99 m", respectively. It is thus evident that the polymer support affects the strength of the binding interaction between the ligand and substrate. Mechanistic studies to understand the basis for this influence and its use in the design of polymers for selective molecular complexation reactions are in progress. 

If precipitation-dissolution reactions dominate (this is sorption by Sposito s definition), there are simply no differentiable isotherms (defined as a functional relationship between S and C) relating solid and aqueous concentrations (Bryant et al., 1987). The above-mentioned retardation factor formulations (equation (10.3)) are not applicable. A more complete form of the ADR equation may be (Fetter, 1993, p.l 15),... 

For the purpose of transport modeling, measured GO2 losses to unsaturated sediments were quantified by Freundlich isotherms for a PGO2 range commonly present in unsaturated zones (400 t 4000 Pa). The isotherms define the moles of dissolved plus adsorbed GO2 /m of sediment surface (GO2) associated with the deposits in which onsite measurements were made ... 

Faujasite zeolite-heavy hydrocarbon systems are characterised by type I isotherms (high slope in the very low pressure area followed by a plateau for higher pressures). The initial linear part of the isotherm is known as the Henry s zone for which the adsorption capacity of die adsorbent for a given adsorptive may be characterised by the Henry constant (slope of the isotherm) defined by Eq. (1) ... 

This formulation is the generalized form for all the DP isotherms. The details of each may be found in the literature [23] along with additional equivalency comparisons to x theory. If 1, this is the special case of the Ereundlich isotherm. Define a quantity Xo ... 

Figure 4.4. Simulation of the displacement train under ideal conditions from a set of individual substance isotherms and the displacer isotherm. The operating line is found by connecting the origin of the plot with the point on the displacer isotherm, defined by the displacer concentration in solution. The intersection points of the operating line with the substance isotherms determined the substance concentration in the displacement train. Substances whose isotherms do not intersect with the operating line elute ahead of the displacement train. Top Multiisotherm-plot with operating line. Bottom Corresponding chromatogram (displacement train).

FIG. 6 Types of isotherms defined and described by the modified FG equation (upper diagram) and the corresponding functions ( r) (lower diagrams). Isotherm Type 1. K= 10,5 =... 

Pressure depletion in the reservoir can normally be assumed to be isothermal, such that the isothermal compressibility is defined as the fractional change in volume per unit change in pressure, or... 

Reservoir fluids (oil, water, gas) and the rock matrix are contained under high temperatures and pressures they are compressed relative to their densities at standard temperature and pressure. Any reduction in pressure on the fluids or rock will result in an increase in the volume, according to the definition of compressibility. As discussed in Section 5.2, isothermal conditions are assumed in the reservoir. Isothermal compressibility is defined as ... 

A somewhat subtle point of difficulty is the following. Adsorption isotherms are quite often entirely reversible in that adsorption and desorption curves are identical. On the other hand, the solid will not generally be an equilibrium crystal and, in fact, will often have quite a heterogeneous surface. The quantities ys and ysv are therefore not very well defined as separate quantities. It seems preferable to regard t, which is well defined in the case of reversible adsorption, as simply the change in interfacial free energy and to leave its further identification to treatments accepted as modelistic. 

It is important to note that the experimentally defined or apparent adsorption no AN 2/, while it gives F, does not give the amount of component 2 in the adsorbed layer Only in dilute solution where N 2 0 and = 1 is this true. The adsorption isotherm, F plotted against N2, is thus a composite isotherm or, as it is sometimes called, the isotherm of composition change. 

Pressure-area isotherms for many polymer films lack the well-defined phase regions shown in Fig. IV-16 such films give the appearance of being rather amorphous and plastic in nature. At low pressures, non-ideal-gas behavior is approached as seen in Fig. XV-1 for polyfmethyl acrylate) (PMA). The limiting slope is given by a viiial equation... 

Sing (see Ref. 207 and earlier papers) developed a modification of the de Boer r-plot idea. The latter rests on the observation of a characteristic isotherm (Section XVII-9), that is, on the conclusion that the adsorption isotherm is independent of the adsorbent in the multilayer region. Sing recognized that there were differences for different adsorbents, and used an appropriate standard isotherm for each system, the standard isotherm being for a nonporous adsorbent of composition similar to that of the porous one being studied. He then defined a quantity = n/nx)s where nx is the amount adsorbed by the nonporous reference material at the selected P/P. The values are used to correct pore radii for multilayer adsorption in much the same manner as with de Boer. Lecloux and Pirard [208] have discussed further the use of standard isotherms. 

Two other important quantities are the isobaric expansivity ( coefficient of themial expansion ) and the isothermal compressibility k, defined as... 

The physical adsorption of gases by non-porous solids, in the vast majority of cases, gives rise to a Type II isotherm. From the Type II isotherm of a given gas on a particular solid it is possible in principle to derive a value of the monolayer capacity of the solid, which in turn can be used to calculate the specific surface of the solid. The monolayer capacity is defined as the amount of adsorbate which can be accommodated in a completely filled, single molecular layer—a monolayer—on the surface of unit mass (1 g) of the solid. It is related to the specific surface area A, the surface area of 1 g of the solid, by the simple equation... 

The kind of results adduced in the present section justify the conclusion that the quantity n calculated by means of the BET equation from the Type II isotherm corresponds reasonably well to the actual monolayer capacity of the solid. The agreement lies within, say, +20 per cent, or often better, provided the isotherm has a well defined Point B. 

To obtain a reliable value of from the isotherm it is necessary that the monolayer shall be virtually complete before the build-up of higher layers commences this requirement is met if the BET parameter c is not too low, and will be reflected in a sharp knee of the isotherm and a well defined Point B. For conversion of into A, the ideal adsorptive would be one which is composed of spherically symmetrical molecules and always forms a non-localized film, and therefore gives the same value of on all adsorbents. Non-localization demands a low value of c as c increases the adsorbate molecules move more and more closely into registry with the lattice of the adsorbent, so that becomes increasingly dependent on the lattice dimensions of the adsorbent, and decreasingly dependent on the molecular size of the adsorbate. 

Examples are provided by the work of Carman and Raal with CF2CI2 on silica powder, of Zwietering" with nitrogen on silica spherules and of Kiselev" with hexane on carbon black and more recently of Gregg and Langford with nitrogen on alumina spherules compacted at a series of pressures. In all cases, a well defined Type II isotherm obtained with the loose powder became an equally well defined Type IV isotherm with the compact moreover both branches of the hysteresis loop were situated (drove the isotherm for the uncompacted powder, but the pre-hysteresis region was scarcely affected (cf. Fig. 3.4). The results of all these and similar... 

A factor militating against the use of other adsorptives for pore size determination at the present time is the lack of reliable r-curves. The number of published isotherms of vapours such as benzene, carbon tetrachloride or the lower alkanes, or even such simple inorganic substances as carbon dioxide, on a reasonable number of well-defined non-porous adsorbents, is very small. 

Striking confirmation of the conclusion that the BET area derived from a Type IV isotherm is indeed equal to the specific surface is afforded by a recent study of a mesoporous silica, Gasil I, undertaken by Havard and Wilson. This material, having been extensively characterized, had already been adopted as a standard adsorbent for surface area determination (cf. Section 2.12). The nitrogen isotherm was of Type IV with a well defined hysteresis loop, which closed at a point below saturation (cf. F, in Fig. 3.1). The BET area calculated from it was 290 5 0 9 m g , in excellent agreement with the value 291 m g obtained from the slope of the initial region of the plot (based on silica TK800 as reference cf. p. 93). 

Experimental findings in the intervening years have tended to support and extend this concept. The results obtained by Ramsay and Avery in their studies of the effect of compaction on the nitrogen isotherms of two finely divided powders, one of zirconia and the other of silica, are especially instructive in the present context. As in earlier studies (cf. Chapter 3) the isotherm on the original powder was of Type II, but on compaction it first became Type IV with a well defined hysteresis loop, which moved... 

The way in which these factors operate to produce Type III isotherms is best appreciated by reference to actual examples. Perhaps the most straightforward case is given by organic high polymers (e.g. polytetra-fluoroethylene, polyethylene, polymethylmethacrylate or polyacrylonitrile) which give rise to well defined Type III isotherms with water or with alkanes, in consequence of the weak dispersion interactions (Fig. S.2). In some cases the isotherms have been measured at several temperatures so that (f could be calculated in Fig. 5.2(c) the value is initially somewhat below the molar enthalpy of condensation and rises to qi as adsorption proceeds. In Fig. 5.2(d) the higher initial values of q" are ascribed to surface heterogeneity. 

Type V isotherms of water on carbon display a considerable variety of detail, as may be gathered from the representative examples collected in Fig. 5.14. Hysteresis is invariably present, but in some cases there are well defined loops (Fig. 5.14(b). (t ), (capillary-condensed water. Extreme low-pressure hysteresis, as in Fig. 5.14(c) is very probably due to penetration effects of the kind discussed in Chapter 4. 

As pointed out earlier (Section 3.5), certain shapes of hysteresis loops are associated with specific pore structures. Thus, type HI loops are often obtained with agglomerates or compacts of spheroidal particles of fairly uniform size and array. Some corpuscular systems (e.g. certain silica gels) tend to give H2 loops, but in these cases the distribution of pore size and shape is not well defined. Types H3 and H4 have been obtained with adsorbents having slit-shaped pores or plate-like particles (in the case of H3). The Type I isotherm character associated with H4 is, of course, indicative of microporosity. 

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