The adsorption isotherm

This is the adsorption isotherm in terms of the absolute activity A of the ligand. To transfer this expression to a more familiar form, we assume that the ligand is in equilibrium with an ideal gas, for which the chemical potential is 

A ad is the adsorption constant that can be obtained from the experimental curve of 6 as a function of the pressure P. 

The quantity denoted by and defined in (2.10.35) will be referred to as the intrinsic binding constant. This quantity is related to the change in the Helmholtz energy for the process of bringing G from a fixed position, say Ro, in an ideal gas to a fixed and empty specific site, say j. The argument leading to this relation is similar to the one given in section 2.8.5. 

Note that in (2.10.40) we returned to the canonical PF. Since the only change that occurs is in the site y, we need to consider only the change in the PF of a single site from empty to occupied. 

We also note that A 4 defined in (2.10.40) is different from a similar quantity defined for the process of bringing the ligand from Ro to the specific site y (empty or occupied), for which we have the relation (see section 2.8.5) 

When a solid such as charcoal is exposed in a closed space to a gas or vapour at some definite pressure, the solid begins to adsorb the gas and (if the solid is suspended, for example, on a spring balance) by an increase in the weight of the solid and a decrease in the pressure of the gas. After a time the pressure becomes constant at the value p, say, and correspondingly the weight ceases to increase any further. The amount of gas thus adsorbed can be calculated from the fall in pressure by application of the gas laws if the volumes of the vessel and of the solid are known or it can be determined directly as the increase in weight of the solid in the case where the spring balance is used. 

In such an experiment the material actually adsorbed by the solid (the adsorbent) is termed the adsorbate, in contradistinction to the adsorptive which is the the general term for the material in the gas phase which is capable of being adsorbed. The adsorption is brought about by the forces acting between the solid and the molecules of the gas. These forces are of two main kinds—physical and chemical—and they give rise to physical (or van der Waals ) adsorption, and chemisorption respectively. The nature of the physical forces will be dealt with in the next section meanwhile it is convenient to note that they are the same in nature as the van der Waals forces which bring about the condensation of a vapour to the liquid state. 

The quantity of gas taken up hy a sample of solid is proportional to the mass m of the sample, and it depends also on the temperature T, the pressure p of the vapour, and the nature of both the solid and the gas. If n is the quantity of gas adsorbed expressed in moles per gram of solid, 

For a given gas adsorbed on a particular solid maintained at a fixed temperature. Equation (1.1) simplies to 

If the temperature is below the critical temperature of the gas, the alternative form 

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The data could be expressed equally well in terms of F versus P, or in the form of the conventional adsorption isotherm plot, as shown in Fig. Ill-18. The appearance of these isotherms is discussed in Section X-6A. The Gibbs equation thus provides a connection between adsorption isotherms and two-dimensional equations of state. For example, Eq. III-57 corresponds to the adsorption isotherm... 

Some data obtained by Nicholas et al. [150] are given in Table III-3, for the surface tension of mercury at 25°C in contact with various pressures of water vapor. Calculate the adsorption isotherm for water on mercury, and plot it as F versus P. 

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... 

It is not necessary to limit the model to idealized sites Everett [5] has extended the treatment by incorporating surface activity coefficients as corrections to N and N2. The adsorption enthalpy can be calculated from the temperature dependence of the adsorption isotherm [6]. If the solution is taken to be ideal, then... 

As discussed in Chapter III, the progression in adsoiptivities along a homologous series can be understood in terms of a constant increment of work of adsorption with each additional CH2 group. This is seen in self-assembling monolayers discussed in Section XI-IB. The film pressure r may be calculated from the adsorption isotherm by means of Eq. XI-7 as modified for adsorption from dilute solution ... 

This type of behavior can cause irreversibility in the adsorption isotherm as well as immobility on the surface [106]. 

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. 

As stated in the introduction to the previous chapter, adsorption is described phenomenologically in terms of an empirical adsorption function n = f(P, T) where n is the amount adsorbed. As a matter of experimental convenience, one usually determines the adsorption isotherm n = fr(P), in a detailed study, this is done for several temperatures. Figure XVII-1 displays some of the extensive data of Drain and Morrison [1]. It is fairly common in physical adsorption systems for the low-pressure data to suggest that a limiting adsorption is being reached, as in Fig. XVII-la, but for continued further adsorption to occur at pressures approaching the saturation or condensation pressure (which would be close to 1 atm for N2 at 75 K), as in Fig. XVII-Ih. 

The following several sections deal with various theories or models for adsorption. It turns out that not only is the adsorption isotherm the most convenient form in which to obtain and plot experimental data, but it is also the form in which theoretical treatments are most easily developed. One of the first demands of a theory for adsorption then, is that it give an experimentally correct adsorption isotherm. Later, it is shown that this test is insufficient and that a more sensitive test of the various models requires a consideration of how the energy and entropy of adsorption vary with the amount adsorbed. Nowadays, a further expectation is that the model not violate the molecular picture revealed by surface diffraction, microscopy, and spectroscopy data, see Chapter VIII and Section XVIII-2 Steele [8] discusses this picture with particular reference to physical adsorption. 

The adsorption isotherms are often Langmuirian in type (under conditions such that multilayer formation is not likely), and in the case of zeolites, both n and b vary with the cation present. At higher pressures, capillary condensation typically occurs, as discussed in the next section. Some N2 isotherms for M41S materials are shown in Fig. XVII-27 they are Langmuirian below P/P of about 0.2. In the case of a microporous carbon (prepared by carbonizing olive pits), the isotherms for He at 4.2 K and for N2 at 77 K were similar and Langmuirlike up to P/P near unity, but were fit to a modified Dubninin-Radushkevich (DR) equation (see Eq. XVII-75) to estimate micropore sizes around 40 A [186]. 

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. 

Discuss physical situations in which it might be possible to observe a vertical step in the adsorption isotherm of a gas on a heterogeneous surface. 

S. Ross and J. P. Olivier, The Adsorption Isotherm, Rensselaer Polytechnic Institute, Troy, NY, 1959, p. 39f. 

The Langmuir-Hinshelwood picture is essentially that of Fig. XVIII-14. If the process is unimolecular, the species meanders around on the surface until it receives the activation energy to go over to product(s), which then desorb. If the process is bimolecular, two species diffuse around until a reactive encounter occurs. The reaction will be diffusion controlled if it occurs on every encounter (see Ref. 211) the theory of surface diffusional encounters has been treated (see Ref. 212) the subject may also be approached by means of Monte Carlo/molecular dynamics techniques [213]. In the case of activated bimolecular reactions, however, there will in general be many encounters before the reactive one, and the rate law for the surface reaction is generally written by analogy to the mass action law for solutions. That is, for a bimolecular process, the rate is taken to be proportional to the product of the two surface concentrations. It is interesting, however, that essentially the same rate law is obtained if the adsorption is strictly localized and species react only if they happen to adsorb on adjacent sites (note Ref. 214). (The apparent rate law, that is, the rate law in terms of gas pressures, depends on the form of the adsorption isotherm, as discussed in the next section.)... 

Influence of the Adsorption Isotherm on the Kinetics of Heterogeneous Catalysis... 

Just as the surface and apparent kinetics are related through the adsorption isotherm, the surface or true activation energy and the apparent activation energy are related through the heat of adsorption. The apparent rate constant k in these equations contains two temperature-dependent quantities, the true rate constant k and the parameter b. Thus... 

Continuing the formal development of the influence of the adsorption isotherm on the apparent reaction kinetics, we next consider the case of a reac-... 

The hydration shell is formed with the increasing of the water content of the sample and the NA transforms from the unordered to A- and then to B form, in the case of DNA and DNA-like polynucleotides and salt concentrations similar to in vivo conditions. The reverse process, dehydration of NA, results in the reverse conformational transitions but they take place at the values of relative humidity (r.h.) less than the forward direction [12]. Thus, there is a conformational hysteresis over the hydration-dehydration loop. The adsorption isotherms of the NAs, i.e. the plots of the number of the adsorbed water molecules versus the r.h. of the sample at constant temperature, also demonstrate the hysteresis phenomena [13]. The hysteresis is i( producible and its value does not decrease for at least a week. 

The adsorption isotherm in the form (7) should be considered as a reasonable approximation, the more so, as our aim here is not to obtain faithful values of the adsorption parameters but to describe as completely as possible the qualitative behaviour of the NA-water system. It is worth to note here that the use of a polynomial of the form ... 

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... 

One application of the grand canonical Monte Carlo simulation method is in the study ol adsorption and transport of fluids through porous solids. Mixtures of gases or liquids ca separated by the selective adsorption of one component in an appropriate porous mate The efficacy of the separation depends to a large extent upon the ability of the materit adsorb one component in the mixture much more strongly than the other component, separation may be performed over a range of temperatures and so it is useful to be to predict the adsorption isotherms of the mixtures. 

Equations (1.2) and (1.3) are expressions of the adsorption isotherm, i.e. the relationship, at constant temperature, between the amount of gas adsorbed and the pressure, or relative pressure, respectively. 

Following the pioneer work of Beebe in 1945, the adsorption of krypton at 77 K has come into widespread use for the determination of relatively small surface areas because its saturation vapour pressure is rather low (p° 2Torr). Consequently the dead space correction for unadsorbed gas is small enough to permit the measurement of quite small adsorption with reasonable precision. Estimates of specific surface as low as 10 cm g" have been reported. Unfortunately, however, there are some complications in the interpretation of the adsorption isotherm. 

Fig. 2.27 Effect of mesoporosily on the adsorption isotherm and the t- (or a,-) plot, (a) (A) is the isotherm on a nonporous sample of the adsorbent (B) is the isotherm on the same solid when mesopores have been introduced into it, (i) being the adsorption, and (ii) the desorption branch. (b) I- (or a,-) plots corresponding to the isotherms in (a) (Schematic only.)...

Fig. 4.26 Low-pressure hysteresis in the adsorption isotherm of water at 298 K on a partially dehydroxy la ted silica gel. O, first adsorption run (outgassing at 200°C) . first desorption A, second adsorption run (outgassing at 200°C) A. second desorption (after reaching p/p = 0-31) X, third adsorption run (outgassing at 25 C).

Fig. 5.10 The adsorption isotherms of n-hexane (A) and of water (B) on graphitized carbon black.Solid symbols denote desorption. (After...

The effect of these factors on the adsorption isotherm may be elucidated by reference to specific examples. In the case of the isotherm of Fig. 5.17(a), the nonporous silica had not been re-heated after preparation, but had been exposed to near-saturated water vapour to ensure complete hydroxylation. The isotherm is of Type II and is completely reversible. On the sample outgassed at 1000°C (Fig. 5.17(h)) the isotherm is quite different the adsorption branch is very close to Type III, and there is extrensive hysteresis extending over the whole isotherm, with considerable retention of adsorbate on outgassing at 25°C at the end of the run. 

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... 

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