Step-like isotherms

Because of the large difference in tp between successive molecular layers, each layer becomes complete at a relative pressure (plp°)g which is determined by the value of rp/kT for that layer, viz pgjkT(here 0 is of course restricted to integral values). Each layer will therefore give rise to a step, such that the riser corresponds to the cooperative build-up of the layer and the tread to the transition between the layer and the next higher one. 

Another spherical, nonpolar molecule is methane its isotherms on both graphite and molybdenite at 77 K have a step-like character. Ethane, whilst slightly less symmetrical, is still nonpolar and it gives two distinct steps on cadmium at 97-4 K, the second step being nonhorizontal.  

With nitrogen, the departure from spherical symmetry combined with the relatively strong quadrupole moment, leads to a blurring of the step-like character of the isotherm in the multilayer region (cf. Fig. 2.29(b)). 

If the surface of the adsorbent is energetically heterogeneous rather than homogeneous each step of the isotherm will be replaced by an assortment of steps, corresponding to the completion of a monolayer on the different homogeneous patches of the surface. If the steps are sufficiently numerous 

Evaluation of the monolayer capacity from a stepped isotherm raises 

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A further complication which not infrequently appears is the occurrence of a phase transition within the adsorbed film. Detailed investigation of a number of step-like isotherms by Rouquerol, Thorny and Duval, and by others has led to the discovery of a kink, or sub-step within the first riser, which has been interpreted in terms of a two-dimensional phase change in the first molecular layer. 

According to a multilayer adsorption theory developed by Champion and Halsey such adsorption on a homogeneous surface leads inevitably to step like isotherms and isotherm continuity should be explained by surface heterogeneity [48]. Differences in the relationship course q°( = f(%PEG) between non-polar hexane, polar chloroform, 5r-complexing benzene, acetone and methanol which are capable of hydrogen bonds formation become comprehensible in view of the fact that specific and non-specific interactions, having different adsorption energies, take part in the adsorption process. Formation of distinct extreme on hexane heat of adsorption curve can be explained by hexane non polarity... 

Figure 5 The six types of International Union for Physical and Applied Chemistry isotherms. The type I isotherm is typical of microporous solids and chemisorption isotherms. Type II is shown by finely divided nonporous solids. Types III and V are typical of vapor adsorption (i.e., water vapor on hydrophobic materials). Types V and VI feature a hysteresis loop generated by the capillary condensation of the adsorbate in the mesopores of the solid. The rare type VI, the step-like isotherm, is shown by nitrogen adsorbed on special carbon.

To obtain the energy distribution analytically, Hobson has chosen a special local adsorption isotherm, namely a step-like isotherm, to describe the local equilibrium. This step-like isotherm has its fractional loading varying linearly with respect to pressure up to a certain pressure, and then beyond which the fractional loading is equal to unity. This step-like isotherm is ... 

At sufficiently low temperatures the isotherm of argon on high-energy surfaces tends to assume a step-like character (cf. p. 86). 

The step-like nature of krypton isotherms on highly uniform surfaces is referred to in Section 2.10. 

The adsorption isotherms for carbon black and graphitised carbon black (graphon) are completely different. For graphitised carbon black a step-like adsorption isotherm is... 

When the surface of a nonporous adsorbent is energetically uniform the isotherm may have a step-like shape (type VI). A good example of a type VI isotherm is found in the adsorption of krypton at 90 K on carbon black, graphitized at 2700°C [3], Type VI isotherms are of theoretical interest only. 

At low pressure, there is hardly any water adsorbed in the pores. At a certain characteristic pressure (specific to each pore width), there is a step-like increase in the amount adsorbed, until the pores are completely filled. All the isotherms, down to the smallest pore width, exhibit this discontinuous transition, which is evidence of capillary condensation. It is... 

Isotherm I is typical of adsorption in micropores, e.g., adsorption on molecular sieves and activated carbons. Isotherm II represents multilayer physisorption on a flat surface (valid for many nonporous substances). Isotherms III and V are characteristics of weak gas-solid interactions, e.g., water adsorption on gold. Isotherm IV is frequently observed in the study of practical heterogeneous catalysts. Its shape is characteristic of multilayer adsorption accompanied by capillary condensation in mesopores. When the surface of a nonporous adsorbent is energetically uniform the isotherm may have a step-like shape (Isotherm VI). A good example of such behaviour is the adsorption isotherm of Rr at 90 K on graphite [5]. 

Like isotherms, the net adsorption rate i]vac,t is defined in quite different ways. One example is given in Eq. 6.33 where a first order equilibrium reaction with two rate constants for the adsorption (kads) and desorption (kdes) step (Ma et ah, 1996) is speci-... 

Figure 9.16 Ne isotherms at temperatures shown, depicting step-like growth and hysteresis at monolayer completion (]V 70). Coverage scale is defined in the text. (Adapted from Ref. [81].)...

For polymers, dielectric spectroscopy is sensitive to fluctuations of dipoles, which are related to the molecular mobility of groups, segments, or the polymer chain as well [38]. The molecular mobility is taken as a probe for structure. The basic quantity is the complex dielectric function e f) = t (f) - it"(f) as a function of the frequency/and the temperature T. s (/) is the real whereas e"(/) is the loss part i = >f ). A relaxation process is indicated by a step-like decrease of s (/) with increasing frequency and a peak in e"(/). From the maximum position of the peak a mean relaxation rate can be deduced, which corresponds to the relaxation time of the fluctuation of the dipole moment of a given structural imit. For details see reference [49]. All shown measurements were carried out isothermally in the frequency range from 10 to 10 Hz by an ALPHA analyzer (NovocontroF). The temperature of the sample is controlled by a Quatro Novocontrol system with stability better than 0.1 K. 

T. Ishikawa, N. Kodaira, K. Kandori, Step-like adsorption isotherms of molecules on y-FeOOH and the surface homogeneity of y-FeOOH, J. Chem. Soc., Faraday Trans. 88 (1992) 719-722. 

Isothermal a—time curves for the decomposition of U02(CH3C02)2 in air (513—573 K) [1018] showed two approximately linear regions, 0.0 < a < 0.2 and 0.2 < a < 0.9, for which the values of E were 107 and 165 kJ mole-1, respectively. In nitrogen, the earlier portion of the curve was not linear and E = 151 kJ mole-1 for the later interval. The zero-order kinetic behaviour was explained by growth of nuclei in thin, plate-like crystals, which were shown by microscopic and surface area measurements to fragment when a > 0.85. The proposed initial step in the decomposition was fission of bonds between the U02+ and the (OCO CH3) species [1018]. 

The final stage in getting a molecule from the bulk phase outside a pellet on to the interior surface is the adsorption step itself. Where adsorption is physical in nature, this step is unlikely to affect the overall rate. An equilibrium state is likely to exist between adsorbate molecules immediately above a surface and those on it. An adsorption isotherm such as equation 17.4 or 17.35 may be applied. 

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