Rate constant of adsorption

The quantitative solution of the problem, i.e. simultaneous determination of both the sequence of surface chemical steps and the ratios of the rate constants of adsorption-desorption processes to the rate constants of surface reactions from experimental kinetic data, is extraordinarily difficult. The attempt made by Smith and Prater 82) in a study of cyclohexane-cyclohexene-benzene interconversion, using elegant mathematic procedures based on the previous theoretical treatment 28), has met with only partial success. Nevertheless, their work is an example of how a sophisticated approach to the quantitative solution of a coupled heterogeneous catalytic system should be employed if the system is studied as a whole. 

The description of molecular adsorption is very similar to that of atoms, provided we account for the molecules internal degrees of freedom. Hence we need to consider how these degrees change in going from the gas phase to the transition state of adsorption. The most general form for the rate constant of adsorption in the transition state theory is... 

The rate constant of adsorption is and the rate constant characterizing the way sites lose adsorbate is kd. From simple kinetics (see Chapter 8), the rate of adsorption depends on the number of sites available N(l - 0), the rate constant of the process fca, and the amount of adsorbate wanting to adsorb, which is proportional to pressure p. Overall, the rate of adsorption is N x (1 — 9) x ka x p. 

Table III. Rate Constants of Adsorption/Desorption of the Phosphate and Chromate...

The difference between the results obtained in the first (124) and the subsequent (126) works consists not only in the region of surface coverages to which the kinetic equation corresponds, but also in the order of magnitude of the constant A. It has been explained in Section XI that A is the ratio of rate constants of adsorption directions of the first and the second stages. According to (350), A < 1 in the temperature range of 400-500°C whereas for the deduction of (352) it must be assumed that A P 1. 

The evaluation of macroscopic rate constants of adsorption (kads) and direct reactions (kr+, kt) involves the following steps ... 

The rate constant of adsorption is determined from the first order rate expression given by Lagergren [6] log (q -q) = logq - k t / 2.303 (1)... 

The equation is true when bp 1 and b is the adsorption coefficient, expressed as the ratio of the rate constants of adsorption and desorption, i.e., ka/kd- When considering n = number of sites occupied and = maximum number of sites, the equation can then express 0 in terms of n/n ... 

Here n is the number of components in the system, coefficients a, and b, are the coefficients of the single-component Langmuir adsorption isotherm for component /. The coefficient bt is the ratio of the rate constants of adsorption and desorption, so it is a thermodynamic constant. The ratio ajb, is the column saturation capacity of component / [13],... 

The simulated profiles of isothermal chromatograms have something in common with the results of Refs. [5, 23, 24], In these studies, the elution chromatography was formally characterized by the rate constants of adsorption and... 

When the rate constants of adsorption and desorption are examined at molecular level, ka and k, respectively, are the probabilities per unit time that a molecule enters or leaves the stationary phase. Thus, during one adsorption-desorption event the average residence times of a molecule in the stationary and the mobile phases are Tg = 1/fc and Xm — i-lka, respectively. Since the molecules spend to time in the mobile phase and on average time in the stationary phase, we can express the average number of adsorption-desorption events as ... 

As a consequence, researchers from different disciplines of the life sciences ask for efficient and sensitive techniques to characterize protein binding to and release from natural and artificial membranes. Native biological membranes are often substituted by artificial lipid bilayers bearing only a limifed number of components and rendering the experiment more simple, which permits the extraction of real quantitative information from binding experiments. Adsorption and desorption are characterized by rate constants that reflect the interaction potential between the protein and the membrane interface. Rate constants of adsorption and desorption can be quantified by means of sensitive optical techniques such as surface plasmon resonance spectroscopy (SPR), ellipsometry (ELL), reflection interference spectroscopy (RIfS), and total internal reflection fluorescence microscopy (TIRE), as well as acoustic/mechanical devices such as the quartz crystal microbalance (QCM)... 

Since adsorption of annexin A1 takes place on a heterogeneous surface with different rate constants of adsorption and desorption, Eq. 4 is no longer valid. The Monte Carlo simulation numerically solves the corresponding master equation for off-lattice adsorption on a heterogeneous surface. 

The simplest case results when a non-localised adsorption is assumed (Baret 1968a, b), so that jjd C(, and r. As the result we obtain Eq. (4.31), where are k j and kj are the rate constants of adsorption and desorption, and c is the bulk concentration of the adsorbing species. On the basis of a localised adsorption the Langmuir mechanism Eq. (4.32) results. Further transfer mechanisms used to describe the kinetics of adsorption are given in Section 4.4, Eqs (4.31) - (4.34). To use these so-called transfer mechanisms for model of dynamic adsorption layers they have to be coupled with the transport process in the bulk. Baret (1969) suggested replacing c by the so-called subsurface or sublayer concentration. This is per definition the bulk concentration adjacent to the adsorption layer c(0,t) localised at x = 0. The following two flux balance equations for the molecular transfer results. 

The definition of k,j and k , the rate constants of adsorption and desorption, respectively,... 

The presence of impurities in surfactant solutions can give very misleading results. In a recent paper, experimental dynamic surface tensions of sodium dodecyl sulphate (SDS) solutions were interpreted by Fainerman (1977) on the basis of a mixed diffusion-kinetic-controlled adsorption model. As the result a rate constant of adsorption k j as a function of time was obtained (cf. Fig. 5.5, ), although this parameter was assumed to be a constant. 

Fig. 5.5 Rate constant of adsorption in dependence on time t for a 10 mol/cm SDS solution, ( ) determined from experimental data by Fainerman (1977), ( ) simulated data (cf Miller Lunkenheimer 1982)...

In the framework of the Temkin theory of biographically nonuniform surfaces discussed in detail in Chapter 3, we introduce X= q-qo widi its highest value =l= i- o. The value of , (equal to the AG°/RT) depends on the surface site. If the Polanyi transfer coefficient a is equal for both steps from the general expressions of the rate constants of adsorption and... 

Where Caji arsenic concentration in the liquid per m (g) of catalyst (pgT ), arsenic concentration of the feed (pgl ), k rate constant of adsorption (h" ), F molar flux (molh" ). 

---