Adsorption rate curves

For comparison, the diffusion coefficients were calculated from each adsorption rate curve at the point where the cumene uptake was half of its equilibrium value. The results are shown in Table II. 

FlO. 1. Adsorption rate curve on raising and lowering temperature (hydrogen on ZnO I). 

Figure P6.3 Typical adsorption rate curve for the adsorption of hydrogen on zinc oxide at 1 atm. [Adapted with permission from H. S. Taylor and S. C. Liang, The Heterogeneity of Catalyst Surfaces for Chemisorption, J. Am. Chem. Soc., 69, 1306 (1947). Copyright 1947 by the American Chemical Society.]...

Equations D3.5.30 and D3.5.32 are both very valuable. They state that the rate of adsorption can be obtained from plots of the interfacial tension versus either tA- (for t—>0) or lth (for the long-term solution f— >). With these two equations the tool to extract the adsorption rate from experimentally obtained surface tension-time curves is at hand. It should be noted that instead of the Gibbs model, one could use one of the previously mentioned adsorption isotherms such as the Langmuir adsorption isotherm to convert interfacial tension to interfacial coverage data. The adsorption isotherms may be obtained by fitting equilibrium surface tension data versus surfactant concentration. 

The rate parameters of importance in the multicomponent rate model are the mass transfer coefficients and surface diffusion coefficients for each solute species. For accurate description of the multicomponent rate kinetics, it is necessary that accurate values are used for these parameters. It was shown by Mathews and Weber (14), that a deviation of 20% in mass transfer coefficients can have significant effects on the predicted adsorption rate profiles. Several mass transfer correlation studies were examined for estimating the mass transfer coefficients (15, jL6,17,18,19). The correlation of Calderbank and Moo-Young (16) based on Kolmogaroff s theory of local isotropic turbulence has a standard deviation of 66%. The slip velocity method of Harriott (17) provides correlation with an average deviation of 39%. Brian and Hales (15) could not obtain super-imposable curves from heat and mass transfer studies, and the mass transfer data was not in agreement with that of Harriott for high Schmidt number values. 

It Is seen that the curves agree only up to the initial 15 minutes of the study. The two-resistance model thus provides a much better representation of the adsorption rates. Furusawa and Smith (25) have found this to be true for even small particle sizes, ranging from 200 to 900 microns In diameter. 

The experimental and predicted profiles for the adsorption rates of phenol and -bromophenol from an equimolar mixture of concentration 5 x 10-4 M are shown in Figure 12. The predicted profile for j>-bromophenol is in excellent agreement with the experimental data. However, for phenol there is some deviation after the Initial time period. The experimental adsorption rate for phenol appears to be faster than predicted for about 60 minutes after the first 15 minutes. Thereafter, the rate Is slightly slower than predicted. From an examination of the binary equilibrium data, this deviation may be attributed to the Inadequate correlation of the mixture equilibrium data in this region. The predicted and experimental total concentration profiles are shown In Figure 13. Initial concentrations of 2.5 x 10 4 M for phenol, and 5 x 10 4 M for j>-bromphenol were used in another rate study, the data from which are shown in Figures 13 and 14. The experimental and predicted curves are in fair agreement. 

The experimental and predicted profiles for adsorption from a mixture 5 x 10 H phenol and 5 x 10 H dodecyl benzene sulfonate are shown in Figure 15. The rate of adsorption of dodecyl benezene sulfonate is faster than predicted, and for phenol, the rate is slower than predicted. However, the shape of the predicted profiles for both solutes closely parallel the experimental curves. Similar trends may be noted in Figure 16 for the adsorption rates from a 10 4 H phenol and 10 4 H dodecyl benzene sulfonate mixture. The mixture equilibrium data for these solutes have been correlated satisfactorily. Thus, it would appear that solute-solute interactions are affecting the diffusional flux of each solute. Moreover, from Figure 17 for the total concentrations, it may be seen that the interaction effects are mutually compensating. The total concentration profiles for both... 

Consider a basic distinction between the transitive state model and the collision model for the adsorption rate Cad(/1). What types of curves could be obtained taking into account the interaction between nearest neighbors and additional contributions from the second neighbors. 

Figure 20. The SECM-induced desorption experiment. Chronoamperometric characteristics for the reduction of H+ on (a) a rutile (001) surface with d — 2.6 pm, and (b) an albite (010) surface with d — 2.8 pm In each case, solid curves from bottom to top represent the theoretical behaviors for an inert substrate and for specified adsorption rate constant values. Adapted with permission from Ref. [79]. Copyright (g) 1992, American Chemical Society.

FIGURE 36 Experimental data and theoretically generated concentration curves for the adsorption of HAS to a weak anion exchange sorbent DEAE-Sepharose FF at two different protein concentrations. As is evident from the data shown in this figure, the BAMcomb model used in this case predicted a slower adsorption rate during the earlier stages of the adsorption process when higher protein concentrations were employed. 

Chase [32] used the adsorption rate-limited model [Eqs. (7 —(11)[ to analyze the experimental breakthrough curves in affinity chromatography. This empirical approach assumes that all the rate-limiting processes can be represented by an apparent single second-order Langmuir adsorption rate equation in which k is considered a lumped" parameter. 

In contrast with the results of Fig. 5. important deviations between the theoretical profile and the experimental one are observed for adsorption studies on the polyclonal antibody [23], even at low HSA feeding concentrations. The frontal elution model given by Eq. (7) with k d = 0 correlates well with the first part of the breakthrough curve, but later a deviation is observed even at very low feeding HSA concentrations. In this case, the simplified model assuming a uniform adsorbent surface is not appropriate. The polyclonal antibody is made of different populations of antibodies of various affinities. With polyclonal immunoadsorbents, the values of ka in Table 2 are to be considered as apparent adsorption rates. 

The results of Renard et al. 23 show that the effective adsorption rate constant can be determined either from the analysis of the breakthrough curves or from the repeated pulse injection mode. The advantage with the latter method is simplicity, because it is based on peak area measurements, with minute amounts of protein consumed. Another advantage is the standard HPLC instrumentation used for such experiments. 

Fig. 2 shows the adsorption kinetics of Pb on ETS-10 together with the pseudo-second-order kinetic curve. It is seen that the adsorption rate is extremely fast. Under the experimental conditions, less than 10 s was required to attain saturation adsorption. When the concentration of Pb was about 2.5 mmol/L, Pb was not detected... 

Maximum temperatures measured in the adsorbent during the adsorption of nitrogen on 4A and propane on 5A zeolite, both at —78°C, were 15° and 50°C above the bath temperature. Finite difference calculations, taking into account the generation and loss of heat and changes in diffusivity and equilibrium adsorption with temperature, reproduced the pertinent features of the rate and temperature data. When the temperature maximum occurs late in the adsorption process, the rate curve is drastically different from that expected for isothermal adsorption. 

Appropriate values of activation energies and heats of adsorption from our laboratory or from the literature (2) were chosen, and diffusivi-ties at —78°C were estimated from our rate curves these values are given in Table I. 

Our data show that large temperature changes can occur during measurements of adsorption kinetics. When the temperature maximum occurs early in the process, no pronounced effect on the rate curve is observed, and the unwary experimenter may conclude that his data were obtained isothermally. 

It was stated, that macroporous carbons of plant origin show better properties in dynamic than in static conditions. It is cormected with relationship of quantity of macropores and diffusion rate. Greater differences in classsification are observed at lower equilibrium concentrations (0,1 and 0,01 mg/dm ) in static adsorption. Therefore direct factors of Freundlich s isotherms influencing adsorption capacities at different concentrations should be considered. It was stated that angle of slope, adsorption rate in static conditions, shape of breaktrough curves in dynamic conditions... 

Breakthrough curves from column experiments have been used to provide evidence for diffusion of As to adsorption sites as a rate-controlling mechanism. Darland and Inskeep (1997b) found that adsorption rate constants for As(V) determined under batch conditions were smaller than those necessary to model breakthrough curves for As(V) from columns packed with iron oxide coated sand the rate constants needed to model the breakthrough curves increased with pore water velocity. For example, at the slowest velocity of 1 cm/h, the batch condition rate constant was 4 times smaller than the rate constant needed to model As adsorption in the column experiment. For a velocity of 90 cm/h, the batch rate constant was 35 times smaller. These results are consistent with adsorption limited by diffusion of As(V) from the flowing phase to sites within mineral aggregates. Puls and Powell (1992) also measured more retardation and smaller rate constants for As(V) at slower flow velocities where there was sufficient time for diffusion to adsorption sites. 

In addition to the two theoretical explanations of S-T rate - p(H20) curves given above (Figure 7.1.), Lyakhov et al. [53], using adsorption data, have ascribed the S-T effect to variations in the sfructure of the adsorbed layer of water on the product solid, caused by systematic changes in polarization with increased coverage. Thomas et al. [54] have provided a kinetic analysis for reactions in which evolved HjO catalyzes the water release step which accounts for the occurrence of a maximum in the rate - curves. A full mechanistic and generally applicable... 

The activation energy obtained from the decreasing region of the growth rate curve in Fig. 4 is about 6 Kcal/mol. This is an appropriate value for the energy of physical adsorption of styrene monomer. 

If the design fluid volumetric flowrate (Q) is sufficiently low that equiUbriiun is rapid in comparison, the Equation (7.17) below is a good approximation of the concentration profile for the breakthrough curve as a function of fluid volume (V) put through the column [11], A Langmuir isotherm is assumed where k is the adsorption rate constant for this isotherm. When q M )g> CoV, the effluent solute concentration is approximately zero. For qoM Co V, the effluent solute concentration is C. See for yourself why this makes sense physically. 

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