Isotherm calculation

When the value of c exceeds unity, the value of n can be derived from the slope and intercept of the BET plot in the usual way but because of deviations at low relative pressures, it is sometimes more convenient to locate the BET monolayer point , the relative pressure (p/p°) at which n/n = 1. First, the value of c is found by matching the experimental isotherm against a set of ideal BET isotherms, calculated by insertion of a succession of values of c (1, 2, 3, etc., including nonintegral values if necessary) into the BET equation in the form ... 

The adsorption isotherms calculated from their data are shown in Figure 3. It is seen that the propanol and butanol rapidly cover the surface at low moderator... 

Thermal structure of lean limit methane flame. Isotherms calculated from the measured gas velocity distribution. 

Figure 3. Adsorption of NP-EO q (a) and SDS (b) on polystyrene latex. The lines are isotherms calculated by fitting of Equation 19.

Figure 15-5 gives the shape of an isotherm calculated with Equation 15-14 at a temperature below the critical temperature. Points a through f are equivalent to points a through f on Figures 15-2 and 15-3. Points e are the values of z-factor that would be measured experimentally. Point f is a nonphysical solution. 

Isotherms calculated for constant pH (Figure 2) show that the amount of selenite adsorbed depends on the equilibrium solution concentration as represented by the equation of the Langmuir form,... 

The qualitative conclusions drawn from Eq. (4.64) are similar to those from Eq. (4.63). Note that in both cases the terms in the square brackets of Eqs. (4.63) and (4.64) in excess to unity are already small at t = 0 and do not exceed 0.15. These terms very soon become negligibly small due to fast growth of h(t). These considerations justify the applicability of the data of the isothermic calculations in Sect. 4.1-4.3 in engineering practice, privided that r is replaced by qn. 

The fit of the theoretical isotherms calculated using the Si and Ei parameters in comparison with the experimental data is satisfactory, as shown in Figures 2 and 3. The sum of squares error calculated by the expression... 

Natural minerals and rocks can be modelled as mixtures of clay-like adsorbents and oxide-like adsorbents. Various isotherms calculated from these assumptions are similar to experimentally observed isotherms for adsorption of Eu(III) on montmorillonite and corundum. Montmorillonite is of course a clay mineral, but it does have oxide-like groups. 

Fig. 5.2. Nonpreference exchange isotherms calculated with Eq. 5.16b for three values of the total moles of adsorptive charge, Q (Eq. 5.7b).

For this work, we [109] proposed an analytical method for the correlation between infinite dilution and finite dilution in adsorbate-adsorbent interaction system. And we reported the results of the experimental adsorption isotherm calculated by a distribution function of adsorption site energies on the basis of Fermi-Dirac s law. 

The procedure is first illustrated in terms of its application to the analysis of an isotherm calculated from an arbitrary assumed site energy distribution, f(Q). For reasons discussed in more detail later, the Langmuir equation—i.e., localized adsorption with no lateral interaction—appears very satisfactory as the form to use for 0(P,T,Q) ... 

Computations were carried through for values of 0.05 < 0 < 0.95 in increments of 0.05 unit, with C — 2, 3, 4, and 5. It was assumed that lateral interactions were due to attractive van der Waals-London dispersion forces, where the leading term in the energy expansion varies with distance as r-1/6 with R = V2 one finds C = C1/8. Calculations were also carried out in the Fowler-Guggenheim approximation this simply requires the determination of the zero-order inputs Po(a 0), Pj(b °K and P/P°. The results are exhibited in Figures 2 and 3 the broken curves refer to isotherms calculated according to Equations 22 and 23. 

Figure 2. Adsorption isotherms calculated as described in text forC = 2 and 3...

Figure 3. Adsorption isotherms calculated as described in text forC = 4 and 5 ----------------Fowler-Guggenheim isotherm...

Fig. 3.74. Oscillatory disjoining pressure (11 ) isotherm calculated from the thinning data in Fig. 3.73 ...

Equation (39) allows the calculation of the surface mole fraction 6b of B-solvent js a function of Nb, for mobile phases A/B (or A/C). The weight of B-solvent in the adsorbed monolayer at saturation (0 = 1) can be calculated from the B-solvent At, value and its molecular weight (see Refs. 1 and 14). The uptake of B-solvent by the adsorbent for some value of Nb is then given as 6 °- Several studies 4, 12) have reported experimental solvent isotherms (plots of 1 ° versus Af ) for silica, and it is of interest to compare these data with isotherms calculated as above. 

The adsorption isotherm calculated is of type IV in lUPAC classification, showing a rapid increase at low pressure as expected for hydrophilic surfaces. The steep rise in adsorption arround P/P°=0.7 is due to capillary condensation in the mesoporous solid. The result is comparable to two available experimental adsorption isotherm of water measured by very different techniques (gravimetry and calorimetry). This result, and the good agreement of the simulated isosteric heat of adsorption at very low coverage (75 kJ/mol) with experimental data, show that the model presented is able to describe quantitatively the hydrophilicity of the vycor surface with no adjustable parameters. 

Textural Parameters. Adsorption-desorption isotherms of N2 at 77K were determined in a Micromeritics ASAP 2010 with a micropore system. Prior to measurement, the samples were outgassed at 140 C for at least 16 h. The specific surface area was determined by the BET method, assuming that the area of a nitrogen molecule is 0.162 nm [12]. Micropore volume was calculated by the t-plot method using the Harkins and Jura [13] thickness. We used model isotherms calculated from density functional theory (DFT) to determine the pore size distributions and cumulative pore volume of the pillared samples by taking the adsorption branch of the experimental nitrogen isotherm, assuming slit-like pores [14]. 

Explain the term adsorption isotherm. Given adsorption equilibrium data or an expression for an adsorption isotherm, calculate the maximum quantity of adsorbate that can be removed from a gas by a specified quantity of adsorbent or, conversely, the minimum quantity of adsorbent needed to remove a specified quantity of adsorbate. 

In Figure 1, the continuous curves depict adsorption isotherms calculated from Equation 4. Experimental points are denoted by circles. The calculation and experimental results are in good agreement. A similar example is illustrated in Figure 2, showing experimental and calculated (from Equation 4) adsorption isotherms of carbon dioxide on Na,K-erionite. The data used in calculation were E = 5250 cal/mole and = 12.4% at to = 80°C. Thus, the general nature of gas and vapor adsorption on zeolites at weak electrostatic interactions is similar to adsorption on active carbons with the finest micropores (3). 

Other thermodynamic quantities that can be evaluated equally well by Monte Carlo and by MD simulations include the molar energy of adsorption, which is just the total potential energy of the adsorbed particles divided by their number, for a classical system [7] the surface tension of the adsorbed fUm [3] and the pressure normal to the surface. In principle, the dependence of the normal component of the pressure tensor upon amount adsorbed could be used to construct an adsorption isotherm since this pressure must be independent of distance from the surface in order to maintain mechanical equilibrium [3,7]. Thus, fer from the surface it must be equal to the bulk gas pressure. However, in practice the normal pressure is hard to evaluate with sufficient accuracy to be useful in an isotherm calculation, especially at the temperatures at or below the normal boiling point of the bulk... 

Figure 9.34. Typical exchange isotherms calculated for the reactions Ca + 2...

Figure 30. Mean adsorption isotherms calculated from uppermost curve...

Figure 3. (a) Isotherm calculated by density functional theory for a 4.1 run wide cylindrical pore with uniform surface potential (solid points). The solid line is the reconstructed isotherm for a flat surface having the adsorptive potential distribution of MCM-41. (b) Normalized isotherm for a 4.1 run MCM-41 (solid points) compared to the composite model for the same pore size. Note that the height of the pore-filling step is accurately accounted for. 

In this equation, z is the normal distance from the adsorbent surface, p is the particle density, U(z) the fluid - fluid potential and V(z) the wall potential, all at position z. Eq. 6 therefore defines the total integral heat of adsorption at any pressure point on a model isotherm calculated by DFT. 

All isothermal calculations discussed here employ Lennard-Jones potential functions and, unless otherwise stated, simulate free-boundary conditions. The neglect of three-particle interactions for a similar (Barker-Fisher-Watts) isolated pair potential has been shown to produce effects that are quite small for Ar systems. For clusters of more than three particles, the third-order potential energy terms 3 increase as the number of three-particle interactions increases. In the limit of zero temperature, where the third-order effects are most prevalent, 3 of the 13-particle Ar cluster (although already 60% of its bulk value) is less than 4.5% of the cluster s total potential energy. For a five-particle Ar cluster, 3 is less than 3% of the total potential energy. 

The second method of Ys determination is based on the interpretation of adsorption isotherms of either the total isotherm (calculation of the spreading pressure) or the initial or linear part of the isotherm. IGC readily provides the necessary information (2). 

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