Isotherm comparison

Kowalczyk, P. et al., Estimation of the pore size distribution function from the nitrogen adsorption isotherm. Comparison of density functional theory and the method of Do and co-workers. Carbon. 2003,41(6), 1113-1125. 

Harhck, R J. E. Tezel, F. H. Use of CoiKentiation Pulse Chromatograplty for Determining binary Isotherms comparison with Statistically Determined Binary Isotherms, Adsorption 9 (2003), p. 275/286... 

The difficulty in the case of microporous materials stems from the porefilling mechanism. For this reason, the surface area of such materials is often determined by other methods than BET, which is based on layer formation. From the Dubinin equation the micropore volume Wo can be converted to the surface area. The as isotherm comparison method is an independent method for estimating the micropore volume and the surface area (20). The reference isotherm is a plot of the measured isotherm normalized by the amount of gas adsorbed at a fixed relative pressure, typically at p/po = 0.4. High resolution as analysis (21) yields more information about the characteristic texture of the adsorbent. Further methods (MP (22), -plot (23), Dubinin-Astakhov (11), Dubinin-Stockli (12), and so on) are also available for more reliable estimates of the micropore volume and surface area. 

Table 2 shows, not surprisingly, a clear correlation between the breakthrough times for pentane vapour and the level of porosity in the samples. The uptakes derived from the breakthrou curves show an excellent match to those read from the adsorption isotherm comparison with the uptakes at p/pO = 0.95 show how little of the microporosity is utilised at this test concentration (a result of the low partial pressure). Lillo-R6denas et al have recently shown [6] a correlation between benzene adsorption at 200ppmv and the amount of narrow microporosity as determined by CO2 adsorption at 273K it is assumed that only the narrower micropores are occupied by pentane in these tests. 

Adsorption of pentane from a dry flowing airstream reveals that the dynamic uptakes are consistent with the uptake from adsorption isotherms. Comparison with the isotherm data shows that only a limited amount of the microporosity is utilised. Increasing the micropore volume results in an increase in breakthrough time and uptake. 

Table 3 shows results obtained from a five-component, isothermal flash calculation. In this system there are two condensable components (acetone and benzene) and three noncondensable components (hydrogen, carbon monoxide, and methane). Henry s constants for each of the noncondensables were obtained from Equations (18-22) the simplifying assumption for dilute solutions [Equation (17)] was also used for each of the noncondensables. Activity coefficients for both condensable components were calculated with the UNIQUAC equation. For that calculation, all liquid-phase composition variables are on a solute-free basis the only required binary parameters are those for the acetone-benzene system. While no experimental data are available for comparison, the calculated results are probably reliable because all simplifying assumptions are reasonable the... 

Fig. XVI-2. Comparison of the pore volume distribution curves obtained from porosimeter data assuming contact angles of 140° and 130° with the distribution curve obtained by the isotherm method for a charcoal. (From Ref. 38.)...

Make a plot of Eq. XVII-69 as 6 versus P, and, for comparison, one of a Langmuir adsorption isotherm of same limiting or Henry s law slope. Comment on the comparison. 

Comparison of BET areas from nitrogen isotherms with independent values... 

Fig. 2.26 Comparison of a number of standard isotherms of nitrogen at 77 K, plotted as n/n against pip . O, Shnll x, Pierce , de Boer el , Cranston and Inkley. ...

The comparison plot offers a particularly simple and direct means of comparing the shapes of a pair of isotherms but for more general applications which involve a numt>er of samples of a solid covering a wide range of specific surface, the a,-method is preferable. The j-curve represents a convenient way of recording and using the reference isotherm. 

Fig. 2.29 Comparison of nitrogen adsorption at 78 K on a carbon black (Sterling FT) before and after graphitization (a) The amount adsorbed on the ungraphitized sample plotted against the amount x, adsorbed on the graphitized sample, at the same pressure, b) The corresponding isotherms O, adsorption, , desorption on the ungraphitized sample (4 runs) A. adsorption A desorption, on the graphitized sample (4 runs).

It was noted earlier (p. 115) that the upward swing in the Type IV isotherm characteristic of capillary condensation not infrequently commences in the region prior to the lower closure point of the hysteresis loop. This feature can be detected by means of an a,-plot or a comparison plot (p. 100). Thus Fig. 3.25(a) shows the nitrogen isotherm and Fig. 3.25(h) the a,-plot for a particular silica gel the isotherm is clearly of Type IV and the closure point is situated around 0 4p° the a,-plot shows an upward swing commencing at a = 0-73, corresponding to relative pressures of 013 and therefore well below the closure point. 

Figure 3.26(a) and (h) gives results for nitrogen on a compact of silica. Curve (a) is a comparison plot in which the adsorption on the compact (ordinates) is plotted against that on the uncompacted powder (abscissae) there is a clear break followed by an increased slope, denoting enhanced adsorption on the compact, at p/p° = 0-15, far below the lower closure point ( 0-42) of the hysteresis loop in the isotherm (curve (b)). A second compact, prepared at 64 ton in" rather than 130 ton in", showed a break, not quite so sharp, at p/p° = 0-35. 

Fig. 3.26 Comparison plots for compacts of silica and magnesia. In each case the adsorption of nitrogen at 78 K on the compact is plotted against that on the uncompacted powder, (a) and (b), comparison plot and adsorption isotherm for silica powder compacted at 130 ton in (c) and (d), comparison plot and adsorption isotherm for precipitated magnesia compacted at 10 ton in. Note that the upward sweep of the comparison plot commences at a relative pressure below the inception of the loop.

The results of a comparison between values of n estimated by the DRK and BET methods present a con. used picture. In a number of investigations linear DRK plots have been obtained over restricted ranges of the isotherm, and in some cases reasonable agreement has been reported between the DRK and BET values. Kiselev and his co-workers have pointed out, however, that since the DR and the DRK equations do not reduce to Henry s Law n = const x p) as n - 0, they are not readily susceptible of statistical-thermodynamic treatment. Moreover, it is not easy to see how exactly the same form of equation can apply to two quite diverse processes involving entirely diiferent mechanisms. We are obliged to conclude that the significance of the DRK plot is obscure, and its validity for surface area estimation very doubtful. 

When the values of the BET monolayer capacity calculated from Type III isotherms are compared with independent estimates (e.g. from nitrogen adsorption) considerable discrepancies are frequently found. A number of typical examples are collected in Table 5.1. Comparison of the value of the monolayer capacity predicted by the BET equation with the corresponding value determined independently (columns (iv) and (v)) show that occasionally, as in line 6, the two agree reasonably well, but that in the majority... 

The Carnot refrigeratiou cycle is reversible and consists of adiabatic (iseutropic due to reversible character) compression (1-2), isothermal rejection of heat (2-3), adiabatic expansion (3-4) and isothermal addition of heat (4-1). The temperature-entropy diagram is shown in Fig. 11-70. The Carnot cycle is an unattainable ideal which serves as a standard of comparison and it provides a convenient guide to the temperatures that should be maintained to achieve maximum effectiveness. 

We developed a sensor for determination of content of phosphorars in metallurgical melts. In quality of ion conductor used orthophosphate of calcium which pressed in tablets 010 mm. Tablets (mass 1-2 g) annealed at a temperature 400°C during 7-10 h. Tablets melts then in a quartz tube and placed the alloy of iron containing 1 mass % P. Control of sensor lead on Fe - P melts. Information on activities (effective concentration) of phosphorars in Fe - P melts was received. It is set that the isotherm of activity of phosphorars shows negative deviations from the Raouls law. Comparison them with reliable literary inforiuation showed that they agree between itself. Thus, reliable data on activities (effective concentration) of phosphorars in metallic melts it is possible to received by created electrochemical sensor for express determination. 

Figure 1.6.1 Comparison of asymptotic and exact solutions for a first order, non-isothermal reaction in a spherical catalyst pellet. ...

The Freundlich liquid phase isotherm can be used to determine the effect of solubility on the adsorptive capacity of activated carbon over a range of different concentrations. Phenol is highly soluble due to its polar nature whilst, in comparison, tetrachloroethylene (PCE) has a low solubility due to being non-polar. In the isotherms illustrated, the concentration of phenol is low relative to its solubility limit and consequently, the adsorptive capacity peaks at 18% maximum (see Figure 9). In comparison the concentration of tetrachloroethylene is relatively close to its solubility limit and, accordingly, the adsorptive capacity is exceptionally good. 

Fig. 5.28 shows a comparison of the isothermal and non-isothermal situations for polypropylene. When pressure is plotted as a function of injection rate, it... 

Fig. 17 shows the adsorption isotherms of all (undimerized and dimerized) particles. Except for a very fast increase of adsorption connected with filling of the first adlayer, the adsorption isotherm for the system A3 is quite smooth. The step at p/k T 0.28 corresponds to building up of the multilayer structure. The most significant change in the shape of the adsorption isotherm for the system 10, in comparison with the system A3, is the presence of a jump discontinuity at p/k T = 0.0099. Inspection of the density profiles attributes this jump to the prewetting transition in the... 

The equlibrium between the bulk fluid and fluid adsorbed in disordered porous media must be discussed at fixed chemical potential. Evaluation of the chemical potential for adsorbed fluid is a key issue for the adsorption isotherms, in studying the phase diagram of adsorbed fluid, and for performing comparisons of the structure of a fluid in media of different microporosity. At present, one of the popular tools to obtain the chemical potentials is an approach proposed by Ford and Glandt [23]. From the detailed analysis of the cluster expansions, these authors have concluded that the derivative of the excess chemical potential with respect to the fluid density equals the connected part of the fluid-fluid direct correlation function (dcf). Then, it follows that the chemical potential of a fluid adsorbed in a disordered matrix, p ), is... 

However, before proceeding with the description of simulation data, we would like to comment the theoretical background. Similarly to the previous example, in order to obtain the pair correlation function of matrix spheres we solve the common Ornstein-Zernike equation complemented by the PY closure. Next, we would like to consider the adsorption of a hard sphere fluid in a microporous environment provided by a disordered matrix of permeable species. The fluid to be adsorbed is considered at density pj = pj-Of. The equilibrium between an adsorbed fluid and its bulk counterpart (i.e., in the absence of the matrix) occurs at constant chemical potential. However, in the theoretical procedure we need to choose the value for the fluid density first, and calculate the chemical potential afterwards. The ROZ equations, (22) and (23), are applied to decribe the fluid-matrix and fluid-fluid correlations. These correlations are considered by using the PY closure, such that the ROZ equations take the Madden-Glandt form as in the previous example. The structural properties in terms of the pair correlation functions (the fluid-matrix function is of special interest for models with permeabihty) cannot represent the only issue to investigate. Moreover, to perform comparisons of the structure under different conditions we need to calculate the adsorption isotherms pf jSpf). The chemical potential of a... 

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