Isothermal plots

The data could be expressed equally well in terms of F versus P, or in the form of the conventional adsorption isotherm plot, as shown in Fig. Ill-18. The appearance of these isotherms is discussed in Section X-6A. The Gibbs equation thus provides a connection between adsorption isotherms and two-dimensional equations of state. For example, Eq. III-57 corresponds to the adsorption isotherm... 

In describing the various mechanical properties of polymers in the last chapter, we took the attitude that we could make measurements on any time scale we chose, however long or short, and that such measurements were made in isothermal experiments. Most of the experimental results presented in Chap. 3 are representations of this sort. In that chapter we remarked several times that these figures were actually the result of reductions of data collected at different temperatures. Now let us discuss this technique our perspective, however, will be from the opposite direction taking an isothermal plot apart. 

Adsorption Plots. Isotherm plots are the most common method of presenting adsorption data. An isotherm is a curve of constant temperature the adsorbed water content of the adsorbent is plotted against the water partial pressure in equiHbrium with the adsorbent. An isostere plot shows curves of constant adsorbed water content the vapor pressure in equiHbrium with the adsorbent is plotted against temperature. Figure 13 shows isosteres for the three primary adsorbents described previously. In this case, the dew points for the three adsorbents are plotted at 0.5, 5, and 10 kg... 

Ideally, straight-line isotherm plots are obtained. However there may occasionally be departures from linearity. A curve as shown in Figure 30 (A) may be obtained if a non-adsorbable impurity is present in the liquid being treated. For such simations subtracting Cj from Q and re-plotting the isotherm will usually yield a straight line. 

It would then be possible to verify obedience of data to eqn. (18) from tabulated values of k and T. This has the advantage that plots of log k against T-1 can be considered and these Arrhenius plots allow the calculation of A and E in exactly the same manner as data collected from a series of isothermal plots. 

At different types of adsorption isotherms plotted for adsorption of donor particles on oxides (see section 1.5) expressions (1.112) - (1.115) provide the rise in and decrease in with the growth of partial pressure of gas P, the functions themselves being different. Thus, in case of applicability of the Henry isotherm at small P we have the function oi - exp const-P becoming a power function <7s P with the rise in P which is often observed in experiments [154, 155, 169]. 

An isotherm test can determine whether or not a particular contaminant can be adsorbed effectively by activated carbon. In very dilute solutions, such as contaminated groundwater, a logarithmic isotherm plot usually yields a straight line represented by the Freundlich equation62 63 ... 

The TGA system was a Perkin-Elmer TGS-2 thermobalance with System 4 controller. Sample mass was 2 to 4 mgs with a N2 flow of 30 cc/min. Samples were initially held at 110°C for 10 minutes to remove moisture and residual air, then heated at a rate of 150°C/min to the desired temperature set by the controller. TGA data from the initial four minutes once the target pyrolysis temperature was reached was not used to calculate rate constants in order to avoid temperature lag complications. Reaction temperature remained steady and was within 2°C of the desired temperature. The actual observed pyrolysis temperature was used to calculate activation parameters. The dimensionless "weight/mass" Me was calculated using Equation 1. Instead of calculating Mr by extrapolation of the isothermal plot to infinity, Mr was determined by heating each sample/additive to 550°C under N2. This method was used because cellulose TGA rates have been shown to follow Arrhenius plots (4,8,10-12,15,16,19,23,26,31). Thus, Mr at infinity should be the same regardless of the isothermal pyrolysis temperature. A few duplicate runs were made to insure that the results were reproducible and not affected by sample size and/or mass. The Me values were calculated at 4-minute intervals to give 14 data points per run. These values were then used to... 

The validity of Johnston s interpretation of the experimental facts in terms of the simple unimolecular dissociation (1) has been questioned by Lindars and Hinshelwood120 and by Reuben and Linnett121. These workers maintain that isothermal plots of k versus p are not smooth curves, but consist of a number of straight lines linked by markedly curved portions. To explain such behaviour they incorporate into their mechanism a collision-induced crossover of vibrationally excited N20 (XS) to repulsive 3II and 3E states. While we incline towards the simpler view held by Johnston105 and others106-116, we feel that this feature of the decomposition kinetics merits further investigation. 

From the adsorption isotherm [plotted in accordance with (4.10a)], the equilibrium constant K and the adsorption capacity, rmax, is obtained by plotting r 1 versus the reciprocal activity of the sorbate (Eq. 4.10c). 

Isotherm plots of TOC data for only Bottom Ash Solid waste and isotherm equations for the different solid phases are shown in Fig. 12, and the isotherm parameters determined from statistical regression analyses with their coefficients are given in Table 4. 

Figure 11.16 Relationship between sorbed and dissolved amphi-phile concentrations (upper isotherm plot). These different parts of the isotherm reflect changes in the solid surface as sorption proceeds, possibly explainable by the following in portion (I) with low dissolved concentrations, sorption occurs via ion exchange and related mechanisms. At some point, sufficient near-surface concentration enhancement occurs that micelles form there (Ha) and rapid coagulation between oppositely charged micelles and the surface follows (lib). When the surface becomes fLilly coated with such micelles, additional sorption is stopped (III). In portion III, the solid surface charge is converted from one sign to the other, implying sorbates must become physically associated with the particle surface, as opposed to simply being present in the diffuse double layer or the vicinal water layer.

Isotherm Plot Slope Intercept Corr. Coeff. 

The abscissa of an adsorption isotherm plot is usually the pressure. For gases it is given in Pa. If we are studying the adsorption of a vapor in equilibrium with its liquid the relative vapor pressure P/Po is plotted. 

Figure 9.9 Left BET adsorption isotherms plotted as total number of moles adsorbed, n, divided by the number of moles in a complete monolayer, ri7non, versus the partial pressure, P, divided by the equilibrium vapor pressure, Po. Isotherms were calculated for different values of the parameter C. Right Adsorption isotherms of water on a sample of alumina (Baikowski CR 1) and silica (Aerosil 200) at 20°C (P0 = 2.7 kPa, redrawn from Ref. [379]). The BET curves were plotted using Eq. (9.37) with C = 28 (alumina) and C = 11 (silica). To convert from n/nmo to thickness, the factors 0.194 nm and 0.104 nm were used, which correspond to n-mon = 6.5 and 3.6 water molecules per nm2, respectively.

Fig. 1. (a) Raw calorimetric data obtained by the titration of 0.57 mM cysteine with 1.59 nM gold nanorod solution and (b) binding isotherm plot obtained by integrating each peak in raw data and normalizing with cysteine concentration. 

FIG. 18.8 Typical isotherm plots of sorbed concentration versus ambient vapour pressure. (I) Henry s law, S is a constant (II) Langmuir equation (III) Flory-Huggins equation (IV) BET equation, site saturation at point B (from C.F. Rogers, 1985). 

It may be pointed out that the isotherms plotted in the figure given above are based on theoretical calcula tions of Vcorresponding to different values of P obtained by using the van der Waals equation. The isotherms for carbon dioxide, obtained by Andrews experimentally, were in close resemblance with these curves, with the difference that the wavelike portion LMNOQ was replaced by a horizontal line. Since then more careful experiments have shown that small portions corresponding to curves LM and OQ can be realised in practice also. These represent supersaturated vapour and superheated liquid, respectively. 

In the work of Isirikyan and Kiselev (1961), adsorption isotherms of nitrogen were determined at 77 K in considerable detail on four different graphitized thermal blacks (with BET areas in the range 6.5-29.1 m2g 1). The isotherms are plotted in Figure 9.3 in a normalized form, as the amount adsorbed per unit area (in pmol m-2) against the relative pressure, p/p°. Kiselev and his co-workers referred to such isotherm plots as absolute adsorption isotherms , but of course they are not stricdy absolute since they are dependent on the validity of the BET-nitrogen areas - with the usual assumption that o(N2) = 0.162 nm2. 

As with iodine, starch forms an inclusion complex with bromine vapor.205 Depending on the starch variety, different colors are developed by the complex. Maize and wheat produce an ochre color, rice produces a light-buff color, potato and sago develop a pale-yellow color, and cassava forms a cream color.69 Iodine cyanide (and bromine)-amylose complexes are brown-black and dark brown, respectively.206 The adsorption of chlorine and iodine proceeds according to the Freundlich isotherm. A discontinuity on the Freundlich isotherm plot is reported, which possibly results from the swelling of starch granules454... 

In all isotherms plotted within the region of low surfactant concentrations (2.5 10 6 to 3 I O 6 mol dm 3) and at A a - 0.5 mN m l there is a linear part corresponding to r = kC dependence (r is the adsorption). This part of the isotherm for curve 1 is presented in linear co-ordinates on the top left side of Fig. 3.77. A short plateau follows where dAo/d gC = 0. The further increase in surfactant concentration leads to a parabolic increase in Act until the next flexion of the curve is reached at Act - 8-10 mN m 1. Similar change in the course of Ao(0 isotherms has been found also for potassium and sodium oleates solutions [369,370], decane and undecane acid solutions [366] and aqueous solutions of saturated fatty alcohols [367]. It is worth to note that the measurements were carried out with purified substances so that any inoculations by a second surfactant are excluded. With the increase in surfactant concentration the parabolic part gradually transforms into a second linear part of the isotherm. 

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. 

Figure 2Id and e show similar isotherm plots for two B-solvents that localize (ethyl acetate and isopropanol). The ethyl acetate data of Fig. 2Id (circles) show reasonable adherence to the calculated curve, with the exception of some scatter at higher concentrations of ethyl acetate. Again, this may reflect experimental error. 

A representative isotherm plot is shown for FeTi in Figure 12. ... 

It is important to realize that useful zeolites have large internal surfaces, that is, a reminder of the sponge analogy, and it is these surfaces that control their observable surface properties. Normally, surface areas of inorganic materials are quantified by standard gas sorption techniques, for example, N2 uptake analyzed by Brunauer, Emmett, and Teller (BET) isotherm plots, and zeolites have nitrogen surface areas in the approximate region 100-1000m g. These estimates should be considered with caution because ... 

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