Normalized isotherm

Inspection of the common normalized isotherm in Figure 9.3 reveals a number of distinctive features. At very low plp°, the isotherm is slightly convex with respect to the p/p° axis and it is evident that the linear, Henry s law region does not extend above p/p° 5x 10-4. Although the isotherm is not truly stepwise (i.e. not a true Type VI isotherm), it does exhibit a characteristic monolayer step. This is followed by a wavy second layer region and then a smooth multilayer curve. Thus, as the multilayer coverage is increased, the isotherm appears to conform to the normal Type II shape. 

The strategy of plotting normalized isotherms may be particularly useful if homologues are compared 1. ... 

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. 

Therefore, it is important to plot the adsorption isotherms obtained for adsorptives with different solubilities against the concentration relative to the saturated solution. This normalization eradicates the differences in hydrophobicity between the adsorptive molecules, so that the normalized isotherms more truly reflect the affinity for the surface [3, 6]. 

At 20 °C the phase transition areas vary over 6 A2 per molecule. Although this range is outside the experimental error for the normalized isotherms, such small variation must be considered to be on the borderline of significance. More striking however is the approximately 10 dyne/ cm range of variation in surface pressure that results from the differences in hydrocarbon solvent effects on the work of compression. 

According to the discussion above, we can define Ca also from the normal isotherm equation where the virtual pressure, P, is used in place of the aetual partial pressure Pa-... 

Fig. 22. The normal isotherm for the alternate type VI isotherm (VIA) where one can observe the steps due to different E s.

Fig. 11.16 Plot of normalized isothermal crystallization of poly( 1,4-butylene adipate) in blends with poly(vinylidene fluoride) as a function of log time. Blend composition indicated. Crystallization temperature 43 °C. (From Pennings and...

THE SUBROUTINE ACCEPTS BOTH A LIQUID FEED OF COMPOSITION XF AT TEMPERATURE TL(K) AND A VAPOR FEED OF COMPOSITION YF AT TVVAPOR FRACTION OF THE FEED BEING VF (MOL BASIS). FDR AN ISOTHERMAL FLASH THE TEMPERATURE T(K) MUST ALSO BE SUPPLIED. THE SUBROUTINE DETERMINES THE V/F RATIO A, THE LIQUID AND VAPOR PHASE COMPOSITIONS X ANO Y, AND FOR AN ADIABATIC FLASHf THE TEMPERATURE T(K). THE EQUILIBRIUM RATIOS K ARE ALSO PROVIDED. IT NORMALLY RETURNS ERF=0 BUT IF COMPONENT COMBINATIONS LACKING DATA ARE INVOLVED IT RETURNS ERF=lf ANO IF NO SOLUTION IS FOUND IT RETURNS ERF -2. FOR FLASH T.LT.TB OR T.GT.TD FLASH RETURNS ERF=3 OR 4 RESPECTIVELY, AND FOR BAD INPUT DATA IT RETURNS ERF=5. 

In the production of hydrocarbon reservoirs, the process of isothermal depletion is normally assumed, that is reducing the pressure of the system while maintaining a constant temperature. Hence, a more realistic movement on the pressure-temperature plot is from point A to A . 

Pressure depletion in the reservoir can normally be assumed to be isothermal, such that the isothermal compressibility is defined as the fractional change in volume per unit change in pressure, or... 

Fig. XVII-22. Isosteric heats of adsorption for Kr on graphitized carbon black. Solid line calculated from isotherms at 110.14, 114.14, and 117.14 K dashed line calculated from isotherms at 122.02, 125.05, and 129.00 K. Point A reflects the transition from a fluid to an in-registry solid phase points B and C relate to the transition from the in-registry to and out-of-registry solid phase. The normal monolayer point is about 124 mol/g. [Reprinted with permission from T. P. Vo and T. Fort, Jr., J. Phys. Chem., 91, 6638 (1987) (Ref. 131). Copyright 1987, American Chemical Society.]...

Fig. XVII-27. Nitrogen adsorption at 77 K for a series of M41S materials. Average pore diameters squares, 25 A triangles, 40 A circles, 45 A. Adsorption solid symbols desorption open symbols. The isotherms are normalized to the volume adsorbed at Pj = 0.9. (From Ref. 187. Reprinted with kind permission from Elsevier Science-NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands.)...

The nitrogen adsorption isotherm is determined for a finely divided, nonporous solid. It is found that at = 0.5, P/P is 0.05 at 77 K, gnd P/F is 0.2 at 90 K. Calculate the isosteric heat of adsorption, and AS and AC for adsorption at 77 K. Write the statement of the process to which your calculated quantities correspond. Explain whether the state of the adsorbed N2 appears to be more nearly gaslike or liquidlike. The normal boiling point of N2 is 77 K, and its heat of vaporization is 1.35 kcal/mol. 

However, it is common practice to sample an isothermal isobaric ensemble NPT, constant pressure and constant temperature), which normally reflects standard laboratory conditions well. Similarly to temperature control, the system is coupled to an external bath with the desired target pressure Pq. By rescaling the dimensions of the periodic box and the atomic coordinates by the factor // at each integration step At according to Eq. (46), the volume of the box and the forces of the solvent molecules acting on the box walls are adjusted. 

It is sufficient, as Sing has pointed out, merely to replace as normalizing factor by the amount adsorbed at some fixed relative pressure (p/p ), in practice taken as (p/p°), = 0-4. The normalized adsorption n/ o (= j). obtained from the isotherm on a reference sample of the solid, is then plotted against p/p°, to obtain a standard a,-curve rather than a t-curve. The a,-curve can then be used to construct an a,-plot from the isotherm of a test sample of the solid, just as the t-curve can be used to produce a t-plot. If a straight line through the origin results, one may infer that the isotherm under test is identical in shape with the standard the slope b, of the linear branch of the j-plot will be equal totio 4 Just as the slope b, of the t-plot was equal to nja (cf. Equation (2.34)). 

If the region FGH of the isotherm represents the filling of all the pores with liquid adsorbate, then the amount adsorbed along to plateau FGH, when expressed as a volume of liquid (by use of the normal liquid density) should be the same for all adsorptives on a given porous solid. This prediction is embodied in a generalization put forward many years ago by Gurvitsch and usually known as the Gurvitsch rule. 

The steps may be so chosen as to correspond to consecutive points on the experimental isotherm. In practice it is more convenient to divide the desorption process into a number of standard steps, either of relative pressure, or of pore radius, which is of course a function of relative pressure. The amount given up during each step i must be converted into a liquid volume i , (by use of the normal liquid density) in some procedures the conversion is deferred to a late stage in the calculation, but conceptually it is preferable to undertake the conversion at the outset. As indicated earlier, the task then becomes (i) to calculate the contribution dv due to thinning of the adsorbed film, and thus obtain the core volume associated with the mean core radius r by the subtraction = t ... 

As noted earlier, control of the column s temperature is critical to attaining a good separation in gas chromatography. For this reason the column is located inside a thermostated oven. In an isothermal separation the column is maintained at a constant temperature, the choice of which is dictated by the solutes. Normally, the tern-... 

Favorable and unfavorable equihbrium isotherms are normally defined, as in Figure 11, with respect to an increase in sorbate concentration. This is, of course, appropriate for an adsorption process, but if one is considering regeneration of a saturated column (desorption), the situation is reversed. An isotherm which is favorable for adsorption is unfavorable for desorption and vice versa. In most adsorption processes the adsorbent is selected to provide a favorable adsorption isotherm, so the adsorption step shows constant pattern behavior and proportionate pattern behavior is encountered in the desorption step. 

During the adsorption or occlusion of various molecules, the micropores fill and empty reversibly. Adsorption in zeoHtes is a matter of pore filling, and the usual surface area concepts are not appHcable. The pore volume of a dehydrated zeoHte and other microporous soHds which have type 1 isotherms may be related by the Gurvitch rule, ie, the quantity of material adsorbed is assumed to fill the micropores as a Hquid having its normal density. The total pore volume D is given by... 

Separation of Norma/ and Isoparaffins. The recovery of normal paraffins from mixed refinery streams was one of the first commercial appHcations of molecular sieves. Using Type 5A molecular sieve, the / -paraffins can be adsorbed and the branched and cycHc hydrocarbons rejected. During the adsorption step, the effluent contains isoparaffins. During the desorption step, the / -paraffins are recovered. Isothermal operation is typical. 

Thermodynamic Properties. The thermodynamic melting point for pure crystalline isotactic polypropylene obtained by the extrapolation of melting data for isothermally crystallized polymer is 185°C (35). Under normal thermal analysis conditions, commercial homopolymers have melting points in the range of 160—165°C. The heat of fusion of isotactic polypropylene has been reported as 88 J/g (21 cal/g) (36). The value of 165 18 J/g has been reported for a 100% crystalline sample (37). Heats of crystallization have been determined to be in the range of 87—92 J/g (38). 

Adsorption of dispersants at the soHd—Hquid interface from solution is normally measured by changes in the concentration of the dispersant after adsorption has occurred, and plotted as an adsorption isotherm. A classification system of adsorption isotherms has been developed to identify the mechanisms that may be operating, such as monolayer vs multilayer adsorption, and chemisorption vs physical adsorption (8). For moderate to high mol wt polymeric dispersants, the low energy (equiUbrium) configurations of the adsorbed layer are typically about 3—30 nm thick. Normally, the adsorption is monolayer, since the thickness of the first layer significantly reduces attraction for a second layer, unless the polymer is very low mol wt or adsorbs by being nearly immiscible with the solvent. 

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