Isotherms combined

Talhout, R. Villa, A. Mark, A.E. Engeberts, J.B.F.N., Understanding binding affinity a combined isothermal tritration calorimetry/molecular dynamics study of the binding of a series of hydrophobically modified benzamidinium chloride inhibitors to trypsin, J. Am. Chem. Soc. 2003,125, 10570-10579. 

There are several ways to form combined isotherms. In this paper we will simply assume that we can combine them in a linear fashion such that each isotherm is completely independent of the other. The independence of the isotherms on the different adsorbents implies the following equations ... 

It should be noted that the Tafel slopes just given were calculated for the combined isotherm under Langmuir conditions, namely at very low coverage. The same type of calculation can be repeated to obtain the kinetic parameters for different mechanisms both at low and at intermediate values of the coverage. The effect on the Tafel slope of competition with water is rather small for small molecules. Thus, for n = 1 the Tafel slope changes only by about 2 mV for the two mechanisms just discussed. This is within experimental error in most cases, perhaps explaining why the need to use the combined adsorption isotherm is not... 

Figure 7 (left hand panel) The input pore size distribution A (points), and the recalculated distribution from the fit to the isotherm (full line), the isosteric enthalpy curve (dash) and combined isotherm and enthalpy (dotted line). 

It should be stressed that the adsorption of isotherms accompanied by charge transfer differ from those measured in the absence of the latter even on uniform surfaces. In Refs. [70-74], the isotherm of adsorption with charge transfer are considered, and the field effect is introduced by two different ways via the Nemst Equation and by considering the displacement of water molecules whose adsorption changes with potential. This is why the author refers to this as combined isotherm. 

Di Lorenzo, M.L. (2001) Determination of spherulite growth rates of poly(L-lactic acid) using combined isothermal and non-isothermal procedures. Polymer, 42, 9441-9446. 

In the view of equations describing the nonisothermal crystallization in detail, kc has no physical meaning. Liu et al. [68] combined isothermal Avrami equation (Eq. 7.10) with the nonisothermal Ozawa equation into a single equation ... 

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. 

Equation XI-27 shows that F can be viewed as related to the difference between the individual adsorption isotherms of components 1 and 2. Figure XI-9 [140] shows the composite isotherms resulting from various combinations of individual ones. Note in particular Fig. XI-9a, which shows that even in the absence of adsorption of component 1, that of component 2 must go through a maximum (due to the N[ factor in Eq. XI-27), and that in all other cases the apparent adsorption of component 2 will be negative in concentrated solution. 

Stahlberg has presented models for ion-exchange chromatography combining the Gouy-Chapman theory for the electrical double layer (see Section V-2) with the Langmuir isotherm (. XI-4) [193] and with a specific adsorption model [194]. 

With nitrogen, the departure from spherical symmetry combined with the relatively strong quadrupole moment, leads to a blurring of the step-like character of the isotherm in the multilayer region (cf. Fig. 2.29(b)). 

Adsorption is invariably an exothermic process, so that, provided equilibrium has been established, the amount adsorbed at a given relative pressure must diminish as the temperature increases. It not infrequently happens, however, that the isotherm at a given temperature Tj actually lies above the isotherm for a lower temperature Ti. Anomalous behaviour of this kind is characteristic of a system which is not in equilibrium, and represents the combined effects of temperature on the rate of approach to equilibrium and on the position of equilibrium itself. It points to a process which is activated in the reaction-kinetic sense and which therefore occurs more rapidly as temperature is increased. 

The relationship between the BET monolayer capacity of physically adsorbed water and the hydroxyl content of the surface of silica has been examined by Naono and his co-workers in a systematic study, following the earlier work by Morimoto. Samples of the starting material—a silica gel—were heated for 4 hours in vacuum at a succession of temperatures ranging from 25 to 1000°C, and the surface concentration of hydroxyl groups of each sample was obtained from the further loss on ignition at 1100°C combined with the BET-nitrogen area. Two complete water isotherms were determined at 20°C on each sample, and to ensure complete... 

Gas—solids fluidization is the levitation of a bed of solid particles by a gas. Intense soflds mixing and good gas—soflds contact create an isothermal system having good mass transfer (qv). The gas-fluidized bed is ideal for many chemical reactions, drying (qv), mixing, and heat-transfer appHcations. Soflds can also be fluidized by a Hquid or by gas and Hquid combined. Liquid and gas—Hquid fluidization appHcations are growing in number, but gas—soHds fluidization appHcations dominate the fluidization field. This article discusses gas—soHds fluidization. 

Combining equation 6 and 7 with the Van t Hoff isotherm the Nemst equation for electrochemicA reactions is obtained ... 

The effective therm conductivity values generally obtained in practice are at least a factor of two greater than the one-dimensional thermal conductivity values measured in the laboratoiy with carefully controlled techniques. This degradation in insulation thermal performance is caused by the combined presence of edge exposure to isothermal boundaries, gaps, joints, or penetrations in the insulation blanket required for structure supports, fill and vent hnes, and high lateral thermal conductivity of these insulation systems. 

The calculation for a point on the flash curve that is intermediate between the bubble point and the dew point is referred to as an isothermal-flash calculation because To is specified. Except for an ideal binary mixture, procedures for calculating an isothermal flash are iterative. A popular method is the following due to Rachford and Rice [I. Pet. Technol, 4(10), sec. 1, p. 19, and sec. 2, p. 3 (October 1952)]. The component mole balance (FZi = Vy, + LXi), phase-distribution relation (K = yJXi), and total mole balance (F = V + L) can be combined to give... 

Several isotherms combine aspects of both the Langmuir and Freudhch equations. One that has been shown to be effective in describing data mathematically for heterogeneous adsorbents is the Toth isotherm [Acta Chim. Acad. Sci. Hung., 69, 311 (1971)]... 

Axial Dispersion Effects In adsorption bed calculations, axial dispersion effects are typically accounted for by the axial diffusionhke term in the bed conservation equations [Eqs. (16-51) and (16-52)]. For nearly linear isotherms (0.5 < R < 1.5), the combined effects of axial dispersion and mass-transfer resistances on the adsorption behavior of packed beds can be expressed approximately in terms of an apparent rate coefficient for use with a fluid-phase driving force (column 1, Table 16-12) ... 

External Mass Transfer and Intraparticle Diffusion Control With a linear isotherm, the solution for combined external mass transfer and pore diffusion control with an infinite fluid volume is (Crank, Mathematics of Diffusion, 2d ed., Clarendon Press, 1975) ... 

It is also possible to have a combined wave, which has both gradual and abrupt parts. The general rule for an isothermal, trace system is that in passing from the initial condition to the feed point in the isotherm plane, the slope of the path must not decrease, if it does, then a shock chord is taken for that part of the path. Referring to Fig. 16-19, for a transition from (0,0) to (1,1), the dashes indicate shock parts, which are connected by a simple wave part between points Pi and Pg. 

FIG. 16-19 Path in isotherm plane for a combined wave (After Tudge). 

The simplest isotherm is /if = cf corresponding to R = 1. For this isotherm, the rate equation for external mass transfer, the linear driving force approximation, or reaction kinetics, can be combined with Eq. (16-130) to obtain... 

FIG. 23-17 Multiple steady states of CSTRs, stable and unstable, adiabatic except the last item, (a) First-order reaction, A and C stable, B unstable, A is no good for a reactor, the dashed line is of a reversible reaction, (h) One, two, or three steady states depending on the combination Cj, Ty). (c) The reactions A B C, with five steady states, points 1, 3, and 5 stable, (d) Isothermal operation with the rate equation = 0 /(1 -I- C y = (C o Cy/t. 

The turboexpander in combination with a compressor and a heat exchanger functions as a heat pump and is analyzed as follows In Fig. 29-44 consider the compressor and aftercooler as an isothermal compressor operating at To with an efficiency and assume the working fluid to be a perfect gas. Further, consider the removal of a quantity of heat by the tumoexpander at an average low temperature Ti-This requires that it dehver shaft work equal to Q. Now, make the reasonable assumption that one-tenth of the temperature drop in the expander is used for the temperature difference in the heat exchanger. If the expander efficiency is and this efficiency is mul-... 

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