Non-ideality parameter

Figure 1. Theoretical dependence of / 0 as calculated from equations (4) and (8) solid lines ideal monolayers dashed lines non-ideal. Parameter is q.

In either equations (1) or (2) the non-ideality parameter w (sometimes written w/RT) arises from the difference between the inter-molecular attraction of unlike species as compared to the mean of the intermolecular attraction for pairs of like species. The second parameter in equation (1), is sometimes ascribed... 

Figure 3.32 Background-corrected voltammetric feature of Co2FTF4 adsorbed on carbon centered at E = 0.27 V versus SCE (solid line) and the calculated classical Nerstian response (dashed line, see text for details). The scattered points represent best fits to the data using the model proposed by Anson for a non-ideality parameter, /T = —0.7 [82],...

In most plasmas of interest Fe(i) 1, which means they are ideal. In a dusty plasma (Fridman Ketmedy, 2004) with a dust particle density n, charge Z e, and temperature 7d, the non-ideality parameter (which is also called the Coulomb coupling parameter) is... 

The Debye radius gives the characteristic plasma size scale required for the shielding of an external electric field. The same distance is necessary to compensate the electric field of a specified charged particle in plasma. In other words, the Debye radius indicates the scale of plasma quasi-neutrality. There is the correlation between the Debye radius and plasma ideality. The non-ideality parameter F is related to the number of plasma particles in the Debye sphere, For plasma consisting of electrons and positive ions,... 

Biological membranes and the lipid mixtures extracted from them are far too complex to gain suitable infonnation on their mixing behavior by calorimetric methods. Therefore, the systematic investigation of mixtures of two or more lipids, preferentially synthetic ones, in aqueous dispersion is the method which has led to important conclusions about lipid miscibility in these quasi-two-dimensional lamellar phases. The analysis of the phase diagrams obtained from the investigation of the thermotropic transitions of suspensions with different lipid compositions in excess water in tenns of the curvature and the location of the phase boundaries provides the necessary information about non-ideality parameters. 

Pi are the non-ideality parameters describing the deviations from ideal mixing beliavior. Positive non-ideality parameters indicate a tendency towards cluster fonnation of like mixing and with increasing p the system finally shows a miscibility gap and phase separation into two phases of different composition. Negative non-ideality parameters lead to complex fonnation of unlike molecules. 

Because the simulation of the cy -curves indicated a non-symmetric, non-ideal mixing behavior, we recalculated the phase diagram using a regular solution model which accounts for this non-symmetric mixing behavior in both phases, yielding four non-ideality parameters ... 

The non-ideality parameters shown in Figure 26 were obtained by fitting the equations describing the coexistence curves to the T(-) and T( i) values (up and down triangles). Fitting to the empirical temperature values would lead to larger non-ideality parameters. 

Figure 26. Left Phase diagram obtained for DMPC/DPPC mixtures. The points for onset and end of melting obtained by the usual empirical procedure are indicated by open and filled dots, the triangles were obtained from the simulation of the DSC curves. The solid lines are fit curves through the triangles using the four parameter model described in the text. Right Non-ideality parameters as a ftinc-tion of composition as obtained from the simulation of phase diagrams of DMPC/DPPC and DMPE/DPPE (not shown) [90].

The mixing behavior in systems, where one of the compounds is charged can be strongly influenced by varying the pH of the suspension and thus the head-group charge. In mixtures of DMPA with DPPC, partial protonation of the PA component leads to phase diagrams with an upper azeotropic point or even a miscibility gap in the liquid-crystalline L -phase [85], The non-ideality parameter for... 

Figure 29. Left DSC curves of DMPA/DPPC mixtures at pH 1 and at pH 4, in the partly protonated form of DMPA. Right Phase diagrams and non-ideality parameters obtained from calculations based on the T(-) and 7 C7-values obtained from the simulation of the c/7-curves (left dotted lines). At pH 4, the shape of the phase diagram indicates lipid iminiscibility in the liquid-crystalline phase (adapted from reference [85]).

Real gases follow the ideal-gas equation (A2.1.17) only in the limit of zero pressure, so it is important to be able to handle the tliemiodynamics of real gases at non-zero pressures. There are many semi-empirical equations with parameters that purport to represent the physical interactions between gas molecules, the simplest of which is the van der Waals equation (A2.1.50). However, a completely general fonn for expressing gas non-ideality is the series expansion first suggested by Kamerlingh Onnes (1901) and known as the virial equation of state ... 

However, the reality is considerably more complicated than the ideal model. Non-ideality of the system causes that segregation extends over more than one layer. Further, when the size of the atoms is not equal one has to consider it in the calculations. This all has been done by A.D.van Langeveld (10), who made the following assumptions on the Pt/Cu system i) segregation extends over the two outmost layers, ii) when the binding energy of pairs of atoms is e the non-ideality of an alloy AB is described by the parameter ... 

Thermodynamic models are widely used for the calculation of equilibrium and thermophysical properties of fluid mixtures. Two types of such models will be examined cubic equations of state and activity coefficient models. In this chapter cubic equations of state models are used. Volumetric equations of state (EoS) are employed for the calculation of fluid phase equilibrium and thermophysical properties required in the design of processes involving non-ideal fluid mixtures in the oil and gas and chemical industries. It is well known that the introduction of empirical parameters in equation of state mixing rules enhances the ability of a given EoS as a tool for process design although the number of interaction parameters should be as small as possible. In general, the phase equilibrium calculations with an EoS are very sensitive to the values of the binary interaction parameters. 

Monitor short- and long-term nutritional status through evaluation of height, weight, and body mass index. Ideally, parameters should be near the normals for non-CF patients. 

Figure 5.7 Comparison of four-parameter fy-maxi mum, v-minimum. IC50, and h) and two-parameter (IC50 and h) fits of non-ideal concentration-response data. In panels A and B the data indicate a nonzero plateau at low inhibitor concentration that might reflect a low-amplitude, high-affinity second binding interaction. In panels C and D the data indicate a plateau at high inhibitor concentration that does not achieve full inhibition of the enzyme. There could be multiple causes of behavior such as that seen in panels C and D. One common cause is low compound solubility at the higher concentrations used to construct the concentration-response plot. Note that the discordance between the experimental data and the expected behavior is most immediately apparent in the plots that are fitted by the two-parameter equation.

This procedure of lumping all non-idealities into a few adjustable parameters is unsatisfactory for many reasons. Thermodynamic rigor is lost if experimentally determined dissociation constants or vapor pressures are disregarded. Also the parameters determined in this way are accurate only over the range of variables fitted and usually the model cannot be used for extrapolation to other conditions. The attractive feature of these models in the past was their need for little input information and the simple equations could often be solved algebraically. 

The object of this work was to extend the field of application of the equation-of-state method. The method was applied to aqueous systems in conjunction with a model that treats water as a mixture of a limited number of polymers, an approach similar to that previously adopted for the carboxylic acids (2). Association is calculated by the law of mass action corrections for non-ideal behaviour are made by means the equation of state. A major problem of the method is the large number of parameters needed to describe the properties and concentrations of the polymers together with their interaction with molecules of other substances. The Mecke-Kemptner model (15) (also known as the Kretschmer-Wiebe model (16) and experimental values for hydrogen-bond energies were usecT for guidance in fixing these parameters. 

Calculations including the vapour phase were ftien made to determine the extent of release of various components during the reaction. Two types of calculation were made, one where ideal mixing in the solution phases was considered and the other where non-ideal interactions were taken into account. For elements such as Ba, U and, to a certain extent. Si, the calculations were relatively insensitive to the model adopted. However, the amoimt of Sr in the gas was 24 times higher in the full mo r in comparison to the ideal model. This led to the conclusion that sensitivity analysis was necessary to determine the extent to which accuracy of the thermodynamic parameters used in die model affected die final outcome of the predictions. 

Sections 3.2.1—3.2.3 have referred specifically to the system illustrated in Fig. 6. However, the approach in these sections is quite general and can therefore be used in situations where the system transfer function G(s) is other than that given by eqn. (7). For the case of the ideal PFR responses, G(s) is exp(— st) and impulse, step and frequency responses are simply these respective input functions delayed by a length of time equal to r. The non-ideal transfer function models of Sect. 5 may be used to produce families of predicted responses which depend on chosen model parameters. 

One of the approaches to calculating the solubility of compounds was developed by Hildebrand. In his approach, a regular solution involves no entropy change when a small amount of one of its components is transferred to it from an ideal solution of the same composition when the total volume remains the same. In other words, a regular solution can have a non-ideal enthalpy of formation but must have an ideal entropy of formation. In this theory, a quantity called the Hildebrand parameter is defined as ... 

It may be conjectured that collective behavior implies that the surfactants that make up the mixture are not too different, the presence of an intermediate being a way to reduce the discrepancy. When the activity coefficient is calculated from non-ideal models it is often taken to be proportional to the difference in solubihty parameters [42,43], which in case of a binary is the difference (3i - if the system is multicomponent, then the dil -ference is - Sm) y which is often less, because the mean value exhibits an average lower deviation. In other terms, it means that for a ternary in which the third term is close to the average of the two first terms, then the introduction of the third component reduces the nonideahty because (5i - 53) + ( 2 - < (5i - 52) -... 

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