Regular solution model temperature

The micellar composition shown in Figure 3 agrees well with predictions based on Equation 1 and reasonably well with the regular solution model. Similar agreement has been found at 37 and 50°C for the 3< Ciq/NPE] q system without any modification of the parameters. This is reflected by the values shown in Table II for the 3itiCjQ/NPEio system which apply over the temperature range between 27 and 50°C. Both the mixture CMC and the micellar composition are well fit using these parameters. 

An interesting result has been obtained after the introduction of the regular solution model [C. Wagner (1952)] into Eqn. (13.12). The free energy f(NA, T)m of the alloy under irradiation equals the free energy / of the alloy without irradiation but at an increased temperature, that is, f(NA, T)kl = f(NA, T+ AT). AT is found to be... 

Repeat the calculations of the previous problem with the regular solution model. Compare the two results. Develop an expression for the activity coefficient of a species in a mixture from the Peng-Robinson equation of state with the van der Waals one-fluid mixing rules, a. Show that the minimum amount of work, W , necessary to separate 1 mole of a binary mixture into its pure components at constant temperature and pressure is... 

In ideal solutions, the pure solute and solvent mix with no heat of mixing, AH"" = 0, and the heat of dissolution is numerically equal to the heat of fusion. However, only a limited number of systems form ideal solutions. A less restrictive assumption is that the solution is represented by a regular solution model. This model assumes the heat of mixing is nonzero, but independent of solution composition and temperature (i.e., AH= constant) (Hildebrand and Scott 1950). For a regular solution the differential entropy of mixing is also assumed ideal (i.e., AS = —R In x). 

The sodium-lithium phase system has been studied by thermal analysis in the liquid and solid regions to temperatures in excess of 400°C. Two liquid phases separate at 170.6°C. with compositions of 3.4 and 91.6 atom % sodium. The critical solution temperature is 442° zt 10°C. at a composition of 40.3 atom % sodium. The freezing point of pure lithium is depressed from 180.5°C. to 170.6°C. by the addition of 3.4 atom % sodium, and the freezing point of pure sodium is depressed from 97.8° to 92.2°C. by the addition of 3.8 atom % lithium. From 170.6° to 92.2°C. one liquid phase exists in equilibrium with pure lithium. Regardless of the similarity in the properties of the pure liquid metals, the binary system deviates markedly from simple nonideal behavior even in the very dilute solutions. Correlation of the experimentally observed data with the Scatchard-Hildebrand regular solution model using the Flory-Huggins entropy correction is discussed. 

For the cases where fn is independent of both temperature and composition, Eq. (5.119) corresponds to the regular solution model... 

Here, Wg and Wq2 represent temperature and composition independent constants called Margules parameters. The substitution of Eq. (11) into Eq. (7) yields the equilibrium potential of the less noble component in the alloy. If this expression is subtracted from the equilibrium potential of the elemental phase defined by Eq. (1), the relation between the underpotentially co-deposited alloy composition and corresponding value of underpotential can be obtained. The example of this approach is shown in Fig. 6 where the composition of UPCD CoPt and FePt is measured as a function of deposition underpotential. The solid lines in the plot indicate the fit of the asymmetric regular solution model. It is important to note that Eq. (11) combined with Eqs. (7) and (1) suggest that composition of UPCD AB alloys (CoPt and FePt) is not dependent on A and B (Co and Fe) deposition kinetics (concentrations in the solutions). 

Matsumoto (l97l). Binary and ternary regular solution models (without ternary coefficients) are used to treat Fe-Mg-Co exchange in coexisting olivine and pyroxene. On the basis of his theoretical formulation and natural composition data, the author attempts to estimate the temperature and pressure of equilibration for selected peridotite inclusions. 

Saxena and Nehru (1975) Independent equations of state for diopside and orthoenstatite solutions are assumed to obey regular solution models (one-constant Margules equations where Wqi= Wg2) Tbe W parameter for each solution is calculated from assumed differences in the standard free energies of fonnation for end-member phases and published experimental data on the compositions of coexisting phases. Temperatures for natural assemblages of enstatite and diopside are estimated from Margules expressions for homogeneous mixtures in which activities are presumed to be related to individual site occupancies. 

We have therefore devised a new method for the determination of the temperatures for onset and end of melting, which is based on a simulation of the experimental DSC-curves. This procedure uses a regular solution model for nonideal mixing in both, the ordered and the liquid-crystalline phase, and incorporates the additional broadening by assuming a simple two-state transition of limited cooperativity, with the cooperative unit size c.u. as an adjustable parameter. This model is still a simplification of the real situation as it is based on the assumption that the cooperativity does not depend on temperature [84]. Nevertheless, this procedure seems to be more reliable for the determination of the phase boundaries as the arbitrariness in detennining the onset and end of melting temperatures is replaced by a more objective procedure. 

Ruckenstein and Li proposed a relatively simple surface pressure-area equation of state for phospholipid monolayers at a water-oil interface [39]. The equation accounted for the clustering of the surfactant molecules, and led to second-order phase transitions. The monolayer was described as a 2D regular solution with three components singly dispersed phospholipid molecules, clusters of these molecules, and sites occupied by water and oil molecules. The effect of clusterng on the theoretical surface pressure-area isotherm was found to be crucial for the prediction of phase transitions. The model calculations fitted surprisingly well to the data of Taylor et al. [19] in the whole range of surface areas and the temperatures (Fig. 3). The number of molecules in a cluster was taken to be 150 due to an excellent agreement with an isotherm of DSPC when this... 

It is generally observed that as the temperature increases, real solutions tend to become more ideal and r can be interpreted as the temperature at which a regular solution becomes ideal. To give a physically meaningful representation of a system r should be a positive quantity and larger than the temperature of investigation. The activity coefficient of component A for various values of Q AB is shown as a function of temperature for t = 3000 K and xA = xB = 0.5 in Figure 9.3. The model approaches the ideal model as T - t. 

This solid solution still makes up the bulk of the solid particles after equilibration in an aqueous solution (59), since solid state diffusion is negligible at room temperature in these apatites (60), which have a melting point around 1500°C. These considerations and controversial results justify a thermodynamic analysis of the solubility data obtained by Moreno et al (58 ). We shall consider below whether the data of Moreno et al (58) is consistent with the required thermodynamic relationships for 1) an ideal solid solution, 2) a regular solid solution, 3) a subregular solid solution and 4) a mixed regular, subregular model for solid solutions. 

To the donor cell, 3 mL of the model drug solution was added, and to the receptor half-cell 3 mL of the solvent was added. Timing of the diffusion of the solute began as soon as both half-cells were filled. At regular intervals (i.e., 15, 30, and 60 minutes) the contents of the receptor cell were moved and replaced with fresh solvent, which in this case was pH buffer solution. To ensure constant temperature of the iimer cell solution, constant-temperature water was pumped through the outer half-cells. 

When gas solubility data are lacking or are unavailable at the desired temperature, they can be estimated using available models. The method of Prausnitz and Shair (1961), which is based on regular solution theory and thus has the limitations of that theory. The applicability of regular solution theory is covered in detail by Hildebrand et al. (1970). A more recent model, now widely used, is UNIFAC, which is based on structural contributions of the solute and solvent molecular species. This model is described by Fredenslund et al. (1977) and extensive tabulations of equilibrium data, based on UNIFAC, have been published by Hwang et al. (1992) for aqueous systems where the solute concentrations are low and the solutions depart markedly from thermodynamic equilibrium. 

The Chao-Seader and the Grayson-Streed methods are very similar in that they both use the same mathematical models for each phase. For the vapor, the Redlich-Kwong equation of state is used. This two-parameter generalized pressure-volume-temperature (P-V-T) expression is very convenient because only the critical constants of the mixture components are required for applications. For the liquid phase, both methods used the regular solution theory of Scatchard and Hildebrand (26) for the activity coefficient plus an empirical relationship for the reference liquid fugacity coefficient. Chao-Seader and Grayson-Streed derived different constants for these two liquid equations, however. 

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