Van Laar expression

This is just the well-known van Laar expression for the heat of mixing in any two-component system. Thus, within the limits of the approximations used, the polymeric character of the solute does not alter the form of the heat of mixing expression. To generalize somewhat, let the solvent molecule contain Xi segments instead of only one. Eq. (19) is then replaced by... 

Usually, the Flory—Huggins interaction parameters %ij are positive quantities smaller than unity. " " " There are, however, cases in which %ij has negative values. Because the interaction of a protein with water or urea is exothermic, the above parameters are expected to be negative. Indeed, according to the van Laar expression for the heat of mixing in a two component system, the Flory—Huggins interaction parameter 7 is proportional to the heat of mixing in the binary system i—j. Furthermore, as demonstrated in a previous section, for the water + lysozyme + urea mixture J2i > 0, and eq 22 in which 2.5 leads to... 

For a multicomponent mixture, it is common to neglect ternary and higher interactions and assume a pseudobinary system. The resulting van Laar expression for the activity coefficient depends only on composition and the binary constants. The following form given by NulF is preferred. 

This equation is the van Laar expression derived for regular solutions and shows that this approach can be applied to polymer systems. To eliminate z, a dimensionless... 

In the case of an ideal solution, = 0, and since A5m is always positive, the value of AG is negative. When the solution does not behave ideally, a simple relation for Mlm has been proposed the van Laar expression for any two-component system based on the cell model for a liquid mixture [8,9],... 

Application of the algorithm for analysis of vapor-liquid equilibrium data can be illustrated with the isobaric data of 0th-mer (1928) for the system acetone(1)-methanol(2). For simplicity, the van Laar equations are used here to express the activity coefficients. 

Van der Waals and especially van Laar simplified these expressions by assuming a geometric mean for and an aritlmietic mean for... 

M refer to the density and molecular weight of /, and R is the gas constant. For simplicity, we assume each component to be monodisperse mote complex expressions result when polydispersity is considered (6). This model also assumes the heat of mixing pet unit volume follows a van Laar-type relation where B is... 

Outlined below are the steps required for of a X T.E calciilation of vapor-phase composition and pressure, given the liquid-phase composition and temperature. A choice must be made of an equation of state. Only the Soave/Redlich/Kwong and Peng/Robinson equations, as represented by Eqs. (4-230) and (4-231), are considered here. These two equations usually give comparable results. A choice must also be made of a two-parameter correlating expression to represent the liquid-phase composition dependence of for each pq binaiy. The Wilson, NRTL (with a fixed), and UNIQUAC equations are of general applicabihty for binary systems, the Margules and van Laar equations may also be used. The equation selected depends on evidence of its suitability to the particular system treated. Reasonable estimates of the parameters in the equation must also be known at the temperature of interest. These parameters are directly related to infinite-dilution values of the activity coefficients for each pq binaiy. 

In the case of a solution of moderate concentration we may perhaps assume the same expression (cf. van Laar, Thermodynamik und Cliemie Thermodyn. Potential Planck, loc. cit.)t whenever the solutions can legitimately be considered as brought, by suitable changes of temperature and pressure with unchanged composition, into ideal gas mixtures ( 185). 

We have a considerable body of knowledge to help us to say something about the third coefficient, the variation of fugacity with composition. Many empirical and semiempirical expressions (e.g., Margules, Van Laar, Scat-chard-Hildebrand) have been investigated toward that end. Most of our experience in this regard is limited to liquid mixture at low pressures, where... 

Following the idea of van Laar, Chueh expresses the excess Gibbs energy per unit effective volume as a quadratic function of the effective volume fractions. For a binary mixture, using the unsymmetric convention of normalization, the excess Gibbs energy gE is found from6... 

From the dilated van Laar model, Chueh obtains two-parameter expressions for the activity coefficients. They are ... 

These simple expressions may also be obtained from the chemical potentials according to Eqs. (XII-26) and (XII-32) by appropriately changing subscripts and recalling that x in these equations represents the ratio of the molar volumes, which in the present case is unity. Owing to the identity of volume fractions with mole fractions in this case, Eqs. (18) and (19) are none other than the chemical potentials for a regular binary solution in which the heat of dilution can be expressed in the van Laar form. The critical conditions (see Eqs. 2)... 

The roots of the CALPHAD approach lie with van Laar (1908), who applied Gibbs energy concepts to phase equilibria at the turn of the century. However, he did not have the necessary numerical input to convert his algebraic expressions into phase diagrams that referred to real systems. This situation basically remained unchanged for the next 50 years, especially as an alternative more physical approach based on band-structure calculations appeared likely to rationalise many hitherto puzzling features of phase diagrams (Hume-Rotheiy et al. 1940). 

The van Laar parameters A = 5135.9 J/mol and = 43213.7 J/mol yield a good fit. The observed variables are only slightly corrected to satisfy the model equations. The quantity "equation error after correction" is expressed in Pascals, hence the above values are negligible small. 

Figure 2 shows that with octadecane the theoretical curves fit the experimental values well if Xo = 0.8. If X2 = 0, the swelling of the seed would be lower than that for all cases where x = 0.8. The interaction parameter, Xo> was expressed by Huggins (2CQ using the van Laar -Scatchard - Hildebrand approximations as... 

The problem of expressing the excess Gibbs energy as a function of composition has been researched extensively, and many methods of varying accuracy and usefulness have been proposed. An extensive discussion of these methods is given by Hala et al. (2), who show that many common expressions—e.g., those of van Laar and Margules—are deduced from the general expression of Wohl (3). 

Equation 19 can, however, be integrated using any of the anal5rtical expressions available for the activity coefficient In such as the van Laar, Margules,... 

This effect of pressure on the equilibrium position of a reaction is a particular case of Le Chatelier s principle which will be discussed in chap. XVII. The quantitative expression (7.36) was given later by Planck and van Laar. 

Derive expressions for log 7, where 7 is the activity coefficient a/N, on the basis of (i) the simple Margules equation (35.9), (ii) the van Laar equation (35.10). Show that for a liquid mixture exhibiting positive deviations from Raoult s law, the activity coefficient of each constituent, on the basis of the usual standard state, must be greater than unity, whereas for negative deviations it must be less than unity. 

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