Mixing rule

It is to the extension of this approach for pure fluids and the estimation of the thermodynamic properties of mixtures that we now turn. This requires the introduction of mixing rules to provide a(x) and b(x), that are now functions of mole fraction x, from the values of a and b for pure substances. The expressions for a x) and b x) include the interactions between unlike molecules, and methods are then required to determine the parameters Oy and by for molecules i and j from the values of a and b for the pure fluids. This step is achieved using combining rules. 

The van der Waals one-fluid theory for mixtures assumes the properties of a mixture can be represented by a hypothetical pure fluid. Thus the thermodynamic behaviour of a mixture of constant composition is assumed to be isomorphic to that of a one-component fluid this assumption is not true near the critical point where the thermodynamic behaviour of a mixture at constant thermodynamic potential is isomorphic with that of a one-component fluid and this is discussed further in Chapter 10. 

The molecular basis of the van der Waals one-fluid approximation was developed by Reid and Leland and is discussed in Chapter 6. In this case, the intermolecular potential was assumed to be composed of a hard-sphere 

The summations in eqs 5.13, 5.14 and 5.15 are over all eomponents C. For three-parameter cubic equations of state, the parameter c(x) is usually assumed to be given by 

Equations 5.13 and 5.14 (and in some cases also eq 5.16) provide the means of estimating the parameters required for a mixture by recourse solely to the critical properties of the pure substances (see Chapter 4). 

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Eijuillbrium. Among the aspects of adsorption, equiUbtium is the most studied and pubUshed. Many different adsorption equiUbtium equations are used for the gas phase the more important have been presented (see section on Isotherm Models). Equally important is the adsorbed phase mixing rule that is used with these other models to predict multicomponent behavior. 

Although PVT equations of state are based on data for pure fluids, they are frequently appHed to mixtures. 7h.e virial equations are unique in that rigorous expressions are known for the composition dependence of the virial coefficients. Statistical mechanics provide exact mixing rules which show that the nxh. virial coefficient of a mixture is nxh. degree in the mole fractions ... 

Most of the assumptions are based on idealized models, indicating the limitations of the mathematical methods employed and the quantity and type of experimental data available. For example, the details of the combinatorial entropy of a binary mixture may be well understood, but modeling requires, in large measure, uniformity so the statistical relationships can be determined. This uniformity is manifested in mixing rules and a minimum number of adjustable parameters so as to avoid problems related to the mathematics, eg, local minima and multiple solutions. 

For both hydrocarbons and nonhydrocarbon organic defined mixtures, the method of Li " is used with a relatively simple volumetric average mixing rule as shown in Eq. (2-5) to calculate the true critical temperature. 

No specific mixing rules have been tested for predicting compressibility factors for denned organie mixtures. However, the Lydersen method using pseudocritical properties as defined in Eqs. (2-80), (2-81), and (2-82) in place of true critical properties will give a reasonable estimate of the compressibihty faclor and hence the vapor density. 

The constants Cj and C9 are both obtained from Fig. 2-40 Ci, usually from the saturated liquid line and C2, at the higher pressure. Errors should be less than 1 percent for pure hydrocarbons except at reduced temperatures above 0.95 where errors of up to 10 percent may occur. The method can be used for defined mixtures substituting pseiidocritical properties for critical properties. For mixtures, the Technical Data Book—Fehvleum Refining gives a more complex and accurate mixing rule than merely using the pseiidocritical properties. The saturated low pressure value should be obtained from experiment or from prediction procedures discussed in this section for both pure and mixed liquids. 

The mixing rule is given by Eq. (2-100) with the interaction parameter Q for each pair of components defined by Eq. (2-101). 

A mixing rule developed by Kendall and Monroe" is useful for determining the liquid viscosity of defined Iiydi ocai bon mixtiai es. Equation (2-119) depends only on the pure component viscosities at the given temperature and pressure and the mixture composition. 

For estimating the liquid viscosity of defined nonliydi ocai bon mixtui"es, a mixing rule shown hy Eq. (2-120) was recommended hy the Technical Data Manual. 

This mixing rule is used to determine the diffiisivity of any component in a / -I-1 component mixture and requires binary diffiisivities of component i with all other components. It has been estimated that errors are about 5 percent greater than the greatest error in the binary diffiisivities. Fairbanks and Wilke, using the same Eq. (2-154), made the same recommendation with essentially the... 

For concentrated binary nonpolar liquid systems (more than 5 mole percent solute), the diffiisivity can be estimated by a molar average mixing rule developed by Caldwell and Babb, " Eq. (2-156). 

For multiconmonent nonpolar liquid systems, Leffler and Ciil-linan developea a mixing rule, Eq. (2-157). 

Although the virial equation itself is easily rationalized on empirical grounds, the mixing rules of Eqs. (4-183) and (4-184) follow rigorously from the methods of statistical mechanics. The temperature derivatives of B and C are given exactly by... 

Although developed for pure materials, this correlation can be extended to gas or vapor mixtures. Basic to this extension is the mixing rule for second virial coefficients and its temperature derivative ... 

The application of cubic equations of state to mixtures requires expression of the equation-of-state parameters as func tions of composition. No exact theory like that for the virial coefficients prescribes this composition dependence, and empirical mixing rules provide approximate relationships. The mixing rules that have found general favor for the Redhch/Kwong equation are ... 

These are general equations that do not depend on the particular mixing rules adopted for the composition dependence of a and b. The mixing rules given by Eqs. (4-221) and (4-222) can certainly be employed with these equations. However, for purposes of vapor/liquid equilibrium calculations, a special pair of mixing rules is far more appropriate, and will be introduced when these calculations are treated. Solution of Eq. (4-232) for fugacity coefficient at given T and P reqmres prior solution of Eq. (4-231) for V, from which is found Z = PV/RT. 

Application of these equations requires specific mixing rules. For example, if... 

The first of the Wong/Sandler mixing rules relates the difference in mixture quantities b a.nda/RT to the corresponding differences (identified by subscripts) for the pure species ... 

The second Wong/Sandler mixing rule relates ratios of a/RT to b ... 

Flammability limits for pure components and selected mixtures have been used to generate mixing rules. These apply to mixtures of methane, ethane, propane, butane. 

The mixture cohesive energy density, coh-m> was not to be obtained from some mixture equation of state but rather from the pure-component cohesive energy densities via appropriate mixing rules. Scatchard and Hildebrand chose a quadratic expression in volume fractions (rather than the usual mole fractions) for coh-m arid used the traditional geometric mean mixing rule for the cross constant ... 

Mashuga, C. V. and Growl, D. A. 2000. Derivation of Le Chateler s Mixing Rule for Flammable Process Safety Progress, 19(2), 112-117. 

Tests by Roe et al. [63] with unidirectional jute fiber-reinforced UP resins show a linear relationship (analogous to the linear mixing rule) between the volume content of fiber and Young s modulus and tensile strength of the composite over a range of fiber content of 0-60%. Similar results are attained for the work of fracture and for the interlaminate shear strength (Fig. 20). Chawla et al. [64] found similar results for the flexural properties of jute fiber-UP composites. 

The chemical literature is rich with empirical equations of state and every year new ones are added to the already large list. Every equation of state contains a certain number of constants which depend on the nature of the gas and which must be evaluated by reduction of experimental data. Since volumetric data for pure components are much more plentiful than for mixtures, it is necessary to estimate mixture properties by relating the constants of a mixture to those for the pure components in that mixture. In most cases, these relations, commonly known as mixing rules, are arbitrary because the empirical constants lack precise physical significance. Unfortunately, the fugacity coefficients are often very sensitive to the mixing rules used. 

For a mixture, we assume the essentially arbitrary mixing rules... 

In their original paper, Redlich and Kwong also used the mixing rules given by Eqs. (17) and (18) in addition, however, they made the important simplification a,j = (OjOj)112. This simplification may introduce appreciable error (C3b, J2), and we do not use it here. Instead, we first rewrite Eq. (15) in the form... 

The variation of In generalized function /. The variation of TCM, PCM, and coM with composition is arbitrary and must be fixed by some mixing rule. For example, Pitzer and... 

By adopting mixing rules similar to those given in Section II, Chueh showed that Eq. (55) can be used for calculating partial molar volumes in saturated liquid mixtures containing any number of components. Some results for binary systems are given in Figs. 7 and 8, which compare calculated partial molar volumes with those obtained from experimental data. 

Several authors, notably Leland and co-workers (L2), have discussed vapor-liquid equilibrium calculations based on corresponding-states correlations. As mentioned in Section II, such calculations rest not only on the general assumptions of corresponding-states theory, but also on the additional assumption that the characterizing parameters for a mixture do not depend on temperature or density but are functions of composition only. Further, it is necessary clearly to specify these functions (commonly known as mixing rules), and experience has shown that if good results are to be obtained, these... 

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