Quadratic mixtures

Gas mixtures are subject to the same degree of non-ideality as the one-component ( pure ) gases that were discussed in the previous section. In particular, the second virial coefficient for a gas mixture can be written as a quadratic average... 

Van der Waals (1890) extended his theory to mixtures of components A and B by introducing mole-fraction-dependent parameters a and b defined as quadratic averages... 

Mixing mles for the parameters in an empirical equation of state, eg, a cubic equation, are necessarily empirical. With cubic equations, linear or quadratic expressions are normally used, and in equations 34—36, parameters b and 9 for mixtures are usually given by the following, where, as for the second virial coefficient, = 0-. 

Binary interaction parameters are determined for each pq pair p q) from experimental data. Note that = k and k = k = 0. Since the quantity on the left-hand side of Eq. (4-305) represents the second virial coefficient as predicted by Eq. (4-231), the basis for Eq. (4-305) lies in Eq. (4-183), which expresses the quadratic dependence of the mixture second virial coefficient on mole fraction. 

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 ... 

This is referred to as quadratic law of mixture shown in curve 4. The parameter K involves an interaction between components A and B and provides an expression for the interfacial effect. 

Specific conductivity (%) of all investigated mixtures showed quadratic dependence on the absolute temperature (T), which can be presented as follows ... 

This is a quadratic equation in [H + ] and may be solved in the usual manner. It can, however, be simplified by introducing the following further approximations. In a mixture of a weak acid and its salt, the dissociation of the acid is repressed by the common ion effect, and [H + ] may be taken as negligibly small by... 

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... 

The experimental designs discussed in Chapters 24-26 for optimization can be used also for finding the product composition or processing condition that is optimal in terms of sensory properties. In particular, central composite designs and mixture designs are much used. The analysis of the sensory response is usually in the form of a fully quadratic function of the experimental factors. The sensory response itself may be the mean score of a panel of trained panellists. One may consider such a trained panel as a sensitive instrument to measure the perceived intensity useful in describing the sensory characteristics of a food product. 

Figure 5.15 Mixture properties modeled by quadratic interaction parameters k...

MeCN/H,0 mixtures of volume fractions from ip = 0.41 to ip = 0.45 containing 0.1% TFA and butylsilica at temperatures T= 278-338 K. The plots show the experimental data and the lines of best fit according to the quadratic form of the relationship given in eq 7 and the polynomial form of the van t Hoff relationship. b Adapted from Jong, Boysen, and Hearn)24 with permission. 

Mixture variables, expressing the composition of the mobile phase as fi ac-tions, have the property that they add up to one (the mixture restriction). The consequence is that no intercept can be estimated when the effects of the solvents are evaluated [10,19]. Moreover interactions and quadratic effects, such as used when the independent variables are process variables, can not be estimated independently. Mathematically it is better to use blending effects only. Interpretation of these blending effects, i.e. explicitly stating what components are responsible for the non-linear effects, is not possible. 

The sulphate, [Rh(NH3)5I]S04, is obtained by treating the chloride with concentrated sulphuric acid. The mixture is diluted with water and left to stand, when orange-yellow efflorescent crystals of composition [Rh(NH3)sI]S04.6H20 separate. The anhydrous salt is prepared from this on drying at 100° C., or on precipitation of the mother-liquor from the hydrated salt. It crystallises in quadratic prisms of an orange-yellow colour. 

If a mixture is prepared with only A and B, in an initial ratio ro equal to Aq/Bq, the reactions of Equations (21) and (22) will occur, to establish the equilibrium. From stoichiometry, it can be shown that at equilibrium the ratio R = kr/kf= Ain /Bin of unlabeled materials must satisfy Equation (23). This is a quadratic that could be solved to evaluate R if AW were known. 

It can be shown that the composition of the permeate gas, P 11, for permeation of a binary gas mixture is given by the following quadratic equation ... 

The model Mr+1 contains the r-th degree term in the mixture components only along with the product of this term with the first degree terms in the Zj s. For example, a planar or first-degree model in the mixture components, and a main effects only model in the process variables, is y=M1+i+e. A planar model in the Xj s, combined with a main effect plus first-order interaction effects model in the Zj s, would be y=Mi+i+Mi+2+ . The model containing up to quadratic blending terms by main effects in the Zfs is defined as y=Mi+i+M2+l+H. This continues, up to the complete 2q+n-2n term model that is defined as ... 

The additional Helmholtz energy responsible for this secondary lattice can be expressed by using Equation (12) for binary Ising mixture with Xi, x2 replaced by and (1 — Finally, we obtain the temperature-dependent interchange energy that is quadratic to the inverse temperature. 

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