Linear mixture rule

In a long series of papers on the master equation, Pritchard and his coworkers elucidated for the first time the effects of rotational and vibrational disequilibrium on the dissociation and recombination of a dilute diatomic gas. Ultrasonic dispersion in a diatomic gas was analyzed by similar computational experiments, and the first example of the breakdown of the linear mixture rule in chemical kinetics was demonstrated. A major difficulty in these calculations is that the eigenvalue of the reaction matrix (corresponding to the rate constant) differs from the zero eigenvalue (required by species conservation) by less than... 

The thermal conductivities of liquid mixtures were found by Lees not to follow the linear mixture rule the best formula was found to be ... 

For mixture properties, use the following linear mixing rules ... 

These equations may be generalized to multicomponent systems with a linear mixing rule as in Eq. 1. It is worth remarking that the goodness of the fit tends to improve when the number of components of the mixture increases. For instance, the deviation shown in Fig. 5 (right) for the mixture of C9 and C15 ortho-xylene sulfonates, disappears if 20% of C12 ortho-xylene sulfonate is added. It is conjectured that the presence of intermediate species improves the collective behavior. 

Nonionic surfactants can be mixed as well, but in practice most of them, at least the ethoxylated ones, are already a mixture because of the polycondensation mechanism in ethylene oxide adduction. Isomerically pure ethoxylates are extremely expensive and are exclusively reserved for research work. Little work has been carried out with mixture of isomerically pure nonionics and the bulk of the work on mixture deals with mixtures of commercial products, i.e., mixture of mixtures, which obey a linear mixing rule on EON, provided that the base mixtures are not too different [8,35]. 

Since this phenomenon takes place with all ethoxylated surfactants, the mixture of two commercial products would exhibit a similar loss of lipophilic oligomers. As a consequence, a mixture of two (commercial) mixtures, would approximately exhibit an apparent linear averaging rule which may be stated as ... 

It is worth noting here that this difference between the interface and in the bulk is not specific to surfactant mixtures. While oil mixtures of very similar substances, such as n-alkanes, exhibit a linear mixing rule written in terms of equivalent alkane carbon number or EACN [62-64], mixtures of oils containing substances with very different polarities behave in a non-ideal way and exhibit a segregation near the interface, which results in an accumulation most polar oil components close to the interface [65]. 

However, if the trend of variation of In S with composition is extrapolated (dashed lines in Fig. 16), it is found that the two lines starting from pure anionic and pure cationic, cross exactly at an equimolar mixture composition. It may be said that on the left side of the diagram (more anionic than cationic) there is a linear mixing rule between the pme anionic and the equimolar compound, whereas in the right part of the diagram the Unear mixing rule is between the cationic and the equimolar compoimd [85,86]. 

Nishikido (21) has done a systematic study o-f mixed sur-factant solubilization. In that study, solubilization in mixed systems was compared to that predicted by application o-f a linear mixing rule to the solubilizations in the pure surfactant component micelles. For example, in this "ideal case, a micelle composed of a 50/50 molar mixture of two surfactants would have a solubilization capacity which is an average of that of the two pure surfactants involved. A system showing negative deviation from ideality would have less solubilization than this ideal system a system having positive deviation from ideality would have more. 

Linear mbcing rules for solubility parameters enable the design of solvent mixtures. In a design space a binary mixture s solubility... 

For binary mixtures, assuming that the volumes of constituent powders do not change during compression, the tensile strength at zero porosity can be calculated using a linear mixing rule ... 

In the absence of a theory to prescribe the composition dependence of parameters for cubic equations of state, empirical mixing rules are used to relate mixture parameters to pure-species parameters. The simplest realistic ejq)ressions are a linear mixing rule for parameter b and a quadratic mixing rule for parameter a, as shown by Eqs. (4-113) and (4-114). A common combining rule is given by Eq. (4-115). The general mole fraction variable Xi is used here because application is to both liquid and vapor mixtures. These equations, known as van der Waals prescriptions, provide for the evaluation of mixture parameters solely from parameters for the pure constituent species. They find application primarily for mixtures comprised of simple and chemically similar molecules. 

Here, C°P ni, C°e i, and C°e 2 are the optimum salinities of the mixture, surfactant components 1 and 2, respectively. The surfactant mole fractions are Xi and X2. Salager et al. also proposed that other characteristic parameters could follow a linear mixing rule. 

Crude oils were found to behave as an equivalent alkane as far as the attainment of optimum formulation was concerned. The equivalent alkane carbon number or EACN was then introduced to characterise pure hydrocarbons or mixtures [26]. The EACN of an oil phase is defined as the ACN of the alkane that results in the satisfaction of the correlation in the same conditions of surfactant, salinity, alcohol and temperature. EACN has been experimentally determined for n-alkanes mixtures, resulting in a linear mixing rule on a molar fraction basis, namely... 

The last technique, which is the best one when there is very little information on the unknown component, is based on the mixing of a pair of known components with the unknown one, and the use of a linear mixing rule. For instance, if the characteristic parameter ((3 = a - EON) of an unknown non-ionic surfactant is to be determined, the correlation to be used for the mixture of the two base products, such as two ethoxylated nonylphenols with different EONs, e.g. EONi and EON2, so that the mixture that results in three-phase behaviour is EONm is as follows ... 

Mixing anionic and cationic surfactants results in the formation of an equimolar catanionic species, which is likely to precipitate even at very low concentration, because it is more hydrophobic (two tails) and less ionic (the charges cancel out at least partially). It was shown, however, that this equimolar catanionic surfactant tends to behave as a hydrophobic amphoteric, i.e. ionic surfactant, which is able to exhibit a linear mixing rule with either of the ionic species provided its proportion remains small, say, less than 20% [57]. For instance, if 5 wt.% of a cationic surfactant is added to 95 wt.% of anionic surfactant, the actual mixture behaves as if it were a mixture of 90 wt.% anionic and 10 wt.% catanionic surfactant. In practice, the pure catanionic species precipitates and hence does not exist as a soluble substance in the microemulsion. Hence, its characteristic parameter has to be estimated by extrapolating the linear trends of the 1 1 mixture, as seen in Fig. 3.10(c). 

Flash point is an important solvent blend performance property. Mixture flash points cannot be predicted using linear mixing rules, and to date no literature describing a satisfactory method of predicting flash... 

The chain-of-rotators (COR) eos has been presented in Equation (4.183). For mixtures, using van der Waals mixing rules for a and b and linear additive rule for c,... 

Oil mixtures were found to exhibit an EACN that can be calculated according to a linear mixing rule on a mole fraction basis [50,65],... 

The churuct eristic parameter of ionic surfactant mixture can be readily calculated from a linear mixing rule on its olK so that i65,67 ... 

Let us now come back to the correlation and turn the attention to the effect of the oil phase nature. If the oil phase is not an alkane, its equivalent ACN. i.e.. the ACN of the alkane that exhibits the same behavior, is used instead (II. 68). Since the EACN of an oil mixture can be deduced from its composition by a linear mixing rule bused on a molar fraction, it has been possible to measure the EACN of many oil.s. 

For the application to mixtures, the volume translation parameter c can be calculated by a simple linear mixing rule ... 

For mixtures, the liquid density can be derived by a linear mixing rule of the specific volume, where the excess volume (see Chapter 4) is neglected ... 

The errors caused by this simplification are usually small The largest error has been reported for the ethanol-water mixture, where the contribution of the excess volume can arise up to 3.5% [17]. It is worth mentioning that Eq. (3.60) yields better results than a linear mixing rule for the density, which is often applied due to its simplicity, especially when the pure component densities are significantly different. 

---