Weakly Compressible Liquid Mixtures

In practice, Uquid mixtures are weakly compressible. That is, in Uquid mixtures the pressure does not induce fluctuations directly but indirectly, since the critical parameters in AfL2, Ap, and AT in eq 10.61 depend on the pressure  

We note that the total derivatives are taken along the critical locus. 

From eq 10.60, we note that, in contrast to the mole fraction, the mass density does depend on and b. Hence, the temperature dependence of the coexisting densities in liquid-liquid equilibria is affected by the pressure dependence of the critical parameters. Specifically, 

We conclude that for a non-vanishing value of the expansion for Ap contains a term proportional to cpi. Hence, the pressure dependence of the critical parameters causes a singular term proportional to Ar in the temperature expansion of the density diameter Ap. Such a singular term has been detected from experimental density data for liquid mixtures.Traditionally the presence of a term proportional to Af was considered an artifact because the density was not a correct order parameter for liquid mixtures. We now see that this term is a consequence of the pressure dependence of the critical parameters. 

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From eq 10.59, we see that the relationship of the mole fraction x with the scaling densities tpi and (pi is independent of either or b. Hence, the theoretical expressions for the temperature dependence of the mole fraction along the two phase boundaries, developed in the previous section, remain equally valid for weakly compressible liquid mixtures. This is the physical reason why eq 10.65 yields an excellent representation of the behaviour of the mole fraction x for liquid-liquid equilibria. As an example we show in Figure 10.5 closed solubility loops in 2-butanol + waterAs we explained earlier, closed solubility loops can be represented by the expansion of eq 10.65 provided that Ar is replaced by IATulI in accordance with eq 10.66. The closed solubility loops collapse into a double critical point at P = 85.6 MPa and T = 340 K. The implications of the theory for the behaviour near such a double critical point have been elucidated by Wang et Both near the upper critical solution temperature Ju and near the lower critical solution temperature Tl, Axcxc varies as A7 il in accordance with eq 10.65a. Near the double critical point both 7 j and 7 approach the temperature I d of the double critical point. Hence, near the double critical point... 

As shown in Section 10.4.3, in weakly compressible liquid mixtures the temperature and the chemical potentials of solvent and solute contribute to the critical fluctuations directly, while the pressure could be treated as a nonordering field, ° ° whose influence only manifests itself through the... 

In this chapter, we have presented a survey of the major theoretical approaches that are available for dealing with the effects of critical fluctuations on the thermodynamic properties of fluids and fluid mixtures. Special attention has been devoted to our current insight in the nature of the scaling densities and how proper relationships between scaling fields and physical fields account for asymmetric features of critical behaviour in fluids and fluid mixtures. We have discussed the application of the theory to vapour-liquid critical phenomena in one-component fluids and in binary fluid mixtures and to liquid-liquid phase separation in weakly compressible liquid mixtures. Because of space limitations this review is not exhaustive. In particular for the interesting critical behaviour of electrolyte solutions we refer the reader to the relevant literature. 

Then, the Korteweg-deVries equation describes the asymptotic behavior of nonlinear waves in liquid-gas mixtures. This equation can be thought of as the canonical" equation for weak evolution of long waves (for exemple, solitary waves). This type of wave motion is important because it describes how compressive pulses evolve in the mixture. Evolution equations for higher-order quantities can be obtained in a similar fashion. 

The weak effect of hydrogen additives on liquid kerosene atomized in oxygen has been proved by direct measurements [73]. The data from those measurements are summarized in Fig. 6.20. The value bands 1,2, 3 denote the atomized propellant RG self-ignition in shock-compressed oxygen at various pressure levels. Curve 4 denotes the spray C10H22 self-ignition. Curves 5 and 6 and band 7 were obtained in experiments with 7.5% H2 + 92.5% O2 and 15% H2 + 85% O2 mixtures. Curves 8,9 are calculated data for 7.5% H2 + 92.5% O2 mixture at 1 MPa and 4 MPa pressures respectively. 

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