Dense gases and liquids

Hirschfelder, J. O., C. F. Curtiss, R. B. Bird, and E. L. Spotz. The transport properties of dense gases and liquids, pp. 611-667. In Molecular Theory of Gases and Uquids. New York John Wiley and Sons, Inc., 1954. 

Further development of the kinetic theory of dense gases and liquids. 

A structure of the obtained set of equations derived by us in [81, 86] is very close to the famous BBGKI set of equations widely used in the statistical physics of dense gases and liquids [76]. Therefore, we presented the master equation of the Markov process in a form of the infinite set of deterministic coupled equations for averages (equation (2.3.34)). Practical use of these equations requires us to reduce them, retaining the joint correlation functions only. 

The analogy just mentioned with the BBGKI set of equations being quite prominent still needs more detailed specification. To cut off an infinite hierarchy of coupled equations for many-particle densities, methods developed in the statistical theory of dense gases and liquids could be good candidates to be applied. However, one has to take into account that a number of the... 

To explain the very different behavior of real gases, the model must be modified. Suppose the molecular volume is small but not negligible. In stales of high compression, where the total molecular volume becomes of the order of the volume available 10 the gas. the free space available to the molecules is only a Traction of what it would be in a perfect gas. and thus the real gas is much harder to compress than Ihe perfect gas. This explains the low compressibility of dense gases and liquids (diagram). 

Iakubov, I.T. and Khrapak, A.G. (1982). Self-trapped states of positrons and positronium in dense gases and liquids. Rep. Prog. Phys. 45 697-751. 

A more rigorous derivation of these relations were given by Curtiss and Hirschfelder [16] extending the Enskog theory to multicomponent systems. FYom the Curtiss and Hirschfelder theory of dilute mono-atomic gas mixtures the Maxwell-Stefan diffusivities are in a first approximation equal to the binary diffusivities, Dgr Dsr- On the other hand, Curtiss and Bird [18] [19] did show that for dense gases and liquids the Maxwell-Stefan equations are still valid, but the strongly concentration dependent diffusivities appearing therein are not the binary diffusivities but merely empirical parameters. 

This balance equation can also be derived from kinetic theory [101], In the Maxwellian average Boltzman equation for the species s type of molecules, the collision operator does not vanish because the momentum mgCs is not an invariant quantity. Rigorous determination of the collision operator in this balance equation is hardly possible, thus an appropriate model closure for the diffusive force is required. Maxwell [65] proposed a model for the diffusive force based on the principles of kinetic theory of dilute gases. The dilute gas kinetic theory result of Maxwell [65] is generally assumed to be an acceptable form for dense gases and liquids as well, although for these mixtures the binary diffusion coefficient is a concentration dependent, experimentally determined empirical parameter. 

For the description of inter-diffusion in dense gases and liquids the expression for ds is further modified introducing a fugacity (i.e., a corrected pressure) or an activity (i.e., a corrected mole fraction) function [76]. The activity as(T,ct 2, 3,a q i) for species s is defined by ... 

In applying this relationship to dipolar molecules in dense gases and liquids the field E is understood to be the local or effective field E. In order to estimate E one must solve the statistical mechanical problem which relates the local quantity to the macroscopic applied field E. The problem is solved by estimating the local field in a spherical cavity within the dielectric (fig. 4.1). The cavity is assumed to have molecular dimensions with diameter a. The material within the sphere is considered in terms of individual molecules, whereas that outside the sphere is... 

This survey begins with a summary of exact results for the thermal-force problem. Then approximate developments will be discussed in the light of available data. Finally, some developments for the thermal force in dense gases and liquids as well as new effects in dilute host gas will be described. 

Finally, just as thermophoresis has as a limit thermal diffusion- in dilute gas mixtures, so one would expect a thermophoretic effect on particles suspended in dense gases and liquids, whose limit would be thermal diffusion of mixtures in these media. The photophoretic effect may have been observed by BARKAS [2.145] in aqueous solutions of colloids. More recently, McNAB and MEISEN [2.146] have reported experimental evidence of thermophoresis in liquids for 1.011 and 0.79 ym spheres in water and n-hexane. They report that their data for the thermophoretic velocity are described by an empirical equation... 

There is no known PVTx equation of state that is suitable for calculation of fiigacity coefficients for alt mixtures at all possible conditions of interest. The choice of an equation of state for an engineering calculation is therefore often made on an ad hoc basis. Guidelines are available, but they reflect the inevitable compromise between simplicity and accuracy. We treat in the remainder of this section three popular classes of equations of state commonly employed for practical calculations the virial equations, u for gases at low to moderate densities the cubic equations of state (exemplified by the R lich-Kwong equations), used for dense gases and liquids and equations inspired by the so-called chemical theories, used for associating vapors and vapor mixtures. 

The approximation wherein one retains the first term in the memory function expression (105) is of special interest because it leads to a kinetic equation that is closely related to the Boltzmann-Enskog equation in transport theory. In this section we will investigate this particular approximation in some detail not only from the standpoint of further analytical analysis but also from the standpoint of practical calculations. We will see that within certain limitations the approximation results in a reasonably realistic description of dense gases and liquids, and in this sense represents the first step in a systematic microscopic calculation. 

We might just as easily have made use of some other statistical theory of dense gases and liquids to describe G(0) and /(0). However, the Carnahan - Starling model seems preferable since it does not involve unwieldy numerical calculation and leads to simple analytical expressions. In expressing these functions for low-concentration systems, it is also possible to use the standard technique of virial expansions. 

It is not difficult to also derive formulae for the mean square of concentrational fluctuations on the basis of any other statistical model proposed for dense gases and liquids (for a review, see reference [18,40,41]). 

It suffices to say that extensions of the reduced Liouville equation to higher orders, necessary for dense gases and liquids, become extremely cumbersome. Approximate solutions to the reduced Liouville equation for nonequilibrium dense gas and liquid systems will be considered in more detail in Chap. 6. In the next chapter, however, we will show that exact analytical solutions to the Liouville equation, and its reduced forms, are indeed possible for systems at equilibrium. 

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