Other Equations of State

In addition to the Van der Waals and virial equation of state, other equations obtained from experiments are presented in Table 19.5. 

When pure substances are found to deviate from ideal gas behavior, equations of state other than the ideal gas law can be used to describe the substances. Some of the equations of state used for substances whose molecules are small are described in the following sections. 

The equation of state for an ideal gas, that is a gas in which the volume of the gas molecules is insignificant, attractive and repulsive forces between molecules are ignored, and molecules maintain their energy when they collide with each other. 

Various other non-ideal-gas-type two-dimensional equations of state have been proposed, generally by analogy with gases. Volmer and Mahnert [128,... 

On the other hand, as applied to the submonolayer region, the same comment can be made as for the localized model. That is, the two-dimensional non-ideal-gas equation of state is a perfectly acceptable concept, but one that, in practice, is remarkably difficult to distinguish from the localized adsorption picture. If there can be even a small amount of surface heterogeneity the distinction becomes virtually impossible (see Section XVll-14). Even the cases of phase change are susceptible to explanation on either basis. 

It is a universal experimental observation, i.e. a law of nature , that the equations of state of systems 1 and 2 are then coupled as if the wall separating them were diathemiic rather than adiabatic. In other words, there is a relation... 

Although the previous paragraphs hint at the serious failure of the van der Waals equation to fit the shape of the coexistence curve or the heat capacity, failures to be discussed explicitly in later sections, it is important to recognize that many of tlie other predictions of analytic theories are reasonably accurate. For example, analytic equations of state, even ones as approximate as that of van der Waals, yield reasonable values (or at least ball park estmiates ) of the critical constants p, T, and V. Moreover, in two-component systems... 

In reeent years global phase diagrams have been ealeulated for other equations of state, not only van der Waals-like ones, but others with eomplex temperature dependenees. Some of these have managed to find type VI regions in the overall diagram. Some of the reeent work was brought together at a 1999 eonferenee [4]. 

Other pressure—volume—temperature (PVT) relationships may be found in the Hterature ie, Benedict, Webb, Rubin equations of state (4—7) the Benedict, Webb, Rubin, Starling equation of state (8) the Redlich equation of state (9) and the Redlich-Kwong equation of state (10). 

Gamma/Phi Approach For many XT E systems of interest the pressure is low enough that a relatively simple equation of state, such as the two-term virial equation, is satisfactoiy for the vapor phase. Liquid-phase behavior, on the other hand, may be conveniently described by an equation for the excess Gibbs energy, from which activity coefficients are derived. The fugacity of species i in the liquid phase is then given by Eq. (4-102), written... 

Evaluation of 9 is usually by Eq. (4-196), based on the two-term virial equation of state, but other equations, such as Eq. (4-200), are also applicable. The activity coefficient Jj is evaluated by Eq. (4-119), which refates In Jj to G /RT as a partial proper. Thus, what is required for the hquid phase is a relation between G /BT and composition. Equations in common use for this purpose have already been described. 

Other volume-explicit equations of state are sometimes required, such as the compressibility equation V = zRT/P or the truncated virial equation V= (1 -i- B P)RT/P. The quantities z a.ndB are not constants, so some land of averaging will be required. More accurate equations of state are even more difficult to use but are not often justified for kinetic work. 

A variety of equations-of-state have been applied to supercritical fluids, ranging from simple cubic equations like the Peng-Robinson equation-of-state to the Statistical Associating Fluid Theoiy. All are able to model nonpolar systems fairly successfully, but most are increasingly chaUenged as the polarity of the components increases. The key is to calculate the solute-fluid molecular interaction parameter from the pure-component properties. Often the standard approach (i.e. corresponding states based on critical properties) is of limited accuracy due to the vastly different critical temperatures of the solutes (if known) and the solvents other properties of the solute... 

This equation is not particularly useful in practice, since it is difficult to quantify the relationship between concentration and ac tivity. The Floiy-Huggins theory does not work well with the cross-linked semi-ciystaUine polymers that comprise an important class of pervaporation membranes. Neel (in Noble and Stern, op. cit., pp. 169-176) reviews modifications of the Stefan-Maxwell approach and other equations of state appropriate for the process. 

The available isentropic head is usually calculated by computer, using any of the various equations of state. In the absence of such facihty, a quick and reasonably reliable calculation follows. In fact, this calculation is valuable as a cross-check on other methods because it is likely to be accurate within a few percent. 

In an ideal fluid, the stresses are isotropic. There is no strength, so there are no shear stresses the normal stress and lateral stresses are equal and are identical to the pressure. On the other hand, a solid with strength can support shear stresses. However, when the applied stress greatly exceeds the yield stress of a solid, its behavior can be approximated by that of a fluid because the fractional deviations from stress isotropy are small. Under these conditions, the solid is considered to be hydrodynamic. In the absence of rate-dependent behavior such as viscous relaxation or heat conduction, the equation of state of an isotropic fluid or hydrodynamic solid can be expressed in terms of specific internal energy as a function of pressure and specific volume E(P, V). A familiar equation of state is that for an ideal gas... 

Arrival time gauges alone can lead to equation-of-state data in other ways, as well, but they are most often used in conjunction with other gauges to be described later. Three different arrival time gauges are discussed below. 

In this chapter we define what is meant by a shock-wave equation of state, and how it is related to other types of equations of state. We also discuss the properties of shock-compressed matter on a microscopic scale, as well as discuss how shock-wave properties are measured. Shock data for standard materials are presented. The effects of phase changes are discussed, the measurements of shock temperatures, and sound velocities of shock materials are also described. We also describe the application of shock-compression data for porous media. 

The motion of disloeations under eonditions of shoek-wave eompression takes plaee at sueh high veloeities (approaehing the elastie sound speed) that many vaeaneies and interstitials are left behind. However, these point defeets ean anneal out at room temperature and are thus diflieult to study by shoek-reeovery teehniques. The presenee of point defeets has little effeet on the material eompressibility and other properties related to equation of state. While they also have little direet influenee on the relief of shear stresses, point defeets do influenee the mobility and multiplieation of disloeations. This, in turn, affeets most of what happens under shoek-wave loading eonditions. 

The chapter on equation-of-state properties provides the basic approaches used for describing the high-pressure shock-compression response of materials. These theories provide the basis for separating the elastic compression components from the thermal contributions in shock compression, which is necessary for comparing shock-compression results with those obtained from other techniques such as isothermal compression. A basic understanding of the simple theories of shock compression, such as the Mie-Gruneisen equation of state, are prerequisite to understanding more advanced theories that will be discussed in subsequent volumes. 

Zudkeviteh, D. and Joffe, J., Conelation and Predietion of Vapor-Liquid Equilibria with the Redlieh-Kwong Equation of State, AlChE Journal, Vol. 16, No. 1, pp. 112-119, January 1970. Mohsen-Nia, M., Moddaress, H., and Mansoori, G. A., A Simple Cubie Equation of State for Hydroearbons and Other Compounds, SPE Paper No. 26667, Proeeedings of the 1993 Annual Teehnieal Conferenee and Exhibition of the Soeiety of Petroleum Engineers, Houston, Texas, 1993. 

The other factors mentioned come into play with the inherent mismatch that occurs when an overall factor must be appUed to correlate the flange-to-flange method. The correlation proposed in the code is based on work by Schultz [8] and was quite good for its day. When combined with modem calculation methods and equations of state, the philosophy is still valid. 

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