Simple Models for Equations of State

As the density of a fluid is decreased, the effects of forces between molecules weaken, and the fluid behaves more like an ideal gas that is, the behavior of real fluids may simplify under extreme conditions. Another extreme occurs by making the temperature high, for then many simple fluids behave as if they were composed of hard spheres  

In a hard-sphere fluid each molecule occupies space and the molecules exert forces on one another, but those forces are purely repulsive and act only when spheres collide the spheres act like billiard balls. So the hard-sphere substance is an extreme model, but under certain conditions it is more realistic than the ideal-gas model. 

In a pure hard-sphere fluid, all spheres have the same diameter a and the compressibility factor Z depends on only the fluid density p = 1 /V, where is the number of spheres held in a vessel of volume V. For hard spheres, the density is conventionally cited in terms of the packing fraction q, which is the ratio of the volume of the spheres to the volume of their container  

Over the years numerous functional forms have been devised for the hard-sphere compressibility factor Z(q). A simple yet accurate expression has been devised by Carnahan and Starling [13]  

Since rj 1, the hard-sphere Z is always greater than unity and as ii - 0 this expression reduces to the ideal-gas value, Z = 1. 

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We then developed equations for computing the difference and ratio measures from the measurables P, v, T, and x. Data for these measurables are correlated by some volumetric equation of state, usually an analytic equation explicit in pressure Pip, T, x ) or explicit in volume v P, T, x ). So the equations we derived for the conceptual all involve integrals over appropriate functions of the equation of state. Then, in the last section of the chapter, we presented a few simple models for equations of state these models are sufficient to illustrate the problems that arise both in trying to use simple analytic functions to represent volumetric data and in evaluating the integrals that provide values for conceptual properties. 

The calculations reported in this paper and a related series of publications indicate that it is now quite feasible to obtain reasonably accurate results for phase equilibria in simple fluid mixtures directly from molecular simulation. What is the possible value of such results Clearly, because of the lack of accurate intermolecular potentials optimized for phase equilibrium calculations for most systems of practical interest, the immediate application of molecular simulation techniques as a replacement of the established modelling methods is not possible (or even desirable). For obtaining accurate results, the intermolecular potential parameters must be fitted to experimental results, in much the same way as parameters for equation-of-state or activity coefficient models. This conclusion is supported by other molecular-simulation based predictions of phase equilibria in similar systems (6). However, there is an important difference between the potential parameters in molecular simulation methods and fitted parameters of thermodynamic models. Molecular simulation calculations, such as the ones reported here, involve no approximations beyond those inherent in the potential models. The calculated behavior of a system with assumed intermolecular potentials is exact for any conditions of pressure, temperature or composition. Thus, if a good potential model for a component can be developed, it can be reliably used for predictions in the absence of experimental information. 

However, the description of gas solubility using activity coefficient models does require some explanation, and this is what is discussed in this section. The activity coefficient description is of interest because it is applicable to mixtures that are not easily describable by an equation of state, and also because it may be possible to make simple gas solubility estirnates using an activity coefficient model, whereas a computer program is required for equation-of-state calculations. 

To compute values for the deviation measures, we need volumetric data for the substance of interest such data are usually correlated in terms of a model PvTx equation of state. In 4.4 we develop expressions that enable us to use equations of state to compute difference and ratio measures for deviations from the ideal gas. Finally, in 4.5 we present a few simple models for the volumetric equation of state of real fluids. These few models are enough to introduce some of the problems that arise in attempting to analytically represent the PvTx behavior of real substances, and they allow us to compute values for conceptual, using the expressions from 4.5. However, more thorough expositions on equations of state must be found elsewhere [1-4]. 

Excitable media are some of tire most commonly observed reaction-diffusion systems in nature. An excitable system possesses a stable fixed point which responds to perturbations in a characteristic way small perturbations return quickly to tire fixed point, while larger perturbations tliat exceed a certain tlireshold value make a long excursion in concentration phase space before tire system returns to tire stable state. In many physical systems tliis behaviour is captured by tire dynamics of two concentration fields, a fast activator variable u witli cubic nullcline and a slow inhibitor variable u witli linear nullcline [31]. The FitzHugh-Nagumo equation [34], derived as a simple model for nerve impulse propagation but which can also apply to a chemical reaction scheme [35], is one of tire best known equations witli such activator-inlribitor kinetics ... 

Ruckenstein and Li proposed a relatively simple surface pressure-area equation of state for phospholipid monolayers at a water-oil interface [39]. The equation accounted for the clustering of the surfactant molecules, and led to second-order phase transitions. The monolayer was described as a 2D regular solution with three components singly dispersed phospholipid molecules, clusters of these molecules, and sites occupied by water and oil molecules. The effect of clusterng on the theoretical surface pressure-area isotherm was found to be crucial for the prediction of phase transitions. The model calculations fitted surprisingly well to the data of Taylor et al. [19] in the whole range of surface areas and the temperatures (Fig. 3). The number of molecules in a cluster was taken to be 150 due to an excellent agreement with an isotherm of DSPC when this... 

The much slower convergence of the entropy relative to the net free energy can be understood from the following simple model for calculating entropy. From the same equation of state that leads directly to Equation 13, it follows that... 

The PVT properties of aqueous solutions can be determined by direct measurements or estimated using various models for the ionic interactions that occur in electrolyte solutions. In this paper a review will be made of the methods presently being used to determine the density and compressibility of electrolyte solutions. A brief review of high-pressure equations of state used to represent the experimental PVT properties will also be made. Simple additivity methods of estimating the density of mixed electrolyte solutions like seawater and geothermal brines will be presented. The predicted PVT properties for a number of mixed electrolyte solutions are found to be in good agreement with direct measurements. 

In the present article, we focus on the scaled particle theory as the theoretical basis for interpreting the static solution properties of liquid-crystalline polymers. It is a statistical mechanical theory originally proposed to formulate the equation of state of hard sphere fluids [11], and has been applied to obtain approximate analytical expressions for the thermodynamic quantities of solutions of hard (sphero)cylinders [12-16] or wormlike hard spherocylinders [17, 18]. Its superiority to the Onsager theory lies in that it takes higher virial terms into account, and it is distinctive from the Flory theory in that it uses no artificial lattice model. We survey this theory for wormlike hard spherocylinders in Sect. 2, and compare its predictions with typical data of various static solution properties of liquid-crystalline polymers in Sects. 3-5. As is well known, the wormlike chain (or wormlike cylinder) is a simple yet adequate model for describing dilute solution properties of stiff or semiflexible polymers. 

Finally, it appears that the kinetic models of complex reactions contain two types of components independent of and dependent on the complex mechanism structure [4—7]. Hence the thermodynamic correctness of these models is ensured. The analysis of simple classes indicates that an unusual analog arises for the equation of state relating the observed characteristics of the open chemical system, i.e. a kinetic polynomial [7]. This polynomial distinctly shows how a complex kinetic relationship is assembled from simple reaction equations. 

A Statistical-Mechanics based Lattice-Model Equation of state (EOS) for modelling the phase behaviour of polymer-supercritical fluid mixtures is presented. The EOS can reproduce qualitatively all experimental trends observed, using a single, adjustable mixture parameter and in this aspect is better than classical cubic EOS. Simple mixtures of small molecules can also be quantitatively modelled, in most cases, with the use of a single, temperature independent adjustable parameter. 

To get an idea about the relative volatilities of components we proceed with a simple flash of the outlet reactor mixture at 33 °C and 9 bar. The selection of the thermodynamic method is important since the mixture contains both supercritical and condensable components, some highly polar. From the gas-separation viewpoint an equation of state with capabilities for polar species should be the first choice, as SR-Polar in Aspen Plus [16]. From the liquid-separation viewpoint liquid-activity models are recommended, such as Wilson, NRTL or Uniquac, with the Hayden O Connell option for handling the vapor-phase dimerization of the acetic acid [3]. Note that SR-Polar makes use of interaction parameters for C2H4, C2H6 and C02, but neglects the others, while the liquid-activity models account only for the interactions among vinyl acetate, acetic acid and water. To overcome this problem a mixed manner is selected, in which the condensable components are treated by a liquid-activity model and the gaseous species by the Henry law. 

Integral equation methods provide another approach, but their use is limited to potential models that are usually too simple for engineering use and are moreover numerically difficult to solve. They are useful in providing equations of state for certain simple reference fluids (e.g., hard spheres, dipolar hard spheres, charged hard spheres) that can then be used in the perturbation theories or density functional theories. 

Plot the Leimard-Jones potentials for each of the gases studied. Obtain ft from Eqs. (16)-(18) by numerioal integration and compare the values from this two-parameter potential with those from the van der Waals and Beattie-Bridgeman equations of state. Optional A simple square-well potential model can also be used to eradely represent the interaction of two molecules. In place of Eq. (18), use the square-well potential and parameters of Ref. 6 to ealeulate /t. Contrast with the results from the Lennard-Jones potential and comment on the sensitivity of the calculations to the form of the potential.]... 

The model based on S-E Case 2 kinetics has been quite successful in handling steady-state data for styrene emulsion polymerizations in a CSTR. One or more of the mechanisms described above, however, generally cause other monomer systems to deviate from this simple model. The nature of these deviations varies among the different monomers. If published literature data are fitted to equations of the type listed below one can obtain values for the exponents a, h, and c. 

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