# Equation of state

A rigorous relation exists between the fugacity of a component in a vapor phase and the volumetric properties of that phase these properties are conveniently expressed in the form of an equation of state. There are two common types of equations of state one of these expresses the volume as a function of [c.15]

This chapter presents a general method for estimating nonidealities in a vapor mixture containing any number of components this method is based on the virial equation of state for ordinary substances and on the chemical theory for strongly associating species such as carboxylic acids. The method is limited to moderate pressures, as commonly encountered in typical chemical engineering equipment, and should only be used for conditions remote from the critical of the mixture. [c.26]

The fugacity coefficient is a function of temperature, total pressure, and composition of the vapor phase it can be calculated from volumetric data for the vapor mixture. For a mixture containing m components, such data are often expressed in the form of an equation of state explicit in the pressure [c.26]

The fugacity coefficient can be found from the equation of state using the thermodynamic relation (Beattie, 1949) [c.26]

Numerous empirical equations of state have been proposed but the theoretically based virial equation (Mason and Spurling, 1969) is most useful for our purposes. We use this equation for systems which do not contain carboxylic acids. [c.27]

The virial equation of state is a power series in the reciprocal molar volume or in the pressure [c.27]

This chapter uses an equation of state which is applicable only at low or moderate pressures. Serious error may result when the truncated virial equation is used at high pressures. [c.38]

The Virial Equation of State, Pergamon Press, Oxford (1969) [c.38]

Enthalpies are referred to the ideal vapor. The enthalpy of the real vapor is found from zero-pressure heat capacities and from the virial equation of state for non-associated species or, for vapors containing highly dimerized vapors (e.g. organic acids), from the chemical theory of vapor imperfections, as discussed in Chapter 3. For pure components, liquid-phase enthalpies (relative to the ideal vapor) are found from differentiation of the zero-pressure standard-state fugacities these, in turn, are determined from vapor-pressure data, from vapor-phase corrections and liquid-phase densities. If good experimental data are used to determine the standard-state fugacity, the derivative gives enthalpies of liquids to nearly the same precision as that obtained with calorimetric data, and provides reliable heats of vaporization. [c.82]

For a real vapor mixture, there is a deviation from the ideal enthalpy that can be calculated from an equation of state. The enthalpy of the real vapor is given by [c.84]

Given the estimate of the reactor effluent in Example 4.2 for fraction of methane in the purge of 0.4, calculate the.actual separation in the phase split assuming a temperature in the phase separator of 40°C. Phase equilibrium for this mixture can be represented by the Soave-Redlich-Kwong equation of state. Many computer programs are available commercially to carry out such calculations. [c.113]

TABLE 4.3 Vapor-Liquid Phase Split Using the Soave-Redlich-Kwong Equation of State [c.114]

Assuming that the phase split operates at 40 bar and 40 C, a rigorous solution of the phase equilibrium using the Soave-Redlich-Kwong equation of state and the recycle equations using flowsheet simulation software gives a composition of the reactor effluent given in Table 4.4. [c.114]

For reduced temperatures higher than 0.98, a second type of method must be used that is based on an equation of state such as that of Lee and Kesler. [c.114]

Each fluid is described by a BWR equation of state whose coefficients are adjusted to obtain simultaneously the vapor pressure, enthalpies of liquid and gas as well as the compressibilities. The compressibility z of any fluid is calculated using the equation below [c.119]

There have been several equations of state proposed to express the compressibility factor. Remarkable accuracy has been obtained when specific equations for certain components are used however, the multitude of their coefficients makes their extension to mixtures complicated. [c.138]

Hydrocarbon mixtures are most often modeled by the equations of state of Soave, Peng Robinson, or Lee and Kesler. [c.138]

Utilization of equations of state derived from the Van der Waals model has led to spectacular progress in the accuracy of calculations at medium and high pressure. [c.152]

In the calculation of vapor phase partial fugacities the use of an equation of state is always justified. In regard to the liquid phase fugacities, there is a choice between two paths [c.152]

In 1972, Soave published a method of calculating fugacities based on a modification of the Redlich and Kwong equation of state which completely changed the customary habits and became the industry standard. In spite of numerous attempts to improve it, the original method is the most widespread. For hydrocarbon mixtures, its accuracy is remarkable. For a mixture, the equation of state is [c.154]

The Soave equation of state, which is a third order equation for V, and which can be put in the form + aV + /3V + y = 0, has, according to the [c.155]

Kabadi, V.N. and R.P. Danner (1985), A modified SRK equation of state for water-hydrocarbon phase equilibria . Ind. Eng. Chem. Proc. Des. Dev., Vol. 24, No. 3, p. 537. [c.457]

Peng, D.Y. and D.B. Robinson (1976), "A new two-constant equation of state". Ind. Eng. Chem. Fund, Vol. 15, No. 1, p. 59. [c.458]

Rackett, H.G. (1970), "Equation of state for saturated liquids". J. Chem. Eng. Data, Vol. 15, No. 4, p. 514. [c.459]

Soave, G. (1972), Equilibrium constants from a modified Redlich-Kwong equation of state . Chem. Eng. Sci., Vol. 27, p. 1197. [c.459]

RKS Redlich-Kwong-Soave equation of state [c.503]

SRK Soave-Redlich-Kwong equation of state [c.503]

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. [c.105]

The function of thermodynamics is to provide phenomenological relationships whose validity has the authority of the laws of thermodynamics themselves. One may proceed further, however, if specific models or additional assumptions are made. For example, the use of the van der Waals equation of state allows an analysis of how P - p in Eq. III-40 should vary across the interface Tolman [36,37] made an early calculation of this type. There has been a high degree of development of statistical thermodynamics in this field (see Ref. 47 and the General References and also Sections XV-4 and XVI-3). A great advantage of this approach is that one may derive thermodynamic properties from knowledge of the intermolecular forces in the fluid. Many physical systems can be approximated with model interaction potential energies a widely used system comprises attractive hard spheres where rigid spheres of diameter b interact with an attractive potential energy, att( )- [c.61]

The gradient model has been combined with two equations of state to successfully model the temperature dependence of the surface tension of polar and nonpolar fluids [54]. Widom and Tavan have modeled the surface tension of liquid He near the X transition with a modified van der Waals theory [55]. [c.62]

The data could be expressed equally well in terms of F versus P, or in the form of the conventional adsorption isotherm plot, as shown in Fig. Ill-18. The appearance of these isotherms is discussed in Section X-6A. The Gibbs equation thus provides a connection between adsorption isotherms and two-dimensional equations of state. For example, Eq. III-57 corresponds to the adsorption isotherm [c.86]

This derivation has been made in a form calculated best to bring out the very considerable and sometimes inconsistent approximations made. However, by treating the surface region as a kind of solution, an avenue is opened for employing our considerable knowledge of solution physical chemistry in estimating association, interionic attraction, and other nonideality effects. Another advantage, from the writers point of view, is the emphasis on the role of the solvent as part of the surface region, which helps to correct the tendency, latent in the two-dimensional equation of state treatment, to regard the substrate as merely providing an inert plane surface on which molecules of the adsorbed species may move freely. TTie approach is not really any more empirical than that using the two-dimensional nonideal gas, and considerable use has been made of it by Fowkes [137]. [c.89]

VPLQFT is a computer program for correlating binary vapor-liquid equilibrium (VLE) data at low to moderate pressures. For such binary mixtures, the truncated virial equation of state is used to correct for vapor-phase nonidealities, except for mixtures containing organic acids where the "chemical" theory is used. The Hayden-0 Connell (1975) correlation gives either the second virial coefficients or the dimerization equilibrium constants, as required. [c.211]

IF BINARY SYSTEM CONTAINS NO ORGANIC ACIDS. THE SECOND VIRTAL coefficients ARE USED IN A VOLUME EXPLICIT EQUATION OF STATE TO CALCULATE THE FUGACITY COEFFICIENTS. FOR ORGANIC ACIDS FUGACITY COEFFICIENTS ARE PREDICTED FROM THE CHEMICAL THEORY FOR NQN-IOEALITY WITH EQUILIBRIUM CONSTANTS OBTAINED from METASTABLE. BOUND. ANO CHEMICAL CONTRIBUTIONS TO THE SECOND VIRIAL COEFFICIENTS. [c.266]

Firstly, one have to develop a numerical model (the forward problem) able to regenerate the responses supplied by the sensor. Unfortunately, the relationship between the object function and the observed data which is used to invert eddy current data is inherently non linear because it consists in a pair of coupled integral equations, which involves the product of two unknowns the flaw conductivity and the true electric field within the flawed region. Several methods have been developed to solve this problem. Some sophisticated methods [12], [11], [5] seek to reconstruct simultaneously the object function and the diffracted electric field. They involve a non-linearized iterative process which leads to minimize a cost-functional depending on two terms the error between the computed scattered field at the present iteration and the measured data, and the error in satisfying the equation of state. So, this way requires to solve the direct-scattering problem in each step of iterations and such methods need more computations in the three-dimensional problem. In this work, we assume that the hypothesis using Born approximation is fuUfilled for solving the linearized inverse problem[4]. We assume therefore that the perturbation of the electric field within the flawed region is small and that the flawed region is uniformely illuminated by the incident field. We consider that the linearized model makes a relatively good compromise between the fidelity to measured data and the simplicity of the model. [c.326]

The first equation (1) is the equation of state and the second equation (2) is derived from the measurement process. Finally, G5 (r,r ) is a row-vector that takes the three components of the anomalous ciurent density vector Je (r) =

Various other non-ideal-gas-type two-dimensional equations of state have been proposed, generally by analogy with gases. Volmer and Mahnert [128, [c.83]

See pages that mention the term

**Equation of state**:

**[c.15] [c.137] [c.220] [c.417] [c.109] [c.109] [c.152] [c.154] [c.493] [c.493] [c.499] [c.82]**

See chapters in:

** High pressure shock compression of solids
-> Equation of state
**

Physical chemistry of surfaces (0) -- [ c.0 ]

Solids under high-pressure shock compression - mechanics, physics, and chemistry (1992) -- [ c.3 , c.7 , c.37 ]