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Virial

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. [Pg.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. [Pg.27]

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

For a pure vapor the virial coefficients are functions only of temperature for a mixture they are also functions of composition. An important advantage of the virial equation is that there are theoretically valid relations between the virial coefficients of a mixture and its composition. These relations are ... [Pg.28]

Equation (10a) is somewhat inconvenient first, because we prefer to use pressure rather than volume as our independent variable, and second, because little is known about third virial coefficients It is therefore more practical to substitute... [Pg.28]

Equations (7b) and (8) into Equation (6), neglecting all third virial coefficients. We then obtain... [Pg.28]

At moderate densities. Equation (3-lOb) provides a very good approximation. This approximation should be used only for densities less than (about) one half the critical density. As a rough rule, the virial equation truncated after the second term is valid for the present range... [Pg.29]

To use Equation (10b), we require virial coefficients which depend on temperature. As discussed in Appendix A, these coefficients are calculated using the correlation of Hayden and O Connell (1975). The required input parameters are, for each component critical temperature T, critical pressure P, ... [Pg.29]

Figure 1 shows second virial coefficients for four pure fluids as a function of temperature. Second virial coefficients for typical fluids are negative and increasingly so as the temperature falls only at the Boyle point, when the temperature is about 2.5 times the critical, does the second virial coefficient become positive. At a given temperature below the Boyle point, the magnitude of the second virial coefficient increases with... [Pg.29]

Figure 3-1. Second virial coefficients for four fluids. Figure 3-1. Second virial coefficients for four fluids.
The virial equation is appropriate for describing deviations from ideality in those systems where moderate attractive forces yield fugacity coefficients not far removed from unity. The systems shown in Figures 2, 3, and 4 are of this type. However, in systems containing carboxylic acids, there prevails an entirely different physical situation since two acid molecules tend to form a pair of stable hydrogen bonds, large negative... [Pg.31]

A component in a vapor mixture exhibits nonideal behavior as a result of molecular interactions only when these interactions are very wea)c or very infrequent is ideal behavior approached. The fugacity coefficient (fi is a measure of nonideality and a departure of < ) from unity is a measure of the extent to which a molecule i interacts with its neighbors. The fugacity coefficient depends on pressure, temperature, and vapor composition this dependence, in the moderate pressure region covered by the truncated virial equation, is usually as follows ... [Pg.37]

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. [Pg.38]

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

To illustrate calculations for a binary system containing a supercritical, condensable component. Figure 12 shows isobaric equilibria for ethane-n-heptane. Using the virial equation for vapor-phase fugacity coefficients, and the UNIQUAC equation for liquid-phase activity coefficients, calculated results give an excellent representation of the data of Kay (1938). In this case,the total pressure is not large and therefore, the mixture is at all times remote from critical conditions. For this binary system, the particular method of calculation used here would not be successful at appreciably higher pressures. [Pg.59]

Figure 13 presents results for a binary where one of the components is a supercritical, noncondensable component. Vapor-phase fugacity coefficients were calculated with the virial... [Pg.59]

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. [Pg.82]

Individual contributions to the second virial coefficient are calculated from temperature-dependent correlations ... [Pg.130]

As discussed in Chapter 3, the virial equation is suitable for describing vapor-phase nonidealities of nonassociating (or weakly associating) fluids at moderate densities. Equation (1) gives the second virial coefficient which is used directly in Equation (3-lOb) to calculate the fugacity coefficients. [Pg.133]

Equilibrium constants,, for all possible dimerization reactions are calculated from the metastable, bound, and chemical contributions to the second virial coefficients, B , as given by Equations (6) and (7). The equilibrium constants, K calculated using Equation (3-15). [Pg.133]

The total free contribution to the second virial coeffi-F... [Pg.133]

The Lewis fugacity rule is used for calculating the fugacity coefficients of the true species, and (2) the second virial co-... [Pg.134]

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. [Pg.211]

Subroutine BIJS2. This subroutine calculates the pure-component and cross second virial coefficients for binary mixtures according to the method of Hayden and O Connell (1975). [Pg.220]

CALCULATE URE COMPONENT ANO CROSS SECOND VIRIAL COEPFICfENTS C... [Pg.262]

PARAMETER USED TO CALCULATE PART OF CHEMICAL CONTRIBUTION TO THE SECOND VIRIAL COEFFICIENT. CALCULATED ONE OF TWO WAYS DEPENDING ON THE VALUE OF ETA(IJ). [Pg.262]

CALCULATE THE MODIFIED REDUCED DIPOLE TO BE USED IN CALCULATING THE FREE-POLAR CONTRIBUTION TO THE VIRIAL COEFFICIENT. [Pg.264]

CALCULATE THE TEMPERATURE DEPENDENT SECOND VIRIAL COEFFICIENTS. [Pg.264]

CALCULATE THE FREE CONTRIBUTION TO THE SECOND VIRIAL COEFFICIENT,... [Pg.264]


See other pages where Virial is mentioned: [Pg.15]    [Pg.16]    [Pg.27]    [Pg.28]    [Pg.28]    [Pg.28]    [Pg.29]    [Pg.31]    [Pg.31]    [Pg.33]    [Pg.34]    [Pg.36]    [Pg.37]    [Pg.84]    [Pg.134]    [Pg.137]    [Pg.220]    [Pg.262]    [Pg.263]    [Pg.266]   
See also in sourсe #XX -- [ Pg.309 ]

See also in sourсe #XX -- [ Pg.57 , Pg.58 , Pg.155 , Pg.297 ]

See also in sourсe #XX -- [ Pg.190 , Pg.191 , Pg.192 , Pg.193 , Pg.194 , Pg.195 ]

See also in sourсe #XX -- [ Pg.77 ]

See also in sourсe #XX -- [ Pg.309 ]

See also in sourсe #XX -- [ Pg.220 ]

See also in sourсe #XX -- [ Pg.220 ]

See also in sourсe #XX -- [ Pg.1044 ]

See also in sourсe #XX -- [ Pg.591 ]

See also in sourсe #XX -- [ Pg.234 ]




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A2 Second virial coefficient

Absorption virial expansion

Acoustic virial coefficients

Adsorption isotherm virial

Adsorption isotherms Virial isotherm

An example the second virial coefficient

Apparent second virial coefficient

Apparent second virial coefficient definition

Application of the Virial Equation

Atomic force and virial theorems in the presence of external fields

Atomic virial

Atomic virial definition

Atomic virial theorem

Bader virial partitioning method

Basin virial

Burnett second virial coefficients

Calculation of Interaction Virial Coefficients

Chain virial coefficients

Clausius, virial theorem

Clausius’ virial

Coefficient of the virial

Coefficients, molar virial

Coil molecules virial coefficient

Convergence of the Virial Series

Cross second virial coefficient

D-INTERPOLATION OF VIRIAL COEFFICIENTS

Density Virial Series for Nonpolar Fluids

Determination of Molecular Weight and Second Virial Coefficient

Dielectric virial coefficients

Differential virial theorem

Diffusion second virial coefficient

Diffusion virial term

Dimensionless virial coefficients

Dispersion virial expansions

Distribution function virial expansion

Effective interaction and second virial coefficient

Electronic kinetic energy, virial

Electronic kinetic energy, virial theorem

Energy quantum virial theorem

Equation of state virial

Equation virial

Equation virial isotherm

Equation virial-type expansion

Equations virial equation

Equations virial expansion equation

Equipartition and the Virial

Ethylene second virial coefficients

Evaluation of Virial Coefficients

Example 8.5. Estimation of Second Virial Coefficients Nonpolar Compounds

Example 8.6. Estimation of Second Virial Coefficients Polar Compounds

Exchange-correlation potential virial theorem

Experimental Methods for Determining Virial Coefficients

Factor virial

First virial coefficient

Flory theory of the second virial coefficient

Force Field, Virial Calculation of Stress

Force constants virial theorem

Force virial, intermolecular

Formal expression for the second virial coefficient

Friction virial coefficient

Fugacities with the Virial Equation

Fugacity Coefficient of the Virial Equation (Leiden Form)

Fugacity coefficient from virial equation

Gas-solid virial coefficients

Gases virial coefficients

General properties virial relation

Generalized correlations second virial coefficient

Generalized virial functions

Hard-sphere fluid second virial coefficient

Helium second dielectric virial coefficient

Hellmann-Feynman and virial theorems

Helmholtz energy and development of the virial

Ideality, deviation from virial function

Interaction parameter from virial coefficient

Interatomic surface virial

Intermolecular potential, effect second virial coefficient

Intermolecular virial

Isotherm virial

Kerr virial coefficient, polarizability

Kerr virial coefficients

Kinetic energy components, virial theorem

Line shape virial expansion

Local virial relationship

Local virial theorem

Low Densities The Virial Equation

Measuring virial coefficients

Mixed Second Virial Coefficients

Mixture virial equation

Mixtures virial coefficients, calculation

Models virial equations

Molecular oxygen second virial coefficients

Molecular virial

Molecular weight and second virial

Molecular weight and second virial coefficient

Mutual virial coefficient

Nonspherical molecules, second virial coefficient

Onsager virial expansion

Operator virial

Osmotic pressures virial expansion

Osmotic virial

Osmotic virial coefficients

Osmotic virial expansion

Poly second virial coefficients

Polymer virial coefficient

Polymers solutions, virial coefficients

Polystyrene second virial coefficient

Pressure surface flux virial

Pressure virial

Pressure virial coefficients

Pressure virial equation

Pressure virial equation of state

Pressure virial equation state

Pressure virial expansion

Pure virial equation

Quantum mechanics virial theorem

Quantum virial theorem

Reduced virial coefficients

Relationship between Second Virial Coefficients of Leiden and Berlin Form

Residual properties from virial equation

Second Virial Coefficients of Polymer

Second Virial Coefficients of Polymer Solutions

Second and third virial coefficients at given temperature

Second osmotic virial coefficient

Second thermal conductivity virial coefficient

Second transport virial coefficient

Second virial approximation

Second virial coefficient Flory-Huggins

Second virial coefficient comparison

Second virial coefficient definition

Second virial coefficient determination

Second virial coefficient estimated

Second virial coefficient experimental

Second virial coefficient from potential function

Second virial coefficient light scattering

Second virial coefficient of mixtures

Second virial coefficient of osmotic

Second virial coefficient of osmotic pressure

Second virial coefficient of water

Second virial coefficient osmotic pressure

Second virial coefficient reduced

Second virial coefficient table

Second virial coefficient value errors

Second virial coefficient, enthalpy

Second virial coefficient-molecular weight

Second virial coefficients

Second virial coefficients excluded-volume

Second virial coefficients for

Second virial coefficients) from chromatography

Second virial coefficients) from light scattering

Second virial coefficients) temperature dependence

Second virial coefficients) units

Second virial coefficients, dependence

Second virial coefficients, dependence molecular weight

Second viscosity virial coefficient

Second viscosity virial coefficient water

Solution micelles second virial coefficient

Spectral moment virial expansion

Sphere virial coefficient

Statistical mechanics second virial coefficient

Statistical mechanics virial equations

Stress tensor virial expression

Surface flux virial

Tensor virial theorem

The Virial Equation

The Virial Equation Explicit in Pressure

The Virial Equation of State

The Virial Theorem

The Virial Theorem and Chemical Bonding

The Virial Theorem for Atoms and Diatomic Molecules

The molecular electronic virial theorem

The second virial coefficients

The third virial coefficient

The virial coefficients

The virial expansion

Theoretical treatment using virial

Theoretical treatment using virial expansions

Third Virial Coefficients of Mixtures

Third osmotic virial coefficient

Third virial coefficient

Third virial coefficient reduced

Variational derivation of the atomic virial theorem

Virial Coef cients

Virial Coefficient Prediction

Virial Coefficients of Selected Gases

Virial Terms Links

Virial Theorem condition

Virial Tsonopoulos method

Virial acoustic

Virial activity models

Virial charges

Virial coefficient approach

Virial coefficient classic fluid

Virial coefficient configuration integral

Virial coefficient determination

Virial coefficient equations for

Virial coefficient isotope effect

Virial coefficient mean-field

Virial coefficient system

Virial coefficients (Pitzer

Virial coefficients For water

Virial coefficients asymptotic behavior

Virial coefficients calculation

Virial coefficients definitions

Virial coefficients estimation

Virial coefficients evaluation

Virial coefficients for mixtures

Virial coefficients from statistical mechanics

Virial coefficients measurement

Virial coefficients mixing rules

Virial coefficients of hard spheres

Virial coefficients of mixtures

Virial coefficients of polymer solutions

Virial coefficients of water

Virial coefficients state dependence

Virial coefficients, renormalization

Virial coefficients, second mixtures

Virial coefficients, use

Virial coefficients, values

Virial coefficients/series

Virial composition dependence

Virial convergence

Virial correction

Virial cross-coefficients

Virial derivation

Virial dimensionless

Virial energy

Virial equation limitations

Virial equation methods

Virial equation of state for gases

Virial equation of state truncated

Virial equation state

Virial equations of state for pure gases

Virial ethylene

Virial expansion

Virial expansion model

Virial expansion nonideal solutions

Virial expansion of line shape

Virial expansion of spectral moments

Virial expansion reduced

Virial expansion validity

Virial expression

Virial expression for the bulk pressure

Virial field

Virial formulation

Virial fragments

Virial lines

Virial methods

Virial mixtures

Virial partitioning

Virial partitioning method

Virial ratio

Virial regime

Virial relation

Virial remainder

Virial second

Virial series

Virial state

Virial temperature dependence

Virial tensor

Virial theorem

Virial theorem 414 Subject

Virial theorem applications

Virial theorem derivation

Virial theorem for atoms

Virial theorem of Clausius

Virial theorem system

Virial theorem, definition

Virial theorem, electronic kinetic

Virial theorem, of mechanics

Virial third

Virial value errors

Virial, coefficients

Virial-type expansion

Water, virial coefficient

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