Coefficients, virial

Equation (P3.6.3) shows the difference in chemical potential of the solvent in the solution and in the pure state to be a power series in solute concentration. Such equations are called virial equations (Rudin, 1982). 

Equation (P3.6.3) is the key to the application of colligative properties to polymer molecular weights. Thus, insertion of experimental values of ci, Pi, and iHi n ) into Eq. (P3.6.3) wouldprovide a measure of the solute molecular weight 

Equation (P3.6.3) derived above for ideal solutions is invalid for real solutions and polymer solutions are rarely ideal even at the highest dilution that can be used in practice. It is, however, useful to retain the form of the ideal equation and express the deviation of real solutions in terms of empirical parameters. Thus, for a real solution Eq. (P3.6.3) is expressed in a parallel form as 

In a bicomponent solution the subscript 2, referring to solute, is often deleted. Equation (3.68) may thus be written as (Rudin, 1982) 

To obtain virial expressions for colligative properties, Eq. (3.69) may be combined with respective Eqs. (3.56)-(3.59). If, for example, osmotic data are used, Eq. (3.69) is combined with Eq. (3.59) to write the virial equation as 

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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 ... 

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... 

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

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, ... 

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... 

Figure 3-1. Second virial coefficients for four fluids.

Individual contributions to the second virial coefficient are calculated from temperature-dependent correlations ... 

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. 

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). 

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. 

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). 

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). 

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

CALCULATE THE TEMPERATURE DEPENDENT SECOND VIRIAL COEFFICIENTS. 

CALCULATE THE FREE CONTRIBUTION TO THE SECOND VIRIAL COEFFICIENT,... 

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. 

CALCULATE SECONO VIRIAL COEFFICIENTS UNLESS TEMPERATURE IS... 

GET SECOND VIRIAL COEFFICIENTS BIJ IN /VIRIAL/ llO CALL BlJS(N,ID,KEY,T,eR8)... 

CALCULATE EFF SECOND VIRIAL COEFFICIENT FOR COMP I IN MIXTURE, SS(I)... 

BUS calculated second virial coefficients for pure compoments and all binary pairs in a mixture of N components (N 20) at specified temperature. These coefficients are placed in common storage /VIRIAL/. 

Second virial coefficients are calculated using the equations for the Hayden-0 Connell correlation (see Appendix A). 

CALCULATION OF TOTAL VIRIAL COEFFICIENT FOR CASES WITHOUT ASSOCIATING 4> VAPORS ... 

ERROR RETURN FOR FAILURE TO FIND VALID VIRIAL COEFFICIENTS ... 

Key = 1 represents an initial calculation for a new system Key 2-5 are subsequent calculations not differing significantly in time requirements Key = 6,7 require temperature derivatives of virial coefficients. 

Stigter and Dill [98] studied phospholipid monolayers at the n-heptane-water interface and were able to treat the second and third virial coefficients (see Eq. XV-1) in terms of electrostatic, including dipole, interactions. At higher film pressures, Pethica and co-workers [99] observed quasi-first-order phase transitions, that is, a much flatter plateau region than shown in Fig. XV-6. 

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