Virial coefficients, use

Young, C.L. Cruickshank, A.J.B. Gainey, B.W. Hicks, C.P. Letcher, T.M. Moody, R.W. Gas-Liquid Chromatographic Determination of Cross-Term Second Virial Coefficients using Glycerol, Trans. Far. Soc., 65, 1014(1969). 

For a theta solvent (V2 = 0) the relevant interaction is described by the third virial coefficient using a simple Alexander approach similar to the one leading to Eq. 13, the brush height is predicted to vary with the grafting density as h pa in agreement with computer simulations [65]. 

The free energy of a slightly imperfect gas may be evaluated easily in terms of the second virial coefficient. Using (11.14) in (11.7) we obtain... 

The tie lines in Figures 1, 2, and 3 were calculated from the phase equilibrium relations represented by Equations 21 which upon substitution of the chemical potential models become Equations 22. There are three equations and four unknown molalities. For each tie line we therefore set a value for one of the molalities, which in our case was that of dextran in the bottom phase, and simultaneously solved Equations 22 for the remaining three molalities. The numerical algorithm used was the same one that Edmond and Ogston (6, 9) used for their model. The virial coefficients used in Equations 22 for all the calculations were predicted from the scaling expressions of Equations 19. 

For the example application of a loaded polymersome, we require Xnab 30.5. The determination of the coefficients of the hydrophilic moieties is motivated by reproducing for them good solvent conditions. In the following, we set all the third order virial coefficients between the hydrophilic units to zero. The second order coefficients should have a positive value, which in the case of the B monomers determines the size of the hydrophilic head of the amphiphile. The coefficient vbb has to be empirically determined to comply with the bilayer stability (e.g., large values of vbb gives rise to micelle formation instead of bilayers). In this work, we use vbb = vqc = 0.1. Substitution of vbb and into (17) yields vMi = -6. Table 1 summarizes the values of the virial coefficients used to study the loaded polymer-somes. 

Table 1 Values of the virial coefficients used in (12) to study loaded vesicles VA4 = —15.15, vBB = vCc = vBC = 0.1, vAB = vAC = —6 WAAA = WAAB = WABB = WAAC = wAcc = 0.564375 WBBB = WCCC = WBBC = WBCC = WABC = 0...

Here a is the microscopic surface free energy of the cluster, ai the area of a cluster of i molecules, and t and c constants. [Note that if x is, as for spherical particles, for suitable values of r and c Eq. (48) looks like either the classical drop model or the drop model modified to include rotation, translation, etc. Therefore, Eq. (48) is, in a sense, a generalization of the classical drop model.] The Fisher drop model, Eq. (48), was originally developed to describe properties of gases very near the critical point, x and r can be obtained from critical-point indices and are found to be x = and t = 2.333. Hamill showed that c could be obtained from the density of the gas and cr from the second virial coefficient. Using the same equation for the free energy of a... 

The agreement with experimental data which may be obtained for the second virial coefficient using any of these three simple inter-molecular potentials is of the same order (Guggenheim and Me Glashan [1951],... 

Determine the second and third virial coefficients using the van der Waals equation of state. Hint Begin by writing the van der Waals equation in compressibility factor form and performing a power-series expansion. The following mathematical relation is useful ... 

Develop an expression for the Joule-Thomson coefficient using the pressure-based expansion of the virial equation truncated at the second virial coefficient. Use the corresponding state relationships presented in Chapter 4 for the temperature dependence of B to develop a generalized correlation for fiyr... 

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... 

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, ... 

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. 

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. 

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. 

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

If /i = 1 ahn, it is sufficient to retain only the first temi on the right. However, one does not need to know the virial coefficients one may simply use volumetric data to evaluate the integral. 

The same result can also be obtained directly from the virial equation of state given above and the low-density fonn of g(r). B2(T) is called the second virial coefficient and the expansion of P in powers of is known as the virial expansion, of which the leading non-ideal temi is deduced above. The higher-order temis in the virial expansion for P and in the density expansion of g(r) can be obtained using the methods of cluster expansion and cumulant expansion. 

The nth virial coefficient = < is independent of the temperature. It is tempting to assume that the pressure of hard spheres in tln-ee dimensions is given by a similar expression, with d replaced by the excluded volume b, but this is clearly an approximation as shown by our previous discussion of the virial series for hard spheres. This is the excluded volume correction used in van der Waals equation, which is discussed next. Other ID models have been solved exactly in [14, 15 and 16]. ... 

By integrating over the hard cores in the SL expansion and collecting tenns it is easily shown this expansion may be viewed as a correction to the MS approximation which still lacks the complete second virial coefficient. Since the MS approximation has a simple analytic fomi within an accuracy comparable to the Pade (SL6(P)) approximation it may be more convenient to consider the union of the MS approximation with Mayer theory. Systematic improvements to the MS approxunation for the free energy were used to detemiine... 

A graphical method, proposed by Zimm (thus tenned the Zinnn plot), can be used to perfomi this double extrapolation to detemiine the molecular weight, the radius of gyration and the second virial coefficient. An example of a Zinnn plot is shown in figure Bl.9.6 where the light scattering data from a solution of poly... 

Thus one must rely on macroscopic theories and empirical adjustments for the determination of potentials of mean force. Such empirical adjustments use free energy data as solubilities, partition coefficients, virial coefficients, phase diagrams, etc., while the frictional terms are derived from diffusion coefficients and macroscopic theories for hydrodynamic interactions. In this whole field of enquiry progress is slow and much work (and thought ) will be needed in the future. 

In the next section we shall describe the use of Eq. (8.83) to determine the number average molecular weight of a polymer, and in subsequent sections we shall examine models which offer interpretations of the second virial coefficient. 

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