Second virial coefficient for

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.

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). 

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/. 

Gas mixtures are subject to the same degree of non-ideality as the one-component ( pure ) gases that were discussed in the previous section. In particular, the second virial coefficient for a gas mixture can be written as a quadratic average... 

STEP 4. The second virial coefficients for cation-anion pairs are... 

Second virial coefficient for the mutual interactions of species i and ... 

The averaged second virial coefficients for a polydisperse system are given by (Schaink and Smit, 2007) ... 

Neal, B.L., Lenhoff, AM. (1995). Excluded-volume contribution to the osmotic second virial coefficient for proteins. AlChE Journal, 41, 1010-1014. 

These manipulations may appear to add little except for needless complication to an interpretation of the second virial coefficient for random coils. Recall, however, that Equation (81) allows the variation of solvent goodness caused by temperature changes to be described quantitatively. Thus the interaction parameter x is used to describe how B changes when a polymer is dissolved in different solvents. By contrast, 9 is used to describe the variation in B when a given polymer-solvent system is examined at different temperatures. This has been done for the polystyrene-cyclohexane system at three different temperatures the results are discussed in Example 3.4. 

For a multicomponent mixture, a virial coefficient is needed to account for each possible interaction. The second virial coefficients for a two component mixture are Bpi, Bp2> and B22 where B- represents the interaction between two molecules of component 1, Bp2 represents the interaction between a molecule of 1 and a molecule of 2, and B22 represents interaction between two molecules of 2. A tabulation of some compounds whose virial coefficients have been measured by GC is given in Table 11.6. 

As mentioned in Section 2.1, the usual Boltzmann equation conserves the kinetic energy only. In this sense the Boltzmann equation is referred to as an equation for ideal systems. For nonideal systems we will show that the binary density operator, in the three-particle collision approximation, provides for an energy conservation up to the next-higher order in the density (second virial coefficient). For this reason we consider the time derivative of the mean value of the kinetic energy,12 16 17... 

Shielding Surfaces for Two Interacting Molecules. In the Maryland meeting Jameson and de Dios reported the first ab initio calculations of the rare gas pair intermolecular shielding surfaces for Ar2, ArNe, Ne2, and NeHe (55). With these shielding surfaces it was possible to calculate the second virial coefficient for nuclear shielding as a function of temperature, using the well established intermolecular potential functions V(R) for the pair. 

Zhang, H.-L., Sato, H., Watanabe, K. (1995a) Vapor pressures, gas-phase PVT properties, and second virial coefficients for... 

The possibility of occurrence of instability of colloidal dispersions in the presence of free polymer was first predicted by Asakura and Oosawa (5), who have shown that the exclusion of the free polymer molecules from the interparticle space generates an attractive force between particles, DeHek and Vrij (1) have developed a model in which the particles and the polymer molecules are treated as hard spheres and rederived in a simple and illuminating way the interaction potential proposed by Asakura and Oosawa. Using this potential, they calculated the second virial coefficient for the particles as a function of the free polymer concentration and have shown that... 

To have an idea about the range of the repulsion required to provide such a high virial coefficient, it should be noted that, if the hard-core repulsion, infinite in magnitude, is extended with 15 A (above the 2a separation), 2 increases from 4 to only 5.6. If the range of the hard-core repulsion is extended with 30 A, 2 increases to 7.55, while 60 A leads to 12.8. From these simple estimations one can infer that the repulsion needed to explain the measured second virial coefficient for apoferritin molecules should have a much longer range than that typically observed for the traditional hydration force. 

Using eqs 7,10,11, and 13, one can determine the charge on an apoferritin molecule at any pH, provided that the values of the dissociation constants Ku and An, are known. Further, using eqs 1 and 2, one can evaluate the second virial coefficient for the interaction between two particles at constant surface charge. 

For the values of the parameters employed (a relatively large Hamaker constant), the potential barrier is only a few kT or less hence, the apoferritin should coagulate at almost all the concentrations studied. Since experiment shows that the proteins did not coagulate, another repulsion should be present, at least al low separation distances. This repulsion, while essential for the stability of the system, did not affect much, because of its short range, the behavior of the second virial coefficient. In the calculation of the second virial coefficient, it was assumed that the distance of closest approach between apoferritin proteins cannot be less than 8 A. This value leads to a dimensionless second virial coefficient for the hard spheres repulsion of 4.8 instead of 4. 

The second virial coefficient for apofcrritin molecules, obtained from experiment in Ref. [18] is compared with calculations based on the polarization model for hydration/double layer interactions for various values of the effective surface dipole moment The details of the calculations are provided in Ref. [19]. 

Figure 1-16. Theoretical (full line) and experimental (open triangles) second virial coefficient for gaseous water at various temperatures...

Pack RT (1983) First quantum corrections to second virial coefficients for anisotropic interactions Simple, corrected formula. J Chem Phys 78 7217-7222... 

Moszynski R, Korona T, Heijmen TGA, Wormer PES, Van der Avoird A, Schramm B (1998) Second virial coefficients for atom-molecule complexes from ab initio SAPT potentials. Polish... 

The second virial coefficient for acetonitrile is given approximately by the equation,... 

In this section we will calculate the second virial coefficient for the solution of rods interacting as described in Sect. 2.2 and we will find the point of inversion of this coefficient, i.e. the 8 point. As noted above, the Flory theory91 gives the incorrect value for the 6 temperature. 

Bn = second virial coefficient for pure component 1, cubic meters per kilomole. 

Interaction parameter in Flory-Huggins treatment of polymer mixtures after normalization on a per monomer basis this becomes Xay also susceptibilities in discussion of density functional theories. Applied external field acting on species a in conformation X". Applied external field as a function of position acting on species a. Contribution of attractive interactions to the second virial coefficient for species pair ay also van der Waals coefficient. 

Second virial coefficient for species pair ay, as a function of temperature T, e.g. K ] T) =... 

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