Virial coefficients calculation

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). 

The partition coefficients predicted by the theory for the four globular protein, lysozyme, chymotrypsin, albumin and catalase were determined by inserting into Equation 31 the second virial coefficients calculated for each... 

Comparison of the ab initio and experimental second virial coefficients. The solid and dashed lines represent second virial coefficients generated from the original and scaled a6 initio potentials, respectively, while the dotted line shows the second virial coefficient calculated using the isotropic component of the ab initio potential. Circles and squares are experimental points from Refis. (74) and (73), respectively. 

It should be mentioned that the difference between Equations 37 and 27 is very small. The second virial coefficient calculated from Equation 25 using the new expression of f0(T) still agrees very well with the correlation of Pitzer and Curl (Equation 26). 

Virial coefficients calculated from the correlation are in good agreement with corresponding states values calculated according to the method of Prausnitz [ ]. The comparison is... 

At 17.4°, 20.4° and 21.8°K, there appear to be no critical points for the helium-hydrogen mixtures this is indicated by phase equilibrium data calculated at these temperatures at pressures up to 7000 psia. If a critical point were reached, thepylx curves for helium and hydrogen could converge to the same value at the critical point, but there appears to be no tendency toward convergence even at pressures of 7000 psia. Unfortunately, no experimental data exist to verify this calculation. Virial coefficients calculated from the correlation are in good accord with the experimental data of Varekamp and Beenakker [ ], as shown in Table V. 

Thus, pressure-explicit equations of state for pure substance 1 (for the first integral) and for the gas mixture (the second integral) are required. Five different equations of state have been used in the analysis of this system (1) the five-constant Beattie-Bridgeman equation (2) the eight-constant Benedict-Webb-Rubin equation (3) the twelve-constant modified Martin-Hou equation and (4) and (5), the virial equation using two sets of virial coefficients. The first of these uses pure-substance second and third virial coefficients calculated from the Lennard-Jones 6-12 potential with interaction coefficients determined by the method of Ewald [ ]. The second set differs only in the second virial coefficients and interaction coefficient, these being found using the Kihara potential Solutions of the theoretical equa-... 

Figure 1 shows reduced virial coefficients calculated for a more realistic two-parameter potential, the Lennard-Jones 6—12 potential. The only significant difference between the Lennard-Jones and square-well second virial coefficients occurs at high reduced temperatures, where experimentally B is observed to have a maximum value. This maximum is related to the softness of the repulsive portion of the potential it therefore cannot be reproduced by the infinitely steep square well. The third virial coefficients calculated from the square well are qualitatively correct but show significant departures from the Lennard-Jones values. 

Isenthalpic Joule-Thomson measurements on Ng + CH4 -1- CaHe mixtures have been reported by Ahlert and Wenzel. They compared their results with the predictions of the virial equation of state with virial coefficients calculated from the Lennard-Jones 6—12 potential. 

Stark, A.C., Andrews, C.T., Elcock, A.H. Toward optimized potential functions for protein-protein interactions in aqueous solutions osmotic second virial coefficient calculations using the MARTINI coarse-grained force field. J. Chem. Theory Comput. 9, 4176 185... 

We have implemented the PIMC method with two approaches based on semi-classical beads (SCB-QFH, SCB-TI) and MSMC to compute more precise quantum virial coefihdents for helium. The SCB results agree well with CB results as they are within statistical uncertainties of each other. The decomposition algorithm of Shaul et al. [18] was implemented to achieve better efficiency of quantum virial coefficient calculations. We observed similar trends in decompositions of simulations in our SCB based approaches as was the case for the CB approach. For lower temperatures, the approximation to u for finite P is... 

We now apply these results for the values of the virial coefficient calculated in the previous section in a common 0-solvent, a selective solvent and a common good solvent. 

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, ... 

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. 

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). 

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 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... 

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 ... 

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. 

Hu C H and Thakkar A J 1996 Potential energy surface for interactions between N2 and He ab initio calculations, analytic fits, and second virial coefficients J. Chem. Phys. 104 2541... 

Figure A2.1.7. The second virial coefficient 5 as a fiinction of temperature T/T. (Calculated for a gas satisfying the Leimard-Jones potential [8].)...

---