Mixtures virial coefficients, calculation

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-... 

Here ns is the amount of substance of stationary liquid, pi is the saturated vapour pressure of the solute at temperature r. Bag is the mixture virial coefficient for solute 4- carrier gas interaction, Bcc is the virial coefficient of the carrier gas, Fjj is the partial molar volume of the solute at infinite dilution in the solvent, is the molar volume of pure liquid A, and pi and po are the column inlet and outlet pressures. The chemical potential at infinite dilution can be calculated by measuring the retention volume of an infinitely small sample for various inlet and outlet pressures and extrapolation to zero pressure drop across the column. Everett and Stoddart proposed using equation (33) to determine the mixture second virial coefficients. The precision in Bag from this method is nearly equivalent to the best static methods. The assumptions required to derive the above equation have been examined by a number of authors. - ... 

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. 

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. 

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/. 

The model presented here develops these ideas and introduces features which make the calculation of mixture properties simple. For a polar fluid with approximately central dispersion forces together with a strong angle dependent electrostatic force we may separate the intermolecular potential into two parts so that the virial coefficients, B, C, D, etc. of the fluid can be written as the sum of two terms. The first terms B°, C°, D°, etc, arise from dispersion forces and may include a contribution arising from the permanent dipole of the molecule. The second terms contain equilibrium constants K2, K, K, etc. which describe the formation... 

Dymond and Smith [11] give an excellent compilation of virial coefficients of gases and mixtures. Cholinski et al. [12] provide second virial coefficient data for individual organic compounds and binary systems. The latter book also discusses various correlational methods for calculating second virial coefficients. Mason and Spurling [13] have written an informative monograph on the virial equation of state. 

The estimation of the jamming coverage for the RSA of monodisperse disks is not an important issue, because its value is already accurately known from Monte Carlo simulations [12], However, it is of interest to develop a procedure that can predict the available area and the jamming coverage for a mixture of disks, for which much Less information is available. Even at equilibrium, for which reasonable accurate equations of state for binary mixtures of hard disks are known for low densities [ 19,20], the available area vanishes only for the unphysical total coverage 9 = 9 +0p = 1 (where the subscripts S and L stand for small and large disk radii, respectively), hence there is no jamming . Exact analytical expressions are known only for the first three virial coefficients of a binary mixture of disks [21], The fourth and fifth coefficients were computed numerically for some diameter ratios and molar fractions for an equilibrium gas [22], However, there are no such calculations for the RSA model. 

Another test to which the Ar-CH4 potential was subjected, is the calculation of the second virial coefficients at various temperatures. The agreement between the theory and experiment is illustrated on Figure 1-13. An inspection of this figure show that the agreement between the measured and computed second virial coefficients is good, so similarly as in the case of He-C02 mixtures, the volume of the interaction potential well for Ar-CH4 is correct. 

The binary mixture parameter has been fitted to VLE data for 29 systems its values are in Table 1. It should be noted that is independent of temperature and always very close to unity. The calculation of phase equilibria was performed by means of the algorithm of Deiters [8, 9], The reproduction of VLE data and the predictions of LLE data, of excess volumes, of virial coefficients are very good for all 29 binary mixtures investigated [3]. 

In the above equations denotes the vapor mole fraction of component i, Pf is the vapor pressure of the pure component i, Bu is the second virial coefficient of component i, dn = 2Bn — Bn — B22 and B 2 is the crossed second virial coefficient of the binary mixture. The vapor pressures, the virial coefficients of the pure components and the crossed second virial coefficients of the binary mixtures were taken from [32], The Wilson [38], NRTL [39] and the Van Ness-Abbott [40] equations were used for the activity coefficients in Eq. (17). The expressions for the activity coefficients provided by these three methods were differentiated analytically and the obtained derivatives were used to calculate D = 1 -I- Xj(9 In Yil 2Ci)pj. There is good agreement between the values of D obtained with the three expressions for the systems V,V-dimethylformamide-methanol and methanol-water. For the system V,V-dimethylformamide-water, the D values calculated with the Van Ness-Abbott equation [40] were found in good agreement with those obtained with the NRTL equation, but the agreement with the Wilson expression was less satisfactory. 

Figure 3. Dependence of the second virial coefficient on the concentration of cosolvent in the water (1)—lysozyme (2)—arginine (3) mixture. , experimental data solid line, values predicted using eq 10 with722 (see Table 2) as an adjustable parameter , protein—solvent interaction contribution B " calculated using eq IIB O, ideal mixture contribution B (eq 11 A) A, protein—protein interaction contribution B " P predicted by eq 11C with 722 as an adjustable parameter from Table 2. See details in Figure 1.

Figure 4. Dependence of the second virial coefficient on the concentration of NaCl in a water (1)—lysozyme (2)—NaCl (3) mixture. B22 (broken line) was calculated using eq 10 with bothyri andyri used as adjustable parameters found by fitting the experimental data ( , ref 48 O, ref 19 and , ref 28). ideal mixture contribution predicted using eq 11 A protein—solvent interaction contribution calculated...

We begin by calculating the Gibbs free energy of an imperfect gas mixture. Employing the van der Waals equation and inserting the virial coefficient (11.35) into (11.21), remembering that B =B/RT, we have... 

Data are readily available for pure component and binary interaction second virial coefficients for a large number of components and binaries. Binary interaction coefficients are required for extending the equation to mixtures. The simplicity of the equation, the availability of coefficient data, and its ability to represent mixtures are some of the reasons the virial equation of state is a viable option for representing gases at densities up to about 70% of the critical density. It may be used for calculating vapor phase properties at these conditions but is not applicable to dense gases or liquids. 

Derive equations to calculate component fugacity coefficients in a binary mixture using the virial equation of state truncated after the second virial coefficient. The mixture second virial coefficient is given as... 

Using eqs. (l)-(9), along with empirical pure-electrolyte parameters 3 ), 3 > 3 and and binary mixture parameters 0, one can reproduce experimental activity-coefficient data typically to a few percent and in all cases to + 20%. Of course, as noted above, the most accurate work on complex, concentrated mixtures requires that one include further mixing parameters and also for calculations at temperatures other than 25°C, include the temperature dependencies of the parameters. However, for FGD applications, a more important point is that Pitzer1s formulation appears to be a convergent series. The third virial coefficients... 

Pitzer s formulation offers a satisfactory and desirable way to model strong electrolyte activity coefficients in concentrated and complex mixtures. When sufficient experimental data are available, one can make calculations which are considerably more accurate than those presented in this paper. Attaining high accuracy requires not only experimentally-based parameters but also that one employ third virial coefficients and additional mixing terms and include explicit temperature dependencies for the various parameters. 

The great appeal of the virial equations derives from their interpretations in terms of molecular theory. Virial coefficients can he calculated from potential fonctions describing interactions among moleculas. More importantly, statistical mechanics provides rigorous expressions for the composition dependeace of ihe virial coefficients. Thus, the nth virial coefficient of a mixture is nth order in the mole fractions ... 

In low- to moderate-density vapors, mixture nonidealities are not very large, and therefore equations of state of the type discussed in this text can generally be used for the prediction of vapor-phase fugacities of all species. [However, mixtures containing species that associate (i.e., form dimers, trimers, etc.) in the vapor phase, such as acetic acid, are generally described using the virial equation of state with experimentally determined virial coefficients.] The Lewis-Randall rule should be used only for approximate calculations it is best to use an equation of state to calculate the vapor-phase fugacity of vapor mixtures. 

Equation 25 was developed from an empirical representation of thg second virial coefficient correlation of Pitzer and Curl (I) parameter b was left unchanged at its classical value of 0.0866. Because of the substantial improvement in the prediction of and its temperature derivatives for nonsimple fluids, the Barner modification of the RK equation gave improved estimates of enthalpy deviations for nonpolar vapors and for vapor-phase mixtures of hydrocarbons. However, the new equation was unsuitable for fugacity calculations. 

The result of a calculation can be quite sensitive to the values for the k. Although these quantities can be correlated at times against combinations of properties for pure species i and / (e.g., critical-volume ratios), they are best treated as purely empirical parameters, values of which are (ideally) backed out of good experimental mixture data for the type of property which is to be represented. Thus, if accurate calculation of low-to-moderate-pressure volumetric properties is required, then the kif could be estimated from available data on mixture second virial coefficients for the constituent binaries. Alternatively, if application to multicomponent VLE calculations is envisioned, then the ki would be best estimated from available VLE data on the constituent binaries. (It... 

We could, of course, attempt more sophisticated simulations of scale formation. Since the fluid mixture is quite concentrated early in the mixing, we might use a virial model to calculate activity coefficients (see Chapter 7). The Harvie-M<)>ller-Weare (1984) activity model is limited to 25°C and does not consider barium or strontium, but Yuan and Todd (1991) suggested a similar model for the Na-Ca-Ba-Sr-Mg-SO4-Cl system in which the virial coefficients can be extrapolated to typical reservoir temperatures in the North Sea. 

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