Virial composition dependence

Higher virial coefficients are defined analogously. AH virial coefficients depend on temperature and composition only. The pressure series and density series coefficients are related to one another ... 

Although PVT equations of state are based on data for pure fluids, they are frequently appHed to mixtures. 7h.e virial equations are unique in that rigorous expressions are known for the composition dependence of the virial coefficients. Statistical mechanics provide exact mixing rules which show that the nxh. virial coefficient of a mixture is nxh. degree in the mole fractions ... 

The density-series virial coefficients B, C, D,. . . , depend on temperature and composition only. The composition dependencies are given by the exact recipes... 

The application of cubic equations of state to mixtures requires expression of the equation-of-state parameters as func tions of composition. No exact theory like that for the virial coefficients prescribes this composition dependence, and empirical mixing rules provide approximate relationships. The mixing rules that have found general favor for the Redhch/Kwong equation are ... 

The two values kp and k are usually not very different, and kp is not strongly composition dependent. Nevertheless, the quadratic dependence of Z — a/RT) on composition indicated by Eq. (4-305) is not exactly preserved. Since this quantity is not a true second virial coefficient, only a value predicted by a cubic equation of state, a strict quadratic dependence is not required. Moreover, the composition-dependent kp leads to better results than does use of a constant value. 

However, these new mixing rules (based both to infinite- or zero pressure limit) give, for the composition dependence of the second virial coefficient, results that are inconsistent with those obtained from statistical mechanics. 

The mixture second virial coefficient 5 is a function of temperature and composition. Its exact composition dependence is given by statistical mechanics, and this makes tire virial equation preeminent among equations of state where it is applicable, i.e., to gases at low to moderate pressures. The equation giving this composition dependence is ... 

An analysis of the cosolvent concentration dependence of the osmotic second virial coefficient (OSVC) in water—protein—cosolvent mixtures is developed. The Kirkwood—Buff fluctuation theory for ternary mixtures is used as the main theoretical tool. On its basis, the OSVC is expressed in terms of the thermodynamic properties of infinitely dilute (with respect to the protein) water—protein—cosolvent mixtures. These properties can be divided into two groups (1) those of infinitely dilute protein solutions (such as the partial molar volume of a protein at infinite dilution and the derivatives of the protein activity coefficient with respect to the protein and water molar fractions) and (2) those of the protein-free water—cosolvent mixture (such as its concentrations, the isothermal compressibility, the partial molar volumes, and the derivative of the water activity coefficient with respect to the water molar fraction). Expressions are derived for the OSVC of ideal mixtures and for a mixture in which only the binary mixed solvent is ideal. The latter expression contains three contributions (1) one due to the protein—solvent interactions which is connected to the preferential binding parameter, (2) another one due to protein/protein interactions (B p ), and (3) a third one representing an ideal mixture contribution The cosolvent composition dependencies of these three contributions... 

To impart a quadratic composition dependence to the second virial coefficient of the eos... 

Parameters B, C, D,. . . are density-series virial coefficients, and B, C, D, , , . ate preseure-series virial coefficients. Virial coefficients depend only on temperature and composition they are defined through the usual prescriptions for coefficients in a Taylor expansion. Thus, the second virial coefficients are given as... 

Since 4,2 = 2B,2 - Bu - die details of the composition dependence of and fa are directly influenced by the magnitude of the interaction coefficient Ba. The effect is illustrated in Fig. 1.3-1. which shows values of versus y, computed from Eq. (1.3-23a) for a representative binary syslant for which the pure-component virial coefficients are SM = —1000 cnrVmol and B22 = -2000 cm /mol. The temperature is 300 K and the pressure is 1 bar the curves correspond to different values of Bn, which range from —300 to —2500 cm /mol. All curves approach asymptotically die pure-componear value fa = 0.9607... 

FIGURE 1.3-1 Composition dependence of fugacity coefficient i>i of component 1 in a binary gas mixture ai 300 K aed I bsr, Curves correspoed to different values of iha interaction recond virial coefficient Bl2, (See text for discussion.)... 

One useful boundary condition is that at low density the composition dependence of the second virial coefficient obtained from an equation of state should agree with the theoretically correct result of Eq. 9.4-5. 

Here B is the second virial coefficient, C is the third, etc. For a mixture containing specified substances, the virial coefficients depend on T and composition only moreover, the composition dependence of mixture virial coefficients is known. 

Since the virial coefficients depend on T and composition only, the equa-tion-of-state parameters can depend at most on T and composition, as already noted. The second virial coefficient B is the only one for which a decent data base and reliable estimation procedures are available according to Equation 3a, values for B (as implied by our equation of state) are determined completely by specification of parameters b and . 

This approach in fact has a reasonable basis, a basis suggested by Equation 3a. If the cubic equation is to be useful at low densities, then the composition dependence of its parameters should be compatible with that of the second virial coefficient, which is known to be quadratic in the mole fractions. Thus, if bm is assumed linear in composition, then ro should be quadratic in composition. 

In addition to pure gases, the Taylor expansion (4.5.8) can be applied to gaseous mixtures. The resulting form is still (4.5.11), but the virial coefficients now depend on both temperature and composition. The composition dependence is rigorously obtained from statistical mechanics here we are interested only in the results. For a mixture containing n components. 

The factor 1.443 VT changes when other model equations of state are used. Note that (6.5.4) involves only pure component parameters, fl and fo,- so, no combining rules are needed. However, these mixing rules do not reproduce the known composition dependence of the second virial coefficient (4.5.18). 

The separation factors mainly depend on composition and temperature. The correct composition dependence is described with the help of activity coefficients. Following the Clausius-Clapeyron equation presented in Section 2.4.4 the temperature dependence is mainly influenced by the slope of the vapor pressure curves (enthalpy of vaporization) of the components involved. But also the activity coefficients are temperature-dependent following the Gibbs-Helmholtz equation (Eq. (5.26)). This means that besides a correct description of the composition dependence of the activity coefficients also an accurate description of their temperature dependence is required. For distillation processes at moderate pressures, the pressure effect on the activity coefficients (see Example 5.7) can be neglected. To take into account the real vapor phase behavior, equations of state, for example, the virial equation, cubic equations of state, such as the Redlich-Kwong, Soave-Redlich-Kwong (SRK), Peng-Robinson (PR), the association model, and so on, can be applied. 

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