Models virial equations

The Langmuir model is discussed in reference 19 the Volmer in reference 20 and the van der Waals and virial equations in reference 8. 

A. Milchev, K. Binder. Osmotic pressure, atomic pressure and the virial equation of state of polymer solutions Monte Carlo simulations of a bead-spring model. Macromol Theory Simul 5 915-929, 1994. 

Can the species activity coefficients be calculated accurately An activity coefficient relates each dissolved species concentration to its activity. Most commonly, a modeler uses an extended form of the Debye-Hiickel equation to estimate values for the coefficients. Helgeson (1969) correlated the activity coefficients to this equation for dominantly NaCl solutions having concentrations up to 3 molal. The resulting equations are probably reliable for electrolyte solutions of general composition (i.e., those dominated by salts other than NaCl) where ionic strength is less than about 1 molal (Wolery, 1983 see Chapter 8). Calculated activity coefficients are less reliable in more concentrated solutions. As an alternative to the Debye-Hiickel method, the modeler can use virial equations (the Pitzer equations ) designed to predict activity coefficients for electrolyte brines. These equations have their own limitations, however, as discussed in Chapter 8. 

Even though the van der Waals equation is not as accurate for describing the properties of real gases as empirical models such as the virial equation, it has been and still is a fundamental and important model in statistical mechanics and chemical thermodynamics. In this book, the van der Waals equation of state will be used further to discuss the stability of fluid phases in Chapter 5. 

Hm for steam + n-heptane calculated by the above method is shown by the dashed lines in figure 6. Considering the simplicity of the model and the fact that no adjustable parameters have been used, agreement with experiment is remarkable. For mixtures of steam + n-hexane, benzene and cyclohexane agreement with experiment is much the same. At low densities the model reproduces the curvature of the lines through the results better than the virial equation of state. The method fails to fully reproduce the downward turn of the experimental curves at pressures near saturation, but does marginally better in this region than the P-R equation with k. = -0.3. At supercritical temperatures the model seems to... 

Flow calorimetric measurements of the excess enthalpy of a steam + n-heptane mixture over the temperature range 373 to 698 K and at pressures up to 12.3 MPa are reported. The low pressure measurements are analysed in terms of the virial equation of state using an association model. An extension of this approach, the Separated Associated Fluid Interaction Model, fits the measurements at high pressures reasonably well. 

Let us now consider some actual numerical data for specific mixed biopolymer systems. Table 5.1 shows a set of examples comparing the values of the cross second virial coefficients obtained experimentally by static laser light scattering with those calculated theoretically on the basis of various simple excluded volume models using equations (5.32) to (5.35). For the purposes of this comparison, the experimental data were obtained under conditions of relatively high ionic strength (/ > 0.1 mol dm- ), i.e., under conditions where the contribution of the electrostatic term (A if1) is expected to be relatively insignificant. 

It is evident that the strength ofthe interaction depends on the distance between the elementary particles. This model is equivalent to the compression of a real gas, that is, of molecules or atoms, that is, elementary particles that exhibit an intense interaction as a function ofthe interparticulate distance. This process can be well simulated with the help ofthe virial equation (Equation 20.7), which corresponds to the van derV feals equation (see Figure 20.11). 

For higher ionic strength, e g. highly saline waters the PITZER equation can be used (Pitzer 1973). This semi-empirical model is based also on the DEBYE-HUCKEL equation, but additionally integrates virial equations (vires = Latin for forces), that describe ion interactions (intermolecular forces). Compared with the ion dissociation theory the calculation is much more complicated and requires a... 

It is generally agreed that a virial form of isotherm equation is of greater theoretical validity than the DA equation. As explained in Chapter 4, a virial equation has the advantage that since it is not based on any model it can be applied to isotherms on both non-porous and microporous adsorbents. Furthermore, unlike the DA equation, a virial expansion has the particular merit that as p — 0 it reduces to Henry s law. 

This is a virial equation, the word virial being taken from the Latin word for force and thus indicating that forces between the molecules are having an effect. It turns out that statistical mechanical models also give equations that can be written in this form with the virial coefficients, B C > etc., being related to various interaction parameters. 

Also note that at low pressures the lines are almost linear and the data could be modeled by an equation involving just the first two or three terms of the virial equation. 

Milner s contribution (1912) was direct. He attempted to find out the virial equation for a mixture of ions. However. Milner s statistical mechanical approach lacked the mathematical simplicity of the ionic-cloud model of Debye and Hiickel and proved too unwieldy to yield a general solution testable by experiment. Nevertheless, his contribution was a seminal one in that for the first time the behavior of an ionic solution had been linked mathematically to the interionic forces. 

Due to lack of space, the few results presented here are primarily intended to demonstrate the validity of the proposed method. The pore space of the adsorbent is assumed to consist of slit-shaped pores of width 15 A, with parameters chosen to model activated carbon. The porosity values are fixed at q = 0.45 and qp = 0.6. The feed stream is atemary gas mixture of H2/CH4/C2H6. The vtqx>r-phase fugacities were computed from the virial equation to second order, using coefficients taken from Reid et al ... 

Fig. 2. Adsorption isotherms in all four model systems. F is the Gibbs excess adsoiption. The pressures corresponding to the three configurations shown in Figure 3 are marked with arrows. The pressure is plotted relative to the vapor pressure of the model fluid, as determined by independent Gibbs Ensemble Monte Carlo simulations. Chemical potentials were converted to pressures using a virial equation of state.

Evaluation of d is usually by Eq. (4-243), based on the two-term virial equation of state. The activity coefficient y is ultimately based on Eq. (4-251) applied to an expression for G /RT, as described in the section Models for the Excess Gibbs Energy. ... 

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. 

In both solid and gaseous solutions, virial equation-based Raoultian coefficients have often been proposed. For example, the Margules equations, often used in binary and sometimes in ternary solid solutions and which have a virial equation basis, were proposed originally for gaseous solutions. However, there is no satisfactory general model for Raoultian coefficients in multi-component solid solutions, and the tendency in modeling has been to treat these solutions as ideal (i.e., to use the mole fraction of a solid solution component as its activity see Equation (3.13)). 

Fig. 21. Entropy versus log-temperature diagram for the hard-sphere model. The solid curves give the computer simulation values for the supercooled fluid, glass, and crystal. The dashed curves have the following bases (a) a calculation from the virial equation using the known first seven coefficients and higher coefficients obtained from the conjectured closure (the plot corresponds quite closely with that calculated from the Camahan-Starling equation ) and (i>) an extrapolation of higher temperature behavior such as that used by Gordon et al., which implies a maximum in the series of virial coefficients. The entropy is defined in excess of that for the ideal gas at the same temperature and pressure. Some characteristic temperatures are identified 7, fusion point 7 , upper glass transition temperature T/, Kauzmann isoentropic point according to closure virial equation.

Aqueous solutions can be modeled by writing a virial equation such as (17.37) in which osmotic pressure replaces pressure. Friedman (1962) describes applications of cluster expansion theory, which include long-range Coulombic potentials as well as short-range square-well potentials that operate when unlike ions approach within the diameter of a water molecule. These models are mathematically quite cumbersome and are not easily used for routine calculations. They do predict the non-ideal behavior of simple electrolytes such as NaCl quite admirably at moderate concentrations however, they use the square-well potential as an adjustable parameter and so retain some of the properties of the D-H equation with an added adjustable term. For this reason these are not truly a priori models. 

In the 1970s, Kenneth Pitzer and his associates developed a theoretical model for electrolyte solutions combined the D-H equation with additional terms in the form of a virial equation. This has proven to be extraordinarily successful at fitting the behavior of both single- and mixed-salt solutions to high concentrations. Recent summaries of this model are provided by Pitzer (1979, 1987), Harvie and Weare (1980), and Weare (1987), and much of the following discussion is adapted from these articles, particularly those by Harvie and Weare (1980) and Pitzer (1987). 

In 1973 Kenneth Sanborn Pitzer (1914-1997) imdertook an attempt to take into accoimt these interactions in the solution s composition. He included binary interaction cation-anion, anion-anion, cation-cation, cation-neutral component, anion-neutral component, neutral component-neutral component and triple interaction cation-cation-anion, anion-anion-cation, etc., for which he expanded first member of equation (1.78) into a series of addends with virial coefficients (Pitzer, 1973). Each of these addends characterizes one type of interaction. His model of more detailed accounting of the interaction between components of water solution is sometimes called the Pitzer model. According to it, equation (1.78) acquired the format of a virial equation of the state of solution, or Pitzer equation with virial coefficients ... 

Such substances represent solutions of nonelectrolytes with minuscule content of polar compounds. As well as water solutions, they can be ideal or real. As ideal (diluted) are treated nonpolar solutions dominated by one component - solvent in conditions of relatively low pressure. It is believed that the behaviour of individual components in their composition is subject to the laws of diluted solutions, namely, Raoult s law (equation (1.60)) for the solvent and Henry s law (equation (2.280)) for dissolved substances. However, in the overwhelming majority of cases these are complex nonideal solutions, whose state is determined by various semiempiric models, which represent equation of state, i.e., correlation of the composition vs. temperature, pressure and volume. They are subdivided into three basic groups virial, cubic and complex. Virial equations are convenient for modeling properties and composition of noncondensable gaseous media... 

Figure 4.12 Dimensionless residual properties for gaseous CH4-SF6 mixtures at 60°C, 20 bar, from the virial equation (4.5.31). It is an artifact of the model that u lRT = P, a )/R.

That is, at low pressures we ignore the pressure dependence of all activity coefficients and all standard-state fugacities. In the 3 phase, values for the activity coefficients depend on the choice made for the standard-state fugacity for example, if the Lewis-Randall standard state is chosen for all components (5.1.5), then the y,- would be obtained from a model for the excess Gibbs energy. Common choices for the standard state are discussed in 10.2. In the a phase, values for the fugacity coefficients are obtained from a volumetric equation of state now, either pressure-explicit or volume-explicit models may be chosen. Fortunately at low pressures, either the ideal-gas law or a virial equation may be sufficiently accurate. 

It requires eight parameters (Ao, Bo, Co, a, b, c, a, and y) that are specific to the fluid. The Benedict-Webb-Rubin equation is modeled after the virial equation and expresses pressure as a finite sum of powers of i/V, up to the sixth power. The exponential term on the right is meant to account for the higher terms of the series that have been dropped. A modified form of this equation was used by Lee and Keslera in the calculation of Zm and Zw. This equation is not cubic but its subcritical isotherms have the same general behavior as those in Figure 2-12. namely, they exhibit an unstable part where the isotherm has a positive slope. 

---