Virial equation methods

This chapter presents a general method for estimating nonidealities in a vapor mixture containing any number of components this method is based on the virial equation of state for ordinary substances and on the chemical theory for strongly associating species such as carboxylic acids. The method is limited to moderate pressures, as commonly encountered in typical chemical engineering equipment, and should only be used for conditions remote from the critical of the mixture. 

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. 

The same result can also be obtained directly from the virial equation of state given above and the low-density fonn of g(r). B2(T) is called the second virial coefficient and the expansion of P in powers of is known as the virial expansion, of which the leading non-ideal temi is deduced above. The higher-order temis in the virial expansion for P and in the density expansion of g(r) can be obtained using the methods of cluster expansion and cumulant expansion. 

Correlation Methods Vapor densities are not correlated as functions of temperature alone, as pressure and temperature are both important. At high temperatures and very low pressures, the ideal gas law can be applied whde at moderate temperature and low pressure, vapor density is usually correlated by the virial equation. Both methods will be discussed later. 

Since reduced pressure is below 0.4, use virial equation (2-67 ). Calculate B by tbe Tsonopoulos method, Eq. (2-68). 

Although the virial equation itself is easily rationalized on empirical grounds, the mixing rules of Eqs. (4-183) and (4-184) follow rigorously from the methods of statistical mechanics. The temperature derivatives of B and C are given exactly by... 

In their classic review on Continuous Distributions of the Solvent , Tomasi and Persico (1994) identify four groups of approaches to dealing with the solvent. First, there are methods based on the elaboration of physical functions this includes approaches based on the virial equation of state and methods based on perturbation theory with particularly simple reference systems. For many years... 

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. 

Unlike the Debye-Hiickel equations, the virial methods provide little or no information about the distribution of species in solution. Geochemists like to identify the dominant species in solution in order to write the reactions that control a system s behavior. In the virial methods, this information is hidden within the complexities of the virial equations and coefficients. Many geochemists, therefore, find the virial methods to be less satisfying than methods that predict the species distribution. The information given by Debye-Hiickel methods about species distributions in concentrated solutions, however, is not necessarily reliable and should be used with caution. 

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... 

One may sometimes have access to the parameters required for the Pitzer approaches, e.g., for some hydrolysis equilibria and for some solubility product data, cf. Baes and Mesmer [3] and Pitzer [4]. In this case, the reviewer should perform a calculation using both the B-G-S and the P-B equations and the full virial coefficient methods and compare the results. 

An efficient method of solving the Percus-Yevick and related equations is described. The method is applied to a Lennard-Jones fluid, and the solutions obtained are discussed. It is shown that the Percus-Yevick equation predicts a phase change with critical density close to 0.27 and with a critical temperature which is dependent upon the range at which the Lennard-Jones potential is truncated. At the phase change the compressibility becomes infinite although the virial equation of state (foes not show this behavior. Outside the critical region the PY equation is at least two-valued for all densities in the range (0, 0.6). 

The thermodynamic functions for the gas phase are more easily developed than for the liquid or solid phases, because the temperature-pressure-volume relations can be expressed, at least for low pressures, by an algebraic equation of state. For this reason the thermodynamic functions for the gas phase are developed in this chapter before discussing those for the liquid and solid phases in Chapter 8. First the equation of state for pure ideal gases and for mixtures of ideal gases is discussed. Then various equations of state for real gases, both pure and mixed, are outlined. Finally, the more general thermodynamic functions for the gas phase are developed in terms of the experimentally observable quantities the pressure, the volume, the temperature, and the mole numbers. Emphasis is placed on the virial equation of state accurate to the second virial coefficient. However, the methods used are applicable to any equation of state, and the development of the thermodynamic functions for any given equation of state should present no difficulty. 

The virial equation of state, first suggested by Kammerlingh-Ohnes, is probably one of the most convenient equations to use, and is used in this chapter to illustrate the development of the thermodynamic equations that are consistent with the given equation of state. The methods used here can be applied to any equation of state. 

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. 

Many other equations of state have been proposed for gases, but the virial equations are the only ones having a firm basis in theory. The methods of statistical... 

All of the methods used to determine molecular weight involve dilute solutions and all apart from SEC (or GPC) use virial equations. So, before we start discussing the details of each method you need to know what these are, in case you erased your memory banks after taking P. Chem. We will start our revision not with dilute solutions, but the ideal gas law (Equation 12-1) ... 

Virial Coefficient Method.—One of the most valuable methods of studying intermolecidar forces consists in direct measurements of the deviations of real gases from the equation of state of the perfect gas. In 1873, van der... 

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 ... 

In this section we introduce several more complex but more accurate equations of state for single species the virial equation, the van der Waals equation, and the Soave-Redlich-Kwong equation. In Section 5.4 we introduce another approach to nonideal gas analysis that makes use of compressibility factors, and we describe Kay s rule, a method for performing PVT calculations on gas mixtures. 

A related method is typified by Pitzer and Weltner s paper on methanol (2156). They used heat capacity expressions corresponding to the virial equation to calculate virial coefficients. They use the calculated fourth virial coefficient to argue that H bonded tetramers arc present. A corresponding interpretation of B is not given. 

Figure 1 shows the representation of the experimental isotherm (B. G. Aristov, V. Bosacek, A. V. Kiselev, Trans. Faraday Soc. 1967 63, 2057) of xenon adsorption on partly decationized zeolite LiX-1 (the composition of this zeolite is given on p. 185) with the aid of the virial equation in the exponential form with a different number of coefficients in the series i = 1 (Henry constant), i = 2 (second virial coefficient of adsorbate in the adsorbent molecular field), i = 3, and i = 4 (coefficients determined at fixed values of the first and the second coefficients which are found by the method indicated for the adsorption of ethane, see Figure 4 on p. 41). In this case, the isotherm has an inflection point. The figure shows the role of each of these four constants in the description of this isotherm (as was also shown on Figure 3a, p. 41, for the adsorption of ethane on the same zeolite sample). The first two of these constants—Henry constant (the first virial constant) and second virial coefficient of adsorbate-adsorbate interaction in the field of the adsorbent —have definite physical meanings. 

Ross and Olivier s Method From Virial Equation Experimental [Extrapolated)... 

Mairy otlrer equations of state Irave been proposed for gases, but the virial equations are tire oirly ones having a finrr basis in theory. Tire methods of statistical mechanics allow derivation of tire virial equations and provide physical sigirificance to the virial coefficients. Thus, for the expansionin 1/ F, the term B/V arises on account of interactions between pairs of molecules (Sec. 7d.2) the C / term, on account of tlrree-body interactions etc. Since two-... 

In later work, Ross and Morrison [7, 8] were able to make several advances. The van der Waals equation of state for real gases, which is the basis of the Hill-de Boer equation, is known to be rather inaccurate. Ross and Morrison based their kernel function on a two-dimensional form of the much better virial equation of state. But more importantly, advances in computing resources made it possible to solve Eqn (7.10) for the unknown distribution function using a nonnegative least squares method, rather than assuming a form a priori [9]. 

Derive equations to calculate the enthalpy departure using each of the following methods (a) the ideal gas equation, (b) the virial equation of state truncated after the second virial coefficient, (c) the Soave-Redlich-Kwong equation of state. 

The use of the virial equation of state and the correlations of Pitzer and Curl15,48,49 for the prediction of fugacity coefficients for nonpolar mixtures is presented first and then the method is extended to polar gas mixtures. 

These results are similar to those obtained using either the Lewis-Randall rule or the virial equation of state. However, greater differences between the methods occur for gases as the pressure is increased. At higher pressures the results from the Peng-Robinson equation of state are expected to be the most accurate. ... 

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