The Virial Equation of State

Department of Chemical Engineering, Imperial College London, South Kensington Campus, London SW7 2AZ, U.K. 

The virial equation of state is a power series expansion for the pressure p of a real gas in terms of the amount-of-substance density p  

T is the thermodynamic temperature, R is the universal gas constant, p = njV, n is the amount of substance, V is the volume, and I, B, C, are called virial coefficients. The virial series is also conveniently written in terms of the compression factor Z  

Since the leading term on the right of eq 3.1 is the pressure of a perfect gas, the second and high-order virial coefficients B, C, ) describe gas imperfections. These coefficients depend upon temperature and upon the nature of the gas so in a mixture, they depend upon the composition. 

The virial equation is limited to gases at low or moderate densities, while many other equations exist that purport to cover the entire fluid region. Nevertheless, the virial equation possess a number of key strengths including  

The vdrial equation of state has a sound theoretical foundation it can be derived from first principles using statistical mechanics. This equation is given by a power series expansion for the compressibility factor in concentration (or the reciprocal of molar volume) about l/v = 0  

Here B, C. are called the second, third. virial coefficients these parameters depend only on temperature (and composition for mixtures). An alternative expression for the virial equation is a power series expansion in pressure  

By solving Equation (4.26) for P and substituting into Equation (4.27), it is straightforward to show the two sets of coefficients are related by  

A common question is What power series expansion do I use Well, Equation (4.26) is explicit in pressure and Equation (4.27) is explicit in volume, so if you need an expression that is explicit in one of these variables (so you can take a derivative, for example), use the appropriate form. The next issue is a question of accuracy. R turns out that at moderate pressures (up to about 15 bar) when you keep only the second virial coefficient, the power series expansion in pressure is better  

Erom 15 to 50 bar, the virial equation should contain three terms, and the expansion in concentration is more accurate  

In 1901, H. Kamerlingh Onnes introduced a fundamentally new and improved description of real gas PVT properties in terms of the virial equation of state. [The word virial, deriving from the Latin word viris ( force ) was introduced into physics by R. Clausius, whom we shall meet later.] This equation expresses the compressibility factor Z(Vm, T) in terms of a general power series expansion in inverse molar volume Vm. The starting point for the virial expansion is the ideal limiting behavior (2.12), which can also be expressed as 

Taylor series concept (1.23) to develop the V7, dependence of Z(Vm, T) around the limit (2.29) as the infinite power series 

Here B(T) is the second virial coefficient, C(T) the third virial coefficient, and so forth. Formally, the virial coefficients can be defined as successive partial derivatives of Z with respect to inverse molar volume (density) under isothermal conditions for example, B(T) is given by 

For low density (large Vm), the series (2.30) is expected to achieve useful accurary with only a few terms. Higher densities within the domain of convergence require additional terms to achieve a desired accuracy. For some densities, the virial series may not converge at all. 

The virial coefficients B(T), C(T), D(T),. .. are functions of temperature only. Although these coefficients might be treated simply as empirical fitting parameters, they are actually deeply connected to the theory of intermolecular clustering, as developed by J. E. Mayer (Sidebar 13.5). More specifically, the second virial coefficient B(T) is related to the intermolecular potential for pairs of molecules, the third virial coefficient C(T ) to that for triples of molecules, and so forth. For example, knowledge of the intermolecular pair potential V(R) (see Sidebar 2.8) allows B(T) to be explicitly evaluated by statistical mechanical methods as 

Although we shall not pursue such specific molecular formulas in the present thermodynamic exposition, they form the basis for modem molecular-level understanding of gases and liquids [J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids (Wiley, New York, 1954)] and support the conviction that the virial equation (2.30) has deeper fundamental significance than other proposed equations of state. 

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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. 

The Virial Equation of State, Pergamon Press, Oxford (1969)... 

Enthalpies are referred to the ideal vapor. The enthalpy of the real vapor is found from zero-pressure heat capacities and from the virial equation of state for non-associated species or, for vapors containing highly dimerized vapors (e.g. organic acids), from the chemical theory of vapor imperfections, as discussed in Chapter 3. For pure components, liquid-phase enthalpies (relative to the ideal vapor) are found from differentiation of the zero-pressure standard-state fugacities these, in turn, are determined from vapor-pressure data, from vapor-phase corrections and liquid-phase densities. If good experimental data are used to determine the standard-state fugacity, the derivative gives enthalpies of liquids to nearly the same precision as that obtained with calorimetric data, and provides reliable heats of vaporization. 

The virial equation of state, first advocated by Kamerlingh Oimes in 1901, expresses the compressibility factor of a gas as a power series in die number density ... 

Real gases follow the ideal-gas equation (A2.1.17) only in the limit of zero pressure, so it is important to be able to handle the tliemiodynamics of real gases at non-zero pressures. There are many semi-empirical equations with parameters that purport to represent the physical interactions between gas molecules, the simplest of which is the van der Waals equation (A2.1.50). However, a completely general fonn for expressing gas non-ideality is the series expansion first suggested by Kamerlingh Onnes (1901) and known as the virial equation of state ... 

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. 

SemiempiricalRelationships. Exact thermodynamic relationships can be approximated, and the unknown parameters then adjusted or estimated empirically. The virial equation of state, tmncated after the second term, is an example of such a correlation (3). 

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. 

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... 

The next level of approximation is valid to higher pressures. It assumes that the gas mixture obeys the virial equation of state, with the third, fourth and higher, virial coefficients equal to zero. Thus... 

Purely phenomenological as well as physically based equations of state are used to represent real gases. The deviation from perfect gas behaviour is often small, and the perfect gas law is a natural choice for the first term in a serial expression of the properties of real gases. The most common representation is the virial equation of state ... 

Its precise basis in statistical mechanics makes the virial equation of state a powerful tool for prediction and correlation of thermodynamic properties involving fluids and fluid mixtures. Within the study of mixtures, the interaction second virial coefficient occupies an important position because of its relationship to the interaction potential between unlike molecules. On a more practical basis, this coefficient is useful in developing predictive correlations for mixture properties. 

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. 

At moderate pressures, the virial equation of state, truncated after the second virial coefficient, can be used to describe the vapor phase. As suggested by Hirschfelder, et. al. (1 3) the temperature dependence of the virial coefficients is expressed... 

With this historical background given by Mader (Addnl Ref N,. pp 7-8), it becomes apparent that the BKW equation of state is based upon a repulsive potential applied to the virial equation of state ... 

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). 

How does the equation of state look in a two-dimensional system Lets consider, for example, the virial equation of state. If II is written instead of P, and A instead of V, then the corresponding equation of state in two dimensions is (Parsons, 1961),... 

The virial equation of state represents the pressure as a polynomial series in the inverse molar volume as... 

Kammerlingb-Onnes (1901) proposed the virial equation of state.7... 

The mathematical relationship between pressure, volume, temperature, and number of moles of a gas at equilibrium is given by its equation of state. The most well-known equation of state is the ideal gas law, PV=RT, where P = the pressure of the gas, V = its molar volume (V/n), n = the number of moles of gas, R = the ideal gas constant, and T = the temperature of the gas. Many modifications of the ideal gas equation of state have been proposed so that the equation can fit P-V-T data of real gases. One of these equations is called the virial equation of state which accounts for nonideality by utilizing a power series in p, the density. 

The virial equation of state is especially important since its coefficients represent the nonideality resulting from interactions between two molecules. The second coefficient represents interactions between two molecules.Thus,a link is formed between the bulk properties of the gas (P,V,T) and the individual forces between molecules. 

The mutual dependence of the pressure, volume, and temperature of a substance is described by its equation of state. Many such equations have been proposed for the description of the actual properties of substances (and mixtures) in the gaseous and liquid states. The van der Waals expression is just one of these and of limited applicability. The virial equation of state ... 

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. 

The virial equation of state discussed in Section 7.2 is applicable to gas mixtures with the condition that n represents the total moles of the gas mixture that is, n = f= l n,. The constants and coefficients then become functions of the mole fractions. These functions can be determined experimentally, and actually the pressure-volume-temperature properties of some binary mixtures and a few ternary mixtures have been studied. However, sometimes it is necessary to estimate the properties of gas mixtures from those of the pure gases. This is accomplished through the combination of constants. 

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