Equation of state virial expansion

The Vina/Expansion. Many equations of state have been proposed for gases, but the virial equation is the only one having a firm basis in theory (1,3). The pressure-explicit form of the virial expansion is 

Equation (95) is obtained from the virial expansion of the equation of state for rigid spheres for higher densities the rigid-sphere equation of state obtained from the radial distribution function by Kirkwood, Maun, and Alder has to be used (K10, Hll, p. 649). When Eq. (95) is substituted in Eqs. (92), (93), and (94) one then obtains the rigorous expressions for the coefficients of viscosity, thermal conductivity, and selfdiffusion of a gas composed of rigid spheres. 

Partial parameter, cubic equation of state 2d virial coefficient, density expansion Partial molar second virial coefficient Reduced second virial coefficient 

The virial expansion (2.30) can also be used to expand empirical equations of state in power series form, allowing term-by-term comparisons of one equation with another. To illustrate this procedure, let us obtain the virial expansion for the Van der Waals equation (2.26), which can be rewritten as 

It is well known that the equation of state for a gas at a temperature T is given by the virial expansion of the form, 

The deviations of a real gas from ideal behavior are accounted for by the virial expansion, so that the equation of state reads 

The most general of the equations of state is the virial equation, which is also the most fundamental since it has a direct theoretical connection to the intermolecular potential function. The virial equation of state expresses the deviation from ideality as a series expansion in density and, in terms of molar volume, can be written 

Molar area, adsorbed phase Parameter, cubic equations of state Partial parameter, cubic equations of state Second virial coefficient, density expansion 

A classical equation of state is normally composed of a truncated Taylor series in the independent variables, normalized to the critical point conditions (e.g., van der Waals, virial expansion, etc.). All these sorts of equations yield similar (so-called classical ) asymptotic behavior in their derivative properties at the critical point. 

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. 

If the gas of charges (plasma) is sufficiently dilute, we could hope a priori that its equation of state would be described by the virial expansion  

Table 4 compares different theoretical approaches with respect to the equations of state and the second and third virial coefficients (B2, B3) for a hard rod solution in the isotropic state B2 and B3 are the parameters appearing in the expansion 

Derive an equationfor the work of mechanically reversible, isotliemial compressionof 1 mol of a gas from an initial pressure Pi to a final pressure P2 when the equation of state is the virial expansion [Eq. (3.11)] trancated to  

The volumetric properties of fluids are conveniently represented by PVT equations of state. The most popular are virial, cubic, and extended virial equations. Virial equations are infinite series representations of the compressibiHty factor Z, defined as Z = PV/RT having either molar density, p[ = V ), or pressure, P, as the independent variable of expansion  

Here the quantity PV/nRT is often called the virial and the quantities 1, B(T), C(7T), etc., the coefficients of its expansion in inverse powers of the volume per mole, F/n, are called the virial coefficients, so that B(T) is called the second virial coefficient, C(T) the third, etc. The experimental results for equations of state of imperfect gases are usually stated by giving B(T), C(T), etc., as tables of values or as power series in the temperature. It now proves possible to derive the second virial coefficient B T) fairly simply from statistical mechanics. 

Find a formula for the change of Gibbs free energy of 1 mol of a gas that expands from a volume V to a volume V2 at constant temperature T. Use terms in the virial expansion up to and including the second virial coefficient to describe the equation of state of the gas. 

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 

This is a simple and important result. It equates VPIE to the isotopic difference of standard state free energies on phase change, plus a small correction for vapor phase nonideality, here approximated through the second virial coefficient. Therefore Equation 5.8 is limited to relatively low pressure. As T and P increase third and higher virial corrections may be needed, and at even higher pressures the virial expansion must be abandoned for a more accurate equation of state. 

The equilibrium between a pure solid and a gaseous mixture is one of very few classes of solution for which an exact treatment can be made by the methods of statistical mechanics. The earliest work on the theory of such solutions was based on empirical equations, such as those of van der Waals,45 of Keyes,44 and of Beattie and Bridgemann.3 However, the only equation of state of a gas mixture that can be derived rigorously is the virial expansion,46 66 

Long ago, Langmuir suggested that the rate of deposition of particles on a surface is proportional to the density of particles in the vicinity of the surface and to the available area on the surface [1], However, the calculation of the available area is still an open problem. In a first approximation, one can assume that the available area is the total area of the surface minus the area already occupied by the adsorbed particles [1]. A better approximation can be obtained if the adsorbed particles, assumed to have the shape of a disk, are in thermal equilibrium on the surface, either because of surface diffusion and/or of adsorption/desorption kinetics. In this case, one can use one of the empirical equations available for the compressibility of a 2D gas of hard disks, calculate the chemical potential in excess to that of an ideal gas [2] and then use the Widom relation between the area available to one particle and its excess chemical potential on the surface (the particle insertion method) [3], The method is accurate at low densities of adsorbed particles, where the equations of state are accurate, but, in general, poor at high concentrations. The equations of state for hard disks are based on the virial expansion and only the first few coefficients of this 

---