Virial expression

Statistical mechanics may be used to derive practical microscopic fomuilae for themiodynamic quantities. A well-known example is tire virial expression for the pressure, easily derived by scaling the atomic coordinates in the canonical ensemble partition fiinction... 

In practice, it is often found that compressing or decompressing a gas does not follow closely to the ideal-gas equation, particularly at high p or low T, as exemplified by the need for equations such as the van der Waals equation or a virial expression. The equation above is a good approximation, though. 

Isobaric ensembles can also be generated in this way by requiring that df>/dt=0. In an inhomogeneous fluid, the three diagonal elements of the pressure tensor p should be considered separately, which means that one could have up to three constraints. Usually, only the two pressure elements px (perpendicular) and pn (parallel) to the surface need be considered. The virial expression for the pressure element in an inhomogeneous fluid can be written as [7] ... 

In the next section we first introduce the main ingredients of a FT by mimicking as far as possible what is done in QFT. To be illustrative, in Sec. 3 we show how it is possible to derive the virial expression for the pressure in a FT. In Sec. 4 we show that we have at our disposal some relations between field-field correlation functions that are obtained from symmetry arguments or from the fact that fields are dummy variables. In Sec. 5 on two examples, we derive some results starting from Dyson-like equations. Finally, in Sec. 6, leaving the purely microscopic level we introduce a model showing the existence of a demixion transition at a metal-solution interface using a mean field approximation. Brief conclusions are presented in Sec. 7. 

In the next section we will show that p introduced in the case of the ideal system verifies the usual virial expression in presence of a potential. 

Together Eqs. (E.70), (E.73), and (E.74) constitute tlie virial expression for the bulk pressure, which parallels Eq. (5.63) for the stress exerted by a con-fined fluid on the planar substrates of a slit-pore. 

To obtain virial expressions for colligative properties, Eq. (3.74) may be combined with respective Eqs. (3.61)-(3.64). If, for example, osmotic data are used, Eq. (3.74) is combined with Eq. (3.64) to write the virial equation... 

This is the pressure formula of Nieminen and Hodges [7.18] and of Pettifor [7.16], who derived it from the virial expression given by Liberman [7.17]. 

But this is not a simple task theoretically (especially since the hydrodynamic interactions that are important for higher concentrations are very difficult to account for), and even for the second-order term B quite different values can be found in the literature (e.g., 6.2 [26] and 14.1 [27]). However, for the description of concentrated dispersions, such a virial expression is not really very useful because it would require a fairly large number of coefficients that are theoretically not easily accessible. For spheres that are not correlated spatially the hydrodynamic interactions can be taken into consideration and yield the... 

To add effects of specific interactions, the Gordon-Taylor expression can be expanded into a virial expression, as in the Schneider equation, listed in Fig. 7.69 [30]. The variable W2c is the expansivity-corrected mass fraction of the Gordon-Taylor expression W2c = kW2 / (Wj + kWj). The Schneider equation can be fitted with help of the constants Kj and K2 to many polymer/polymer solutions, as is illustrated in Sect. 7.3.2. The parameter Kj depends mainly on differences in interaction energy between the binary contacts of the components A-A, B-B, and A-B, while Kj accounts for effects of the rearrangements in the neighborhood of the contacts. 

While linking structure and thermodynamics based on the virial expression is not straightforward, this link can in fact be established using an alternative desciiptiOTi based on Kirkwood-Buff (KB) theory [76], Whereas the virial route requires information on the effective potential, the KB description does not make any assumption on the nature of the potentials, is exact, and its central quantities can be interpreted in terms of local solution structure. To this end, we consider the derivatives of the salt activity with respect to the density at constant pressure p and temperature T. For the systems shown in Fig. 5 these derivatives show the same order as the osmotic coefficients/salt activities for the different ions [70]. Hence, the microscopic mechanism explaining the order among the derivatives of the salt activity for the different ions also explains the Hofmeister series for the activities obtained by integration of the derivatives. Based on this, the relation between... 

For very accurate measurements the non-ideal behaviour of CO2 has to be taken into account, Le., fugadty has to be used instead of partial pressure. This is the case if the results are to be used to calculate other parameters of the CO2 system in seawater. The fugacity can be calculated from a knowledge of the virial expression of the equation of state for CO2. For the binary mixture C02-air the fugacity of CO2 (/(CO2)) is given by... 

The virial expression for the surface tension (4.104)-(4.106) has been extended in various ways. For a c-component mixture of spherical molecules, we sum the right-hand side of (4.104) over the ic(c + l) pain of interactions. Carey, Scriven, and Davis have extended the expressions for the pressure tensor to mixtures. If contains multi-body potentials then o- can be written as a sum of virial terms, eadi of the same general form as (4.104). If is pair-wise additive, but if the pair potentials depend also on the orientations oit and the ij-pair, thra o-comprises two terms o, and angle between i 2 and the z-axis. ... 

With these preliminary results we can now convert the virial expression for direct correlation function. We start with the variant (4.105), use (4.192) and (4.179), and then the identity... 

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