Mechanical expressions for the grand potential

The simplest case that we shall be discussing hen in some detail is that of a fluid confined to a nanoscopic. slit-pore with homogeneous (infinitesimally) smooth substrate surfaces. For this prototypical model, it was shown in Section 1.6.1 that a mechanical expression for the grand potential exists. However, in what follows, it is more convenient to focus on the grand-potential density rather than on il itself. The former is defined through the relation... 

We now turn to a microscopic treatment of the. Toule-Thomson effect and begin with the limit of vanishing density. Th(j treatment below is very sinrilar to the one presented in Section 3.2.2 where we derived molecular expressions for the first few virial coefficients of the one-dimensional hard-rod fluid. Here it is important to realize that a mechanical expression for the grand potential exists for a fluid confined to a slit-pore with chemically structured substrate surfaces as we demonstrated in Section 1.6.1 [see Eq. (1.65)]. Combining this cxpres.sion with the tnolocular expression given in Eq. (2.81) we may write... 

The existence of mechanical expressions for the grand potential introduces an additional equation for fl. Take as an example Exi- (1.62) whose exact differential may be cast as... 

This statistical mechanical expression for surface tension depends explicitly on the potentials of inteimolecular force and molecular distribution functions. Upon recognition that the two-phase system under consideration is thermodynamically open, it follows that the distribution functions must be represented in the grand canonical ensemble. Thus, the dependence of y on temperature, T, and chemical potentials, enters through the implicit dependence of the distribution func-... 

The statistical mechanical verification of the adsorption Equation 11 proceeds most conveniently with use of the expression for y given by Equation 5. An identical starting formula is obtained via the virial theorem or by differentiation of the grand partition function (3). We simplify the presentation, without loss of generality, by restricting ourselves to multicomponent classical systems possessing a potential of intermolecular forces of the form... 

Abstract Fluctuation Theory of Solutions or Fluctuation Solution Theory (FST) combines aspects of statistical mechanics and solution thermodynamics, with an emphasis on the grand canonical ensemble of the former. To understand the most common applications of FST one needs to relate fluctuations observed for a grand canonical system, on which FST is based, to properties of an isothermal-isobaric system, which is the most common type of system studied experimentally. Alternatively, one can invert the whole process to provide experimental information concerning particle number (density) fluctuations, or the local composition, from the available thermodynamic data. In this chapter, we provide the basic background material required to formulate and apply FST to a variety of applications. The major aims of this section are (i) to provide a brief introduction or recap of the relevant thermodynamics and statistical thermodynamics behind the formulation and primary uses of the Fluctuation Theory of Solutions (ii) to establish a consistent notation which helps to emphasize the similarities between apparently different applications of FST and (iii) to provide the working expressions for some of the potential applications of FST. 

We round out this introduction to the virial equation of state by reference to its theoretical foundation. Thus statistical mechanics permits deduction of an expression for pVin terms of either the grand partition function or the radial distribution function. The leading term in the expansion of the latter function corresponds to pairwise interaction between molecules, and indicates the following relation between the second virial coefficient and the potential energy (r) of the interacting pair, when this depends only on the distance r between molecular centres ... 

---