Chemistry Reference
In-Depth Information
Abstract
: Fluctuation Theory of Solutions or Fluctuation Solution Theory
(FST) combines aspects of statistical mechanics and solution thermodynam-
ics, with an emphasis on the grand canonical ensemble of the former. To under-
stand 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 pro-
vide experimental information concerning particle number (density) fluctua-
tions, 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.
1.1 BACKGROUND AND THEORY
The Fluctuation Theory of Solutions—also known as
Fluctuation Solution Theory
,
Kirkwood-Buff Theory
, or simply
Fluctuation Theory
—provides an elegant
approach relating solution thermodynamics to the underlying molecular distribu-
tions or particle number fluctuations. Here, we provide the background material
required to develop the basic theory. More details can be found in standard texts on
thermodynamics and statistical mechanics (Hill 1956; Münster 1970). Indeed, the
experienced reader may skip this chapter completely, or jump to Section 1.2. A list
of standard symbols is also provided in the Prolegomenon to aid the reader, and we
have attempted to use the same set of symbols and notations in all subsequent chap-
ters. Throughout this work we refer to a collection of species (1, 2, 3,…) in a system
of interest. We consider this to represent a primary solvent (1), a solute of interest (2),
and a series of additional cosolutes or cosolvents (3, 4,…) which may also be present
in the solution. However, other notations such as A/B or u/v is also used in the vari-
ous chapters. All summations appearing here refer to the set of thermodynamically
independent components (
n
c
) in the mixture unless stated otherwise. Derivatives of
the chemical potentials with respect to composition form a central component of the
theory. The primary derivative of interest here is defined as
∂
∂
βµ
i
µ
=
(1.1)
ij
x
j
pT
,
although the most convenient derivative often depends on the exact application. A
general fluctuation in a property
X
is written as δ
X
=
X
− ⟨
X
⟩, where the angular
brackets denote an ensemble or time average in the grand canonical ensemble unless
