Fluctuation Solution Theory - Fluctuation Theory of Solutions - page 3

stated otherwise. The Kronecker delta function (δ ij ) is used consistently during the
theoretical development and is equal to unity when i = j , and zero otherwise.
Finally, throughout the various chapters the contributors have generally used
the terms species , component , and molecule type interchangeably. This is perfectly
acceptable for most applications. However, there are some applications for which a
distinction between species and components must be made, and extra care should be
taken to avoid confusion in these cases. Examples include the treatment of electro-
lytes provided in Section 1.3.6 and in Chapter 8, the treatment of solute association
in Section 1.3.8, as well as the general treatment of reactive equilibria outlined in
Chapter 9.

1.1.1 B asic T Thermodynamics

Let us consider the fundamental equation of Gibbs describing the dependence of the
internal energy on entropy, volume, and composition, that is, U ( S , V ,{ N }) such that

=− + ∑ µ dU TdS pdV dN (1.2)

i i

i

for systems in equilibrium with only pV work (Münster 1970). More convenient
relationships, which express the above relationship in terms of alternative sets of
variables, can be obtained from the above equation by a series of Legendre transfor-
mations. Consequently, a series of thermodynamic potentials are obtained with the
most common being

∑

dH =++ TdSVdp µ dN

i i

i

∑

dA =− SdT −+ pdV µ dN

i i

i

(1.3)

∑

dG =− Sd TTVdp ++ µ dN

i i

i

∑

dpV =+ + SdT pdV Nd µ

i i

i

where we have used the usual definitions; H = U + pV , A = U - TS , and G = H - TS .
One is then free to choose the most appropriate variables of interest for a specific
application. The set of variables of most interest in this work are { N }, p , and T for
which the characteristic function is the Gibbs free energy, and {μ}, V , and T for which
the characteristic function is pV . The primary aim is to use statistical thermodynam-
ics for an open system to describe the properties of an equivalent closed system that
are amenable to experiment.