Page 180 - Thermodynamics of Biochemical Reactions

180 Chapter 11 Use of Semigrand Partition Functions

for each choice of natural variables. Gibbs (1903) introduced the grand canonical
partition function for a system containing a single species that is in contact with
a large reservoir of that species through a permeable membrane. Semigrand
partition functions were introduced later by statistical mechanicians to treat
systems with several species, one or more of which are available from a large
reservoir through a semipermeable membrane. The idea of holding the chemical
potential of a species constant was often used in statistical mechanics long before
it was used in thermodynamics.
In this chapter the usual convention in statistical mechanics of using numbers
N, of molecules (rather than amounts q), the Boltzmann constant k (rather than
the gas constant R), and [j = l/kT have been used, but the same symbols have
been used for thermodynamic properties as in thermodynamics. Thus the proper
interpretation of these latter symbols depends on context. Detailed information
on various partition functions is provided by textbnooks on statistical mechanics
(McQuarrie, 2000; Chandler, 1987; Greiner, Neise, and Stocker. 1995; di Cera,
1995; Widom, 2002).

11.1 INTRODUCTION TO SEMIGRAND PARTITION
FUNCTIONS

Gibbs considered the statistical mechanics of a system containing one type of
molecule in contact with a large reservoir of the same type of molecules through
a permeable membrane. If the system has a specified volume and temperature and
is in equilibrium with the resevoir, the chemical potential of the species in the
system is determined by the chemical potential of the species in the reservoir. The
natural variables of this system are 7: V, and p. We saw in equation 2.6-12 that
the thermodynamic potential with these natural variables is U[7; ,HI using Callen's
nomenclature. The integration of the fundamental equation for U[7; p] yields
- PV (see equation 2.6-20), and so - PV can be considered a state function of the
system under these conditions.
Statistical mechanics is based on the use of ensembles (collections of systems
under specified conditions) that lead to partition functions. Partition functions are
sums of exponential functions. Gibbs referred to the ensemble for a system at T
and V in contact with a large reservoir of that species through a permeable
membrane as the grand canonical ensemble. The corresponding grand canonical
partition function is represented by Z(7; V, p). Since thermodynamic potentials are
given by - kT times the natural logarithm of a partition function, the value of PV
can be calculated using
PV = kTlnE(7: V, p) (1 1.1-1)

where k is the Boltzmann constant (R/NA) and N, is the Avogadro constant.
If a system contains two types of species, but the membrane is permeable only
to species number 1, the natural variables for the system are 7; V, pi, and N,,
where N, is the number of molecules of type 2 in the system. The thermodynamic
potential for this system containing two species is represented by U[7: pi]. The
corresponding ensemble is referred to as a semigrand ensemble, and the semigrand
partition function can be represented by "(7; V, p,, NJ. The thermodynamic
potential of the system is related to the partition function by

U[T. pl] = -kTln "(7: r! pi, N,) (11.1-2)
The Gibbs energy for a system at constant T and P containing a single species
is given by
G(7: P, N) = -kTlnA(T P, N) (I 1.1-3)

where A is the isothermal-isobaric partition function.