Page 118 - Modern physical chemistry

6

Relationships between

Phases

6. 1 Intensive Variables
IN A GIVEN SYSTEM, THE INTENSIVE THERMODYNAMIC properties may vary from
point to point. But each part of the system through which these properties are constant
or vary continuously constitutes a phase. A system may contain gaseous, liquid, and solid
phases. A chemical constituent whose mass can vary independently in any small region
of the system is called a component. Because of reactions, the number of chemical species
present may be greater than the number of components.
As thermodynamic variables, a person may choose pressure P, temperature T, and
concentrations of all but one component. The equation of state for each phase yields its
specific volume V (and the reciprocal, the density p) from P and T. Since all the mole
fractions add up to one in each phase, the mole fraction of the last component is obtain-
able from the other mole fractions.

6.2 The Gibbs Phase Rule
In studying systems, a person needs to know not only the various intensive variables
but also how many are independent.
Consider a macroscopic system at equilibrium. Each phase is then uniform. Let p be
the number of phases, c the number of components.
If we consider the pressure P, the temperature T, and c - 1 concentrations in each
phase to be variables, the total is 2 + p(c - 1). But partition of the components between
the phases would have proceeded until equilibrium was set up. There are p - 1 indepen-
dent distribution constants for each component. For c components, we have c(p -1) inde-
pendent conditions.
The total number of independent intensive variables is thus

f = 2+ p(c-1)-c(p-1) = c- p+2. [6.1 ]
We callfthe number of degrees of freedom for the system. Formula (6.1) is called the
Gibbs phase rule.
Since a system must contain at least one phase, the maximum number of degrees of
freedom is given by
fmax = c -1 + 2 = c + l. [6.2]
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