CHAP. 7: PHASE EQUILIBRIA [CONTENTS] 179
7.3 Gibbs phase rule
7.3.1 Independent and dependent variables
The thermodynamic state of ak−component one-phase system is fully characterized by the
temperature, pressure and amount of substance of all its components. We will call these
quantitiesindependent variables. Other thermodynamic quantities, e.g. the volume, Gibbs
energy or enthalpy, are thedependent variablesof a system and they are the functions of
T, p, n 1 , n 2 ,... nk.
Note:The above classification of thermodynamic quantities into independent and depen-
dent variables is most usual but it is not unambiguous. We may, e.g., choose enthalpy,
pressure and amount of substance as independent variables, in which case the system’s
temperature becomes a dependent variable.7.3.2 Intensive independent variables
We are often interested solely in relations between the intensive variables of a system [see
1.3.1] and not in the system’s size, i.e. its total amount of substance. In this case the number of
independent variables drops by one.T, p, x 1 ,... , xk− 1 (the molar fraction of thekthcomponent
is calculated from relation (1.7)),xk= 1−
∑k− 1
i=1xi) are usually chosen as independent variables,
and we will call themintensive independent variables. Otherintensive thermodynamic
quantities, e.g. the molar volume, molar Gibbs energy or molar enthalpy, are the functions of
intensive independent variables.
In ak−component system formed byf separatephases, i.e. phases which do not coexist,
the intensive independent variables are the temperatures, pressures and compositions of all
phases. The other intensive thermodynamic quantities in a given phase are the functions of the
temperature, pressure and composition (i.e. the molar fractions) of that particular phase.
When studying a system under certain fixed conditions, the number of intensive independent
variables drops by the number of these conditions.
Example
Give the number of intensive independent variables in a system formed by gaseous nitrogen,
oxygen and argon, and in an isobaric system formed by the same gases.