CHAPTER 2 SYSTEMS AND THEIR PROPERTIES
2.4 THESTATE OF THESYSTEM 47
We could make other choices of the independent variables for the aqueous sucrose sys-
tem. For instance, we could choose the setT,p,V, andxB, or the setp,V,, andxB.
If there are no imposed conditions, thenumberof independent variables of this system is
always four. (Note that we could not arbitrarily choose just any four variables. For instance,
there are only three independent variables in the setp,V,m, andbecause of the relation
Dm=V.)
If the system has imposed conditions, there will be fewer independent variables. Sup-
pose the sucrose solution is in a closed system with fixed, known values ofnAandnB; then
there are only two independent variables and we could describe the state by the values of
justTandp.
2.4.3 More about independent variables
A closed system containing a single substance in a single phase has two independent vari-
ables, as we can see by the fact that the state is completely defined by values ofTandpor
ofTandV.
A closed single-phase system containing a mixture of several nonreacting substances,
or a mixture of reactants and products in reaction equilibrium, also has two independent
variables. Examples are
air, a mixture of gases in fixed proportions;
an aqueous ammonia solution containing H 2 O, NH 3 , NH 4 C, HC, OH , and probably
other species, all in rapid continuous equilibrium.The systems in these two examples contain more than one substance, but only onecom-
ponent. The number of components of a system is the minimum number of substances or
mixtures of fixed composition needed to form each phase.^10 A system of a single pure sub-
stance is a special case of a system of one component. In anopensystem, the amount of
each component can be used as an independent variable.
Consider a system with more than one uniform phase. In principle, for each phase we
could independently vary the temperature, the pressure, and the amount of each substance
or component. There would then be 2 CCindependent variables for each phase, whereC
is the number of components in the phase.
There usually are, however, various equilibria and other conditions that reduce the num-
ber of independent variables. For instance, each phase may have the same temperature and
the same pressure; equilibrium may exist with respect to chemical reaction and transfer
between phases (Sec.2.4.4); and the system may be closed. (While these various condi-
tions do nothaveto be present, the relations amongT,p,V, and amounts described by an
equation of state of a phase arealwayspresent.) On the other hand, additional independent
variables are required if we consider properties such as the surface area of a liquid to be
relevant.^11
We must be careful to choose a set of independent variables that defines the state
without ambiguity. For a closed system of liquid water, the setpandV might be a(^10) The concept of the number of components is discussed in more detail in Chap. 13.
(^11) The important topic of the number of independentintensivevariables is treated by the Gibbs phase rule, which
will be discussed in Sec.8.1.7for systems of a single substance and in Sec.13.1for systems of more than one
substance.