Static indifferent states and Duhems theorem

We found in chapter XIII that the equilibrium state of a closed system, corresponding to given initial conditions, is completely determined by two variables (Duhem s theorem). For polyvariant systems 

We see therefore, that if a system is in an indifferent state, the variables T and p are no longer sufficient to determine its state completely. 

The name indifferent state seems to have had its origin in the fact that certain equilibrium states, even though completely defined by the temperature, pressure and composition of each phase, are indifferent to the mass of the phases present. Monovariant and invariant systems, which we have already seen to be indifferent, exhibit in all their equilibrium states this indifference to the masses of the phases For example, in the very simplest case of a monovariant system, a pure liquid in the presence of its vapour, we find just these properties. For at constant temperature, and hence at constant pressure, a closed system containing a pure substance can have an infinite range of equilibrium states which depend simply on the volume in which the system is confined. These equilibrium states differ only in the amounts of the two phases. 

Quite generally, if we fix the temperature of a monovariant closed system, the equilibrium conditions fix the pressure and composition of the phases, but the masses of the separate phases are not determined, since in the system of equations of enclosure (13.13) the number of equations (c) is less than cf + r ) the total number of unknowns. .. 

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