Thermodynamic State for Multiphase Systems

1 we discussed the thermodynamic state for closed systems composed of a single homogeneous phase we now extend that discussion to heterogeneous systems, especially, systems containing more than one phase. The fundamental questions addressed in 3.1 are revisited here How many interactions are available for manipulating the state ( 9.1.1) How many property values are needed to identify the state ( 9.1.2) Even when we specify the correct number of properties for identifying the state, is there still a possibility of encountering computational difficulties ( 9.1.3)  

Consider a system composed of C components in a single homogeneous phase. The system can interact with its surroundings through the thermal interaction, a PV work mode, and the exchange of any of the components. For such a system, we found in 3.1.1 that the number of interactions available for changing the state is given by 

For a single homogeneous phase containing C components, we found in 3.1.2 that the number of properties needed to identify the extensive state is given by 

Again this assumes only the thermal interaction and a single work mode are present. For a heterogeneous system containing T homogeneous phases, (9.1.4) applies to 

But at equilibrium we also have internal constraints imposed by Nature. For example, if the T homogeneous phases are all open to one another through ((P-l) different interfaces, then each interface imposes the (C + 2) phase-equilibrium constraints given in 7.3.5. For the one interface between phases a and (1 in equilibrium, these constraints are 

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