The Fundamental Equations Open Systems

In the case of an open homogeneous system, i.e. one that can exchange mass with its surroundings, the number of moles present is not constant. Eq.9.3.3 becomes, therefore  

From this relationship we obtain the following expression for the differential change in U resulting from differential changes in S, V, and N.  

We notice that the first two derivatives, taken while holding constant the mole number of all components, correspond to those for a closed system. It follows from Eqs 9.3.5 and 9.3.6 respectively, that  

Notice that the multiplier of the first differential in Eq.9.4.5, the temperature, is the potentisd for heat transfer and that of the second one, the pressure, is the potential for P-V work. By analogy, the multiplier of each of the remaining differentials must be also a potential. It is indeed the potential for mass transfer from one phase to an other, or for the feasibility of chemical reactions in a mixture of compounds, and it is called, appropriately, the Chemical Potential of component . 

It is apparent that Eq.9.4.1, combined with Eqs 9.4.3,9.4.4 and 9.4.6, provides complete description of the thermodynamic state of an open homogeneous system, i.e. it is a fundamental equation. 

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