Effects of External Constraints on System States

In previous chapters we have tried to convince you that if we have a complete equation of state for a one-phase substance, then we can compute values for all thermodynamic properties. Up to now, much of our attention has focused on volumetric equations of state, P(T, v, x ) or v T, P, x ), because these equations contain only mea-surables. But those forms are not the only possibilities. For example, our fairy godmother might present us with a complete functional form for the Helmholtz energy 

From either of these we could use relations presented in earlier chapters to obtain all remaining thermodynamic properties. 

To determine the number of independent properties required to completely define an equation of state, we use the procedure introduced in 3.1. There we made a distinction between V, the number of orthogonal interactions available to manipulate a state, and number of independent properties needed to identify a state. We 

Consider two systems, 1 and 2. System 1 is a one-phase mixture of C components, with mole numbers N. This mixture fills a rigid vessel of volume Vj, and the vessel is immersed in a heat bath maintained at temperature Tj. System 2 is another sample of the same mixture, having the same C components and the same mole numbers N. System 2 fills the cylinder of a piston-cylinder apparatus. The cylinder is immersed in a heat bath at T2. A constant external pressure is imposed on the mixture at equilibrium the system pressure P2 balances that external pressure. Therefore, system 2 is at constant pressure, while system 1 is at constant volume. 

To identify each state (with S = 0), we must by (3.1.7) specify values for (C + 2) properties. We have met this requirement Ty Vy and C mole numbers. Moreover, the two sets of values are identical. Hence, the two states are identical, and consequently, all thermodynamic properties are exactly the same in the two systems, even though the external constraints differ. (This assumes relations among properties are monotonic if they are not, we can still adjust T, V, and N so that the two states are identical.) 

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