Conceptuals from Measurables Using Equations of State

4 CONCEPTUALS FROM MEASURABLES USING EQUATIONS OF STATE 

In this section we present explicit expressions that allow us to use the measurables P, V, T, and x to compute values for u, h, s, a, and g. We can only obtain values for these conceptuals relative to some well-defined reference state, so here we choose the reference to be the ideal gas. As a result, the expressions obtained below provide the residual properties. In addition, from the expression for the residual chemical potential we can readily obtain expressions for the fugadty coefficient. All the relations derived below involve integrals over functions of P, V, T, and x, and to exploit those relations, we need a volumetric equation of state for our substance. 

However, if the independent variable (P or v) in our equation of state differs from the one that is appropriate for a particular conceptual, then a more involved computational procedure must be followed. This procedure appears on the rhs of the figure. For example, if we need Ah T, P, x ), but our equation of state uses (T, v, x ), then we have an incompatible situation to resolve. It should be resolved not by tampering with the expression given in 4.4.1 for the residual enthalpy, but instead by computing Ah indirectly via a Legendre transform. Therefore the procedure should be this  

When we have an equation of state in the form v T, P, x ), then pressure is the independent variable (rather than v), and the relevant forms for conceptuals are the iso-baric residual properties introduced in 4.2.1. To evaluate those quantities, we start with the observation that the residual properties are all state functions this allows us to write 

The lower limit is zero because the residual properties of ideal gases are zero. Our objective is to write the rhs of (4.4.1) as an integral over a measurable, and since our equation of state has P independent, we choose P to be that measurable. Therefore, we 

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