Residual properties isometric

Instead of using T and P as the independent variables, as we did in 4.2.1, we could choose T and V. Therefore, we define another set of residual properties in which the real substance and the ideal gas each occupy a container of the same volume V. Extensive isometric residual properties are defined by... 

We caution that (4.2.31) cannot be derived in a simple way by applying the partial molar derivative to the difference in residual Gibbs energies given in (4.2.30). The difficulty is that the partial molar derivative imposes a fixed pressure, but when the Ihs of (4.2.30), g (T, V, x ), is changed at fixed pressure, the mixture and ideal-gas volumes are no longer the same. Consequently, the isobaric derivative of the Ihs of (4.2.30) is not an isometric residual property in particular, it is not the Ihs of (4.2.31). 

When we have an equation of state in the form P(T, v, x ), then the isometric residual properties are easier to compute than are the isobaric ones. However, in applications, we usually need the isobaric residual properties, not the isometric ones. We foUow a two-step procedure to obtain values for isobaric residual properties (1) evaluate the isometric residual properties from the integrals presented in this section, then (2) convert those isometric properties to isobaric ones using the relations given in 4.2.3. 

The procedure for obtaining computationally useful expressions for isometric residual properties parallels that used in 4.4.1 for the isobaric properties. That is, analogous to (4.4.2), we can obtain each residual property by starting from... 

The model (4.5.31) is simple enough that it can be written explicitly in both v and P, so we can compare results for the isobaric residual properties with those for the isometric residual properties. 

To evaluate other isometric residual properties, we will need the thermal pressure coefficient (3.3.5). Applying its definition to the equation of state (4.5.38), we obtain... 

In this chapter we have developed ways for computing conceptual thermodynamic properties relative to well-defined states provided by the ideal gas. We identified two ways for measuring deviations from ideal-gas behavior differences and ratios. Relative to the ideal gas, the difference measures are the isobaric and isometric residual properties, while the ratio measures are the compressibility factor and fugacity coefficient. These differences and ratios all apply to the properties of any single homogeneous phase (liquid or gas) composed of any number of components. 

In this subsection we consider those situations in which our mixture is described by a pressure-explicit equation of state, P = P(T, v, x ). Our objective is still to relate excess properties to residual properties and to the equation of state, but with v as an independent variable, we would prefer to express those relations in terms of the isometric residual properties, rather than the isobaric ones. However, the development is not as simple as what we did in the previous section because now we have an inconsistency the equation of state and the isometric residual properties use (T, v, x ) as the independent variables, but the excess properties defined by (5.2.1) use (T, P, x)). 

To illustrate how these equations are applied, we repeat the calculations in 5.3.1 to obtain excess properties for gaseous mixtures of methane and sulfur hexafluoride at 60°C, 20 bar. Values for the isometric residual properties of this mixture have already been determined in 4.5.5. We continue to use the virial equation of state (5.3.2), but now we write it in a pressure-explicit form. 

We define two classes of residual properties isobaric ones and isometric ones. The isobaric residual properties ( 4.2.1) are the traditional forms and use P as the independent variable. The isometric ones ( 4.2.2) use v as the independent variable and thereby simplify computations when our equation of state is explicit in the pressure such equations of state are now commonly used to correlate thermodynamic data for dense fluids. Although isometric property calculations may be more complicated than those for isobaric properties, with the help of computers, tiiis is not really an issue. 

For first-law conceptuals u and h) this inconsistency poses no problem because values of first-law isometric and isobaric residual properties are the same see (4.2.24) and (4.2.25). However, for second-law conceptuals (g, a, and s) the two kinds of residual properties differ (see 4.2.3), so we must exercise care when using residual properties to evaluate second-law conceptuals. We need to evaluate only three quantities v, u, and s ) then the remaining three Qi, g, and a ) can be obtained from Legendre transforms. We also obtain the expression for the excess chemical potential in terms of isometric residual chemical potentials. 

It may seem that the residual properties offer additional flexibility because we defined two kinds— isobaric ones and isometric ones—while we introduced only iso-baric excess properties. But this difference is mainly one of historical significance. The two kinds of residual properties allow us to perform calculations using both pressure-explicit and volume-explicit equations of state. In contrast, the excess properties were originally applied only to liquids, for which pressure and volume effects are often ignored. We could certainly define isometric excess properties, but in practical applications involving liquids, there seems to be little advantage to doing so. Differences between isometric and isobaric excess properties are discussed by Rowhnson and Swinton [13], but for condensed phases, those differences are usually small. 

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