Property partial

This definition is the means by which partial properties are calculated from solution properties. Equation 115 can now be written as equation 117 ... 

This summability equation, the counterpart of equation 116, provides for the calculation of solution properties from partial properties. Differentiation of equation 122 yields equation 123 ... 

This result, known as the Gibbs-Duhem equation, imposes a constraint on how the partial molar properties of any phase may vary with temperature, pressure, and composition. In particular, at constant T and P it represents a simple relation among the Af/ to which measured values of partial properties must conform. 

For the Gibbs energy of an ideal gas mixture, — T the parallel relation for partial properties is equation 149 ... 

Equation 163, written as = G- /-RT, clearly shows that In ( ) " is a partial molar property with respect to G /KT. MultipHcation of equation 175 by n and differentiation with respect to at constant T, P, and in accord with equation 116 yields, after reduction, equation 179 (constant T,x), where is the partial molar compressibiUty factor. This equation is the partial-property analogue of equation 178. 

All other thermodynamic properties for an ideal solution foUow from this equation. In particular, differentiation with respect to temperature and pressure, followed by appHcation of equations for partial properties analogous to equations 62 and 63, leads to equations 191 and 192 ... 

Foi an ideal solution, G, = 0, and tlieiefoie 7 = 1- Compatison shows that equation 203 relates to exactiy as equation 163 relates ( ) to GG Moreover, just as ( ) is a partial property with respect to G /E.T, so In y is a partial property with respect to G /RT. Equation 116, the defining equation for a partial molar property, in this case becomes equation 204 ... 

The basis for calculation of partial properties from solution properties is provided by this equation. Moreover, the preceding equation becomes... 

Equation (4-49) is merely a special case of Eq. (4-48) however, Eq. (4-50) is a vital new relation. Known as the summahility equation, it provides for the calculation of solution properties from partial properties. Thus, a solution property apportioned according to the recipe of Eq. (4-47) may be recovered simply by adding the properties attributed to the individual species, each weighted oy its mole fraction in solution. The equations for partial molar properties are also valid for partial specific properties, in which case m replaces n and the x, are mass fractions. Equation (4-47) applied to the definitions of Eqs. (4-11) through (4-13) yields the partial-property relations ... 

Equation (4-47), which defines a partial molar property, provides a general means by which partial property values may be determined. However, for a.hinary solution an alternative method is useful. Equation (4-50) for a binaiy solution is... 

Thus for a binary solution, the partial properties are given directly as functions of composition for given T and P. For multicomponent solutions such calcufations are complex, and direc t use of Eq. (4-47) is appropriate. 

Comparison with Eq. (4-20) provides an example of the parallelism that exists between the eqnaOons for a constant-composition sohiOon and those for the corresponding partial properties. This parallelism exists whenever the sohidon properties in the parent equation are related hnearly (in the algebraic sense). Thus, in view of Eqs. (4-17), (4-18), and (4-19) ... 

This is expressed mathematically for generic partial property M/ by the equation... 

This equation demonstrates that In (()j is a partial property with respect to G /RT The partial-property analogs of Eqs. (4-83) and (4-84) are therefore ... 

This definition is analogous to the definition of a residual property as given by Eq. (4-67). However, excess properties have no meaning for pure species, whereas residual properties exist for pure species as well as for mixtures. In addition, analogous to Eq. (4-99) is the partial-property relation,... 

In apphcatious to equilibrium calculations, the fugacity coefficients of species iu a mixture are required. Given au expression for G /RT as aetermiued from Eq. (4-158) for a coustaut-compositiou mixture, the corresponding recipe for In is found through the partial-property relation... 

A graphical integration of the Gibbs-Duhem equation is not necessary if an analytical expression for the partial properties of mixing is known. Let us assume that we have a dilute solution that can be described using the activity coefficient at infinite dilution and the self-interaction coefficients introduced in eq. (3.64). 

In general, thermodynamic properties of the components in a solution vary with composition because the environment of each type of atom or molecule changes as the composition changes. The change in interaction force between neighbouring atoms or molecules with the change in composition results in the variation of the thermodynamic properties of a solution. The thermodynamic properties that components have in a solution are called partial properties. 

Given equations that represent partial properties M, Mf, or Mf as functions of composition, one may combine them by the summability relation to yield a mixture property. Application of the defining (or equivalent) equations for partial properties then regenerates the given equations if and only if the given equations obey the Gibbs/Duhen equation. 

These clearly are not the same as the suggested expressions, which are therefore not correct. Note that application of the summability equation to the derived partial-property expressions reproduces the original equation for H. Note further that differentiation of these same expressions yields results that satisfy the Gibbs/Duhem equation, Eq. (11.14), written ... 

If the answers to part (a) are mathematically correct, this is inevitable, because they were derived from a proper expression for ME. Furthermore, for each partial property Mf, its value and derivative with respect to Xj become zero at x, = 1. 

The Lee-Kesler (7) generalized equation of state, which also applies to both phases, is the basis for the sixth thermodynamic properties method. As originally developed, the Lee-Kesler equation was for predicting bulk properties (densities, enthalpies etc.) for the entire mixture and not for calculating partial properties for the components of mixtures. Phase equilibrium was not one of the uses that the authors had in mind when they developed the equation. Recognizing the other possibilities of the Lee-Kesler equation, Ploecker, Knapp, and... 

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