Thermodynamics

Chapter 13 | 699ing processes, negative for endothermic mixing processes, and zero forisothermal mixing processes during which no heat is absorbed or released.Note that mixing is an irreversible process, and thus the entropy of mixingmust be a positive quantity during an adiabatic process. The specific volume,enthalpy, and entropy of a mixture are determined fromv aiy i v i h aiy i h iands aiy i s i(13–35)where y i is the mole fraction of component i in the mixture.Reconsider Eq. 13–29 for dG. Recall that properties are point functions, andthey have exact differentials. Therefore, the test of exactness can be applied tothe right-hand side of Eq. 13–29 to obtain some important relations. For the differentialdz Mdx Ndy of a function z(x, y), the test of exactness isexpressed as (M/y) x (N/x) y . When the amount of component i in a mixtureis varied at constant pressure or temperature while other components (indicatedby j ) are held constant, Eq. 13–29 simplifies todG S dT m i dN i 1for P constant and N j constant2 (13–36)dG V dP m i dN i 1for T constant and N j constant2 (13–37)Applying the test of exactness to both of these relations givesa 0 m i0T b a 0S b s P,N 0Nianda 0 m ii T,P,N j0P b a 0V b viT,N 0N i T,P,N j(13–38)where the subscript N indicates that the mole numbers of all components(and thus the composition of the mixture) is to remain constant. Taking thechemical potential of a component to be a function of temperature, pressure,and composition and thus m i m i (P, T, y 1 , y 2 , . . . , y j . . .), its total differentialcan be expressed asdm i dg i a 0 m i0P b dP a 0 m iT,y 0T b dT aiP,y(13–39)where the subscript y indicates that the mole fractions of all components(and thus the composition of the mixture) is to remain constant. SubstitutingEqs. 13–38 into the above relation givesdm i v i dP s i dT aia 0 m i0y ibP,T,y jdy ia 0 m i0y ibP,T,y jdy i(13–40)For a mixture of fixed composition undergoing an isothermal process, it simplifiestodm i vi dP1T constant, y i constant2 (13–41)Ideal-Gas Mixtures and Ideal SolutionsWhen the effect of dissimilar molecules in a mixture on each other is negligible,the mixture is said to be an ideal mixture or ideal solution and thechemical potential of a component in such a mixture equals the Gibbs function