To MEEG344 students Summary Notes for Chapter 13: Gas Mixtures Prof. Lian-Ping Wang, University of Delaware A. General definitions and notations The composition of a gas mixture can be specified either by mole fractions yi or by mass fractions mfi of the components, Ni , Nm yi mi ; mm mfi where the subscript “i” denotes a property associated with the ith component in the mixture and the subscript “m” for the mixture as a whole. N represents the mole number and m the mass. By definition, y i mf 1, i 1. Since mi N i M i ( M i is the molar mass for the i component), the molar mass of the mixture can be defined, similar to a single-component gas, as th Mm mm yi M i , Nm and the specific gas constant of a mixture can be calculated, similar to a single-component gas, as Rm Ru , Mm where Ru = 8.314 kJ/kmol.K is the universal gas constant. In terms of M m , mass and mole fractions can be related as mfi yi Mi . Mm Our overall goal is to find ways to compute the properties of the mixture in terms of known properties of the components (when each exists alone as a pure substance). B. The equation of state for a mixture The equation of the state for an ideal-gas mixture remains unchanged if the mixture mole number N m and the mixture specific gas constant Rm are used, namely, PmVm N m RuTm mm RmTm . The equation of the state for a real-gas mixture can be formally written as, PmVm Z m N m RuTm Z m mm RmTm . The question is how to determine the apparent compressibility factor Z m for the mixture. We first observe that the following statements are true at least for an ideal-gas mixture, Pm N m RuTm Vm Vm N m RuTm Pm N R T i u m Vm N R T i Pm u m N i Ru Tm Pi (Tm ,Vm ) Vm Dalton' s law N i RuTm Vi (Tm , Pm ) Pm Amagat' s law where the component pressure Pi (Tm ,Vm ) is the pressure that one would realize if the ith component were to exist alone at Tm and Vm ; the component volume Vi (Tm , Pm ) is the volume that the ith component would take if it alone exists at Tm and Pm . For an ideal-gas mixture, the component pressure is equal to the partial pressure ( yi Pm ) , and the component volume is equal to the partial volume ( yiVm ) . If we assume that the above laws are approximately valid for a real-gas mixture, then we have two ways of determining the compressibility factor of a mixture: PmVm Vm V Pi (Tm ,Vm ) yi Z i (Tm , m ) N m RuTm N m RuTm mi PV Pm Based on the Amagat' s law Z m m m Vi (Tm , Pm ) yi Z i (Tm , Pm ) N m RuTm N m RuTm Based on the Dalton' s law Z m The third way is to treat the mixture as if it is a pure substance with the following pseudocritical temperature and pseudocritical pressure: Z m Z (TR Tm P , PR ' m ) ' Tcr , m Pcr , m wher e Tcr' , m yiTcr ,i and Pcr' , m yi Pcr ,i Kay' s Rule This third method is known as the Kay’s rule. C. Properties of an ideal-gas mixture Here u m , hm , Cv , m , and C p , m are only function of the mixture temperature Tm and, therefore, um mfi ui (Tm ), hm mfi hi (Tm ), Cv,m mfi Cv,i (Tm ), or for molar analysis, um yi ui (Tm ), hm yi hi (Tm ), Cv,m yi Cv,i (Tm ), C p,m mfi C p,i (Tm ) C p,m yi C p,i (Tm ) The mixture entropy change, however, should be determined, as Tm , 2 C p ,i (T ) y P y P sm mfi dT Ri ln i , 2 m, 2 mfi si0 (Tm, 2 ) si0 (Tm,1 ) Ri ln i , 2 m, 2 yi ,1Pm,1 yi ,1Pm,1 Tm ,1 T where the subscripts “1” and “2” refer to the initial and final state, respectively. For molar analysis, Tm , 2 C p ,i (T ) y P y P sm yi dT Ru ln i , 2 m, 2 yi si0 (Tm, 2 ) si0 (Tm,1 ) Ru ln i , 2 m, 2 . yi ,1Pm,1 yi ,1Pm,1 Tm ,1 T The important point is that the component entropy change is computed with the mixture temperature Tm and the component partial pressure ( yi Pm ) (i.e., the spirit of Dalton’s law). D. Properties of a real-gas mixture In this case, the departures from the ideal gas behavior need be determined. For example, the enthalpy change of a real-gas mixture can be determined as hm mfi hi mfi hi (Tm )ideal RiTcr ,i Zh,i (Tm, 2 , Pm, 2 ) Zh,i (Tm,1, Pm,1 ) and hm yi hi yi hi (Tm ) ideal RuTcr ,i Zh,i (Tm, 2 , Pm, 2 ) Zh,i (Tm,1, Pm,1 ) And similarly, the entropy change can be determined as sm mfi si mfi si ideal Ri Z s,i (Tm, 2 , Pm, 2 ) Z s,i (Tm,1, Pm,1 ) and sm yi si yi si ideal Ru Z s ,i (Tm, 2 , Pm, 2 ) Z s,i (Tm,1, Pm,1 ) The important point to note is that the component departure factors should be evaluated at the mixture temperature Tm and the mixture pressure Pm (i.e., the spirit of Amagat’s law). The ideal-gas contributions are computed following Part C above. Alternatively, the mixture may be treated as a pseudo pure substance with pseudocritical properties given by the Kay’s rule.