Thermodynamics

690 | ThermodynamicsSimilarly, the specific heats of a gas mixture can be expressed asc v,m akc p,m aku m aki1i1i1h m aki1s m aki1mf i u imf i h i 1kJ>kg21kJ>kg2andu m akmf i s i 1kJ>kg # K2 ands m akmf i c v,i 1kJ>kg # K2 andc v,m akmf i c p,i 1kJ>kg # K2 andc p,m ak(13–19)(13–20)(13–21)(13–22)(13–23)Notice that properties per unit mass involve mass fractions (mf i ) and propertiesper unit mole involve mole fractions (y i ).The relations given above are exact for ideal-gas mixtures, and approximatefor real-gas mixtures. (In fact, they are also applicable to nonreactingliquid and solid solutions especially when they form an “ideal solution.”)The only major difficulty associated with these relations is the determinationof properties for each individual gas in the mixture. The analysis can besimplified greatly, however, by treating the individual gases as ideal gases,if doing so does not introduce a significant error.Ideal-Gas MixturesThe gases that comprise a mixture are often at a high temperature and lowpressure relative to the critical-point values of individual gases. In such cases,the gas mixture and its components can be treated as ideal gases with negligibleerror. Under the ideal-gas approximation, the properties of a gas are notinfluenced by the presence of other gases, and each gas component in themixture behaves as if it exists alone at the mixture temperature T m and mixturevolume V m . This principle is known as the Gibbs–Dalton law, which isan extension of Dalton’s law of additive pressures. Also, the h, u, c v , and c p ofan ideal gas depend on temperature only and are independent of the pressureor the volume of the ideal-gas mixture. The partial pressure of a component inan ideal-gas mixture is simply P i y i P m , where P m is the mixture pressure.Evaluation of u or h of the components of an ideal-gas mixture duringa process is relatively easy since it requires only a knowledge of the initialand final temperatures. Care should be exercised, however, in evaluating thes of the components since the entropy of an ideal gas depends on the pressureor volume of the component as well as on its temperature. The entropychange of individual gases in an ideal-gas mixture during a process can bedetermined fromi1i1i1andh m aki1i1y i u i1kJ>kmol2y i h i1kJ>kmol2y i s i1kJ>kmol # K2y i c v,i1kJ>kmol # K2y i c p,i1kJ>kmol # K2or¢s i s° i,2 s° i,1 R i ln P i,2P i,1 c p,i ln T i,2T i,1 R i ln P i,2P i,1¢ s i s ° i,2 s ° i,1 R u ln P i,2P i,1 c p,i ln T i,2T i,1 R u ln P i,2P i,1(13–24)(13–25)