Thermodynamics

686 | ThermodynamicsP m V m = Z m N m R u T mZ m = ∑ y i Z ii = 1FIGURE 13–8One way of predicting the P-v-Tbehavior of a real-gas mixture isto use compressibility factor.kPseudopure substancekP ' cr,m = ∑ y i P cr,ii = 1kT ' cr,m = ∑ y i T cr,ii = 1FIGURE 13–9Another way of predicting the P-v-Tbehavior of a real-gas mixture is totreat it as a pseudopure substance withcritical properties P cr and T cr .from ideal-gas behavior. One way of doing that is to use more exact equationsof state (van der Waals, Beattie–Bridgeman, Benedict–Webb–Rubin,etc.) instead of the ideal-gas equation of state. Another way is to use thecompressibility factor (Fig. 13–8) as(13–9)The compressibility factor of the mixture Z m can be expressed in terms of thecompressibility factors of the individual gases Z i by applying Eq. 13–9 to bothsides of Dalton’s law or Amagat’s law expression and simplifying. We obtain(13–10)where Z i is determined either at T m and V m (Dalton’s law) or at T m and P m(Amagat’s law) for each individual gas. It may seem that using either lawgives the same result, but it does not.The compressibility-factor approach, in general, gives more accurate resultswhen the Z i ’s in Eq. 13–10 are evaluated by using Amagat’s law instead ofDalton’s law. This is because Amagat’s law involves the use of mixturepressure P m , which accounts for the influence of intermolecular forcesbetween the molecules of different gases. Dalton’s law disregards the influenceof dissimilar molecules in a mixture on each other. As a result, it tendsto underpredict the pressure of a gas mixture for a given V m and T m . Therefore,Dalton’s law is more appropriate for gas mixtures at low pressures.Amagat’s law is more appropriate at high pressures.Note that there is a significant difference between using the compressibilityfactor for a single gas and for a mixture of gases. The compressibility factorpredicts the P-v-T behavior of single gases rather accurately, as discussedin Chapter 3, but not for mixtures of gases. When we use compressibilityfactors for the components of a gas mixture, we account for the influence oflike molecules on each other; the influence of dissimilar molecules remainslargely unaccounted for. Consequently, a property value predicted by thisapproach may be considerably different from the experimentally determinedvalue.Another approach for predicting the P-v-T behavior of a gas mixture isto treat the gas mixture as a pseudopure substance (Fig. 13–9). One suchmethod, proposed by W. B. Kay in 1936 and called Kay’s rule, involves theuse of a pseudocritical pressure P cr,m and pseudocritical temperature T cr,mfor the mixture, defined in terms of the critical pressures and temperaturesof the mixture components asP ¿ cr,m aki1PV ZNR u TZ m aky i P cr,i andT ¿ cr,m ak(13–11a, b)The compressibility factor of the mixture Z m is then easily determined byusing these pseudocritical properties. The result obtained by using Kay’srule is accurate to within about 10 percent over a wide range of temperaturesand pressures, which is acceptable for most engineering purposes.Another way of treating a gas mixture as a pseudopure substance is touse a more accurate equation of state such as the van der Waals, Beattie–Bridgeman, or Benedict–Webb–Rubin equation for the mixture, and to determinethe constant coefficients in terms of the coefficients of the components.i1y i Z ii1y i T cr,i