Basic principle II Second class Dr. Arkan Jasim Hadi 1. Real gas Real gases – as opposed to a perfect or ideal gas – exhibit properties that cannot be explained entirely using the ideal gas law. To understand the behavior of real gases, the following must be taken into account: compressibility effects; variable specific heat capacity; van der Waals forces; non-equilibrium thermodynamic effects; Issues with molecular dissociation and elementary reactions with variable composition. Critical state and Reduced conditions Critical point: The point at highest temp. (Tc) and Pressure (Pc) at which a pure chemical species can exist in vapour/liquid equilibrium. The point critical is the point at which the liquid and vapour phases are not distinguishable; because of the liquid and vapour having same properties. Reduced properties of a fluid are a set of state variables normalized by the fluid's state properties at its critical point. These dimensionless thermodynamic coordinates, taken together with a substance's compressibility factor, provide the basis for the simplest form of the theorem of corresponding states The reduced pressure is defined as its actual pressure divided by its critical pressure : The reduced temperature of a fluid is its actual temperature, divided by its critical temperature: The reduced specific volume ") of a fluid is computed from the ideal gas law at the substance's critical pressure and temperature: This property is useful when the specific volume and either temperature or pressure are known, in which case the missing third property can be computed directly. 1 Basic principle II Second class Dr. Arkan Jasim Hadi In Kay's method, pseudocritical values for mixtures of gases are calculated on the assumption that each component in the mixture contributes to the pseudocritical value in the same proportion as the mol fraction of that component in the gas. Thus, the pseudocritical values are computed as follows: where yi, is the mole fraction, ppc is the pseudocritical pressure and Tpc, is the pseudo- critical temperature The values so obtained are pseudocritical temperature and pressure, Tpc and ppc which replace T, and P, to define pseudoreduced parameters:: 푇 푇푝푟 = 푇푝푐 푝 푝푝푟 = 푝푝푐 Compressibility factor (Z): is the ratio of the molar volume of a gas to the molar volume of an ideal gas at the same temperature and pressure. It is a useful thermodynamic property for modifying the ideal gas law to account for the real gas behavior. Or - For an ideal gas or perfect gases, the compressibility factor Z=1 - For real gas Z≠ 1 2 Basic principle II Second class Dr. Arkan Jasim Hadi 3 Basic principle II Second class Dr. Arkan Jasim Hadi 4 Basic principle II Second class Dr. Arkan Jasim Hadi 5 Basic principle II Second class Dr. Arkan Jasim Hadi 14.6 Calculating the Compressibility Factor Using the Pitzer Factors zo and z1 Several methods have appeared in the literature and in computer codes to cal- culate z via an equation in order to obtain more accurate values of z than can be obtained from charts. Equation (14.4) employs the Pitzer acentric factor, w where zo and z1 are listed in tables in Appendix C as a function of Tr and Pr, and co is unique for each compound, and can be found in the CD that accompanies this book. Table 14.1 is an abbreviated table of the acentric factors from Pitzer. The acentric factor 흎 indicates the degree of acentricity or nonsphericity of a molecule. For helium and argon, 흎 is equal to zero. For higher molecular weight hydrocarbons and for molecules with increased polarity, the value of 흎 increases. 14.7 Real Gas Mixtures How can you apply the concept of compressibility to problems involving gas mixtures? Each component in the mixture will have different critical properties. Numerous ways have been proposed to properly weigh the critical properties so that at appropriate reduced temperature and pressure can be used to obtain z. Refer to the references at the end of this chapter for some examples. One simple way that is rea sonably accurate, at least for our purposes, is Kay's method? 6 Basic principle II Second class Dr. Arkan Jasim Hadi In Kay's method, pseudocritical values for mixtures of gases are calculated of the assumption that each component in the mixture contributes to the pseudocritica value in the same proportion as the mol fraction of that component in the gas. Thus the pseudocritical values are computed as follows: where yi is the mole fraction, 푝푐́ is the pseudocritical pressure and 푇́푐 is the pseudo critical temperature. EXAMPLE 14.3 Calculation of p-V-T Properties for a Real Gas Mixture A gaseous mixture has the following composition (in mole percent): Methane, CH4 20 Ethylene, C2H4 30 Nitrogen, N2 50 at 90 atm pressure and 100°C. Compare the volume per mole as computed by the methods of: (a) the ideal gas law and (b) the pseudoreduced technique (Kay's method). Solution Basis: 1 g mol of gas mixture Additional data needed are: Component Tc (K) Pc (atm) from App. CH4 191 45.8 C2H4 283 50.5 N2 126 33.5 a. Ideal gas law 7 Basic principle II Second class Dr. Arkan Jasim Hadi 푅푇 (1)(82.06)(273+100) 푉 = = = 340 푐푚3/푔푚표푙 푝 90 b. According to the Kay’s method, first calculate the pseudo critical values of mixture 푝푝푐 = 푝푐퐴푦퐴 + 푝푐퐵푦퐵 + 푝푐퐶푦퐶 = (45.8)(0.2) + (50.5)(0.3) + (33.5)(0.5) = 41.1 atm 푇푝푐 = 푇푐퐴푦퐴 + 푇푐퐵푦퐵 + 푇푐퐶푦퐶 = (191)(0.2) + (283)(0.3) + (126)(0.5) = 186 K Then calculate the pseudocritical values of the mixture 푝 푇 푝푝푟 = 푇푝푟 = 푝푝푐 푇푝푏 = 90/41.1 = 2.19, = 373/186 = 2.01 With the aid of these two parameter and from fig. 14.4 b that z Tpr =1.91, and thus z = 0.95 푧푅푇 (1)(0.95)(82.06)(273 + 100) 푉 = = = 323 푐푚3/푔푚표푙 푝 90 8 .