( ) ( ) Volumetric Properties of Pure Fluids, part 4. The generic cubic equation of state:

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1 CE304, Spring 2004 Leture 6 Volumetri roperties of ure Fluids, part 4 The generi ubi equation of state: There are many possible equations of state (and many have been proposed) that have the same general form as the van der Waals equation of state. An important group of these have the form at ( ε )( σ ) RT = V b V + b V + b In this generi form for a ubi equations of state, a is a funtion of temperature (as shown expliitly above) and b is a onstant. a and b are both substane speifi (different for eah fluid). The other two onstants, ε and σ are pure numbers (they don't have units) and are the same for all fluids for a given equation of state, but different for different equations of state. If a is taken to be onstant with respet to temperature and if ε and σ are both set to zero, this redues to the van der Waals equation of state. Several other ommonly used ubi equations of state are obtained by speifying different funtional forms for a(t) and different values of ε and σ. Determination of equation of state parameters: If one has a good set of VT data for a substane, then the parameters in the equation of state an be determined diretly by fitting the equation to the data (non-linear regression). However, often the equation of state parameters are estimated based on the ritial properties of the substane. To do this, we note that at the ritial point the ritial isotherm has a point of infletion. That is, its first and seond derivatives are both zero: = = 0 at the ritial point 2 d d dv 2 T dv T Taking these derivatives of the generi ubi equation of state gives at at d RT = + + dv T ( V b) V + εb V + σb V + εb V + σb 2 d 2RT 2aT 2aT 2aT = dv V b V + εb V + σb V + εb V + σb V + εb V + σb T Setting these equal to zero at the ritial point gives p. 1 of 6

2 CE304, Spring 2004 Leture 6 ( + ε ) + ( + σ ) ( ε ) ( σ ) RT at V b V b + = 0 V b V b V b ( + ε ) + ( + ε )( + σ ) + ( + σ ) RT a T V b V b V b V b = 0 V b V b V b ( + ε ) ( + σ ) Now, if T and V are known, then this gives us two equations that would allow us to determine the two substane speifi parameters, a(t ) and b. Remember that the other onstants, as well as the funtional form of a(t) are all speified a priori for a given equation of state (that is, we already know them). If, as is often the ase, T and are known, rather than T and V then we would also use the equation of state itself to determine V from T and. RT at ( ) = V b V + εb V + σb As an example, let's apply this general proedure to the van der Waals equation of state. In that ase, we have RT a = V 2 b V RT 2a + = ( V b) V RT 3a = V b V Sparing you the algebra, we an solve these (lik here for a Maple worksheet that solves them) to get a b V 27 = 64 1 = 8 3 = R T RT RT This implies that if the van der Waals equation of state were stritly orret, then all substanes would have a ritial ompressibility (Z = V /(RT ) = 3/8). In fat, different substanes have different values of Z, and almost all substanes have values of Z that are somewhat smaller than 3/8. We ould similarly solve these equations for other equations of state, and would get, for a(t ) and b expressions like p. 2 of 6

3 CE304, Spring 2004 Leture R T at ( ) =Ψ RT b =Ω where Ψ and Ω are numbers that are different for eah equation of state, but independent of the substane. Values of these (and other parameters, yet to be disussed) are tabulated for several equations of state on p. 99 of SVA. The Theorem of Corresponding States The two-parameter theorem of orresponding states, whih is based on experimental observations, says that all fluids, when ompared at the same redued pressure and temperature, have about the same ompressibility. Note that this is an unusual use of the word theorem. It might better be termed a 'oarse generalization of experimental observations' or something like that. It is not a statement that is mathematially provable or that is always true. The redued temperature and redued pressure are defined as T T r That is, if we sale the temperature and pressure by the ritial temperature and pressure, all fluids are about the same. This is nearly true for simple fluids (monatomi gases exept helium), but less nearly true for other substanes. The desription of other fluids an be improved by introduing a third parameter. The most popular 3rd parameter is the aentri fator, ω. The aentri fator of a pure substane is defined in terms of its vapor pressure. If we plot the log of the vapor pressure vs. inverse temperature, we get approximately a straight line. If we plot this as redued pressure vs. inverse redued temperature, we get a plot like the one shown below. p. 3 of 6

4 CE304, Spring 2004 Leture 6 Vapor ressure Curves Argon (blue) -1 Methane (magenta) log 10 ( r,sat ) Hexane -3 =0.7 Ethanol / (1/K) The aentri fator is defined as the differene in the value of this urve from the value of -1.0 that is observed at T r = 0.7 for simple fluids (argon, neon, krypton). Thus, to find it one needs one vapor pressure measurement (at T r = 0.7) as well as the ritial properties. This is the basis of the three-parameter theorem of orresponding states: all fluids that have the same aentri fator, when ompared at the same redued pressure and temperature, have about the same ompressibility. This is more generally true than is the 2-parameter theorem of orresponding states. That is, it agrees with reality for more ases, beause it ontains one more parameter and is therefore more flexible. The aentri fator is used in some equations of state, like the Soave-Redlih-Kwong equation of state and the eng-robinson equation of state, where it shows up in the funtion a(t). It is also used in the generalized orrelations for the ompressibility of gases that we will disuss shortly. Volume roots of Cubi Equations of State: Cubi equations of state annot easily be written in a form where volume is an expliit funtion of temperature and pressure. Usually, they are solved iteratively. To solve for the vapor (largest) volume at a given pressure, it works well to rearrange the equation as: p. 4 of 6

5 CE304, Spring 2004 Leture 6 ( + ε )( + σ ) RT at V b V = + b V b V b Starting from an initial guess of V = RT/ and iterating will usually result in rapid onvergene to the largest value of V for a given T and, whih represents the vapor phase (for temperatures below the ritial temperature where there are 3 volumes orresponding to eah pressure). One an also re-write this to solve iteratively for ompressibility as shown on p. 97 of SVA: Z β Z = 1+ β qβ ( Z + εβ )( Z + σβ ) In this equation, the dimensionless parameters β and q are defined as b r β =Ω RT at Ψα ( ) q = brt Ω Using this form of the equation, one finds the ompressibility by iteration, starting from an initial guess of Z = 1. When there are 3 volumes orresponding to a pressure (when we are below the ritial temperature), we will also want to solve for the liquid (smallest) volume root. A rearrangement that works well for this ase is RT b V V b ( V εb)( V σb) + = at In this ase, the initial guess for the volume an be taken to be the parameter b. This an also be written in terms of the ompressibility and the dimensionless parameters β and q as: Z ( Z )( Z ) 1 + β β εβ σβ Z = qβ To find the ompressibility from this equation, one iterates starting from an initial guess of Z = β. Generalized orrelations for gases: There are a variety of generalized orrelations that an be used to estimate the VT behavior of real (non-ideal) gases. These are often of the form Z = Z 0 + ωz 1 where Z is the ompressibility of the gas (or superritial fluid), ω is the aentri fator that we just disussed, and Z 0 and Z 1 are both funtions of the redued temperature and redued pressure. The most popular of these orrelations is the Lee/Kesler orrelation. For it, values of Z 0 and Z 1 are found in appendix E of the textbook. To use these, you first look up the ritial temperature, ritial pressure, and aentri fator for the substane of interest, then ompute the redued pressure and temperature, then look up Z 0 and Z 1 in the tables (interpolating between entries) and then finally put them in the above equation for Z. For nonpolar or slightly polar gases, this orrelation should give the ompressibility to with 2 or 3 perent. p. 5 of 6

6 CE304, Spring 2004 Leture 6 Over a limited range of redued temperatures and pressures (shown on p. 103 of SVA), this orrelation an be represented by a two-term trunated virial equation of state as follows. The ompressibility is written as B B r Z = 1+ = 1+ RT RT with the orrelation B 0 1 = B + ωb RT whih gives 0 r 1 r Z = 1+ B + ωb or, in the form of the original Lee/Kesler orrelation: 0 1 Z = Z + ωz 0 0 r Z = 1+ B T r 1 1 r Z = ωb T r The oeffiients B 0 and B 1 are funtions only of temperature (remember that the virial oeffiients themselves are funtions only of temperature). They are well represented by: B = B = T Generalized orrelations for liquids r The Lee/Kesler orrelation disussed above for gases also inludes data for subooled liquids. Figure 3.14 on p. 101 of SVA shows its preditions for both liquids and gases. For saturated liquids (those in equilibrium with the vapor) some simpler orrelations work. The Rakett equation gives good results and only requires the ritial parameters as inputs: sat ( 1 T V = V Z ) r Another graphial orrelation for liquid densities is shown in figure 3.17 on p. 109 of SVA. So, now we have a variety of equations of state and orrelations to ompute the VT behavior of fluids. In the homework and in-lass examples, we will pratie applying these for suh diret alulations (i.e. ompute volume given temperature and pressure), and then we will go on to use them throughout the rest of the semester to solve more ompliated and more interesting problems. p. 6 of 6