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35 Cubic Equations o State f
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pressure This behavior, shown by the dashed line in Fig 312, is nonanalytic, and we accept as inevitable the unrealistic behavior of equations of state in the two-phase region
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Figure 312 Isotherms as given by a cubic equation of state
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Actually, the P V behavior predicted in this region by proper cubic equations of state is not wholly fictitious When the pressure is decreased on a saturated liquid devoid of vapornucleation sites in a carefully controlled experiment, vaporization does not occur, and the liquid phase persists alone to pressures well below its vapor pressure Similarly, raising the pressure on a saturated vapor in a suitable experiment does not cause condensation, and the vapor persists alone to pressures well above the vapor pressure These nonequilibrium or metastable states of superheated liquid and subcooled vapor are approximated by those portions of the P V isotherm which lie in the two-phase region adjacent to the saturated-liquid and saturated-vapor states Cubic equations of state have three volume roots, of which two may be complex Physically meaningful values of V are always real, positive, and greater than constant b For an isotherm at T > T,, reference to Fig 312 shows that solution for V at any positive value of P yields only one such root For the critical isotherm (T = T,), this is also true, except at the critical pressure, where there are three roots, all equal to V, For isotherms at T < T,, the equation may exhibit one or three real roots, depending on the pressure Although these roots are real and positive, they are not physically stable states for the portion of an isotherm lying between saturated liquid and saturated vapor (under the "dome") Only the roots for P = P Sat, namely Vsat(liq)and Vsat(vap), stable states, connected by the horizontal portion of the true are isotherm For other pressures (as indicated by the horizontal lines shown on Fig 312 above
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CHAPTER 3 Volumetric Properties of Pure Fluids
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and below P Sat), the smallest root is a liquid or "liquid-like" volume, and the largest is a vapor or "vapor-like" volume The third root, lying between the other values, is of no significance
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A Generic Cubic Equation of State
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Since the introduction of the van der Waals equation, scores of cubic equations of state have been proposed All are special cases of the equation:
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Here, b, 8, K , A and r] are parameters which in general depend on temperature and (for mix, tures) composition Although this equation appears to possess great flexibility, it has inherent limitations because of its cubic form7 It reduces to the van der Waals equation when q = b, O=a,and~=h=O An important class of cubic equations results from the preceding equation with the assignments:
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It is thus transformed into an expression general enough to serve as a generic cubic equation o f state, which reduces to all others of interest here upon assignment of appropriate parameters:
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p=--
V- b
+ cb)(V + a b )
For a given equation, E and a are pure numbers, the same for all substances, whereas parameters a(T) and b are substance dependent The temperature dependence of a(T) is specific to each equation of state For the van der Waals equation, a(T) = a is a substance-dependent constant, and = 0 = 0
Determination of Equation-of-State Parameters
The constants in an equation of state for a particular substance may be evaluated by a fit to available P VT data For cubic equations of state, however, suitable estimates are usually found from values for the critical constants T and PC , Since the critical isotherm exhibits a horizontal inflection at the critical point, we may impose the mathematical conditions:
where the subscript "cr" denotes the critical point Differentiation of Eq (341) yields expressions for both derivatives, which may be equated to zero for P = PC,T = T,, and V = V, The equation of state may itself be written for the critical conditions These three equations contain five constants: PC,V,, T,, a(T,), and b Of the several ways to treat these equations, the
7 ~M Abbott, AIChE J, vol 19, pp 596601, 1973; Adv in Chern Series 182, K C Chao and R L Robinson, Jr, eds, pp 47-70, Am Chem Soc, Washington, DC, 1979
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