Volumetric Properties of Pure Fluids
3.6 CUBIC EQUATIONS OF STATE
Superimposed is the “dome” representing states of saturated liquid and saturated vapor.10 For the isotherm T1 > Tc, pressure decreases with increasing molar volume. The critical isotherm (labeled Tc) contains the horizontal inflection at C characteristic of the critical point. For the isotherm T2 < Tc, the pressure decreases rapidly in the subcooled-liquid region with increasing V; after crossing the saturated-liquid line, it goes through a minimum, rises to a maximum, and then decreases, crossing the saturated-vapor line and continuing downward into the superheated-vapor region.
10Though far from obvious, the equation of state also provides the basis for calculation of the saturated liquid- and vapor-phase volumes that determine the location of the “dome.” This is explained in Sec. 13.7.
Figure 3.9: PV isotherms as given by a cubic equation of state for T above, at, and below the critical temperature. The superimposed darker curve shows the locus of saturated vapor and liquid volumes.
T1 > Tc
Vsat(liq)
V
Psat
P
C
Vsat(vap)
Tc T2 < Tc
Experimental isotherms do not exhibit the smooth transition from saturated liquid to saturated vapor characteristic of equations of state; rather, they contain a horizontal segment within the two-phase region where saturated liquid and saturated vapor coexist in varying proportions at the saturation (or vapor) pressure. This behavior, shown by the dashed line in Fig. 3.9, cannot be represented by an equation of state, and we accept as inevitable the unrealistic behavior of equations of state in the two-phase region.
Actually, the PV behavior predicted in the two-phase region by proper cubic equations of state is not wholly fictitious. If pressure is decreased on a saturated liquid devoid of vapor nucleation sites in a carefully controlled experiment, vaporization does not occur, and liquid 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 vapor persists
3.6. Cubic Equations of State 97 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 PV isotherm which lie in the two-phase region adjacent to the states of saturated liquid and saturated vapor.11
Cubic equations of state have three volume roots, of which two may be complex, i.e., may have an imaginary component. Physically meaningful values of V are always real, posi- tive, and greater than b. For an isotherm at T > Tc, reference to Fig. 3.9 shows that solution for V at any value of P yields only one such root. For the critical isotherm (T = Tc), this is also true, except at the critical pressure, where there are three roots, all equal to Vc. For isotherms at T < Tc, 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 iso- therm lying between saturated liquid and saturated vapor (under the “dome”). Only for the saturation pressure Psat are the roots, Vsat(liq) and Vsat(vap), stable states, lying at the ends of the horizontal portion of the true isotherm. For any pressure other than Psat, there is only a single physically meaningful root, corresponding to either a liquid or a vapor molar volume.
A Generic Cubic Equation of State
A mid-twentieth-century development of cubic equations of state was initiated in 1949 by publication of the Redlich/Kwong (RK) equation:12
P = ____RT
V − b − ______a ( T)
V ( V + b) (3.40) Subsequent enhancements have produced an important class of equations, represented by a generic cubic equation of state:
P = _RT
V − b − ___________V + εab () ( TV) + σb) ( (3.41) The assignment of appropriate parameters leads not only to the van der Waals (vdW) equation and the Redlich/Kwong (RK) equation, but also to the Soave/Redlich/Kwong (SRK)13 and the Peng/Robinson (PR) equations.14 For a given equation, ɛ and σ are pure numbers, the same for all substances, whereas parameters a(T) and b are substance dependent. The tempera- ture dependence of a(T) is specific to each equation of state. The SRK equation is identical to the RK equation, except for the T dependence of a(T). The PR equation takes different values for ɛ and σ, as indicated in Table 3.1.
11The heating of liquids in a microwave oven can lead to a dangerous condition of superheated liquid, which can
“flash” explosively.
12Otto Redlich and J. N. S. Kwong, Chem. Rev., vol. 44, pp. 233–244, 1949.
13G. Soave, Chem. Eng. Sci., vol. 27, pp. 1197–1203, 1972.
14D.-Y. Peng and D. B. Robinson, Ind. Eng. Chem. Fundam., vol. 15, pp. 59–64, 1976.
Determination of Equation-of-State Parameters
The parameters b and a(T) of Eq. (3.41) can in principle be found from PVT data, but suffi- cient data are rarely available. They are in fact usually found from values for the critical con- stants Tc and Pc. Because the critical isotherm exhibits a horizontal inflection at the critical point, we can impose the mathematical conditions:
( ∂ P___
∂ V )
T;cr = 0 (3.42) ( ____ ∂ 2P
∂ V 2 )
T;cr
= 0 (3.43)
Subscript “cr” denotes the critical point. Differentiation of Eq. (3.41) yields expressions for both derivatives, which are set equal to zero for P = Pc, T = Tc, and V = Vc. The equation of state itself can also be written for the critical conditions. These three equations contain five constants: Pc, Vc, Tc, a(Tc), and b. Of the several ways to treat these equations, the most suitable is elimination of Vc to yield expressions relating a(Tc) and b to Pc and Tc. The reason is that Pc and Tc are more widely available and more accurately known than Vc.
The algebra is intricate, but it leads eventually to the following expressions for parame- ters b and a(Tc):
b = Ω _RT c
P c (3.44)
and
a ( T c ) = Ψ _____R 2 Tc2 P c
This result is extended to temperatures other than Tc by the introduction of a dimensionless function α(Tr; ω) that becomes unity at the critical temperature:
a ( T) = Ψ α ( Tr ; ω ) R 2 Tc2
___________
P c (3.45)
In these equations Ω and Ψ are pure numbers, independent of substance but specific to a particular equation of state. Function α(Tr; ω) is an empirical expression, wherein by defini- tion Tr ≡ T/Tc, and ω is a parameter specific to a given chemical species, defined and discussed further below.
This analysis also shows that each equation of state implies a value of the critical com- pressibility factor Zc that is the same for all substances. Different values are found for different equations. Unfortunately, Zc values calculated from experimental values of Tc, Pc, and Vc dif- fer from one species to another, and they agree in general with none of the fixed values pre- dicted by common cubic equations of state. Experimental values are almost all smaller than any of the predicted values.
3.6. Cubic Equations of State 99
Roots of the Generic Cubic Equation of State
Equations of state are commonly transformed into expressions for the compressibility factor.
An equation for Z equivalent to Eq. (3.41) is obtained by substituting V = ZRT/P. In addition, we define two dimensionless quantities that lead to simplification:
β ≡ ___bP
RT (3.46) q ≡ ____a ( T)
bRT (3.47) With these substitutions, Eq. (3.41) assumes the dimensionless form:
Z = 1 + β − qβ __________Z − β
( Z + εβ) ( Z + σβ) (3.48) Although the three roots of this equation can be found analytically, they are usually calculated by iterative procedures built into mathematical software packages or implemented on a hand calculator. Convergence problems are most likely avoided when the equation is arranged to a form suited to the solution for a particular root.
Equation (3.48) is particularly adapted to solving for vapor and vapor-like roots because it takes the form of Z = 1 plus other terms that will be small at gas-like densities, where Z is not far from 1. Iterative solution starts with the value Z = 1 substituted on the right side. The calculated value of Z is returned to the right side and the process continues to convergence.
The final value of Z yields the volume root through V = ZRT/P = ZVig.
An alternative equation for Z is obtained when Eq. (3.48) is solved for the Z in the numerator of the final fraction, yielding:
Z = β + ( Z + εβ) ( Z + σβ) ( 1 + _β − Z
qβ ) (3.49)
This equation is particularly suited to solving for liquid and liquid-like roots because it takes a form in which Z = β plus another small term that is always positive. Iterative solution starts with the value Z = β substituted on the right side. Once Z is known, the volume root is again V = ZRT/P = ZVig.
Experimental compressibility-factor data show that values of Z for different fluids exhibit similar behavior when correlated as a function of reduced temperatureTr and reduced pressurePr, where by definition Tr ≡ T/Tc and Pr ≡ P/Pc. Equation-of-state parameters are therefore commonly computed in terms of these dimensionless variables. Thus, Eq. (3.46) and Eq. (3.47) in combination with Eqs. (3.44) and (3.45) yield:
β = Ω __P r
Tr (3.50) q = _______Ψα ( T r ; ω )
Ω Tr (3.51)
With parameters β and q evaluated by these equations, Z becomes a function of Tr and Pr and the equation of state is said to be generalized because of its general applicability to all gases and liquids. The numerical assignments for parameters ε, σ, Ω, and Ψ for the equations of interest are summarized in Table 3.1. Expressions are also given for α(Tr; ω) for the SRK and PR equations. Many other expressions for α(Tr; ω) have been published over the years, but those in Table 3.1 are the original formulations for these equations.15
15See, for example, A. F. Young, F. L. P. Pessoa, and V. R. R. Ahón, Ind. Eng. Chem. Res., vol. 55, pp. 6506–6516, 2016. This paper compares the performance of 20 different α(Tr; ω) functions.
Example 3.9
Given that the vapor pressure of n-butane at 350 K is 9.4573 bar, find the molar vol- umes of (a) saturated-vapor and (b) saturated-liquid n-butane at these conditions as given by the Redlich/Kwong equation.
Solution 3.9
Values of Tc and Pc for n-butane from App. B yield:
T r = _____350
425.1 = 0.8233 and P r = ______9.4573
37.96 = 0.2491
Parameter q is given by Eq. (3.51) with Ω, Ψ, and α(Tr) for the RK equation from Table 3.1:
q = _______ΨTr−1/2 Ω T r = __Ψ
Ω Tr−3/2 = 0.42748________
0.08664 ( 0.8233 ) −3/2 = 6.6048 Parameter β is found from Eq. (3.50):
β = Ω __P r
Tr = ( 0.08664 ) _______________( 0.2491 )
0.8233 = 0.026214
(a) For the saturated vapor, we write the RK form of Eq. (3.48) that results upon substitution of appropriate values for ε and σ from Table 3.1:
Z = 1 + β − qβ ______( Z − β) Z ( Z + β) or
Z = 1 + 0.026214 − ( 6.6048 ) ( 0.026214 ) _____________Z( ( ZZ − 0.026214 + 0.026214 ) )
Eqn. of State α(Tr) σ ε Ω Ψ Zc
vdW (1873) 1 0 0 1/8 27/64 3/8
RK (1949) T r −1/2 1 0 0.08664 0.42748 1/3
SRK (1972) α SRK ( T r ; ω) † 1 0 0.08664 0.42748 1/3 PR (1976) α PR ( T r ; ω) ‡ 1 + √__ 2 1 − √__ 2 0.07780 0.45724 0.30740 † α SRK ( T r ; ω) = [ 1 + (0.480 + 1.574 ω − 0.176 ω 2 )( 1 − Tr 1/2 ) ] 2
‡ α PR ( T r ; ω) = [ 1 + ( 0.37464 + 1.54226 ω − 0.26992 ω 2 ) ( 1 − Tr 1/2 ) ] 2 Table 3.1: Parameter Assignments for Equations of State
3.6. Cubic Equations of State 101 Solution by iteration starting from Z = 1 yields Z = 0.8305, and
V v = ____ZRT
P = ( 0.8305 ) _________________( 83.14 ) ( 350 ) 9.4573 = 2555 cm 3 ·mol −1 An experimental value is 2482 cm3·mol–1.
(b) For the saturated liquid, we apply Eq. (3.49) in its RK form:
Z = β + Z ( Z + β) ( _1 + β − Z qβ ) or
Z = 0.026214 + Z ( Z + 0.026214 ) ________________( 1.026214 − Z) ( 6.6048 ) ( 0.026214 ) Solution by iteration yields Z = 0.04331, and
V l = ____ZRT
P = __________________ ( 0.04331 ) ( 83.14 ) ( 350 ) 9.4573 = 133.3 cm 3 ·mol −1 An experimental value is 115.0 cm3·mol–1.
For comparison, values of Vv and Vl calculated for the conditions of Ex. 3.9 by all four of the cubic equations of state considered here are summarized as follows:
Vv/cm3·mol–l Vl/cm3·mol–l
Exp. vdW RK SRK PR Exp. vdW RK SRK PR
2482 2667 2555 2520 2486 115.0 191.0 133.3 127.8 112.6
Corresponding States; Acentric Factor
The dimensionless thermodynamic coordinates Tr and Pr provide the basis for the simplest corresponding-states correlations:
All fluids, when compared at the same reduced temperature and reduced pressure, have approximately the same compressibility factor, and all deviate from ideal-gas behavior to about the same degree.
Two-parameter corresponding-states correlations of Z require the use of only two reducing parameters Tc and Pc. Although these correlations are very nearly exact for the simple fluids (argon, krypton, and xenon), systematic deviations are observed for more complex fluids.
Appreciable improvement results from the introduction of a third corresponding-states param- eter (in addition to Tc and Pc), characteristic of molecular structure. The most popular such parameter is the acentric factor, ω, introduced by K. S. Pitzer and coworkers.16
16Fully described in K. S. Pitzer, Thermodynamics, 3rd ed., App. 3, McGraw-Hill, New York, 1995.
The acentric factor for a pure chemical species is defined with reference to its vapor pressure. The logarithm of the vapor pressure of a pure fluid is approximately linear in the reciprocal of absolute temperature. This linearity can be expressed as
d log P r sat
_________
d ( 1 / T r ) = S
where P r sat is reduced vapor pressure, Tr is reduced temperature, and S is the slope of a plot of log P r sat vs. 1/Tr. Note that “log” here denotes the base 10 logarithm.
If two-parameter corresponding-states correlations were generally valid, the slope S would be the same for all pure fluids. This is observed not to be true; within a limited range, each fluid has its own characteristic value of S, which could in principle serve as a third corresponding-states parameter. However, Pitzer noted that all vapor-pressure data for the sim- ple fluids (Ar, Kr, Xe) lie on the same line when plotted as log P r sat vs. 1/Tr and that the line passes through log P r sat = −1.0 at Tr = 0.7. This is illustrated in Fig. 3.10. Data for other fluids define other lines whose locations can be fixed relative to the line for the simple fluids (SF) by the difference:
log P r sat ( SF ) − log P r sat
The acentric factor is defined as this difference evaluated at Tr = 0.7:
ω ≡ −1.0 − log ( P r sat ) T r = 0.7 (3.52) Therefore ω can be determined for any fluid from Tc , Pc , and a single vapor-pressure measurement made at Tr = 0.7. Values of ω and the critical constants Tc , Pc , and Vc for a num- ber of substances are listed in App. B.
Figure 3.10: Approximate temperature dependence of the reduced vapor pressure.
1.0 0
1.2 1.4 1.6 1.8 2.0
1
2
1 Tr
1
0.7 1.43 Slope 3.2
(n-Octane)
Slope 2.3 (Ar, Kr, Xe) 1/Tr
logPrsat
The definition of ω makes its value zero for argon, krypton, and xenon, and experimen- tal data yield compressibility factors for all three fluids that are correlated by the same curves when Z is plotted as a function of Tr and Pr. This is the basic premise of three-parameter corresponding-states correlations: