3838
Ind. Eng. Chem. Res. 2003, 42, 3838-3844
Saturation Properties from Equations of State Philip T. Eubank* and Xiaonian Wang Department of Chemical Engineering, Texas A&M University, College Station, Texas 77843-3122
Following a review of cubic equations-of-state (EoS), a new identity linking the heat of vaporization, for pure components, to the EoS is derived. The identity is applied to simple cubic EoS to allow tuning of both the attraction (a) and repulsion (b) constants to experimental vapor pressures and heats of vaporization at subcritical temperatures. These new tuning procedures are compared with previous procedures for benzene and for methanol. The results show marked improvements to the point where the simpler Redlich-Kwong EoS provides better results with our procedures than does the usually superior Peng-Robinson EoS with conventional procedures. Introduction
III. Soave’s modification of II (S-R-K) (1972):
For pure components, the Maxwell Equal-Area-Rule (MEAR) is a well-known thermodynamic identity for vapor-liquid equilibria (VLE):
Pσ(VσV - VσL) )
∫
VσV VσL
a)
PEoS(dV)T
(1)
Equation 1 provides universal vapor pressure curves in reduced coordinates for simple cubic equations of state (EoS) such as those of van der Waals1 and of Redlich-Kwong2 and further for the Soave-RedlichKwong3 and Peng-Robinson4 EoS for fixed acentric factors ω when, in all cases, the two constants a and b are set by the critical point constraints.5 We summarize the results from these four cubic EoS below where the values of the two constants a and b are given for pure components from the critical isotherm constraints of (∂P/ ∂V)TCPc ) 0 ) (∂2P/∂V2) TCPc . Additional relations for vari-
I. van der Waals (vdW) (1873): P)
a)
(
(
) ( )
)
64Pc
;
( ) RTc 8Pc
b)
II. Redlich-Kwong (R-K) (1949): P)
a)
(
( ) (
)
RT a V-b xTV(V + b)
)
0.4274R2Tc5/2 c
P
;
b)
(
(
(
) (
RT a V-b V(V + b)
)
0.4274R2Tc2s2
(
)
)
0.08664RTc ; P Pc Tr ≡ (T/Tc); s ≡ 1 + (1 - xTr)fs; c
b)
;
fs ≡ 0.480 + 1.574ω - 0.176ω2; ω ≡ Pitzer’s acentric factor ≡ -1 log10 Pσr |Tr)0.7 IV. Peng-Robinson (P-R) (1976): P)
a)
(
(
) (
RT a - 2 V-b V + 2bV - b2
)
0.45724R2Tc2t2
(
)
)
0.07780RTc ; Pc Pc Tr ≡ (T/Tc); t ≡ 1 + (1 - xTr)ft; ;
b)
ft ≡ 0.37464 + 1.54226ω - 0.2699ω2; ω ≡ Pitzer’s acentric factor ≡ -1 log10 Pσr |Tr)0.7
RT a - 2 V-b V
27R2Tc2
P)
)
0.08664RTc Pc
* To whom correspondence should be addressed. Tel.: (979)845-3339. Fax: (979)845-6446. E-mail:
[email protected].
ousproperties from these cubic EoS are given in Table 1, where ZC ≡ (PCVC/RTC) is the critical compressibility factor, ψC is the Riedel factor defined as (Tc/Pc) (∂P/∂T) CP V with the slope of the critical isochore at the critical point being identical to the vapor pressure slope at the same point, B(T) is the second virial coefficient in the density expansion, C(T) is the third virial coefficient in the density expansion, [PCBC/RTC] is the reduced second virial coefficient at the critical temperature (which is -0.34 ( 0.01 for all compounds measured), (∂2P/∂T2) σV|liq is the second derivative of liquid isochores as they emerge from the vapor pressure curve on a P/T diagram (positive from experiment), and similarly, (∂2P/∂T2) σV|vap is the second derivative of vapor isochores (negative from experiment). Conversely, in commercial practice, eq 1 allows for the attraction constant a to be tuned to experimental vapor pressures when these EoS are used with only the repulsion constant b taken from the critical point
10.1021/ie030068y CCC: $25.00 © 2003 American Chemical Society Published on Web 07/09/2003
Ind. Eng. Chem. Res., Vol. 42, No. 16, 2003 3839 Table 1. Various Thermophysical Properties Consistent with the Various Cubic EoS R-K
S-R-K
P-R
PC
prop./EoS:
vdW (a/27b2)
[(0.08664/b)5(a/0.42747)2]1/3
(0.017563a/b2)
TC VC ZC ψC B(T) C(T) (VC/b) -[PCBC/RTC] (∂2P/∂T2)σV|liq (∂2P/∂T2)σV|vap
(8a/27bR) 3b (3/8) 4 b - (a/RT) b2 3 0.2969 0 0
[0.20268a/bR]2/3 3.847b (1/3) 5.5804 b - (a/RT3/2) b2 + (ab/RT3/2) 3.847 0.3408
