20 An Equation for Liquid-Vapor Saturation Densities as a Function of Pressure PHILIP A. THOMPSON
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1
Max-Planck-Institut für Strömungsforschung, D 34 Göttingen, Federal Republic of Germany
λ
β
The explicit formula /ρ r — 1/ = (1 — P ) for reduced satu ration density as a function of reduced pressure is proposed for the entire liquid-vapor saturation boundary. The expres sion λ ~ 1 depends on P ; β ~ 0.35 depends weakly on P , corresponding at P = 1 to the critical exponent β . The parameters λ and β can be related to the Pitzer factor ω. Special cases include the power law /ρ — 1/ = C(1 — T ) c . . . and the low-pressure vapor equation ρr 0 = β Ρ . The function λ — λ = g(P ) is found from data to be a universal function for nonpolar substances. If λ is correlated with ω, the formula takes on the corresponding-states form ρ = ρ (P , ω). This form predicted the density of saturated liquid and vapor with 0.4% and 0.9% accuracy, respectively, for 38 substances. r
r
r
r
c
β
r
r
λ
0
c
r
r
c
r
r
r
H P h e smooth curve passing through the critical point a n d b o u n d i n g the two-phase
l i q u i d - v a p o r region
i n a pressure-volume
d i a g r a m is
f a m i l i a r to every student of t h e r m o d y n a m i c s . T h e m a t h e m a t i c a l d e s c r i p t i o n p(P)
of this coexistence c u r v e or s a t u r a t i o n b o u n d a r y is t h e subject
of this c h a p t e r . D e s c r i p t i o n s i n terms of t e m p e r a t u r e r a t h e r t h a n pressure are w e l l known.
T h e n o t a b l e equations of G u g g e n h e i m ( J ) a r e , f o r l i q u i d a n d
vapor, respectively, Pr(l)
- 1 +
-1
(1
- T
r
) +
(1
-
Tr) / 1
3
(1)
Present address: Department of Mechanical Engineering, Rensselaer Polytechnic Institute, Troy, N Y 12181. 1
0-8412-0500-0/79/33-182-365$05.00/l © 1979 American Chemical Society In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
366
EQUATIONS
OF
STATE
and
+ -f- d -
Pr(T> - 1
-
τ) τ
d - T y*
\
(2)
r
w h e r e the s u b s c r i p t r denotes a r e d u c e d p r o p e r t y , e.g., T = r
T/T .
a n d the near-critical power l a w
— p
Ρ { ϊ )
( v )
\τ
Benzene n-Pentane n-Hexane n-Heptane n-Octane
.299 .316 .307 .309 .308
.330 .327 .325 .323 .322
1.079 1.070 1.068 1.058 1.058
1.068 1.064 1.061 1.056 1.053
Fixing the Value of λ.
(
(υ)
T h e f u n c t i o n λ ( Ρ ) has t w o b r a n c h e s c o r r e Γ
s p o n d i n g to s a t u r a t e d v a p o r a n d s a t u r a t e d l i q u i d : these b r a n c h e s m e e t at the c r i t i c a l p o i n t w h e r e the c o m m o n v a l u e of λ is d e s i g n a t e d as A . c
T h e t w o b r a n c h e s c a n be c a l c u l a t e d f r o m e x p e r i m e n t a l p ( P ) d a t a f r o m r
r
E q u a t i o n 3 r e w r i t t e n i n the f o r m
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
370
EQUATIONS
OF
STATE
w h e r e t h e u p p e r a n d l o w e r signs c o r r e s p o n d to t h e v a p o r a n d l i q u i d b r a n c h e s , r e s p e c t i v e l y . F r o m E q u a t i o n 3, t h e v a p o r v a l u e λ
( ν )
and liquid
v a l u e λ ΐ ) are r e l a t e d b y (
PTM ( P ) W r
V +
M
L
)
(P )\»«V = 2
(14)
r
F o r c o n v e n i e n c e , t h e pressure scale is e x p a n d e d b y t h e substitute i n d e p e n d e n t v a r i a b l e x, - l n P
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x= ( N o t e that χ «
1 -
P for χ <
9 χ
