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20 An Equation for Liquid-Vapor Saturation Densities as a Function of Pressure PHILIP A. THOMPSON

Downloaded by UNIV OF ARIZONA on December 6, 2012 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch020

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 )





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

Downloaded by UNIV OF ARIZONA on December 6, 2012 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch020

x= ( N o t e that χ «

1 -

P for χ <
9 χ