Equations of State in Engineering and Research

11 Applications of the Augmented van der Waals Theory of Fluids: Tests of Some Combining Rules for Mixtures

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ALEKSANDER KREGLEWSKI and STEPHEN S. CHEN Thermodynamics Research Center, Texas A&M University, College Station, TX 77843

An accurate equation of state of fluids is used to test the combining rules for the interaction energy ū of mixtures derived from the theories of London energy between small or large (chain) molecules. The tests are based mostly on comparisons with the thermodynamic excess functions of binary systems at Ρ = 0. For long chains the parameter η/k, determining the dependence of ū/k on the temperature, depends on the reduced density ρ* = V°/V of the system (where V° is the close-packed volume) and η/k->0 when ρ* > 0.75 (solid state). ij

H P h e t h e r m o d y n a m i c p r o p e r t i e s of m i x t u r e s are e x t r e m e l y sensitive to -•-the values of m i x e d - p a i r i n t e r a c t i o n energies Uu(r) r e l a t i v e to those a c t i n g b e t w e e n t h e m o l e c u l e s of t h e p u r e c o m p o n e n t s , Uu(r) a n d

Ujj(r),

a n d to t h e average c o l l i s i o n d i a m e t e r σ o f t h e m i x t u r e o b t a i n e d f r o m σ « , χ

a,» a n d σ ·. A r e l a t i v e l y s m a l l error i n ί;

obtained.

g r e a t l y affects t h e p h a s e d i a g r a m

I f t h e p r e d i c t e d v a l u e o f ι*# is too l o w b y 2 % , t h e p h a s e d i a

gram may, for example, exhibit a n azeotropic mixture or even a separation i n t o three phases, l i q u i d +

liquid +

gas, w h i l e t h e r e a l d i a g r a m shows

o n l y t w o phases a n d n o azeotrope. T h e p u r p o s e of this c h a p t e r is to test t h e c o m b i n i n g rules t h a t r e s u l t f r o m three a p p r o x i m a t i o n s f o r r a n d o m m i x t u r e s of fluids. T h e s q u a r e - w e l l (SW)

a p p r o x i m a t i o n y i e l d s v e r y g o o d results f o r systems of s m a l l m o l e

cules. T h e c o m p l e x i t y of t h e p r o b l e m f o r l a r g e m o l e c u l e s , a g g r a v a t e d b y t h e d e p e n d e n c e of u(r)

o n t h e r e d u c e d d e n s i t y of t h e system, is o u t l i n e d .

0-8412-0500-0/79/33-182-197$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.

198

EQUATIONS O F S T A T E

Systems

of

Small

Molecules

W e s h a l l l i m i t o u r c o n s i d e r a t i o n to systems w i t h L o n d o n energies o n l y .

F o r s u c h systems

London

(dispersion)

derived

(1,2)

the

famous

relation


i n w h i c h p» a n d p

;

are the p o l a r i z a b i l i t i e s of t h e t w o m o l e c u l e s , r is the

d i s t a n c e b e t w e e n t h e i r centers, a n d hv are c e r t a i n energies w h i c h p r o b a b l y are r e l a t e d to the i o n i z a t i o n potentials I . E x c e p t for H e a n d H , hv 2

is not e q u a l or p r o p o r t i o n a l to 7. T h e s u p e r s c r i p t ' V i n

indicates that

the r e l a t i o n w a s d e r i v e d for s m a l l s p h e r i c a l m o l e c u l e s . T a k i n g i n t o a c c o u n t that a h a r d r e p u l s i o n exists at close distances r , we may write =

Uij(r)

oo

at r ^

σ

(2)

at r > σ

(3)

and Uij (r) = where « ûij =

- ïïy ( σ , / Λ )

6

is the m i n i m u m v a l u e of Uij(r).

y

Uij . 0

F o r small spherical molecules

B y c o m p a r i n g E q u a t i o n s 1 a n d 3 for the p u r e c o m p o n e n t s ,

one obtains 4 hvi = Upon

4

— u°

substitution into

u

u*u/v i a n d h 2

Equation

Vj

=

_ u°jj

aVP ; 2

1 the

unknown

quantities

+

fife) ]

hv

are

eliminated.

° [Àf (S)

U {i= 2

T h i s c o m b i n i n g r u l e w a s d e r i v e d i n a different m a n n e r b y K o h l e r I t is n o w w e l l k n o w n (4)

(3).

that the so-called r a n d o m m i x i n g approxi-

m a t i o n , w h i c h is b a s e d o n the L e n n a r d - J o n e s p o t e n t i a l , g r e a t l y gerates t h e effect of size differences of the m o l e c u l e s o n « ·. ί;

m i x t u r e s are r a n d o m i n a l l the m o d e l s c o n s i d e r e d m o r e a p p r o p r i a t e to c a l l that i n w h i c h u(r)

exag-

Since the

i n this c h a p t e r , i t is

varies w i t h ( σ / r )

Lennard-Jones ( L J ) approximation.


6

the

11.

KREGLEwsKi A N D C H E N

The van der Waals

Theory

199

of Fluids

I n t h e S W a p p r o x i m a t i o n a n d at a constant w i d t h ( σ / r ) o f t h e w e l l for a l l substances w e have ^o( ) = c o n s t a n t l y

(r > σ) ;

r

(5)

a n d instead of E q u a t i o n 4 w e o b t a i n

ΰ°

υ

= 2 [\z^-— + ^


A l s o , t h e average ù

m

(6)

( s m a l l molecules)

1

of t h e m i x t u r e b e c o m e s i n d e p e n d e n t of σ m — Σ Σ i j

u

where i7 =

- 1

u jj Pi J

[_w a Vj

ij

X

(7) \ f

ij

X

U

f o r s m a l l molecules.

i;

I n t h e theory of v a n d e r W a a l s ( V D W ) t h e p a r a m e t e r a is p r o p o r t i o n a l to ΰσ , w h e r e ù is t h e m i n i m u m v a l u e of a n u n k n o w n p o t e n t i a l 3

u(r).

L e l a n d et a l . (3) h a v e s h o w n that t h e V D W a p p r o x i m a t i o n is

much

better

combining where k

tj

than the L J approximation.

I n this a p p r o x i m a t i o n t h e

rules a l w a y s a r e a p p l i e d to a

ijf

u s u a l l y α# =

&ϋ(αϋα ·) ,

is a constant b e c a u s e n o t h i n g better m a y b e d e r i v e d

;;

1/2

when

is n o t k n o w n .

u(r)

Kac

et a l . (5) have s h o w n that t h e a t t r a c t i o n t e r m i n t h e V D W

e q u a t i o n of state, a/RTV,

is o b t a i n e d w h e n the i n t e r m o l e c u l a r forces a r e

infinitely w e a k , du(r)/dr

= 0. T h i s is also t h e c h a r a c t e r i s t i c feature of

the S W p o t e n t i a l w h i c h differs f r o m t h e K a c p o t e n t i a l i n that t h e attrac t i o n is c u t off at a c e r t a i n ( r / σ ) , u s u a l l y at 3/2. H e n c e , E q u a t i o n 6 i s the p r o p e r c o m b i n i n g r u l e f o r u j i n a . {

{j

O n t h e c o n t r a r y , i f w e insist o n c o u p l i n g ΰ · w i t h σ · a n d u s i n g t h e ί;

VDW

approximation for a

m>

ί;

t h e effect o n t h e c a l c u l a t e d properties of

m i x t u r e s is t h e same as i f W y ( r ) w e r e expressed b y a h y p o t h e t i c a l p o t e n t i a l ùij(aij/r) . z

T h i s leads to a c o m b i n i n g r u l e g i v e n b y E q u a t i o n 4 i n

w h i c h a l l of t h e σ ' a r e r e p l a c e d b y σ ' . T h i s is c a l l e d t h e r u l e f o r t h e 6

VDW

8

3

8

approximation.

T h e rules f o r t h e V D W a n d S W a p p r o x i m a t i o n s a r e c o m p a r e d w i t h the e x p e r i m e n t a l d a t a f o r m i x t u r e s o f n o b l e gases i n T a b l e I . T h e e x p e r i m e n t a l values of % are selected b y S m i t h et a l . (6) f r o m m o l e c u l a r b e a m scattering, s e c o n d v i r i a l coefficient, a n d viscosity d a t a . T h e values of ù/k of t h e p u r e c o m p o n e n t s a r e : H e , 10.5 K ; A r , 140 K ; K r , 196 K ; a n d X e , 265 K . T h e values of σ g i v e n b y C h e n et a l . (6) a r e : H e , 2.65 A ; N e , 2.75 A ; A r , 3.34 A ; K r , 3.64 A ; a n d X e , 3.81 A . T h e p o l a r i z a b i l i t i e s w e r e t a k e n f r o m t h e L a n d o l t - B o r n s t e i n T a b e l l e n (1960).

B y comparing the


200

EQUATIONS OF STATE Table I.

Values of û°ij for Mixed Interactions between Noble Gases ( K )

Experimental

30.2 30.2 28.0

He + Ar He + K r He + Xe 1

6

Equation

VDW

a

Equation

33.0 35.2 33.8

38.6 45.6 54.3

29.0 28.8 25.3

Equation 4 with all σ 'β replaced by σ . The σ α 3

6

8

is calculated from Equation 10.

3

values of iïy g i v e n b y S m i t h et a l . w i t h those of C h e n et a l . ( w h i c h are Downloaded by CORNELL UNIV on December 6, 2012 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch011

lower)

w e estimate t h e i r u n c e r t a i n t y to be a b o u t

s h o w that E q u a t i o n 6 is the best a p p r o x i m a t i o n .

±10%.

These

data

T h e last c o l u m n i n

T a b l e I gives the g e o m e t r i c m e a n values (û«%)

%i—

(8)

1 / 2

T h e c o m p a r i s o n s w i t h other e x i s t i n g c o m b i n i n g rules w e r e m a d e

by

S m i t h et a l . a n d are not r e p e a t e d here. T h e t w o most c o m m o n l y

u s e d rules for σ»,, n a m e l y that o w i n g

to

Lorentz,

σ « — (σ« + σ „ ) / 2 ,

(9)

a n d to v a n d e r W a a l s

σ\}=

(σ * + σ , , ) / 2 3

(10)

3

are c o m p a r e d w i t h the e x p e r i m e n t a l d a t a g i v e n b y C h e n et a l . (7) T a b l e II.

E q u a t i o n 10 appears

to be m u c h better

a l t h o u g h it is o v e r s i m p l i f i e d i n that it neglects the effect of w average a

m

i;

o n σ ·. ί;

The

of the m i x t u r e , d e r i v e d f r o m E q u a t i o n 10, is

Table II.

Collision Diameters for Mixed between Noble Gases (in A ) Experimental

He He He He

in

than Equation 9

+ Ne + Ar + Kr + Xe

2.73 3.09 3.27 3.61

Equation

2.70 3.00 3.14 3.23

9

Interactions Equation

2.70 3.03 3.22 3.33


10

11.

KREGLEWSKI AND C H E N

The van der Waals

201

Theory of Fluids

I n a d d i t i o n to t h e a b o v e d i r e c t tests, t h e r u l e g i v e n b y E q u a t i o n 6 w a s tested b y c a l c u l a t i n g the t h e r m o d y n a m i c excess f u n c t i o n s of m i x t u r e s . T h e relations a r e g i v e n b y K r e g l e w s k i a n d C h e n ( 8 ) . T h e b a s i c o n e f o r the r e s i d u a l H e l m h o l t z energy of a m i x t u r e A

r

m

RT

-=

/ v

ο

" (a

m

z

m

ι \ -ι ;

V i

η Downloaded by CORNELL UNIV on December 6, 2012 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch011

t. \ ι

ζ.*

~f" 3 a ) £ (1 -

— 1) In (1 — im) + i

is

1

m

3a £

m

U

m

w

2

m

T h e first t w o terms w e r e d e r i v e d f r o m B o u b l i k ' s ( 9 ) r e l a t i o n f o r t h e c o m p r e s s i b i l i t y factor of h a r d c o n v e x bodies. H e r e α is a constant d e p e n d i n g o n t h e shape of t h e m o l e c u l e s (a = m i x t u r e s a s s u m e d to b e a =

]T i

m

T h e t e r m U = 0.74048 V °/V m

close-packed volume, N

0

1 for spherical molecules), for

Xi«t.

m

=

1/6 7rN a */V , 0

m

m

where V ° m

is t h e A v o g a d r o n u m b e r , a n d V

m

is t h e

is t h e m o l a r

v o l u m e of t h e system. V ° is a s i m p l e f u n c t i o n of t h e t e m p e r a t u r e ( T ) (10)

w i t h a c h a r a c t e r i s t i c v a l u e V ° ° at Τ = 0 K . T h e last t e r m i n E q u a

t i o n 12 w a s i n t r o d u c e d b y A l d e r et a l . (11).

D

nm

a r e 24 u n i v e r s a l c o n

stants c o m m o n f o r a l l substances w h o s e r a d i a l a n d h i g h e r d i s t r i b u t i o n f u n c t i o n s are t h e same f u n c t i o n s of û/kT a n d t h e r e d u c e d d e n s i t y p* V°/V.

A s s h o w n b y C h e n a n d K r e g l e w s k i (10)

C h a o (12),

=

a n d Simnick, L i n , a n d

E q u a t i o n 12 is t h e most a c c u r a t e k n o w n e q u a t i o n w i t h f o u r

c h a r a c t e r i s t i c constants: a, V ° ° ( V ° at Τ =

0 K ) , û°/k, a n d η/k

(see

E q u a t i o n s 13 a n d 14). T h e y also h a v e s h o w n ( J O ) t h a t i n o r d e r t o o b t a i n a g r e e m e n t w i t h s e c o n d v i r i a l coefficient d a t a of t h e gas a n d t h e i n t e r n a l e n e r g y o r t h e e n t h a l p y of t h e l i q u i d , i t is necessary t o assume t h a t u(r) is a f u n c t i o n of Τ as r e q u i r e d b y t h e t h e o r y of n o n c e n t r a l forces b e t w e e n nonspherical molecules

(13)

^ - δ Μ Ι

+ Λ Α Γ )

(13)

w h e r e L i n d i c a t e s L o n d o n i n t e r a c t i o n s . F o r a p u r e fluid ( J O )

7? /fc « 0.60 ωΤ° L

w h e r e ω is the a c e n t r i c factor a n d T

c

(14)

is t h e c r i t i c a l t e m p e r a t u r e i n K .

T h e y also h a v e c o n c l u d e d t h a t η\ί~

(ιΛι + Λ > ) / 2


(15)

202

EQUATIONS O F S T A T E

T h e values of the c h a r a c t e r i s t i c constants for A r , K r , N , a n d C H 2

are g i v e n i n R e f . 10.

4

F o r X e , 0 , a n d a l l t h e substances c o n s i d e r e d i n 2

T a b l e I V the constants w e r e e s t i m a t e d as f o l l o w s .

T h e effect of t h e v a l u e

of a o n t h e excess f u n c t i o n s is n e g l i g i b l e ; therefore, a w a s set e q u a l to u n i t y . T h e η/k w a s e s t i m a t e d f r o m E q u a t i o n 14 a n d V ° ° f r o m t h e l i q u i d m o l a r v o l u m e at t h e r e d u c e d t e m p e r a t u r e T/T T h e e n e r g y û°/k

^

o n l y if η/k

T

=

0.6.

=

c

c

0.

I n o t h e r cases i t is n o t

r e l a t e d so d i r e c t l y to a m a c r o s c o p i c p r o p e r t y b u t i t c a n b e e s t i m a t e d b y r e l y i n g o n the a c c u r a c y of E q u a t i o n 12.

W h e n a l l of t h e c h a r a c t e r i s t i c

constants are k n o w n , E q u a t i o n 12 y i e l d s a c c u r a t e m o l a r v o l u m e s of

the


s a t u r a t e d l i q u i d a n d v a p o r phases o b t a i n e d b y a n i t e r a t i o n at constant Τ a n d Ρ f r o m the G i b b s e q u i l i b r i u m c o n d i t i o n . H e n c e , i f a, V ° ° , a n d η/k h a v e b e e n e s t i m a t e d a l r e a d y , ù°/k

is f o u n d b y v a r y i n g its v a l u e u n t i l t h e

l i q u i d v o l u m e c a l c u l a t e d f r o m E q u a t i o n 12 or f r o m the c o m p r e s s i b i l i t y f a c t o r Ζ agrees w i t h the o b s e r v e d v a l u e at a c e r t a i n T , say, T/T

c

=

0.6.

T h e values of η/k f o r t h e c o m p o n e n t s g i v e n i n T a b l e I I I are 1 Κ for C H , 2 Κ for 0 , 3 Κ for N , a n d zero for A r , K r , a n d X e . T h e v a l u e s of 4

u

i ;

L

2

2

i n E q u a t i o n 13 w e r e c a l c u l a t e d f r o m E q u a t i o n s 6 a n d 15. C o n s i d e r i n g

t h a t a s m a l l error i n the c o m b i n i n g rules leads to l a r g e errors i n G

and

E

the a g r e e m e n t w i t h t h e o b s e r v e d v a l u e s is v e r y satisfactory.

H, B

t i o n a l l y , for the N

+

2

C H

4

system E q u a t i o n 15 y i e l d s ? / i

v a l u e s of the excess f u n c t i o n s

t h a t are too

(8)

small.

q u a d r u p o l a r m o l e c u l e a n d t h i s effect varies w i t h (kT)' If the v a l u e of ηη/k

actions, 7 7 n A , w e h a v e N : /k Q

— V22 /k =

(T7I

2

L

+

2

1,77

L

vi2 )/k Q

22

Q

/fc =

V11

=

0.5 Κ i n s t e a d of

=

Q

~ /k

V12

=

2 K

and to

is a s c r i b e d to q u a d r u p o l e i n t e r

r;n /fc =

0. S i n c e η^

L

N i t r o g e n is a analogously

1

the L o n d o n energy.

2

Excep

(*?u

=

Q

η^

3, ηη^/k V22 ) > Q

=

Table IV.

1/2

=

0; C H : r? /fc 4

22

w e o b t a i n η^/k

=

2 Κ f o r the t o t a l .

Excess F u n c t i o n s

of

Components (1) n-C n-C cy-C cy-C

6

e

5

5

+ + + +

(2)

Ύ/Κ

n-Ci2

293 293 293 293 298 298 298 298

n-C

1 6

cy-C

8

OMCTS

218' 0 305" 0 101* 0 194' 28

Binary systems at χ = 0.5. * T h e values kept constant are η/k = 88 of η-Ce, TJ//C = u°/k and V° of all the components. Weighing of interactions : mole fractions.

t 1.33

1 0.91

f 1.275 1.11 0.834

0

59 Κ of c y c l o - C and

0


5

11.

KREGLEWSKI A N D C H E N

Table III.

The

van

der

Waals

Theory

of

203

Fluids

Excess Functions of L i q u i d Systems of Small Molecules 0

Observed System

T / K

A r + Kr

103.94 115.77 115.77 161.36 161.36 83.82 91 115.77 115.77 90.7 91.5 105.0

Kr +

Xe

Ar + 0 Ar + C H


2

4

Kr + C H N + C H 2

4 4

G

Predicted Equation

Values'

E

82.5 83.9 82.4 103 114 37 72 76 29 169.6

JJE

V

60 103

-0.53 -0.464 -0.459 -0.695 +0.14 +0.18

— 55 — 138

— —

E

-0.025

— — -1.16

101

QE

H

87 93 93 135 135 35 79 79 22 166" 166" 169"

47 26 26 64 64 50 100 69 22 190" 186" 86"

Values, 6 V

E

E

-0.19 -0.51 -0.51 -0.51 -0.51 +0.12 +0.15 -0.12 -0.02 -0.71" -0.75" -1.95"

° Binary systems at χ = 0.5. G Ε and H are in joules mole" and V is in cubic centimeters mole" . ° The references to the observed values are given in Refs. 8 and 17. Also, some of the calculated values were taken from Ref. 8. Values calculated with the assumption that η/k of nitrogen is caused by the quadrupole moment (see the text). b

1

E

1

E

d

Systems

of Large

Molecules

W h e n one or m o r e c o m p o n e n t s are l a r g e m o l e c u l e s , E q u a t i o n b e g i n s to f a i l .

D e p a r t u r e s a l r e a d y are n o t e d for ethane (η/k =

6

19 K ) .

T h e r e a s o n m a y be that o n l y a p a r t of t h e l a r g e m o l e c u l e is i n v o l v e d i n the m i x e d interaction ù . i}

A c c o r d i n g l y , t h e c o n c e p t of i n t e r a c t i o n s b e -

t w e e n segments of m o l e c u l e s is p o p u l a r . S i n c e for m o l e c u l e s w i t h L o n d o n

L i q u i d Systems of Large Molecules

0

Calculated '

Observed

b 0

G

H

E

E

d

V

E

— — -0.20

—65

+129

-0.48

+2

-40

-0.28

+2.36 +0.06

-208

+212

+0.05

-50

+95

-0.46

+1 -209 -215

-34 +516 +334

e 1

E

-0.31

-0.31

d

H

+46

+31

—

E

—25 -20

—

G

The references to the experimental data are given in Ref. 17. From Equation 14. Wrong results for any value of 112.


204

EQUATIONS O F

energies ( o n l y ) p< is a n a p p r o x i m a t e l y l i n e a r f u n c t i o n of σ «

STATE

(expressed

3

b y t h e c r i t i c a l v o l u m e V / or t h e c l o s e - p a c k e d v o l u m e V»° ), t h e c o m b i n i n g r u l e for i n t e r a c t i o n s € ° b e t w e e n segments of e q u a l size a n d , i m p l i c i t l y , {

e q u a l p o l a r i z a b i l i t i e s is

c°«-2[(l/e°

4 l

) +

(ΙΑ ;;)]" 0

(16)

1

S i m u l t a n e o u s l y , the w e i g h i n g of i n t e r a c t i o n s b y means of t h e m o l e f r a c tions i n E q u a t i o n 7 m u s t b e r e p l a c e d b y a m o r e p r o p e r m e a s u r e of the Downloaded by CORNELL UNIV on December 6, 2012 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch011

p r o b a b i l i t i e s of en, c#, a n d b y F l o r y (14),

i n t e r a c t i o n s . T h e site f r a c t i o n s , as defined

are u s e d f o r systems of c h a i n m o l e c u l e s b u t t h e y c o n t a i n

the p a r a m e t e r Si/sj t h a t has to b e fitted to a g i v e n system. K r e g l e w s k i and K a y (25)

o b t a i n e d v e r y g o o d results for t h e c r i t i c a l constants

m i x t u r e s b y u s i n g E q u a t i o n 16 a n d c r u d e l y d e f i n e d surface

of

fractions.

T h e agreement w i t h o b s e r v e d d a t a is so g o o d that t h e c o n c e p t s h o u l d n o t be a b a n d o n e d o n l y b e c a u s e the surface fractions h a v e a p o o r t h e o r e t i c a l f o u n d a t i o n . T h e r e m u s t exist a b r i d g e b e t w e e n t h e m o l e fractions (appropriate for small spherical molecules)

a n d t h e site fractions

for

l o n g chains. T h e c o n c e p t of segment interactions has not b e e n u s e d i n o u r present w o r k . O u r considerations are l i m i t e d to the exact t r e a t m e n t b a s e d o n t o t a l i n t e r a c t i o n s tt# a n d m o l e fractions. A c c o r d i n g to S a l e m (16)

the L o n d o n

e n e r g y b e t w e e n t w o l o n g straight or c i r c u l a r c h a i n s at close distances is p r o p o r t i o n a l to m u°(r)

w h e r e u°(r)

is g i v e n b y E q u a t i o n 1 a n d m is

t h e n u m b e r of c h a i n units. T h e same t r a n s f o r m a t i o n t h a t l e d to E q u a t i o n 6 yields

2 [ ( Ι / υ * ) (l/q,) +

û°n =

0

w h e r e ty =

Vi il'(p^n*).

values of G

m

E

system

are

HjY

1

(large molecules)

T h e values of ^ r e q u i r e d to fit t h e

of t h e n-hexane +

cyclopentane +

(1/ΰ°„)

n - d o d e c a n e or n - h e x a d e c a n e

cyclooctane or octamethyl-cyclotetrasiloxane

close to u n i t y a n d so

they

differ v e r y

(17)

observed and the

(OMCTS)

much from

the

theoretical values. T h e r e is a m o r e serious p r o b l e m t h a n the necessity of u s i n g e m p i r i c a l v a l u e s of m. T h e values of û°/k,

η/k, a n d V ° of t h e c o m p o n e n t s

e s t i m a t e d f r o m t h e p r o p e r t i e s of the l i q u i d state.

T h e values of

e s t i m a t e d b y means of E q u a t i o n 14, r a n g e f r o m 59 Κ f o r to 305 Κ f o r h e x a d e c a n e .

were η/k,

cyclopentane

T h e v a l u e f o r η^ w a s e i t h e r c a l c u l a t e d f r o m

E q u a t i o n 15 or d e l i b e r a t e l y v a r i e d w i t h i n reasonable l i m i t s . A l s o HJ w a s v a r i e d . W e h a v e e s t a b l i s h e d t h a t n o n e of t h e c o m b i n a t i o n s of ηα a n d HJ


11.


The van der Waals

y i e l d s i m u l t a n e o u s l y g o o d values of G ,

H

E

E

205

Theory of Fluids

, and V .

T h e results are not

E

i m p r o v e d w h e n m o l e fractions i n E q u a t i o n 7 are r e p l a c e d b y surface or site fractions. T h e results m a d e it clear that the values of η/k of the p u r e ponents m u s t b e c h a n g e d .

com

T h i s c o n c l u s i o n is s t r o n g l y e n d o r s e d b y the

f o l l o w i n g results o b t a i n e d r e c e n t l y b y one

of the authors.

For

both

s p h e r i c a l a n d c h a i n m o l e c u l e s u p to n-pentane or c y c l o p e n t a n e the same values of η/k, o b t a i n e d f r o m E q u a t i o n 14, y i e l d a c c u r a t e values of t h e second

v i r i a l coefficients

β

r e s i d u a l i n t e r n a l energies U

r

(Τ)

(reduced

density ρ*

of the l i q u i d ( a t p* >

=

0)

and

the

0.52). T h e acentric


factors are d e t e r m i n e d at ρ * ^ 0.52. F o r l a r g e r m o l e c u l e s , b e g i n n i n g w i t h b e n z e n e or n - C , η/k values r e q u i r e d to fit β (Τ)

data become larger than

6

E q u a t i o n 14 suggests w h e r e a s t h a t fitting U d a t a of t h e l i q u i d b e c o m e s r

s l i g h t l y s m a l l e r ( a t constant ù°/k

a n d V ° ). T h a t is, η/k b e g i n s to d e p e n d

o n p*. T h e v a l u e of η/k varies w i t h p* m o r e r a p i d l y for n-alkanes ( n - C , 7

etc.)

t h a n for c y c l o a l k a n e s

( c y - C , etc.). 7

The phenomenon

is r e l a t e d

c l e a r l y to r e s t r i c t e d r o t a t i o n at h i g h densities. W h e n t h e c h a i n is suffi c i e n t l y l o n g a n d p* >

0.75 ( s o l i d state) w e c a n expect η/k -> 0 ( c o m

p l e t e l y r e s t r i c t e d rotations ). A c c o r d i n g l y , w e have r e t a i n e d the values of η/k of n - C

and c y - C

6

3

c a l c u l a t e d f r o m E q u a t i o n 14, b u t the values for n - C i , n - C i , c y - C , a n d 2

6

8

O M C T S w e r e decreased to a n a p p r o p r i a t e l e v e l . S i m u l t a n e o u s l y , η

=

(vu +

V

ι}

Vn)

/2

a n d Hj w e r e v a r i e d u n t i l the three p r o p e r t i e s G ,

H

E

E

, and

E

a g r e e d w i t h i n ± 20 J m o l " a n d ± 0 . 0 5 c m m o l ' w i t h the o b s e r v e d v a l u e s . 1

3

1

T h e results are c o m p a r e d i n T a b l e I V . T h e first r o w for e a c h system shows the errors i n H

E

when L

and V

E

12

is fitted to G

and η

E

22

is t h a t

c a l c u l a t e d f r o m E q u a t i o n 14. T h e second r o w shows the i m p r o v e m e n t of the results w h e n η

22

η

22

=

is p r o p e r l y d i m i n i s h e d . F o r three of t h e

0 is r e q u i r e d . I n the f o u r t h system, a decrease of η /k

would improve H 112 appears

E

; however, V

E

w o u l d b e c o m e negative.

to v a r y i n a n u n p r e d i c t a b l e m a n n e r .

fractions are u s e d ( w i t h the same values of η )

systems

b e l o w 28 Κ

22

T h e v a l u e of

When

the

surface

t h e n a l w a y s 112 >

22

q u a l i t a t i v e agreement w i t h the t h e o r y of S a l e m . H o w e v e r , L

12

1 in

cannot

be

p r e d i c t e d w h e n the systems are t r e a t e d as r a n d o m m i x t u r e s . I t is s h o w n elsewhere

(18)

t h a t the p r o p e r t i e s of m i x t u r e s of l a r g e m o l e c u l e s

can

b e p r e d i c t e d w i t h n e a r l y the same a c c u r a c y as those of s m a l l m o l e c u l e s b y i n t r o d u c i n g a n a p p r o x i m a t e c o r r e c t i o n to A

r

m

o w i n g to

nonrandom

mixing. I n a l l theories of p o l y m e r solutions u^ or c a l w a y s are a s s u m e d to b e i;

i n d e p e n d e n t of the t e m p e r a t u r e , a p p a r e n t l y c o n t r a d i c t i n g the t h e o r y of n o n c e n t r a l forces.

O u r results s h o w that there is n o c o n t r a d i c t i o n a n d

that this a s s u m p t i o n is a l l o w e d f o r l o n g straight or c i r c u l a r c h a i n s at h i g h densities ( l i q u i d s b e l o w t h e i r n o r m a l b o i l i n g t e m p e r a t u r e s ).


206

EQUATIONS OF

Glossary

of

STATE

Symbols

Τ = temperature i n Κ Ρ = pressure i n b a r V = v o l u m e of the system V ° = close-packed volume

( V ° = V

o

o

a t r = 0 )

A = H e l m h o l t z free energy, A ( T , V ) U = i n t e r n a l energy, U( Τ, V) G = G i b b s free energy, G ( T , P) Ή = enthalpy,

H(T,P)

I = ionization potential Downloaded by CORNELL UNIV on December 6, 2012 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch011

k = B o l t z m a n n constant ( R =

N k is the gas c o n s t a n t ) 0

r = intermolecular distance = pair interaction energy

u(r)

û, ύ° = m i n i m u m v a l u e of u(r)

(ù/k i n K )

η/Jc = p a r a m e t e r of n o n c e n t r a l energy ( i n K ) σ

= c o l l i s i o n d i a m e t e r of a m o l e c u l e

ρ = ω

=

m e a n p o l a r i z a b i l i t y of a m o l e c u l e a c e n t r i c factor

x = mole fraction p* = r e d u c e d d e n s i t y ,

V°/V

Superscripts r = r e s i d u a l p r o p e r t y ( r e a l fluid m i n u s p e r f e c t gas at the same Γ a n d Ρ or Γ a n d V ) Ε = excess p r o p e r t y ( r e a l m i x t u r e m i n u s i d e a l m i x t u r e at the same Τ, P , a n d * ) c = g a s - l i q u i d c r i t i c a l constants Subscripts i, j = i n t e r a c t i n g species ( i =

1 , 2 , . . . m; j =

i or /

%)

m = m o l a r p r o p e r t y of a fluid or p e r m o l e of a m - c o m p o n e n t m i x t u r e , e

-g-> V / c m m

3

m o l " ; also a n average v a l u e of a m o l e c u l a r p r o p e r t y 1

of a m i x t u r e , e.g., ù

m

or

E x c e s s p r o p e r t i e s , e.g., G , E

a. m

are also m o l a r b u t the s u b s c r i p t is

deleted. Acknowledgments D r . C h e n p a r t i c i p a t e d i n a n e a r l y stage of this w o r k . H e d i e d i n S e p t e m b e r , 1977. T h a n k s to B . J . Z w o l i n s k i , f o r m e r D i r e c t o r of t h e T h e r m o d y n a m i c s R e s e a r c h C e n t e r , f o r e n c o u r a g i n g this r e s e a r c h a n d t o C . C h e n f o r h e r


11.


The van der Waals Theory

of Fluids

207

h e l p i n c o m p u t e r p r o g r a m m i n g . P a r t i a l s u p p o r t of t h e T e x a s E n g i n e e r i n g E x p e r i m e n t S t a t i o n is g r a t e f u l l y a c k n o w l e d g e d .

Literature Cited 1. 2. 3. 4. 5.


6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

London, F. Z. Phys. 1930, 63, 245. London, F. Z. Phys. Chem., Abt. Β 1930, 11, 222. Kohler, F. Monatsh. Chem. 1957, 88, 857. Leland, T. W., Jr.; Rowlinson, J. S.; Sather, G. A. Trans. Faraday Soc. 1968, 64, 1447. Kac, M.; Uhlenbeck, G. E.; Hemmer, P. C. J. Math. Phys. 1963, 4, 216, 229. Smith, K. M.; Rulis, A. M . ; Scoles, G.; Aziz, R. A. J. Chem. Phys. 1977, 67, 155. Chen, C. H.; Siska, P. E.; Lee, Y. T. J. Chem. Phys. 1973, 59, 601. Kreglewski, Α.; Chen, S. S. J. Chim. Phys. 1978, 75, 347. Boublik, T. J. Chem. Phys. 1975, 63, 4084. Chen, S. S.; Kreglewski, A. Ber. Bunsenges. Phys. Chem. 1977, 81, 1048. Alder, B. J.; Young, D. A.; Mark, M. A. J. Chem. Phys. 1972, 56, 3013. Simnick, J. J.; Lin, H. L.; Chao, K. C.; Chapter 12 in this book. Rowlinson, J. S. "Liquids and Liquid Mixtures"; Butterworths: London, 1969. Flory, P. J. J. Am. Chem. Soc. 1965, 87, 1833. Kreglewski, Α.; Kay, W. B. J. Phys. Chem. 1969, 73, 3359. Salem, L. J. Chem. Phys. 1962, 37, 2100. Kreglewski, Α.; Wilholt, R. C.; Zwolinski, B. J. J. Phys. Chem. 1973, 77, 2212. Kreglewski, A. J. Chim. Phys. 1979, 76 (in press).

RECEIVED

August 8, 1978.


Equations of State in Engineering and Research

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