Phase Equilibria and Fluid Properties in the Chemical Industry

19

Phase Equilibria and Fluid Properties in the Chemical Industry Downloaded from pubs.acs.org by NANYANG TECHNOLOGICAL UNIV on 06/10/16. For personal use only.

Criteria of Criticality MICHAEL M O D E L L Massachusetts Institute of Technology, Cambridge, MA 02139

Over a century ago, Gibbs (1) developed the mathematical c r i t e r i a of c r i t i c a l i t y . He defined the c r i t i c a l phase as the t e r m i n a l s t a t e on the b i n o d a l s u r f a c e and reasoned that i t has one l e s s degree of freedom than the b i n o d a l s u r f a c e ( i . e . , f = n-1). He then developed two equations as the c r i t e r i a of c r i t i c a l i t y ; these two equations, when imposed upon the Fundamental Equation, reduce the degrees of freedom from n+1 t o n-1. Gibbs presented three a l t e r n a t e sets of the two c r i t i c a l i t y criteria. I n terms of the v a r i a b l e s e t T , V , y i , . . . , y _ i , N , they a r e : n

= 0

n

(1)

n-1

= 0 n-1

369

(2)

P H A S E EQUILIBRIA

370

AND FLUID

PROPERTIES IN C H E M I C A L

INDUSTRY

To ensure s t a b i l i t y of the c r i t i c a l phase, Gibbs noted that the f o l l o w i n g r e l a t i o n must a l s o be s a t i s f i e d :

(3)

> 0 T,V,y ,...,y Phase Equilibria and Fluid Properties in the Chemical Industry Downloaded from pubs.acs.org by NANYANG TECHNOLOGICAL UNIV on 06/10/16. For personal use only.

1

n-1

Gibbs noted that Eq. (1) may lead to an indeterminant form. When t h i s occurs, he suggested permutting the component s u b s c r i p t s or using "another d i f f e r e n t i a l c o e f f i c i e n t of the same general form." He noted that Eq. (1) i s the c r i t e r i o n of the l i m i t of s t a b i l i t y ( i . e . , the s p i n o d a l s u r f a c e ) . I n an e a r l i e r d i s c u s s i o n on that s u b j e c t , Gibbs showed that a l l of the f o l l o w i n g p a r t i a l d e r i v a t i v e s v a n i s h on the s p i n o d a l s u r f a c e .

(») V-'

.f-h)

V, ...,y V

V

n

/

1 ,v,u ,...,p T

2

V n

n /

T,V,p ,..., 1

V l

(4) Thus, any one of these forms could be used i n place of Eq. ( 1 ) , w i t h corresponding changes made i n Eqs. (2) and ( 3 ) . As an a l t e r n a t i v e to Eqs. (1) and ( 2 ) , Gibbs suggests " f o r a p e r f e c t l y r i g o r o u s method there i s an advantage i n the use of S^V, N^,...,N as independent v a r i a b l e s . " I n t h i s v a r i a b l e s e t the c r i t i c a l i t y c o n d i t i o n s become: n

(5)

R , = 0 n+1

(6) SS

where

R

s

N S n

N

N Nn 1

N

i

N

i i

(7)

n+1

SN

N N n n

1 n

and S i s the determinant formed by r e p l a c i n g one of the rows of R

n 1 ? n lJ h

+

(R

+

(I

W

R

N

< n l> +

19.

Criteria of Criticality

MODELL

SS

371

s

N S n

N

N HL n 1

N

i

N

i i


e.g., S

(8) SN

( R

N.N

N N , n n-1

1 n-1

n-1 ( R

n l> +

( R

n l> +

n+l\

2

where the n o t a t i o n i s = ( 3 U / 3 i 8 j ) and the v a r i a b l e s held constant i n the d i f f e r e n t i a t i o n are n+1 of the s e t _S,V,N]_,... ,N . Similarly, O R / ) Gibbs s t a t e s that Eqs. (7) and n

N

+

1

9

N

L

S , V , N N

L n (8) " w i l l hold true of every c r i t i c a l phase w i t h o u t e x c e p t i o n , " implying that t h i s s e t of c r i t i c a l i t y c r i t e r i a i s more s t r i n g e n t than Eqs. (1) and ( 2 ) . Gibbs presents a t h i r d a l t e r n a t i v e s e t w i t h T,P, N . ,N as independent v a r i a b l e s : ^ n

N

N

2 1

N n-1 -N-1

N

N N

n-1 2

N

i i

N

12

N

2

2

0 (9)

N

and

C

=

i

N N 2 n-1

1 n-1i N

0

n-1 n-1

(10)*

where C i s the determinant formed by r e p l a c i n g one of the rows of the B-determinant by B B„ ... B. . Gibbs does not s p e c i f y i f N N N 1 2 n-1 Eqs. (9) and (10) are t r u e w i t h o u t e x c e p t i o n or i f there a r e r e s t r i c t i o n s t o t h e i r use. T

T

N

Development of the General C r i t i c a l i t y C r i t e r i a Each of the c r i t i c a l i t y c r i t e r i a presented by Gibbs can be shown *Gibbs uses the symbols U and V f o r the B- and C-determinants. We use B and C here t o avoid c o n f u s i o n w i t h i n t e r n a l energy and volume.


372


A N D F L U I D PROPERTIES IN C H E M I C A L

INDUSTRY

to be s p e c i a l cases of more general forms. The development of the general form f o l l o w s from the unique f e a t u r e of the c r i t i c a l phase which d i s t i n g u i s h e s i t from other phases i n the v i c i n i t y ; namely, the c r i t i c a l phase i s the s t a b l e c o n d i t i o n on the s p i n o d a l s u r f a c e . This statement i s e q u i v a l e n t t o Gibbs argument that the c r i t i c a l phase l i e s on the b i n o d a l surface and, t h e r e f o r e , i s s t a b l e ( w i t h respect to continuous changes) and a l s o s a t i s f i e s the c o n d i t i o n of the l i m i t of s t a b i l i t y (which d e f i n e s the s p i n o d a l curve). S t a r t i n g w i t h the Fundamental Equation i n the i n t e r n a l energy representation, U = f (S,V,N,...,N ) u l n

(11)

we can t e s t the s t a b i l i t y of a substance by expanding U i n a Taylor s e r i e s and examining the change i n IJ f o r s m a l l p e r t u r b a t i o n s w h i l e h o l d i n g the t o t a l entropy, volume, and mass c o n s t a n t ( 2 ) . That i s , AU = 6U + ~j 6 U + ~r 6 U + ... 2

3

(12)

For e q u i l i b r i u m to e x i s t , 6U = 0

(13)

and f o r the system to be s t a b l e ,

where 6 U i s the lowest order non-vanishing v a r i a t i o n . The s p i n o d a l surface represents the c o n d i t i o n s where the second-order v a r i a t i o n f i r s t l o s e s the p o s i t i v e , d e f i n i t e character of the s t a b l e , s i n g l e phase (3). The second-order v a r i a t i o n can be expressed as

2

ss^> n 2 £ j=i n n

j = l k=l

+ 2U (6S)(6V) gv

(U

SN. J

6

^

+ U

VN. J

+ U

(

)2

w^

6V)6N. J (15)

IT „ 6N. 6N N.N, j k J

k

or, i n c l o s e d form, as a sum of squares, u

u

ss sv



MODELL

373

SS

'sv

SN,

VS

vv

VN,

N V

N

N

11 6Z

2 3

+

'SS "VS

U

U

°SV

'SS

U

VS

l S

N l

N S n

V

VV

U

\

SN,

V

N

N V n

J

SN

VN

1 n

N

i i

N n 1

N N n n 6Z

'SS

J

U

VS

N

n-1

_S

U

'SV

SN.

SN

VV

VN,

VN

N V

N

N

,V n-1

U

N

n-1

n-1

i i

N N 1 n-1

-Nn-1 1

N

N

n-1

n

N

n-1

n+2

(16)

374


A N D F L U I D PROPERTIES

IN C H E M I C A L

INDUSTRY

Using the shorthand n o t a t i o n , A

u

i - ss


SS

i+2

sv

SN.

SS

SV

VS

vv

VN,

VN.

NV

N

N- N. 1 l

N

i

s

N.V

N.S

N.N, l 1

1

1

N

i i

N.N.

Eq. (16) takes the form,

9

9

6^U = A , 6 Z /

n

+

+

Z

2

A

i

9

6Z.

(17)*

The l a s t term i n Eqs. (16) or (17) can be shown to be zero as a v i r t u e of the f a c t that only n+1 i n t e n s i v e v a r i a b l e s are independent f o r a s i n g l e phase. The proof f o l l o w s . Let us expand each of the n+2 i n t e n s i v e v a r i a b l e s T, P, u-^,..., u as f u n c t i o n s of S^,V,N^,. .. ,N . n

dT =

n

(ff) ^V,N

dS +

dV + ^S,N

Z 1=1

(fjU 8 N

iS,V,N[i]

d N 1

*For the form of the 6Zj v a r i a t i o n s , see ( 3 ) ; s i n c e they always appear as squared terms, we need not consider them f u r t h e r here.

19.

MODELL

or

dT = U


dS + U d V + I

s g

U

gv

375

i=l


dP = U

d

dS + U

v s

d i+

"i-V

Vl

% = U

U

N

d

n-1 l

S

d S

S

V N

dN.

(19)

i

+

d N

1

= N

(18)

(20)

V - ' i V- i

1

d

±

1

dV + I U i=l

w

dN

g N

+

*

+

U

D

N

n

1=1

V I n-1

H

V

d

1

d V

+

n

Z

+

1

?

1

° H ,N. l i = l n-1 1 dN

1

U i=l n

dN.

(

2

1

)

(22)

1

Since the n+2 d i f f e r e n t i a l s of T,P,ui_,. . . , u are r e l a t e d by the Gibbs-Duhem equation, any one from t h i s set can be expressed as a f u n c t i o n of the other n+1. Thus, i f P,y]_,...,y are h e l d constant ( i . e . , dP = d u i = ... = d y = 0 ) , then T must a l s o be constant ( i . e . , dT = 0 ) . I t f o l l o w s that the determinant of the m a t r i x of c o e f f i c i e n t s i n Eqs. (18) to (22) must be zero. Since t h i s d e t e r minant i s A +2 i t f o l l o w s that A 2 0. Therefore, the general form of Eq. (17) i s n

n

n

=

n

S

n+

2 2 6 U = A 6Z + Z n

Z

+

1

Z

1

1

2

A 2j

2 6Z.

Z

(23)

Vl

The l i m i t of s t a b i l i t y i s d e f i n e d as the c o n d i t i o n under which 6^U_ l o s e s i t s p o s i t i v e , d e f i n i t e c h a r a c t e r . That i s , one of the c o e f f i c i e n t s i n Eq. (23) vanishes. As has been shown p r e v i o u s l y (_3), when the s p i n o d a l s u r f a c e i s approached from a s t a b l e s i n g l e phase r e g i o n , the c o e f f i c i e n t of 6 Z i i s among the f i r s t to reach zero ( i . e . , i f another c o e f f i c i e n t a l s o v a n i s h e s , i t does so simultaneously w i t h the c o e f f i c i e n t of 6 Z ^ ) . Therefore, on the s p i n o d a l s u r f a c e n +

n +

A , = 0 n+1

(24)

This equation i s one of the c r i t e r i a of c r i t i c a l i t y . I t i s equival e n t to reducing the degrees of freedom from n+1 f o r a s t a b l e s i n g l e phase to n on the s p i n o d a l s u r f a c e . Since A +i i s the determinant n

376

P H A S E EQUILIBRIA A N D F L U I D PROPERTIES IN C H E M I C A L

INDUSTRY

of the m a t r i x of c o e f f i c i e n t s of Eqs. (18) to (21) a t constant N , the requirement of A +i = 0 i s e q u i v a l e n t to dT = dP = dy^ = ... = d y _ i = 0. That i s , i f n v a r i a b l e s from the set T,P,y-[, . . . , y - l are held constant, the remaining v a r i a b l e w i l l a l s o be constant. n

n

n

n

The second c r i t e r i o n of c r i t i c a l i t y i s that the d i f f e r e n t i a l of A 2 must v a n i s h f o r a l l p o s s i b l e v a r i a t i o n s . Thus, Phase Equilibria and Fluid Properties in the Chemical Industry Downloaded from pubs.acs.org by NANYANG TECHNOLOGICAL UNIV on 06/10/16. For personal use only.

n+

dA

dN.=0 l S,V,N [ i ]

n+1

(25) A l t e r n a t i v e l y , Eq. (25) can be s a t i s f i e d s i m u l t a n e o u s l y w i t h n equations from the set.Eqs. (18) to (21) a t constant N . Choosing dT = dP = dy^ = ... = dy _2 = 0 from t h i s s e t , the second c r i t e r i a of c r i t i c a l i t y becomes: R

n

SS

VS

u SN,

vv

VN,

VN

N

N-N . 1 n-1

N S

N

( A

S n-2 0

n-2

n l>

( A

+

u SN n-1

u sv

n l\ +

N

1 1

N

( A

N

N. n-2 1 0

( A

n l>

n-1

N n-2 n-1 9

n l> +

+

n-1 (26) A l t e r n a t e C r i t e r i a i n the U-Representation The two c r i t e r i a of c r i t i c a l i t y developed i n the l a s t s e c t i o n are s i m i l a r t o , but no i d e n t i c a l w i t h Eqs. (5) and (6) which were s t a t e d by Gibbs. We s h a l l now show that the two are equivalent and that they are p a r t i c u l a r s e t s from a more g e n e r a l form of the c r i t e r i a . We began the d e r i v a t i o n of Eq. (24) by expanding U i n terms of S,V,N ,...N Eq. (15), and then closed the sum of squares, Eq. (16). In the process, we maintained the o r d e r i n g of the independent v a r i a b l e s as S_,V,Nj_, . . ,N . I f we had chosen t h e ^ r d e r as S_,Ni, . . .N ,V, then the numerator of the c o e f f i c i e n t of &Z +i, which was A +i, would have been R + i , as expressed by Eq. ( 7 ) . Thus, Eqs. (5) and (24) d i f f e r only i n the o r d e r i n g of v a r i a b l e s i n c l o s i n g the sum of squares and, hence, are of equal v a l i d i t y . The same statement a l s o a p p l i e s to 1

n

n

n

n

n

n

19.


MODELL

377

Eqs. (6) and (26). I t f o l l o w s immediately that there a r e n+2 e q u i v a l e n t forms of Eqs. (5) or (24) and (6) or (26), each formed by o m i t t i n g one of the n+2 v a r i a b l e s

, x

S_,V,NT_, . .. ,N . N

The general form can be s t a t e d i n terms of y =f (z^, . . ., z ) where y(°) i s U and z ^ , . . . , z i s any o r d e r i n g of the n+2 v a r i a b l e s jS,V,N ... ,N ( i . e . , m=n+2). The general form of Eq. (23) becomes m


m

L5

N

2

6 y

= V

( o )

1

6Z

1 1

n+1 V. JL + E •

9

V

-

i

(27)

6Z/ J

J =2 j-1 ( o )

y 12 y

( o )

y 22

21

f

where V, =

2k

y

(28)

k

'kl and y
J

H

rH

H

CO

25

o

II

25 CM O

>^

u

cd a

•H

II

CN CO CO

w

H

3

CO

25

CN 3-

we could t e s t the f i r s t c r i t i c a l i t y c r i t e r i o n by using Eq. (61): T

2

3

=

a n a

3

G?

=

i

s

!

3

N

a

A

p . k v

O

\

R

=

O) Y

4 4

G

' N

e

(2)

N

G

2

N

r

n

a

t

(1)

3 3

Y

2

t

2 2

Y

I

N

y

\

I

v

n

c

a

z

e

r

o

e

(o) l l

4 K

V °SS

)

'

(

6

5

)

(2) (3) That i s , we can t e s t f o r y ^ when i s indeterminate. As pointed out by Heidemann ( 4 ) , i n t e s t i n g the s t a b i l i t y of l i q u i d - l i q u i d c o e x i s t i n g phases i n a t e r n a r y system, there a r e c o n d i t i o n s w i t h i n the unstable r e g i o n where y^y vanishes ( i . e . , where yffl i s n e g a t i v e ) . Thus, when t e s t i n g transforms of lower order than y^ ^\ / > s t be sure that we have approached the c o n d i t i o n uAcfer wRicn y ( ) i s indeterminate from a r e g i o n of proven stability. ( >< ) For a b i n a r y system, Eqs. (42), (62) and (63) are truncated a f t e r the t h i r d term. I f we a r e d e a l i n g w i t h the s p e c i a l case i n which #2 " ^3 v a n i s h s i m u l t a n e o u s l y , then Ayy = ^2/^1 0 * ^NiNi P3/P2 i s indeterminate. T h i s s p e c i a l case i s known to occur when a b i n a r y mixture e x h i b i t s a z e o t r o p i c behavior i n the c r i t i c a l r e g i o n ; that i s , when the locus of a z e o t r o p i c c o n d i t i o n s i n t e r s e c t s the l o c u s of c r i t i c a l c o n d i t i o n s ( 5 ) . For such a system, the r e o r d e r i n g of v a r i a b l e s might take the f o l l o w i n g form. S t a r t i n g w i t h U = f(^,N^,N2,V), we would have v ( D = f (T,Ni,N ,V) and y ( ) = f (T,y]_,N2,V) . The truncated b i n a r y form of Eq. (62) would then become n

w

e

m u

n

l N /

y

n + 1

1 1 N

n + 1

a n a

=

a n c

=

2

2

2

« £ =U

s s

^

+

6Z

2 2

+ A ' ^

6Z

2 3

(66)

19.

385


MODELL

I f t h i s form i s s u c c e s s f u l i n removing the indeterminate c o n d i t i o n , then we should f i n d A ^ ^ ^ non-vanishing and A ^ ^ ®** I f i t i s u n s u c c e s s f u l , then another choice of reordered v a r i a b l e s should be pursued. As an a l t e r n a t i v e to r e o r d e r i n g the independent v a r i a b l e s , we can use Eq. (61) f o r the b i n a r y : Phase Equilibria and Fluid Properties in the Chemical Industry Downloaded from pubs.acs.org by NANYANG TECHNOLOGICAL UNIV on 06/10/16. For personal use only.

1

V

m

y

3

°

r

(2) (1) (o) 33 22 l l y

"\

3

V

=

( 6 7 )

y

V

^

U

K

'

SS

(

6

8

)

Thus, when a z e o t r o p i c behavior i s suspected i n the c r i t i c a l r e g i o n , we could t e s t Ayy ^ d G ^ N - I simultaneously or use the product of the two to determine the v a n i s h i n g of £ 3 . This l a t t e r procedure was a p p l i e d s u c c e s s f u l l y by Teja and K r o p h o l l e r ( 6 ) . a

C r i t e r i a i n Gibbs Free Energy The t h i r d c r i t e r i a put f o r t h by Gibbs, Eqs. (9) and (10), a r e the forms f r e q u e n t l y quoted i n the c u r r e n t l i t e r a t u r e . These c r i t e r i a , which were presented by Gibbs f o r G = f G ( T , P , N ] _ , . . . , N ) , a r e the s p e c i a l case of y ( ) = f ? 2 > Z 3 > • • • > Z n + 2 ) • The form o f f e r e d by Gibbs can be developed d i r e c t l y from the procedure of the l a s t s e c t i o n . Using Eq. (61) t o evaluate P » N

2

2

V

=

(1) (o) 22 l l y

2

y

(

K

6

9

)

J

Eq. (61) can be r e w r i t t e n as

2

k=3

*Note that f o r a pure m a t e r i a l , the f i r s t c r i t i c a l i t y c r i t e r i o n i s y^l) = 0 where y(Jp = Ayy or A ^ N O ) • A r e o r d e r i n g of v a r i a b l e s does not remove the c r i t e r i o n of mechanical i n s t a b i l i t y . On the other hand, f o r the b i n a r y , Ayy may v a n i s h w h i l e some other form of 22^ ( " * > N j N i ) should not v a n i s h . Thus, to t e s t mechanical s t a b i l i t y of a B i n a r y , one would have to evaluate Ayy. However, i n the general case the r e g i o n of mechanical i n s t a b i l i t y w i l l l i e w i t h i n the r e g i o n of m a t e r i a l i n s t a b i l i t y and, thus, the c o n d i t i o n of mechanical i n s t a b i l i t y w i l l be academic. I t i s always the v a n i s h i n g of the l a s t term i n Eq. ( 2 7 ) , or i t s e q u i v a l e n t , Eq. ( 4 2 ) , that d e f i n e s the l i m i t of i n t r i n s i c s t a b i l i t y . y

e

g

A

386


A N D F L U I D PROPERTIES

IN C H E M I C A L INDUSTRY

Each term i n the product of Eq. (70) can be expressed i n terms of d e r i v a t i v e s of y ( w by u s i n g the step-down operator [Eq. (20) of Beegle, et a l . , ( 3 ) ] :


Y

(k-1) kk

m y

(k-2) _ kk

y

(k-2) k(k-l)

2

(k-2) (k-l)(k-l)

/ y / y

K

'

± J

Repeated use of the step-down operator leads to the f o l l o w i n g equation: (2) 33

y

34

y

34

y

44

v y

( 2 )

3k

y

J> 2

y

v y

( 2 )

4k

y

3k

y

4k

( 2 )

v kk y

(k-D kk

(72) v 33

(2) 3(k-l)

( 2 )

y

v y

( 2 )

34

( 2 )

v 3(k-l) y

y

34

v y

( 2 )

44

( 2 )

v 4(k-l) y

y

..(2) *4(k-l)

..(2) (k-1)(k-1) y

S u b s t i t u t i n g Eq. (72) f o r each of the terms i n the product of Eq. (70), we o b t a i n : (2) ^33 n+1

=

( 2 )

y 34

^3k

^44

'4k

y

^34

(73)

( 2 )

( 2 )

y 3(n+l)

y4 ( n + l ) y

v

( 2 )

y

y

( n + l ) (n+1) Eq. (73) i s the determinant of Eq. (9) when y i s taken as G_. f o r a quarternary system (n=4), Eq. (73) i s ,(2)

Thus,

19.

387


MODELL


(74)

G,

G,

G,

(75)

G,

G,

G

N N 3

3

From t h i s a n a l y s i s , i t should be c l e a r that the c r i t e r i o n o f f e r e d by Gibbs i n Eq. (9) i s e q u i v a l e n t to the requirement that the product of Eq. (70) should v a n i s h . That i s , f o r a quarternary system, f o r example,

= 0

(76)

F o l l o w i n g the d i s c u s s i o n i n the l a s t s e c t i o n , t h i s form may be indeterminant i f y ^ v a n i s h e s . Thus, f o r a system i n which an azeotrope i n t e r s e c t s the c r i t i c a l l o c u s , we would expect Eq. (76) to be indeterminant. I n such cases, the a l t e r n a t e procedures outl i n e d i n the preceding s e c t i o n should be f o l l o w e d . Concluding Remarks In t h i s paper, we have developed the c r i t e r i a of c r i t i c a l i t y f o r multicomponent, c l a s s i c a l systems. The development u t i l i z e d Legendre transform theory described i n the f i r s t two papers of t h i s set (Beegle, et_ al., 1974a,b). I t was shown that the c r i t e r i a can be expressed i n terms of (n+1)-order determinants i n v o l v i n g second order p a r t i a l d e r i v a t i v e s of U [Eqs. (29) and ( 3 0 ) ] , s i n g l e second order p a r t i a l d e r i v a t i v e s of n - v a r i a b l e Legendre transforms of U [Eqs. (43) and ( 4 4 ) ] , or determinants of t w o - v a r i a b l e Legendre transforms [Eq.(73)]. The a l t e r n a t e s e t s presented by Gibbs a r e s p e c i a l cases of the g e n e r a l c r i t e r i a presented h e r e i n .

388

P H A S E EQUILIBRIA AND

F L U I D PROPERTIES IN

Abstract The general criteria of criticality for a classical thermodynamic system are developed in terms of the internal energy representation of the Fundamental Equation, U, (S,V,N,...,N). Alternative criteria in other variable sets are derived using Legendre transformations. The criteria of criticality as stated by Gibbs are shown to be special cases of the general criteria. The development utilizes extensive potential functions with mole number variables rather than mole fractions. 1


C H E M I C A L INDUSTRY

n

Notation A = t o t a l Helmholtz f r e e energy A-£_|_2 determinant d e f i n e d above Eq. (17) B = determinant defined i n Eq. (9) 8 = determinant defined i n Eq. (26) #k = determinant defined i n Eq. (28) E_j_2. determinant d e f i n e d i n Eq. (30) G_ = t o t a l Gibbs f r e e energy H = t o t a l enthalpy Nj = moles of component j n = number of components i n mixture P = pressure R +1 = determinant d e f i n e d i n Eq. (7) S = determinant d e f i n e d i n Eq. (8) Si = t o t a l entropy T = temperature IJ = t o t a l i n t e r n a l energy V_ = t o t a l volume y(k) k - v a r i a b l e Legendre transform Zk = e x t e n s i v e parameter d e f i n e d i n Eq. z^ = e x t e n s i v e independent v a r i a b l e =

=

n

n

=

(16)

Greek L e t t e r s Uj £j

= chemical p o t e n t i a l of component j = i n t e n s i v e independent v a r i a b l e that i s the conjugate coordinate of Z j

Literature Cited 1. Gibbs, J. W., "On the Equilibrium of Heterogeneous Substances," Trans. Conn. Acad. I I I , 108 (1876). 2. Modell, M., and R. C. Reid, Thermodynamics and Its Applications,


19.

MODELL


389

Ch. 5, 7, Prentice-Hill, Englewood C l i f f s , N. J. (1974). 3. Beegle, B. L., M. Modell and R. C. Reid: (a) "Legendre Transformations and Their Application in Thermodynamics," AIChE J., 20, 1194 (1974); (b) "Thermodynamic Stability Criterion for Pure Substances and Mixtures," Ibid., 1200 (1974). 4. Heidemann, R. A., "The Criteria for Thermodynamic Stability," AIChE J., 21, 824 (1975). 5. Rowlinson, J. S., Liquids and Liquid Mixtures, 2nd Ed., Ch. 6. Butterworth, London (1969). 6. Teja, A. S., and H. W. Kropholler, "Critical States of Mixtures in which Azeotropic Behaviour Persists in the Critical Region," Chem. Engng Sci., 30, 435 (1975).

Phase Equilibria and Fluid Properties in the Chemical Industry

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