Equations of State - American Chemical Society

1 Equations of State and Classical Solution Thermodynamics Survey of the Connections Michael M. Abbott and Kathryn K. Nass

Downloaded by LMU MUENCHEN on March 25, 2013 | http://pubs.acs.org Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch001

Department of Chemical Engineering and Environmental Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180-3590

Many contemporary researches in equation-of-statery focus on the representation of properties of liquid mixtures. Here, the connection with experiment is made through excess functions, Henry's constants, and related quantities. We present in this communication a review and discussion of the apparatus linking the equation-of-state formulation to that of classical solution thermodynamics, and illustrate the key ideas with examples. The c o r r e l a t i o n and prediction of mixture behavior are central topics i n applied thermodynamics, important not only i n their own right, but also as necessary adjuncts to the c a l c u l a t i o n of chemical and phase e q u i l i b r i a . Two major formalisms are available for representation of mixture properties: the PVTx equation-of-state formul a t i o n , and the apparatus of c l a s s i c a l solution thermodynamics. I t i s well known that the two formalisms are related, that a PVTx equation of state i n fact implies f u l l sets of expressions for the quantities employed i n the conventional thermodynamics of mixtures. Only with advances i n computation, however, has i t become possible to take advantage of these relationships, which are now used i n building and testing equations of state. The formulations d i f f e r i n at least two major respects: i n the choices of independent variables, and i n the d e f i n i t i o n s of s p e c i a l functions used to represent deviations of r e a l behavior from standards of " i d e a l i t y " . In the equation-of-state approach, temperature, molar volume, and composition are the natural independent variables, and the residual functions are the natural deviation functions. In c l a s s i c a l solution thermodynamics, temperature, pressure, and comp o s i t i o n are favored independent variables, and excess functions are used to measure deviations from " i d e a l i t y " . Thus translations from one formulation to the other involve both changes i n independent variables and conversions between residual functions and excess functions. 0097-6156/86/0300-0002$ 10.75/0 © 1986 American Chemical Society

In Equations of State; Chao, K., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

1.

ABBOTT AND NASS

3 Classical Solution Thermodynamics

Simple as a l l this sounds, we have found i t f r u s t r a t i n g not to have available a single source i n which the connections between the two formulations are neatly l a i d out i n purely c l a s s i c a l terms. P a r t i c u l a r l y vexing i s the lack of a f l e x i b l e but clean notation. To meet our own needs, we have synthesized a system of d e f i n i t i o n s and notation, p a r t i a l l y described i n the following pages. The system seems adequate for both research and classroom use.


Deviation Functions Rationale and Definitions. It i s rarely p r a c t i c a l to work d i r e c t l y with a mixture molar property M. For example, M may not be defined unambiguously, and may therefore not admit, even i n p r i n c i p l e , d i r e c t experimental determination. Thus, neither U nor S nor H nor G i s defined at a l l , i n the s t r i c t sense of the word. Both U and S are primitive quantities, and H and G are "defined" i n terms of one or both of them. Moreover, property M by i t s e l f may not admit physi c a l interpretations, except i n a loose sense (e.g., entropy as a "measure of disorder"). For these and other reasons, one finds i t convenient to introduce such quantities as residual functions and excess functions. These quantities, themselves thermodynamic prope r t i e s , are examples of a general class of functions which we c a l l deviation functions. Deviation functions represent the difference between actual mixture property M and the corresponding value for M given by some model of behavior: M(deviation) = M(actual) - M(model) The choice of a model i s to some extent arbitrary, but to be useful the model must have certain a t t r i b u t e s . I t s molecular implications should be thoroughly understood, so that deviation functions defined with respect to i t can be given clean interpretations. Real behavi o r should approach model behavior i n well-defined l i m i t s of state variables or substance types, so that the deviation functions have unambiguous zeroes. F i n a l l y , to f a c i l i t a t e numerical work, i t i s desirable that the properties of the model be capable of concise a n a l y t i c a l expression. Once a model i s chosen, the conditions at which the comparison ( r e a l vs. model) i s made must be s p e c i f i e d . There are several p o s s i b i l i t i e s here, but two are p a r t i c u l a r l y f e l i c i t o u s . Thus, we may define deviation functions at uniform temperature, pressure, and composition: M° = M -

od

rf" (T,P,x)

(1)

Here, the notation signals that mixture molar property for the model i s evaluated at the same T,P, and composition as the actual mixture property M; superscript (capital) "D" i d e n t i f i e s the devia t i o n function as a constant - T,P,x deviation function. Alternat i v e l y , we may define deviation functions at uniform temperature, molar volume (or density), and composition:


4

EQUATO INS OF STATE: THEORIES AND APPLICATIONS mod, M (T,V,x)

M

M

(2)

mod Here, M i s evaluated at the same T,V, and composition as the actual mixture property M; superscript (lower-case) "d" d i s t i n guishes this class of functions from that defined by Equation 1. The two kinds of deviation function are related. Subtraction of Equation 1 from Equation 2 gives M

d

mod, M (T,V,x)

D mod, - M - M (T,P,x)

v


whence we find that

Here, pressure P i s the pressure for which the mixture molar volume of the model has the same value V as that of the r e a l solution at the given temperature and composition. According to Equation 3, deviation functions and M are i d e n t i c a l for those properties M for which M * i s independent of pressure. D

moc

Residual Functions. ideal-gas mixture: m

M °

d

The simplest model of mixture behavior i s the

= M

ig

Deviation functions defined with respect to the ideal-gas model are c a l l e d residual functions, and are i d e n t i f i e d by superscript R or r . Thus, as special cases of Equations 1 and 2, we define i

(4)

i

(5)

M

R

= M - M 8(T,P,x)

M

r

= M - M 8(T,V,x)

and

Residual functions M and M are related by Equation 3, with the assignments mod = i g and P* = RT/V. Thus R

r

(6)

Ideal-gas properties U*§, H*8, of pressure. Hence M

R(

r .M

M

C^S, and

are a l l independent

(7)

U,H,C ,C ) v

p

i

On the other hand, ideal-gas properties S^S, A*8, and G 8 are functions of pressure. In p a r t i c u l a r ,


1.

ABBOTT AND NASS

5

Classical Solution Thermodynamics

R P 1£ 3P

RT P

T,x

and hence, by Equation 6, (8)

R£nZ A


G

r

G

R

+ RTfcnZ

(9)

+ RT£nZ

(10)

where Z = PV/RT i s the compressibility factor. Expressions f o r either M or M are found from a PVTx equation of state by standard techniques: see e.g. Van Ness and Abbott (1_)« If the equation of state i s e x p l i c i t i n pressure, then T,V (or molar density p), and composition are the natural independent variables and the M are the natural residual functions. If the equation of state i s e x p l i c i t i n volume, then T,P, and composition are the natural independent variables and the M are the natural residual functions. Tables I and II summarize formulas f o r computing the M from a pressure-explicit equation of state, and the M from a volume-explicit equation of state. Conversion from M to M , or vice versa, i s done by Equations 7 through 10. Note that V and P are i d e n t i c a l l y zero. R

r

r

R

r

R

R

r

r

R

r

Residual Function A : a Generating Function. Most r e a l i s t i c equations of state are e x p l i c i t i n pressure; T,V, and composition are the natural independent variables. These are also the canonical variables f o r the Helmholtz energy A, so the constant - T,V,x r e s i d ual Helmholtz energy A plays a special role i n equation-of-state thermodynamics. I t can be considered a generating function, not only f o r the other constant - T,V,x residual functions, but also f o r the equation of state i t s e l f . The relevant working equations assume a pretty symmetry when the independent variables are chosen as reciprocal absolute temperature x ("coldness" = T ~ l ) , r e c i p r o c a l molar volume p (molar density = V~*), and composition. They are summarized i n this form i n Table I I I . Application of these formulas may be i l l u s t r a t e d by a simple example. We choose f o r this purpose the van der Waals equation of state, f o r which r

r

A (vdW) = - RT£n(l-bp) - a

p

(33)

where parameters a and b depend on composition only. The residual pressure P (= P-pig = P-pRT) i s found from Equation 27: r

2

r

P (vdW)

bp RT

2

so the equation of state i s


6

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

P(vdW)

pRT 1-bp

(34)

ap

The other residual functions follow from Equations 28 through 32, Two results are r

(35)

r

(36)

S (vdW) = R£n(l-bp)


U (vdW) - - ap

Equations 35 and 36 support the common interpretations of the hard-sphere (repulsive) term as representing an "entropic" contribut i o n to the equation of state, and of the van der Waals "a" as an energy parameter. The material of this section can i n fact be taken as a point of departure for the development of a c l a s s i c a l l y inspired generalized van der Waals theory, motivated by Equations 33 through 36, but unrestricted by the assumptions attendant to the o r i g i n a l van der Waals equation of state (2_). Excess Functions, The conventional standard of mixture behavior for condensed phases i s the ideal solution. Deviation functions reckoned against this model are c a l l e d excess functions, and are i d e n t i f i e d by superscript E or e. We define id

(37)

id

(38)

M - M (T,P,x) and M - M (T,V,x)

where superscript i d i d e n t i f i e s the i d e a l solution. Equations 37 and 38 are special cases of Equations 1 and 2, with the assignment mod = i d . Excess functions M and M are related by e

M

6

= M

E

+ /

3Ml*T,x

(39)

dP

P

which i s a special case of Equation 3. Unlike the ideal-gas case,^ no simple general closed-form expression can here be written for P . By d e f i n i t i o n , P i s i n this case the pressure for which the i d e a l solution has the same molar volume V as the real solution at the given temperature and composition. Since

t h i s pressure must be found as a solution to the equation Jx V (T,P*) - V(T,P,x) i 6

(40) E

C l e a r l y , numerical r e l a t i o n of M to M v i a Equations 39 and 40 requires equation-of-state information, both for the real pure com-


1.

ABBOTT AND NASS

Table I.

Residual Functions from a Pressure-Explicit Equation of State

P


1 Classical Solution Thermodynamics

r

(11)

- pRT(Z-l)

„r

_ 2

P

RT

=

;

(

|Z

d^ (12)

H

r

- -RT

Z

^

+ RT(Z-l)

(13)

dp. p

A

r

G

r

- RT / (Z-l)

(14)

dp. (15)

= RT / ( Z - l ) ^

+ RT(Z-l)

(16)

C„ =

C

r p

= C

(17)

r v

-R

+

R[

Z

+

T(|f)

p ) X

]

2

[Z + p ( { f ) . ] T

_ 1

x


(18)

8

EQUATO INS OF STATE: THEORIES AND APPLICATIONS Table I I .

Residual Functions from a Volume-Explicit Equation of State


(19)

"

Rt2

Z

/ (H) x f " ^ ' »

(20)

i UTJP.X P

(21)

Pi

RT

R

T

+

" / [ (|f)p, ^

f

X

A*

=

(22)

RT / ( Z - l ) ! ^ - RT(Z-l)

(23)

RT / ( Z - r f

(24)

P

2

-RT ; [T(^)P. - (H)P,] f X

R .

C

+ R

R[Z

+

T (

||

) p ) x ]

(25)

x

2

[ z

.

p j!) j-i (

T j


(26)

1.

9

ABBOTT AND NASS


]

Table I I I . Residual Functions from A

2

r

=Ρ

r

_


r

S

"

„Γ

G

r

(|4 r

2 ,3A , τ

(ϊΓΪρ,χ

|~3( Α ) 1 L 3τ Jp,x Γ

=

τ

l~3(pA )] r

.

Γ r H

=

r A

Γ

(fM

+ τ f-^-) + ρ 3 τ -'ρ,χ *-3ρ τ,χ ν

c{

-

-

,

μ

'

;

Μ


10


e

ponents and for the r e a l mixture. Hence the r e l a t i o n between M and M i s not as "clean" as that between M and M . Rough closed-form approximations to Equation 39, appropriate for applications to condensed phases, may be found however. We write Equation 39 as E

r

M

e

» M

+ (!|H

E

T)X

R

(P-P*)

(41)

where the derivative i s evaluated at the pressure Ρ of the r e a l mix ture. An expression for Ρ follows from Equation 41 by the assign ment M = V, because V = 0 i d e n t i c a l l y . Thus we f i n d that


e

p

"

p

< 4 2 )

Τ ^ Λ

'

J ι ι i where

i s the isothermal compressibility of pure i : 1

3

V

i

A l l quantities on the right side of Equation 42 are evaluated at pressure P. According to Equation 42, the sign of V determines whether Ρ i s less than or greater than P. Combination of Equations 41 and 42 gives e

i which i s the required approximation to Equation 39. Particular cases of Equation 43 are generated on s p e c i f i c a t i o n of M and of the corresponding derivative (3Μ^^/3Ρ)χ . Table IV summarizes expressions f o r this derivative i n terms of the volumetric proper t i e s of the species composing the mixture. To demonstrate i t s application, l e t us take M = A. Then, by Equations 43 and 48 we f i n d that x

A

e

« A

E

G

E

- A

E

+ PVΕ ]

But +

PV

E

and hence we have rationalized the approximation A

e

« G

E

a result frequently used i n molecular modeling of the constant Τ,Ρ,χ excess Gibbs energy.


1.

A B B O T T A N D Ν ASS

11


. ,id 3M — x


Table IV.

Expressions for

;

^3P

V

T,x

V

" [¥i i

p

u

x

v

(44)

T

I in

111]

. . (i2L]

9n

\ i>X',Y',n.

/ax 1^1 9n

fâlL] a

^ ' χ ' , χ \ Vx',X,n.

f

- - ^ require e x p l i c i t expressions for the mixing rules for parameters a and b.


1.

ABBOTT AND NASS

21


Mixture Fugacity Behavior


Fugacity Coefficients, A c t i v i t y C o e f f i c i e n t s , and Henry's Constants. Component fugacity c o e f f i c i e n t s are readily obtained from a PVTx equation of state. For developing and testing equations of state for phase-equilibrium applications, however, i t i s sometimes useful to deal d i r e c t l y with quantities conventionally used for description of the l i q u i d phase, e.g., a c t i v i t y c o e f f i c i e n t s and Henry's constants. We review i n the following paragraphs the connections among these measures of component fugacity behavior, and i l l u s t r a t e how they are determined from pressure-explicit equations of state. The fugacity c o e f f i c i e n t i s defined as

(89)

where f^ i s the fugacity of species i i n solution. coefficient is

ι

x t 1

The a c t i v i t y

1

where f° i s the standard-state fugacity. Two standard states are popularly employed: Lewis-Randall ("Raoult's-Law") standard stat< for which f? i s the fugacity of pure i at the mixture Τ and P,

f°

±

(91)

(LR) - f .

and Henry's-Law standard states, for which f J (HL) =

%

where Henry's "constant"

(92)

±

. i s defined as

(93)

the l i m i t being taken at the mixture Τ and P. If we write Equation 90 as

i

Ρ

Ρ

then we see that a l l that i s required to "convert" a fugacity coef f i c i e n t to an a c t i v i t y c o e f f i c i e n t i s an expression for the r a t i o


22

f°/P.


For Lewis-Randall standard states, we have

where φ± i s the fugacity c o e f f i c i e n t of pure i . Thus the conven t i o n a l Lewis-Randall a c t i v i t y c o e f f i c i e n t i s

(94)


For Henry*s-Law standard states, we have

1

±

1·

— Ρ

lim — x. χ +0 ι n

But

—

lim X

i*°

=

Ρ l i m φ.

X j L

X

±*°

1

where φ ^° i s the fugacity c o e f f i c i e n t at i n f i n i t e d i l u t i o n .

9f.

= ?l °°

Thus

(95)

±

1

and the Henry s-Law a c t i v i t y c o e f f i c i e n t

is

(96)

Conversion from fugacity c o e f f i c i e n t s to a c t i v i t y c o e f f i c i e n t s and Henry's constants i s thus straightforward. One needs i n addi tion to the component fugacity c o e f f i c i e n t one or another of i t s l i m i t i n g values, v i z . , φ

ί

φ

1

Ξ

Ξ

lim

φ

±

lim φ^^ x^+0

For a pressure-explicit equation of state, both of these are found as l i m i t s of Equation 86.


1.

ABBOTT AND NASS


23

Consider as an example the van der Waals equation of state, for which φ^ i s given by Equation 88.

For pure i , this equation yields

as a special case

b

^ i

(

v

d

W

)

i i

p

2

i

a

i i

p

i

IP

=T ^ T ii i

"

( 9 ? )

^ " " i l P i * !


H

where the doubly-subscripted parameters refer to pure i . Equations 94, 88, and 97 when combined produce an expression for the Lewis-Randall a c t i v i t y c o e f f i c i e n t implied by the van der Waals equation. Evaluation of φ ι " (and hence of or y± ) requires a l i t t l e more care, because the state of i n f i n i t e d i l u t i o n for a species i n a multicomponent mixture can i n p r i n c i p l e be defined i n many ways. The natural d e f i n i t i o n of this state i s as that state for which x^ approaches zero as the i - f r e e mole fractions x^ remain constant. (H. ere, χ'. Ξ X./^X,

J 88 yields

J

£ηφ

(vdW) =

±

, where j,k ψ i.)

By this d e f i n i t i o n , Equation

k

b V ^

(a'+â^V gj±

,

in ^22> 12> * 1 2 The f i r s t four are found from information on the pure components; the interaction parameters a\2 and b^2 estimated from combining rules or, pre ferably, from mixture data. Since the application i s to the representation of excess func tions for l i q u i d mixtures, i t i s reasonable to determine the and b±± from data on pure l i q u i d s . Various combinations of properties are employed for this purpose; we choose here the liquid/vapor a

a

r

at

anc

b

e

e

saturation pressure P^ , the molar density

of the saturated

ον l i q u i d , and the molar heat of vaporization ΔΗ^ .

ον A molar property change of vaporization ΔΜ^ of pure f l u i d i i s defined as


26


AMj

Ε M^(T,P* ) - M*(T,P* )

V

at

at

(108)

where the terms on the right side denote molar properties of pure i as saturated vapor and as saturated l i q u i d . Since temperature and pressure are uniform, ΔΜ* Τ,Ρ,χ

i s just a difference between constant -

residual functions: ΔΜ

Aν

(109)


Here, the terms on the right are constant - Τ,Ρ,χ residual functions for pure i as saturated vapor and as saturated l i q u i d : V

M*'

at

8

at

at

8

at

= M^(T,P* ) - M j ( T , P ^ )

M*>* = M*(T,P* ) - M j ( T , P ^ )

Residual functions are readily determined from a PVTx equation of state by procedures reviewed e a r l i e r . Hence, by Equation 109, so also are property changes of vaporization. By Equation 109, Consider the molar heat of vaporization Δ# £v i

Η

ΔΗ'

i

" i

and, by the d e f i n i t i o n of H, H* = U* + PvJ where, by Equation 7, ,R U i

= u;i

Thus, by the l a s t three equations,

(110)

Equation 110 expresses the molar heat of vaporization i n a form con venient for use with a pressure-explicit equation of state. Constant - T,V,x residual i n t e r n a l energies are found from Equation sat α ν i s related to p. and ρ by the equation of state 12, and Ρ itself. For the van der Waals equation of state, we have by Equations 110, 36, and 34 that 1

1

1


1.

ABBOTT AND NASS


27

a ( p j " ρ*) + ?\ \-\ - -j-) Λ

V

AHj (vdW) =

(111)

u

Pi

Pi

where Ρ

1-b

a (pp

112

< >

tl

l l P l

and

;Rv T

Pi

1

b

a^Cpp"

(113)

N

v

1-K 1


, v 92

K

pSat

p

ii i

Given experimental values f o r P*

, p*, and ΔΗ~

at specified T, one

solves Equations 111, 112, and 113 f o r p^, a ^ , and b ^ .

The value

γ

of p^ so obtained i s merely an intermediary

quantity; because the

equal-fugacity requirement f o r liquid/vapor equilibrium i s not invoked, p^ i s not necessarily the "true" saturation vapor density implied by the equation of the state at the specified T. (The equal-fugacity requirement would provide a fourth equation; vapor pressure P^ would then be treated as an unknown, to be determined at

along with p^, a ^ , and b ^ . ) The above-described procedure when applied separately to pure 1 and to pure 2 provides values for ^H> 22> * ^22' ^° ^*- * a\2 and b j ^ assume the a v a i l a b i l i t y of data f o r H and V , each at a single composition. The working equations follow from Equations 60, 34, and 102, applied to the l i q u i d phase: a

w

a n c

e

n(

E

E

E

E

H (vdW) - - ( a p - x a p - x a p ) + PV (vdW) 1

PivdW) =

-

E

V (vdW) - ρ"

1

b =

x

b

i n

+

x

a

b

ap

- x^"

2 2 a - x ^ a ^ + 2 22 x

1 1

2 22

+

+

2 x

2 x

1

1

2

2 2

2

(34)

- χ ρ 2

x

Χ

(115)

2

a

l 2 12 x

(114)

2

(116)

b

i 2 i2

(117)

Here, l i q u i d molar densities p i , p2> and ρ are found as solutions to Equation 34, under appropriate assignments f o r the equation-of-state parameters. Agreement with experiment i s forced f o r H^ and V at the single states from which a^2 * 12 determined; calculations E

a n (

b

a

r

e


28


of H and V at other states, or of other excess functions at any states, constitute extrapolations. When compared with experiment, these extrapolations provide tests of the van der Waals mixing rules and (to a lesser degree) of the a b i l i t y of this equation of state to represent properties of the l i q u i d phase. The l i t e r a t u r e abounds with such comparisons and i t i s not our purpose to survey them here. Instead we show i n Figure 1 predicted values of the scaled excess functions V /xiX2» G /x^X2RT, H /x^X2RT, S /xiX2R> and Cp /x x R for the system argon(l)/krypton(2). Purecomponent parameters were estimated as described above from satura t i o n data compiled by Vargaftik (5_). Parameters a i 2 and b ^ ^ estimated from equimolar values of H and V at zero pressure, as given by the correlations of Lewis et a l . (6_) for H at 116.9 Κ and Da vies et a l . (7_) for V at 115.77 K. Both pressure and temperature effects are i l l u s t r a t e d i n Figure 1. P a r t i c u l a r l y to be noted are the e s s e n t i a l equivalence of the 0 bar and 1 bar isobars at 120 K. This j u s t i f i e s the frequent use of the zero-pressure l i q u i d state i n calculations of excess functions from equations of state. For example, one obtains for the van der Waals equation at zero pressure an e x p l i c i t expression for the l i q u i d density: E

E

E

E

E

1

2

w

E

r

e

e

E


e

P*(vdW;P=0) - ^ ( 1 +

7

f ^)

Moreover, the expression for the excess enthalpy s i m p l i f i e s to H (vdW;P=0) - - ( E

a

p

-

ï

x

a 1

p

i i i

)

Next we examine the composition dependence of Henry's constant ^1;2,3 f ° solute species 1 i n a mixed solvent containing species r

2 and 3.

Here, the solute-free mole fractions x^ and x^ are

appropriate measures of composition.

Subtleties of behavior are

n i c e l y displayed through the "excess" quantity ^ r r ^ . 2 3» defined as n

n

X

x

( 1 1 8

* ^ ; 2 , 3 -= *-«%Z,3 " 2 ^ 1 ; 2 " 3 ^ 1 ; 3 where'^^2

a n c

* ^1.3

solvents 2 and 3. T,P,

a

r

e

>

Henry's constants for species 1 i n pure

The comparison i n Equation 118 i s made at uniform

and composition, and £η^/ί „ i s i d e n t i c a l l y zero i f the three * ' Ε species form an i d e a l solution; however, £n^f. ^ i s not a true excess function as defined by Equation 37, because ζηΊψ^^ 3 ia mixture molar property. '' According to Equation 95, Henry's constant i s proportional to the fugacity c o e f f i c i e n t at i n f i n i t e d i l u t i o n . Combination of Equations 118 and 95 thus y i e l d s the general result 9

# 9

s


n

o

t




I.

Figure 1 (A,B). Scaled van der Waals excess functions f o r l i q u i d mixtures of argon( 1 ) and krypton(2). (The 10,000-bar isobar i s not shown i n A) .


29

30

EQUATO INS OF STATE: THEORIES AND APPLICATIONS

G /x^x^RT v s . x^ P=0 bar_

a t T = 120 Κ

"p^Too"


P=1000

P=10000

0.0

0.2

0.4

0.6

0.8

1.0

(C)

Figure 1 (C,D). Scaled van der Waals excess functions f o r l i q u i d mixtures of argon( 1 ) and krypton(2).


1.

A B B O T T A N D ΝA S S

31


0.30

0.25

0.20


0.15

0.10 P=10000 0.05

H /x^x^RT v s . x^ at T = 120 Κ

0.00 0.0

0.2

0.4

0.6

0.8

1.0

(Ε) 0.20
33> 33> 12> 12> 13> 13> 23> 23* Ideally, the purecomponent parameters would be estimated from liquid-phase data (e.g., as i n the excess-function example), and the interaction para meters from liquid/vapor e q u i l i b r i a and g a s - s o l u b i l i t y data. We adopt f o r this example a more straightforward approach. Parameters an * ii estimated from c r i t i c a l constants Τ ^ and V ^ v i a the c l a s s i c a l relations b

a

b

a n c

a

b

b

a

r

a

b

a

a

n

d

b

e

0

c


1.



a,, = ii

*jT RT 8

35

V c.^

(125)

(126)

Interaction parameters are determined from the conventional com bining rules

a

ij

b

- "-'ijX'ii-j/' (1

)(b

+

2

( 1 2 7

2

>

(128)

ij - ^i ii V'


3

where parameters k^j and £jj are pure numbers, of absolute value less than unity. We wish to i l l u s t r a t e the effects of varying •Ε k . and £ on the "excess" quantity fcn^f-, ~, i . e . , the sens i t i v i t y of mixed-solvent Henry's constants to the numerical values of the interaction parameters. The parametric study i s done for a simulated ternary system at 300 Κ and 1 bar, i n which hydrogen(l) i s the solute, and n-heptane(2) and n-decane(3) compose the mixed solvent. Parameters for the pure f l u i d s are obtained from Equations 125 and 126 and parameters a23 and b23 are fixed once and for a l l by setting k23 = £23 0 Equations 127 and 128. Assignment of numerical values to 9

=

k^

3

and £ ^

then permits c a l c u l a t i o n of An^f-j^ 3 from Equations 119

through 124. Numerical results are displayed on Figure 2. Figures 2A and 2B i l l u s t r a t e the effects of independently varying the energy interac t i o n parameters; here, we have set £ 1 2 * &13 ~ 0· Figures 2C and 2D s i m i l a r l y show the effects of varying £ 1 2 and £ 1 3 , with k\2 ^13 0. The results confirm that mixed-solvent Henry's constants, l i k e excess functions for l i q u i d mixtures, can serve as probes for assessing mixing rules and combining rules for PVTx equations of state. =

=

Closure Connections between the PVTx equation-of-state formalism and the conventional apparatus of c l a s s i c a l solution thermodynamics are cleanly exposed through a few unifying concepts, e.g., generalized deviation functions, generalized p a r t i a l properties, and component fugacity c o e f f i c i e n t s . We have found the notion of p a r t i a l equation-of-state parameters to be p a r t i c u l a r l y helpful, because i t allows one to postpone questions r e l a t i n g to composition dependence u n t i l they r e a l l y need to be addressed. Much of the substance of this communication resides i n d e f i n i tions and generalizations, and i n the summaries of working formulas collected i n the tables. To keep the paper to a reasonable length, we have provided examples and i l l u s t r a t i o n s for but a single






Figure 2 (C,D). van der Waals Henry's constants at 300K and 1 bar, f o r hydrogen( 1 ) i n mixed solvents containing n-heptane(2) and n-decane(3).


38


equation of state: the van der Waals equation. The p r i n c i p l e s and procedures are of course easily applied to other, more r e a l i s t i c equations of state. Exercises i n synthesis necessarily build on precedents, i n this case too diffuse and numerous to c i t e i n d e t a i l . We are however pleased to acknowledge as general sources of i n s p i r a t i o n the published researches of P.T. Eubank, K.R. H a l l , the late A. Kreglewski, M.L. McGlashan, K.N. Marsh, J. Mollerup, S.I. Sandler, R.L. Scott, K.E. Starling, and J. V i d a l . To these and to other equation-of-state enthusiasts we acknowledge our indebtedness.


Acknowledgments This work was p a r t i a l l y supported by the National Science Foundation under Grant No. CPE-8311785. K.K.N, i s pleased to acknowledge support provided by an AAUW Educational Foundation Fellowship, i n the form of Conoco Engineering/Mobil Awards. Eugene Dorsi assisted with the calculations. Legend of Symbols a,b

=

parameters i n van der Waals equation of state

a^

=

a c t i v i t y of species i

A

-

molar Helmholtz energy

Cp,Cy

=

molar heat capacities

f^

=

fugacity of pure i

f^

-

fugacity of species i i n solution

f°

=

standard-state fugacity of species i

G

=

molar Gibbs energy

H

=

molar enthalpy

=

Henry's constant for species i

=

arbitrary molar, or intensive (e.g. Μ = Ρ ) , property

=

constant - T,V,x

=

constant - Τ,Ρ,χ deviation function

=

constant - T,V,x

=

constant - Τ,Ρ,χ excess function

M d M

deviation function

D M e M

excess function

Ε M Μ

Γ

M

s

constant - T,V,x constant - Τ,Ρ,χ

residual function residual function

M*"

=

t o t a l property Ξ nM

M^

=

generalized p a r t i a l property

M^

=

constant - T,P p a r t i a l property

»

constant - T,V p a r t i a l property

ΔΜ

-

constant - Τ,Ρ,χ

property change of mixing


1.

A B B O T T A N D Ν ASS 0

γ

ΔΜ^

=

molar property change of vaporization of pure i

η

•

amount of substance ("mole number")

Ρ pSat

• _

pressure liquid/vapor saturation pressure of pure i

R

=

universal gas constant

S

=

molar entropy

Τ

-

absolute temperature

U V



= a

molar internal energy molar volume

χ

-

mole f r a c t i o n

X,Y

-

arbitrary intensive variables

Ζ

=

compressibility factor Ξ PV/RT

Greek Letters 3

=

γ κ

= =

π ρ

volume expansivity = V ^ a V / a T ) r ,x a c t i v i t y c o e f f i c i e n t of species i isothermal compressibility = -V (3V/3P) ι ,x

=

absolute a c t i v i t y of species i

=

chemical potential of species i

= s

arbitrary equation-of-state parameter molar density

τ

=

"coldness" = Τ *

=

fugacity c o e f f i c i e n t of pure i

φ^

-

fugacity c o e f f i c i e n t of species i i n solution

Superscripts id

=

denotes an i d e a l - s o l u t i o n property

ig

•

denotes an ideal-gas property

mod

=

denotes a model mixture property

«

=

denotes a property of a species at i n f i n i t e d i l u t i o n

Literature Cited 1. Van Ness, H.C.; Abbott, M.M. "Classical Thermodynamics of Nonelectrolyte Solutions: With Applications to Phase Equilibria," Appendix C, McGraw-Hill, New York, 1982.


39


40

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

2. Abbott, M.M.; Prausnitz, J.M. "Generalized van der Waals Theory: A Classical View," manuscript in preparation. 3. Reis, J.C.R. "Theory of Partial Molar Properties," J. Chem. Soc., Faraday Trans. II 1982, 78, 1595. 4. Abbott, M.M. "Higher-Order Partial Properties," seminar presented at the University of California, Berkeley, 21 Sept. 1983; unpublished notes. 5. Vargaftik, N.B. "Tables on the Thermophysical Properties of Liquids and Gases," 2nd Edition, Wiley, New York, 1975. 6. Lewis, K.L.; Lobo, L.Q.; Staveley, L.A.K. "The Thermodynamics of Liquid Mixtures of Argon + Krypton," J. Chem. Thermodynamics 1978, 10, 351. 7. Davies, R.H.; Duncan, A.G.; Saville, G.; Staveley, L.A.K. "Thermodynamics of Liquid Mixtures of Argon and Krypton," Trans. Far. Soc. 1967, 63, 855. RECEIVED November 5, 1985


Equations of State - American Chemical Society

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