Thermodynamics of associated solutions: Henry's constants for nonpolar solutes in water

Fluid Phase Equilibria, 17 (1984) 303-321 Elsevier Science Publishers B.V., Amsterdam

303 -

Printed

in The Netherlands

THERMODYNAMICS OF ASSOCIATED SOLUTIONS: HENRY’S CONSTANTS FOR NONPOLAR SOLUTES IN WATER YING

HU *, EDMUND0

JOHN

PRAUSNITZ

AZEVEDO

**, DOROTHEA

LODECKE

*** and

Molecular and Materials Research Division, Lawrence Berkeley Laboratory Engineering Department, Universit)a of California, Berkeley, CA 94720 (U.S.A.) (Received

September

27, 1983; accepted

in final form

February

and

Chemical

2, 1984)

ABSTRACT Hu,

Y., Azevedo, E., Ludecke, solutions: Henry’s constants 303-321. A systematic

derivation

D. and Prausnitz, J., 1984. Thermodynamics for nonpolar solutes in water. FIuid Phase

is presented

mixture whose nonideality derivation, applicable to all chemical equilibrium constants a and b. Attention is given simplifying assumptions that to correlate Henry’s constants

for the Helmholtz

energy

of associated Equilibria, 17:

of a van der Waals

fluid

is ascribed to both chemical and physical interactions; this fluid densities, leads to an equation of state which contains in addition to the customary physical van der Waals constants to the need for simplifying assumptions and to the variety of can lead to useful results. A particular equation of state is used for nonpolar solutes in water over a wide temperature range.

The correlation, however, is only partly successful, because a one-fluid van der Waals theory of mixtures is not satisfactory for mixtures containing molecules that differ appreciably in size, especially

in the dilute

region,

Many authors have explained fluid nonideality by supposing that a fluid contains not only the apparent molecular species but, in addition, polymers of such species which are in chemical equilibrium; for example, for acetic acid vapor, deviations from ideal-gas behavior at low or moderate pressures are readily explained by assuming that acetic acid dimerizes, in part, to form acetic acid dimers in addition to acetic acid monomers. Similarly, the thermodynamic properties of liquid solutions of hydrocarbons and alcohols can be correlated by postulating that alcohols polymerize, in part, to form dimers, trimers, etc., in addition to alcohol monomers; finally, the thermody* East China Institute of Chemical Technology, ** Instituto Superior Tecnico, Lisbon, Portugal. *** Institut

fir

Metallphysik,

0378-3812/84/$03.00

Gbttingen,

0 1984 Elsevier

Shanghai,

China.

W. Germany.

Science

Publishers

B.V.

304

namic properties of liquid solutions of chloroform and acetone are readily interpreted by assuming that these two species solvate to form a 1:l chloroform-acetone complex. When these “chemical” theories of nonideal fluids were first proposed early in this century (Dolezalek, 1908) it was customary to assume that the “true” species form an ideal solution; in other words, all deviations from ideal behavior were ascribed to chemical effects. It was not until about 1950 that some authors also took into account physical interactions between the true species (Scatchard, 1949; Kretschmer and Wiebe, 1954). However, theories which used both chemical and physical contributions to nonideality were confined either to gases at moderate densities (second virial coefficients) or to liquid solutions containing only subcritical components at conditions remote from critical. In 1976 Heidemann and Prausnitz (1976) proposed an equation of state, applicable to all fluid densities, for fluids which exhibit chemical and physical contributions to nonideal behavior; Heidemann’s work uses the continuous-association model to account for chemical effects on nonideahty. In the work of Rupp (1976) and Baumgaertner et al. (1979), dimerization and trimerization were considered. Since then, other authors, notably Gmehling et al. (1979), Baumgaertner et al. (1980) and Wenzel et al. (1982), have proposed equations of state combined with appropriate association models, coupled with a variety of simplifying assumptions. In this work, we present a systematic derivation, applicable to all fluid densities, which is based on a model for the Helmholtz energy; we indicate several choices of simplifying assumptions. We then illustrate the applicability of a chemical-physical equation of state in the correlation of Henry’s constants for nonpolar solutes in water over a wide temperature range. THERMODYNAMIC

FRAMEWORK

Consider a binary mixture of components A and B. In the absence of any association or solvation, the number of moles of A is nAO and the number of A and B can associate moles of B 1s n BO.We assume that both components continuously to form linear polymers (dimers, trimers, etc.) according to A, + A, = A, A,+A,=A,

B, + B, = B, B, + B, = B,

Ai+A,

B,,+B,

=A;+,

=B,+r

where the index i goes from unity to infinity. As shown by the double arrows, it is assumed that the molecular species indicated are in chemical

30.5

equilibrium,

characterized by equilibrium constants KA2,. . . ,K, ,+I and KW. -* J$*, which depend only on temperature. For simplicity, we assume that, for each component, all association constants are equal and independent of i, and may be designated by K, and K,, respectively. For solvation, there are many stoichiometric possibilities: the general formula for a solvated species is A, B,, where the indices n and m can be any integers between one and infinity. To simplify, we assume that solvation equilibria are of the form A, + B, AB+AB

=AB =(AB),

(AB), + AB = (AB)j+, where the index i goes from one to infinity. Further, we assume that one equilibrium constant, KIB, applies to all solvation equilibria, except the first one; the latter’s equilibrium constant is KAB. Given this physical picture, our aim is to establish an expression for the Helmholtz energy A of a mixture at temperature T and volume I/ containing monomers, associated species and solvated species which are in chemical equilibrium. In this work, we do not assume that this mixture is ideal; subject to simplifying assumptions, we take into account physical interactions (repulsion and attraction) between the chemically equilibrated molecular species. To obtain the desired expression for A we consider four steps, all at temperature T. Our reference states are pure component A and pure component B, each at T and 1 bar, in the ideal-gas state. In this reference state, the pure components A and B exist only as monomers. Step I. Formation of pure associated and soluated species in the ideal-gas state at T and 1 bar. For this process, 00 Cc

where Agi,, , is the standard Gibbs energy of formation of one mole of species A ;+, . Similar definitions hold for Agi,, , and AggB),. The number of moles of species j is designated by ni. Step II. The pure monomers and the pure associated and solvated species at T and I bar in the ideal-gas state are mixed to form an ideal-gas mixture at T and V. For this process, AA,, = xn,RT

ln( n,RT/V)

(2)

306

where the gas constant R in the logarithm is chosen to give units of bars. We now introduce nr, the total number of moles, given by nT

nA, + i=l

:

nB, + ?

,=I

r=l

AA,, is then obtained -nT

n(AB),

as

In V+ nT In RT+

cc +

c nA, In nA, + c nB8In nB, ,= 1 ,= 1

1

c ,=I

‘(ABJ,

In

nCAB),

(3) J

Step III. Each molecule in the ideal-gus mixture at T and V (a point) is inflated to a hard sphere to form a hard-sphere mixture. For this step, AA,,, = n,RTA*(&)

(4)

where 5 = n-$/V is the reduced density with & a size parameter for the mixture, and A*( 6) is obtained from a suitable hard-sphere equation of state. Equation (4) uses a one-fluid approximation. While some other approximation could be used, only the one-fluid approximation gives a feasible algebraic form. For example, using the original van der Waals equation, A*(c) = -ln(l

- 45)

(5)

or, using the Carnahan-Starling A*(S) = -t(36

equation,

- 4>,‘(1 - 6)2

(6)

Step IV. The hard spheres at T and V are charged with an attractive potential to form a real mixture. For this step, AA,, = nT( ii/b)A**(&)

(7)

where 6/i) is a parameter characterizing the attractive potential energy of the equation of state of mixture, and A**([) is obtained from a semiempirical the van der Waals form. For example, if the original van der Waals equation is used, A**([)=

-5

or, adopting

(8)

the Redlich-Kwong

A**( 6) = - (l/4)

equation

of state,

ln( 1 + 40

The desired Helmholtz

energy A of the real mixture

A = nA,,ai + nBoaE + AA, + AA,, + AA,,, + AA,,

(9) is given by (10)

307

where ai is the molar

Helmholtz energy of pure A in the standard similar definition holds for a:. The equation of state for pressure P follows from

P = - (~A/‘~~) r.n,

state; a

(11)

which leads to P = nr[ N,/V-

RTaA*( c$‘X’-

(i$@3A**(t;1/W]

(12)

The chemical potentials for the components A and B are, respectively, equal to those for the monomers A, and B, (for a simple proof of this general result, see Prigogine and Defay (1954)): thus PA

=

PA,

PB=PB,

=

(aA/anA,)T

=

(aA,a~~,i...,,,*,~

pLAis then obtained PA

=

ai

-

RT[ln(

’ l’n. i (]#A

(13)

)

(141

as V/n,,&)

- l] + RTA*(t)

+n,RT[aA~(~)/aE]a~/an,,

+(ii,‘z,)A**(~)

+n.(Li/$)(aA”*/aE)a~/an,,

+ +4**(5)a(ci/i,)/an,,

(15)

There is a similar relationship for pa. In eqn. (15) the first three terms are ideal-gas contributions, the fourth and fifth terms are hard-sphere contributions, and the remaining terms are contributions from attractive forces. We consider four types of chemical equilibria: Association of A : &4 = (+A,+,/+A,+A,p)zA,+

06)

,/‘zA,zA,

Association of B:

Here x is the true mole fraction (xi = ni/nr), cient given by the well-known thermodynamic RT In +k =

j‘]0‘xI (ap/ank)r.V.,,,,+,,

where z = PV/n,RT

and $J is the fugacity relation

-RT/V]dC’-

is the compressibility

factor.

RTlnz

coeffi-

(20)

308

The four chemical equilibria (eqns. (16)-(19)) contain ratios of fugacity coefficients. In general, these ratios depend on temperature, density and composition, resulting in tremendous mathematical complexity. However, as shown by Heidemann and Prausnitz (1976). this complexity can be reduced greatly by a proper choice of mixing rules for ci and 6. Several choices are available such that the ratios of the fugacity coefficients are independent of i (the degree of association and/or solvation) and, further, are the same for all four equilibria, as shown below. For example, Heidemann assumed a, = i2a, b, = ib,

(21)

and then derived

the mixing rules

(22) and 6 = ~%Wt%b,,

(23)

+X&B,)

where n, = nAO + n,,. However, experimental values of a, for normal alkanes indicate that eqn. (21) overestimates the effect of i on a,. In this work, we suggest a generalized mixing rule to replace eqn. (22): zi = ( rl&zT)‘+w

[ +‘4,

+ ~~,~,(~*,~,,)“‘*0

where 0 G w < 1 and k,, is a binary interaction unchanged. These mixing rules yield

= @‘RT/@

exp/:j

-aA*(S)/aS-

+ b)

(24)

+ h,]

parameter.

Equation

(23) is

[(l - w)/RT](ci/2,)aA**(~)/a5}d~

The detailed derivation is presented in Appendix I. Note that when w = 1 (as in Heidemann’s work), only repulsive forces contribute to (Y. it is necessary to solve simultaneously the To obtain nA,, n,, and n,/n,, chemical equilibria and material balances, as discussed in Appendix II. HENRY’S CONSTANT SOLVENT

FOR A NONASSOCIATING

SOLUTE

IN AN ASSOCIATED

The most sensitive test for a theory of solutions is in the dilute region. Therefore, we now turn attention to Henry’s constants. To obtain an

309

expression associated H A,B

=

,liyo 4

for HA,B, Henry’s constant for a nonassociating solvent B, we first recall the definition (fAbA

(26)

>

where f,, the fugacity potential pa by

of the solute,

is related

to the residual

lnfA = [pi + RTln(Px,/z)]/RT of the liquid

~La=~*(T,x,V)-~.‘adea’gas(T,x,V) constant

mixture;

the residual

(28)

is then obtained

H A.9 = lim_ [P exp( pa/RT-

chemical

(27)

where z is the compressibility factor chemical potential is defined by

Henry’s

solute A in an

as

lnz)]

(29)

In the limit xA j 0, z is the compressibility factor of pure liquid B. Because A is nonassociating, K, = 0, and the general formula for solvated species is AB,. Data

reduction

To illustrate eqn. (29), we take water as solvent B in the temperature range 20-300°C. At any temperature, it is necessary to obtain the parameters a and b and the association constant K,. To do so, we use experimental data for vapor pressures and liquid densities over this range of temperatures (Bain, 1964). Since it is difficult to obtain three unique parameters, we use, in addition, experimental Henry’s constants for a particular nonassociating solute A, namely, methane (Crovetto et al., 1982) and experimental P-V-T and vapor-pressure data for pure methane (IUPAC, 1976). We use as mixing rules eqns. (23) and (24) with w = 0, and for methane-water we arbitrarily set k,, = 0. Table 1 shows the results of data reduction for water. True mole fractions (up to zg) are also shown at each temperature for saturated liquid water. Figures 1 and 2 indicate that, using the parameters indicated, an excellent fit is obtained. To facilitate calculations, the parameters for pure water may be represented by a = (- 612.42 + 67.545T)/T’.4

(bar l2 mole2)

b = (0.13672 + 2.7457 X 10-5T1.2)/T0.5 In K=

-AH/RT+AS/R

(1 mall’)

(30) (31) (32)

1

b (1 mol-‘)

0.00946

0.00917

0.00881

0.00853

0.00831

0.00814

0.00805

a (bar I2 molW2)

6.757

6.507

6.168

5.884

5.641

5.426

5.244

20

50

100

150

200

250

300

t

(“Cl

--

0.0001998

0.0007025

0.002881

0.01357

0.08563

0.9830

6.511

K(bar-‘)

-

*

__ _

9.64x 10-l

9.38 x 10 - ’

9.00x10-’

8.60x10-’

8.20 x 10-l

7.67 x lo-

7.27 x10-’

z,

3.64x lop3 1.27~10-~

3.50x10-*

-

9.01 x1o-3

2.66 x lop2 1.69X IO_’

5.84~10-~

-

5.42 x 1O-2 4.16~10-~

9.oox19-2

1.48x 10-l 1.21xlO~:

1.79x10-’

1.99x10-’

Parameters and true mole fractions of associated species in saturated liquid water

TABLE

.__

4.62~10-~

1.68~10-~

1.41~10-~

9.02x 1O-5

9.01 x 1O-4 2.27~10-~

8.63x 1O-4 3.33~10~”

2.38x IO-’

2.25x1O-3 4.79x10-3

4.04x10K3

1.48~10-~

25

9.68~10-~

24

_.. ._

8.79x lo-’ 6.11x10-*

9.03 x lo-”

4.68~10-~

1.55 x 1O-4

5.24~10-~

l.lO~lO-~

26

311

,o-2i

600

T/K Fig. 1. Vapor pressure of water: --,

calculated;

300

\

500

400

\i

t/*c

Q

_ I: _ _

0.1.

\ 1

0 50 + 100 Xl50 a200 ~250 l 300

1"

40

Fig. 2. Pressure-liquid

c

\

0-z

0, Bain (1964).

';,

'\ \ \ \ \ \

”

”

50 p/mole

’

“1

60

1-l

density diagram for water: p,

calculated;

symbols, Bain (1964).

312

where A H/R = - 5.890 X lo3 K and AS/R = - 18.22 for temperatures below 200°C (for temperatures greater than 200°C we use the association constants shown in Table 1). The parameters a and b for methane (and other nonassociating solutes) are given in Appendix III. Figure 3 shows calculated and experimental Henry’s constants for methane in water. Also shown are Henry’s constants calculated using the Carnahan-Starling-van der Waals model (no association) and different values of k,,. It is evident that if the latter model is used to fit Henry’s constants, it is necessary to assign a large empirical temperature dependence to k,,. In this latter model, a single value of k,, cannot reproduce the observed maximum when Henry’s constant is plotted versus temperature. We also reduced experimental Henry’s_constant data in water for 14 nonassociating solutes in addition to methane. The results are shown in Figs. 4-9. Good fits are obtained for argon, krypton, nitrogen, oxygen, carbon monoxide, carbon dioxide and hydrogen sulfide using a single value of kAB, as given in Table 2. Note that the molecular sizes of these solutes are close to that of water. However, the results are not good for xenon, pentane, hexane, cyclohexane and benzene, whose molecular sizes are appreciably larger than that of water; the results are also not good for hydrogen, helium and neon,

Measured

0

cannot reproduce maximum -Association Mod1

0

I

I

I

I

I

50

loo

150

200

250

Temperature,

I

300

I

35(

“c

Fig. 3. Henry’s constant for methane in water. Methane is used as a probe to determine the chemical equilibrium constant for water; k,, is arbitrarily set equal to zero: 0, Crovetto et al. (1982).

313

Fig. 4. Henry’s constants for Ar, Kr and Xe in water: o, a, @ , Potter and Clynne (1978); A, Wilhelm et al. (1977); Cl, n, 8, Crovetto et al. (1982); A, Benson and Krause (1976).

A, v,

16.C

1

I

I

-r-

,

I

t2.c

Fig. 5. Henry’s constants for 0, and N, in water: Himmelblau (1960); 0, Pray et al. (1952).

A, A,

Wilhelm

et al. (1977);

q, n,

t PC Fig. 6. Henry’s constants for CO, CO, and H,S in water: A, Winkler (1901); 0, Jung et al. (1971); A, Schulze and Prausnitz (1981); 0, 0, Gas Processors Association (1982).

Fig. 7. Henry’s constants for n-C5HIz an d n-C,.H:, (1982); A, Tsonopoulos and Wilson (1983).

in water: n , Gas Processors

Association

315

Fig. 8. Henry’s

constants

for C,H,

and cycle-C,H,*

in water:

0, n . Tsonopoulos

(1983).

TABLE k,,

2

for aqueous

Solute

binary

systems

(relative k,,

to k,,

at 100°C

= 0 for methane) dk,,/‘dT

Ne

0.277

He

0.082

- 0.00284

H* 0, Ar

- 0.104 - 0.089 0.082 - 0.100 0.070

- 0.00125 - 0.00023 - 0.00010 -0.00019 0.00018

- 0.246 - 0.007

- 0.00069 0.00009 - 0.00011

N, Kr co CO, H,S Xe GH, n-C,H1z cycle-C,H,, n-GH,,

0.002 0.010 -0.170 - 0.221 - 0.309 - 3.323

- 0.00096

0.00032 0.00057 0.00078 0.00112 0.00061

and Wilson

316

Fig. 9. Henry’s constants for He, Ne and H, in water: o, A, Benson and Krause (1976); 0, 0, Pray et al. (1952); V, 0, Potter and Clynne (1978); V, A, Wilhelm et al. (1977); I, Crovetto et al. (1982).

whose molecular sizes are appreciably smaller than that of water. Our inability to fit Henry’s constants for these asymmetric systems is probably due to the use of a one-fluid approximation (eqn. (4)) and also to the inadequacy of the mixing rules (eqns. (23) and (24)). It appears that, while the chemical theory of associated fluids takes into account the effect of strongly oriented intermolecular forces, the one-fluid approximation with simple mixing rules fails to take into account normal intermolecular forces (repulsion and attraction) in mixtures where the molecules differ significantly in size. The present mixing rules are, essentially, generalizations of the van der Waals one-fluid theory of mixtures; the work of Shing and Gubbins (1983) has shown that a one-fluid theory leads to significant errors for asymmetric mixtures, especially in the dilute region. To improve the chemical theory of solutions, it will be necessary to focus attention on better relations for calculating A*(& ii and &. Proceeding empirically, the above fits of Henry’s constants can be improved markedly by assigning a small linear temperature dependence to k,, as shown in Table 2. Note that for large solutes, k,, increases with temperature, while for small solutes, k,, decreases with temperature. Figures 4-9 show the calculated results using a temperature-independent k,, (dashed

317

lines) and a slightly temperature-dependent k,, (solid lines) according to Table’2. This empirical procedure is clearly not satisfactory but may have some utility for estimating solubilities in water when no experimental data are available.

CONCLUSIONS

For associated fluids and their mixtures, it is possible, as shown here, to develop a generalized expression for the Helmholtz energy as a function of temperature, volume and composition. The generalized expression leads to an equation of state for an assembly of molecules that interact chemically to form dimers, trimers, etc., all at chemical equilibrium; this chemical interaction reflects strong orientational forces, such as hydrogen bonding. Normal forces of repulsion and attraction are taken into account by application of the classical van der Waals theory of fluids. If complete generality is to be maintained, the mathematical complexity is hopelessly large. To reduce the theory to practice, simplifying assumptions must be made, as discussed here. While the introduction of chemical equilibria is useful for taking into account strong orientational forces of attraction, the chemical theory of real fluids is nevertheless limited at present by our inadequate understanding of mixtures containing molecules of different size. The conventional procedure for such mixtures is to use a one-fluid van der Waals approximation, but this procedure is insufficiently precise, especially in the dilute region. Future progress in mixture thermodynamics can be attained only if the customary one-fluid approximation can be replaced by theoretically significant expressions for the potential energy and for the configurational entropy of mixtures containing molecules of different size.

ACKNOWLEDGMENTS

This work was supported by the Director, Office of Energy Research, Office of Basic Energy Sciences, Chemical Sciences Division of the U.S. Department of Energy. For partial financial support, the authors are grateful to the National Science Foundation, to DECHEMA (Frankfurt), to the Donors of the Petroleum Research Fund administered by the American Chemical Society, to Exxon Research and Engineering Company, and to the Alexander von Humboldt Stiftung.

318 LIST OF SYMBOLS

A, B A

u

a b 8

H H A.B K k n P R s T I/ x

components Helmhol tz energy van der Waals attraction parameter molar Helmholtz energy van der Waals size parameter fugacit) molar Gibbs energy enthalpy Henry’s constant of A in B equilibrium constant binary parameter number of moles pressure gas constant entropy temperature volume liquid-phase mole fraction compressibility factor true mole fraction fugacity-coefficient ratio fugacity coefficient reduced density chemical potential

Superscripts 0

* **

r

standard state hard-sphere contribution attractive-potential contribution mixture residual

Subscripts

A, B i, n, m I, 11, III, IV

; 0

component index degree of association steps of model construction reduced property total absence of any association or solvation

319 APPENDIX

I. DERIVATION

Equation

OF EQN. (25)

(12) is rearranged

P = n,RT/V-

as follows:

n,RTtJA*(&‘W

- n$- “( n,b)“‘( 8,‘b*+w)llA**(~)/W (Al)

According to eqns. (23) and (24), n,b and L?/&“~~ are functions of nAO, n,, and parameters for the monomers only, and an.,,/anAj, ~n,,/tln,, all equal i. The following equations are obtained %40/~~,m,8~ ~%l,/~~~,,,, upon differentiating eqn. (Al): T3Y’ nI (j,A,) = RT/V-

@p/an*,)

-

(af’/anB,)

RTaA*(&hW’

(1 - w)(ci,‘b)~A**( f),/W

T.v,n,~,ZB,~= RT/V-

CAB),>T,V,~,(J+(AB),)

=

(AZ)

RTaA*(t)/aV

- (1 - w)( a/&)aA**( w/an

- Fai

RT,‘V-

<),‘aV - F,i

(A31

RTaA*(~)/W

- (1 - w)(B,&)~A**(

<)/W-

FABi

W)

where

F, = M.RTa[aA*(5)/aV],/anB, +n:-~a[(n~~)w(ci/~‘+w)aA**(S)/aV]/an,,

(A@

FAB= FA + FB Substitution

(AT)

of eqns. (A2)--(A4) into eqn. (20) yields

(y = p+A,+A,/+A, F1= P+B,+B,/+B,+, = P+4,+B,/+AB = =

P exp(Ja[ V

=

(

[RT/L)

- aA*( &GVep~‘{&4*(

[(l - w)/RT]

&Gh$ + [(I -

w)/RT]

P+(AB),+AB/$AB),

(&‘?$A**( (iii/b)k4**(

?,

<)/aV]dV)/z [)/at

)dt (25)

320 APPENDIX

II. SOLUTION

Substitution

IE

zA,

+

E

= %3,+,/%,%,&3

= %B/%,%,KL

(A@

(ABp.‘dG

The following

i=l

equations ‘B,

+

i=l

: i=l

can then be written

z(AB),

=

‘A,/@

-

+z*&

i ,=l

=n

nT

T

[

=

zA,,’

izB,

T [ xB,/

Solution ZAB

+ ZB,/(l

KAazA,)

-

K;BaZAB)

=

-

balance: KBazB,)

1

(A91

‘i

I=1

E i i=l =n

from material

izA, + cO” ixtAB),

E

nT

n,,,‘nT

of eqn. (25) into eqns. (16).~(19) yields

cy = ZA,-1I%,%,&4 = ZW), +,/z

FOR nA,. nB, AND

( +

1 -

K,uz,,)~

+ +,B/(l

- K&x~,,)~]

= MOXA

(AlO)

KB~~Bl)2 + zAB/(l

- KL+sAB)‘]

=

(All)

E iz(AB), !==I

(

1 -

:i

of eqns. (A9)-(All),

together

nOxB

with (A12)

Kk3aZA,ZB,

from eqn. (A@, yields zA,, zB,, zAB and n/n,, APPENDIX

III.

PARAMETERS

[b = (b(O) + b”)T,‘)/(l Solute

a (bar I2 K’/’

Ne He H*

a/@,

b (‘)

+ ‘Tr2),where T, = T/T,

3.916 6.470 4.228

mole2)

AND

and also nA,/nO and nB,/no. b”’

FOR

PURE

COMPONENTS

(T, ==critical temperature)] b(O) (1 mol

-’)

b’” (1 mol-‘)

0.03594 0.01119 0.08378

0.00433 0.00609 0.00507

Ar N, Kr

20.73 22.24 21.77 43.24

0.01529 0.01685 0.02067 0.01900

0.00780 0.00789 0.01023 0.00970

co CO, H,S

17.84 71.95 105.6

0.01940 0.01967 0.02052

0.00990 0.01004 0.01047

Xe GH,

86.97 454.7

0.02458 0.05371

0.01254 0.01559

419.5

621.9

0.05041 0.07835

0.02693 0.01735

598.9 39.22

0.07587 0.02061

0.03359 0.01052

02

n-C,H,, cycle-C, n-C,H,, CH,

H ,z

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