THERMODYNAMICS OF GAS SOLUBILITY IN MIXED SOLVENTS J. P. O'CONNELL AND J. M. PRAUSNITZ Defiarimrnt of Chrmical EnginPertng, C n i c r r s i t y of Caitjornia, Bri k e l e j , Caizf.
This work considers the thermodynamic description of a liquid solution containing one gaseous solute and two or more miscible liquid solvents. For the solvents, the reference fugacity i s that of the pure components but for the solute, the reference state fugacity i s set equal to Henry's constant in one of the solvents; thus, the unsymmetric convention for activity coefficients i s employed. A transformation of Wohl's method i s used to predict the properties of a multicomponent solution from information on the binary pairs formed b y the solution's components. Equations are derived for the case where the properties of each binary pair can b e described b y a one-term Margules equation; therefore, the results presented here are applicable only to simple, nonpolar solutions. However, the method can b e extended to higher-order equations, i f sufficient experimental information i s available to justify the use of additional parameters. In nonpolar systems, the logarithm of the gas solubility at low pressures is, to a good approximation, a linear function of the solvent composition.
recognizrd many years ago (5) that the thermotreatment of liquid solutions which contain a supercritical component differs from that for solutions Lvherein all components can exist as pure liquids at the solution temperature. T h e reference fugacity for the activity coefficient of the subcritical component-i.e., the liquid solvent- is usually chosen to be that of the pure liquid at a specified pressure, but the reference fugacity for the activity coefficient for a supercritical component is most conveniently given by Henry's constant as obtained from thermodynamic data for the dilute solution (7). Sumerous textbooks on thermodynamics have discussed this point but they have also limited the discussion to binary mixtures. or else to very dilut: solutions. In this work kve consider the thermodynamics of a solution consisting of one supercritical component-i.e., a gasand t\vo or more liquid solvents. In particular, \\'e are concrrned \vith a technique for predicting the thermodynamic properties of a multicomponent mixture using only information on the properties of the various possible binary mixtures which can be formed from the components in the multicomponent solution. Lye consider first the case wherein a gas is dissolved in a mixture of t\vo miscible solvents and then generalize our results to the case Lvhere a single gas is dissolved in a mixture of any number of miscible solvents. T
\VAS
I dynamic
Definition of Activity Coefficients
I n order to be consistent \rith the notation introduced by Le\vis and Randall (.5)and by Hildebrand (,?) (which is still used in later editions of these pioneering books), we designate the light component-i.e., the gas-by subscript 2. T h e sol\rents are designated by subscripts 1 and 3, subscript 1 being arbitrarily assigned to the solvent having the lower vapor pressure a t the solution temperature. T h e activity coefficients of the solvents are both referred to the saturation (vapor) pressure, PIs, of solvent 1. They are defined by
In Equation 1 f l stands for the fugacity, x 1 for the mole fraction, and c 1 for the partial molar volume of component 1 in the liquid solution. Analogous definitions for the corresponding symbols in Equation 2 hold for component 3. T h e solution temperature is T and the total pressure is P. T h e reference fugacity fla(Pls) is the fugacity of pure liquid 1 a t temperature T and pressure PI', and the reference fugacity J3O(Pl8) is the fugacity of pure liquid 3 a t temperature T and a t pressure P I s (not P3'). From Equation 1 it follows that the activity coefficient for solvent 1 goes to the desired limit. y1 + 1
as
x1 + 1
(3)
Since the lower limit of integration in Equation 2 is PIs rather than Pa', the activity coefficient of solvent 3 also goes to the desired limit. 73 -+ 1
as
+1
x3
(4)
T h e activity coefficient of the gaseous solute is defined by
(5) In Equation 5 f2, X Z . and 2'2 have definitions analogous to those given for Equations 1 and 2. T h e reference fugacity H ~ , ~ ( P is I ' )Henry's constant for solute 2 in solvent 1 in the absence of solvent 3. For a given solute-solvent pair it is a function of temperature only; it is defined by
x 3 = o
"
T h e asterisk in Equation 5 serves as a reminder that yz* is normalized in a manner different from that used for y 1 and 73. From Equations 5 and 6 it follows that y2* -+
1
as
Y Z -+
0
(7)
provided that .x3 = 0. T h e activity coefficient for the solute brcomes unity only in the ease lvhen it is infinitely dilute in pure solvent 1. T h r activity coefficient y 2 * will not, in general, approach unity when the solute is infinitely dilute in a mixture of solvents 1 and 3 or in pure solvent 3. VOL. 3
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347
T h e normalization of the activity coefficients for the solvents is clearly diffcrent from that for the solute; this sort of normalization for binary syytems has been designated .'the unsymmetric convention" by Prigogine and Defay (8)and it appears appropriate to use this designation for ternarb- systems as ivell, By contrast the "symmetric convention" designates a normalization which is the same for all components and is gi\,en by yz--t 1
as
yi
+1
( z = 1.
2. 3.
, ,
)
(8)
The activity coefficients defined by Equations 1. 2, and 5 all refer to the same reference pressure: P?. Therefore, for a ternary system at constant temperature T (but not necessarily at constant total pressure P) the three activity coefficients are related exactly by the Gibbs-Duhem equation. SI
d In
y1
f x? d In yn* f x3 d In y3 = 0
(9)
(8) for binary solutions and their results are readily extended to the ternar? case (see 'Appendix). TVe find that y1 (un>ymmetric) = 71(symmetric)
(181
y 2 * (unsymmetric) = y ? (symmetric) exp ( y3
(unsymmetric)
= y3
el?)
(symmetric)
(19)
[ X )
Substituting these last three equations into Equations 15, 16, and 17. Lve have for the activity coefficienrs as defined by Equations 1. 2. and 5 the results: In
y1
=
cull.u2(l
In y?* = cyln[xl(I in
7 3
=
al3x1(1
-
x2)
-
xl)
+
- 11 x3)
cyl3.~3(1
-
XI)
+ a23X3(1 -
+ a?sx?(l -
-
x?)
x3)
-
cy23xZx3
-
ct13Ylx3
CYIGIX?
(21) (22) (23)
Excess Free Energy
Evaluation of Parameters
I t was sho\vn by T'r'ohl (70) tha: if the excess free energy is knoivn at a given temperature for the three binaries 1-2: 1-3: and 2-3. a good approximation can be \vritren for the excess free energy of the ternary solution at the same temperature. HoIve\.er. TVohl's definition of excess free energy is based on the sb-mmetric convention for activity coefficients and thus. for the present case) it is necessary to transform TVohl's results into a form conGstent rvith the unsymmetric convention for activity coefficients. Vsing the s)-mmetric convention as employed by TVohl we \\.rite the molar excess free energies of the three binaries in a one-parameter Margules expansion :
Equations 21, 22, and 23 give the thermodynamic properties of the ternary solution in terms of the binary parameters a l l , a 1 3 : and a?3. These parameters can be obtained from data for the three binary solutions 1-2, 1-3. and 2-3. T o obtain c y 1 2 consider Equations 21 and 22 for the case x3 = 0. TZ'e obtain
where the cy's are empirical coefficients obtained from binary data and are referred to the reference pressure PI' T h e molar excess free energy for the ternary solution is then avsumed to be
T h e acti\,ity coefficients (symmetric convention) follo\v from differentiating Equation 13 For any component z
In y l
(1-2 binary) (1-2 binary)
In y ? * =
= alnx2* alp(,r1*
- 1)
(21a) (22a)
Therefore solubility data for the gas (component 2) in pure solvent 1 may be used to obtain cy]?. This parameter is the self-interaction coefficient for solute 2 tvhich gives the deviation from Henry's law only due to composition effects. T h e effect of deviations due to pressure (Krichevsky-Kasarnovsky equation) is accounted for b!- the integral in the defining Equation 5. may be obtained from solubility data for T h e parameter the gaseous solute 2 in solvent 3. Ho\ve\Ter, from the relationship betiveen symmetric and unsymmetric activity co. efficients (see Appendix) it can sholvn that
where H2,3(PlS) is Henry's constant for solute 2 in solvent 3 at pressure PIs. Experimentally. Henry's constant for solute 2 in solvent 3 is found at the saturation (vapor) pressure of solvent 3 according to the definition '
H2,3(P38) = Limit x2
+0
x: = 0
T h e effect of pressure on Henry's constant is given by where n i stands for number of moles of component z and
c
flT =
I1 i.
t
Performing the differentiation \ve obtain for the activity coefficients in the symmetric convention:
+ cy13~3(1In y 2 = - \ ? j + cyy.13Y3(1 I n yn CY13tl(l + In y l = c u l ? u r ( l -
tl)
- k.1
cyl.\~(l
=
- Y3)
CY?3\?(1
-
~ ~ 2 3 1 2 . ~ 3
(15)
-
0113YlY3
(16)
-
cyI?Ylt.'
(17)
.XI)
t3)
TI-e no\v must relate the activity coefficients of the unsymmrtric convention to those of the symmetric convention. .I'his problem has been considered by Prigogine and Defay
348
l&EC FUNDAMENTALS
where p 2 : is the partial molar volume of solute 2 in solvent 3 at infinite dilution. The use of Equation 24 yields txvo equivalent expressions for each of th: Equations 21 to 23.
Substitution of Equation 24 yields
Recalling the definirion of y L * ah given by Equdtion 5. we have
I11 7 3 = c y 1 3 Y 1 ( 1
-
x3j
+
ff12.d
+ Henry's constant for solute 2 in the mixed solvent referred to
PIs is given by
H Q , , ~ ~ ~ , -=~ i? ( P(when I ~ ) x p is very small) ~
solvent
\chew in the a equations, the binary parameter for the light component is from the 1-2 mixture while in the b equations the parameter is from the 2-3 mixture. 'I'hese equations should be used in place of Equations 21 to 23 only in the case Lvhere one of the parameters cy12 or a23is not available. Parameter o(13 is found from vapor-liquid equilibrium data for the binary solution of solvents 1 and 3 in the absence of solute 2. \Vhen .x2 = 0. Equations 21 and 23 become
(1-3 binary)
In y 1 =
013h.3'
(21b)
(1-3 binary)
In y3 =
cyy13x12
(23b)
Parameter 0113 can therefore be determined from the activity coefficients of the binar)- liquid solution a t the temperature of interest. Holvever cy13 is a function of pressure as \vel1 as of temperature and in Equations 21, 22. and 23 a13 must be corrected to the reference pressure PI'. If the total pressure over the binary solution 1-3 is designated by PI-3, then
\vhere k is a constant related to the molar excess volume uE of the binary 1-3 solution by uE =
(36)
.Y 2
Combination of Equations 35 and 36 gives the desired result: In
ff2,m,sed
(PI') =
,xl
+
In H?:I(P,S)
solvent
.t3
In H!.3(P1') a1311Y3
(37)
Equation 37 gives Henry's constant in the mixed solvent as a function of the t\vo Henry's constants in the pure solvents a n d of the nonideality of the solute-free solvent mixture. This equation shows the interesting result that even if the t\vo solvents form an ideal mixture (a13 = 0) Henry's constant for the solute in the mixed solvent is a n exponential rather than a linear function of the solvent composirion. Generalization of Results for Solvent Containing Three or More Components
It can be readily shown that the excess free energy for a system of n components (component 1 is the reference component and component 2 is the light component) is given by
Then n-1
n
n
(31)
kXlX3
j#i
I n most cases the pressure correction given by Equation 30 is completely negligible. Thus. from the solubility of the solute in one of the solvents, Henry's constant for the solute in the other solvent, and the activity coefficients for the mixture of solvents alone. the activity coefficients in Equations 21. 22: and 23 may be determined. If data for the binary solution of the t\yo solvents are nor available. cy13 can be estimated for solutions of nonpolar solvents by a simplified form of Hildebrand's equation ( 3 ) . giving
n-1
n
n
where y2* is defined in Equation 5 and the reference fugacity Hz.l(Pls)is defined as
= Limit
H?,I(pl')
xr-0 x,
=o
(e)
(41
If1
For the binary parameters. it can be sho\vn that \there 61 and z I stand. respectivel). for the solubilitv parameter and liquid molar volume of pure liquid 1 i\ith similar definitions for 6 3 and z 3 Henry's Constant for a Solute in a Binary Mixed Solvent
\Ve consider a solution of solute 2 in a mixture of solvents 1 and 3 under conditions where the solute i4 extremely dilute. In that case \ve can let .Y? = 0 in Equation 2 2 and. after rearrangement. obtain
-
Limit In y 2 * == ( a y 3- al?)uS il
n
a13\I\R
Equations 39 and 40 may now be expressed in terms of only one binary paramcter invol\.ing the light component. the parametrr obtained from the binary mixture of solvent 1 and solute 2.
n-1
n
(3.3)
VOL. 3
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349
n-1
n
n
2000 TOLUENE- HEPTANE (a
800 Again. it should be emphasized that rather than Henry's constant, the binary interaction parameters for the gas with a solvent should be used whenever possible. T o obtain Henry's constant for the solute in a mixed solvent. Equation 45 is used mith 4 2 = 0 ; after rearrangement this is:
z
1 Ideal
(b)
Calculated from Ternary Margules Equation
600
8
-400 : 200
tSO-OCTANE -PERFLUOROHEPTANE
But from the definition of yf*(Equation 5)
I Figure
which is equivalent to n
72-t
0
1
1.0
Henry's constants in solvent mixtures a t
n
Equation 37 indicates that a plot of In H2, mired against solvent
mole fraction of either solvent may go through a maximum or minimum. Using the relation for the extremum
we obtain that
Provided that 0 < ~1 < 1, if ala > 0 (positive deviation from Raoult's law) the solubility goes through a maximum. while if a13 < 0 (negative dei.iation from Raoult's law) the solubility goes through a minimum. Applicability and Limitations
T h e key assumption in the derivation of Equations 21. 22. 23: and 37 is the validity of Equation 13. This equation assumes that only t\vo-body interactions contribute significantly to the excess free energy of the ternary mixture and that the characteristic coefficients for these interactions are a13.and a23 as determined given bl- the empirical constants ai?, from binary data. Such a simplified picture of a liquid solution cannot be expected to have general validity but it should be a good approximation for solutions consisting of simple. nonpolar molecules. .An equation similar to Equation 13 but containing higher terms would be required for polar solvents such as alcohols, ketones. etc. Ynfortunately there do not seem to be available at this time any suitable nonpolar ternary solubility data of sufficient acI&EC FUNDAMENTALS
1.
O
0.4 0.6 0.8 MOL F R A C T I O N OF SECOND SOLVENT
0.2
25" C.
M a x i m u m or Minimum Solubility in Binary Solvenl
350
0
curacy to test the validity of the derived equations. Holyever. to obtain some quantitative insight on the effect of solvent composition on gas solubility \ve have calculated Henry's constants for t\vo typical cases. First. \ve have considered the solubility of hydrogen in mixtures of toluene and heptane at 25' C. Toluene and heptane form a moderately- nonideal solution ; the excess free energ)for the equimolar mixture is 48 cal. per gram mole (9). Henry's constants in pure toluene and in pure heptane have been reported by Cook. Hanson, and Alder (I). Using Equation 37, Henry's constant in the solvent mixture \vas calculated as a function of solvent composition ; the results. sho\\.n in Figure 1, indicate only very small departure from the straight-line (ideal solvent mixture) relationship. Second, \ve consider the solubility of oxygen in mixtures of iso-octane and perfluoroheptane at 25' C. These ttvo sol\rents form a highly nonideal mixture; the excess free energy for the equimolar solution is 330 cal. per gram mole (6). In fact, 25' C. is just 1' above the critical solution temperature for these t\vo solvents. Ox)-gen solubilities in the two pure solvents have been reported ( 2 , J ) , Henry's constants for the solvent mixture have again been calculated using Equation 37 and the results are also shown in Figure 1. In this case there is a significant difference between the calculated results and those predicted by assuming that the tivo solvents form an ideal mixture. 'The solubility of oxygen is enhanced by as much as nearly 20Yc by the large nonideality of the solvent mixture. Solubility of Solids
T h e expressions developed in this paper can also be directly applied to solutions containing solids. Although the usual con\rention for activity coefficients of solids in solution is the symmetric convention referrcd to the subcooled liquid. often the extrapolation to obtain the fugacity of the reference state is uncertain. In such cases. the definition of a constant analogous
to Henry's constant can be made and the unsymmetric convention for activity coefficients should be used. T h e most useful application of this theory is to calculate the solubility of solids in mixed sol\,ents. If the solubility of a solid in both of t\yo solvents is very small, then the solubility of that solid in a mixture of the two solvents is given by In
.ks,nrir
= XI
In
+
ss,l
x3
ln
x,,3
+
O(13w3
If the excess free energy for the 1-2 binary is expressed in the form
then
(49)
\\.here 21 and 1 3 are the mole fractions of the tbvo solvents and r S , l and x s , 3 are the solubilities of the solid in each of the pure solvents. -4s in the case for gas solubility, this relation is expected to apply only to a nonpolar solid dissolved in a mixture of simple, nonpolar liquids.
and from Equation A-5
I n the same manner. for the 2-3 binary In Y L , = ~
('4-1 1)
aL3~3*
Substituting Equations -4-9 and '4-1 1 into '4-8 yields: Appendix. Relation between Symmetric and Unsymmetric Activity Coefficients
Prigogine and Defay (8)show that in a binary solution, the activity coefficient in rhe symmetric convention, y2. is related to that in the unsymmetric convention, ? I * . by
(.4-1) xvhere H1 is Henry's constant for component 2 in the mixture and O.'j is the (hypothetical) fugacity of pure "liquid" 2. [In this scction it is understood that Henry's constant is always referred to some (constant) reference pressure such as PI'.] From
Limit y 2
T h e authors are grateful to the donors of the Petroleum Research Fund and to the Sational Science Foundation for financial support. Nomenclature
J 8
H
Hz
= -
x2-0
Acknowledgment
(A-2)
J20
k n
P R T
substitution gives Y?
L;
= Limit y2
YZ*
(A-3)
x2-0
0 X
Also *-
(A-4)
Limit y2* x2+1
[Equation 21.39 of Prigogine and Defay ( 8 ) takes the wrong limit.]
YZ
-
Y2*
~ 2 , i - H2,i
-
Y?,l>C
-
Limit y ? , ~
('4-5)
xz-0
f20
For the binary-containing components 2 and 3. the limit of the activity coefficient of the light component 2 as defined in Equation A-2 is different from that of the ternary (Equation A-5) : 7_2 ,_s Y2.3*
-- _H2,3 f20
-
Limit
y2.3
('4-6)
X?+O
But
('4-7) or
= =
= =
Y*
=
6
=
fugacity molar Gibbs free energy Henry's constant constant number of moles pressure gas constant temperature molar volume partial molar volume mole fraction Margules constant activity coefficient: referred to pure component activity coefficient. referred to infinitely dilute solution solubility parameter
SCPERSCRIPTS E = excess property s
For the binary-containing components 1 and 2, the definition of the activity coefficient of the light component is the same as that for the ternary, so that
= = = =
Y
(Y
~ 2 Y2
= = = = =
0
= =
saturation conditions pure component
SUBSCRIPTS S
= solid
1 2 3, 4.
=
reference component (solvent)
= solute =
other solvents
Literature Cited
(1) Cook, H. LV., Hanson. I). N...\lder! E. J.. J . Chern. Phys. 26, 748 (1957). (2) Gjaldbaek, J. C., A c l n Chzm. Scnnd. 6 , 623 (1952). (3) Hildebrand, J. H.. "Solubility:'' Keinhold. New York; 1 9 2 4 ;
later editions, 1936 and (with K. I,. Scott) 1950. (4) Kobatake, Y.>Hildebrand. .J. H., J . P h y Chprn. 65, 331 (1961). (5) Lev.%. G. N.:Randall, M.. '"Thermodynamics and the Free Energ)- of Chemical Substances," McGra\v-Hill, Ne\v York, 1923; 2nd ed. (revised by K . S. Pitzer and L. Brelver). 1961. (6) Mueller, C. K.. Lewis. .J. E.. J . Chpm. Phjs. 26, 286 (1957). M . ?Cheni. EnS. Scz. 18, 613 (1963). I . . Defay. I
