I n d . Eng. Chem. Res. 1990, 29, 560-564
560
= osmotic pressure = ideal osmotic pressure 7rp = osmotic pressure of salt-free polyelectrolyte solution rS = osmotic pressure of polyelectrolyte-free s a l t solution x1 = osmotic pressure of polymer solution T:, = elastic c o m p o n e n t of osmotic p r e s s u r e T , = osmotic p r e s s u r e of mobile ions = osmotic coefficient of gel phase 4 = osmotic coefficient of e x t e r n a l solution 4p = osmotic coefficient of salt-free polyelectrolyte solution & = osmotic coefficient of polyelectrolyte-free s a l t solution x = polymer-solvent interaction parameter K
Tideal
Registry No. N a N 0 3 , 7631-99-4; AgNO,, 7761-88-8; KMnO,, 7722-64-7; KzS04, 7778-80-5; Ca(NO,),, 10124-37-5; CaCI,, 10043-52-4; CoCl,, 7646-79-9; CU(NO,)~,3251-23-8; La(NO&, 10099-59-9; NaCl, 7647-14-5; Na2S04, 7757-82-6; N a 2 H P 0 4 , 7558-79-4; (acrylamide)(N,N'-methylenebis(acry1amide))(sodium acrylate (copolymer), 33882-67-6; (acrylamide (N,N'-methylenebis(acrylamide))([ 3-(methacrylamido)propyl]trimethylammonium chloride) (copolymer), 98587-56-5.
Literature Cited Alexandrowicz, Z. Results of Osmotic and of Donnan Equilibria Measurements in Polymethacrylic Acid-Sodium Bromide Solutions. Part 11. J . Polym. Sci. 1960, 43, 337-349. Alexandrowicz, Z. Calculation of the Thermodynamic Properties of Polyelectrolytes in the Presence of Salt. J . Polym. Sci. 1962,56, 97-115. Brannon-Peppas, L.; Peppas, N. A. Structural Analysis of Charged Polymeric Networks. Polym. Bull. 1988, 20, 285-289. Cussler, E. L.; Stokar, M. R., Varberg, J. E. Gels as Size Selective Extraction Solvents. AIChE J . 1984, 30, 578-582. Dolar, D.; Kozak, D. Osmotic Coefficients of Polyelectrolyte Solutions with Mono- and Divalent Counterions. Proc. Leiden S y m p . 1970, 2 2 , 363-366. Dolar, D.; Peterlin, A. Rodlike Model for a Polyelectrolyte Solution with Mono- and Divalent Counterions. J . Chem. Phys. 1969,50. 3011-301 5. Flory, P. J. Phase Equilibria in Polymer Systems: Swelling of Network Structures. Principles of Polymer Chemistry; Cornell University: Ithaca, NY 1953. Freitas, R. F. S.; Cussler, E. L. Temperature Sensitive Gels as Extraction Solvents. Chem. Eng. Sci. 1987, 42, 97-103. Gehrke, S. H. Kinetics of Gel Volume Change and its Interaction with Solutes. Ph.D. Thesis, University of Minnesota, Minneapolis, 1986. Gehrke, S. H.; Andrews, G. P.; Cussler, E. L. Chemical Aspects of Gel Extraction. Chem. Eng. Sci. 1986, 41, 2153-2160. Goldberg, R. N. Evaluated Activity and Osmotic Coefficients for Aqueous Solutions: Thirty-Six Uni-Bivalent Electrolytes. J . Phys. Chem. Ref. Data 1981, 20, 671-764. Guggenheim, E. A. Regular Solutions: Nature of Quasi-Chemical Treatment. Mixtures; Clarendon: Oxford, 1952. Hamer, W. J.; Wu, Y.-C. Osmotic Coefficients and Mean Activity Coefficients for Uni-Univalent Electrolytes in Water at 25OC. J . Phys. Chem. Ref. Data 1972, 2 , 1047-1099. Hasa, J.; Ilavsky, M.; Dusek, K. Deformational, Swelling, and Potentiometric Behaviour of Ionized Poly(Methacry1ic Acid) Gels.
I. Theory. J . Polym. Sci., Polym. Phys. Ed. 1975, 13, 253-262. Helfferich, F. Equilibria: Swelling. Ion-Exchange; McGraw-Hill: New York, 1962. Hirokawa, Y.; Tanaka, T. Volume Phase Transition in a Non-Ionic Gel. J . Chem. Phys. 1984, 81, 6379-6380. Ilavsky, M.; Hrouz, J. Phase Transition in Swollen Gels: 4. Effect of Concentration of Crosslinking Agent a t Network Formation on the Collapse and Mechanical Behaviour of Polyacrylamide Gels. Polym. Bull. 1982, 8 , 387-394. Katchalsky, A. Polyelectrolytes. Pure Appl. Chem. 1971, 26, 325-373. Katchalsky, A.; Michaeli, I. Polyelectrolyte Gels in Salt Solutions. J . Polym. Sci. 1955, 15, 69-86. Katchalsky, A,; Zwick, M. Mechanochemistry and Ion Exchange. J. Polym. Sci. 1955, 16, 221-234. Katchalsky, A.; Cooper, R. E.; Upadhyay, J.; Wasserman, A. Counter-ion Fixation in Alignates. Chem. Soc. J . 1961,5198-5204. Kern, W.Der Osmotische Druck Wasseriger Losungen Polyvalenter Sauren und Ihrer Salze. Z . Phys. Chem. 1939, A184, 197-210. Kitchener, J. A. Recent Developments: Ion-Exchange Resin Membranes. Ion-Exchange Resins; Methuen: London, 1957. Konak, C.; Bansil, M. Swelling Equilibria of Ionized Poly(methacrylic acid) Gels in the Absence of Salt. Polymer 1989, 30, 677-680. Kou, J. H.; Amidon, G. L.; Lee, P. I. pH-Dependent Swelling and Solute Diffusion Characteristics of Poly(Hydroxyethy1 Methacrylate-Co-Methacrylic Acid) Hydrogels. Pharm. Res. 1988,5, 592-597. Panayitou, C. G.; Vera, J. H. The Quasi-Chemical Approach for Non-Randomness in Liquid Mixtures. Expressions for Local Surfaces and Local Compositions with an Application to Polymer Solutions. Fluid Phase Equilib. 1980, 5,55-80. Peppas, N. A. Dynamically Swelling Hydrogels in Controlled Release Applications. In Hydrogels in Medicine and Pharmacy; Peppas, N. A., Ed.; CRC: Boca Raton, FL, 1987; Vol. 3. Prange, M.; Hooper, H. H.; Prausnitz, J. M. Thermodynamics of Aqueous Systems Containing Hydrophilic Polymers or Gels. AIChE J . 1989, 35,803-813. Otake, K.; Inomata, H.; Konno, M.; Saito, S. A New Model for the Thermally Induced Volume Phase Transition of Gels. J . Chem. Phys. 1989, 91, 1345-1350. Ricka, J.; Tanaka, T. Swelling of Ionic Gels: Quantitative Performance of the Donnan Theory. Macromolecules 1984,17,2916-2921. Ricka, J.; Tanaka, T. Phase Transition in Ionic Gels Induced by Copper Complexation. Macromolecules 1985, 18, 83-85. Siegel, R. A,; Firestone, B. A. pH-Dependent Equilibrium Swelling Properties of Hydrophobic Polyelectrolyte Copolymer gels. Macromolecules 1988, 22, 3254-3259. Staples, B. R.; Nuttal, R. L. The Activity and Osmotic Coefficients of Aqueous Calcium Chloride at 298.15 K. J . Phys. Chem. Ref. Data 1977, 6, 385-407. Sun, C.-L. Thermodynamics of Polyelectrolyte Gels. Ph.D. Thesis, University of Southern California, Los Angeles, 1986. Tanaka, T. Collapse of Gels and the Critical Endpoint. Phys. Reu. Lett. 1978, 40, 820-823. Treloar, L. R. G. Non-Gaussian Chain Statistics and Network Theory. The Physics of Rubber Elasticity; Clarendon: Oxford, 1958. Gill, S. J. Interaction of Cupric Ions with Polyacrylic Wall, F. T.; .4cid. J . Phys. Chem. 1954, 58, 1128-1130. Received for reuiew J u l y 11, 1989 Accepted October 20, 1989
Prediction of Equilibrium Data of Adsorptions from Liquid Mixtures Wei-Rong Ji* and Y. C. Hou Chemical Engineering Thermodynamics Laboratory, Zhejiang University, H a n g t h o u , Zhejiang, PRC
On the basis of solution theory and surface thermodynamics, an activity coefficient model for the adsorbed phase with no adjustable parameters is presented. In combination with the phase-exchange adsorption model, it can predict the equilibrium data of nonideal adsorption from binary liquid mixtures. Five systems with nonideal adsorbed phases are tested, The results are satisfactory. Many efforts have been made on the nonideality of adsorbed solutions. A few models d e a l i n g with n o n i d e a l 0888-5885/90/2629-0560$02.50/0
adsorbed s o l u t i o n s of a d s o r p t i o n f r o m gas m i x t u r e s (AFGM) have been published (Suwanayuan and Danner, G 1990 American Chemical Society
Ind. Eng. Chem. Res., Vol. 29, No. 4, 1990 561
1980a,b; Lee 1973), but little progress has been made on the nonideality problem of adsorption from liquid mixtures (AFLM). It is well-known that the experiments of AFLM are difficult and time consuming, so the prediction of the equilibrium data of AFLM has great significance and is a challenging task. Bering and Serpinskii (1973), Myers and Sircar (1972a,b), Sircar and Myers (1970, 1973), and Siskova et al. (1974) have proposed ideal adsorbed solution models. Among them, the work of Sircar and Myers was very important. They proposed the analogy between adsorption from liquids and adsorption from vapors. According to their theory, the surface excess amount of AFLM could be approximated by that of adsorption from corrosponding saturated gas mixtures: ns(n)
= lim
From this equation and assuming the adsorbed solution is ideal, a model was obtained: =
X ? X $ (-~K )
xVm1 + xVm2 K =
2
ex.(
-)
cp-cbz
m2RT + mlRT cp-cp1
Y2
m,RT
where A,, is an interchange energy parameter and its physical meaning is similar to that of quasichemical theory (Prausnitz et al., 1986); O1 and 8, are volume fractions that can be calculated by 81
=
x!
(5)
+ Bx$
x;
where the superscript b represents properties of the bulk phase, yp is the activity coefficient of component i, mi is the saturated adsorbed amount of pure substances, and cp and cpl are the free energies of immersion of mixtures and pure components, respectively. The values of m iand pi can be estimated from adsorption isotherms of pure unsaturated vapors, so this model can predict the equilibrium data of AFLM. Nevertheless, it could only be suited to ideal adsorptions due to failing to consider the nonideality of adsorbed phases. Larionov and his co-workers (1977) used the Flory-Huggins equation to calculate the activity coefficients of adsorbates:
( -3
In 77 = In ( O , / x ; )
+
In y; = In (O,/X$)
+ (c - 1.0)0,
where B is the molar volume ratio of the two pure adsorbates (VS,/ VS,). For the adsorption on microporous adsorbents, the volume of the adsorbed phase is assumed to be equal to the pore volume of the adsorbent (V,). Then the molar volumes of pure adsorbates can be estimated by the saturated adsorbed amounts I 7
m2RT
1
(4)
(1)
ns(n)v
p'p'l.
,dn)
and obtained an equation of the Gibbs free energy of mixing, which was used by Lee (1973) successfully to describe the Gibbs free energy of mixing of AFGM. Now the equation is adopted to calculate the Gibbs free energy of mixing of the adsorbed solution of AFLM:
- O2
"P V ?= (7) mi and the value of the molar volume ratio can be obtained as B = ml/m2 (8)
From eq 4, the excess Gibbs free energy can be calculated:
-AGSE - - - A- -G=S RT
Theory It is assumed, in this paper, that the adsorbed solution of AFLM is a separate phase with finite volume, so the adsorbed phase could be treated similarly as a three-dimensional solution. Moelwyn-Hughs (1961) made a statistical treatment for binary solutions formed from molecules of different sizes
RT 81 x ~ In , x!
+ x;
02
A12
x;
RT
In - + (x~x;~,O,)'/~-
(9)
where G id is the free energy of mixing of ideal solutions. The partial derivatives of the excess Gibbs free energy with respect to the numbers of moles of adsorbates a t constant temperature and pressure give the activity coefficient expressions
I
(3)
where c is the molecule size ratio and Bi are volume fractions. Because in this model only the size contribution to the nonideality of adsorbed solutions was taken into account, it could not be applied to systems in which the thermal contribution is large. So, to date, there is still a pressing need for the activity coefficient model of nonideal AFLM. In this work, an activity coefficient model for the adsorbed phase is presented. By combining it with the phase-exchange adsorpt,ion model, a new method is obtained that can predict the equilibrium data of nonideal adsorption systems from adsorption isotherms of pure unsaturated vapors.
AGid
RT
In 77 = - In (x7
1
+ Bx!) + ( 1 - B)x; - ( X ; ) ~ B ~ / ~ A , ~ x i + Bx; ( x ; + Bx$12RT (10)
In y; = r
in which there exists only one unknown parameter (Al2). The basic equation of the surface thermodynamics is d G s = -S9 d T - K dA
+ V 9 d P + C p ? dx?
(12)
At constant temperature and pressure, d G 9 = -K dA
+ Cp?x?
(13)
By use of the Euler theorem, the Gibbs free energy of the adsorbed solutions at constant temperature and pressure equals
G 9 = -KA
+ Zpfx?
(14)
562 Ind. Eng. Chem. Res., Vol. 29, No. 4, 1990
For pure adsorbates, the Gibbs free energy is Gs = -n,A,
+ (p$)O
(15)
Taking the pure adsorbate at the same temperature and pressure as the standard state, the Gibbs free energy of mixing can be described as AG' = G s - x ~ G -! x ; G ~
(16)
adsorbed phase and a bulk phase. For binary adsorption systems with finite adsorbed phase volume, this can be expressed as 1 1 -(l)S + -1( 2 ) b F= -1( l ) b + -(2), (27) 01
"2
L'1
u2
where u1 and u2 are molecular volumes of component 1 and 2, respectively. The condition of equilibrium of (27) is
Substituting (14) and (15) in (16) gives
A G 5 = -aA
(28)
+ T ~ A , +x ~X ~ A ~+X ; x ~ ( P S - ( P S ) ~ ) + x@l
- (p!Io)
(17)
According to the equilibrium criterion, the chemical potentials of component i in the adsorbed phase and in the bulk phase are equal: P?
(18)
= P;
(19)
(P$P = ( P P P
So the chemical potentials of adsorbates could be evaluated from the properties of the bulk phase: pf - (@)O
= pp - (p:)O = RT In $x!
(20)
Substituting (20) in (17) yields
A G S = -xA
+ nlA,xT + a2A2x;+ xtRT In (ryx?) +
where the chemical potentials of the adsorbates are similar to these used by Everett (1965) p? = (p$)O
where A, is the surface area per unit mass of adsorbent. If the excess molar area could be neglected, Le., AE = 0, then the molar area of the adsorbed solution is
The saturated amount adsorbed from the liquid mixture is m=
1
(29)
Substituting (19), (20),and (29) in (28), the equilibrium equation of AFLM is obtained:
(-r( E)
= exp(
s)
(30)
where the ratio of molecular size is approximated as the molar volume ratio of the adsorbates. A t constant temperature and pressure, the Gibbs-Duhem equation for binary adsorbed solutions becomes A d.rr = x i dg!
x;RT In (ybx)) (21) The occupied area of adsorbent by 1 mol of adsorbate i can be estimated as AI = A s / m , (22)
+ RT In ~ 7 x 7- (ai- a ) A i
+ X;
dp;
(31)
Substituting (18)in (31) and dividing through by dx5)gives
(
A -d n ) dx!
= xT( T,P
9) "!
2 ) ax!
+XI(
TP
(32)
T,fJ
From the bulk-phase Gibbs-Duhem equation,
Substituting (33) and (24) in (32) gives (34)
(24)
xs/m1 + xVm2 Substituting (22) and (23) in (21) and letting pl = x,A, (pl is known as the free energy of immersion), another equation of the Gibbs free energy of mixing is obtained:
If the Wilson equation is used to calculate the activity coefficients in the bulk phase, then 1\21
x!
+ A12X$
xi
+ h2,4
+
Comparison of (4) with (25) yields are Wilson parameters. where .Il2and Substituting (35) in (34), a differential equation is obtained:
Equations 10, 11, and 26 constituted the activity coefficient model for the adsorbed solutions. However, additional relationships among p, xy, and x! are also required for solution. Here the phase-exchange model (Everett, 1964, 1965; Adamson, 1976) is employed to describe the process of AFLM. In this model, the adsorption equilibrium is considered as the phase-exchange equilibrium between an
which describes the relationship between the free energy of immersion and the bulk-phase composition. Now, a set of equations has been established composed of ( I O ) , (ll),(26), (30), and (36). The parameters (mL,pl, riv) in those equations can be estimated respectively from
Ind. Eng. Chem. Res., Vol. 29, No. 4, 1990 563
-, h
0.0
6.1
0.2
01
3.t:
0.J
0.6
I
E )
C J
f',j/.o
Figure 1. n'(")vs xy for the benzene (1) + n-heptane (2) system.
adsorption isotherms of pure unsaturated vapors and VLE data. If the bulk-phase composition ($1 is given, the activity coefficients of bulk phase can be evaluated by the Wilson equation. There are only five unknown quantities left. They are xy (x; = 1 - xi), cp, yt, 792,and A12. This set of equations has a unique solution. The adsorbed-phase compositions can be obtained by solving this set of equations, and the surface excess amount could be calculated by n$n) = (x; - x ) ) / ( x ; / m l + x ; / m z )
(37)
Thus, the equilibrium data of AFLM could be predicted from adsorption isotherms of pure unsaturated vapors and VLE data.
Results of the Calculation and Discussion Five systems with nonideal adsorbed phases have been calculated by the new model. They are benzene + 1,2dichloroethane, n-heptane + cyclohexane, benzene + cyclohexane, benzene + n-heptane, and cyclohexane + 1,2dichloroethane/on silica gel a t 30 "C. It is of interest to note that, for systems with a small difference in the free energy of immersion between pure components, the first term of the numerator of (26) is small. If this term is neglected, (26) becomes
Calculations show that (26) and (26a) have nearly the same accuracy for the systems mentioned above. Nevertheless, the computation time with (26a) is much shorter than that with (26), so (26a) is recommended for systems with small differences in the free energy of immersion. In order to test the effectiveness of the new model, calculations have been made by using Sircar and Myers' model and also by (30) with the assumption ( ~ ~ / $ ) B ( y ~ / -=& 1. J In both of these models, the adsorbed phase was assumed to be ideal. They are ideal adsorbed solution models. Figures 1-5 show the results of the calculations. In these figures, the solid lines represent the predicted results of the new model, the dash lines represent the results of Sircar and Myers' method, the dash-dot lines re resent = the results of (30) with the assumption ( $ / $ ) B ( & $ ) 1, and the points represent experimental data measured
CJQ
4 1 o r
0 )
b &
0.5 Lit)
Figure 3. ns(")vs xy for the benzene (1) system.
6 ' 7
/
1'8
I ' f
/.o
+ 1,2-dichloroethane (2)
I
I
-
7
~
Figure 4. ns(")vs xp for the cyclohexane (2) + n-heptane (1) system. Table I component DroDerties pI, mmol/g
m,,
mmol/g
benzene 14.5 3.91
cyclohexane
n-heutane
7.90 3.06
7.90 2.37
1,2-dichloroethane 15.5 3.97
1
564
Ind. Eng. Chem. Res., Vol. 29, No. 4, 1990 G = Gibbs free energy m = saturated adsorbed amounts n = surface excess amount, mmol/g P = pressure, Pa
R = gas contant, J/(mol.K) T = temperature, K x = mole fraction Greek Symbols y = activity coefficients A = properties changes p = free energy of immersions T = spreading pressure A = Wilson parameters H = volume fractions
I 1 Figure 5. ns(")vs x! for the cyclohexane (1)+ 1,2-dichloroethane (2) system. T a b l e I1 Wilson parameters benzene + 1,2-dichloroethane n-heptane + cyclohexane benzene + cyclohexane benzene + n-heptane cyclohexane + 1,2-dichloroethane
1.778 1.573 0.664 0.515 0.292
0.441 0.563 0.901 0.887 0.777
by Sircar and Myers (1973) and Myers and Sircar (1972a,b). In the calculations, the pure component parameters needed were measured by Sircar and Myers (1970) and are shown in Table I. The Wilson parameters of the bulk phase are shown in Table I1 (Gmehling et al., 1980). As shown in these figures, when the difference in the free energy of immersion between two pure components is small, these three models give almost the same accuracy. Nevertheless, as the difference increases, the deviations of the dashed lines and dash-dot lines becomes obvious because of the ideal adsorbed solution assumption. These results indicate that the nonideality of the adsorbed phase increases with the difference in the free energy of immersion, and it might be much different from that of bulk phases because of the existence of adsorbents. Conclusions A new activity coefficient model with no adjustable parameters is proposed to describe the nonideality of adsorbed phases. In combination with the phase-exchange adsorption model, it can predict the equilibrium data of nonideal AFLM from pure adsorption isotherms and VLE data. The results of the five systems calculated are in good agreement with the experimental data, and these results also show that the effect of the adsorbent on the properties of adsorbed phases may be taken into account with the free energy of immersion. Nomenclature A = molar area of adsorbates or interchange energy parameter of adsorbed solutions B = molar volume ratio
Superscripts b = bulk phases e = excess amounts s = adsorbed phases s ( n ) = excess amount of adsorbates 0 = pure state Subscripts 1 = component i in the bulk or adsorbed phase p = pores of adsorbents s = surface
Literature Cited Adamson, A. M. Physical Chemistry of Surface; 3rd ed.; Wiley: New York, 1976. Bering, B. P.; Serpinskii, E. Zzo. Akad. N a u k , Ser. Khim. 1973,22, 3; Bull. Acad. Sei. C.S.S.R., Chem. Ser. 1973, 22, 1. Everett, D. H. Thermodynamics of adsorption from solutions. part 1: perfect systems. Trans. Faraday SOC.1964, 60, 1803. Everett, D. H. Thermodynamics of adsorption from solutions. part 2: imperfect systems. Trans. Faraday SOC. 1965, 61, 2478. Gmehling, iJ.; Onken, U.; Arlt, W. \'LE Data Collection-Aliphatic Hydrocarbons. DECHEMA, Chem. Data Ser. 1980, 1 (6a). Larionov, 0. G.; Chmutov, K. V.; Tudilevich, M. D. Adsorption from pure vapors and liquid mixtures on silica gel. Zh. Fiz. Khim. 1977, 51, 188. Lee, A. K. K. Lattice theory correction for binary gas adsorption equilibria on molecular sieves. Can. J . Chem. Eng. 1973,51,687. Moelwyn-Hughs, T. Physical Chemistry, 2nd revised ed.; Wiley: Oxford-London-New York, 1961, 828. Myers, A. L.; Sircar, S.A thermodynamic consistency test for adsorption from liquids and vapors on solids. J . Phys. Chem. 1972a, 76, 3412. Myers, A. L.; Sircar, S. Analogy between adsorption from liquids and adsorption from vapors. J . Ph3s. Chem. 1972b, 76, 3415. Prausnitz, J . M.; Lichtenthaler, R. N.; Azevedo, E. G. Molecular thermod.ynamics of fluid phase equilibria; Prentice-Hill: Englewood Cliffs, NJ, 1986. Sircar, S.;Myers, A. L. Statistical thermodynamics of adsorption from liquid mixtures on solids. I. Ideal adsoebed phases. J . Phys. Chem. 1970, 7 4 . 2818. Sircar, S.;Myers, .4.L. Predicted of adsorption at liquid-solid interface from adsorption isotherms of pure unsaturated vapors. AZChE J . 1973, 19, 159. Siskova, M.; Erdos, E.; Kadlec, 0. Relation between the adsorption from solutions and the adsorption of pure components from the gases phase on solid adsorbents. Collet. Czech. Chem. Commun. 1974, 39, 1954. Suwanayuan, S.; Danner, R. P. A gas adsorption isotherm equation based on vacancy solution theory. AZChE J . 1980a, 26, 68. Suwanayuan, S.; Danner, R. P. A gas adsorption isotherm of adsorption from gas mixtures. AZChE J . 1980b, 26, 76.
Received for reuiew January 31, 1989 Revised manuscript received November 6, 1989 Accepted November 19, 1989
