-
P
.\
+
Y
CY
(1 - x j
=
system characteristic for in-
terphase polycondensation, dimensionless.
SUBSCRIPTS
A A , AB, BB = terminal groups = continuous phase d = dispersed phase n = chain length = initial condition (0) W = molecular weight C
literature Cited
Abraham, W. H., IND.ENG.CHEM.FUNDAMENTALS 2, 221 (1963). Flory, P. J., “Principles of Polymer Chemistry,” p. 153, Cornel1 University Press, Ithaca, N. Y . , 1939. Goodrich,F. C., J . Chem. Phys. 35,No. 6,2101 (1961). Kilkson, H., IND.ENG.CHEM.FUNDAMENTALS 3, 281 (1964). Morgan, P. W., “Condensation Polymers. By Interfacial and Solution Methods,” Chap. 111, Intersciexe, New York, 1965. Scanlan, J. J., Trans. Faraday SOC. 52, 1286 (1956). Zeman, R. J., Amundsen, N. R., Chem. Eng.Sci.4,363 (1965). RECEIVED for review February 14, 1967 ACCEPTED April 12, 1963
IMPROVED TWO-DIMENSIONAL EQUATION OF STATE T O PREDICT ADSORPTION OF PURE AND MIXED HYDROCARBONS HOUSTON
K. P A Y N E , ’ G E O R G E A. S T U R D E V A N T , 2 A N D T H O M A SW . L E L A N D
Department of Chemical Engineering, Rice University, Houston, Tex. 77001
A two-dimensional equation of state has been derived from the Hirschfelder-Eyring modification of the van der Waals equation and applied to the prediction of high pressure adsorption of pure and mixed hydrocarbons on charcoal. Equations are derived to express the thermodynamic equilibrium between a mixed adsorbate and gas, assuming the validity of the mobile fluid model for the surface phase. Solution of these equations permits the total adsorption and surface composition to be calculated for a mixed isotherm from pure component adsorption data. New experimental data are presented for high pressure adsorption of methane, propane, butane, and their mixtures on charcoal.
ITH
the growing importance of industrial adsorption proc-
W esses there is a need for more extensive information on physical adsorption over \vide ranges of conditions and better correlations for describing the data. Ideally, this description should permit the extrapolation of low pressure isothermal adsorption data to high pressures. I t should be applicable to porous solids with heterogeneous surfaces and should allow for the extension of pure component data to the prediction of mixture adsorption. Models for Physical Adsorption
A great many analytical equations have been proposed to describe isothermal physical adsorption equilibrium. These are summarized in recent monographs by Ross and Olivier (1964) and Young and Crowell (1962). However, these equations can be conveniently classified into two main groups according to the basic physical model used in their origin. One group consists of equations based on localized adsorption or a Langmuir model. Although subject to many modifications, the basic premise is that a local site or small region on the surface interacts stoichiometrically with a fixed number of molecules from the gas. The adsorbed molecules on the surface may move over the surface and may have complicated interactions with each other; the gas phase may be highly nonPresent address, E. I . du Pont de Nemours & Co., Inc., Chattanooga, Tenn. Present address, Chevron Research, La Habra, Calif.
ideal. However, during the approach to equilibrium and after its attainment there is always a stoichiometric relation of the type
nA
+ mX e A,X,
(1)
where A represents a gas molecule, X represents an unoccupied element of the surface, and m and n are constants. The isotherm is obtained by minimizing the free energy of a gas plus solid a t a fixed T , V , and total mass, subject to the additional restriction of Equation 1. If this procedure is carried out by setting m and n equal to unity, assuming ideal gas conditions and no lateral surface interactions, the result is Langmuir’s equation. I t is unnecessary to make any assumptions regarding the kinetics of adsorption or desorption. A second group of adsorption equilibrium equations may be considered to be based on a mobile fluid model. I n this model there are no restrictions in the form of Equation 1 and a variable number of molecules may be compressed into a small element of surface. T h e equilibrium equation is obtained in a manner analogous to that for a multiphase system with components distributed among several fluid phases. Properties of the mobile adsorbate phase in this model may be described by means of a two-dimensional equation of state, as pointed out by de Boer (19531, who demonstrated a two-dimensional form of the van der Waals equation. Both of these basic models might be required for the same gas-solid system a t varying conditions, the localized adsorption model being more appropriate a t low temperatures and the VOL.
7 NO. 3 A U G U S T 1 9 6 8
363
Table 1.
Gibbs Adsorption, iMg.-Moles/ G. Charcoal C. on Charcoal 3,668 5.748 6.479 6,960 7.148 7.187 7.111 6.903 C. on Charcoal 1.860 3.122 4,321 5.240
5
10
Figure 1.
Schematic diagram of apparatus 1.
0- to 200-p.s.i.a. Heise g a g e
2. Vacuum-pressure gage 3.
4. 5. 6.
7.
a. 9.
10. 11. 12. 13. 14. 15. 16. 17.
Sample bomb Gas supply Dryer Three-way valve Diaphragm gage separator Hg vessel Adsorption column Volumetric Hg pump Reservoir Monometer Mcteod gage Vacuum pump Magnetic pump Vent Thermostat bath
mobile fluid model at high temperatures. The transition from one to the other has been discussed by Hill (1952). The approach used in this study is to use a fluid model ibith an improved t\vo-dimensional equation of state but to retain some allowance for a degree of ordering or localization on the surface. This is very much analogous to a cell model for a three-dimensional liquid, 11hich allows for fluidity but retains some of the order of a cell structure with an average coordination number. The equation of state with this facility is a t\vodimensional form of an equation of state proposed by Eyring and Hirschfelder (Hirschfelder et ai., 1954) as an improvement over the van der Waals equation at high densities. Experimental Data
Experimental measurements were obtained for the high pressure adsorption of the pure components methane. propane, and n-butane and the adsorption of methane-propane mixtures and methane-n-butane mixtures on activated charcoal. The experimental method was a direct measure of the Gibbs adsorption. The apparatus, shown in Figure 1, consists of two main sections: the injection system and the adsorption loop. The adsorption loop consists of a closed coil packed with a Lveighed amount of adsorbent. The void volume in the loop was determined by expanding helium into it as discussed by Payne (1964). Measured quantities of gas are admitted to the loop from the injection system and circulated in the closed loop with a magnetic pump. All parts of the apparatus containing gas are enclosed in a thermostated temperature bath. The moles adsorbed, according to the Gibbs definition, are the total moles injected less the number of moles filling the void volume a t equilibrium conditions. 364
Experimental Equilibrium Isotherms
I&EC FUNDAMENTALS
Pressure, P.S.I.A. CH4 at 10' 75.0 248 .O 400.2 598.2 799 .0 999.5 1250.5 1499.0 CHI a t 20' 26.5 72.0 150.0 253.0 400 0 549 0 702 5 896 5 1150 5 1400.5 1702.5 1993 5 CHI a t 30' C. on 23 5 67.0 108.0 202.5 348 0 511 5 704 0 950 5 1201 5 1454.5 1?06,5 1994.0
5 975
6 388 6 640 6 789 6 773 6.690 6.452 6.154 Charcoal 1 435 2,678 3.416 4.478 5 388 5 929 6 274 6 485 6 505 6.400 6.244 5,974
Pressure, P.S.I.A. CH4 at 40" 23.5 73 5 199 5 399 0 603 0 804.0 1003.0 1251 . o 1501 . O 1810.5 CH4 a t 50' 24.5 75 . O 149.5 200.2 399 . O 605.5 799.8 999 .O 1249.0 1500.0 1751 0 1998 5 1304 0
Gibbs Adsorption, Mg.-Moles/ G. Charcoal C. on Charcoal 1 110 2 417 3 983 5 134 5 694 5.982 6.155 6,180 6.105 5.923 C. on Charcoal 0.978 2.153 3,152 3.623 4.722 5.285 5.576 5.730 5,809 5 761 5.665 5.545 5.791
For the mixture isotherms a small sample of the nonadsorbed gas is removed for analysis by a gas chromatograph. For the mixture isotherms a measured quantity of the heavier hydrocarbon is injected first and the methane added afterward in measured increments. The gas composition and total equilibrium pressure are measured when equilibrium is established after each methane injection. From these measurements along with the known void volume and the compressibility factor for the mixture. the total Gibbs adsorption with the equilibrium composition of adsorbate and gas phases can be determined by a material balance on each component. The necessary compressibility factor data were obtained by interpolation in the tables presented by Sage and Lacey (1 950) and Din (1961). Additional experimental details are given by Sturdevant (1966) and Payne (1964). Experimental adsorption results for methane. propane, and n-butane on charcoal are presented in Tables I , 11: and 111. Data for mixtures of methane and propane and methane and n-butane are given in Tables IV and V. Adsorbents Studied
Two different charcoal adsorbents were studied. Each charcoal sample had a measured BET surface area of 1157 i 30 sq. meters per gram. The charcoal used by Sturdevant (1966) was activated by heating in air a t 500' F. for 7 days. The charcoal used by Payne (1964) was activated in a different manner. I t was heated for about 1 2 hours in vacuum at 275' C. until the vacuum gage indicated a pressure less than 15 microns. Both charcoals were originally Columbia G grade 8/14 mesh. Experimental data for adsorption of hydrocarbons obtained by Payne and Sturdevant were used to evaluate constants for the 2-D Eyring equation. Data of
~
Table II.
Experimental Equilibrium Adsorption Isotherms
Gibbs Adsorption, M g .-Moles / G. Charcoal
Pressure, P.S.I.A.
CaHs at 20' C. on Charcoal 0.123 0.344 0.815 2,004 5.093 10.850 19.416 39.500 65.000 88,000
0.892 1.810 2.740 3.673 4.556 5.223 5,694 6.291 6,683 6.982
CJHa at 30" C. on Charcoal 0,049 0,241 0,889 2.711 6,660 10.948 19.509 36.000 61.500 93,000 131.000
0,687 1.614 2.602 3,600 4,402 4.828 5.305 5.837 6.215 6.530 6.828
Pressure, P.S.I.A.
Gibbs Adsorption, Mg.-Moles/ G. Charcoal
C3Ha at 50" C. on Charcoal 0.476 1.817 5.756 9.946 19.253 40.000 72.000 106,500 137 ,000 167.000 198.500
1.234 2.396 3.548 4.074 4.675 5.265 5.747 6,034 6.186 6.293 6.415
CaHs at 60' C. on Charcoal 0.349 1.528 4.558 10.221 20.614 41.000 76.500 116.000 162.500 196.500
0.946 1.983 3.015 3.791 4.431 4.987 5.494 5.767 5.941 6,048
Table IV.
Experimental Equilibrium Adsorption Isotherms
(CHd-CBH,a t 30' C. on Charcoal) Gibbs Adsorption, M g . Moles/G. Charcoal C~HB Total
Pressure, P.S.I.A.
Mole Fraction C B H Bin Gas
52.0 101.5 201.5 401.5 601.5 802.0 1001 0 1251 0
0.0766 0,0480 0.0332 0.0285 0,0296 0.0300 0 0316 0 0324
3.895 3.884 3 860 3.796 3.707 3.615 3 500 3 358
4.377 4.777 5.326 6 036 6 512 6 857 7 131 7 410
102 5 205 5 302.5 403.0 501.5 749.0 1002 5 1303 5 1604 0 1902 5
0 0237 0 0189 0 0167 0 0173 0 0176 0 0190 0 0187 0 0221 0 0247 0 0249
3 3 3 3 3 3 3 2 2 2
4 5 5 6 6 6 7 7 7 7
0 0132 0 00950 0 00730 0 00746 0,01200 0,00895
2 219 2 215 2 208 2 198 2.148 2.136
50 0 102 0 201 0 301 5 500.0 751.5
279 260 244 218 193 116 047 893 719 588
442 151
io;
004 266 789 203 370 557 677
1 -270 __
3 956 4 817 5 425 6.185 6.810
CiH8 at 40" C. on Charcoa 0.241 0.997 3.188 9.813 19.941 40.500 72,500 101 ,500 128.500 158 500
Table 111.
Pressure, P.S.I.A.
1.132 2.263 3.385 4.433 5.056 5.619 6.081 6.322 6.478 6.638
Gibbs Adsorption, iMg.-MoIes/ G. Charcoal
4.354 4.532 4.696 4.903
n-CaHlo at 50' C: on Charcoal 0.5 1.o 1.5 4 0 14 0 35.5 59.5
Pressure, P.S.I.A.
1.479 2.112 2.818 3.452 3.976 4,324 4.629
Experimental Equilibrium Adsorption Isotherms
.+foleFraction n-CIHlo in Gas
Gibbs Adsorbtion. Mg.-.Moles/G.'Charcoai n-C4HIo Total
CH4-n-C4Hloat 40' C. on Charcoal
Experimental Equilibrium Adsorption Isotherms
n-C4HIoat 40" C. on Charcoal
19.0 28.5 39.5 50.5
Table V.
Pressure, P.S.I.A.
Gibbs Adsorption, Mg.-Moles/ G. Charcoal
n-CdHlo at 60' C. on Charcoal 0.5 1.5 8.0 19.0 31.5 41 . O 50.5 60.5 70.5 80.5 86.0
1.875 2.788 3.610 3.925 4.099 4.195 4.287 4.371 4,441 4.551 4.608
n-C4H10 at 70" C. on Charcoa 0.5 2.0 9.5 18.0 30.0 40.0 51 . O 60.5 71 . O 80.5 90.5 100.0 109.5
1.871 2.816 3,500 3.787 3.956 4.059 4.136 4.197 4.262 4.316 4.375 4.441 4.557
2.0 190.0 397 . O 692.0 992.0 1243.5 1497.0 1747.5 1985.5
1.000 0.0192 0.0172 0.0164 0.0171 0.0186 0.0199 0.0211 0.0225
3.467 3.445 3.402 3 318 3.255 3.161 3,056 2,944 3.819
3,467 3.904 3.978 3.938 3.808 3,644 3.454 3.210 2.977
257.5 500.0 752.0 999.5 1249.5 1498.0 1749,O 1990,5
0,000920 0 , 0 0 1398 0.001593 0 002118 0 002597 0 003334 0 003843 0 00459
1.743 1.737 i 710 1 7ij 1 700 1 674 1 648 1 611
3.681 4.154 4.363 4.376 4,360 4.231 4.011 3.826
5.5 248.5 502.0 751.5 994.5 1247.0 1496.5 1750.0 1986.5
CH4-n-C4HIoat 60' C. on Charcoal 1,000 3.426 0.0352 3.386 0.0278 3.318 0.0260 3.245 0.0259 3.159 0.0263 3.060 0.0264 2.964 0.0270 3.853 0.0271 2,759
3.426 3.727 3.751 3.684 3.572 3,428 3.242 3.039 2,844
9.5 249.5 490.5 750.0 1001.5 1244.5 1497 . O 1746.0
CH4-n-CdHlo at 70' C. on Charcoal 1.000 3.383 0.0462 3.358 0.0348 3.288 0.0309 3.210 0.0298 3.122 0.0296 3.029 0.0294 2,932 0.0294 2.830
3.383 3.635 3.622 3.512 3.399 3.266 3.059 2.869
VOL. 7
NO. 3
AUGUST
1968
365
1
I
*.
I
i 0
I 500
0
1500
1000
2000
PRESSURE, (PSIA)
Figure 2.
L
c
O
8
E
7
\ P
M
-al
Gibbs definition of methane adsorption
6
0
E l
5
-
4 3 I-
n
a 0 cn D
4
EYRING CORRELATION
A 10. C, EXPERIMENTAL 0 2ooc. 0 30% 0 4OOC. v 5OOC.
2
I
Y
I
I
200
0
400
I
600 GAS
Figure 3.
I
I
000
I
I
I
1000
I200
1400
FUGACITY (PSlA)
Methane absolute adsorption data and Eyring correlation
Haydel (1965) for adsorption of CHd and C3Hg on silica gel to 2-D Eyring equation constants. The silica gel was Davidson Chemical Co. Grade 15 with a measured BET surface of 803.5 sq. meters per gram. The charcoal surface area was similar to that of the charcoal used by Ray and Box (1950), which had a reported BET surface of 1152 sq. meters per gram.
I t is easy to show (Sturdevant, 1966) that the number of moles of absolute adsorption, ns, is related to the Gibbs adsorp-
by ns =
[nSIGibba -
(1 Experimental Gibbs Adsorption Data and Absolute Adsorption
The experimentally determined Gibbs adsorption is defined as the number of moles which must be injected into a fixed evacuated volume containing the adsorbent to bring the equilibrium pressure and temperature to the desired values, less the number of moles which \ \ o d d be injected into the same evacuated apparatus to produce these equilibrium conditions if no adsorption occurred. The absolute adsorption, as defined by Young and Crowell (1962), is the total number of moles of gas in the range of the operative surface molecular forces between gas and solid. The absolute adsorption and Gibbs adsorption differ because the number of moles not affected by the surface forces is actually smaller than the number of moles which would occupy the container if there xvere no adsorption. The difference is due to the volume of the adsorbed phase. 366
l&EC FUNDAMENTALS
1600
(2)
);-
Isotherms of Gibbs adsorption data (Figure 2 ) often show a maximum when plotted against pressure, whereas the absolute isotherms (Figure 3) do not. Adsorbed Phase Volumes
Recently, Haydel (1965) developed a method for the direct experimental measurement of v" using perturbation gas chrowas matography. I n this work, howwer, a fitted value of used to relate the experimental Gibbs adsorption to the absolute molal adsorption area which appears in the two-dimensional equation of state. This relationship is as follows:
va
.=(
BET area
g. adsorbent
)(
g. adsorbent) ( I - ; ) = ;
(3)
nSGibba
The value of Y" can be expressed as a product of the range of the effective potential of the solid surface into the gas phase,
designated by h, and the molal adsorption area, 01. For a mobile fluid model, where adsorbed molecules are not 10calized a t a fixed point or vibrating relative to a fixed location, a realistic form of the effective solid potential is given by the concept of “mechanical adsorption” proposed by Hill (1948), which defines the absolute adsorbed layer as the region in which the translational energy normal to the surface is less than the interaction potential between a gas molecule and the entire solid. Using the analytical form of this potential proposed by Hill, the absorbate region has a n effective thickness, h, of , c is the distance for a zero value in approximately 2 ~ where the Lennard-Jones potential between a gas molecule and a single atom of the solid. The value of h should therefore be independent of pressure a t pressures up to the limit of monolayer adsorption, provided the lateral interaction on the surface does not affect the potential function for the interaction normal to the surface. If one uses the c value for methane of 3.81 A , , the value of h is then 7.6 X cm. This agrees well with the value of 6 X lop8 proposed by Kemball and Rideal (1946). The value of is therefore on the order of 7.6 X l o p 8 CY. Although h may be constant with pressure, v“ on a fixed surface area will change greatly with pressure because of the variation in the moles adsorbed. At high pressures in the region of a n isotherm where the amount adsorbed changes but little with pressure, v“ is relatively constant. At low pressures where the surface phase for the mobile fluid model becomes an ideal twodimensional (2-D) gas, the value of v“ approaches an infinitely large value numerically equal to h R T / a where T is the lateral 2-D pressure. I t is entirely possible that some fluids which are described well by the mobile fluid model a t high surface coverages may become highly localized at very small coverages. As an indication of this, large abrupt increases in the isosteric heat of adsorption as surface coverages approach zero have been reported for some systems (Aston et al., 1961). I t is possible that systems of this type would not approach a n ideal 2-D gas limit and would not behave as a mobile fluid a t all a t low coverages. This is probably not the case for the charcoal studied here. Adsorption on graphite shows no sudden increase in heat of adsorption and actually has a decreasing heat of adsorption as coverages approach zero.
vs
Adsorption Thermodynamics for a Mobile Fluid Model on a Heterogeneous Surface
The fluid model lends itself readily to a thermodynamic analysis. Consider a gas and solid system with temperature T , total volume V , and total moles S remaining fixed lvhile the molecules of various components distribute themselves between a bulk gas phase and an adsorbed phase until a n equilibrium distribution is attained. The process is described thermodynamically by (dA)T,,~,.Y2 0
(4)
with the equality applying a t the equilibrium state. I n order to express the Helmholtz free energy, A , as an exact total differential of the variations in properties of the system, it is necessary to subdivide the system into the various homogeneous regions which comprise it and write the total d A as the sum of the variations of A for each homogeneous portion. T h e bulk solid region is considered to have no variation at all in its properties during the approach to equilibrium. The homogeneous regions which undergo variation are considered to be the bulk gas phase plus small patches or regions on the solid surface which have a uniform interaction potential
between the solid and adsorbing molecules and uniform properties of the adsorbate in the region. Sanford and Ross (1954) have used the term “homotattic” to define small homogeneous subregions of this type. T h e concept of dividing a complex surface into many homogeneous surface regions has been discussed by Ross and Olivier (1964). Neighboring patches may differ appreciably in the value of the potential between solid and adsorbate, in the density of the adsorbate, and in the area of the small region. However, within any one subregion or patch, the thermodynamic properties are considered to be uniform throughout. The total free energy change of the system is then r
1
(5) where subscript j refers to a single homogeneous patch on the surface, superscript g represents the bulk gas phase, and s represents the adsorbate layer on the surface. I t is easy to show that the assumption of a lateral equilibrium between adjacent homogeneous patches in Equation 5 implies, for directions parallel to the surface, a common two-dimensional pressure and a common two-dimensional chemical potential for the entire heterogeneous surface. The equilibrium adsorption isotherm for n components on a heterogeneous surface \\.hen the mobile fluid model is applicable may then be obtained from Equation 5 in the form r
I-adK
1
- V S d P Q+
= 0
nisdp,l] i=l
T
The symbol K represents the common 2-D pressure, Po is the 3-D gas phase pressure, a is the total surface area, V sis the total volume of the adsorbed phase, nt is the number of moles of component i on the surface, and p i is the equilibrium chemical potential for component i common to all regions. The amount adsorbed per unit area and the adsorption potential between adsorbate and surface may vary widely between one subregion of the surface and another. Ho\cever, the common 2-D pressure and common fugacity of each component for all regions allow a single 2-D equation of state to define the value of T in Equation 6 for the entire heterogeneous surface. This one equation of state expresses the common surface 2-D pressure in terms of the total moles adsorbed on the entire surface and parameters \vhich depend on average solidadsorbate interactions for the composite surface. These averages may be obtained by assuming a Gaussian or other type of distribution function for the vertical potential behveen the solid and adsorbate (Ross and Olivier, 1964) or may be determined experimentally from low pressure adsorption data. I f the adsorbate and gas consist of one component only, Equation 6 contains only nsdpS in place of c n i g d p i s . Since 1
dps = dpO = V Q d Pat equilibrium, the equation may then be written [ v g
dP0 = Otdr
+ v” d P ] ] ,
(7)
Equation 7 may also be rearranged by using Equation 2 to give the entirely equivalent form [ c i ~ i b b . dT
=
dp’]~
(8)
Surface Fugacities
The equilibrium adsorption isotherm for a pure component is obtained by integration of Equation 7 after one assumes a particular form for an equation of state to express the 2-D pressure, T , as a function of T and a. At low pressures, where V O L . 7 NO. 3
A U G U S T 1968
367
the difference between Gibbs adsorption and absolute adsorption is negligibly small, the v” dP” term in Equation 7 can be neglected. However, a t pressures where the Gibbs adsorption decreases with pressure, the v“ dPQterm is important and cannot be neglected. Since the two sides of Equation 7 represent dpo and dp’ at a constant temperature, the left side may be expressed in terms of the gas-phase fugacity and the terms on the right may be thought of as parallel and vertical components of a total surface fugacity. The terms in Equation 7 correspond to RTdlnfQ= RTdlnf=sfRTdInfl’
(9)
T h e terms in Equation 9 may be integrated relative to a set of arbitrarily chosen pure states at the system temperature chosen as a reference or standard for each term. If this is done, Equation 9 expresses the following relationship among the fugacities
f=S
=
[(e-
g)(g)]Yb)
=
Kf*
c$)
= K , (IO)
The f Qand f=’ terms may be evaluated by conventional methods from 3-D and 2-D equations of state, respectively. The value off=’, the fugacity parallel to the surface, is given by
The integral is evaluated isothermally from an analytical 2-D equation of state relating n and cy. The ( e -
g)term in Equation 10 is completely analogous
to the thermodynamic “equilibrium constant” in the law of mass action for a reaction equilibrium. I n this case the “reaction” is one mole of an adsorbed substance changing to one mole of a gaseous product when each is a t the reference conditions. The Kf*term is a function only of temperature, \vith its numerical values depending on the reference states chosen for f E 0 and f,”. The ratio (f,”/f=O) gives dimensions of reciprocal length to Kf*. Setting the ratiofLS/fio equal to unity a t all conditions is equivalent to neglecting the v“ dP” term entirely in Equation 7. If this is done, Equation 10 may be approximated by
-i”
f=’ -
K,*
(12)
Equation 1 2 is a very good approximation over the pressure range for which Gibbs adsorption and absolute adsorption are in close agreement. The high pressure correction to Equation 12 can be evaluated by integration of d lnf=’ as it is defined by Equations 9 and 11 from the reference state pressure, Po,used in the definition of AGO to any gas pressure PO. This evaluates the ratio so that Equation 12 becomes
(flS,fLo)
Equation 13 may be regarded as the complete mobile fluid adsorption isotherm for a pure component at high pressure. The exponential in Equation 13 acts as a high pressure correction to the K,* value which applies a t low pressures. The choice of the reference state pressure Po,in Equation 13 is completely arbitrary, although the value of 1 atm. is probably most convenient for computational. purposes. 368
I&EC FUNDAMENTALS
Selection of Reference States
Although any reference states may be chosen for the standard state fugacities in Equation IO, the most convenient are as follows : f,” = pure gas at the system temperature and a unit fugacityof 1 atm. f-0 = pure adsorbed component at a horizontal surface fugacity of 1 atm.-cm. f O = pure adsorbed component in a reference state which exerts a pressure of Po atm. normal to the surface. The value of Po is conveniently chosen as either an average over the pressure range of the isotherm, or 1 atm. A@ = (p: - p-0 - p10) = free energy change for the isothermal desorption of one mole of pure adsorbate from a surface where it exerts a vertical pressure of PO and has a horizontal fugacity of unity, followed by a compression or expansion of the desorbed material to form a gas at a fugacity of l atm.
\Vith the reference states of unity for f,” and f=O, the value of K,* is numerically equal to K , and the K’, value thus gives a direct measure of A@ for the change between the reference states. Although a standard state of unity for f=O is convenient numerically, the 1 atm.-cm. surface fugacity corresponds to a high surface density of the adsorbed molecules with a great deal of lateral interaction on the surface. I f one wishes to obtain a standard free energy change between the gas reference and a n adsorbate reference state which is due primarily to adsorbate-solid interaction \+ith almost no lateral molecular interaction on the surface, it is better to use the adsorbate reference state recommended by Kemball and Rideal (1 946). This reference is a standard surface fugacity corresponding to approximately the same molecular spacing on the surface as in an ideal gas at 1 atm. This reference state may be shown to be (f=o) = (5.15 X lo-* atm.-cm.
T1I3
(14)
Evaluation of Adsorption Isotherm
In evaluating the integral in the exponent of Equation 13 the reference state Po of 1 atm. is high enough to assure that acts as the molal volume of a condensed phase lvhich might be replaced with a constant average over the pressure range. Equation 13 then becomes
vs
The original evaluation off=’ in Equation 13 was developed by de Boer from Equation 11 using the two-dimensional analog of the van der Waals equation. This equation and the twodimensional van der Waals constants a2 and 6 2 have been discussed by de Boer and Kruyer (1958), Cassel (1944), Hill (1952), and Ross and Olivier (1964). de Boer and Kruyer used the value of 62 derived in two dimensions in the same manner as the three-dimensional van der Waals 6 constant is obtained for hard spheres. The result is
where n’ is 3.1415 and A’ is Avogadro’s number. Using this 62 with the adsorption data of Ray and Box for methane on charcoal they found that the isothermal data could not be fitted with an a2 value which was independent of pressure.
100.0
The a2 value obtained was actually negative a t low pressures and positive a t high pressures. This would indicate an intermolecular repulsion a t low densities changing to attraction a t high densities, which is difficult to understand for a molecule like methane. None of the isotherms obtained in this work could be fitted by the two-dimensional van der Waals equation with a single a2 value.
o
4OOC.
A
5OOC.
Two-Dimensional Eyring Equation
As a n improvement, this study uses a modification of the van der Waals equation derived originally by Eyring and Hirschfelder from a simplified cell model for liquids. The three-dimensional Eyring equation is
N )_I
a
+
x E Y
I L
Constants a and b have roles similar to the van der Waals a and b constants. The c constant depends on the shape of the cell assumed for the fluid model. The two-dimensional analog is
The a2 and 62 constants are analogous to the corresponding twodimensional van der Waals constants. The c z constant depends ideally on the shape of a n average two-dimensional cell structure assumed for the fluid model and should vary with the adsorbent used. O n a heterogeneous surface these constants must represent averages over the various homogeneous subregions making up the surface. From this equation the value of fZs may be found from Equation 11. Substituting this f=*value into Equation 13 produces the complete isotherm for high pressures. Vav. (PO- PO)
RT [Kf*c j o =
RT
(a1/' -~2b2~/~)~
Evaluation of 2-D Equation Constants
T h e Eyring isotherm, Equation 13, can be rearranged in linear form by defining
Y
=
X=
Xaz f In ( R T K,)
-
(A)
The slope of a plot of I' us. X is the value of a2 and the intercept is In ( R T K,*). T h e value of b2 was taken in every case from Equation 16 using the 3-D van der Waals b constant for the gas. The value of a is the absolute adsorption which is obtained from the experimental Gibbs adsorption as shown in Equation 3. For analysis of data a t high surface coverages an the adsorbed volume on the surface, initial trial value of was taken as the molal liquid volume. T h e subscript 2 has been deleted from the c term in Equation 20 and this term without the subscript will henceforth be considered to apply always to a two-dimensional phase. An optimization procedure \cas used to determine c, by obtaining a least squares fit of I' us. X with several values of c. The value of c
v",
3
h a
0.00lI
I
0 .I
1 I I I I I I I I ,2 .3 .4 .5 -6 .7 .8 -9 1.0 1.1
C
Figure 4.
Evaluation of c for n-butane
Sum of squares of deviations b y Eyring isotherm (this study)
which gave the minimum sum of squares deviations of Y from the straight line was chosen. Frequently a very sharp indication of the value of c could be obtained, as shoivn in Figure 4. I n general, the smaller the gas molecule and the higher the temperature, the more insensitive the fit becomes to the value of c. Fugacity data presented by Din (1961) and by Sage and Lacey (1950) were used for the hydrocarbon gases. The experimental results for c are given in Table VI. For high pressure data this procedure was then repeated with a different trial value of 7'.The value of c and other constants accompanying the value giving minimum sum of squares of deviations were then adopted. This precise determination of was necessary only a t the highest pressures. At pressures below roughly 35 atm. for methane through butane a t the temperatures studied here the results are almost completely insensitive to the value used for p'. Only the methane data on charcoal were taken a t pressures high enough to permit a n accurate determination of 7'.The results obtained for CHI on charcoal were
where 7'has the units of cubic centimeters per gram mole and t is in degrees centigrade. For the other hydrocarbons the molal liquid volume was used, although the pressure was low enough for the results to be insensitive to the actual numerical value used for v". Only a t the highest pressure does the v"(P - P o ) ) / ( R Tterm ) contribute significantly to the value of Y in Equation 20. T h e plot of Y us. X for methane is made in Figure 5 by neglecting this correction completely. Considering the reference pressure as Po = 1 atni., the P ( P g - P O ) / ( R Tterm ) for methane is only about 1% of the value of I' a t pressures of about 2000 p.s.i.a. This causes a very slight deviation from linearity a t the highest VOL. 7 NO. 3
AUGUST 1 9 6 8
369
Table VI.
Average 2-D Eyring Parameters
Adsorbate
0.81 0.76 0.95
10-40
0.70
SILICAGEL 2.04 x 1013
30-40
0.50
1.10
Ref.
O
CH4 This work C3Hs This work CIHlo This work
c.
G
10-50 30-60 40
CH4
Haydel (1965) C3Hs Haydel (1965)
x
Ergs Sq. Cm./ ( M g . Mole)=
bz, Sq. Cm./ M g . Mole
Mean Square Error, d%=a/(No.)
2.04 5.30 6.72
x x
1013 1013 1013
9 . 9 x 106 15.6 X 106 20.0 x 106
0.041 0.076 0.154
2.04
x x
1013
x
106
0.023
15.6 X 106
0.038
(aZ)Thear.
(a2)Exptll
Ergs Sq. Cm ./ ( M g . Mo1e)l CHARCOAL 2.04 x 1013 -2.40 x 1013 9.06 x 1013
Temp. Range,
5.30
1013
x
9.9
1013
13.8
-
.Y
I -IN
I
%
W
n
-IN
u I
l7*5
'd
0)
n 0 I
N I '
v
13.4
13.2
-
13.0
-
12.8
-
3 Y
I S J
C
-c
II
).
II
>
I zo -60
-40
-50
-30
-2 x =- 8 3170 TOC
Figure
5.
-20
-10
0
10'~
-40
Linearized Eyring isotherm for methane at
X =
30' C. In
Figure 6.
= 0.810
c
= 2.04 X 10'3 X lo5
Significance of a2 Term
If there were no effect of the solid on surface molecular interactions, the va1u.e of a2 could be calculated directly from the intermolecular potential obtained from bulk gas properties. T h e Eyring model assumes that the average lattice structure on the heterogeneous surface influences the free volume, through the c term, but the molecular interactions are described by the usual van der Waals method. Using the conventional method for relating a van der Waals interaction term to a potential function by assuming a random distribution a t separation distances greater than u, the value of the two-dimensional a2 term for a Lennard-Jones potential is
370
3 - N2?r'eu2
(22)
5
l&EC FUNDAMENTALS
0
x IO'*
Linearized Eyring isotherm for propane at = 0.760 bz = 159 X l o 3
pressures. For the other hydrocarbons the p ( P Q- P o ) / ( R T ) term is negligible a t all pressures. An example of a plot of Equation 20 for propane adsorption on charcoal is shown in Figure 6. The negative slope indicates a n intermolecular repulsion on the surface, in contrast to the positive slope shown for methane in Figure 5.
where N is Avogadro's number and s' is 3.1415
-2
83170. T*
-10
E
bz = 9.9
a2 =
-20
30" C.
Kf* = 18.21 6 5
a2
-30
. , ,.
Discrepancies between the calculated a2 of Equation 22 and the experimental value given by the slope of plots such as Figure 6 may be considered as a measure of the effect of the solid on the molecular interactions on the surface. de Boer and Kruyer (1958) considered this disagreement to be due entirely to an induced dipole moment caused by the surface potential. The magnitude of this dipole moment is proportional to the difference in a2 calculated theoretically and the value obtained by fitting experimental data. If the experimental result is less than the calculated, de Boer and Kruyer explain the reduction in interaction by assuming induced dipoles aligned on the surface to cause a lessening of the free molecular van der Waals attraction. Similarly, an experimental a2 greater than the result calculated for molecules uninfluenced by the surface indicates induced dipoles aligned in a way to cause additional attraction. I n the 2-D van der Waals analysis of the Ray and Box isotherms of methane on charcoal, de Boer and Kruyer explained the variation in a2 value from negative a t low coverage to positive a t high coverages by postulating an initial surface polarization followed by a depolarization caused by the increasing adsorbate density. I n applying the 2-D Eyring equation to methane on charcoal this situation is not found a t all. Instead, for a coverage range even greater than that
studied by Ray and Box, the Eyring equation a2 value found by optimizing the fit of the experimental data agreed exactly with the theoretical a2 value. This same value of u z also produced a n effective correlation of CHd adsorption on silica gel using the 2-D Eyring equation to check the data of Haydel (1965). Results are shown in Table V I . T h e values of a2 obtained for hydrocarbons other than methane all disagreed with the theoretically calculated values. Only propane showed a lessening of the gas-phase pair interactions by the surface. O n charcoal the propane showed repulsion over the entire range, as shown by the negative slope in Figure 6. O n silica gel the magnitude of the interaction was lessened but remained a n attraction. Butane interactions showed enhanced attraction on the surface. All of the a2 values except those for methane showed some temperature dependence and the values reported in Table V I represent averages over a temperature range.
Table VII. Structure
Calculated Values of c
3-D Simple cubic Body-centered cubic Face-centered cubic 2 -D
Square Hexagonal
C
Ref.
0.7816 0.7163 0.6962
Hirschfelder et al. (1954)
0.798 0.743
Sturdevant (1966)
gas when the gas and adsorbate are in a standard state. interaction is described thermodynamically by AGO = RT In K,*
A P = R Significance of c Term
The third constant, c, takes into account the effect of the solid surface on the free area available to molecular motion. Its theoretical numerical value for simple molecules can be calculated from the shape of a cell structure assumed for the various subregions of the surface. The theoretical values of c for 3-D and 2-D cases have been calculated in Table V I I . O n an irregular heterogeneous surface the value of c must be regarded as a n average and may not correspond exactly to any of the calculated results in Table V I I . The experimental c values are given in Table V I . For complex molecules the c parameter serves also as a means of modifying the 2-D analog of the 3-D van der Waals 6 term to account for orienting effects of the surface on individual molecules. For example, the van der Waals 6 term for a 3-D gas accounts for encounters between more or less randomly oriented molecules. When converted to a corresponding 2-D value this will need modification if the surface produces encounters only between molecules lying flat on the surface or only between molecules standing on end. High values of c indicate a loose packing and a smaller coordination number. The lower the value of c, the more dense the packing. As c becomes very small, the adsorption isotherm approximates the Fowler-Guggenheim (Fowler and Guggenheim, 1949) result for a n immobile and completely localized adsorption with a van der Waals type of interaction between molecules on the surface. When the fraction of the available surface covered is small, both the Fowler-Guggenheim isotherm and Equation 19 with c = 0 have the form
jo
=
(a> RT K ,
exp
(h) R Ta
where k is a dimensionless constant and K , is given by Equation 10. T h e c term thus allows for the fact that the transition between localized and mobile fluid adsorption may not have a sharp dividing line. The c terms obtained also showed some temperature dependence. The values in Table V I are averages over the range shown. Values of c on silica gel showed the largest temperature dependence and varied by almost 10% over the range studied. I n general, the c values were significantly lower on silica gel than on charcoal, indicating a greater tendency toward localized adsorption on the silica gel. Experimentally Measured Valuer of K,*
The K,* term obtained by fitting the isotherm to the experimental data is related to the interaction between solid and
This
b In K,* b 1/T
These K,* values could be correlated very accurately by the equation (25) where A and B are constants obtained from fitting the experimental K,* values. From Equation 24 these constants are also related by AF?' -_ - A -
RT
T
T
where AHo and ASo are molal enthalpy and entropy changes between the gas and adsorbate a t unit fugacities. If the reference state for the adsorbate,f,O, is changed to that given by Equation 14, constants A and B are related to
where AH" and AS+ are, respectively, the molal enthalpy and entropy changes between a pure gas a t unit fugacity and a n adsorbate with a mean molecular spacing on the surface similar to that in the gas. T h e functions in Equation 27 represent changes on desorption of molecules not strongly interacting with their surface neighbors into the gas a t 1 atm. Values of A and In B for Equation 25 and derived values a t 30' C. are presented in Table V I I I . The entropy change, AS+. for desorption into a gas a t 1 atm. from a surface state with similar intermolecular spacing agrees well with the entropy change of vaporization a t 1 atm. for methane. However, the corresponding values for propane and butane are appreciably greater than the changes on vaporization a t 1 atm. These standard entropy changes for desorption are essentially the same on both silica gel and charcoal. This standard enthalpy change AB+ is about tirice the latent heat of vaporization a t 1 atm., as shown in Table V I I I . VOL. 7
NO. 3
AUGUST
1968
371
Table VIII.
Evaluation of (RT K,) and Standard Entropy and Enthalpy Changes Valiier -...- - a -t. .?Oo - . C. -
AH+,
A, O K. In B ( R T K,') kcal./g. mole kcal./g. mole 2190 24.47b 18.216b 3.95 1.95 8.90 8.8 1560 24.47 20.414 2.70 1.95 8.90 8.8 C.H. Charcoal 411 25.99 12.740 8.15 4.48 10.42 9.7 _n. _ 4.48 10.57 9.7 5.46 17.404 2950 26 14 GiFa gel C3Ha 5.16 14.04 9.5 11 . l l 11.46 5800 29.61 Charcoal C4HlO AH, = Latent heat of liquid at I atm. AS, = Entropy of vaporization a Standard state of adsorbate with same au. molecular separation as in gas ut 7 atm. o j liquid at 1 atm. b K,* has units ojcm.-l, withf,Q andf,0 dejined at unit fugacities, T = K., R = (0.08206)(14.7)p.s.i.a. cc.lmg. mole a K . Absorbate
CHI CH, - 1
Adsorbent Charcoal Silica gel
I
Mobile Fluid Adsorption Isotherms for Mixtures
The mobile fluid model can be extended to mixtures by writing Equation 6 in terms of the individual fugacities of the various components present. As in the treatment of pure components, these fugacities will be divided into those applying to the gas phase, those which depend on the intensive properties of the adsorbate exerted parallel to the surface, and fugacity terms depending on the total gas-phase pressure exerted normal to the surface. At low and moderate pressures the adsorbate phase fugacity terms which depend on the gas-phase pressure normal to the surface can be neglected and only the parallel surface fugacity terms need to be considered. The symbol ( f t ) - is used here to represent the fugacity of component i, depending on the lateral or 2-D properties of the surface phase and the symbol ( f i ) represents the adsorbate fugacity terms arising from the total gas pressure for component i. Equation 8 may be expressed in terms of these individual component fugacities and then integrated isothermally and a t constant surface composition from the low pressure equilibrium state corresponding to P Q and n values approaching zero to an equilibrium state corresponding to higher values of Po and a. The gas-phase composition is allowed to vary during the integration in order to keep the equilibrium adsorbate a t the same surface composition. At the lower limit the equilibrium is described by
where ( K f * ) ris the low pressure equilibrium constant for pure i defined in Equations 14 and 15. Because the effect of the gas pressure on the vertical adsorbate fugacity normal to the surface is relatively small, it is useful to approximate this effect on a component in a mixed adsorbate by assuming that it is the same as that for the component in the pure state when adsorbed a t the same T and n as the mixture. With this approximation, the expansion of Equation 6 in terms of individual component fugacities followed by integration frum a limit of PQ.--, 0 and n .-,0 gives the following
The degree superscript indicates the pure state a t the same T and as the mixture, so that the exponential is the high pressure correction to Kf* for pure component i. The first term represents the total 2-D fugacity of the entire surface phase a t the equilibrium composition and is designated as f=', so that Equation 29 may be written
rn
372
I&EC FUNDAMENTALS
n
1
The In (f=') term is evaluated from a mixed 2-D equation of state in terms of the total specific adsorption area, a, and the mixed 2-D equation of state constants. K I is the high pressure corrected equilibrium constant for pure adsorbate i evaluated a t the P and n value of the mixture. Equation 30 can be solved directly for the total adsorption a if the adsorbate composition and pure component isotherms are known. Composition of Adsorbed Phase
The adsorbate composition can be found from Equation 30 and a 2-D equation of state for the mixed adsorbate. Since the individual chemical potentials of each component must be equal in the gas and on the surface, a relation is specified between the fugacities in each phase
L
J
The f i g term is the 3-D fugacity of component i in the gas phase. The (fins) term can be found from the total adsorbate phase fugacity expressed by a 2-D equation of state. The result is completely analogous to similar results in three dimensions (Joffe, 1948)
The value of (ft=') is obtained from Equation 31 when f i Q is known and K r is obtained from pure component adsorption data. Substitution into Equation 32 then permits the equation to be solved for x:. Although the total moles of adsorbate, ns, appear in Equation 32, it cancels after the differentiation. For multicomponent systems, sets of equations such as 31 and 32, written for each component, may be solved simultaneously for n - 1 independent mole fractions of surface components. 2-D Eyring Equation for Mixtures By crossplotting the experimental mixed adsorption data for C3Hg and CHI on charcoal, two constant composition isotherms were obtained a t 30' C. At a constant adsorbate composition the mixed equation of state constants can be obtained in the manner discussed for pure components (Table I). The values of b2 and v" used for the mixture were taken as molal averages of the pure component results. Other combinations were tried, but the errors of the correlation were no less. The n
(a2)12 values were obtained by equating a2 to i
n
cxtxI i
(a~)~,
and solving for the unlike pair interaction result. If this quadratic form is valid, the u12 result should be independent of composition, Results are shown in Table IX. The errors in fit would not be significantly less if an average a12 value of 1.8 X l O I 3 were used.
n
Table IX.
Experimentally Evaluated Constants in the 2-D Eyring Equation for a Mixed Adsorbate
C3H8-CH, on charcoal, T Ergs H Sq. ~ Cm./Mg.-Mole 1 , 2 4 9 x 1013 8 . 4 7 3 X 1OI2
= 30'
a2,
X ~ C ~
0.4 0.5
c
0.785
E 0
(61221
(33)
where the 6 terms are the solubility parameters which represent an ideal van der Waals u term divided by the molal volume. This same type of combination rule will predict closely the 2-D unlike pair interactions between a nonpolar adsorbate, such as methane, and a heavier hydrocarbon which is considered to be polarized by the solid surface. No firm conclusion as to a general combining rule for ( U ~ ) can ~ Z be obtained, however, without further experimental data. I n summary, the best combining rules which can be obtained from these data, applicable only to methane mixed with a heavier hydrocarbon, are as follows: =
XiSCI
i=l
n
u2
n
=
X;XjS(az)ij
i=l j=1 (az)i+j CY
= 0.26 [(eu2)i =
+
(Q/C ntS)
I
1.0
t
0.0
w
2
B 0.6
$ z
2 I-
0,4
O
f
a
a LL
w
0.2
-
IDEAL SOLUTION APPROX
0
EXPERIMENTAL
J
0
=*
m
X
I
0
.02
I
I
.04
.06
I
.08
Y3,MOLE FRACTION OF PROPANE IN GAS
Figure 7. Ideal solution approximation of methane-propane mixture at 30" C. and 200 p.s.i.a. compared with data
Although derived for methane-propane mixtures, the combination rules in Equation 34 are also effective in predicting the methane-n-butane adsorption data in Table V. Surface Activity Coefficients
The use of a two-dimensional equation of state for a mixed adsorbate in Equations 30, 31, and 32 eliminates the need for direct evaluation of a n activity coefficient in predicting adsorption equilibria for mixtures. However, in order to study the behavior of a component in a mixed adsorbate, its activity coefficient for lateral nonideality on the surface may be evaluated from
n C
I
a
Ergs Sq. Cm ./Mg.-Mole 1 . 8 7 0 X 1013 1 . 7 1 2 x 1013
An attempt was made to predict the (u2)12 value by assuming a n induced dipole in the methane due to the presence of a propane neighbor with a surface-induced dipole. This, however, does not even roughly approximate the numerical values of u12. I n a n earlier study (Leland e t al., 1955) 3-D van der Waals type interactions were used to estimate the very slight solubility of water in normal paraffin hydrocarbons by approximating the polar-nonpolar pair interaction with
+
I
n
C.
nu
(6)iz = 0.26 [(6)ii
I
v)
(Q2)12,
0.790
w m
(37)
(e~~)jI(~/ N2n') ~
(34)
I
The combination rules in Equation 34 may be substituted in the 2-D Eyring equation for n and for ln(f='). Adsorbate phase compositions and the total moles adsorbed are then obtained from Equations 30, 31, and 32. The expression for ln(f=') from the 2-D Eyring equation is
where ln(ft=s)ois the parallel surface fugacity of the pure component at the same T and ?r as the mixture. This fugacity is also given by Equation 35 but with pure component 2-D parameters. The contribution of the activity coefficients for methanepropane adsorbates is indicated in Figure 7 : in which the assumption of an ideal surface solution proposed by Myers and Prausnitz (1965) is compared with experimental results. Conclusions and Summary
The partial differential ofln (f=') with respect to n; which is required in Equation 32 is obtained by expressing the relations in Equation 34 in terms of moles instead of mole fractions and substituting into Equations 18 and 35. Partial differentials are then evaluated as follows:
r
I n general, the mobile fluid model using the two-dimensional Eyring equation is effective in developing high pressure adsorption isotherms on heterogeneous surfaces for both pure components and mixtures. Constants in the equation may be obtained from low pressure data and, for simple molecules, can be calculated from van der Waals-type interaction parameters. The 2-D Eyring equation is a significant improvement over the 2-D van der Waals equation. The analysis of the constants suggests a more localized adsorption character to propane adsorption than for either methane or butane on both adsorbents. The propane adsorption mechanism appears to differ greatly from that of methane and butane. Propane molecules apparently repel one another on the charcoal surface and have surface intermolecular attractions reduced below the VOL. 7
NO. 3
AUGUST 1968
373
expected 2-D interaction derived from bulk gas interactions alone. Butane, on the other hand, appears to have a greater intermolecular attraction on the surface. The constants in the 2-D Eyring equation can serve as simple characterizing parameters which indicate the ordering properties of the surface, through the c term, and the distortion of the intermolecular potential on the surface, through the a2 term. One cannot make conclusive statements about adsorption mechanisms from fitting equilibrium isotherms alone and alternative models can predict isotherms with about equal success using fitted parameters which have values consistent with the model chosen. The model of Koble and Corrigan (1952) is a n example. Adsorption isotherms for mixtures can be calculated from the isotherms for pure components using relatively simple combining rules for the mixture equation of state constants. This approach makes the assumption of an ideal surface solution unnecessary and allows direct computation of the total moles adsorbed and the surface composition from pure component adsorption data. Nomenclature
a
= surface area
a2 = constant in 2-D equation of state indicating lateral
interaction between adsorbate molecules
A = Helmholtz free energy b2 = constant in 2-D equation of state indicating effective molecular size on surface c2
= constant in 2-D equation of state indicating type of
f
= fugacity = Gibbs free energy
idealized cell structure on surface
G
G
eH
= molal Gibbs free energy
= enthalpy = molal enthalpy
h
= effective thickness of adsorbed phase K , = equilibrium constant expressing equilibrium ratio o
fugacities
N = Avogadro’s number m
= stoichiometric coefficient in reaction between gas and
n
= number of moles in surface phase, stoichiometric coeffici-
P
= = = =
surface
R S_
S T
=
t- =
V = V =
ent in reaction between gas and surface pressure gas constant entropy molal entropy absolute temperature temperature, degrees centigrade molal volume total volume
GREEKLETTERS = molal adsorption area, reciprocal of moles adsorbed (absolute) per unit area = maximum attractive potential between two molecules E y = activity coefficient p = chemical potential
374
l&EC FUNDAMENTALS
T
= two-dimensional (2-D) pressure exerted parallel to the
gas-solid interface
n ’ = 3.1415.. . u
= interaction distance corresponding to zero intermolec-
6
= solubility parameter
ular potential SUPERSCRIPTS * = ideal gas conditions s = surface phase g = bulk gas phase 0 = pure reference state = pure reference state for an adsorbed phase with molecular spacing the same as in a n ideal gas at system temperature
+
SUBSCRIPTS = = direction parallel to gas-solid interface = direction normal to gas-solid interface z = component index j = index indicating homogeneous subregion on surface literature Cited
Aston, J. G., Tomezoko, E. S.J., Chon, Hakze, Advan. Chem. Ser. NO. 33, 325-31 (1961). de Boer, J. H., “The Dynamical Character of Adsorption,” Ciarendon Press, Oxford, 1953. de Boer, J. H., Kruyer, S., Trans. Faraday Soc. 54, 540 (1958). Cassel, H. M., J . Phys. Chem. 48, 195 (1944). Din, F., Ed., “Thermodynamic Functions of Gases,” Vols. 2 and 3, Butterworths, London, 1961. Fowler, R. H., Guggenheim, E. A., “Statistical Thermodynamics,” p. 431, Cambridge University Press, Cambridge, 1949. Haydel, J. J., “Development of Gas Chromatography to Study High Pressure Adsorption,” Ph.D. dissertation, Rice University, Houston, Tex., 1965. Hill, T. L., J . Chem. Phys. 16, 181 (1948). Hill, T. L., AdLsan. Catalysis 4, 211 (1952). Hirschfelder, J. O., Curtiss, C. F., Bird, R. B., “The Molecular Theory of Gases and Liquids,” p. 5, Wiley, New York, 1954. Joffe, J., Ind. Eng. Chem. 40, 1738 (1948). Kemball, C., Rideal, E. K., PYOC. Roy. Soc. A187, 53 (1946). Koble, R. A., Corrigan, T . E., Ind. Eng. Chem. 44, 383 (1952). Leland, T. LV., McKetta, J. J., Kobe, K. A., Ind. Eng. Chem. 47, 1265 (1955). Myers, A. L., Prausnitz, J. M., A.I.Ch.E. J. 11, 121 (1965). Payne, H. K., “Adsorption of Methane and n-Butane on Charcoal at High Pressures,” Ph.D. dissertation, Rice University, Houston, Tex., 1964. Ray, G. C., Box, E. O., Ind. Eng. Chem. 42,1315 (1950). Ross, S., Olivier, T. P., “On Physical Adsorption,” Interscience, New York, 1964. Sanford, C., Ross, S., J . Phys. Chem. 58, 288 (1954). Sage, B. H., Lacey, W. N., “Thermodynamic Properties of the Lighter Paraffin Hydrocarbons and Nitrogen,” API Project 37, American Petroleum Institute, New York, 1950. Sturdevant, G. A,, “High Pressure Adsorption of Light Hydrocarbons and the Two-Dimensional Equation of State,” Ph.D. dissertation, Rice University, Houston, Tex., 1966. Young, D. M., Crowell, A. D., “Physical Adsorption of Gases,” Butterworths, Lt’ashington, D. C., 1962. RECEIVED for review January 24, 1967 ACCEPTEDFebruary 29, 1968 Part of a research program on solid surfaces supported by the National Aeronautics and Space Administration under Grant XsG-6-59 for research on physics of solid materials.
