Helium-Nitrogen-Activuted Carbon System

reaction, both with and without a Lewis acid catalyst. If all isomers are formed from a common transition state, the reaction should yield the same product distribution a t all pressures, whereas if two or more transition states are involved, the distribution would be pressure dependent. For example, if the system studied b y Thompson and Melillo (1970) proceeds through multiple transition states, the product distribution should vary with pressure, as i t does for chloroprene (Stewart, 1971, 1972). Moreover, if any two or more of the products appear in the same ratio, regardless of variations in pressure or solvent, they probably proceed through the same transition state. Finally, and perhaps most important, the application of high-pressure kinetics has been demonstrated as an experimental tool to study the detailed role of the catalyst in a homogeneously catalyzed liquid-phase reaction. Used in conjunction with other types of data, a high-pressure kinetics experiment is clearly capable in many cases of yielding new and valuable information about mechanisms of catalysis.

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literature Cited

Allen, C. F. H., Ryan, R. W., Jr., Van Allen, J. A., J . Org. Chem. 27, 778 (1962). “Beilsteins Handbuch der Organischen Chemie,” Springer-Verlag, Berlin, 1968. Benson, S. W., Berson, J. A., J . Amer. Chem. SOC.84, 152 (1962). Benson, S. W., Berson, J. A., J . Amer. Chem. SOC.87,40 (1965). Coillet, D. W., Hamann, S. D., ?vIcCoy, E. F., Aust. J . Chem. 18. 1911 (1965). “Faraday’s ‘Encyclopedia of Hydrocarbon Compounds,” Butterworths, London, 1957. Favorskaya, I. A., Auvinen, E. II.,Zh. Obshch. Khim. 33, 2795 (1963). Fray, G. I., Robinson, R., J . Amer. Chem. SOC.83,249 (1961).

Gonikberg, hi. G., “Chemical Equilibria and Reaction Rates at High Pressures,” 2nd ed, National Science Foundation, Washington, D.C., 1963. Grieger, R. A., Chaudoir, C., Eckert, C. A., IND.ENG.CHEM., FUNDAY. 10, 24 (1971). Grieger, R. A., Eckert, C. A.,A.1.Ch.E. J . 16, 766 (1970a). Grieger, R. A., Eckert, C. A., J . Amer. Chem. SOC.92, 7149 (1970b). Inukai, T., Kasai, AI., J . Org. Chem. 30,3567 (1965). Inukai, T., Kojima, T., J . Org. Chem. 31, 1121 (1966a). Inukai, T., Kojima, T., J . Org. Chem. 31, 2032 (1966b). Inukai, T., Kojima, T., J . Org. Chem. 32,869 (1967a). Inukai, T., Kojima, T., J . Org. Chem. 32, 872 (196713). Jahn, H., Goetzky, P., 2.Chem. 2, 311 (1962). Kojima, T., Inukai, T., J . Org. Chem. 35, 1342 (1970). Lappert, M. F., J . Chem. SOC.(London)817 (1961). Leffler, J. E., Grunwald, E., “Rates and Equilibria of Organic Reactions,” p 162, Wiley, Yew York, K.Y., 1963. LIcCabe, J. R., Grieger, R. A , , Eckert, C. A,, IND. ENG.CHEM., FCNDAM. 9, 156 (1970). “Organic Syntheses,” Collect. Vol. 111, Wiley, New York, N.Y., 1965. Poling, B. E., Ph.D. Thesis, University of Illinois, Urbana, 1971. Rubin, hi., Steiner, H., Wasserman, A., J . Chem. SOC.(London) 3046 (1949). \ - - ~ - ,

Soula,- J. C., Lumbroso, D., Hellin, M., Coussement, F., Bull. SOC.Chim. Fr. 2059, 2065 (1966). Stewart, C. A., Jr., J . Amer. Chem. SOC.93, 4815 (1971). Stewart, C. A., Jr., J . Amer. Chem. SOC.94,635 (1972). Thompson, H. W., Melillo, D. G., J . Amer. Chem. SOC.92,3218 f 1970). Walling; C., Peisach, J., J . Amer. Chem. SOC.80,5819 (1958). Walling, C., Schugar, H. J., J . Amer. Chem. SOC.85, 607 (1963). Weale, K. E., “Chemical Reactions at High Pressures,” Spon, London, 1967. Williamson, K. L., Hsu, Y. L., J . Amer. Chem. SOC.92, 7385 (1970). Yates, P., Eaton, P., J . Amer. Chem. SOC.82,4436 (1960). RECEIVED for review December 30, 1971 ACCEPTED August 2, 1972 Work supported financially by the National Science Foundation and by the U. S. Army Research Office, Durham, N. C.

Adsorption Equilibria at High Pressures in the Helium-Nitrogen-Activuted Carbon System John M. Fernbacher*l and Leonard A. Wenzel Department of Chemical Engineering, Lehigh Cniversity, Bethlehem, Pa. 18015

Adsorption equilibria in the helium-nitrogen-activated carbon system were measured volumetrically at 100 and 150”K, at pressures up to 100 atm, for gas-phase compositions of 0.000327 and 0.001059 mole fraction of nitrogen. The corresponding pure component isotherms were also determined. A general method for the prediction of multicomponent adsorption equilibria is proposed. This method is based on the calculation of constituent fugacities in pure and mixed equilibrium gas phases at constant temperature and adsorbate volume, and the assumption of an ideal mixed adsorbate. Deviations from an ideal mixed adsorbate can b e described in terms of activity coefficients. The method predicts the total volumes adsorbed and the adsorbate composition within experimental error for the system studied, except at the highest coverages, where the assumption of an ideal mixed adsorbate apparently becomes invalid.

T h e growing application of new adsorption processes for the separation and purification of gas mixtures requires useful methods for predicting and correlating mixed adsorption equilibria. These methods should strike a balance between 1 presentaddress, Park, N. J. 07932.

~~~~~~~h& Engineering c0.,~

’

l

complexity and ease of engineering application. I n addition, such methods should be based on fundamental principles whenever possible, so t h a t some physical interpretations of the adsorption process are possible. I n the past 50 years, numerous methods have been proposed of mixed gas adsorption equilibria. The ~ for ~ the h prediction ~ ~ earliest of these approaches was based on extensions of the Ind. Eng. Chem. Fundam., Vol. 1 1 ,

No. 4, 1972 457

olumetric ratus otarneter

Feed Mixture

\

\ "

\


Adsorbent

1

Bath T.C.

Feed Cooling COl I

Figure 1. Apparatus for mixed adsorption

work of Polanyi (1916), Langmuir (1918), and Brunauer, et aZ. (1938). These and other earlier approaches are discussed in the monograph of Young and Crowell (1961). Sewer methods have been developed during the past decade; among these are the contributions of Cook and Basmadjian (1965), Myers and Prausnitz (1965), Grant and Manes (1966), and Payne, et al. (1968). These recent approaches can be applied to a variety of commercially important adsorption systems, including some a t high pressures. The present work was undertaken to obtain a better understanding of mixed adsorption at high pressures, particularly for gas mixtures containing components with widely differing adsorptive properties. The particular objectives were to obtain a representative body of adsorption data for pure components and their mixtures, and t o investigate methods for predicting or correlating the mixed adsorption equilibria (Fernbacher, 1968). Experimental Section

Adsorption data were obtained for the adsorption system helium-nitrogen-activated carbon, using a volumetric openflow method. Mixed adsorption data were measured at 100 and 150"K, over a pressure range of about 2 to 100 atm, for two constant gas-phase concentrations of 327 and 1059 pym (molar) of nitrogen in helium. Pure nitrogen isotherms a t these temperatures were determined between about and 0.9 a t m using standard volumetric methods. Pure helium isotherms mere determined by a volumetric series desorption method between about 1 and 100 atm. Materials. T h e adsorbent used was T y p e BPL activated carbon (12 X 30 mesh), obtained from the Pittsburgh Chemical Co. This adsorbent is made from steamactivated bituminous coal combined with suitable binders, and has a n ash content of 8%. A surface area of 1123 mz/g was determined by use of the standard B E T method. Fresh samples were initially prepared by outgassing a t 10-4 mm and 350°C for 14-16 hr. The samples were then put through three adsorption-desorption cycles, each consisting of ad458

Ind. Eng. Chem. Fundam., Vol. 11, No.

4, 1972

sorbing nitrogen at 78°K and th& outgassing at room temperature until the pressure was less than mm; this treatment was necessary before reproducible low-pressure nitrogen isotherms could be obtained. The sample thus treated was stored in vacuo. All subsequent samples for pure and mixed adsorption measurements were taken from this initial sample. Before each pure or mixed adsorption run, the adsorbent was outgassed again a t about mm and 130°C. The skeleton density of the adsorbent was determined very carefully, since this density was used in subsequent calculations. The density measurements were made volumetrically by helium expansion a t 24, 175, and 350°C; bhe apparent densities were, respectively, 2.23, 1.83, and 1.81 g/cm3. The higher apparent density a t 24OC indicated that helium adsorption is significant enough (in this case, ca. 0.03 cm3 STP/g at 300 mm) to cause a n erroneous density determination. Similar observations were made by Kini and Stacey (1963) in their extensive work on the densities of porous solids. For the present work, an average value of 1.82 g/cm3 was adopted as the adsorbent skeleton density. C P grade helium and prepurified grade nitrogen obtained from Air Products and Chemicals, Inc., were used for the pure component isotherms. Before use, the nitrogen was further purified by passing it through Drierite, over copper shot a t 330°C, and finally through a molecular sieve adsorber a t 0°C. The helium was passed through an activated carbon adsorber a t 150 a t m and 78°K before use. The gas mixtures were also obtained from the above supplier. Before use, the cylinders were rolled and heated for several days to assure homogeneous mixtures. At the end of the experimental program, the two mixtures were analyzed by slowly passing large samples of known size through an activated carbon adsorber a t 78OK, and then determining by gas chromatography the amount of nitrogen thus collected. The results were 320, 341, and 318 ppm (for an average of 326.6 ppm) and 1061, 1078, and 1039 ppm (for a n average of 1059 ppm). Apparatus. Pure nitrogen isotherms were determined by the usual series adsorption method, using a calibrated volumetric apparatus constructed of borosilicate glass. The adsorbent for the high-pressure helium desorption measurements was contained in a 6411. length of 5,'g-h. 0.d. stainless steel tubing, plugged on one end and connected to the charginglines and pressure gauge a t the other end. Mixed adsorption data were obtained using an open flow method in which equilibrium is attained by cooling a constant composition gas mixture to the desired temperature, passing it through the adsorbent, and throttling it t o the atmosphere. The advantages of this method in the present work include the minimization of dead volume associated with the adsorbent tube, definite knowledge of the approach to equilibrium, and the ability to obtain isothermal adsorption data a t constant gas-phase compositions. X schematic diagram of the apparatus for mixed adsorption is shown in Figure 1. The adsorbent (about 1.5 g) was packed into a 7-in. length of 0.25-in. 0.d. X 0.18-in. i.d. stainless steel tubing, and held in place by small glass wool plugs and brass screens. This C-shaped tube was connected with Swagelok fittings directly to two stainless steel bellows seal valves (Hoke HY441) which were fitted with extensioii handles. All excess volume bounded by the bellows valves and shutoff valve D was carefully minimized. Flow rates and pressures were controlled by manipulation of ball valves A, B, and C, needle valve E, and pressure regulator R. The pressure a t various points in the system was measured by a 12-in. precision Heise bourdon-tube gauge. Temperatures were measured

Table 1. Adsorption of Nitrogen on Type BPL Activated Carbon

r Pressure, atm


1.80 X 3.83 x 1.03 x 2.38 x 8.91 x 4.32 x 9.87 x 1.97 x 3.42 x 6.19 X 0.0217 0,0866 0.304 0.602 0.839

10-5 10-5 10-5 10-4 10-4 10-3

10-3

=

r

IOOOK

Amount adsorbed, mg-mole/g

Volume adsorbed, cm3/g

0.081 0.163 0.345 0.711 1.45 2.65 3.37 3.87 4.53 5.27 6,78 8.21 9.56 10.5 10.9

0.0030 0.0060 0.0126 0.0260 0.0531 0.0970 0.123 0.142 0.166 0.193 0.248 0,300 0.350 0.384 0.399

using copper-constantan thermocouples calibrated against an KBS standard platinum resistance thermometer. Thermocouple EhlF's were measured with a Type K-3 Leeds & Northrup potentiometer. The constant-temperature bath (which iws also used for the pure component adsorption work) was contained in a 4-1. stainless steel flask insulated by polyurethane foam. Bath liquids used were Freon 12 a t 150°K and Freon 13 a t 100°K. Refrigeration was provided by vaporizing liquid nitrogen in a finned-tube cooling coil; temperature was controlled to *0.02"K by balancing a slight excess of refrigeration with a 125-W (at l l O S r ) resistance heater. Current t'o the heater was controlled by a Bayley proportional temperature coctroller (Model 237) which sensed the bath temperatures with a nickel resistance thermometer. The void volumes of adsorbent tubes used for t h e highpressure pure component and mised adsorption measurements were determined indirectly by taking the difference between the empty tube volumes (determined by helium expansion) and the volumes of the adsorbent samples charged to the tubes. These sample volumes were calculated (after weighing the samples carefully) from the previously determined skeleton densities. The effects of temperature and pressure on the sample tube volumes were negligible. Procedures. T h e high-pressure helium isotherms were obtained by charging the adsorbent tube to about 110 a t m and then bleeding successive portions of gas from the tube into a calibrated gas holder until the pressure was about 2 atm. The adsorbent was then heated, and the remaining gas was transferred to the gas holder by positive displacement using a variable volume mercury-filled burette. The amounts adsorbed a t each step were calculated by material balance from the measured pressures and amounts of gas removed, the known void volume of the adsorbent tube, and the volumetric properties of helium. Absolute adsorption was calculated using the adsorbate densities suggested by Dubinin (1960). Each mixed adsorption data point was determined in t h e following manner. *kt the desired temperature and pressure,

Pressure, atm

3.42 x 5.66 x 9.08 x 1.29 x 2.37 x 4.47 x 9.21 x 1.45 x 2.50 x 6.05 x 0.0117 0.0236 0.0399 0.0637 0.0975 0.205 0.366 0.537 0.757 0,872

10-5 10-5 10-5 10-4 10-4 10-4 10-4 10-3

10-3 10-3

=

1 5 0 ~ ~

Amount adsorbed, mg-mole/g

Volume adsorbed, cm3/9

0.018 0.028 0.040 0,053 0.078 0.124 0.214 0.300 0.431 0.721 1.02 1.42 1.82 2.25 2.75 3.84 4.60 5.12 5.27 5.48

0.00070 0.0011 0.0016 0.0021 0,0030 0.0048 0.0084 0.0117 0.0169 0.0282 0.0400 0,0555 0.0712 0.0880 0.108 0.150 0.180 0.200 0.206 0.214

the gas mixture was passed through the adsorbent a t 250 to 500 em3 (25'C, 1 atm) per minute, while the effluent composition was constantly monitored by a Gow-Mac (Model 21 OL) thermal conductivity analyzer. When the breakthrough curve of nitrogen was complete, the adsorbent was isolated by closing the two bellows valves. After a period of 1 to 3 hr of isolation, flow was started again, and the equilibrium adsorption conditions were checked. (In all runs, this check procedure showed t h a t the equilibrium, after completion of initial breakthrough, was in fact the true equilibrium.) .kt this point, the pressure gauge and the adsorbent were isolated b j closing valves D and F (Figure I ) , the lines were evacuated, and the adsorbent tube was bled into a n evacuated 7-1. calibrated expansion tank. The cold bath was removed, and the adsorbent was heated to 130°C. The gas remaining in the tube and the lines was transferred to the expansion tank by the positive-displacement mercury burette. After a mixing period of about 12 hr, the pressure in the tank was measured, arid a sample was taken and analyzed in triplicate by gas chromatography. The raw experimental data were reduced to adsorption data by material balance. Results. T h e nitrogen adsorption d a t a are listed in Table I. T h e absolute adsorption d a t a for helium, mhich were calculated from the raw data by using the method of estimating adsorbate densities suggested by Dubinin (1960), are shown in Table 11. The volumetric properties of helium were calculated from a three-term virial equation, using Keesom's (1942) coefficients at 100°K and those of Canfield, et al. (1963), at 150°K. The mixed adsorption data are summarized in Table 111. The experimental data were interpreted in terms of absolute adsorption, which is the most appropriate approach for highpressure adsorption systems of the type studied in this work. When the data were treated in terms of Gibbs adsorption, the calculated mixed helium isotherms all showed maxima, with the 100°K isotherms actually becoming negative a t pressures much above 50 atm. I n calculating the absolute mived adsorption isotherms, the adsorbate volumes were estimated by assuming that the pure component adsorbate Ind. Eng. Chem. Fundam., Vol. 11, No. 4, 1972

459

Table 11. Adsorption of Helium on Type BPL Activated Carbon Pressure, otm

Amount adsorbed," mg-mole/g

Volume adsorbed cmS/g

T = lOOOK


2.53 9.03 17.26 32.16 52.29 71.54 91.75 111.0 1.92 5.15 11.34 21.85 35.22 48.14 62.09 75.83 89.71 103.2 Absolute adsorption.

0.191 0.655 1.18 1.97 2.87 3.66 4.46 5.22 T = 150°K 0.045 0.125 0.261 0.491 0.785 1.036 1.32 1.57 1.83 2.06

0.00453 0.0155 0.0280 0.0467 0.0680 0.0867 0.106 0.124

Nitrogen concentration in gas phose, mole fraction

Interpretation of Gas Adsorption at High Pressures

The interfacial region in a gas-solid system is not well defined, since the extent and properties of this region are not directly measurable. All adsorption measurements in gassolid systems thus require a n a priori assumption concerning the nature of this interfacial region. Thermodynamic treatment of the resulting adsorption data should be consistent with such assumptions. The results of experimental adsorption measurements at low gas-phase densities are essentially independent of the assumptions about the interfacial region. At high gas-phase densities, however, the assumptions made concerning the interfacial region are of primary importance, and the physical and thermodynamic interpretations of experimental observations depend directly on these assumptions. Two interpretations of the interfacial region in gas-porous solid systems are possible. The first of these follows from

Pressure, atm

Total amount adsorbedp mg-mole/g

T

0.000327

0.001059 0.00107 0.00296 0.00619 0.0116 0.0186 0.0246 0.0313 0.0372 0.0434 0.0488

volumes were additive. The vapor-phase densities were calculated from the three-term virial equation of state. The uncertainties in the experimental data were estimated by the method of piopagation of errors. The variances of the actual measured quantities (temperature, pressure, volume, composition) were estimated and then used with the material balance equations for representative runs. The relative minimum and maximum uncertainties (at the 90% confidence level) in the amounts adsorbed were: pure helium, h 2 . 9 and 6.0%; pure nitrogen, k 1 . 2 and 2.3%; helium from mixtures, h12.1 and 17.1%; nitrogen from mixtures, *2.3 and 2.8%. For t h e adsorbate mole fractions calculated in the mixture runs, the relative uncertainties a t the 90% confidence level for nitrogen were *4.7 t o 7.1%, and for helium were 1 1 6 . 7 to 21.0%. I n the pure nitrogen isotherms these Uncertainties were due largely to the uncertainties in measuring pressure and in volume calibration. For the helium isotherms and all mixture runs, the uncertainties in the data were due mostly to uncertainties in volume calibration.

460 Ind. Eng. Chem. Fundam., Vol. 1 1 , NO. 4, 1972

Table 111. Experimental M i x e d Adsorption Data for the Helium-Nitrogen-BPL Carbon System

0.000327

1.29 9.98 22.02 45.42 72.70 94.33 104.27 1.18 3.08 5.97 14.67 35.29 62.84 102.7

1.04 3.10 8.86 10.05 19.84 42.84 62.43 82.22 102.1 1.21 0.001059 6.03 14.88 29.37 50.45 69.10 102.5 Absolute adsorption.

lOOOK 2.49 4.79 5.88 7.13 7.87 8.56 8.31 3.61 4.58 5.30 6.53 8.78 9.56 10.4 T = 150°K 0.146 0.332 0.645 0.730 1.25 1.88 2.39 3.00 3.21 0.325 0.952 1.36 1.91 2.64 3.03 3.82

Total adsorbed: cm3/g

Nitrogen concentrotion in adsorbate, mole fraction

0.0900 0.170 0.207 0.247 0.268 0.286 0.281 0.131 0.165 0.191 0.233 0.304 0.323 0.353

0.968 0.917 0.896 0.848 0.798 0.771 0.760 0.967 0 959 0.955 0.929 0.849 0.782 0.762

0.00556 0.0122 0.0235 0.0258 0.0416 0.0613 0.0763 0.0938 0.100 0.0127 0.0350 0.0495 0.0693 0.0914 0.109 0.132

0.933 0.850 0.813 0.759 0,633 0.576 0.532 0.492 0.485 0.996 0.850 0.817 0.819 0.780 0.712 0.696

volume

=

I

Gibbs' classical treatment of interfaces (Gibbs, 1948). He proposed t h a t a n interface could be represented by a hypothetical mathematical surface which separates the two bulk phases, and further assumed t h a t the individual properties of these two bulk phases exist unchanged right up to this surface. The properties assigned to this surface include the free energy, entropy, and quantity of each component, but of course no volume. Such a n approach allows rigorous mathematicai treatment of the resulting two-dimensional interface. Gibbs' basic idea has been applied widely t o gas-solid adsorption systems, and the thermodynamic relations between the two-dimensional interface and the bulk gas phase are well known and have useful applications (Hill, 1949, 1950; Myers and Prausnitz, 1965; Van Ness, 1969). This classical approach is less than satisfactory in a physical sense, however. As Guggenheim (1940) has pointed out, the idea of a mathematical surface with zero volume is a t variance with our physical picture of a n interfacial region. The definition of gas-solid adsorption inherent in Gibbs' treatment of a n interface also presents difficulties in the interpretation of highpressure volumetric gas adsorption measurements, as i t is possible for calculated isotherms to exhibit maxima; in mixed adsorption systems such as the one studied in the present work, calculated coverages can even become negative. An alternate treatment of the interfacial region was discussed by Guggenheim (1940), based 011 the work of Ver-


schaffelt (1936) and others. I n this approach, the interfacial region is assumed to have a definite (but arbitrary) thickness, bounded by two parallel planes. The properties of this region are uniform in directions parallel to the boundary planes, but vary in the perpendicular direction between the properties of the two bulk phases. Guggenheim then derived the fundamental thermodynamic equations for this interfacial region, as well as for curved interfaces. The nature of the highly complex surface of a heterogeneous adsorbent is such that the adsorbate and its properties are irregularly distributed over the surface. Guggenheim's idea of a n interfacial region which has a n actual volume can be extended to gas-porous solid systems if, instead of considering the thickness of the adsorbate, we consider a n adsorbate molar volume averaged over the surface. An estimate of this average adsorbate molar volume then can be used t o reduce raw volumetric adsorption measurements to absolute adsorption data. Such an approach to the gas-porous solid interfacial region is more realistic physically than the Gibbs interpretation, especially a t high gas-phase densities. In high-pressure adsorption studies, the concept of absolute adsorption, n-liich follows from Guggenheim's treatment of interfaces, has been more useful than the approach baaed on Gibbs' work. 'The adsorption measurements in the present work have been interpreted in terms of absolute adsorption. Some further discussions and comparisons of these two interpretations of adsorption have been given by recent workers in t'his area (Haydel and Kobayashi, 1967; Masukawa aiid Kobayashi, 1968; Payne, et al., 1968).

dimensional equation of state and a correction t'erm involring the partial molar volumes of adsorbate compoiieiits. While this approach is somewhat complex for engineering purposes, i t appears to provide useful insights into the actual mechanism of physical adsorption. These methods are all based on thermodynamic arguments, although with varying degrees of rigor. The general thermodynamic functions for an adsorbed phase have been discussed by Guggenheim (1940) and Hill (1949, 1950). When restricted to a two-dimensional adsorbate, these fuiictioiis are directly analogous to those for a n ordinary three-dimensioiial fluid. The pririciples of classical solution thermodynamics thus can be applied by analogy to adsorpt'ion equilibria in terms of a two-dimensional adsorbate; this approach was used by AIyers and Prausnitz (1965) and has been discussed iu detail by Van Xess (1969). 1rigorous thermodynamic method for predicting or correlating adsorption equilibria in terms of a threedimensional adsorbate does not exist. X method to treat' adsorption equiiibria in terms of a tlireedimensional adsorbate is outlined in the follo~viiigsection. The equilibrium criterion of uniform temperature aiid coilstituent fugacity is applied, and activity coefficients are defined in terms of a standard state fugacity for the piire coniponent a t constant temperature aiid adsorbate volume. The method, while not thermodynamically coiisistent, provides a useful tool for predicting or correlating adsorption equilibrium data.

Prediction of Mixed Gas Adsorption Equilibria

An auxiliary thermodynamic function, the activity coefficient, can be defined in the case of mixed adborption as

I n engineering applications of physical adsorption, methods are needed for the prediction of mixed adsorption equilibria from pure component data or, when this is not possible, for the correlation of experimental mixed adsorption data. The main features of t h e most recently proposed methods are summarized below. Cook and Basmadjiaii (1965) have developed a semienipirical method based on the earlier work of Basmadjian (1960) and Broughton (1948) which relates the adsorpt,ion relative volatility for a binary system to the pure-component pressure ratio a t constant volume adsorbed. Myers arid Prausnitz (1966) proposed a method of treating mixed adsorption using solution thePmodyaamics in conjunction with the classical Gibbs adsorption isotherm. They defined a n ideal adsorbed solution a t constant temperature and two-dimensionai adsorbate spreading pressure, and derived a n expression for adsorpt,ion analogous t o Raoult's law. They also introduced the useful concept of an activity coefficient for adsorbed solutions. I n a n approach based on earlier extensions of the classical Polanyi potential theory to mixtures, Grant and Manes (1966) developed a useful prediction method which they applied to a number of binary hydrocarbon niistures a t moderate and high pressures. Their approach ii: based on the assumption that the adsorbate behaves as an ideal liquid mixture, and t h a t the total adsorbate volume determines the Polanyi adsorption potential of each pure component and thus its adsorption pressure as a pure gas. h recent contribution to the prediction of mixed gas adsorption is due to Payne, et d. (1968), who used a twodimensional equation of state based on a mobile-fluid model of the adsorbed phase. They derived expressions to predict mixed adsorption equilibria, in rrhich fugacities in three-dimeiisional adsorbates are calculated using the two-

Proposed Method

where f i 3 is the fugacity of constituent i having a mole fraction in a mixed adsorbat>e,and fi* is the fugacity of i in some convenient etandard state. This standard state mill be defined arbitrarily as pure component i adsorbed a t the same temperature, 011 the same total adsorbent surface area, A , and at' the same adsorhate volume, V", as the mixture. The restriction of coiistaiit surface area is easily met in adsorption work by the usual practice of reducing all experimental coverages to a unit mass of adsorbent. The fugacity of constituent i in the absorbate is then

zi

fia =

y. ]xl..f.130

( T , Ira,A are constant)

(2)

The fugacitj of constituent i in a mixed gas phase is given b y

where y i is the mole fraction, & is the fugacity coefficient, aiid P is the pressure. -kt adsorption equilibrium, the fugacitjof each constituent is equal throughout both the gas arid adsorbed phases, so t h a t y i # i P = yizijiaO

( T , V3, -4 are constant)

(4)

This expression provides a basis for correlating mixed adsorption equilibrium in terms of the adsorption activity coefficient. If ideal mised adsorbate behavior is assumed, so that the activity coefficients are unity, the mised adsorption equilibria can be predicted from the pure-component isotherms The equations necessary for these calculations a t constant T , V", and -4 are discussed and listed below. First, the espression for adsorption equilibrium (eq 4) is written as

Ind. Eng. Chem. Fundam., Vol. 11, No. 4, 1972

461

t o indicate t h a t the standard state fugacity is a function of the adsorbate volume, Vi"0. This function is simply the purecomponent adsorption isotherm expressed in terms of adsorbate volumes and fugacities, which can be written in general form as

and may be represented by any convenient graphical or algebraic relation which gives flao as a function of VIaO. The fugacity of the pure adsorbate in the standard state, flaO, is equal to its pure equilibrium gas phase fugacity flgo, which is easily calculated from generalized tables or equations of state. The adsorbate volume VIao is calculated from the experimental amount adsorbed (in moles) by estimating the adsorbate molar volume at the conditions of interest, as discussed later. Since the pure and mixed adsorbate volumes are assumed equal in the above definition of the standard state


v1a0

=

V,"

(7)

where the subscript m refers to the mixed adsoibate. From complete mixed isothermal adsorption data (including gas and adsorbate compositions and amounts adsorbed), and the corresponding pure-component adsorption data, eq 5, 6, and 7 can be used to calculate the activity coefficients. If the assumption of an ideal mixed adsorbate is made, so that the activity coefficients are unity, the mixed adsorption equilibrium compositions and the total volume of the adsorbate can be predicted from the pure-component isotherms through the use of eq 5, 6, and 7 , and the restrictions that N

c x i = 1 1

N &/i=1

i

(9)

For example, assume t h a t the adsorbate composition in a multicomponent system is to be calculated a t a known temperature, pressure, and gas-phase composition. The pure component isotherms a t this temperature are first obtained in the form SIao= gl(Vlao). Then, by trial and error, the value of the adsorbate volume is found for which the adsorbate mole fractions add t o unity. This final adsorbate volume is then the mixed adsorbate volume at the conditions of interest. When the method discussed above is applied to the prediction of adsorption equilibrium, the compositions and total volume of the mixed adsorbate are the calculated quantities of interest. The amounts adsorbed in mass or mole units can be calculated if the molar volume of the mixed adsorbate is known or can be estimated. If it is assumed t h a t the mixed adsorbate molar volume is a linear function of the pure adsorbate molar volumes, then

Application of Method

i

The total moles adsorbed is

and the number of moles of each component adsorbed is given by nima = xlnrna

(12)

Adsorbate Molar Volumes. In the above calculations, and in the reduction of the experimental adsorption data (discussed earlier), i t is necessary to have estimates of the pure and mixed adsorbate molar volumes. A number of 462 Ind. Eng. Chem. Fundom., Vol. 11, No. 4, 1972

methods for the estimatiok of adsorbate molar volumes have been proposed. Dubinin (1960), drawing on his extensive work in gas adsorption on porous solids, proposed that a t temperatures below the normal boiling point, the density of a pure adsorbate is equivalent to the ordinary liquid volume; a t temperatures above the ordinary critical temperature, the adsorbate density is constant and is equal to the quadrupled volume proper of the molecules, which is the van der Waals b value; and finally, at, temperatures between the critical and the normal boiling point, the adsorbate density varies linearly with temperature between the density a t the normal boiling point and the van der Waals b value. Dubinin's proposed adsorbate density is thus dependent upon temperature only. Haydel and Kobayashi (1967), in their work on chromatographic measurements of high-pressure adsorption, estimated adsorbate volumes by taking the difference between the void volume of a chromatographic column and the measured retention volume of a helium sample injected into a pure elution gas a t high-pressure column conditions. For methane adsorption on silica gel in the temperature range of 0 to 4OoC (well above the methane critical of -82.5OC), they reported a n average adsorbate molar volume nearly identical to the van der Waals b value of 42.9 em3 per moie. For propane over the same temperature range (which is about midway between the boiling point of -44.7OC and the critical point of 95.6OC), they reported a n average adsorbate molar volume of 85.8 c,m3per mole. This agrees fairly well with the propane adsorbate molar volume of 79.3 em3 per mole calculated by Dubinin's method a t 20OC. Masukawa and Kobayashi (1968) have developed a novel chromatographic method for determining adsorbate volumes. The method, which requires the assumption of an average thickness of the adsorption zoiie, involves a trial-and-error extrapolation of the retention volumes of a series of perturbing gases t o a hypothetical "perfect gas." This gives a n estimate of the free or dead gas volume of the column, from which the adsorbate volume can be calculated. They reported adsorbate volumes for methane and ethane between 5 and 35°C as functions of temperature and amount adsorbed. I n general, the values for methane were higher than the value predicted by Dubinin's method except a t higher coverages. For ethane, the adsorbate volumes are all lower than t'hose predicted by Dubinin's method. I n the present work, adsorbat,e molar volumes were estimated by Dubinin's method. The adsorbate molar volumes for nitrogen were 36.6 and 39.1 em3 per mole a t 100 and 150°K, respectively, and 23.7 em3 per mole for helium a t both temperatures. I n t,he reduction of the experimental mixed adsorption data by mat'erial balance, the mixed adsorbate molar volume was estimated as a molar average of the purecomponent molar volumes.

Calculations. T h e equations necessary for calculating isothermal adsorption equilibria for the binary system studied in the present work are given below (nitrogen is component 1, helium is component 2). Equilibrium relations YdlP =

ylxlflao

(13)

Y2hP

Y2x2f2a0

(14)

Pure-component isotherms

=

1.0

A

A

-

2

A

yN=0.001059

I

I

I

I

I

I

I

1

I

W

c m a

8 Ln

Predicted

0.9

0

a

-z

0.30

w m

z

E

W 8

Ln 0

n

0.8

c? !=

a

z

E

$

z

0 IV

n W

1

T = 150'K

1 0 0 D K 150'K 0 y~=0.000327

0.7

a

LL 0:

0.20

v) 0

w -1

n

0

a

Z

w

0.6

z x

5

1

0

>

J

a

0.5

O I-

0.10

0

20

40

60

EO

100


PRESSURE, A T M O S P H E R E S

Figure 3. Comparison of experimental and predicted adsorbate compositions at 150°K

0.0 0

60

40

20

80

100

PRESS b R E, A TLI OS PH E R E S

Figure 2. Comparison of volumes adsorbed

f2&0

=

experimental and

g2(VP)

predicted

(16)

Equality of adsorbate volumes in t h e standard state = v2ao =

T/'ma

(17)

Composition restrictions zj y1

+

x2 =

+ y2

=

1

(18)

1

(19)

If the adsorbate is assumed ideal, y1 and y2 are unity, and a n y 8 of the 11 variables in eq 13-19 can be calculated if the other three variables are fixed. I n t h e present work, each mixed adsorption data point was obtained a t constant temperature, pressure, and gas-phase composition. The variables t o be calculated and compared with experimental results were the total mixed adsorbate volume and the adsorbate composition. I n this case, the calculation procedure using eq 13-19 could be carried out in two mays: ( I ) find t h e value of the adsorbate volume, by trial and error, such t h a t the adsorbate mole fractions add t o unity; and (2) find the adsorbate composition, by trial and error, such that VIao= VzaO. The second procedure was used for the present work, so t h a t the pure-component isotherms (eq 15 and 16) had t o be correlated explicitly in terms of Vao. This was done by fitting the data t o a Polanyi potential type curve, although any convenient method of fitting the data can be used. The trialand-error calculations were done by digital computer, and solutions of sufficient accuracy were generally obtained in less than ten iterations. When doing calculations by hand, only two t o three tries were needed. Gas-phase fugacities for the pure-component isotherms and fugacity coefficients in the mixtures mere calculated using a three-term virial equation of state.

Because of the large differences in adsorption characteristics of helium and nitrogen, i t was necessary in some of the calculations t o extrapolate the pure helium isotherms above t'he highest experimentally measured pressure (ca. 110 a t m ) . Since the two helium isotherms are reasonably linear above about 70 atm, the extrapolations were done on this basis. The extrapolation introduces a source of uncertainty in the prediction calculations which can be expected in adsorption systems of this type. Comparison with Experimental Data. T h e predicted values of t h e total volume adsorbed are compared with t h e experimental data in Figure 2. A t 1 5OoIi, the predictions show excellent agreement with the experimental data. .kt 100°K for t h e constant nitrogen gas-phase mole fraction of 0.000327, the predicted volumes adsorbed fall just outside of the esperimental uncertainties in the data, while for the nitrogen gas-phase mole fraction of 0.001059, the predictions fall well below the experimental errors in the data. While the disagreement may be due in part t o uncertainties ill the est~rapolatioii of the pure helium isotherm necessary in the calculations, it is probably due largely t o deviations from the assumption of an ideal mised adsorbate. The experiment,al and predicted adsorbate compositions are compared in Figures 3 and 4. .It 15OOK (Figure 3), the predicted adsorbate mole fractions of nit,rogen agree with the data within the estimated esperimental error, except for three points which show excessive scatter. Xt 100" K (Figure 4), for the nitrogen gas-phase mole fraction of 0.000327, the predicted and esperimental points agree within the estimated error in the data; for the nitrogen gas-phase mole fraction of 0.001059, however, the agreement is not good. I t is concluded, as in the above discussion of Figure 2, that the disagreement is due largely to deviations from the assumed ideal mixed absorbate. Discussion

The method described here provides a useful tool for predicting mixed adsorption equilibria for engineering applications, particularly at high pressures for mixtures with widely different pure component isotherms. At higher coverages, Ind. Eng. Chern. Fundorn., Vol. 1 1 , No.

4, 1972 463

T yN

w

= 100’K = 0.001059

0.9 -

L

a m 0 e v) 0

-2

E 2 w

0.8 I

1

I

l

l

x

0.7

I

0

20

I.

1

I

I

40

I

1

I

I

60

I

SO

I

100


PRESSURE, ATMOSPHERES

Figure 4. Comparison of experimental and predicted adsorbate colmpositions a t 100°K

deviations between predicted and experimental data can occur, because adsorbate molecules begin to interact with each other as well as with the adsorbent surface. When this occurs, the similarity between pure and mixed adsorption which is invoked in the definition of the standard state and in the assumption of a unit activity coefficient begins to break down. A t such conditions, experimental data of sufficient accuracy can be correlated within the framework of the present approach. One advantage of this approach is that the pure component behavior needed in mixture calculations is described directly by the adsorption isotherms expressed in terms of fugacities and adsorbed volumes. These isotherms need not follow any particular type of analytical equation or theoretical model, so that any functional relation between the pure-component fugacity and the volume adsorbed can be chosen. This can be a graphical plot for hand calculations, or a convenient curve-fit for computer calculations. The method is not restricted to binary systems or to high pressures and should be useful for multicomponent mixtures as well as other pressure conditions. A disadvantage of the present method is that i t is not thermodynamically rigorous and cannot be used to test the consistency of experimental mixed adsorption data. In developing the method, the activity coefficient (eq l) and the equilibrium criterion (eq 5) were written directly in terms of fugacities, which in turn are related by definition to the Gibbs free energy. The primary variables in the expression for the Gibbs free energy of a n adsorbed phase (eq 9) are temperature ( T ) , pressure ( P ) , and the two-dimensional spreading pressure ( T ) ; for a gas phase, the primary variables are T and P. On the other hand, the standard state for a component in a mixed adsorbate was defined in the present work as the pure component adsorbed at the same temperature and adsorbate volume (V”)and on the same adsorbent surface area ( A ) as the mixture. These three variables ( T , Va, and A ) are the primary variables in the expression for the Helmholtz free energy of a n adsorbed phase (Guggenheim, 1940). This inconsistent mixture of primary variables is useful in the 464 Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 4, 1972

development of a simple and useful method for predicting or correlating mixed adsorption equilibria; however, the method is not thermodynamically rigorous, because equations which relate the activity coefficients and fugacities to basic thermodynamic functions cannot be written on a consistent basis. Myers and Prausnitz (1965) developed their method of treating mixed adsorption equilibria by combining classical (Gibbs) surface thermodynamics with the concept of adsorbed solutions, and were the first to describe the use of activity coefficients in adsorption equilibria calculations. Their adsorption activity coefficients are related to the Gibbs free energy and the fugacity of adsorbed components in a rigorous manner, while the activity coefficients defined in the present work are not so related. I n addition, the approach taken by Myers and Prausnitz applies when Gibbs’ interpretation of the interfacial region is used, while the present approach applies when the gas-solid interface is interpreted in terms of the concept described by Guggenheim. The term “ideal adsorbed solution” was used by Myers and Prausnitz to describe an adsorbed phase in which their activity coefficients are unity. I n the present work, the term “ideal mixed adsorbate” is used to indicate the differences between the two methods, particularly with respect to the definition of the activity coefficient. For adsorption conditions where the particular interpretation of the interfacial region is not important (e.g., where “Gibbs” and “absolute” adsorption are identical), mixed adsorption equilibria can be predicted or correlated using either the Myers and Prausnitz approach or the present method. While difficulties are sometimes encountered in the former method in evaluating the spreading pressure for gases which are adsorbed strongly a t low gas-phase pressures, the approach is thermodynamically rigorous and can be used to test the consistency of accurate adsorption data in nonideal systems. The present method, while lacking thermodynamic rigor, is easier to use for predicting or correlating mixed adsorption because the pure-component isotherms (in terms of adsorbed volumes) are used directly in the calculations. Grant and hfanes (1966) recently developed a method for predicting binary adsorption equilibria based on the earlier work of Lewis, et al. (1950), who modified and extended the original Polanyi potential theory to mixtures. The approach of Grant and Manes is based on the assumption that a mised adsorbate behaves as a n ideal liquid mixture, and that the total adsorbate volume determines the Polanyi adsorption potential of each pure component and thus its adsorption pressure as a pure gas. This method is a special case of the general approach described in the present work which results when the pure-component isotherms described by eq 6 are correlated in terms of either the classical form or modified forms of the Polanyi potential. Thus, it appears that the various extensions of the potential theory to mixtures have been relatively successful, not because this particular theory was used, but rather because pure and mixed adsorption were compared a t equivalent adsorbate volumes. The particular type of relation between the pure-component adsorbate volumes and the gas-phase pressures (or fugacities) is relatively unimportant, as long as the relation reproduces the pure-component experimental data. Conclusions

The method described here is a useful tool for the prediction or correlation of multicomponent mixed adsorption equilibria, particularly for adsorption systems which are best interpreted in terms of a three-dimensional adsorbate

or “absolute” adsorption. T h e approach should also be useful at conditions where ”absolute” and “Gibbs” adsorption are essentially equivalent. For the helium-nitrogen-activated carbon system studied in this work, the method predicts the total volume adsorbed and the adsorbate mole fraction within experimental error, except a t the highest coverages, where the assumption of a n ideal mixed adsorbate apparently begins to become invalid. In systems which deviate from the ideal mised adsorbate, accurate adsorption equilibrium data can be correlated in terms of the adsorption activity coefficients described in the present rvork. Since these activity coefficients are not rigorously defined, they cannot be used to test the thermodynamic consistency of experimental data; their potential usefulness lies in correlative applications. Nomenclature

A = adsorbent surface area, m 2

f = 9

fugacity, a t m

= general function

n = g-moles adsorbed per gram of adsorbent = number of componeiits P = pressure, a t m T = temperature, O K v = molar volume, cm3/g-mole total volume, cm3 x = adsorbate mole fraction Y = gas-phase mole fraction


lv

v=

GREI:KLLTTLRS y R

4

= = =

adsorption activity coefficient two-dimensional spreading pressure vapor fugacity coefficient

SUBSCRIPTS i

m 1 2

= = = =

any component mixture nitrogen helium

SUPERSCRIPTS a = adsorbed phase g = gas phase 0 = pure component * = standard state literature Cited

Basmadjian, D., Can. J . Chem. 38, 149 (1960). Broughton, D. B., Znd. Eng. Chem. 40, 1506 (1948). Brunauer, S., Emmett, P. H., Teller, E., J . Amer. Chem. SOC.60, 309 11938’1. Canfield, F. ’B., Leland, T. W., Jr., Kobayashi, R., Adaan. Cryog. Eng. 8 , 146 (1963). Cook, W. H., Basmadjian, D., Can. J . Chem. Eng. 43, 78 (1965). Dubinin, AI. M.,Chem. Rev. 60, 235 (1960). Fernbacher, J. AI., Ph.D. Thesis, Lehigh University, Bethlehem, Pa.. 1968. Gibbi, J. W., “The Collected Works of J. W. Gibbs,” Vol 1, p 219, Yale University Press, New Haven, Conn., 1948. 5 , 490 Grant, R. J., Manes, AI., IND. ENG. CHEM.,FUNDAY. (1966). Guggenheim, E. A,, Trans. Faraday SOC.36, 397 (1940). Haydel, J. J., Kobayashi, R., IND.ENG.CHEX.,FUNDAM. 6 , 546 i 1967). Hill, T. ’L., J . Chem. Phys. 17, 520 (1949). Hill, T. L., J . Chem. Phys. 18, 246 (1950). Keesom, W. H., “Helium,” P 241, Elsevier, Amsterdam, 1942 Kini, K. A,, Stacey, W. O., Carbon (Oxford) 1, 17 (1963). Langmuir, I., J . Amer. Chem. SOC.40, 1361 (1918). Lewis, W. K., Gilliland, E . R., Chertow, B., Cadogan, P., Znd. Eng.Chem. 42, 1319 (19S0). LIasukawa, S., Kobayashi, R., J . Gas Chromntogr. 6, 257 (1968). Myers, -4.L., Prausnitc, J. JI., A.I.Ch.E. J . 11, 121 (1965). Payne, H. K., Sturdevant, G. A., Leland, T. W., IND.ENG. CHEY.,F U S D ~7, M363 . (1968). Polanyi, XI., Verh. Deut. Phys. Ges. 15, 55 (1916). . (1969). Van Ness, H. C., ISD.ENG.CHEM.,F U N D ~8,M464 Verschaffelt, Acad. Roy. Belg. Bull. CZ. Scz. 22 (4), 373 (1936). Young, D. M., Crowell, A. D., “Physical Adsorption of Gases,” Butterworths, London, 1961. RECEIVED for review October 17, 1969 RESUBMITTED August 7, 1972 ACCEPTED August 7, 1972 The financial support of Air Products and Chemicals, Inc., and the National Aeronautics and Space Administration is gratefully acknowledged.

Ind. Eng. Chem. Fondom., Vol. 1 1 , No. 4, 1972

465

Helium-Nitrogen-Activuted Carbon System

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