106
F. D. MASUN, M. ALTMAN AND E. R. ABERTH
practice there may be even minute destabilizing ones. The method seems, however, to be completely insensitive to such inversions of density because in two runs in which sodium chloride solutions were used in inverted position ( i . e . , 0.7 N in the upper and 0.2 N in the lower disc), the values of R deviated by -0.6% and t;-1.1% from the calibration line for I1 hr. and 17 hr. experiments, respectively. (Dr. R. J. Williams has found that discs of coarse porosity are no longer insensitive to such extreme inversions of density gradient but are still satisfactory for tracer work). Temperature control is provided by an air-bath whose temperature varies by =l=0.05"with a period of approximately 1 minute a t 25 i 0.1". The mercury cups are supported on rubber in an almost vibrationlexs way although strong vibrat.ions seem to have no effect,' Accuracy and Precision.-The above results show that the method seems to eliminate most sources of methodical errors except that of adsorption and surface diffusion in the particular system studied. The presence of these effects is indicated by a discrepanc between volumes determined using tracer and using N a d a n d may be ascertained by using discs of different porosities. The precision depends on the value of R , showing a flat minimum near R = 0.65. The reproducibility of measurements with sodium chloride corresponds to about f 0.5% in the value of D and with dye solution it corresponds to the analytical error. Thus in 5y0 Aerosol MA, Orange OT in three different discs gave four results averaging 1.089 X 10-8 cm.Z/sec. with a mean deviation of 1.2%, while oil blue (which is appreciably water-soluble) gave three results averaging 1.156 X 10-6 cm.*/sec. with a mean deviation of 1.1%.
DISCUSSION 8. C. LIANQ.-wOUld the dye affect the diffusion of the micelles because of the amount of dye pres'ent in the micelle? K. J. MYsELs.-That is a very good question. However, the amount of dye employed was on the average less than one dye molecule per micelle. Therefore such an effect would be very small. Furthermore the dye was so com-
Vol. 57
pletely water-insoluble and very soluble in oil 'so that it should be located well inside the micelle. Also in electrophoretic measurements we used several different dyes to see if they gave the same results and found that they did. I do not think we are changing things very much but we plan to check this point by comparing different dyes in diffusion.
IRVING REICH.-I would like to just go a stage further on the question just asked. Obviously the molecule or two of dye locked within the micelle will not affect the speed of diffusion very much. But there is so much to be understood about the geometry and the energetics of the micelle. It may be a highly sensitive structure. Might it not be possible for the micelle to let's say, double in size because of the presence of even one molecule of the dye? K. J. MYsELs.-If there was a large effect by one kind of dye, presumably different kinds of dyes would give very different results and we propose to consider this effect a t a later time. MALCOLM DoLE.-Could not dye molecules diffuse from one micelle to another micelle by some kind of chain effect? ' The solubility of the d e in the solvent, even if the dye is very insoluble, may maze this kind of an effect possible. K. J. MYsELs.-If the dye is completely insoluble in water, then in order for them to jump from one micelle to another, the micelles must be very close together, in fact the micelles must touch, which is very unlikely. MALCOLM DoLE.-It is also unlikely that the dye is completely insoluble. K. J. MYsELs.-The solubility of the dye in water is estimated at perhaps 1/10,000of the solubility in the soap solution. Hence for 10 OOO dye molecules which are entrapped inside the micelies there is only one which is in the water. Even if this one were diffusing ten times faster than the micelles it would introduce only an error of 1/10 of 1%.
.
PREDICTION 'OF GAS-ADSORBENT EQUILIBRIA* BY F. D. MASLAN,~ M. ALTMAN .4ND E. R. ABERTH Departnmt of Chemicd Engineering, New Yo& University, New York, N . 1.. Received July 22, 1858
It has been found that a modified Polanyi-Dubinin theory can be used to correlate the adsorption of various gases on activated carbon, silica gel and activated alumina with good accuracy both above and below the critical I n this modification the adsorbate is considered as a highly compressed gas, and its volume and fugacity are cagdl%ed at the adsorption temperature. The method has been tested on nine systems and good checks with experimental results are obtained. Binary gas adsorption can be predicted from single gas adsorption data when N12V1e = NIVI N2V2. This method has been tested on the system oxygen-nitrogen-activated carbon wlth success.
+
It is desirable ttobe able to predict gas adsorptiou equilibria from a minimum of experimental data. The Polanyi-Dubinin theory offers a route to this goal. Single-Gas Adsorption.-The modification of the Polanyi-Dubinin adsorption theory proposed hy Lewis and his co-workers3 is given by when -
NrV'r = N I ~ V ' I I
(2)
(1) Presented before the twenty-sixth N'ational Colloid Symposinin
which was held under the auspices of the Division of Colloid Chemistry of the American Chemical Society in Los Angeles. California, June 1618, 1952. (2) National Research Corp., Cambridge, Mass. (3) W. K. Lewis, E. R. Oilliland, B. Chertow and W. P. Cadogan, I n d . Eng. Chem., 42, 1826 (1950) ,
where f = fugacity of the gas a t adsorption pressure and temperature, f s = fugacity of saturated liquid at adsorption temperature, N = moles adsorbed per unit weight adsorbent, R = gas law constant, T = adsorption temperature,' degrees absolute, and V' = molal volume of saturated liquid at a vapor pressure equal to adsorption pressure. The subscripts I and I1 refer to different sets of adsorption conditions. I n the above relationships the adsorbate is assumed to be a saturated liquid. Lewis, et aZ., found that good correlation could be obtained using the above equations and definitions below the critical point. Data for many hydrocarbons correlated well. Branching of the generalized correlation curves, NV' vs. (RTIV') In fa/f, were obtained for saturated and unsaturated hydrocarbons. Whereas the substitution of fugacities for pres-
*
,Jan., 1853
*
PREDICTION OF GAS-ADSORBENT EQUII,IBRIA
107
sures in the original Polanyi theory is theoretically sound to account for the non-ideality of the gases, no justification can be found on theoretical grounds for assuming that the molal volume of the adsorbate is equal to the saturated liquid volume a t a temperature where the vapor pressure equals the adsorption pressure. While this correlation is useful below the critical points of the gases and has been successful in this region, estrapolation above the critical point has been found to be impractical and the method breaks The difficulty in determining a liquid volume above the critical point is obvious. However, many adsorption temperatures are above the critical temperature of the adsorbed gas. I n order that a generalized correlation may hold over the entire temperature range for a particular gas, a consistent set of assumptions is needed to Frovide a unique relationship between the pressure and volume of the adsorbed gas at any one temperature. If one considers the adsorbate as B highly compressed gas and that the gas has the same escaping tendency from the surface of the adsorbent as from a saturated liquid-vapor interface, then it is possible to respecify the volume ( V ) and fugacity (is)for equation 1. In order to calculate this fugacity, the pressure of the adsorbed gas must be obtained. This is done by plotting the vapor pressure curve for the liquefied adsorbate and (!!‘/V) log (f./f), ‘IC. g. mole extrapolating it to the desired adsorption temperaFig. 1.-Adsorption potential diagram for oxygenture. If the temperature is below the critical activated carbon and nitrogen-activated carbon over the temperature of the gas, then the adsorbate pressure temperature range 0 to -150’. is identical with the vapor pressure, and the fugacity is the same as defined by Lewis, et al. characteristics on activated carbon are very If the temperature is above the critical temperature similar. The method mas tested again successfully by of the gas, the extrapolation appears justified on the basis of experimental data discussed by Brun- plotting adsorption data for oxygen and nitrogen a ~ e r .He ~ reports results that show that no on activated alumina and silica gel as shown in discontinuity OCCUPS in the adsorbed phase on Figs. 2 and 3, respectively.6 The data for activated alumina (Fig. 2) cover a temperature region passing the critical temperature. of -150 to -175’. It can be seen that the data Once the pressure of the adsorbate is lsnowvn, the molal volume of the compressed gas (T’) may be for both of the gases give smooth curves and small calculated from an equation of state or from a deviations. In this case the oxygen and nitrogen compressibility chart. The fugacity ( f s ) may be curves do not coincide. This might be accounted obtained by familiar thermodynamic equations or for by a diffeeence in adsorption characteristics on activated alumina. * from a fugacity coefficient chart. The data for silica gel in Fig. 3 cover the temUsing the method outlined above, adsorption This is part of data for the two systems oxygen-activated carbon perature range -130 to -150’. the same region covered in Fig. 1. However, it will and nitrogen-activated ~ a r b o nare , ~ plotted in Fig. 1. It was found that the experimental data could be noted that the silica gel figure has a separate be correlated over the full temperature range of 0 curve for oxygen and for nitrogen, while one curve to - 150’ with an average deviation of 3%. This correlates the data on activated carbon. No temperature range co\‘ers the region from below explanation for this behavior can be offered at the critical points of the gases to a region con- present other than to assume that ’the difference siderably above them and hence can be regarded is caused by different adsorption characteristics on as a rather critical test of the assumptions. The the silica gel than on the activated carbon. As a further test of this method, data for adsorppoints for both of the gases show a smooth transition as the cur’ve crosses the respective critical tion of hydrogen on activated carbon0 are successtemperatures. This helps justify the assumption fully correlated with a deviation of less than 2% made in the extrapolation of the vapor pressure in Fig. 4. These data cover the temperature region 55 to 69’K. The thermodynamic data, the vapor curve above the critical point. pressure and the PVT data for hydrogen were obThe fact that the two gases can be correlated by a single curve indicates that their adsorption tained from a recent National Bureau of Standards publication.’ The curve for the hydrogen-acti(4) 5. Brunauer, “Adsorpt~onof Ganes and Vapors.” Vol 1 , Princeton University Press, Princeton, N. J., 1946. ( 5 ) M. Altman, D Sc. Tliesis, New York University, Univers~ty Heights. New Yo1 k, 1452.
(6) W. V. Dingenin and A. V. Itterbeck, Physica, 6, 49 (1939). (7) H. W. Woolley, “Natl. Bur, Stds., Researrh Paper R P 1932,” F’ol. 41, 1948.
J 08
( T / V ) log cf./f), O K . g. mole cm.+. Fig. 2.-Adsorption potential diagram for oxygen-activated alumina and nitrogen-activated alumina over the temperature range - 150 to - 175". 10.0
N V , cm.8 g.-1 Fig. $.-Adsorption potential diagram for hydrogen-activated carbon over the temperature range 55 to 69°K.
8.0 6.0
4.0
2.0
_*
I. 0 0.8 0.6
I
0.4
d
j 0.2
2
. .IO
. .20
.40 .60.801
2
4
6 8 IO
( T / V ) log (js/f), O K . g. mole cm.-S Fig. 3.-Adsorption potential diagram for oxygen-silica gel and nitrogen;silica gel over the temperature range - 130 to -150".
vated carbon system cannot be directly compared with that of Fig. 1because the carbon was different. The excellent results obtained with these seven systems, especially above the critical points, appear t o justify this correlation method. It is hoped that in the future data on other gases will be available for further testing. Data on the adsorption of propane and propylene on activated carbons a t temperatures of 0 to -30' did not correlate well by use of this method. Since (8) E R. Aberth, M.Ch.E. Thesis, New York University, University Heights, New York, 1952.
these data were a t temperatures about 120° below the respective critical points and near the normal boiling points, it is believed that the assumption of a compressed gas is no longer justified as most of the adsorbate is undoubtedly present as liquid. This probably can account for the breakdown of the correlation method in these cases. Binary Gas Adsorption.-The method used above for single gas adsorption can be extended to multicomponent adsorption. In the following discussion the case of binary gas adsorption will be considered. The adsorption potential theory states that the work required to bring a gas from the gas phase into the adsorbed phase is equal to its free energy change. Extending this to binaries, if two gases are adsorbed a t the same time, the total work is equal to the sum of the work done on each gas, provided there is no energy change on mixing. Hence
+ [NRT Inf./flz
lNRT lnfs/f112 = [NRT l n f d f l ~
(3)
where the subscript 12 represents the binary gas mixture and subscript,s 1 and 2 represent the pure components. The other quantities in equation 3 are as defined previously. Applying the law of additive volumes to the adsorbate, the assumption is made that the total volume of the adsorbed gas is equal to the sum of the volumes of the individual components of the gas mixture NEVI2 = NlVl
+ N*V*
(4)
I n order for the final relationship to be specific and give only a single result, the adsorption potential corresponding to the left side of equation 3 must be constant for a certain value of N I ~ V I ~ , no matter what are the values of NlVl and N2V2.
c
I'REDICTION OF GAS-ADSORBENT EQTXLIBILIA
Jan., 1953
Dividing equation 3 by N12V12 and cancelling R gives ~ ' / V IIn Z (fi lz/fd= T / V d l n (fel/fi)"lIn (faz/jdn*l(5)
+
where n1 =
Nl/Nlr and nz
Since
=
N2/N12
+
Niz = Ni Nz n2 = 1 - nl
(6)
(7)
Itearranging equation 5 and taking the antilogarithm gives j;!f;-"l = j2!f:2-"l/j8l?/fin
(8)
Equation 8 is a general relationship between the pure gas and binary gas fugacities. For a given amount of adsorbate of a certain composition, the cleiiominator of the right side of the equation may be found from a generalized adsorption potential curve at ordinate NI2Vl2.The two fugacities in the numerator are calculated as described previously for pure gases. This leaves the gas composition in equilibrium with the adsorbate, as represented by the left side of the equation, unknown. The solution for the two gas fugacities fl and f2 can be handled by trial and error or a t low pressures by the following method. At low pressures the fugacity can be taken as being equal to the partial pressure of the gas with only a slight error. If the total pressure is one atmosphere, then fl
+ jz = 1
(9)
and j2 = 1
- fl
Therefore equation 8 may be written for one atmosphere as jI"l(1- jp-"' = j.7 !f:, -m1/js12/j12
(10)
In this equation alL values are known for a specified adsorbate amount and composition except SI.' Once the correct value of f l is obtained, the gas phase composition can be readily calculated.
101)
The gas-adsorbent equilibrium is then completely established. Equation 10 was tested on the system oxygennitrogen-activated carbon for which experimentally determined values of relative volatilities are available.6 The results are given in Table I. The calculated results compare quite closely with the experimental ones. Even though the temperatures range from below to above the critical points, the average deviation is only 10% and the maximum is 19%. This system is a good one for such a test as the relative volatilities are small and the adsorption characteristics of oxygen and nitrogen are similar. Since the prediction method gives good results on this system, a similar or better accuracy might be expected with other systems having larger relative volatilities. Lewis and co-worltersa used an equation derived by Broughton for the prediction of binary equilibria from single component data with only fair results. This equation was tested also on the oxygennitrogen-activated carbon system and results are given in Table I. They are about the same as those from equation 10 except for the -150' case. This is considerably in error.
.
TABLE I COMPARISON OF EXPERIMENTAL AND CALCULATED RELATIV~ VOLATILITIES FOR OXYGEN-NITROQEN-ACTIVATED CARBON SYSTEM Relative volatilitiea
Temp, OC.
- 150 - 130 - 110
Lewis, el at.,
Exptl.
Eq. 10
eq.
1.7 1.5 1.3
2.1 1.3 1.2
2.9 1.3
'
1.3
The explanation for this large deviation is that integration of the Broughton equation requires extrapolation of the pure isotherm data to very low pressures which can introduce errors. These errors become larger as more gas is adsorbed, and in these results the largest deviation occurred a t -150' where the largest amount of gas is adsorbed. (9) W. IC. Lewis, E. R. Gilliland, B. Chertow and W. P. Cadogan, Ind. Enp. Ohem., 42, 1319 (1950).
