Equilibrium Properties for Gas Mixtures on Homogeneous Solid Surfaces from a Statistical Mechanical Model for Partially Mobile Adsorption Chul So0 Lee and John P. O'Connell* Department of Chemical Engineering, University of Florida, Gainesviile, Fla. 3267 7
A previously developed statistical mechanical model for partially mobile adsorption of pure adsorbates on homogeneous solid surfaces is extended to mixture adsorption equilibria. Although the model is applicable to both thermodynamic and transport properties of adsorbed molecules on homogeneous surfaces, only the adsorption isotherm data are presently available. Comparison with experimental data gives good agreement when simple mixing rules are used, while model calculations show that deviations from "ideality" are important only if the molecular parameters for adsorbate interactions are widely different.
Introduction The application of thermodynamics to the study of the adsorption of binary gas mixtures on solid surfaces is well summarized by Van Ness (1969). The ideal adsorbed solution theory developed by Myers and Prausnitz (1965) has been compared favorably with experimental data in Myers and Prausnitz (1967) and Friedrich and Mullins (1972), while the use of a two-dimensional equation of state by Hoory and Prausnitz (1967) indicates that the deviation from "ideal solution behavior" can be significant. The latter extension of an earlier model by Ross and Olivier (1964) is based on the assumption of completely mobile adsorption. However, this is not likely to be an accurate description of the physical situation as Ross and Steele (1961) show. The development of a single model for correlation of both adsorption and surface diffusion data on homogeneous surfaces (Lee and O'Connell, 1972) indicates that adsorption should be considered as only partially mobile since the surface potential field is structured. The present work is an extension of that model to equilibrium properties of multicomponent systems. The brief analysis given here is intended to show the first application of a completely molecular model to these properties and to indicate the effects of varying molecular parameters on mixed adsorbate properties, particularly comparing results of the macroscopic "ideal solution" approach and the completely mobile approach with those from our microscopic expressions. This is also preparatory to prediction of transport properties of mixtures on homogeneous surfaces and of all properties on heterogeneous surfaces. Molecular Model a n d Adsorption Equilibria The potential energy of an admolecule of species i a t position r can be given by (Lee and O'Connell, 1972)
uf(r)= u , s ( r m )+ nL,kj)
(1)
1
where
and
For unlike interactions,
follow the common mixing rules
From the van der Waals and Bragg-Williams mixture theories for a t-component mixture, the system partition function of the Lee-O'Connell model is
where f
*N=
EN, f-1
and M is the total number of sites. The expressions for the molecular partition functions, q , are the same as those for single components given in Lee and O'Connell (1972). The configurational part of the partition function is calculated in the free-area approximation
Z,y = A,.'' exp( -4) (11) where AT is the free area and 1c, is the total potential energy. The potential energy is given by
where T is the temperature, k is the Boltzmann constant, A is the surface area, and R B1~is given by
tggij
and
UHij
are assumed to
(13) Here, uAvis the nearest neighbor distance between sites Ind. Eng. Chern., Fundam., Vol. 13, No. 3, 1974
165
and the integer n is taken such that n.uM is larger than U H L j , but ( n - l ) u is ~ not. For graphitized carbon black U M is 2.5 8, and n is generally 2. The free area part is calculated from
where P, = N , I A
and the contact radial distribution function g i j ( U H i j ) is from the two-dimensional scaled-particle theory of Lebowitz, et al. (1965).
The binary adsorption equilibrium is well described using an x-y diagram as in vapor-liquid equilibria where
Defining the relative volatility Prausnitz (1967)
n2,1
as given in Hoory and
(18) in eq 16 is unity if two molecules a t adjacent sites can interact and zero if not (See Appendix A of Lee and 0'Connell, 1972 and corrigendum for detailed discussion). Inserting eq 17 into eq 14 results in 6ij
a
A(A
- 8.A)
+
0.Ab A(A - 8 . A ) '
('')
where
where
The adsorption isotherm of component i is calculated from
Assuming ideal gas in the bulk phase
Here
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Ind. Eng. Chem., Fundam., Vol. 13, No. 3, 1974
then
The value of n2,1 is calculated using eq 26 and 31. The present expressions, while complex, are programmed for the computer with little difficulty and x-y diagrams can be generated for little cost. There is no convenient way to rearrange our expressions to make direct comparison of our quantities with the "SOlution" quantities of Myers and Prausnitz (1965). Only final results of n2,1 can be compared numerically and we have done some of this below. We are developing expressions from the present model for surface diffusion of mixtures on homogeneous surfaces, but since no data are now available, we plan to report them in the future. Comparison with D a t a and Discussion The adsorption isotherm data of Friedrich and Mullins (1972) are the only ones presently available in the literature for mixtures on homogeneous surfaces. The potential parameters U H , egg, and e and the Henry's constant were fitted to their pure component data. The values of e are estimates within *0.2 kcal/mol since the equilibrium isotherms are not very sensitive to the value chosen. All values of &, in eq 16 were set at 0.5. The values of us = 0.23 8, and p s = 0.192/A2 were set from previous work on graphitized carbon blacks. The specific surface area was 13 m2/g, as for P33 (2700), considerably less than the 89 m2/g we used for Graphon. No mixture parameters were required other than those calculated from the mixing rules (4) and (5). The adsorption equilibrium data for ethylene-ethane mixtures on Sterling FTG-D5 carbon black are compared with theoretical results in Figure 1. Similar comparisons for the systems propylene-propane and ethane-propane are shown in Figures 2 and 3. All of these systems also had been calculated successfully by the ideal adsorbed solution theory and by the two-dimensional equation of state method in Friedrich and Mullins (1972). Here, the deviations in Figure 2 are somewhat smaller than those in the above reference where the calculated values were higher than the data, although all are within the estimated accuracy. This success of the ideal-solution theory at low coverages is expected within the reasoning of Henson and Kabel(l967). The parameters shown in Table I are not related to any
YI-XI
0 05
0 4
02
G O
06
08
IO yI-xl
Yl
Figure 1. Binary adsorption equilibria for ethylene (1)-ethane (2) at 25°C and 700 mm Hg on SterlingFTG-D5
175
15 IO
A
100
20
175
I
0.0 0 IO
0 00
I
A
0.00
00
02
04
06
08
IO
Yl
0.05
1
P= 10-0 Torr
T=25OC
Figure 4. Mixture adsorption model calculations for various adsorbate size and energy parameter ratios when Henry’s constants are equal on Sterling FTG-D5. Component 1 identical with propylene
0.04
0.03 Yl-xl
0 02
0 01 -Calculated
0.00
00
0.2
0.4
08
0.6
1.0
YI
Figure 2. Binary adsorption equilibria for propylene (1)-propane ( 2 ) a t 25°C on Sterling FTG-D5
I
O5 04
Figure 5 . Mixture adsorption model calculations for various total pressures when Henry’s constants are equal on Sterling FTG-D5, component 1 identical with propylene
0.3
y1-5 02
0. I
00 00
02
04
06
08
I O
Yl
Figure 3. Binary adsorption equilibria for ethane (1)-propane (2) at 25°C and 700 mm Hg on Sterling FTG-D5 Table I. Molecular Parameters and Henry’s Constants for Gases on Homogeneous Carbon Black 2 (OO, u ~ A~ ,
kcal/ mol
4.10 4.20 4.51 4.58
2.90 3.10 2.90 3.20
~
Ethylene Ethane Propylene Propane
e? K, kcal/mol mm H g
1.40 1.40 1.50 1.50
7840 4980 750 670
bulk gas potential parameters or to the van der Waals parameters obtained by Friedrich and Mullins (1972). However, the parameters we have obtained appear to be physically reasonable compared to those obtained for other substances on homogeneous surfaces (Lee and O’Connell, 1972, 1974). Since no other data are available, we explore further
consequences of our model with the calculations of Figure 4, showing the effect of the ratios of the parameters tgg and ~ T Hfor two hypothetical components on the value y1 XI (An ideal mixture would be one in which the parameter ratios are unity so that y1 - XI is zero for all values of xl). The conditions are 100 mm total pressure, at a temperature of 298.15”K. Component 1 has the parameters of propylene while component 2 has the same Henry’s constant and surface interaction parameters, t and us, as propylene but different values of the admolecule interaction parameters U H and e g g . Here are found both negative and positive deviations from ideality, the magnitudes being as large as those for the ethane-propane system, where the positive y1 - XI values are primarily due to the differences in Henry’s constants. For fractional surface coverage less than 0.3, the effect of the size parameter OW is less important than is that of the energy parameter e g g . However, the ratios of parameters chosen for Figure 4 are much larger than those found for the systems shown in Figures 1, 2, and 3 so it is not surprising that they were adequately fitted by the ideal adsorption solution theory. The advantage of the present model, like that of the completely mobile model, is that the regions where “nonideality” is important can be delineated. Such knowledge can be of importance in designing and screening adsorption separation processes based on equilibrium (as opposed to rate-controlled or molecular sieve systems). Ind.
Eng. Chem., Fundam., Vol. 13, No. 3,1974
167
Figure 5 shows model calculations for a single set of parameter ratios when the total pressure is varied. The deviations from equal mole fraction in gas and adsorbed phases (y1 - XI = 0) go from positive to negative with increasing pressure. This behavior has been observed previously by Hoory and Prausnitz (1967). Crossing of the diagonal line is anologous to azeotropic behavior in vaporliquid equilibria. Thus the present calculation shows the pressure requirements for a successful separation process. All of these phenomena can be predicted from only pure component properties in the present model. The “ideal solution” theory of Myers and Prausnitz (1965) rigorously holds here only if the parameters CTH and tgg are the same for both components. However, the calculations shown in Figures 4 and 5 indicate that significant deviations occur only for large differences of parameters and that even then, their effect will not be important when the Henry’s constants are quite different, as they are likely to be in most cases. Thus, the ideal solution model gives quite good values of y1 - XI for somewhat different substances (K1/K2 1 1.5) and the results are not sensitive to the mixing rules for parameters. However, azeotropy can occur if the Henry’s constants are sufficiently similar and the interaction parameters are sufficiently different that the “activity coefficient” described by Hoory and Prausnitz can assume a value greater than the ratio of the Henry’s constants. This probably will require K1/K2 to be less than 1.2 and the parameters to be more than 20% different. The propane-propylene system examined here meets the former but not the latter criteria. Although the calculations given here indicate that the Hoory-Prausnitz method, also used by Freidrich and Mullins, can predict the same equilibrium phenomena as the present model with fewer complexities, the present model was developed to predict both adsorption and surface diffusion. It is expected that both kinds of data can be pre-
dicted for mixtures using only the pure component parameters and mixing rules employed here. Unfortunately, only the few data analyzed here are available for comparison at the present time. Summary A statistical mechanical model for partially mobile adsorption of gas mixtures on homogeneous solid surfaces has been developed for equilibrium isotherms. In the absence of surface diffusion data, which better determine the parameters characterizing partial mobility, the choices of some parameters are uncertain, but comparison of experimental data with calculated values shows good agreement. Model calculations indicate that only large differences in adsorbate parameters lead to significant deviations from ideal behavior on graphite surfaces.
Literature Cited Friedrich, R. O., Mullins, J. C., lnd. Eng. Chem., Fundam., 11, 439 (1972). Henson, T. L., Kabei, R. L., Chem. Eng. Progr. Symp. Ser., 6 3 (74), 36 (1967). Hoory, S. E., Prausnitz, J. M., Chem. Eng. Sci., 22, 1025 (1967). Lebowitz, J. L., Helfand, E., Praestgaard, E.. J . Chem. Phys., 43, 774 (1965). Lee, C. S..Dissertation, Unlversity of Florida, 1972. Lee, C. S., O’Connell, J. P., J , Colloid lnterface Sci., 41, 415 (1972). (A corrigendum is available from the authors.) Lee, C. S., O’Connell, J. P., submitted to J. Phys. Chem., 1974. Myers, A. L., Prausnitz, J. M.,A./.C.h.E. J . , 11, 121 (1965). Ross, M., Steele, W . A.. J. Chem. Phys., 35, 850 (1961). Ross, S.. Olivier, J. P., “On Physical Adsorption.” Interscience, New York, N . Y., 1964. Van Ness, H. C., lnd. Eng. Chem., Fundam., 8,464 (1969).
Received f o r review M a r c h 5,1973 Accepted J a n u a r y 28,1974 T h e a u t h o r s are g r a t e f u l to t h e N a t i o n a l Science F o u n d a t i o n for f i n a n c i a l s u p p o r t a n d t o t h e U n i v e r s i t y of F l o r i d a C o m p u t i n g Center f o r use o f i t s facilities.
High-pressure Kinetic Studies of Solvent and Substituent Effects on Diels-Alder Reactions James R. McCabe’ and Charles A. Eckert* Department of Chemical Engmeering, University of llllnois, Urbana, Illinois 61807
The rates of three Diels-Alder reactions of maleic anhydride with chloroprene, with piperylene, and with 2-methoxybutadiene have been studied in six solvents each as a function of temperature and at pressures up to 1700 atm, using the new in situ high-pressure technique. Together with similar data already reported for the same reaction with two other dienes, isoprene and 1methoxybutadiene, these provide a comprehensive data base for studying solvent and substituent effects on the kinetics of a nonpolar reaction, resulting in a detailed understanding of this reaction. This not only permits the observation of many properties of the transition states (such a s dipole moment and sr-interactions) but also leads to a predictive method for reaction design, involving the Hammett equation for substituent effects and regular solution theory for kinetic solvent effects. This method should be applicable in similar form to other nonpolar reactions and, with some modification, to polar reactions as well.
The key to any detailed study of reaction chemistry is the transition state or activated complex. Some knowledge of the physical characteristics of the transition state
’ Present address, Chevron Research Co., Richmond, Calif. 94802. 168
Ind. Eng. Chem., F u n d a m . , Vol. 13, No. 3 , 1974
is essential in determining reaction mechanisms, evaluating and predicting solvent and substituent effects on reaction rates and controlling reaction selectivity. Since, by definition, the transition state is not isolable, its physical properties must be determined indirectly from transition
