THEROLEOF HINDERED ROTATION
1985
The Role of Hindered Rotation in the Physical Adsorption of Hydrogen Weight and Spin Isomers
by P. L. Gant, K. Yang, Central Research Division, Continental Oil Company, Ponca City, Oklahoma
M. S. Goldstein, Organic Chemicals Division, American Cyanamid Company, Stamford, Connecticut
M. P. Freeman, and A. I. Weiss Central Research Division, American Cuanamid Company, Stamford, Connecticut 06904 (Received September 1.9, 1969)
Gas adsorption chromatographic techniques are employed to study the limiting adsorptions of all six weight isomers of molecular hydrogen on Molecular Sieve 4A in the temperature range from 169 to 219'K. Further, the separation factors for 0-Hzlp-H~are also determined in the range from 135 to 160'K. It is found that adsorption models based on the Kirkwood-Wigner partition function (developed for spherical molecules through terms in h4) are inadequate to explain the observed relative adsorptions and of course they predict no separation of the spin isomers. On the other hand, an adsorption model based on a hybrid partition function which takes full account of molecular rotation and libration (developed for adsorption on a plane surface elsewhere) fits the differenced data of the symmetric weight isomers to within experimental precision with reasonable values of the parameters, but for two different energies. The fit of this model to the raw data gives essentially the same parameters with substantially greater residual error, probably reflecting the departure of molecular sieve geometry from a plane surface. No attempt was made to include the asymmetric weight isomers in this analysis because of clear-cut inconsistency with behavior predicted by the hybrid-partition-function model. Furthermore, the observed o-Hz/p-Hz separation factors are accounted for t o within the ambiguity of this hybrid-partition-function model, demonstrating that the effects of hindered rotation are of paramount importance in this temperature regime for the adsorption of molecular hydrogen.
Introduction The physical interaction of gases, far above their normal boiling points, with solid surfaces is of considerable interest. Static P , N , T (pressure, amount, temperature) adsorption measurements, probably the most revealing in this are difficult because of the unwonted precision with which such measurements must be carried out. Gravimetric measurements,4 still in their infancy, show great potential, but thus far most of the work in this area has been done by gas adsorption chromatography (gsc). Kobayashi, et al.,s have compiled a substantial list of workers who have used gsc techniques t o infer thermodynamic information, or its equivalent, about high-temperature physical adsorption. The present work is such a study of the interaction of the variously isotopically substituted isomers, the weight isomers, of molecular hydrogen with a molecular sieve adsorbent. DeMarcus, et have long since shown that the different weight isomers of hydrogen should display different equilibrium adsorption properties because of the differences in the widely spaced negative discrete energy levels in their interactions with the surfade. This theoretical development, based on the KirkwoodWigner expansion (through terms in h2)of the quantum-
statistical partition function, has since been tested by Freeman' and, using the lower temperature data of Constabaris, et U E . , ~ by Yaris and Sams8and found to be functionally correct with reasonable values of fitted parameters. (Note that these latter authors extended the theory, albeit i n c ~ r r e c t l yto , ~terms in h4.) The purpose of the present work is twofold. On the one hand, by using gsc techniques to measure a subtle differential equilibrium effect, it serves as an excellent check on the validity of using these techniques for adsorption studies. On the other hand, b y extending the measurements to tritium and tritium-containing (1) For a summary of earlier work, see M. P. Freeman, J. Phys. Chem., 62, 723, 729 (1958). (2) G . Constabaris, J. R . Sams, Jr., and G. D. Halsey, Jr., ibid., 65, 367 (1961). (3) J. R. Sams, Jr., G . Constabaris, and G . D. Halsey, Jr., J. Chem. Phys., 64, 1689 (1960). (4) Y, Y. Huang, J. E. Benson, and M. Boudart, in press. (6) R. Kobayashi, R. 8. Chappelear, and H, A. Deans, Ind. Eng. Chem., 59, 63 (1967). (6) W. C. DeMarcus, E. H. Hopper, and H. M. Allen, AEC Bulletin K-1222, Carbide & Carbon Chemicals Company, Oak Ridge, Tenn., 1955. (7) M. P. Freeman, J . Phys. Chem., 64, 32 (1960), (8) R. Yaris and J. R . Sams, Jr., J . Chem. Phys., 37, 571 (1962). (9) M. P. Freeman and M. J. Hagyard, ibid., 49, 4020 (1968). Volume 74, Number 9
April 80,1070
1986
P. L. GAXT,K. YANG,M. S. GOLDSTEIN, M. P. FREEMAN, AND A. I. WEISS
isomers (a somewhat hazardous procedure not yet undertaken by static measurement techniques) a much more sensitive test of the theory can be made than has hitherto been possible. Theoretical considerations of Friedmann'O and of White and his coworkers11,12have shown that the early DeXarcus treatment may have been oversimplified in that it treats the molecules as spherical systems. In fact the rotational energy levels of the molecules, greatly perturbed by the presence of the surface, should substantially modify the relative locations of the lowest negative discrete energy levels for the different isomers and consequently modify their relative equilibrium behavior. Where one would expect this effect to be of the greatest significance is in the experimentally observed separation of the nuclear spin isomers of the symmetric weight isomersl3 for which the theory of DeMarcus provides no basis. Although no measurable spin isomer separations occur in the relatively high and restricted temperature interval in which the bulk of the experimental work reported here was done, a special lower temperature study was made to determine separation factors for the spin isomers of Hz; the results constitute an important part of the present study. To interpret the data we have chosen to use the formalism of the virial equation of state' which has been shown by Barker and EverettI4 to be completely equivalent to more conventional formalisms based on models related to the Gibbs boundary condition. Although the "Gibbsian" models have the advantage of greater familiarity (and indeed for some problems such as the lateral interaction of admolecules, greatly simplify numerical calculation,15,16) the virial approach is more nearly tied to gsc and to the area to be examined. That is, the excess retention volume of gas chromatography may be directly identified with the prime measurable of the virial approach, the excess apparent ~olume,'7~~8 while the fitted parameters of this approach are a capacity factor (simply related to the area) and the cogent parameters of the adsorption potential energy expression. Hence the virial approach should be the most sensitive in indicating apparent changes in this potential due to quantum effects.
Experimental Method Materials. Deuterium was obtained from Airco, Allentown, Pennsylvania. 3He was obtained from illonsanto Research Corp., Miamisburg, Ohio. Tritium obtained from Oak Ridge National Laboratories, Oak Ridge, Tenn., was purified by passing it through a short Molecular Sieve 3A column maintained at liquid methane temperature.lg Samples of Hz, HD, and DZ were prepared by heating a mixture of hydrogen and deuterium to 500" in a nickel bomb. Mixtures containing all the hydrogen isotopes and 3He were prepared by transferring 2.2 Ci of Tz gas and 500 ml 3He at 70 Torr into a 500-ml glass bulb. The bulb was then The Journal of Physical Chemistry
pressurized to 700 Torr with Hz, HD, and Dz mixture and allowed to stand for 2 weeks to allow H T and D T to form through p radiolysis from Tz. Chromatographic Conditions. The Linde 4A 50-60mesh Molecular Sieve obtained from Analabs Inc., Hamden, Connecticut, was treated to remove fines20 and actjivatedin the following manner. A 50-mm glass column, 60 cm long, with a coarse glass frit at the bottom was filled about two-thirds full with deionized water. About 100 g of sieve was slowly poured into the column and washed with a gentle stream of deionized water for about half an hour. This removed about 5 g of fine material. The washed 4A sieve was transferred as a slurry to a beaker, allowed to settle, and most of the water decanted. It was then placed in a glass column heating mantle, gently heated to remove the remaining water, and activated at 400" for 24 hr. The column used in this investigation was a 20-ft, l/g-in. 0.d. by 0.075411. i.d. Type 303, stainless steel tubing obtained from Greenville Tube Co., Greenville, Pa. Since this tubing was cleaned and capped, no additional treatment was necessary. The column was packed with house vacuum and gentle tapping. The columns were then coiled on a 3.5-in. mandrel and heated to 400" for 24 hr with about 30 cm3/min helium flow. This long activation was found necessary for optimum resolution. The low-temperature column compartment consisted of a double-walled foamed-polystyrene box cooled with liquid nitrogen. Pressurized liquid nitrogen from a Linde 110-1. vessel was fed to a flat refrigerator coil between the two polystyrene boxes and was controlled by a Teflon-seated solenoid valve. The nitrogen gas emerging from the coil provides even temperature without the added complexity of a blower. Temperature was controlled by a transistorized controller of our own design which used a low-temperature thermistor for sensing. These thermistors were mounted near the gas outlet holes from the refrigerator coil. Using this controller, column temperatures between 0 and - 175" could be maintained to at least & l oinside the inner box. Temperature was continuously recorded on a (10) H. Friedmann, Physica, 30, 921 (1964). (11) D. White and E. N. Lasettre, J . Chem. Phys., 32, 72 (1960). (12) A. Katorski and D. White, ibid., 40, 3183 (1964). (13) For a review, see 8 . Akhtar and H. A. Smith, Chem. Rev., 64, 261 (1964). (14) J. A. Barker and P. H. Everett, Trans. Faraday Soc., 58, 1608 (1962). (15) J. R. Sams, Jr., G. Constabaris, and G. D. Halsey, Jr., J . Chem. Phys., 36, 1334 (1962). (16) J. R. Sams, Jr., ibid.,43, 2243 (1965). (17) J. F. Hanlan and M. P. Freeman, Can. J . Chem., 37, 1575 (1959). (18) J. F. Hansen, H. A. Murphy, and T. C. McGee, Trans. Faraday Soc., 60, 597 (1964). (19) K. Yang and P. L. Gant, J . Phys. Chem., 66, 1619 (1962). (20) F. Farre-Rius and G. Guiochon, J . Gas Chromatogr., 7, 33 (1963).
THEROLEOF HINDERED ROTATION
1987
Table I: Net Retention Volume (cm3/g) of Hz a t Different Sample Pressure and a t Different Temperatures
300 150 75 15 OQ
Standard dev a
1.47 1.55 1.59 1.65 1.65 ic0.01
1.79 1.88 1.90 1.97 1.96 fO.01
2.17 2.25 2.32 2.40 2.39 f0.02
2.50 2.57 2.62 2.69 2.68 rt0.02
3.04 3.11 3.20 3.28 3.27 dz0.03
3.66 3.77 3.93 3.97 3.99 10.03
The retention volume, V N ' , at zero pressure is estimated by assuming a relation, VN = VN'
Foxboro low-temperature recorder. Columns were mounted in the inner box with long, low-volume Swagelok bulkhead connectors. A small volume gas injection system was constructed using '/*-in. Whitey Swagelok type valves, '/s-in. tubing, and a Wilkens Instrument and Research Corp. (Walnut Creek, California) gas injection valve. The sample size was 0.25 ml at 15-300 Torr. Fenwall Type 112 termistors were used for detection of H2, HD, and Dz. These were mounted in a low-volume stainless steel block and were screened in the manner described by Kieselbach. z1 An independent supply of carrier (He) was fed to the reference thermistor at a rate of 5 cm3/ min. The block temperature was maintained at 28" with a water bath. A 0.5-ml ion chamber of our own design was used for detection of HT, DT, and Tz peaks. Potential for this chamber was supplied by a 6-V dry cell. This was connected to an Applied Physics Corp. Model 31 vibrating reed electrometer equipped with a turret switch containing a 101'-ohm resistor, which allowed a response time of approximately 1 sec. Peaks from both detectors were registered on a Brown dual-pen potentiometric recorder.
Experimental Results Table I summarizes a typical set of net retention volumes, VN, at different sample pressures and at different temperatures. VN is calculated from the (1)
where tR = retention time measured at peak maximum, t H e = retention time of 3He, = weight of solid, F , = flow rate measured a t ambient condition with a soap bulb flowmeter, T , = column temperature, T, = ambient temperature, P, = vapor pressure of water a t ambient temperature, Pa = ambient pressure, and Pi = pressure at column inlet. As seen in Table I, VN decreases with increasing sample pressure. By supposing a linear relation,' VN is extrapolated to zero sample pressure. Resulting retention volumes, VN ",a t zero sample pressure are summarized in Table 11.
w,
5.45 5.54 5.69 5.77 5.77 f0.04
+ const P.
Table 11: Ket Retention Volumes (cm*/g) a t Zero Sample Pressurea Temp, OK
Ha
HD
HT
D2
Ta
219 211 203 197 189 181 174 169
1.65 1.96 2.39 2.68 3.27 3.99 4.85 5.77
1.75 2.10 2.57 2.84 3.56 4.37 5.36 6.40
1.91 2.28 2.77 3.12 3.80 4.67 5.75 7.23
1.87 2.26 2.83 3.13 3.91 4.87 6.07 7.30
2.05 2.49 3.07 3.49 4.43 5.38 6.76 8.21
a
For DT, VN" = 6.30 a t 174" and 7.85 at 169°K.
At temperatures above 169"K, where data for the weight isomers are determined, there is no measurable separation of any of the spin isomers. The spin isomer effect in H2 is investigated at lower temperatures from 160 to 135°K Table I11 summarizes the results in terms of the separation factor S
S =
(torch
-
tHe)/(tpara
Table 111: The Separation Factors, S
-t
(tOT1hO
-
(3)
tHe)
=
~ ~ ) / (& ~tae),a t Different Temperatures
-------__
Temp, OK
relationsz2
VN = ( t ~ ~HB)FC/WB
4.58 4.70 4.79 4.83 4.85 fO.09
S
135
141
147
152
160
1.108
1.088
1.073
1.058
1.042
Discussion Kirkwood-Wigner Partition Function. The possibility exists that the spin isomer separations, though ubiquitous, may be due to some form of specific magnetic interaction with the surface and so the first attempt to interpret the adsorption results involved the weight isomer data' only and used the formulation of DeMarcus, et aLJ6t7based on the Kirkwood-Wigner (21) R. Kieselbaoh, Anal. Chern., 32, 1749 (1960). (22) For example:,see 5.D. Nogare and R. S. Jevet, Jr., "Gas Liquid Chromatography, Intarscience Publishers, New York, N. Y . , 1962, pp 73-78.
P. L. GANT,K. YANG,M. S. GOLDSTEIN, M. P. FREEMAN, AND A. I. WEISS
1988
partition function expanded through terms in h2. This is of the form
to somewhat lower temperature dataS2 Therefore, for completeness it was attempted here, where now
V N / A s o = B(#('J)/kT)4
where A is the area, SO is that distance, s, ( u = s/so) of the molecule from a reference plane running through the surface where attractive and repulsive forces exactly cancel, and m is the molecular mass of the weight isomer is question. The integrals B (the classical contribution) and IIare functions of #, the potential energy function of interaction of the molecule with the surface (and its derivatives), while kT has its usual significance. This partition function has been applied successfully to account for the difference in adsorption between H2 and D2 on a fairly homogeneous carbon black over a much wider temperature range' than that studied here. Friedmann23 has shown in connection with vapor pressure isotope effects, it is possible to use eq 4 to apply a necessary, but not sufficient, test of consistency to data that is independent of the actual values of the integrals and indeed of the potential energy expression. That is, given data for three isomers, e.g., Hz, D2, T2,one can write
VN(&)
- V N ( H ~) -
VN(TZ)-
VN(DZ)
A somewhat more complex test of consistency analogous to eq 5 , using the additional weight isomer HD, works quite well and so the data were fitted to the model, eq 6 , using the integrals tabulated for interaction with a plane surface by Yaris and Samss for an attractive third power repulsive ninth power potential energy expression
The parameter ~ i *is the depth of the potential energy well for each isomer, initially taken t o be all equal. The parameters of eq 6, evaluated by the method of least squares, are tabulated in Table V. Two facts emerge immediately. First, the rms deviation is rather large. Second, and more important, the collision parameter so, which should certainly be between 2 and 3 A, is unrealistically small (as indeed it must be to have I2 contribute significantly at such high temperatures).
-
m(Td MD2) m(Hd1 = 3,005 ( 5 ) m(H2) [m(T2) - m ( W 1 Table IV summarizes experimental values of the lefthand side of eq 5. There does not seem to be any systematic variation with temperature, but the average value is only 1.5, only half the value predicted. Thus this simple approach may be discarded forthwith.
Table V:
"Best Fit" Results to Yaris and Sams Functions, Equal EL*
Isomers (joint fit)
A,
r*/k,
ma/g
"K
'4
Root mean square deva
Ha, Dz, Ta Hz, Dz,Tz,H D Hz, Dz, Tz, HD, H T
160 171 187
1290 1282 1261
1.189 1.163 1.17
0.29 0.28 0.26
so,
~~
Table IV : Retention Volume Ratios (Left-Hand Side of Eq 5) a t Different Temperatures Temp, OK
219 211 203 197 189 181 174 169
Ratio
1.2 1.3 1.7 1.3
1.2 1.7 1.8 1.7 Av 1 . 5 = k 0 . 2
Jenkinsz4 showed that the next higher term in the K-W expansion should be negligible at the temperatures involved in the present experiment. However, Yaris and Samss have made some use of it in fitting this model The Journal of Physical Chemistry
a The rms deviation is to be compared to the standard deviations of the data in Table I and the maximum difference in V N " for Hzand Tzof 2.44 at 169°K and the minimum difference of 0.40 a t 219°K.
This approach, though not promising, could not be categorically dismissed without trying various refinements. The first change was to rederive the equations for adsorption in spherical cavities so that they more nearly correspond to the adsorption geometry in molecular sieves.25 When the radius of the sphere was left as an adjustable parameter, however, the computer consistently converged to infinite radius with no (23) H. Friedmann, Advan. Chem. Phys., 4, 225 (1962). (24) C. Jenkins, Jr., Ph.D. Thesis, Department of Physics, University of Kentucky, Lexington, Ky., 1962. (25) D. W. Breck, J. Chem. Educ., 41, 678 (1964).
THEROLEOF HINDERED ROTATION improvement in the collision parameter so. When the radius was fixed at reasonable values (2 to 3 molecular diameters), the converged value of the surface area was an order of magnitude too high (ca. 3500 mz/g) and the collision parameter smaller yet. When the model was refined further, by allowing the isomers to assume difYerent adsorption energies to allow for the fact that the greater centrifugal force on the interatomic bonds of the lighter isotopes causes them to stretch and increase in polarizability,26no further improvement was noted. The computer made only slight adjustments in the energies of the different isamers and more often than not ordered them in the wrong sequence. Further refinements, mostly trivial reformulation attempts, met with no better success, so we are forced to the negative conclusion that an adsorption model based on the Kirkwood-’CVigner partition function does not satisfactorily account for these data. Hybrid Partition Function. Stimulated by these problems, an independent study was initiated on the role of hindered rotation in hydrogen adsorptiongusing a rather more cumbersome approximate partition function based on the work of Hagyarde27 This new partition function, fully described el~ewhere,~ may be generated when the physical situation is sufficiently simple to allow the partition function to be factored into single particle partition functions (the case for high-temperature adsorption) and the lower few energy levels are known for the potential expression which may now include the rotational degrees of freedom. The procedure is to form the usual exponential sum over states for the known energy levels and add to that the classical partition function evaluated over all phase space, again including the rotational degrees of freedom, excluding only those lowest cells containing and corresponding in number to the known energy levels. A single gross feature immediately emerging from this study is sufficient to explain the failure of the Kirkwood-Wigner partition function for these data. Namely, the partition functions for all isomers when logarithmically plotted us. 1/T waver somewhat but are more or less well behaved excepting in the limited temperature range studied in this experiment (Figure 1). Here, a significant “spoon” arising entirely from the rotational contribution of the classical part of the partition function obviates conformity to the simpler K-W model. The functions plotted in Figure 1 are for the combined nuclear spin isomers and have been reduced by the respective gas-phase partition functions so as to give a function directly identifiable with the excess retention volume measurements of gas chromatography, but reduced by the capacity factor As,. They were generated for a depth of potential energy well of 2500 cal/mol (corresponding to a depth of well of 2167 cal/mol for spherically cveraged molecules and a collision parameter of 2.2 A (y= 1.20) in a Morse po-
1989
-4
-9 m > z m
-5
-
.
I
0 v
-6
5
e
1
6
II
IO
9
c*/kT
Figure 1. The revised hybrid partition functions for the weight isomers of hydrogen shown. e * = 2500 cal/mol, so = 2.2 A (y = 1.20).
1.30
.90
-
I
110
I
60
I
80
100
I
I
110
140 T
I60
180
200
120
(’K)
Figure 2. The measured ortho-para separator factors for He together with theoretical separation factors derived from the hybrid partition function ( e * = 2500 cal/mol, SO = 2.2 A) both old (ref 9) and revised (this work).
tential described e l ~ e w h e r e . The ~~~~ partition functions plotted here, the revised values, differ slightly from those published previou~ly,~ called the old values, in that allowance is made for the known empty energy levels above the known filled levels for ortho-H2 and Tz and para-Dz. Although this change is scarcely noticeable in the plot of Figure 1, as shown below it has a rather beneficial effect on the computed spin isomer separation factors (Figure 2). Note that the increased “wavering” of the revised functions is probably due a t least in part to the increased density of computed values. For various reasons, fitting this model to the present experimental data is not straightforward. First, the hybrid function does not work for the asymmetric isotopes in that H T is almost indistinguishable from Dz in its experimental behavior (as predicted by the K-W formulation), whereas the hybrid function says it should be indistinguishable from hydrogen in the temperature regime studied here (see a limited discussion in Conclusions, below). For this reason, only the (26) H. F. F. Knapp and J. J. M. Beenakker, Physica, 27, 523 (1961). (27) M. 8. Hagyard, Ph.D. Thesis, Department of Physics, University of Kentucky, Lexington, Ky., 1967. Volume 7 4 , Number 0 April 80,1070
P. L. GANT,K. YANG,M. S. GOLDSTEIN, M. P. FREEMAN, AND A. I. WEISS
1990
I
Tz DIFFERENCED DATA Tz DI
-- HzD2 - H2
t
A L L ISOMERS H A V E SAME c*/k
c
ISOMERS ALLOWED TO H A V E D I F F E R E N T E*/k'S 1000
I100
I
1200
1300
R\
J
1400
-
KEY
E*/k (H2)
Figure 3. Sections of the sum of squares surfaces (fitting the best capacity parameter at each energy) for data both raw and differenced.
symmetric isomers could be considered. Second, the energy levels were evaluated for a particular depth of potential energy well and collision parameter,g so that the partition function cannot be rigorously used to evaluate independently all the parameters available with the R W models. The area is of course still available as a capacity factor and, as a compromise measure, the partition function is incorrectly expressed as a function of reduced reciprocal temperature factor e*/kT to permit the estimation of an energy parameter. However, it must be understood that the resulting fit is only accurate for e * = 2500 cal/mol; the larger the departure from this figure, the greater the uncertainty in the parameters evaluated. Further, as mentioned before, the plane geometry implicit in the model is inappropriate for adsorption in molecular sieves where adsorption in spherical cavities would be a more meaningful model. A somewhat subtle test of the importance of this consideration is to fit (1) the data as measured, which we will call the raw data, and differencing the partition functions, (2) to fit the differences in retention volumes for the three isomers. Differencing should subtract out the classical contribution (and thus the "spoon") which should be most strongly a function of surface curvature because it involves functions that decay less rapidly with separation from the surface than the quantum effects which are important only when the molecule is close to its potential energy minimum. If there is no bias caused by failure of the classical part of the model and all error is random, it is easy to show that the residual error of the curve-fitting process for the differenced data should be bigger than that for the raw data by a factor of
d.%
The Journal o j Physical Chemistry
t.2
1
-7.
I
1
'
160
I
170
I
I
I
1
180 190 200 210 TEMPERATURE (OK)
I
I
220
Figure 4. Residuals of the best fits to the raw and differenced data shown for both minima (all isomers same energy).
The model was fitted to the raw data for the three symmetric isomers and the differenced model to the differenced data using the redundant set of all three differences so as to weight the isomers equally. In each case this was done for the same value of E* for each of the isomers and then again, allowing the energies to vary independently. In every case a double minimum was found in the sum of squares surface. Table VI shows the fitted parameters together with their asymptotic marginal 95% confidence limits and estimated standard deviation of fit. As anticipated, the fit of the differenced data has but one-third the standard deviation of the best fits of the raw data (instead of being 4 5 times as great) so that bias in the fit of the raw data is clearly demonstrated even though the parameters obtained are essentially the same as those obtained for the difference fits. (Note that the standard deviations of the differenced data are on the order of 2/2 times the standard deviations of Table I, indicating optimum conformity to model.) The double minima may in fact arise from this surface curvature, but it seems more probable that it is the result of incorrectly treating the partition function as a func-
THEROLEOF HINDERED ROTATION
1991
Table VI : Best Fits of the Hybrid Partition Function to the Symmetric Isomer Retention Time Data Estd std Case
dev
Raw Data
a
...
...
(1.007 f 0.006)
(1,009 f 0.005)
(1.018 f 0.009)
(1.027 f 0.009)
Differenced Data 1351 f 40 ... (1,001 i 0.006) (1352 f 62) 1139 f 92 ... (0.994 f 0.002) (1157 i 3)
(1,001 f 0.006)
High energy minimum Low energy minimum
148 f 10 (151 f 9)a 367 f 140 (385 f 90)
1351 f 15 (1339 i 15) 1119 f 88 (1089 f 52)
High energy minimum Low energy minimum
177 f 40 (173 f 40) 548 f 26 (595 f 33)
...
... ...
...
(0.992 f 0.002)
0,155 (0,131) 0.271 (0.165) 0,052 (0.051) 0.086 (0.051)
Parenthetical values are those obtained when the energies were allowed to vary independently.
tion of reduced temperature. One of these minima is certainly spurious. However, picking either of these alternatives as the correct one would be at best arbitrary. Refitting the raw and differenced data, allowing the isomers to have independently adjustable adsorption energies, improves the fit somewhat but makes no appreciable difference otherwise, as shown by the sections through the sum of squares surfaces shown in Figure 3 and the energy ratios in Table VI. One might note, however, that examination of the residuals (Figure 4) shows that for both raw and differenced data the residuals for the lower energy minimum demonstrate a more systematic departure from model. Furthermore, the capacity factor is somewhat more nearly the expected value for the high energy surface in that it is about one-third the known area of these ads or bent^^^ and this is probably about right for these mildly heterogeneous surfaces.28 It remains to compare the spin isomer separation factors predicted by this model with those observed for Hz a t lower temperatures. The measured separation factors are in Figure 4, together with separationfactor functions derived from the previously published hybrid partition functionsg and these revised functions (Figure 1). No attempt has been made to adjust the energy parameters. Clearly the agreement is satisfactory to within the degree of ambiguity and precision of calculation of the functions.
Conclusions It is a t once clear that carefully done gsc techniques are indeed suitable for the detailed study of high temperature physical adsorption phenomena on the one hand, and that hydrogen adsorption in this tempera-
ture regime importantly involves hindered rotation on the other, a conclusion reached previo~sly.~Although the ambiguity and approximate nature of the partition function, together with the compromises made in its application, preclude an exact or even unique assignment of the adsorption parameters, the functional form is more nearly correct than that of the Kirkwood-Wigner formulation for spherically symmetric molecules. Note that this latter formulation may only be applied to data encompassing a much wider temperature range than is usually experimentally convenient, and then only with reservation. The conclusion found before when analyzing other data,g that the polarizability differences of the different molecules is inconsequential, is borne out in the present analysis. In only one of the four fitting procedures where the energies were allowed to vary independently did they differ by more than 1% and in three of the four cases they were ordered in the wrong direction (Le., Hzwas the least energetic). Finally, the dilemma of the asymmetric isomers remains. These data strongly reaffirm that for these isomers the hybrid partition function is wrong. This probably reflects back to the potential energy expression, albeit apparently quite reasonable, used to find the energy levels.12 No serious attempt a t an explanation has as yet been proposed, but it is probably somehow related to the effect of the constraint of the surface on the librational mode. This might be expected to cause the angular motion to center more nearly about the center of the molecule than about the center of mass as it does in free rotation. (28) M. P. Freeman and K. Kolb,
J. Phys. Chem.,
67, 217 (1963).
Volume 74, Number 9 April SO, 1970
