W. ALEXANDER VAN HOOK
3270
phase diagram which is a t least roughly symmetrical in volume fraction. For a phase diagram t,o be symmetrical in surface fraction, one would need interactions proportional to V4”.
Acknowledgments. This research was supported by the Atomic Energy Commission and by the National Science Foundation. We thank a referee for a number of very thought-provoking suggestions.
Isotope Effects on Vaporization from the Adsorbed State. The Methane System
by W. Alexander Van Hook Chemistry Department, University of Tennessee, Knoxville, Tennessee 3791 6
(Receiaed May 1 , 1967)
It is pointed out that the statistical theory of isotope effects in condensed systems‘ is applicable to isotope effects on adsorption. The appropriate theoretical equations and approximations to them are discussed. Adsorption isotope effects for the methane “wet glass” system213are treated in detail as an example. Consistency with the theory is demonstrated and it is concluded that a t least for this system rotation on the surface is strongly hindered and that it is not necessary to invoke different interaction potentials with the surface for different isotopically substituted molecules.
Introduction I n recent years, the statistical theory of isotope effects in condensed systems’ has been successfully applied to processes involving vaporization from condensed liquids4and dilute solution.6 The object of the present paper is to point out that this theory can also be used to treat the isotope effect on adsorption (AIE). After the appropriate formulas and approximations to them have been introduced, we shall proceed to treat the methane system as an example. The essential difference between this treatment and some earlier analyses of methane adsorption6r7is that the present approach takes explicit account of the effect of molecular structure on the AIE. It has been shown that structural considerations are essential in the analysis of the isotope effect on vaporization from pure liquids4 or solution6 and it is to be expected that they are also important in the AIE. I n the second section of the paper, results of Bruner, Cartoni, and Liberti2 and Bruner and Cartoni3 are analyzed. These authors The Journal of Physical Chemiatry
measured separation factors for all of the protio, deuterio and C13 isomers of methane, some as a function of temperature. Their extensive data including the AIE for the intermediate isotopic isomers afford a good test of the theory. In the paper which follows,* some new measurements on the CH4-CD4 system are (1) J. Bigeleisen, J . Chem. Phys., 34, 1485 (1961). (2) F. Bruner, G. P. Cartoni, and A. Liberti, Anal. Chem., 38, 298 (1966). (3) F. Bruner and G. P. Cartoni, J. Chromatog., 18, 390 (1965). (4) (a) J. Bigeleisen, 8. V. Ribnikar, and W. A. Van Hook, J . Chem. Phys., 38, 489 (1963); J. Bigeleisen, M. J. Stern, and W. A. Van Hook, ibid., 38, 497 (1963); M. J. Stern, W. A. Van Hook, and M. Wolfsberg, ibid., 39, 3179 (1963); (b) W. A. Van Hook, ibid., 44, 234 (1966); (c) W. A. Van Hook, ibid., 46, 1907 (1967). ( 5 ) W. A. Van Hook and J. T. Phillips, J. Phys. Chem., 7 0 , 1515 (1966). (6) 9. Ross and J. P. Oliver, “On Physical Adsorption,” Interscience Publishers, Inc., New York, N. Y , 1964,pp 236,270. (7) R.Yaris and J. R. Sams, J. Chem. Phys., 37, 571 (1962). (8) J. T. Phillips and W. A. Van Hook, J. Phys. Chem., 71, 3276 (1967).
ISOTOPE EFFECTS ON VAPORIZATION FROM
THE
ADSORBEDSTATE
presented and these are discussed in the context of the material presented below and then compared with some other earlier measurements of the AIE for the CH,CD, system on various surfaces.
The Isotope Effect on Adsorption The statistical theory of isotope effects in condensed systems leads on the basis of a cell model in the condensed phase and the assumption of harmonic frequencies to 3.1‘ - 6
ln(;)=ln(%)=
interna!
x
frequencies
frequencies
+ ‘/zCoP2)’J -
(BOP
G(g,g’)g
(1)
Here c and g refer to condensed and gaseous phases, u = hv/kT, the prime signifies the lighter isotope, y is the activity coefficient, Po the vapor pressure of the pure liquid, and the last three terms are corrective and will be discussed later. Equation 1 connects the force fields describing the motions of an isolated molecule in the gas phase and those of a condensed molecule in the cell defined by its neighbors with the vapor pressure ratio. These force fields are different owing to the intermolecular forces which become operative in the condensed phase. In the pure liquid they are exclusively “solutesolute” forces, in very dilute solution “solvent-solute,” while in more concentrated solutions both types of forces must be considered. Similarly, for an adsorbed gas at low coverage, the origin of the forces is exclusively surface adsorbate in nature but at higher coverages lateral interactions must also be considered. The former case will be of interest later in this paper; it is nicely approximated in gas chromatography. X o w consider the corrective terms. The third term corrects for the isotope effect on molar volume. In the present context it must be rewritten in terms of the surface tension of the adsorbed film and the molar coverage. We expect the latter to be determined by the surface la1tice constants and be isotopically invarient a t low coverages and to approach (V’/V)’’a only as a limit at high coverages. For CH4-CD4over the liquid range, (V/V’)‘13amounts to only 1.007.9 Similarly, the surface tension of the adsorbed film should
3271
be isotopically invarient at low coverages. The limit at high coverages is difficult to estimate. It is likely of the same order of magnitude as for the liquid, which for representative hydrocarbons is only about O.l%/D atom.’O The third term then is estimated as negligible even at high coverages. The fourth term gives the correction for gas imperfection. Bo and Co are the second and third virial coefficients. For pure samples around 1 atm pressure the correction amounts to about 1% of the vapor pressure isotope for the CH4-CD4 system. At lower pressures, such as were used in the study treated in the latter part of this paper, the corrections are, of course, smaller and may be neglected. (Additionally, it should be remembered that in chromatographic experiments, the sample S is in dilute solution of solvent carrier gas C. The appropriate virial coefficient is then Bcs. One might expect that for a helium or other inert nonpolarizable carrier that the isotope effect on BCS would be less than that on Bss. This would tend to make the fourth corrective term even smaller.) The first two and the last terms in eq 1 then remain to be considered. The former have been derived on the assumption that the harmonic approximation is applicable to both the 3N - 6 ( 3 N - 5 for linear molecules) internal motions and the six (five) external modes which are assumed to be described by a harmonic hindering potential in the condensed phase. For the case where a two-dimensional lattice gas is assumed for the adsorbed system, the partition functions used in deriving’ the form of eq 1 can be modified to include one or more free translations and/or rotations as needed. ,4lternatively, the force constants pertaining to these modes can be set equal to zero. Experiments on the vapor pressure isotope eff ect4bsC have shown that the harmonic approximation is not appropriate for the lattice modes of at least liquid hydrocarbons but it has been found possible to describe these modes in terms of temperature-dependent “effective” harmonic frequencies. The approach is identical with that used in the pseudoharmonic theory which has been successfully applied to both monatomic and molecular solids. l 2 For small or moderate temperature changes such effective frequencies might well be effectively constant for some systems at least to within the precision of the experimental measurement^.^^ We (9) S.Fuks,J. C. Legros, and A. Bellemans, Physica, 31, 606 (1965). (10) L. S. Bartel and R. R. Roskos, J. Chem. Phys., 44, 457 (1966). (11) G. Thomaes and R. van Steenwinkel, J W O ~Phys., . 5, 307 (1962). (12) (a) T. H. K. Barron, Discussions Faraday SOC.,40, 69 (1965); (b) T. H. K. Barron in “Lattice Dynamics,” R. F. Wallij, Ed., Pergamon Press, Ltd., London, 1965, p 247; (c) A. J. Leadbetter, Proc. R o y . SOC.(London), A287, 403 (1965).
V o l u m e i+13Number 10
September 1967
3272
might reasonably expect similar considerations to apply in dealing with adsorbed systems. The last term in eq 1 is the correction for nonclassical rotation in the gas phase. This correction amounts to several per cent of the isotope effect on the vapor pressure for methane but is much smaller for more massive rotors. Any complete theoretical analysis of the methane system must include such a correction and also take proper account of the relative ortho, meta, and para spin isomers in both the gaseous and condensed phases for the symmetrically substituted isomer^.'^ On the other hand, many applications of the theory will be made in an effort to illuminate specific experimental results which are themselves not of sufficient precision to demonstrate spin effects. One such example is the analysis of isotopic separation factors of methane which follows. I n view of the fact that no effects due to spin isomerization were observed in the experiments, such small corrections will be ignored in both the gas and condensed phases in the analysis below. This will have the effect of slightly shifting the force constants used to describe the effective harmonic frequencies. Since the corrections would in any event be small, this will not change the force of our arguments. We are then led to
We choose to call this result derived on the basis of a cell model, neglect of anharmonicities, gas imperfection, isotope effects on surface coverage, etc., the “complete equation.” If the 3N frequencies of the system can be factored into two distinct sets, eq 2 reduces to
The A term is interpreted as a first-order quantum correction for modes u 2n) frequencies on condensation. It is given by
where the sums are over all internal frequencies.
Methane Adsorption on Wet Glass In this section we make an application of the theoretical formulas discussed above. Bruner, Cartoni, and Liberti2 and Bruner and Cartoni3 have made careful measurements of the isotope effects on adsorption for the systems 12CHPCD4, 12CH4-12CHD3,I2CH4l2CH2D2, W H I - W H ~ D ,and 12CH,-13CH4. The first pair was measured over the range 84-153°K and the others were measured around 85°K. A capillary chromatographic technique was used. The columns were prepared2 by drawing soft glass capillary, etching the inner surface at 100” with 20% NaOH solution, xr-ashing, drying, and then treating to constant activity by passing nitrogen saturated with water vapor through the column. This last part of the treatment has prompted us to label the columns as “wet glass.” The ratio of corrected chromatographic retention volumes is readily identified with the inverse ratio of thermodynamic activities5 which in turn is given following the arguments above by the complete equation (eq 2) or the approximation to it (eq 3). The frequencies
(7) which enter the equations must of course have been calculated from proper gas and condensed phase force fields. In the calculations which are reported in this paper, eq 6 will be employed exclusively, but we will find it convenient in the discussion to speak of lattice ( A ) and zero point energy ( B ) terms and thus to orient much of our discussion about eq 7 . (In the temperature region of interest (80-150°K), eq 7 is not a particularly good approximation to eq 6 for the force fields which we employ. The difference is on the order of 10% of the isotope effect itself.) Ideally, one would have independent spectroscopic data for the gas and condensed phases of ordinary (13) See R. F. Curl, Jr., J. V. Kasper, K. S. Pitrer, and K. Sathianandan, J. Chem. Phys., 44, 4636 (1966),for a discussion of the methane case; J. King, Jr., and S.W. Benson, ibid., 44, 1007 (1966); A. Katorski and D. White, ibid., 40, 3183 (1964),for treatments of the hydrogen case.
ISOTOPE EFFECTSON VAPORIZATION FROM THE ADSORBED STATE
methane. These would enable force fields for the two phases to be determined, the frequencies for all isotopic species to be calculated, and the B terms thus defined. Unfortunately, no spectroscopic data for methane adsorbed on wet glass is available. We, therefore, in our first calculation, I (Table I), assume a B term such as would be deduced from the normal mode frequency shifts on condensation to the liquid. These shifts together with literature references are shown in Table I. The values for the gas-phase frequencies are those calculated from the force field of Jones14 which is shown in valence coordinates in Table 11. The condensed phase fields used in the calculations are also shown. In a second calculation, I1 (Table I), we have employed the B term which Bruner, et al., deduced2 by fitting their data for the CH4-CD4 system from 85 to 150°K to an equation of type 7. They quote In [VR(CH~)/VR(CD~)] = (1766/T2) - (13.8/T). We have multiplied the force constant changes employed in calculation I by the ratio B I I / B Ito deduce a force field for calculation I1 which is also listed in Tables I and IJ.
Table I: Gas and Condensed Phase Internal Frequencies of CIIk (cm-1)
yg*sa
(A) 2(E) I
307)
4(F)
-(Gascond)caicd--Field I Field I1
Aqgas-iiq)
3143.9 1573.7 3154.3 1398.1
8.3 1.5 10.0
8b 1.5' 10d 1.5'
1.5
12.8 2.3 15.5 2.3
'
Calculated from force field of ref 14. M. F. Crawford, H. L. Welsh, and J. H. Harrold, Can. J . Phys., 30, 81 (1952). N. Shepard and D. J. C. Yates, Proc. Roy. SOC.(London), A238, 69 (1956). These authors report A(vz v4) = 3 cm-l. See ref 15.
+
It is now necessary to evaluate the A contribution. For methane, the lattice term A can be written4b A =
A(;)'{
3vt,"( 1 -
E)
CI~B''(1 - f)} 3
4-
(8) where vtrt and VR' signify the hindered translational and rotational frequencies of ordinary methane and M , M' and I , I' the masses and moments of inertia of the substituted molecules. One of the most interesting points about the Italian experiments is that the 13CH4measurement enables the relative contributions of the translational and rotational terms to be assessed because the moments of inertia of 13CH4 and WH4
3273
Table I1 : Force Fields Gas'
FI F,I Fa
-
Faat
Far!
fa at^
= Fartf
5.495@ 0.12400 0.54935 0.01855 0.16519 0
Condensed I
Condensed
5.46203 0.12467 0.54821 - 0.01855 0.16519 0
5.44403 0.12504 0.54759 - 0.01855 0.16519 0
I1
b
a
80.75 94.51
77.36 90.55
81.47 106.03
' Field and notation of Jones.14 This author quotes five independent constants which we have expressed as above purely for convenience in using our computational technique. * Units are millidynes per angstrom for stretches and millidyne angstroms per square radian for bends. Not all figures are significant; they are reported only for computational accuracy. Lattice force field is reported in terms of the frequencies calculated for the unsubstituted species. are equal. The term A(l3CH4) is then a direct experimental measure of the translational contribution. Once known, Atran, for any other isotopic isomer can be calculated and the ArOt, contribution for that isotopic isomer determined by difference. In this fashion the importance of hindered rotation in the adsorbed phase can be assessed. Finally, the separation factors can be calculated for the other isomers using the parameters defined above and the results compared with experiment, thus testing the theory. Such a calculation will be outlined below. The available data are shown in Table 111. It is evident that columns a and b are similar but not identical. A 15-20% difference in the effect is observed between the columns for CHaD and CD,. This is not unexpected. Phillips and Van Hook* have shown that the separation factor is a function of the water content on glass columns and it is possible that slight differences in column preparation could account for the observed difference. Unfortunately, data for the 13Ceffect are only available for the b column, whereas we should like to know the 13C effect for both sets of data to facilitate cross-comparisons. In order t o procede, we shall directly apply the 13C-b result to calculat,ions on the a column without making a 20% correction because, to anticipate, most of the deuterium isotope effect is due to hindered rotation (an effect which cannot be present in the 13C case). Secondly, even a 2OY0 correction would be only slightly larger than the bounds on (14)
L. H. Jones, Mol. Spectry., 3 , 632
(1959).
Volume 71, Number IO
September 1967
W. ALEXANDER VANHOOK
3274
Table 111: Observed and Calculated Separation Factors for Isotopic Methanes in Units of 1Oa[ln ( P c H I / P x ) ] Temp,
X
13CH4 12CHsD "CH2Dz "CHD3 '2CD4
OK
84 85 84 85 84 85 84 85 84 85 100 125 150 200
Obsd' col a
Calcd Ia
Obsdd col b
Calcd Ib
Calcd I1
Temp fit/ b
1.2
(1.2)
(1.2)
3.3 3.5
2.1 3.0
3.5 3.3
6.3
6.2
5.2
6.2
8.2
8.1 9.8 (9.4) 5.6 2.2 0.6 -0.8
1.2*
(1.2)c
3.9
9.4
8.1
1.6e
6.8 (8.1) 7.8 4.4 1.5 0.0 -1.1
7.1 (8.1) 7.8 3.7 0.3 -1.3 -2.4
8.6 8.2 3.8 0.3 - 1.4, -1.5e -2.5
a See ref 3. Calculated from reproduction of chromatogram using 4.5 min as correction for residence time in the gas phase (this amounts to a 2% correction on retention times). 'Transferred from column b. "Parentheses denote values used in the fitting process. See ref 2. 152°K. Predicted by the smoothed fit to CH4-CD4 data in ref 2; In ( P c H , / P c D= ~ )( - 13.82/T) (1766/T2).
+
the experimental precision and would not materially affect the development below or the conclusions drawn from it. An outline of the calculation based on field I follows. Substitution of B1(13CH4) and the experimental 13CH4/12CH4ratio (1.012 a t 84°K) into eq 3 gave A = 94("K)2. The translational contribution of the CD4 term is then 321(OK) since
Substitution into eq 7 together with the experimental CD4/CH4 ratio, In (PcHJPcD,) = 0.094 at 85°K (column a), allows the rotational contribution to be estimated as 1094(°K)2. The equivalent lattice frequencies and force constants are deduced from eq 8 assuming three isotopic translations and three isotropic librations. Other assumed frequency distributions would not change the force of our arguments. The deduced force constants were employed as initial guesses for lattice constants in a calculation using the complete equation which itself was fit to the experimental CD4 ratio at 85°K. The final values of the lattice frequencies differed from the initial guesses by only some 2 cm-' out of 90 cm-1 a t 85°K. The difference is due to the failure of approximation 7 in this temperature range. The final values for the lattice frequencies of condensed CHI are reported in lieu of the force constants describing these modes in Table 11. It is interesting to compare the result for hindered rotation with the 63 cm-I suggested by Ewing's for the liquid. Clearly a The Journal of Physical Chemistry
significant increase in the barrier hindering rotation occurs in going from the liquid to the wet glass surface. We have thus employed the 13CH4and the CD, data together with an internal field assumed to be equivalent to the liquid to fix a condensed phase force field for adsorbed methane. Isotope effects for the intermediate isomers are easily calculated using the complete eyuation. The results at 85' are compared with experiment in Table 111. The predicted intermediate separation factors at 85' (Table 111) are in striking agreement with experiment. They unequivocally demonstrate that hindered rotation is occurring on the surface. (In the absence of hindered rotation, the effects would go 2.35, 4.7, 7.0, and 9.4% for CHID, CH2D2,CHD3, and CD4 instead of the observed 3.9, 6.3, 5.2, and 9.4?&.) In a second calculation (Ib), internal field I was fit to the CD4 data on column b at 84°K by the procedure outlined above. The calculated isotope effects for the other isomers are shown in Table 111. The agreement with the available experimental data (CH3D at 84°K) is again satisfactory. The results calculated at other temperatures from field Ib are also shown. These can be compared with the values calculated from Bruner, et al., smoothed fit to the data. The tail off from the calculated values at the higher temperatures reflects the choice of a B term significantly lower than the experimental best fit. In a final calculation, field I1 was fit to the CD4 data at 84°K on column b. The results of calculations with this field are shown in the last column of Table 111. (15) G . E. Ewing, J. Chem. Phys., 40, 179 (1964).
ISOTOPE EFFECTS ON VAPORIZATION FROM
THE
ADSORBED STATE
Agreement is again satisfactory. In particular, the temperature coefficient is better than that for field Ib. This is to be expected from the relative starting points of the two calculations; even so, calculation Ib reproduces the observed temperature variation of the CD4/ CH, data remarkably well and there is actually very little reason to choose between the two fields from this criterion alone. In summation, we should like to reemphasize that our object has been not to deduce a unique force field which describes the molecular motions of adsorbed methane but rather to demonstrate that the isotope effects displayed by this system are consistent with the theory of isotope effects in condensed systems. To do so we employed two different but plausible condensed phase internal force fields and deduced the translational and librational contributions from the data on two isotopes at one temperature. The data on the temperature coefficient and on the intermediate isomers served as a test for the theory. The results indicate that rotation on the surface is strongly hindered. They also show in a broader sense that isotope effects on adsorption can be understood in terms of the adsorbed layer surface interaction and that such interpretation must include the effect of the details of the molecular structure. The present’ approach is then to be contrasted with calculations which are based on virial expansion technique~.’*’~A proper virial calculation is in principle exact, at least in the low-coverage limit. The details of surface-molecule and molecule-molecule interactions
3275
can be treated using effective pair potentials which for polyatomic molecules are angle dependent. Proper inclusion of the angular dependency should lead to treatment of the effects we have labeled as ((due to hindered rotation.” In actual fact, however, the gas surface interaction for structured molecules is so complicated that it becomes necessary to spheracaliee the potential. In that event, the angularly dependent (hindered rotational) properties are hidden under the new spheracalized effective potential which is now, in general, isotope dependent. l7 Such effective potentials are not able to explain details of the interaction due to molecular geometry such as the large deviations from the rule of the mean discussed in the present paper. The point is that details of molecular structure are important and should be considered and that the Bigeleisen approach which is employed here offers a convenient formalism for making such considerations.
Acknowledgment. This research was supported by The Petroleum Research Fund administered by the American Chemical Society. Conversations with Dr. Max Wolfsberg of Brookhaven National Laboratory were helpful in the formulation of this paper. (16) (a) P. L. Gant and K. Yang, J . Am. Chem. Soc., 86, 6063 (1964); (b) G. Constabaris, J. R. Sams, and G. D. Haleey, Jr., J. Phy8. Chem., 65,367 (1961). (17) A reference which also gives many citations to work developing this point of view is R. Wolfe and J. R. Sams, J. Chem. Phye., 44, 2181 (1966).
Volume 71, Number 10 September 1067
