J. Phys. Chem. 1984,88, 232-238
232
CHE81-15109 for partial support. This work was performed under the auspices of the Office of Basic Energy Sciences, Division of Chemical Sciences, US.Department of Energy, under Contract W-3 1-109-Eng-38.
Appendix I As discussed in section IIIA, the experimental values of Pxu, Sxy,and k7/ks are constructed from two or more measured relative rates. The purpose of this appendix is to specifically enumerate the relative rates involved. These directly measured rates will be denoted below in brackets. As mentioned in the text, for each construction, interpolation based on the Arrhenius form was used to obtain relative rates (and their associated errors) at a common set of temperatures. The error bars on the constructed value were made from the relative error which was assumed to be the sum of the relative errors of each rate in the construction. Only ref 11 provided the necessary relative error information. For Pxy,the experimental results obtained by construction are marked by symbols in Figure 4. Those due to ref 11, denoted by symbols with error bars, are constructed as follows: P" from [kl/k7] (Le., directly measured), PDH from [k3/k4]/ ([kCH3+CH3COCH / k 4 I [kCH3+CH3COCH,Ik71 1 7 and PHD from fk4/ kC€l +CH,COCH,] ikCH3+CH3COCH3/k71 Those due to ref l 2 and 3, [ ~ ~ / ~ c H ~ + c [H~ ~~ ]/ /~ c H ~ + c by H ~Symbols ]. without error bars, are constructed as follows: P" from [kl/kCH3+CH3]/ [k7/k~H3+CH3], P D H from [kfkCH,+CH3] / [k7/kCH3+CH31 and PHD from Lk4/ ~ C H , + C H ~/] [k 7 / kcH3+CH,]. Reference 12 was used alone to provide 9
provide P" while ref 13 was used alone to provide P D H and PHD. For k7/k8,the experimental results obtained by construction are indicated by symbols in Figure 5. Those denoted by symbols with error bars are from ref 11 according to [kCD3+cD3cocD,/ Those denoted by symbols k21/ Ik8/k21/ [kCH3+CH$OCH,/k71* without error bars are from ref 12 according to [ k c ~ , + c ~ , / k + / [ k c ~ , + c ~ ~ / k 7 As ] . mentioned in the text, the reference reaction rate constants needed to reduce these constructions to the values in Figure 5 are all taken from ref 3. For Sxy,the experimental results obtained by construction are indicated by symbols in Figure 6 . Those denoted by symbols with error bars are from ref 11 and are obtained as follows. For S H D and SDH, the required ratios (see text for the definition of Sxy) k3/k59 k 7 / k 8 9 and k l / k 2 are from ([k3/k41/[kCH,+CH3COCH 1 k411[kCD3+CD3COCDj/k51? ([kCD,+cD,COCD,/kzl/[k8/kzlf/ [kCH3+CH,COCH,/k71, and ( [ k C D , + C D , C O C D , / k 5 1 / [ k 6 / k 5 1 ) /
[ ~ c H ~ ~ c H , c o c Hrespectively. ,/~~], For S", a more direct construction is available, from [kl/k7][k8/k2]. The experimental results denoted by symbols without error bars are from ref 12 and 13 and are obtained as follows. Reference 12 was used alone to obtain SHH from [kl/kCH3+CH,l/[k2/kCD3+CD31/([k7/ kCH3+CH,l[k8/kCD3+CD,I). For SHD and S D H , ref l 2 provided [ k s / k c ~ ~ + cwhile ~ ~ ] ref 13 provided all the remaining relative rates in the form [k7/kCH3+CH,l [k3/kCH3+CH,], [k5/kCD3+CD31 [ k 4 / kCH3+CH31, and [k6/kCDj+CD31. Registry No. Hydrogen, 1333-74-0;methyl, 2229-07-4;deuterium, 9
9
7782-39-0.
Molecular Statlstical Theory of Adsorption for Hydrocarbons on Graphite. Effect of Polarizability Anisotropy in Adsorption Potential Calculations Claire Vidal-Madjar* and Erika Bekassy-Molnart Laboratoire de Chimie Analytique Physique, Ecole Polytechnique, 91 128 Palaiseau Cedex, France (Received: November 29, 1982; In Final Form: June 21, 1983)
The molecular statistical theory of adsorption taking into account the high anisotropic polarizability tensor of graphite has been applied to the calculation of the second adsorbate/surface virial coefficient and the isosteric heat of adsorption of various hydrocarbons on graphite for zero surface coverage. The anisotropic adsorption potential model carried out for methane and benzene adsorbed on the basal (0001) graphite surface predicts values which are in good agreement with the results of gas-chromatographic and static-adsorption measurements on graphitized carbon blacks. For methane, the C..C distance and the vibration frequencies perpendicular to the graphite surface, calculated at 0 K, are close to the structural data of the adsorbed layer as measured from neutron scattering techniques at low temperature. The major role of the adsorbate polarizability in adsorption potential calculationsis demonstrated in the prediction of naphthalene, phenanthrene, and anthracene relative adsorption on graphite.
Introduction The development of the molecular statistical theory of adsorption based on the atom-atom approximation for the potential energy permits the prediction of the thermodynamic functions of adsorption on Comparison with adsorption experiments obtained on graphitized carbon black is valid as its fairly homogeneous surface exposes the basal face of graphite. It is the dispersion forces which account for most of the interaction energy of nonpolar hydrocarbon molecules on graphite and the models consider an adsorption potential which is an additive function of the adsorbate-adsorbent interactions. The methods based on a priori c a l c ~ l a t i o n s ~determine -~ the attractive constant from the Kirkwood-Muller equations-' or from combination rules derived from the self-interaction atom-atom potentials.lO-ll As too large values for the attractive constant are Present address: Department of Chemical Engineering, Technical University of Budapest, 1521 Budapest, Hungary.
0022-3654/84/2088-0232$01.50/0
~ b t a i n e d ~from - ~ a priori calculations, more empirical models are which adjust the constants of the adsorption potential (1) W.A. Steele,"The Interaction of Gases with Solid Surfaces"; Pergamon Press: Oxford, 1974. (2) D. P. Poshkus, Discuss. Faraday Soc., 40, 195 (1965). (3) A. V. Kiselev and D. P. Poshkus, J. Chem. Soc., Faraday Trans. 2,72, 950 (1976). (4) A. V. Kiselev and D. P. Poshkus, Tram. Faraday Soc., 59,428 (1963). ( 5 ) A. V. Kiselev, D. P. Poshkus, and A. Y . Afreimovich, Rum. J . Phys. Chem. (Engl. Transl.), 42, 1345, 1348 (1968). (6) C . Vidal-Madjar and G. Guiochon, Bull. SOC.Chim. Fr., 3105,3110 (1971). (7) C. Vidal-Madjar, M. F. Gonnord, and G. Guiochon, J . Colloid Interface Sci., 52, 102 (1975). ( 8 ) J. G. Kirkwood, Phys. Z . , 33, 57 (1932). (9) A. Muller, Proc. R. SOC.London, Ser. A , 154,624 (1936). (10)M. La1 and D. Spencer, J . Chem. SOC.,Faraday Trans. 2,70, 910 (1974). (1 1) L.Battezzati, C.Pisani, and F. Ricca, J. Chem. Soc., Faraday Trans. 2,71, 1629 (1975).
0 1984 American Chemical Society
Adsorption of Hydrocarbons on Graphite
The Journal of Physical Chemistry, Vol. 88, No. 2, 1984 233
to the experimental data measured at zero surface coverage on graphitized carbon black. Because of the importance of polarizability in dispersion force interactions, Meyer and diet^'^^^^ have developed an adsorption potential which takes into account the anisotropic nature of graphite polarizability. Ricca et al.16317have considered the graphite polarizability anisotropy for the adsorption of rare gases but have shown that, when combination rules are used for the atom-atom dispersion interaction constants, the adsorption potential is not significantly altered. Carlos and Colei8 have calculated the potential energy of a helium atom adsorbed on graphite and compared both the isotropic and anisotropic pair potentials. They have shown that only in the latter case is the theory in good agreement with He atom scattering experiments. The purpose of this work is to adopt the anisotropic model for the adsorption potential and to use the molecular statistical theory of adsorption to predict the equilibrium constant and the thermodynamic functions of adsorption of hydrocarbons on graphite, with a priori calculations based on the independent properties of the adsorbate and the adsorbent, Le., their respective polarizabilities and molecular structure.
Theory Adsorption Potential. The adsorption potential is the sum of the interaction potentials between each atom i of the adsorbate molecule and each atom j of the ad~0rbent.l~For the adsorption of nonpolar molecules on graphite aiJis the sum of the dispersion forces potential QIJD and the repulsive one QiJR:
The anisotropic model of Meyer14 describing the dispersion interaction potential is qjD
=-
h 3
with
(5)
zi and are the mean diamagnetic susceptibilities, m the mass of electron, and c the velocity of light. Adopting a repulsive term with an inverse 12th-power law of the distance as in the Lennard-Jones potential, the adsorption potential ai,describing the interaction of an adsorbate atom i of the molecule and the whole graphite surface, is for the anisotropic adsorption potential model
xj
3
@ianis
3
= -Cijc I
+
aikajc(Aijki)2 BijCrij-I2 /=1 k = l
(6)
J
The constants of repulsion Bijare calculated for each type of atom i and for each orientation of the adsorbate molecule at the equilibrium distance (sum of the van der Waals radii) for which the adsorption potential is a minimum. Lippincott and StutmanZohave determined the polarizability of the carbon atom parallel to the basal plane of graphite (ajl= ajz= 1.71 X cm3/molecule), assuming that the polarizability aj3perpendicular to that plane is zero, with aj = 1.14 X cm3/molecule. The theoretical calculations of Meyer14 and Ricca et aLi6*17 are based on this approximate assumption. Carlos and Cole18 have used a more realistic value for the anisotropic ratio P ( P = ajj/ajl= a j 3 / a j= 2 0.29) which is derived from the anisotropic polarizability .tensor of aromatic compounds. In this work, we shall use both values of P for the calculations of the anisotropic adsorption potential (eq 6 ) and compare the results with those obtained with the isotropic adsorption potential. In the special case where the adsorbate polarizability is isotropic, the dispersion interaction potential on graphite (eq 4) reduces to QijD
=
+
+
-Cij&i[601j,rij-~ (aj3- a j l ) ( ( ~ i j + i 3() 2A ~ ~ ( A~~ ~~ ~ (7) ) ~~ ) ~ ) ]
-4 k=l c I=1c (Aijk')2---aikajl VikVjl Vikvjl
As
The component polarizabilities of the adsorbate atoms of the molecule are all,ala,a13; those of the adsorbent atoms are aJl, aJ2, aJ3.The corresponding characteristic frequencies are vll, u12, u13for the adsorbate molecule atqms and vJl, vJ2, vJ3 for the adsorbent atoms. The attraction constant A,k' is expressed by a combination of nine terms: r 1
+ (Ai423)2+ ( ~ ~ ~ 3 3=) 4p.4 2 - 3p,.2r..-8 11 IJ
(Aij13)2
IJ
where Zij is the projection of Zijon the basal graphite surface, the total adsorption potential is with P = a j 3 / ~ ~ , 1 : +pianis
=
-Cipiajl[(2
+ 4P)Criy6 + 3 ( 1 - P ) cJ p i j 2 r i j - 8+] Bij?rij-l2 (8) J
Computer calculations have shown that with a good approximation (within 1% relative error) 3Cp..r..2r,.4N Crij-6 IJ IJ
2and
are respectively the directions of the unit vectors of the adsorbate and the adsorbent polarizability tensor. FIJis the vectorial distance between the adsorbate and the adsorbent atoms. As in Meyer's work, the components of the characteristic frequencies are assumed to be equal (ql = v12 = v13; vJl = v12.= vJ3). If the energies hv are put equal to -4mc2 ~ / athe, dispersion ~ interaction potential is (4)
(12) A. V. Kiselev, D. P. Poshkus, and A. J. Grumadas, J . Chem. SOC., Faraday Trans. I , 75, 1281, 1288 (1979). (13) E. S. Severin and D. J. Tildesley, Mol. Phys., 41, 1401 (1980). (14) E. F. Meyer, J . Phys. Chem., 71, 4416 (1967); J. Chem. Phys., 48, 5284 (1968). (15) E. F. Meyer and V. R. Dietz, J . Phys. Chem., 71, 1521 (1967). (16) C. Pisani, F. Ricca, and C. Roetti, J . Phys. Chem., 77, 657 (1973). (17) G. Bonino, C. Pisani, F. Ricca, and C. Roetti, Surf. Sci., 50, 379 (1975). (18) W. E. Carlos and M. W. Cole, Phys. Rev. Lett., 43,697 (1979); Surf. Sci., 91, 339 (1980). (19) N. N. Avgul, A. A. Isirikyan, A. V. Kiselev, I. A. Lygina, and D. P. Poshkus, Izv. Akad. Nauk SSSR, 1314 (1957).
1J
The adsorption potential can thus be reduced to a more simple expression: aianis
+
= -3cijaiajl(l P ) C r i r 6 J
+ BijCrij-12
(9)
J
For the isotropic model with a j , = aj2= aj3= aj and P = 1, the adsorption potential is @.is0
= -6C..a.&.cr..-6 IJ 1 1 IJ + B,.xr,.-12 IJ IJ
(10)
and 6 C f i i a , is equal to the constant of the Kirkwood-Muller expression.d9 Thus, when the adsorbate polarizability tensor is isotropic, the potential expressions derived either from the anisotropic adsorption potential model (eq 9) or from the isotropic adsorption potential model (eq 10) have the same Lennard-Jones potential expression: 0. = -D..'&-b IJ . IJ
+ B..&-'2 1J 1J
(1 1 )
However, the constant Dij describing the dispersion interaction (20) E. R. Lippincott and J. M. Stutman, J . Phys. Chem., 68,2926 (1964).
234
The Journal of Physical Chemistry, Vol. 88, No. 2, 1984
Vidal-Madjar and Bekassy-Molnar
TABLE I: Adsorption of Methane on Graphite DCG: DHG
-aa,
(BCGIDCG)"~
(BHGIDHG)'"~ za, nm
v z , cm"
0.330 t 0.005f
experiment 1.420 0.477 1.600 0.540
anisotropic model (eq 9, P = 0) anisotropic model (eq 9, P = 0.29) isotropic model (eq 10, P = 1) combination of selfinteraction p a r a m e t e d empirical modele
0.340 0.2971
empirical model, Buckingharn-Corner eqc
* 5f
0 "C,
kJ/mol 12.7g
-4.38gh -4.351!k -4.7 16'7
83
12.4
11.58
-4.814
0.337
91
13.9
12.96
-4.296
0'3401 0'3401 0.297
0.337
100
16.4
15.23
-3.450
0'331\ 0.286 0.330 0.298
0.321
125
16.6
15.30
-3.485
0.330
97
13.6
12.70
-4.388
0.337
86
13.7
12.94
-4.250
1 0'338 1 0.294
a Units: J nm6/mol. Units: nanometers. Reference 3,. Reference 11. e Reference 13. f Reference 26. erence 28, for a surface area of 11.3 mZ/g. Reference 29. J Reference 30. Extrapolated a t 0 "C.
+
+
forces is lower, in a ratio of 3(1 P)/2(2 P ) , when the anisotropic adsorption potential model on graphite is considered, as 0rj,(2+ P) = 3aj. With the hypothesis of Lippincott and Stutman20 for graphite, P = 0, and the-dispersion interaction in the adsorption potential expression is 3 / 4 times higher than when an isotropic tensor is assumed for graphite. The adsorption potentials are calculated by direct summation with a Fortran program described in ref 6 over a semispherical volume of the graphite crystal of 2-nm radius. The relative error in neglecting the other carbon atoms is The distance between the nearest carbon neighbors in the basal plane is 0.142 nm and the interlayer spacing 0.335 nm.21 The mean polarizability of the carbon force center of graphite is a, = 1.14 X cm3/ molecule;20 its diamagnetic susceptibility is xJ = 1.20 X ~m~/molecule.~' Molecular Statistical Calculations. In classical approximation the molecular statistical theory of a d ~ o r p t i o nderives ~ * ~ the second adsorbate/surface second virial coefficient BAS or adsorbateadsorbent equilibrium constant at zero surface coverage from the ratio of the partition functions of the molecule in the adsorbed and gaseous phases. For quasi-rigid molecules we have BAS =
1 -j...j [exp(-+/RT) 8r2A
- 11 dx dy dz sin 0 d q dJ/ (12)
where x, y , z are the coordinates of the mass center of the molecule and 8, q, and J/ are the Euler angles of orientation to the surface. A is the adsorbent surface area. Numerical integration is carried out for all the space available to the molecule, with 2% precision. When the gaseous phase is ideal and adsorbate-adsorbate interactions are negligible, BASis equal to the retention volume VA per unit surface area measured by gas-adsorption chromatography. The differential energy of adsorption is derived from the change of BASwith temperature: A T i s related to the isosteric heat of adsorption qst: The differential free energy of adsorption is4] = -RT In B , ~
with a standard reference state chosen so that the number of molecules in the adsorbed state per unit surface area equals the number of molecules in the gaseous phase per unit volume.22 AA ~~~~
In B A s at 0 'C, (wno1/rn2)/ (mol/m3)
0'337
0.297
1.893 0.637 1.486 0.5 94 2.039 0.394 1.483 0.533
100
kJ/mol
qst at
~
(21) P. Pascal, 'Nouveau Trait6 de Chimie Min&rale, Tome VIII, Carbone", Masson, Paris, 1968. (22) R. M. Barrer and L. V. C. R e s , Trans. Faraday Soc., 57,999 (1961).
g
Reference 27.
Ref-
depends on the choice of units for BAS which is expressed throughout this work in (pmol/m2)/(mol/m3).
Results and Discussion Adsorption of Methane on Graphite. The potential function for methane molecule adsorbed on the basal graphite surface has been calculated with the following assumptions. For the adsorbate molecule, two force centers are considered, the carbon and the hydrogen atoms. Their polarizability tensor is assumed to be isotropic and the parameters used to calculate the interactions between each atom of the adsorbate molecule are the following. For the carbon atom, ai= 0.99 X ~ m ~ / m o l e c u and le~~ = 1.22 X ~ m ~ / m o l e c u l e . *For ~ the hydrogen atom, ai = 0.42 X ~ m ~ / m o l e c u land e ~ ~z, = 0.33 X cm3/ molecule.24 The minimum approach distance for C. .C atoms is 0.340 nm and for Cas .H atoms 0.300 nm.5 The geometrical structure of methane is taken from ref 25. The valence angle is tetrahedral and the C-H bond length is 0.109 nm. As the adsorbate atom polarizability tensor is isotropic, the anisotropic adsorption potentiab model (eq 9) can be approximated with a Lennard-Jones potential, within 1% relative deviations in calculations. The corresponding attractive and repulsive constants DiJand B, are listed in Table I. Subscripts CG and HG are respectively for carbon (molecule)/carbon (graphite) and hydrogen (molecule)/carbon (graphite) interactions. The reported constants for the models investigated in this work will permit comparison with the other adsorption potential expressions reported in the literature. Therefore, the conclusions of Battezzati et al." concerning the most stable conformations of methane on graphite still hold for the anisotropic adsorption model. In agreement with ref 11, the equilibrium adsorption conformation of methane on graphite is the tripod configuration with the mass center of the molecule located above the center of graphite carbon hexagons (Figure 1). Figure 1 illustrates the adsorption potential as a function of z, the distance above the graphite surface, the other coordinates of the methane molecule being fixed in the tripod configuraiton, for various hypotheses: an anisotropic polarizability ratio P equal to 0 (Lippincott and StutmanZ0hypothesis) or equal to 0.29 (Carlos and Cole'* hypothesis) and an isotropic model ( P = 1). For a given z value, the adsorption potential is lowest in the isotropic case, its absolute value decreasing with P . For each model, the corresponding minimum adsorption potential is given in Table I, with the equilibrium distance zo of the
z,
(23) K. G. Denbigh, Trans. Faraday SOC.,36, 936 (1940). (24) P. Pascal, A. Pacault, and J. Hoarau, C. R. Hebd. Seances Acad. Sei., 233, 1078 (1951). (25) L. E. Sutton, Ed., "Interatomic Distances", The Chemical Society, London, 1965, Spec. Publ. no. 18.
The Journal of Physical Chemistry, Vol. 88, No. 2, 1984 235
Adsorption of Hydrocarbons on Graphite
t
I
5 (kJ. mol:’)
5
1
Ln BAS(p mol. rn-*/moI. m-3)
-10
2 4 6 6 Figure 2. Second adsorbate/surface virial coefficient of methane adsorbed on graphite as a function of temperature. Dotted line = anisotropic theoretical model; solid lines = experiment. Coefficients a and @ of the equation In BAs = a + @/Tin parentheses: Theory in classical approximation (-9.09,1310) for 220-300 K and (-9.80,1440) for 120-160 K temperature range. Experiment, ref 28: curve 1 (-9.13,1310); ref 29: curve 2 (-9.16,1310); ref 30: curve 3 (-10.09,1420).
Figure 1. Methane adsorbed on graphite. The adsorption potential as a function of distance to graphite basal face for the equilibrium conformation position of the molecule: (1) anisotropic model (eq 9, P = 0); (2) anisotropicmodel (eq 9, P = 0.29); (3) isotropic model (eq 10, P =
The isotropic adsorption potential (eq 10) predicts BAS and qst values which are respectively 2.9 and 1.2 times larger than the experimental ones. The attractive constant D,, is then equal to the Kirkwood-Muller expression. 1). The anisotropic potential models give results which compare more favorably with the experiments. It is the Carlos and Cole methane molecule mass center to the graphite basal plane. Its hypothesis’*with a nonzero value for the component polarizability value (0.337 nm) does not change with the anisotropic polarizaperpendicular to the basal plane which is in best agreement with bility ratio P as it is only dependent on the value assigned to the experiment (Figure 2 and Table I). minimum approach distance of C- .Cand C. .H atoms. It is in However, it is not possible from these calculations to decide good agreement with the carbon-graphite distance, 0.330 nm, which is the “best choice” for P as theoretical calculations are not measured for methane adsorbed on graphite with neutron scatprecise enough and are highly sensitive to the minimum approach tering techniques.26 distance assigned to adsorbate-adsorbent atoms. For example, The vibration frequency v, perpendicular to the graphite surface a lower approach distance of 0.001 nm produces an increase in is calculated from the second derivative of the potential function the adsorption potential and EM values of, respectively, 10% and $22ff0, for the equilibrium distance z = zo assuming harmonic 70%. vibrations around the equilibrium position: The adsorption potential equation of Battezzati et al.,” which is a Lennard-Jones (12-6) expression, is based on combination rules of the self-interaction parameters and predicts an isosteric adsorption heat 20% too high and a BAS twice larger than the experimental values. The predicted vibration frequency at 0 K is between 83 and The attractive constant DCG of the model of Battezati et al. 100 cm-I depending on the value of the polarizability ratio P. It which represents carbon (molecule)/carbon (graphite) interactions can be compared to the experimental frequency (100 cm-I) is close to the one derived in this work from the anisotropic measured at low temperature from the incoherent neutron elastic adsorption potential mode. However, the attractive constant DHG spectra of the methane layer.26 of ref 11 for hydrogen (molecule)/carbon (graphite) interactions The second virial adsorbate/surface coefficient BAS and the is large and most probably the combination rule method is not isosteric heat of adsorption qstat 0 O C have been computed from valid in this case. the change of the adsorption potential for all z distances and The empirical adsorption potential model of Severin and orientations of the adsorbate molecule (Table I and Figure 2). TildesleyI3 is a Lennard-Jones (12-6) expression. The attractive At 0 OC, the quantum corrective factor which is a function of the and repulsive constants which are reported in Table I are calcuvibration frequency2 is only 2% and is considered as negligible. lated in order to obtain the “best” fitting to the static experiments The theoretical calculations are compared with the experimental thermodynamic functions of adsorption obtained from ~ t a t i c ~ ’ - ~ ~ of ref 28. Kiselev and Poshkus3 have used a more complicated form for or gas-chro.matographic measurements3” at zero surface coverage. the adsorption potential, the Buckingham-Corner potential, with an exponential repulsive term and two attractive constants corresponding to dipole-dipole and dipole-quadrupole interactions. (26) G. Bomchil, A. Huller, T. Rayment, S. J. Roser, M. V. Smalley, R. K. Thomas, and J. W. White, Philos. Trans. R. SOC.London, Ser. B, 290, For comparison with the other models the constants D and E , given 537 (1980). in Table I for ref 3, have been adjusted to give the best agreement (27) G. Constabaris, J. R. Sams, and G. D. Halsey, J . Phys. Chem., 65, with the Buckingham-Corner potential law. However, the 367 (1961).
-
-
(28) J. R. Sams, J . Chem. Phys., 43, 2243 (1965). (29) N. N. Avgul, A. G. Bezus, E. S. Dobrova, and A. V. Kiselev, J . Colloid Interface Sei., 42, 486 (1973).
(30) E. V. Kalaschnikova, A. V. Kiselev, R. S. Petrova, and Shcherbakova, Chromutogruphia, 4, 495 (1971).
K. D.
236 The Journal of Physical Chemistry, Vol. 88, No. 2, 1984
Vidal-Madjar and Bekassy-Molnar
TABLE 11: Adsorption of Benzene o n Graphite
experiment
anisotropic nwdel (eq 6, P = 0) anisotropic model (eq 6, P = 0.29) isotropic model (eq 1O,P= 1) combination of selfinteraction parametersd empirical model, Buckingham-Corner eqc Units: J nm6/mol.
2.144 0.637 1.486 0.594 1.583 0.533
0.330
68
37.7
40.9 2 0.3e 41.1 t 0.2e 41.7 t 0.2e 39.41 33.2
0.330
72
42.8
38.6
0.273
92
55.7
52.5
3.719
76
47.8
44.4
1.413
73
44.1
40.3
0.511
o,330
0.340
0.2971 0.331 0.286 0.338 0.294
Units: nanometers.
\ 1
o.320 o.330 Reference 12.
thermodynamic functions of Table I are calculated with the original expression of ref 3. The results of the empirical models of ref 3 and 13 are of course in good agreement with the measured thermodynamic functions of adsorption. They are not better, however, than the predictions of the a priori anisotropic adsorption potential model applied to the thermodynamic statistical calculations. The adsorption potential model discussed in this work which is based on the general properties of the adsorbate and the adsorbent is more helpful than the empirical ones as it may allow prediction of the adsorption behavior of other families of organic compounds and will be used in the next sections to predict the thermodynamic functions of aromatic molecules on graphite. It is to be expected that the anisotropic adsorption potential model will predict the second virial adsorbate/surface interaction of n-alkanes on graphite with as good agreement as the one reported by Kiselev and Poshkus3 with an empirical adsorption potential model, as the attractive and repulsive constants Dijand Bij given in Table I are close for both models. Adsorption of Benzene on Graphite. The adsorption potential models of benzene on graphite derived in this work are calculated with the following parameters. The mean polarizability ai (1.30 X cm3/molecule) and diamagnetic susceptibility zi (1.19 x cm3/molecule) of carbon atom force centers are increment values calculated from the corresponding properties of the benzene m o l e c ~ l e ,the ~ ~same , ~ ~ increment values as with methane being assigned for the hydrogen force center. An anisotropic polarizability tensor has been taken for carbon atom force center with ail = a i2 = 1.63 X cm3/molecule. The anisotropic polarizability ratio ai3/ail= 0.39 is derived from the experimental polarizability data of the benzene molecule. The minimum approach distances for C. 42 and C. .-H atoms are respectively 0.340 and 0.300 nm. The geometrical structure of the benzene molecule is from ref 33 with C-C and C-H bond lengths of 0.1392 and 0.1080 nm, respectively. For all adsorption models, the adsorbate/surface second virial coefficient BASand the isosteric heat of adsorption are calculated with eq 12 and 14 from the change of the adsorption potential in the whole space available to the molecule and for all orientations. Comparison with experiment (Table 11) shows that too high values are predicted with the isotropic adsorption potential model. In this model the attractive constants Dijare calculated with the Kirkwood-Muller expression. qstis 1.3 larger than the experimental data and BAS25 times too high. The adsorption potential model which takes into account the high anisotropic polarizability of graphite with a nonzero polarizability ratio predicts an isosteric adsorption heat which is in good e
(31) Landolt-Bornstein, 'Zahlenwerte und Functionen", Vol. 1, Part 3, Springer, West Berlin, 1951, p 510. (32) P. S . O'Sullivan and H. F. Hameka, J . Am. Chem. SOC.,92, 1821 (1970). (33) E. G. Cox, D. W. J. Cruickshank, and J. A. S . Smith, Proc. R . SOC. London, Ser. A , 247, 1 (1958).
Reference 13.
e
Refereme 34.
0.428e 0.488e 0.533e 0.642f -1.026
Reference 35.
10YT (OK) r
24
2.6
2.6
3.0
Figure 3. Second adsorbate/surface virial coefficient of benzene adsorbed on graphite as a function of temperature. Dotted line = anisotropic theoretical model; solid lines = experiment. Coefficients a and 0 of the equation In BAS = a P / T in parentheses: Theory (-1 1.07,4230);ref 34: curve 1 (-1 1.63,4500); curve 2 (-1 1.70,4550); curve 3 (-1 1.25,4240); ref 35: curve 4 (-1 1.35,4474).
+
agreement with experiments (-5%). The predicted BASvalue is 20-40% lower than the measured values obtained by gas ~ , ~ ~ the BAS chromatography at zero surface c ~ v e r a g e . ~However, values measured with high-precision chromatography in ref 34 are significantly scattered (about 10%). The low experimental precision achieved has been partly explained because of the residual heterogeneity of the surface of the graphitized carbon black samples (Figure 3). As for n-alkanes, the empirical adsorption potential model of Kiselev, Poshkus, et a1.I2 describing the adsorption of aromatic molecules on graphite has been introduced because the isotropic adsorption model is unable to describe experiments. The model of Battezzati et al.' based on combination of self-interaction parameters predicts too high values for BASand qSv The conclusion of their work concerning the most stable conformation for benzene adsorbed on graphite (its plane parallel to the 0001 graphite face and its mass center located above a graphite carbon atom) still holds with the anisotropic adsorption potential model.
'
(34) C. Vidal-Madjar, M. F. Gonnord, M. Goedert, and G . Guiochon, J . Phys. Chem., 79, 732 (1975). (35) E. V. Kalascbnikova, A. V. Kiselev, R. S. Petrova, K. D. Shcherbakova, and D.P. Poshkus, Chromatographia, 12, 799 (1979).
The Journal of Physical Chemistry, Vol. 88, No. 2, 1984 237
Adsorption of Hydrocarbons on Graphite TABLE 111: Adsorption of Aromatic Hydrocarbons on Graphite 1030gMQ 1024-
"M
1030&c
experiment anisotropic model
naphthalene phenanthrene anthracene naphthalene
(eq 6, P = 0.29)
with benzene parameters
phenanthrene anthracene
anisotropic model
naphthalene
(eq 6, P = 0.29)
with adsorbate parameters
phenanthrene anthracene
anisotropic model (eq 6, P = 0.29), with adsorbate parameters
naphthalene phenanthrene anthracene
a
Units: ~cm3/molecule. Reference 6. Reference 41.
1.19 1.19 1.19 15.3d 1.26 21.2d 1.28 21'5d 1.30 15.3d 1.28 21.2d 1.28 21'5d 1.30
In
(1
lOZ4Zic
280 440 440
67 * 1: 67e 93 t 97e 93 t 1,b 96.6e
0.681,' 0.710e 0.897,b 0.335e 1.104,b 0.557e)
0.39
280
60.6
0.208
0.39
440
83.0
0.400
0.39
440
83.1
0.458
0.43f
280
64.4
0.832
o.35g
440
87.1
0.971
0.32g
440
70.0
1.408
0.36'
280
63.6
0.752
o.35i
440
89.1
1.240
0.33'
440
91.2
1.603
ai3c/qlc t, "C
1.30 1.30 1.30 16.6f 1.32 23.6g 1.38 25'3g 1.5 1 17.5h 1.41 24.7h 1.46 25'9h 1.55
Reference 7.
qst, kJ/mol
(wol/m2)/ (mol/m3)
S
1.23,b 1.25e
1.06
1.55
1.44
Referenc:e 32.
e
Reference 35.
f
Reference 38.
g
Reference 39.
Refer-
ence 40.
For this equilibrium conformation, the minimum adsorption potential, the equilibrium distance (carbon-carbon distance), and the vibration frequency of the molecule perpendicular to the adsorbent surface are listed in Table 11. Comparison of these data predicted by theory with experiment is not possible, at present, as to our knowledge no measurements of the minimum approach distance of benzene on graphite have been published, although the benzene adsorbed layer has been studied with neutron scattering technique^.^^,^^ This parameter is important to measure as its determination would allow one to test the validity of the adsorption potential model. For example, the experimental equilibrium distance would permit one to fix the zo distance and with a shorter distance than the one predicted with the anisotropic adsorption model a better agreement with the experimental thermodynamic functions would be obtained. Adsorption of Condensed Aromatic Hydrocarbons. The anisotropic adsorption potential model with a nonzero value for the polarizability component perpendicular to the basal graphite surface ( P = 0.29) has been used to predict the adsorption properties of naphthalene, phenanthrene, and anthracene on graphite (Table 111). The mean polarizability and diamagnetic susceptibility increment of the carbon force center is calculated from the molecular properties of the molecule^^*-^^ listed in this table the increments of the hydrogen force center being those assigned for the methane molecule. The C...C and C...H equilibrium distances are the same as before. The geometric structure of the molecules is taken from ref 42 and 43. The anisotropic adsorption potential model predicts values which are in good agreement with experiment for naphthalene adsorbed on graphite: with BAS (5-15% agreement) and with qst (4% agreement) when the increment values assigned to the carbon force center are derived from the general properties of the whole ~~
(36) P. Meehan, T. Rayment, R. K. Thomas, G. Bomchil, and J. W. White, J . Chem. SOC.,Faraday Trans. 1 , 76, 2011 (1980). (37) M. Monkenbusch and R. Stockmeyer, Ber. Bunsenges Phys. Chem., 84, 808 (1980). (38) R. J . W. Lefevre and L. Radom, J . Chem. SOC.B, 1295 (1967). (39) R. J. W. Lefevre and K. M. S . Sundaram, J . Chem. SOC.,4442 (1963). (40) J. Schuyer, L. Blom and D. W. Van Krevelen, Trans. Faraday SOC., 49, 1391 (1953). (41) N. V. Cohan, C. A. Coulson, and J. B. Jamieson, Trans. Faraday SOC.,53, 582 (1957). (42) D. W. J. Cruickshank, Acta Crystallogr., 10, 504 (1957); 9, 915 (1956). (43) J. Trotter, Acta Crysrallogr., 16, 605 (1963).
molecule (Table 111). Deviation from experiment is larger when the adsorption potential model is used with the same attractive and repulsive constants as for benzene. Similarly, for phenanthrene and anthracene (Table 111) the adsorption potential calculated with the same increments as those used for benzene predicts isosteric adsorption heats which are 10% lower than the experimental ones. The scattering of the experimental data for BAs6935does not permit comparison with predicted values and more reliable experiments should be obtained by gas chromatography. More interesting is to compare the adsorption behavior of phenanthrene and anthracene by calculating the ratio of the respective adsorbate/surface second virial coefficients:
s = BAsAn/BAsPh s is equal to the relative retention of the compounds and is independent of surface area determination. It is measured with good precision by gas chromatography even with a low-precision instrument. It is related to the difference between free energies through the relation The theoretical values for s are compared to the experimental ones in Table 111. With the models which use, for the adsorption potential, C and H force center parameters equal to those of benzene molecule the predicted s value is small: 1.06 for either the anisotropic adsorption potential model (eq 6 ) or the isotropic one (eq 10) instead of 1.23 for the experimental value. In a previous work we have already shown that it is not possible to predict the retention behavior of the geometrical isomers when equal force center adsorption potential functions ai are used (phenanthrenelanthracene; cis-2-butene/trans-2-butene;methyland polymethylbenzenes and -naphthalenes.44 Other adsorption potential expressions were introduced to explain the experimental relative retentions which are based on the first-degree expansions of the adsorption potential near the equilibrium position of the adsorbate molecule on graphite surface. Kiselev et al.'* have discussed the problem of anthracene and phenanthrene relative retention and explained that the lower calculated s values with the adsorption potential model used are (44) M. F. Gonnord, C . Vidal-Madjar, and G. Guiochon, J. Chromatogr.
Sci., 12, 839 (1974).
238
J. Phys. Chem. 1984, 88, 238-243
due to the differences in electronic configuration and thus differences in polarizabilities of these molecules. In this work, the polarizability differences are taken into account in the calculation of the adsorption potential as the mean polarizability and the polarizability ratio of the adsorbate molecule (ail/ais) assigned to each C force center of the molecule, derived from the corresponding molecular experimental data. The predicted s value for phenanthrene/anthracene adsorption on graphite is 1.44 when polarizability data are taken from Lefevre et al.39and 1.55 from Schuyer et al.40 instead of 1.23 for the experimental value as measured by gas-adsorption chromatography. These calculations demonstrate how sensitive the model is to the value accepted for the mean polarizability of the carbon force center of the adsorbate molecule.
Conclusion These results show that the anisotropic adsorption potential model of Meyer and Dietz,14Js when applied to the molecular theory of adsorption, predicts with good agreement, at least as well as the empirical adsorption potential laws, the thermodynamic functions of adsorption of alkanes and aromatic hydrocarbons on graphite.
The adsorption potential which takes into account the high anisotropic polarizability of graphite can be expressed with a good approximation by a Lennard-Jones (12-6) potential. The Kirkwood-Muller constant used for isotropic adsorption potential calculations on graphite is 4/3 larger than the attractive constant which assumes that the polarizability in the direction normal to the basal plane is zero. Therefore, adsorption potential laws which use the Kirkwood-Muller attractive constants to predict BASand qst values for the adsorption on graphite give results which are too large. The relative retention of geometrical isomers are equal when calculated with the anisotropic or isotropic potential models if the same polarizability and diamagnetic susceptibility increments are assigned to the adsorbate force centers. Similarly the conclusions of Battezzati et al.” which cencern the most stable equilibrium adsorbate conformations on graphite still hold with the anisotropic adsorption potential model. Finally, this work shows that correct knowledge of the polarizability of the adsorbate molecule and the structural properties of the adsorbed layers is of great importance in order to derive the constants of the adsorption potential laws. Registry No. Graphite, 7782-42-5.
New Simple Functions To Describe Klnetlc and Thermodynamic Effects of Pressure. Application to Z-E Isomerization of 4-( Dimethylamino)-4’-nitroazobenzene and Other Reactions Tsutomu Asano* and Toshio Okada Department of Chemistry, Faculty of Engineering, Oita University, Oita 870-1 1 , Japan (Received: February 8, 1983; In Final Form: May 18, 1983)
Two simple three-parameter equations are proposed as functions to describe kinetic and thermodynamic effects of pressure. The functions are found to reproduce experimental results more accurately than the most frequently used quadratic equation. The estimated activation volumes at zero pressure are almost independent of the experimental pressure range for most of the reactions examined and their standard deviations are reasonably small. The activation volumes at infinite pressure are, in many cases, in fairly good agreement with the intrinsic activation volumes calculated by an independent procedure.
Introduction Activation volume has been proved to be an effective tool to elucidate organic as well as inorganic reaction mechanisms.’-3 For this purpose, activation volume at zero pressure, AVO*,which is practically equal to the value at 1 bar, is usually employed. To estimate AVO*is a simple task when the value is independent of pressure. The pressure dependence of a rate constant can be described by eq 1 and AVO*is given by -bRT as shown in eq 2, In ( k p / k l )= a + bP (1) AVO*= -RT(B In k P / d P ) , = -bRT
(2)
where k p is the rate constant at pressure P . However, in many reactions studied so far, In kp turned out to be a nonlinear function of pressure. In such instances, we need a suitable function to describe the pressure dependence of In k p Several equations have been proposed and compared. For example, Hyne and his coworkers4 analyzed their data on the hydrolysis of benzyl chloride in water by eq 1 and 3-7 and concluded that all the functions ~~
(1) Asano, T.; le Noble, W. J. Chem. Rev. 1978, 78, 407. (2) van Eldik, R.; Kelm, H. Reu. Phys. Chem. Jpn. 1980, 50, 185. (3) le Noble, W. J. Rev. Phys. Chem. Jpn. 1980, 50, 207. (4) Lohmuller, R.; Macdonald, D. D.; Mackinnon, M.; Hyne, J. B. Can. J . Chem. 1978, 56, 1739.
0022-3654/84/2088-0238$01.50/0
+ bP + cP2 In ( k p / k l )= a + bP + cP2 + dP3 In ( k p / k l )= bP + cPz In ( k p / k l )= aP + bP1s23 In ( k p / k l )= a
(3) (4) (5)
(6)
except eq 1 gave essentially the same activation volume. Kelm and Palmers tried eq 3, 4, 6, 8, and 9 for data on a Diels-Alder In ( k p / k l ) = a b [ l - exp(-cP)] (8)
+
In ( k p / k l ) = a
+ bP/(c + P)
(9)
reaction and a ligand substitution reaction. They noticed that the quadratic equation 3 tends to underestimate the activation volume but none of the functions appears to be superior in describing the experimental data. Because of its mathematical simplicity, the quadratic equation has been used by the over(5) Kelm, H.; Palmer, D. A. “High Pressure Chemistry”; Kelm, H., Ed.; Reidel: Dordrecht, 1978; pp 281-309.
0 1984 American Chemical Society
