On Physical Adsorption XIV. The Nature of the Adsorptive Forces between the Inert Gases and a Graphite Surface SYDNEY ROSS and JAMES P. OLIVIER Department of Chemistry, Rensselaer Polytechnic Institute, Troy, Ν. Y.
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a gas and a solid surface have hitherto not been definitively compared with experimental results, because of difficulty in evaluating the contribu tions that make up the measured heat of adsorp tion.
Methods described in previous papers pro
vide a means of analyzing experimental data in terms of the adsorptive potential of the gas with respect to the most frequent patch of the surface. The adsorptive potential thus obtained is made up of the dispersion interactions plus an energy term due to interaction between the dipole of the adsorbate
and the electric field of the substrate.
For the inert gases adsorbed by graphite enough information is now available for calculation of the necessary quantities. The results derived from experiment are compared with calculations of the dispersion energies using potential functions of different degrees of approximation.
Crowell's
approximation of the lattice sum, in which the graphite lattice is treated as a set of planes of continuous structure and uniform density, when applied to a 6-8-10-12 potential function, is found to correspond most closely to the observed values.
^observations of a large number of adsorption systems disclose that the heat evolved during adsorption may be anywhere from a few hundred calories to over a hundred kilocalories per mole adsorbed. Early workers in the field used a value of 20 kcal. per mole to define the upper limit of physical adsorption, believing that heats of adsorption above this value could arise only from a chemical bonding between the adsorbate and the surface. We are now aware that a wide 309
In SOLID SURFACES; Copeland, L., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1961.
310
ADVANCES IN CHEMISTRY SERIES
range of heats of adsorption could arise from any one of four adsorption mechanisms or combinations thereof; and therefore that a knowledge of the heat of adsorption is not enough, save in extreme cases, to distinguish between adsorption mechanisms.
Origins of Adsorptive Potential
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Four general types of molecular interaction lead to a potential energy for adsorption: 1. London-type dispersion forces resulting from induced-dipole/induceddipole and multipolar attractions 2. Induction forces brought about by the operation of a surface electric field on induced or permanent dipoles of resident molecules 3. Charge transfer (6) between the adsorbed molecule and the surface, resulting in a no-bond resonance state 4. Dative bonding resulting from a chemical reaction between the adsorbate and surface atoms The first two types of interaction are certainly physical adsorption; the fourth is certainly chemisorption; the third, which is a postulated mechanism whose actual occurrence has not yet been adequately demonstrated, exists in a twilight zone that defies classification as either physical or chemical adsorption. Previous papers of this series have not been concerned with the source of the adsorptive potential, but have postulated only that the adsorbed film be mobile; their considerations could apply, therefore, to all adsorption situations other than No. 4. The present paper is devoted to a consideration of the adsorptive potential arising from 1 and 2—that is, physical adsorption properly so called. This general topic has been discussed by de Boer (5) for a number of specific cases; the theory and techniques that we have developed enable us to derive from experimental data a quantity, P , the depth of the adsorptive potential well, which can also be calculated a priori from physical properties of the system. We are therefore able to compare theory and observation and so have an independent test of conclusions reached in previous papers. g
&ds
Expressions for Dispersion Potential The Direct Lattice Sum. Dispersion forces between two atoms can be described by a potential function expressed in terms containing inverse powers of the internuclear separations, s. The simplest function of this sort includes a potential energy of attraction proportional to the inverse sixth power of the separation and a repulsion that is zero at distances of separation greater than a particular value s and infinite at separations less than s . This is the so-called "hard sphere" or van der Waals model. Such an approximate potential function can be improved in two respects. Investigations of the second virial coefficient have revealed that the potential energy of repulsion is best described as proportional to the inverse twelfth power of the separation; and the term in sr , which accounts for the greater part of the total attraction potential, due to the attraction of mutually induced dipoles, should have added to it the dipole-quadrupole and quadrupole-quadrupole attractions, expressed as terms in s^ and s , respectively. The complete potential function for the forces between two atoms is, therefore: e
e
6
8
-10
Pa = CV^ + C s^ + C W ° - (Rj« 2
12
(1)
where P is the potential energy lost by two particles, i and /, on approaching each i;
In SOLID SURFACES; Copeland, L., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1961.
ROSS AND OLIVIER
Adsorption of Inert Gases on Graphite
311
other from an infinite separation to a separation s; C C , and C are dipole-dipole, dipole-quadrupole, and quadrupole-quadmpole coefficients, respectively; and (K is a repulsion constant. To calculate the potential energy lost by a gas molecule on approaching a surface, the potential difference, P has to be summed for the simultaneous interactions of the gas molecule, i, with each of the atoms of the adsorbent. l9
3
2
ij9
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(2)
For a polyatomic adsorbate it may be necessary to sum the interactions over each atom of the adsorbate molecule as well. The various values of in Equation 2 can, for a particular adsorbent of known crystal lattice, be calculated in terms of a single distance, z, which is defined as the distance between the center of an adsorbate atom and the mathematical plane in which lie the centers of the surface atoms of the adsorbent. The constants C C , and C in Equation 2 can be evaluated by means of the following approximate quantum mechanical equations (I, 7) : l9
3
2
C, = emc^i
(3)
1
I*.
it
-f
Xi
45A
2
°
3
"
256TTW
W
In these equations £ represents polarizability and X represents diamagnetic sus ceptibility; subscripts i and ; refer to the adsorbate and the adsorbent, respectively. No theoretical expression for the repulsion constant, , has been developed; it can, however, be evaluated in terms of the equilibrium separation, ζ = z , where the potential function has its maximum value. At ζ = z , dP/dz — 0; hence e
e
Ci
(R =
()
T r - i
1
6
a dz
The derivatives in Equation 6 can be obtained in two ways. The summations may be plotted vs. ζ in the range bracketing the equilibrium separation, z , and the slopes found graphically; or these plots may be expressed by an empirical equation and the slopes obtained analytically. The latter method is more accurate, as an equation of the form e
\j~
^
n
= az~
h
(7)
J
has been found to hold ( 1 ). The method described above has been used by Avgul et al. (I) to calculate the dispersion interactions between a number of adsorbates and the cleavage sur face of graphite. A prime disadvantage of the method of direct lattice sums, apart In SOLID SURFACES; Copeland, L., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1961.
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312
ADVANCES IN CHEMISTRY SERIES
from the laborious calculations, is the dependence of the answer on the lattice position above which the adsorbate molecule is situated (see Figure 1). For a mobile adsorbed film the proper procedure would be to find the weighted average interaction of all possible positions on the surface. In practice, only a few principal positions are ever worked out and a straight arithmetic average of the interactions for these positions would not be characteristic of the mobile molecule. Figure 1 is a schematic diagram of an argon atom on the three principal posi tions of the basal plane of graphite: on top of a carbon atom, between two carbon atoms, and in the center of a ring. The results calculated by Avgul et al. (I) are reported for neon, argon, and krypton in Table I; we do not include a number of other calculations made by the same authors for polyatomic adsorbates, as the writers do not take into account the anisotropics of polarizability and magnetic susceptibility when calculating dispersion interactions for oriented molecules.
Figure 1.
Argon atom in different positions relative to graphite substrate (1) Drawn to scale
Table I. Calculated Values of P by Equation 2 for Inert Gas Molecules Situated at Different Parts of Graphite Substrate dlap
g
(Kcal./mole) Above C Atom
Adsorbent
Neon Argon Krypton
Between Two C Atoms
0.77 1.95 2.59
0.84 2.07 2.78
Center of Ring
Av.
1.11 2.64 3.47
0.91 2.22 2.95
Crowell's Expression for Lattice Sum. T H E ORIGINAL APPLICATION. (2) has shown how to evaluate the lattice sums, ^
Crowell
s~ , for a layer-lattice strucn
ture by an analytical method. The lattice is approximated by a set of layer planes, each with a uniform distribution of matter, and separated by the interplanar dis tance, d. The lattice sum is approximated by integrating over the planes and form ing the sum of the resulting terms. Let the adsorbate, at a distance ζ from the surface, be separated by a distance «y from any point in the m plane below the surface. The vertical distance of the adsorbate from this plane is (z + md); let y be the projection of on the m plane. Then the summation of s~ ior the m plane is tn
n
t h
y^^
ra
j (m)
= 2πρ
Γ JO
[{ζ +
df
m
+ f]-*t*ydy
In SOLID SURFACES; Copeland, L., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1961.
th
(8)
ROSS AND OLIVIER
Adsorption of Inert Gases on Graphite
where ρ is the number of atoms per unit area in the m plane. tice the summation becomes
For the whole lat
th
00
^2
S
ï "
=
^
j
313
00
^ )
=
2
π
^
η
-
)~^ ~
2
2
^
n
m — 0 j (m)
(*
+
m
)
2
~
n
( ) 9
m=0
where χ — z/d. The summation of (x + m ) is the generalized Riemann zeta function of χ and η — 2. These functions have been tabulated (4) as derivatives of ψ (χ) = (d/dx)\n(x\). The fundamental expansion of this expression for the k derivative of Ψ (χ) is 2 _ n
th
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¥*(*)
^
= (-l) k\ k+l
(* + m)- ~ k
l
(10)
r/i = 0
Equation 9 can now be written as 2ΤΓ (-1)» Ρ
{η - 2)d ~\n - 3)! n
* -\x) n
(11)
The functions 4, provided x
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Table II.
re, Z, e
Parameters Involved in Calculating Dispersion Potentials of Inert Gases on Graphite
A. A.
χ Χ 10 , cc. £ X 10 , cc. C i Χ ΙΟ *, kcal. mole" cm. C Χ 10 , kcal. mole" cm. C X 10 , kcal. mole" cm. 4
3
1
f)
60
1
8
76
1
10
Table III.
Neon
Argon
1.70
1.59 3.29 12.0 0.398 0.214 0.026 0.04
1.91 3.61 32.2 1.63 0.77 0.104 0.17
10.5 0.937
30
24
2
Carbon
P , kcal./mole (jP , kcal./mole pdi kcal./mole ν X 10- , sec." clec
ads
(7
sP)
12
Xenon
2.01 3.71 46.5 2.48 1.15 0.157 0.31
2.25 3.95 71.5 4.00 1.82 0.252 0.43
Observed and Calculated Values of Dispersion Potentials G P Vibrational Frequencies for Inert Gases Adsorbed by Graphite Related Equation No.
a
Krypton
1
(27) Observed Observed Observed (2) (15) (17) (19) (20) (23) (24)
^pdisp
ν X 10~12 ^pdisp
ν χ io-12 ν χ io-12
d i s p
) and
Neon
Argon
Krypton
Xenon
0.043 0.737 0.694 1.19 0.91 0.729 1.16 0.764 1.20 0.23 0.55
0.180 2.12 1.94 1.28 2.22
0.256 2.80 2.54 1.00 2.95
0.415 3.69 3.28 0.85
1.84 1.20 1.92 1.23 0.64 0.60
2.48 0.93 2.59 0.96 0.88 0.47
3.11 0.78 3.22 0.80 1 .2 0.40
Acknowledgment The authors gratefully acknowledge a grant in aid of these researches from the Esso Research and Engineering Co. Literature Cited (1) Avgul, Ν. N., Kiselev, Α. V., Lygina, I. Α., Poschkus, D. P., Izvest. Akad. Nauk S.S.S.R., Otdel. Khim. Nauk 1959, 1196. (2) Crowell, A. D., J. Chem. Phys. 22, 1397 (1954). (3) Ibid., 26, 1407 (1957). (4) Davis, H. T., "Tables of Higher Mathematical Functions," Vol. 2, Principia Press, Bloomington, Ind., 1933. (5) de Boer, J. H., Advances in Colloid Sci. 3, 1-66 (1950). (6) Gundry, P. M., Tompkins, F. C., Trans. Faraday Soc. 56, 846 (1960). (7) Lennard-Jones, J. E., Dent, Β. M., Ibid., 24, 92 (1928). (8) Ross, S., Machin, W. D., unpublished results. (9)
Ross, S., Olivier, J. P., ADVANCES IN C H E M . SER., No. 33, 301 (1961).
RECEIVED May 9, 1961.
In SOLID SURFACES; Copeland, L., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1961.
