NOTES
1768
Acknowledgment. The author thanks Dr. D. J. C. Yates of Esso Research and Engineering Co. for allowing use of some of his results and also for valuable discussionsand suggestionson experimental technique.
method which preferably does not use any theoretical equations. Such a method utilizes the isotope effect observed in Vx.5 For a system behaving nearly classically, V x (which is seen to be a configuration integral) can be expanded in an ascending power series in fi.6 The f i s t three terms are7
Determination of Potential Parameters for
V x = d C 1 - (fi2/24pkT)~So-zIq(1) + (7fi4/5760p2k‘T ’) U S ~ - * I , ( ~ ) (4)
Gas-Solid Interactions from Quantum Effects in Physical Adsorption
by K. Yang, P. L. Cant, and D. E. Cooper Central Research, Division, Continental Oil Company, Ponca City, Oklahoma (Received December 16, 1964)
Information on the potential of gas-solid interaction is often required in the investigation of physical adsorption and related phenomena.’ An empirical equation commonly used for this purpose is the general Lennard-Jones potential2
4
=
mlh-4
[ n / ( m -- n) I(n/m>
E
* [(So/S)m - (So/fYI (1)
where So is the gas-solid separation a t net zero interaction, E* is the depth of the potential, and m > n. The parameters E* and So can be determined by using the concept of excess volume V , defined as NkT Vx -- P
n
V,ea
S.S.e.
where is the number Of molecules in at an equilibrium pressure Of p and a temperature Of T, and Vmo is the geometrical W h ~ (the e volume by the minus the the For 8 &Ssical system, to zero pressure is related to 9 as
V,
=
Jvgeo
[exp(--/kT)
v x
excluded by extrapolated
- l]dV = a1,1
=
i=l
( V x obad -
V x caled)i2
(5)
where n denotes the number of observations. Necessary techniques for this nonlinear, least-squares estimationare well known.8,9 The technique, as applied to the present investigation, is briefly described below. Integrals, Iol,Is(’), and I,(z), have been tabulated.7 For the nonlinear estimation, it is convenient to use
(3)
where a is the surface area times So, and 1 , i for a given set of m and n is a function only of e*/kT. The temperature variation of Vx then enables one to determine a and E*. This method, however, does not provide So directly; hence, it is necessary to employ some theoretical equations such as the Kirkwood-Muller formula by means of which SOcan be related to E*. Theoretical potential equations usually incorporate in them Various approximations’; and it is to compare the above So values, as well as the resulting surface areas, with the values obtained by some other The Journal of Physical Chemistry
where p is the molecular mass and and I q ( z )for a given set of m and n are functions only of e*/kT. It has been demonstrated that a comparison of V , between two isotopic species yields So uniquely, provided the following approximations are justified : (1) the first two terms in the right-hand side of (4) are sufficient for V,; (2) E* for the two isotopic species is the same. A recent investigation7 reveals that, for the HgD2 pair to which the above method was applied, there is an appreciable isotope effect in E*. This variation of E* is difficult to take into account in the above method. The present note reports a new attempt to determine So (hence, surface area also) directly. For this, it is noted that the quantum correction terms in (4) decrease sharply with increasing So. With this in mind, V x is regarded as a function of a, E*, and So, and m attempt is then made to find a set of these parameters which minimizes the sum of the squares of error (s.s.e.)
(1) For example, see D. M. Young and A. D. Crowell, “Physical Adsorption of Gases,’’ Buttervorth, Inc., Washington, D. C., 1962. (2). For example, see E. A. Moelwyn-Hughes, “States of Matter,” Ohver and Boyd Ltd., Edinburgh and London, 1961, p. 33. (3) W. A. Steel and G. D. Hdsey, Jr., J. Chem. p h w . , 22, 979 (1954). (4) M P. Freeman and G. D. Halsey, Jr., J . Phys. Chem., 59, 181 (1955): (5) M. P. Freeman, ibid., 64, 32 (1960). (6) For example, see J. 0. Hirsohfelder, C. F. Curtis, and R. B. Bird, “Molecular Theory of Gases and Liquids,” John Wiley and Sons, Inc., New York, N. Y., 1954, p. 401. (7) R. Yitris and J. R. Sams, Jr., J . Chem. Phys., 37, 571 (1962). (8) H. 0.Hartley, Technometrices, 3,269 (1961). (9) D. w. Marquardt, J. SOC.I&. ~ p p z Math., . 11,431 (1963).
NOTES
1769
Table I: Coefficients in Eq. 6 Function
Range
Potential
< 4.0
I,,
1
