Relative energy distribution in physical adsorption on patchwise heterogeneous surfaces

Volume 19, number 2

CHEMICAL PHYSICS LETTERS

RELATIVE

ENERGY

DISTRIBUTION

ON PATCHWISE

IS hIarch 1973

IN PHYSICAL

HETEROGENEOUS

ADSORPTION

SURFACES

Wadyslaw RUDZIIkXI Department

of Physical

Chenristry,

institute

of Chemistry

Received 22 November

A new function is proposed.to culation.

characterize

the heterogeneity

One of the basic problems in the experimental investigation of heterogeneous adsorbents is the determination of the energy distribution function X(E) = aV/ae, where E is the adsorptive energy, and V is the volume of the adsorption space or the-number of adsorption sites, which depends upon the adsorption model assumed. Many attempts have been made to evaluate the function X(E). The procedure most often used is based on the assumption that the surface is patchwise heterogeneous, and the total adsorption isotherm B(p, r) is given by the equation [ 1J , %O-)=~xIWIt~,

i-3 +dE,

(1)

where 8, is the “local” adsorption isotherm describing the adsorption on the patch having adsorptive energy E. The problem now is to resolve the integral equation (1) for a chosen analytical form of O1, and for 0 given in the form of a table (experimental points). The first results are due to Sips [2], who succeeded in solving eq. (I),

when

0,

is the Langmuir

isotherm,

but as

Harris [3] wrote, “since that time little progress has been made in solving the problem for other cases”. Recently computers have been used [ 1,4] , in order to solve eq: (1). Barker and Everett 151, and then Steele [6] have attempted to use the virial formalism in investigating the heterogeneity. However, as Pierotti [7] wrote, “the problem of heterogeneity is still one of the great unresolved problems of physical adsorp.tion”. Apart from the difficulties existing here, I would

UhfCS, Lubfin,

Nowotki

12. Poland

1972

of adsorbents,

and

a method is provided for its cal-

like to discuss the usefulness of x(c), when characterizing the heterogeneity effects. This function characterizes global energetic conditions provided by a surface for a single molecule to be adsorbed, in the absence of any other adsorbate molecules. The simultaneous presence of other adsorbate molecules changes the global energetic conditions for the considered molecule in a very complicated way. This change is realized by modifying the mutual interactions between adsorbed molecules, due to the presence of the surface. These are the so-called ‘higher-order ‘titeraction effects”. Except for the third-order effects [8] : no other information is known concerning the aatdre of these many-body effects. Therefore, the characteristic of an adsorption system, given by x(e) becomes Iess and less valuable, as the amount of the adsorbed phase increases. In my opinion, the more v&able characteristic would be the function x,(p, T, e), called here “the relative distribution function”, and defined as the potential of the average force acting on one chosen molecule by the surface and by all other molecules adsorbed at pressure p, and temperature T. When assuming additionally the inert structure of the adsorbent solid surfaces, then the averaging should be carried out over the positions of alI other adsorbed molecules. Of course, we must have,

$

X*(Pt

Tie) = X(E).

The function xr(p, T, E) is as valuable during the 221’

Volume 19, number

CHEMICAL P~,YtfL$ &TTERS ,j :;...

2

whole adsorption process as the function X(E) in’the zero pressure limit. The purpose of this paper is to develop a general theoretical method for evaluating x,(p, T, E) from experimental isotherms. To this purpose I shall use the theoretical results of Harris 131, which will be the starting point for this method. Following Harris, “if in eq. (1) the assumed form of the local isotherm 0 1 is approximate, then the form of the distribution function x1 deduced from that equation will likewise be only approximate, since in general if 6 1 and B2 are different functions, the equations

(in which the T dependence is suppressed for brevity) are satisfied by two different functions x1 and x2. Now consider the special case in which the adsorption isotherm B(p) is identical to the local isotherm B,(p, E’), where E’ is a particular value of E. In this case the distribution x1 satisfying eq. (3) is a delta function a(~-e’), which describes a surface on which all patches are of the same adsorptive energy; namely E’. On the other hand, for this same case, the distribution x2 (if it exists) satisfying eq. (4) is some particular function, which 1 will call x12(e, E’), which satisfies the equation,

6&b ~‘1=~x12kf’,62(w)de

-

(5)

If e2 were identical to o1 then x~~(E,E’) would be the delta function 6(e-e’). If 19~is an approximation to el, then the better the approximation, the more nearly will xl2 approximate 6(e-e’). Substituting (5) into (3) and interchanging the order of integration, we fmd that,

which by comparison with (4) shows that

(6) In the same way, there may exist a function x21 such ‘222.

..

15 March 1973

/ i

that,

!

ez(~,~‘)=SX21(~,~‘)e,(~,~)de

(7)

and i . ,i\.;;: : ‘a xl~~)=SX2(E)jXZl(f,f’)dE’.

(8)

Thinking now of ~92as an approximate form of 8 1, it is clear that if xZ1 existed and were known, then eq. (8) would be extremely useful, since having determined x2 corresponding to the approximate local isotherm 6,, it would be possible to use eq. (8) to find x1. The main obstacle to the use of eq. (8) is the determination of x2I, which requires solving (7). This solution is difficult for the same reason that the solution (3) is difficult; namely, the appearance of 8r under the integral sign. In fact, x21 may not even exist if the function 82(p, e’) cannot be expressed as a superposition of the form 8,(p, e).” We shall omit the last difficulty by choosing a priori x21 in eq. (7). Let x21 in (7) be of the form

x21=~(~-~‘>[~l(~,~)l-l(p/~)exp(~/~~), (9) where (p/K) exp (e/RT) is Henry’s equation. In fact, there exist two equivalent possibilities in the above procedure. Either to assume a priori the foml of e2 and 0 1, and then to seek xzl and x2, or to assume a priori x21 and 8,) and then seek 8, and x2. The above choice of x21 determines the form of d2, which will be used by us, namely, B,(P, e’) = (P/K) exp (e’/RT) .

(IO)

Thus the 19~to be used later by us is simply Henry’s isotherm. Now, let us discuss in detail the sense of using x21 in the fomr given by eq. (9). By assuming x21 in this form we have stated that Henry’s isotherm is valid throughout the pressure region !O,p). Such a statement is valid only in the case when E’ is the potential of the average force, averaged additionally over the pressure region (0, p>. Thus, the integration with the function x21 from eq. (9) has the meaning of double averaging: (r) the averaging over all degrees of freedom of all molecules, except for the considered molecule, (ii) the averaging over the pressure region (O,p>, since the potential of the average force is in generai a funciion of pressure. . ..

Volume 19, number 2

CHEMICAL PHYSICS LETTERS

15 hkuch 1973

Consequently, if x2 were calculated by means of Henry’s equation (with doubly averaged E’), then x1 calculated from eq. (8) and x21 from eq. (9), would be the “relative energy distribution”, averaged additionally over the pressure region (O,p), since the use of xsl from eq. (9) in eq. (8) has the same meaning’ as double averaging. From this it follows, that the relative energy distribution xr will be, X*(6 P, 0 = WXI )l@ 3

(11)

where x1 is calculated from eq. (8). However, it should be emphasized that the fully correct form of xr would have been obtained only if the function 8 1 in eq. (9) had been fully correct, and the potential e’ in Henry’s isotherm had been a fully correct, doubly averaged one. Thus, the relative distribution obtained, xr> represents the potential of the average force insofar, as .91 and E’ represent the real physical situat,ion. Now the problem which remains, is one of finding ~2 by means of 8, assumed in the form of Henry’s equation with the doubly averaged potential E’. This problem has been solved by Hobson [9] in the ideal gas approximation. The local adsorption isotherm adopted by Hobson has the form,

where

Fig. 1. The ordinary x1 (E) and reiative ,T&. p. T) energy distribution functions for argon adsorbed on rutiie [IO]. The dashed lines denote the ordinary distribution functions calculated by Dormant and Adamson [ I I], and by the author using the Hobson method. The solid lines denote two relative distribution functions, for p = 0.1 torr. and for p = 0.5 torr.

03) eo =-RTIn(p,-,/K)-Q and K is constant for a given temperature and adsorbate. p. is the pressure at which condensation occurs. Thus Q represents the additional potential energy of an adsorbed molecule at the condensation pressure pg. Hobson has solved the integral equation (1) assuming E in eqs. (12) and (13) to be still equal to the gas-solid interaction energy. Therefore x2 obtained from Hobson’s solution represents the function x2 in the ideal gas approximation. Let us additionally remark, that according to relation (13), x&Y~, E) - 0 for E < el, where e1 is equal to -RT In (pl /K) - Q. !t follows that the patches having adsorptive energy greater than el are completely fdled by the adsorbate moiecules at pressure p1 = p(el , Q), and that no other molecule can be adsorbed there. Thus, these patches are no more

available for adsorption, and therefore, they cannot be added to the adsorption space. In order to illustrate the theory developed above, a number of the relative distribution functions have been calculated by the author, from the weLLknown experimental data of Drain and Morrison [IO], describing the adsorption of argon on rutile. The locai adsorption isotherm 81, the Langmuir isotherm, has been used in the form 0 1 = [(~l~)exp(-~I~T)tl]-~. The results of the author’s calculations are presented in fig. 1. The dashed lines denote the ordinary energy distribution functions x1, calculated by Dormlint and Adamson [ 111, and by the author using Ho&on’s method, for the experimental isotherm of Drain and Morrison [lo]. From eqs. (I), (12) and (13) it is cIear that the Hobson method is applicable in a pressure region in which uncovered patches of surface still exist.

I~,(E, p, T) = (p/K) exp (E/RT) = 1

for E < e0 , 02)

for e 2 co ,

.223

Thus, it remains to be decided at which pressure the existence of uncovered patches may be assumed, for a given experimental isotherm. In the author’s calculations this pressure was assumed equal to the adsorbate pressure, at which an inflexion point on the experimental isotherm occurs. This arises from the fact that in the case of homogeneous surfaces it is, to a good approximation, the region of applicability of the Langmuir isotherm (model of monolayer adsorption). Using the value p = 56 torr for the inflexion point, we get simultaneously the lowest adsorption energy occurring on the adsorbent surface. This is the value E = 1850 Cal/mole. Below this value the Hobson function x1 must be assumed equal to zero, i.e., x1(e) = 0 for E < 1850 cal. The last result is in excellent agreement with the result of Dormant and Adamson [I 1J , since their function x1(f) = 0 for e < 1900 Cal. It has been found by Hobson [ 121 that the choice of Q has only a slight influence upon the form of x1 obtained. However, this choice has a strong influence on the form of xr, since it decides the value e1 = E(pl, Q), at which xr(e, pl, T) = 0 for E > el. One of Hobson’s propositions is to chose Q equal to the heat of liquefaction, which is in accordance with his assumption that at 8, = 1 condensation occurs. Such a statement leads us to the rather improbable result that at i?1 = 0.01 torr, el = 1556 cal for argon adsorbed on rutile, whereas the lowest adsorption -energies found by rhe author (using Hobson’s method), and by Dormant and Adamson, are about 1900 Cal/mole. It would simply mean that at p = 0.01 torr, the whole surface is covered by a monolayer, and this is a rather unrealistic result. In the author’s opinion, the completing of a mono.layer is not connected with the gas-liquid phase transition. Instead of Hobson’s assumption that Q is the heat. of liquefaction, I would Iike to propose

another

one. Namely,

15 March 1973

CHEMICAL PHYSICS LETTERS

Volume 19, number 2

that the additional

potential

energy Q at a complete monolayer, i.e., when 8, = 1,

is simply the interaction energy between two admolecules, being adsorbed directly on top of one another. It arises from the fact that multilayer formation starts effectively, after a monolayer is completely formed. From the theory of the third-order interactions [8] it appears that this attractive energy is about 90%-100% of the attractive energy between two adsorbate molecules in the free gas phase. Using a square-well potential for the interaction between two argon atoms in the bulk phase, we get Q equal to abouz 120 Cal/mole [13] . This value was used in the author’s calculations. With this assumption I have calculated two relative energy distributions: One for p = 0.1 torr, and another for p = 0.5 torr. They are shown as solid lines in fig. 1.

References [l] S. Ross and J.P. Olivier, On physical adsorption (Inter-

Science, New York, 1964). [2] R. Sips, J. Chem. Phys. 18 (1950) 1024. [3] LB. Harris, Surface Sci. 10 (1967) 129. [4] A.W. Adamson and 1. Ling, Advan. Chem. Ser. 33 (1961) 51. [5] J.A. Barker and D.H. Everett, Trans. Faraday Sot. 58 (1962) 1608. [6] W.A. Steele, J. Phys. L’hem. 67 (1963) 2016. [ 71 R.A. Pierotti and H.E. Thomas, Physical adsorption; the interaction of gases with solids (Wiley, New York, 1971). [8] 0. Sinanoklu and KS. Pitzer, J. Chem. Phys. 32 (1960) 1279.

-

J.P. Hobson, Can. J. Phys. 43 (1965) 1934. L.E. Drain and J.A. hlorrison, Trans. Faraday (1952) 840. L.M. Dormant and A.W. Adamson, J. Colloid Sci. 38 (1972) 285. J.P. Hobson, Can. J. Phys. 43 (196.5) 1941. J.O. Hirschfelder, C.F. Curtis5 and R.B. Bird, theory of gases and liquids (Wiley, New York, p. 160.

Sot. 48 Interface

Molecular 1954)