PHYSIC-iL .IDSORPTION.
IV
3 83
ON PHYSICAL ADSORPTION. IV'
-4 COMPARISON O F
TIYO
THEORIES OF
n1ULTILAYER A D S o R P T I O S
SI-DNEY ROSS2
Oak Ridge n'alional Laboratory, Oak Ridge, Tennessee
Received June 3, 1948
Since its publication ten years ago the multilayer adsorption t8heoryof Brunauer, Emmett, and Teller (3) has been the source of a prolific literature of both theoret,ical and practical importance. Although in its original and simpler form t,he irheory does not describe experimental adsorption isotherms above relative pressures of 0.3, yarious modifications have been suggested to enlarge its scope (1, 2, 4, 8, 10). I n general, the modifications have dealt either with the summation of the adsorption to a limited number of layers, or with tlie variation of the lieat of adsorption in the second and higher layers. In every case, it has been found necessary to add a third, or a third and a fourth, empirical constant to the t\vo constants of the original equation. Recently another treatment of the multilayer adsorption theory has been published by G. F. Huttig (B), which challenges an implicit assumption of t,he B.E.T. theory not heretofore recognized. A comparison of the two treatments brings into prominence some important theoretical considerations, which it is my present purpose to point out. The following brief derivation follows Huttig's procedure but uses the symbols of the R.E.T. theory which are now familiar to U. S. chemists. As in the B.E.T. theory, Huttig uses Langmuir's treatment of unimolecular adsorption applied to each layer. Let So, 81,82 Xi represent the surface area per gram of layers of adsorbed molecules, adsorbent that is covered by 0, 1, 2, * . . i, containing 0, 7i1, n2, . 'ai,. molecules, respectively. The symbol x represents the inazinzum number of molecules that is adsorbed on the first layer. e
so
= ao(x
SI =
sz
n1)
(1)
uo(n1 - nz)
(2)
- n3)
(3)
= uo(n2
Sa =
-
uO(?L~
- nc), etc.
where uo is the area per adsorbed molecule on a completely filled layer. The conditions for equilibrium are that on each layer the rate of condensation is equal l This paper is based on work performed under Contract No. W-7405 eng 26 for the Atomic Energy Project a t Oak Ridge Kationd Laboratory, Oak Ridge, Tennessee. The material was presented at the Symposium on Adsorption which was held by the Gordon Research Conferences of tlie American Association for the Advancement of Science a t Colby Junior College, New London, New Hampshire, on June 23, 1945. * Present address: Department of Chemistry, Rensselaer Polytechnic Institute, Troy, S e w Pork.
384
SYDNEY ROSS
to the rate of evaporation. In its most general form, for the first layer, this condition is expressed : a l p ( x - *nl) = bl(n1 - ynz)e-E"RT
(4)
where p is the pressure, E , is the heat of adsorption in the first layer, and al, bl, and y are constants. The right-hand side of this equation signifies that the rate of evaporation of molecules on the first layer is proportional to the total number of molecules on that layer minus a certain effect produced by those molecules that are covered by one or more higher layers. The B.E.T. theory assumes that this effect is powerful enough to prevent evaporation of first-layer molecules so covered (y = l), while the theory of G. F. Huttig uses the other extreme viewpoint,-namely, that molecules covered by second and higher layers still play t'heir r61e in evaporation as freely as if the higher layers were not present (7 = 0). That is the only point of difference of the two theories. All the other assumptions made in the B.E.T. development are followed by Huttig.3 Huttig's mechanism for equilibrium on each layer gives:
a d n l - ne) aap(n2
--
Assume E'z = E3 = Ei B.E.T. development. Let
bznne- E ? I R T
=
- n3) = b3n3e-E~'RT,etc. =
*
E,, and assume bz/a2 = b3/a3 *
y = (al/bl)peEl'RT
x
= (?J/g)e
and n, the total amount adsorbed, = nl Equations 5, 6, and 7 then become
n3
= z(n2
=
nl
Hence
n
((9
ELIRT
+ nz + - -
* *
=g, as in the
(8) (9)
- n3),etc.
+ + n3 + -
= Y(X
(7)
n2
- n1) + znl
(13)
a Up t o this point, Huttig's approach is identical with the much earlier paper of E. C. C. Baly (Proc. Roy. Soc. (London) A163, 495 (1937)). Huttig's original contribution is to combine the ideas of Baly with the simplifying assumptions of Brunauer, Emmett, and Teller. It is perhaps because of the disruption of the usual interchange of scientific journals between countries caused by the late war that Huttig is apparently unaware of these earlier workers, so well known in this country. At any rate, he makes no reference to them, although his development runs so closely parallel. The mathematical formalism in the present paper follows that of the B.E.T. theory and will be found to differ from that employed either by Baly or by Huttig.
PHYSICAL ADSORPTION.
IV
385
Combined with equation 1 0 :
For condensation and evaporation on a liquid surface a t its equilibrium vapor pressure po, equation 7 can be applied: apo4
=
be-E~'RT4
(15)
where 4 is the extent of the liquid surface. Combining equations 9 and 15 gives
x
=
P/Po
In the terminology of the B.E.T. theory (3):
Hence g = cx = cp/po
Equation 14 then becomes n - --
x
v =
vrn
(1-+ CPlPO CPlPO
)(l+$
This is an expression of Huttig's theory, using constants whose physical significance is the same as the constants found in the B.E.T. theory. For comparison the B.E.T. equation is given:
For the purpose of experimental testing of equations 18 and 19, they may be written respectively
and
A graphical plot of P (1+ p/po) against p is a straight line according to equa-
V tion 20, with a slope equal to 1 / V , and an intercept equal to po/cVm. According to equation 21, a plot of p / V ( p , - p ) against p / p o is a straight line with a slope equal to (c - l ) / c V r nand an intercept equal to l/cVT,. The experimental data used for testing and comparing equations 20 and 21 are the results of lowpressure ethane adsorption on sodium chloride samples a t the temperature of
38G
SYDKEY ROSS
liquid oxygen, using an apparatus similar to that described by Wooten and Brown (11). These isotherms have not been previously published. In addition, some
PIP.
FIG.1
FIG.2 FIG.1. B.E.T. equation for adsorption of ethane on crystals of sodium chloride at 9O0I
