Application of the BET equation to heterogeneous surfaces

Application of the BET Equation to Heterogeneous Surfaces L E O N M. D O R M A N T ~ AND A R T H U R W. A D A S i S O N

Department of Chemistry, University of Southern California, Los Angeles, California 90007 Received January 31, 1971; accepted March 10, 1971 It is shown that a previously published procedure for obtaining site energy distributions can be applied to data which include multilayer adsorption, with the BET equation used as the local isotherm function. When applied to the adsorption of argon on rutile, the resulting distribution function is essentially the same as that obtained by Drain and Morrison using a nearly thermodynamic procedure. The BET model can thus be applied to adsorption on heterogeneous surfaces. On the other hand, the BET c value obtained from a simple fitting of the equation to the adsorption isotherm reflects only the average energy of those sites being filled in the region of the fit, and is low. A better single-parameter description of the adsorbent surface is given by the average site energy as obtained from the actual distribution function. INTR.ODUCTION An important aspect of the B E T equation is that in order to arrive at a simple annlyrical expression its authors (1) assumed the surface to be homogeneous. The ability of the equation to fit data for heterogeneous adsorbents is a consequence of compensating variations of site adsorption energy and lateral interaction with surface coverage. It has also been pointed out t h a t heterogeneity has only a second-order effect on the value of the surface area of the solid as given by the v,~ parameter, since the fit to the equation is usually limited to the small multilayer region of the isotherm (2, 3). The degree of confidence in this v~ parameter is such that investigators proposing new methods of surface area determination usually take the B E T method as a standard. Such reliability is not found for the energy parameter c of the B E T equation. Here, poor agreement with thermodynamic heats is the rule. I t is the purpose of this paper to show how the B E T equation can be used to characterize the energetic heterogeneity of

an adsorbent and to provide a detailed example of the error incurred in accepting the c parameter as giving a correct average heat of adsorption. Some general considerations are the following. Statistical thermodynamic treatmeats of the extension of the B E T assumptions to heterogeneous surfaces (4:, 5) lead to the same result as obtained by taking as additive the amounts absorbed on sites or patches of various energies. Thus the total amount adsorbed is

V(x) = ~

[1]

= ~

f~x

(1 - x)(1 + c~x -

z)

We use f instead of the usual v~ to simplify

the notation; x has its, usual meaning

P/Po and c -- exp [(E -

of E~)/RT]. If the

vapor-solid interactions are given b y distribution function f(E), then

a

v(z) = f~

Present address: Aerospace Corp., Materials Science Division, P.O. Box 95985, Los Angeles, California 90045. C o p y r i g h t @ I972 b y Academic Press, Inc.

vi(x)

i

~

[2]

f(E)c(E) (1 -

x)(1 + c(E)x -

dE. z)

Journal of Colloid and Interface Science, Vol. 38, No. 1, J a n u a r y 1972

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DORMANT AND ADAMSON

The total area a is given by either ~ i f i or

I f ( E ) dE. We shall call E a site energy, meaning the energy of interaction of an adsorbate molecule with a particular type of adsorption site (note reference 7). Several attempts have been made to solve Eq. [1]. Walker and Zettlemoyer (6) noted that when only two different energy patches are assumed to be present, the equation then has four unknown parameters and four isotherm points suffice in principle for their determination. Some cMculationM difficulties were experienced, however, because the B E T equation is an ill-conditioned equation, 2 and these authors proposed an alternative procedure whereby it is assumed that one of the c values is 100. Tazikawa (5) developed another technique, based on the behavior of slopes and intercepts, and Steele (8) was able to determine up to four unknown fi's by placing the c,'s arbitrarily. ~ I t was at another occasion honoring Brunauer that it was shown that Eq. [2] can be solved b y a graphical iteration technique (9). This equation can be put in the more general form

V(P) =

f f(E)O(P,T, E)

dE.

[3]

The procedure has thus far been used only to treat adsorption in the submonolayer region, using the Langmuir, the FowlerGuggenheim, or the two-dimensionM van der Waals equation for the local isotherm function, O(P, T, E) (9, 10). Much use has been made of the latter function, incidentally, assuming f(E) to be Gaussian (11). We show here how the graphical solution method can be extended to include the B E T equation as the local isotherm function, and thus multilayer adsorption on heterogeneous surfaces. An important point is that once a choice is made as to 0 (P, T, E) no assumption is required as to f ( E ) - - t h e description Ill-conditioned equations are t r e a t e d in numerical analysis; for our purpose, the m a t h e m a t i cal n a t u r e of such equations is t h a t relatively large changes in the solutions m a y still give a good fit to the data. 3 Steele actually used the Langmuir equation b u t showed t h a t the B E T e q u a t i o n may be used as well. Journal of Colloid and Interface Science,

of this last is now implicit in the adsorption isotherm. Nor need any limits of integration be imposed; if there are no sites of a particular energy, f(E) as obtained from the solution to Eq. [3] will be zero for such energy. The range of experimental data required to describe a unique distribution has been discussed before (9), but not in terms of tile B E T equation. The usual region of fit of this last is 0.05 < x < 0.3, and it can be shown that the adsorption isotherm in this region contains little information about highenergy sites. For example, if N2 is used as the adsorbate at 77°K, many refractory solids exhibit isosteric heats in the region of 2-5 kcal mole -I. The c value corresponding to adsorption on a patch of 3 kcal mole -1 is 6.2 X 10-4 and the rate of change of dO/dx will be a maximum for such patches when x = 1/c or 1.6 X 10-5 • Adsorption data should be in this region of pressure to be sensitive to the presence of such sites. Alternatively stated, no discrimination between various high-energy sites can be made from data at pressures such that all are essentially occupied. Thus to obtain f(E) from the empirical application of Eq. [3] it is necessary to have adsorption data down to a surface coverage of about 0.01. Conversely, it should not be surprising that B E T c values are incapable of reflecting the distribution of high-energy sites and therefore cannot give reliable average site energies. A related consequence is that adsorption isotherms for which the B E T c value is over about 200 do not differ greatly in shape in the usual region of fit to the equation. A P P L I C A T I O N OF T H E B E T M O D E L TO OBTAIN S I T E E N E R G Y D I S T R I B U T I O N S

Drain and Morrison (12) obtained data for the adsorption of argon on rutile at 85°K which meet our requirements. The method of Adamson and Ling (9) needs only minor modifications to be used with the B E T equation. ~ The experimental isotherm is 4 The modificati.ons are as follows. (1) The isotheral is p l o t t e d a s V vs. log 1 / x . (2) Use as a first approximation to F vs. c a plot of V(1 = x ) vs. log 1 / x . (3) R e p l o t t i n g a s F v s . E (c = exp ( E / I ~ T ) ) will help to find points in the high-energy region.

¥ol. 88, No. 1, January 1972

287

APPLICATION OF BET EQUATION TO HETEROGENOUS SUI~FACES plotted as V vs. log 1/x in Fig. 1. T h e same scales are used for the integral distribution, F(E) vs. log c • E ) = dF(E)/dE), and the result of the first a n d the last (fourth) iteration are shown in the figure. As a cheek on t h e procedure, the solid line gives the adsorption isotherm calculated f r o m the distribution function; t h e m a x i m u m deviation f r o m the original is 0.8 %. T h e distribution function f(E) is o b t a i n e d b y differentiation. N o t e t h a t E is related to c b y c = e x p [ ( E -- E~)/RT], where E~ is the heat of vaporization, 1.55 kcal mole-L Figure 2 shows f(E). T h e m o n o l a y e r c a p a c i t y is given b y the area u n d e r f(E) b u t is f o u n d more accurately as the intercept of F(E) in the final iteration. This area, measured in cubic centimeters S T P , should agree with the B E T area f o u n d in the usual manner. T h u s we find 760 =i= 4 ec whereas the B E T plot yields 758

co from a least-squares fit to the data in the range 0.02 < x < 0.4. ~Iore important in the present context is the shape of f(E). The distribution function was also found for this system by Drain and Morrison (13). They bypassed the assumption of a local isotherm function by recognizing that at 0°I~2 there is no entropy contribution and that molecules fill sites in order of decreasing energy. Thus L i m d V / d q = f(E),

shapes of the two distribution functions are the same within the a c c u r a c y of either proeedure. T h e r e are two minor differences. T h e first is t h a t the use of the B E T equation provides a n a t u r a l closing point and allows f(E) to be obtained in the low-energy

1200- ~ ~ _

ARGONON RUTILE

IO00 800

\ o,~

m600 = 400

2O0

0

1

I

I

2

S

4

-Log x (Logc)

5

FIG. 1. Adsorption of argon on rutile at 85°I4 (data from reference 11). O : experimental points. O: first approximation to the integral site energy distribution, F. Dashed line: final approximation (fourth) to the integral site energy distribution. Solid line : adsorption isotherm calculated from the final site energy distribution.

T-~0

where q is a differential heat of adsorption. These authors were able to extrapolate their d a t a to close to 0 ° K b y m e a s u r e m e n t s of the h e a t capacity of the adsorbed gas. Their procedure does not separate adsorbateadsorbate interactions in the submonolayer region, b u t neither does the B E T equation so their a n d our f(E) represent the same kind of q u a n t i t y .

Examination

I E (CBET)

I E

/~ /-,,

12

,//\',,

--

o

i08 from \\ G OA]-~ ,'

of Fig. 2 show's that the

/1111 I [

(4) The procedure differs from the isotherm for submonolayer adsorption, where a master 0 vs. bP plot suffices, in that each 0 must now be calculated. The process may be expedited by preparing a table of 0 vs. x for various c values; only the F values change for each iteration. With some practice the procedure requires less than an hour per approximation. It may be partially or completely computerized.

1 E(c)

1,6

\ \\

"~"~'~

2.0

2.4 28 52 56 E (kcal/ mole) FIG. 2. Site energy distributions for argon rutile from data of Drain and Morrison (12, 13). Solid line: calculated from the 85°K isotherm. Dashed line: obtained by Drain and Morrison by extrapolation to 0°K; the light dashed line shows their assumed closure at low energies. Energy averages are defined in Table I.

Journal of Colloid and Interface Science, V o ] . 38, N o . 1, J a n u a r y

on

1972

288

DORMANT

AND ADAMSON

region. Drain and Morrison were obliged to estimate the dashed extension shown for their distribution. The second difference is that our distribution is displaced to lower energies by approximately 150 cal mole-L The most likely explanation for this shift lies in the arbitrary assumption of the B E T theory that the heat of adsorption in the second layer is E~. The two distributions are brought into coincidence either b y assuming that the heat of adsorption in the second layer is 150 cal mole -~ greater than E , or by taking the preexponential factor for c to be slightly greater than unity. The shift can thus be accounted for on relatively trivial grounds and the main point is that the two distributions are essentially identical. It thus appears that in this particular system the B E T model is very nearly correct; the experimental isotherm is obtained when the B E T equation is used in Eq. [3], plus the almost thermodynamic f(E) obtained b y Drain and Morrison. We expect such agreement to be fairly general, and likewise the conclusion that the Adamson and Ling procedure gives a fairly accurate f(E). Furthermore, any small displacements between f(E) so obtained and the true distribution function are not apt to be a source of difficulty when distributions are compared for various solids with the same adsorbate and the same temperature. As an excellent example of such an application, we cite the work of Whalen (14), who studied two forms of silica under different outgassing conditions. His work also shows the advantage of using nitrogen as the adsorbate. Nitrogen can differentiate between sites the heterogeneity of which is electrical in nature, presumably because of its high quadrupole moment.

TABLE

I

~ALCULATED AVERAGE PARAMETERS Average parameters Quantity calculated and definition a

Energy (kcal mole-1)

e

E(c) using c value

2.23

95

from the BET region of fit (85°K) E(c) same as above but from isotherm calculated at 75°K

2.32

177

2.50

E(c)

f r o m (c} at 8 5 ° K

2.98 3.02 3.34 2.55 3.42 1.30 b

at 75°K /~ from (Ec)/(c} ~ from (E2}/(E} E(bc) from (c2}/(c} E~g~ from A/(f)

ha = (E~)/(E2}

=

277 4643 19400 40600 369 6 . 5 X 10 ~

1.02

= 1.08 a ( ) s t a n d s f o r n o r m a l i z e d a v e r a g e ; t h u s (Z) =

ff(E)ZdE/ff(E)dE. b E n e r g y r a n g e if

f(E)

is c o n s t a n t at i t s a v e r a g e

value.

its value is 2.32 kcal mole -1 at 75°K, for example. The distribution function yields several types of average or characteristic site enerties. A rational one is the site average energy, E = (l/a)ff(E)E dE, which is 2.50 kcal mole-l; this value is, of course, temperature independent. Alternatively, the energy corresponding to 5 can be obtained from ( l / a ) f f(E)c(E) dE; this gives E(~) = 2.93 kcal mole -1 at 85°K and 2.98 kcai mole -~ at 75°K. The average heat of adsorption is given by ~ E i V i / ~ Vi or in the limit of x --* 0, b y 1/c f f(E) (Ec) dE/C; this average, ~'c, is 4.00 kcai mole-L Finally, the most probable site energy Ep is 2.4 kcal mole-L The physical explanation for the above AVERAGE SITE ENERGIES differences is easy to find. E(c) is low beThe above example shows that the B E T cause it represents an average of the energies model can yield an essentially correct f(E). of those sites being filled in the region of fit-It remains to consider the relation between these occupy the low-energy portion of the the average E as given b y the c parameter- site energy distribution, their range varying_ and averages obtained from the actual with temperature. E(5) is higher than E distribution. Table I summarizes several because the exponential dependence of c such quantities. First, E(c) as given by a fit on E gives added weight to high-energy sites; to the isotherm at 85°K is 2.23 kcal mole-k the effect decreases with increasing temperaThis quantity is temperature dependent; ture. The limiting calorimetric heat of adJournal of Colloid and Interface Science, Vol. 38, No. 1, January 1972

APPLICATION OF BET EQUATION TO HETEROGENOUS SURFACES sorption E~ is high and t e m p e r a t u r e dependent, for the same type of reason. There is a need for a single-parameter description of the heat of adsorption on a heterogeneous surface. The B E T c value, although useful in specifying the adsorption isotherm in the region of fit, is inaccurate if not actually misleading with respect to the overall nature of the adsorbent-adsorbate system. T h e p a r a m e t e r should also be t e m p e r a t u r e independent if its physical meaning is to be simple. We suggest therefore t h a t / ~ be used whenever possible. Some indication of the spread of the distribution can be given b y a second parameter. Three possibilities are evaluated in Table I; /~range has perhaps the simplest physical interpretation, being the width of a rectangular distribution function centered at J~. ACKNOWLEDGMENT This investigation was supported in part by contract AF-AFOSR-1097-66 between the University of Southern California and the Air Force Office of Scientific Research. REFERENCES 1. BRUNAUER, S., EMMETT, P. H., AND TELLER,

J., J. Amer. Chem. Soc., 60,309 (1938).

289

2. (a) BRUNAUER, S., COPELAND, L. E., AND

KANmRO, D. L., in E. A. FLOOD, Ed. "The Solid Gas Interface," Chap. 3. Marcel Dekker, New York, 1967. (b) KANTRO, D. L., ]~RUNAUER, S., AND COPELAND, L. E., ibid.,

chap. 12. 3. ADAMSON,A. W., "Physical Chemistry of Surfaces." Interscience, New York, 1967. 4. JoY, A. S., Proc. Intern. Contr. Surface Activity, 2nd London 1957 II, 54 (1957). 5. TAZlKAWA,A., Kolloid Z., 222, 141 (1966). 6. WALKER, W. C., AND ZETTLEMOYER, A. C.,

J. Phys. Chem., 52, 47, 58 (1948). 7. BRUNAt~ER, S., Proc. Intern. Contr. Pure Appl. Chem. lOth Moscow 1965, p. 293. 8. STEELE,W. A., J. Phys. Chem. 61, 1551 (1957). 9. ADAMSON,A. W., AND LING, I., Advan. Chem. 33, 51 (1961). 10. ADAMSON, A. W., LING, I., DORMANT, L. M., AND OREM, M., J . Colloid Interface Sei., 21,

445 (1966). 11. Ross, S., AND OLIVIER, J. P., "On Physical Adsorption," Interscience, 1964. 12. DRAIN, L. E., AND MORRISON, J. A., Trans.

Faraday Soc., 48, 840 (1952). 13. DRAIN, L. E., AND ~/IORRISON, J. A., Trans. Faraday Soc. 48,316 (1952). 14. Wm~LEN, J. W., J. Phys. Chem., 71, 1557 (1967).

Journal of Colloidand Interface Science, Vol. 38, No. 1, January 1972