Adsorption on nonporous solids

Adsorption on Nonporous Solids STEPHEN

BRUNAUER,

J A N S K A L N ¥ , A~D E. E. B O D O R

Institute of Colloid and Surface Science and Department of Chemistry, Clarkson College of Technology, Potsdam, New York 13676

Received January 24, 1969 It is a well-known fact that the two-parameter BET equation gives too high adsorption values at x (or P/Po) > 0.35. The reason for this is the assumption that at x = 1 an infinite number of layers are adsorbed. Experimental data for nonporous adsorbents show that the number of adsorbed layers close to x = 1 is far from infinite; in fact, it is in the vicinity of 5 or 6. A modified BET equation is derived, employing a third parameter/¢, which is a measure of the attractive force field of the adsorbent. The equation is identical with one derived by Anderson in 1946, but the model on which it is based, the derivation, and the method of application are different. Examples of the application of the modified BET equation are presented, and certain conclusions are drawn about the BET parameters vm and c. INTRODUCTION

b u t the condensation does not occur in e m p t y pores; the pores have an adsorbed film on their walls prior to condensation. I t was assumed t h a t the thickness of the adsorbed film at a n y value of x was equal to the thickness of the film adsorbed on nonporous solids, and it was believed t h a t in the capillary condensation region, at x > 0.4, the nature of the adsorbent h a d little or no influence on the thickness of the adsorbed film. Composite curves for a v a r i e t y of adsorbents for nitrogen at 77.3°K were published b y Shull (2), Cranston and I n k l e y (3), Pierce (4), and Lippens, Linsen, and de Boer (5). Some of these investigators believe t h a t there is a "universal" nitrogen adsorption isotherm for nonporous solids, b u t the four composite curves are significantly different. For example, at x = 0.4, C r a n s t o n and I n k l e y ' s curve indicates a 15 % thicker film t h a n Shull's. This makes a considerable difference in pore structure analysis. T h e importance of selecting the proper t-curve (statistical thickness vs. relative pressure) was emphasized in an earlier paper (6). Journal of Colloidand InterfaceScience,¥ol. 30, No. 4, August 1969

I n the B E T paper (1), a t w o - p a r a m e t e r equation is derived for adsorption of vapors on free surfaces, i.e., on surfaces of nonporous solids, and a three-parameter equation for adsorption in a parallel-plate pore. T h e former is customarily called the B E T equation, and this designation will be used in the present paper, According to the B E T equation, when the equilibrium pressure p of the adsorption is equal to p0, the v a p o r pressure of the liquid adsorbate, an infinite n u m b e r of layers are adsorbed on a nonporous adsorbent. T h e relative pressure P/po is designated with the letter x, and the theoretical isotherm asymptotically approaches a vertical d r a w n at x = 1. D u r i n g the last two decades, a large n u m ber of adsorption isotherms of nitrogen were obtained for nonporous solids. T h e interest arose f r o m the fact t h a t pore structure analysis b y means of nitrogen adsorption or desorption isotherms has necessitated the knowledge of the thickness of the adsorbed film on nonporous solids. T h e analyses are based on capillary condensation in the pores,

546

ADSORPTION

ON NONPOROUS

Even though the four t-curves for nitrogen differ from each other, they agree in one important respect: the number of adsorbed layers very close to p0 is far from infinite. I t is difficult to obtain accurate adsorption data close to p0 ; nevertheless it is clear from the curves that only five or six layers of nitrogen are adsorbed when p0 is reached. In a recent paper it is shown that the number of layers of water adsorbed on nonporous solids close to p0 is also about six (7). These facts have led to the theoretical considerations and results reported in this paper. THE BET EQUATION

1. The derivation of the B E T equation will not be reproduced here. The terms used below are the same as in the original derivation. Two assumptions were made: (1) the heat of adsorption in the second and higher layers is equal to the heat of liquefaction, E2 = Es . . . .

E~ = EL

[11

and (2) b2

bs

bi

. . . . . . . . . a2 as ai

g = a

constant. [2]

The term x was defined by the equation =

(p/g)e ~L1~.

[3]

The B E T equation is v

v~

_

cz

[4]

(1 -- x)(1 - - x - I - c x ) "

After deriving Eq. [6], the authors stated that on a free surfaee at po an infinite number of layers can build up on the adsorbent. Tt~is means that the value of z must be 1 at p0, or

1 = ( p o / g ) e ~LjRr.

[5]

Dividing Eq. [3] by Eq. [5], one obtains x = p/Po.

[6]

Although it is true that at P0 an infinite number of l i q u i d layers can build up on the adsorbent, it was shown above that an infinite number of a d s o r b e d layers does not

SOLIDS

547

build up when p0 is reached. B E T assumed that liquid and adsorbed layers are indistinguishable at this point, but this is not necessarily true, as was pointed out by de Boer (8). He further pointed out that an adsorption isotherm may approach asymptotically a vertical drawn at a pressure smaller or greater than p0. In the latter ease, the isotherm will cut the vertical drawn at p0 at a finite number of adsorbed layers. The physical meaning of de Boer's virtual asymptote at p > p0, as interpreted by the present authors, is the following. When at p0 the adsorbate liquefies in bulk inside the vessel containing the adsorbent, further adsorption may take place. Some of the liquid will be under the influence of the force field of the adsorbent, and will be distinguishable from the bulk liquid in binding energy or entropy or both. The number of such layers (adsorbed in addition to those adsorbed below p0) may possibly be large. Derjaguin demonstrated experimentally that some of the properties of water in the vicinity of a surface may differ from those of bulk water to distances amounting to hundreds of layers (9). From the point of view of the present discussion the number of layers differing from the bulk liquid is not important; it is important only to realize that additional adsorption occurs after p0 is reached. After the adsorption of a number of additional layers, the bulk liquid and the last adsorbed layer become indistinguishable, and one may then say that there are an infinite number of adsorbed layers on the adsorbent. This model leads to an isotherm which intersects the vertical drawn at p0 at a finite and possibly small number of adsorbed layers, instead of the asymptotic approach of the B E T model. Nevertheless, the total number of adsorbed layers remains infinite, just as in the B E T model; consequently, the summation in the derivation that will follow can be carried to infinity as in that model. The explanation of de Boer for the virtual asymptote at p > p0 was different, tie asJournal of Colloid and Interface Science, Vol. 30, No. 4, August 1969

548

BRUNAUER, SKALNY, AND BODOR

sumed that the heat of adsorption in the second and higher adsorbed layers was smaller than the heat of liquefaction. However, all isosterie and calorimetric heats of adsorption obtained for nitrogen indicate that the heats of adsorption in the second and higher adsorbed layers are equal to or greater than the heat of liquefaction. Although the considerations advanced in the present paper are believed to be valid for most adsorbates, the experimental data to be cited are taken from nitrogen isotherms measured at 77.3°K. 2. For many nitrogen isotherms, the value of the parameter c in Eq. [4] is in the vicinity of 100. The number of adsorbed layers calculated from Eq. [4] at p / p o = 0.9 is 10, and at p / p o = 0.99 is 100. At p / p o = 0.4, the statistical number of adsorbed layers is 1.64. One of the best known facts in surface chemistry is the fact that the BET equation gives too high adsorptions at p / p o > 0.35, and frequently even at lower relative pressures. The assumption shown in Eq. [1] should result in too low adsorption values, because the experimental heats of adsorption for nitrogen are higher than E~. Halsey, therefore concluded that the entropy term in the BET equation was much too high (10). This belief is shared by most investigators. The belief was bolstered by the erroneous pictures published by numerous authors on the BET model of adsorption. In each subsequent publication, the picture of the model became more erroneous. In a book published in 1968, the model is represented by nine adsorbed molecules in the first layer, with empty spaces for five additional molecules; there are six adsorbed molecules in the second layer, four in the third, and two in the fourth. The picture creates the impression that the model is based on absurdly high entropies or, briefly, that the model is absurd. When there are nine adsorbed molecules in the first layer and five empty spaces, the surface coverage is about two-thirds. With c = 100, Eq. [4] indicates that such surface coverage is reached at about x = 0.02. Journal of Colloid and Interface Science, VoL 30, No. 4, August 1969

According to the derivation of Eq. [4], s2 = x s l ,

[7]

which means that the number of molecules in the second layer is 0.02 times the number in the first layer; in other words, there should be no molecules in the second and higher layers. The value of c is seldom less than 40 for nitrogen, but even if a value of 10 is assumed, the number of molecules in the second layer should be only 2, and none in the higher layers. With c = 1, which leads to a Type III isotherm, rarely obtained and never for nitrogen, the number of molecules in the second layer should be 3.6, in the third layer 1.4, and in the fourth layer 0.6. 3. The c constant of the BET equation is given by c

--

al 52

~ a2 e

(E1--EL)/RT

[81

Cassie (11) and Hill (12) derived the BET equation by statistical mechanics, and their derivation shows that the term alb2/bla2 is a function of the entropy of adsorption. Equation [4] can be rearranged to a linear form, and many thousands of linear plots were obtained by investigators in the "BET region," which extends from about x = 0.05 to 0.35, or from v / v ~ values ranging from about 0.5 to 1.5. Such linear plots can be obtained only if c is actually constant in the BET region. Because BET assumed too low heat of adsorption for the second layer, c can be constant only if too high entropy compensates for it. It was not intended to deny in the previous section that the BET model contains too high entropy in the c parameter. The intention was to deny that the model leads to absurdly high entropy values. The entropy term, as will be seen, is not even high enough to compensate for the low enthalpy term in the BET region. Furthermore, the entropy term has nothing to do with the high adsorptions at x > 0.35. With a c constant of 40, the statistical number of adsorbed layers at x = 0.4 according to

ADSORPTION ON NONPOROUS SOLIDS Eq. [4] is 1.61; with c = 4000, it is 1.67. With c = 1, the number of adsorbed layers at x = 0.90 is 9.9, and at x = 0.99 it is 99. The cause of the high adsorptions beyond the B E T region is the third, unannounced, assumption given in Eq. [5]. The value of x when p0 is reached is not 1, as assumed in the B E T model, but less than 1. A value of x < 1 leads to a finite number of adsorbed layers at p0 • MODIFICATION OF THE BET THEORY 1. When a liquid is in equilibrium with its vapor, the rate of condensation is equal to the rate of evaporation, or [9]

aLpo = bLe-~LIRr.

In Eq. [5], the constant g is equal to b J a L . The "adsorption coefficient" c of the B E T equation is equal to bp6, where b is the adsorption coefficient of the Langmuir equation, which characterizes the adsorption in the first layer. The B E T equation can be derived on the basis of the assumptions E2 = E3 . . . .

E~ = E',

[10]

where E ~ is not equal to E L , and b2

b~

b~

a2

a~

al

,

where g' is not equal to bL/aL. The summation can still be carried out to infinity on the basis of the model described earlier; however, an infinite number of adsorbed layers is not reached when p0 is reached. In other words, at a pressure less than p0 by an infinitesimal amount, the number of adsorbed layers will not be close to infinity but will be finite, possibly even a small number. In the terminology of de Boer, the adsorption approaches asymptotically a vertical drawn at a virtual pressure greater than p0. Equation [3] is replaced by /RT,

[11]

]c = (po/ g' )e ~" /'r,

[12]

l

x = (p/g')e ~

and Eq. [5] is replaced by

549

where k is a number smaller than 1. Dividing Eq. [11] by Eq. [12], one obtains [131

x' = k ( p / p o ) = kx.

The modified B E T equation is v

v~

-

ckx

(1 - kx)(1 + ( c -

1)kx)'

[14]

or in linear form kx v(1-1~z5

-

1 vmc

+

c -- 1 v~c

]~z

[15]

An equation identical in form with Eq. [14] or [15] was derived by Anderson (13). His reasoning was different; he assumed, like de Boer, that the heat of adsorption in the second and higher layers was smaller than the heat of liquefaction, but he pointed out also that the same equation could be derived by assuming smaller entropy than that assumed by B E T . With different values of k, he was able to obtain good fits for a number of isotherms up to x = 0.6 to 0.8. Anderson assumed that his equation breaks down in the vicinity of P0, and the B E T equation becomes valid. This would mean that for nitrogen the number of adsorbed layers at x = 0.986 and 0.970 should be 71.5 and 33.3, respectively. One of the present authors (E. E. Bodor) has obtained for Degussa Alumina C a nitrogen adsorptiondesorption isotherm at 77.3°K that was completely reversible to p0. An adsorption point at x = 0.986 and a desorption point at 0.970 indicated 11.6 and 9.3 adsorbed layers. These values are too high because of interparticle condensation. A Lippens-de Boer vz-t plot (14) began to deviate upward at x = 0.90, clearly showing interparticle condensation. The plot showed also that the alumina was nonporous, and St was in good agreement with S z B r , 102 and 105 m2/gm, respectively. Unless one uses larger particles, one cannot obtain adsorption data free of interparticle condensation at x > 0.90. 2. The composite curves, or t-curves, of Shull (2), Cranston and Inkley (3), and Pierce (4) indicate that the statistical humJournal of Colloid and Interface Science, Vol. 30, No. 4,'August 1969

550

\

BRUNAUER, SKALNY, AND BODOlZ

2

5 i

oOf/

°?

/

0.5 0.4

I

0.3 0.2

03 I 0,1

1 0.2

I 0.5

I 0.4

I 0.5

I 0.6

I 0.7

I 0.8

I 0.9

×

0.1

0.2

0.5

0.4

0.5

0.6

0.7

kx

FIG. 1. Shull's composite nitrogen isotherm fitted by the modified BET equation, with k = 0.79. (a) The isotherm at 77.3°K. (b) The modified BET plot. The solid curves ~re theoretical; the points are experimental. ber of adsorbed nitrogen layers at x = 0.90 is 3.50, 3.47, and 3.33, respectively. The average of these values is 3.43. ( T h e statistical number of adsorbed layers on Degussa Alumina C at x = 0.90 was 3.53.) If a c constant of 100 is assumed, a k value of 0.79 in Eq. [14] gives v/v~ = 3.44 at x = 0.90. As was shown earlier, the value of c makes no significant difference at x > 0.4. The adsorption at higher relative pressures is determined b y k. With c = 100 and k = 0.79, Eq. [14] gives the nitrogen adsorption isotherm at 77.3°K shown b y the solid curve in Fig. la. The ordinate is the statistical number of adsorbed layers, v / v ~ . The experimental points represent Shull's composite curve; the d a t a were taken from a paper of Pierce (15). I n the entire pressure range from x = 0.1 to 0.9, the largest deviation of an experimental point from the curve is 2.3 %. The modified B E T plot is shown in Fig. lb. The left-hand side of Eq. [15] was plotted against kx. Since v,~ = 1 and c --- 100, the intercept of Journal of Colloid and Interface Science, Vol. 30, No. 4, August 1969

the straight line is 0.01, and the slope is 0.99. The experimental points were calculated from Shull's data. The modified B E T plot extends through the entire pressure range. The adsorption at p0, or kx = 0.79, is 4.75 layers. The t-curve of Lippens, Linsen, and de Boer (5) indicates greater adsorptions at x > 0.65 than the three other composite curves. The highest reliable point is at x = 0.7; beyond this point probably interparticle condensation occurred. The number of adsorbed layers at x = 0.7 is 2.42, which is 11% higher than Shull's value, but only 2.5% higher t h a n t h a t of Cranston and Inkley. A k value of 0.84, with c = 100, gives 2.42 adsorbed layers at x = 0.7. Equation [14], with these values of/c and c, gives a fairly good fit for the composite curve of Lippens, Linsen, and de Boer, but not as good as the fit shown in Fig. 1 for Shull's curve. T h e number of adsorbed layers at p0 is 6.24. T h e constant k measures the strength of

ADSORPTION ON NONPOROUS SOLIDS the attractive force field of the adsorbent. With k = 0.79, the number of adsorbed layers when p0 is reached is less than five; with k = 0.84, it is more than six. Anderson (13) used k values between 0.57 and 0.715 to fit nitrogen isotherms of various adsorbents, b u t his curves at x > 0.6 to 0.8 fell below the experimental points, whereas the evaluation of the above k values was made from the highest reliable adsorption point of each isotherm. The strength of the force field, or potential field, doubtless varies from adsorbent to adsorbent, but the variation is not very great. With the use of 3.56 A for the thickness of an adsorbed layer of nitrogen, the statistical number of adsorbed layers indicates t h a t the thickness of the adsorption space measured at p0 is 17 A with k = 0.79, and 22 A with k = 0.84. Some adsorbents m a y have a thinner adsorption space than 17 A, but the 22 A thickness m a y be too high. This would be the case if the v/v~ value of Lippens, Linsen, and de Boer at x = 0.7 was too high because of some interparticle condensation. A recent p a p e r on t-curves for water vapor (7) indicates a thickness of 18 A at 25°C. 3. Surface area determinations were made of four nonporous adsorbents: high-surface silica and manganese dioxide, and low-surface titanium dioxide and zirconium silicate. Table I shows the parameters v~ and c evaluated fl'om the regular B E T plot (k = 1), and from the modified B E T plots (k = 0.84 and 0.79). The v~ values are given in cubic centimeters at S.T.P. per g r a m of adsorbent. The B E T surface areas are 165.3, 127.5, 2.34, and 1.23 m2/gm, respectively. As was found also b y Anderson (13), the v~ values increase as k decreases. T h e differences in Table I range from 4.2 % to 6.9 % higher t h a n the B E T values; the average is 5.1%. Although the new v,, values are believed to be correct, the surface areas calculated from the old vm values remain correct. The B E T areas obtained from the old vm values in conjunction with a nitrogen area of 16.2 A 2 have received so m a n y independent

551

TABLE I BET AND MODIFIED BET PARAMETERS SiO2

MnO2

TiO:

ZrSiO4

k v,~ c

1.0 38.0 105

1.0 29.3 68

1.0 0.54 116

1.0 0.28 93

k vm c

0.84 39.6 101

0.84 30.7 72

0.84 0.57 95

0.84 0.30 112

k vr,~ c

0.79 39.9 104

0.79 30.8 79

0.79 0.58 116

0.79 0.30 111

confirmations t h a t there is no reason to doubt their correctness. The new v~ values would give the same answer, if a 5 % smaller area were used for a nitrogen molecule. If, for example, Harkins and Jura in their absolute method (16) had used the modified B E T equation to obtain vm for nitrogen, they would have obtained a nitrogen area of 15.4 A 2. This is exactly the value calculated b y Livingston from a two-dimensional van der Waals equation (17). The critical temperature of nitrogen is 126.1°K; the twodimensional theoretical critical temperature is 63°K; at 77.3°K, therefore, the nitrogen should not be a two-dimensional liquid. According to the above results, nitrogen is a two-dimensional compressed gas, with a 5 % less dense packing on the surface t h a n the packing of the liquid. As was shown above, the B E T areas obtained in the past are correct. I t would make no sense to a t t e m p t to evaluate v~ in the future from a modified B E T equation, and then use 15.4 A 2 for the area of a nitrogen molecule. The result would be the same as using the B E T v~ with the old area. The significance of the previous p a r a g r a p h is theoretical rather t h a n practical. 4. As Table I shows, the modified B E T c values show no definite trend compared to the B E T c values. This was found by Anderson also (13). Actually the new c Journal of Colloid and Interface Science, Vol. 30, No. 4, August 1969

552

BRUNAUER, SKALNY, AND BODOR

values differ only slightly from the old ones. The average of the twelve c values in Table I is 98. The c constant in the modified B E T equation is

ACKNOWLEDGMENT The authors gratefully acknowledge their indebtedness to the National Science Foundation for Grant GP-7746, which supported the work presented iE the paper.

!

al g e(El-~,)/Rr. c - - bl

REFERENCES [16]

I n the B E T equation the exponent contains E1 -- E L , the net heat of adsorption. With the questionable assumption t h a t al g/bl is approximately 1, E1 -- EL could be calculated from c, and it was invariably found to be lower t h a n the value determined by heats of immersion experiments. Nevertheless, the low values could be adequately explained (18). I n the modified B E T equation, the relation between the energy t e r m in the exponent and heats of immersion data disappears. Since E ! is greater than E L , the heat of immersion should be greater than E1 - E'. The theoretical curves for the composite curves of Shull and Lippens, Linsen, and de Boer fell below the experimental values in the B E T region, b y about 2% for the former and about 5 % for the latter. This shows t h a t the compensation for the low energy of adsorption b y a high entropy t e r m in the c constant was not fully achieved. A p a r t of the compensation in the B E T theory was caused b y the extremely strong force field attributed to the adsorbent, b y making ]~ = 1. Although the effect of k decreases with decreasing x, it has an appreciable effect even in the B E T region.

Journal of Colloid and Interface Science, Vol. 30, No. 4, August 1969

1. BRUNAUER, S., EMMETT, P. H., AND TELLER, E., Y. Am. Chem. Soc. 60, 309 (1938). 2. SHv~,L, C. G., J. Am. Chem. Soc. 70, 1405

(1948). 3. CRANSTON, R. W., AND INKLEY, F. AI, Advan. Catalysis 9, 143 (1957). 4. PIERCE, C., Or. Phys. Chem. 63, 1076 (1959). 5. LIPPENS, B. C., LINSEN, B. G., AND DE BOER, J. It., J. Catalysis 3, 32 (1964). 6. MIKHAIL, R.SH., BB,UNAUER, S., AND BODOR, E. E., J. Colloid and Interface Sci. 26, 45

(1968). 7. HAGYMASSY, J.

8. 9. 10. 11. 12. 13. 14. 15. 16.

JR., BRUNAUER, S., AND MI~:HAIL, R.S~I., J. Colloid and Interface Sci. 29, 485 (1969). DE BOER, J. H., "The DynamicM Character of Adsorption." Clarendon Press, Oxford. 1st ed., 1953; 2nd ed., 1968. DERJ~GVlN, B. V., ed. "Research in Surface Forces" (translated from Russian). Consultants Bureau, New York, 1963. HALSEY,G. D., Advan. Catalysis 4, 259 (1952). CASSlE, A. B. D., Trans. Faraday Soc. 41, 450 (1945). HILL, T. L., J. Chem. Phys. 14, 263 (1946). ANDERSON, R. B., J. Am. Chem. Soc. 68, 686 (1946). LIPt'ENS, B. C., AND DE BOER, J. It., J. Catalysis 4,319 (1965). PIERCE, C., J. Phys. Chem. 72, 3673 (1968). HARI(INS,W. D., AND JVRA, G., J. Am. Chem. Soc. 66, 1362, 1366 (1944).

17. LIVINGSTON, ~-L K . , J. Colloid Sci. 4, 447 (1949). 18. ]~RUNAUER,S., COPELAND, L. E., AND KANTRO,

D. L., "The Langmuir and BET Theories," Chap. 3 in E. Alison Flood, ed., "The SolidG a s Interface," Vol. 1. Marcel Dekker, New York, 1967.