The analysis of gas adsorption data to determine pore structure

Surface Technology, 6 (1978) 231 - 258 © Elsevier Sequoia S.A., Lausanne -- Printed in the Netherlands

231

Review paper

THE ANALYSIS OF GAS ADSORPTION DATA TO DETERMINE PORE STRUCTURE

D. DOLLIMORE and G. R. HEAL Department of Chemistry and Applied Chemistry, University of Salford, Salford M5 4WT (Gt. Britain) (Received March 2, 1977)

Summary T h e use o f gas a d s o r p t i o n d a t a in t h e d e t e r m i n a t i o n o f p o r e size a n d p o r e size d i s t r i b u t i o n is reviewed. T h e use o f t h e Kelvin and D u b i n i n equat i o n s is o u t l i n e d . Cylindrical, slit-shaped and p a c k e d s p h e r e m o d e l s are e x a m i n e d . O t h e r t e c h n i q u e s m e n t i o n e d include m o l e c u l a r p r o b e a n d t h e m i c r o p o r e analysis m e t h o d s . T h e d i f f i c u l t y o f d e t e r m i n i n g surface areas and p o r e sizes w h e n t h e p o r e s are o f m o l e c u l a r d i m e n s i o n s is e x p l a i n e d .

Contents Introduction The restrictionof adsorption by pores The capillary condensation theory The pore size distribution calculation Methods of determination of t values Other pore models The packed sphere model The exclusion of adsorbate molecules Microporosity The M P method The Dubinin equation

Introduction In a previous review the application of the Brunauer-Emmett-Teller (BET) equation to adsorption data was considered [ 1] with special reference

232 to the calculations of specific surface area. The imposition of a porous structure can modify the adsorption isotherm and cause a departure from BET behaviour. However, in treatments designed to explain the relationship between the amount adsorbed and the equilibrium pressure, the BET theory still finds a use. The various treatments and their relationship to the BET equation are explained in this review. Whereas in the application of the BET equation to a non-porous surface the emphasis was on the calculation of the specific surface area and a factor related nominally at least to the heat of adsorption, in its use in studying porous adsorbents there is an additional factor to be derived concerning the nature of the porosity. Isotherms showing type Z behaviour (see previous review [1] ) in the higher pressure region (types IV and V on the BET classification) are obtained with samples that contain mesopores or macropores. The BET analysis m a y be applied to give linear plots for both types of low-pressure behaviour (types A and B (see previous review) or types II and III of BET analysis [1] ). In the case of type A, however, micropores may be present and the Langmuir [2] or other equations such as the Dubinin equation [3] may also apply. The BET equation applies despite pores being present when the pores are wide enough for condensation to take place at pressures above the BET range, and also are large enough for the multilayers to build up normally. In this review the same nomenclature for pore sizes is used as in the earlier review [ 1 ], namely that used by Dubinin [ 4].

The restriction of adsorption by pores For smaller pores the BET adsorption isotherm may be modified to deal with a porous structure by limiting the total adsorption volume uptake to a finite value. This is arrived at by the simple assumption that on a concave surface the number of multilayers is terminated at a finite layer thickness. This simple modification to the BET equation derivation leads to the relationship [ 5] V Vm

_ C(P/Po) 1 -- P/Po

1 -- (n +

1) ( P / P o ) n + n ( P / P o ) " + 1

(1)

1 + (c -- 1) (P/Po) -- C(P/Po) n + 1

where n is the number of adsorbed layers at saturation. The presence of pores also causes a variety of effects in addition to the limited adsorption predicted by the modified BET theory. So far, there have been few attempts to adapt the BET model to describe adsorption behaviour for simple curved geometries. This could follow from the introduction of a new term to take account of changes in surface free energy with changes in coverage on a curved surface. Attempts have been made to check the theory relating to the limiting value of surface coverage arising from adsorption proceeding independently on each of two planar walls [6]. There is also a difficulty in that the projection of this modified equation to n = oo provides an anomalous estimate of the fraction of sites occupied at saturation on a free surface.

233 The BET surface area calculation is also used in extracting details of the porosity of an adsorbent in basically simple geometric terms leading to the calculation of the mean pore radius, the pore size distribution and the total volume of the pores from the adsorption isotherm [7]. Assuming a cylindrical pore shape, it can be shown that if the a m o u n t adsorbed is expressed as a volume of condensed liquid, then

VJS

=

~-/2

(2)

where S is the specific surface area, Vs the volume of condensed liquid calculated from the a m o u n t adsorbed near the saturated vapour pressure (say P/Po ~ 0.95) and ~- the mean pore radius of the pores. The method depends heavily on the calculation of the BET surface area. The total pore volume can alternatively be determined by a comparison of the density of the solid in helium (assumed to penetrate all pores) and that measured by displacement of mercury (assumed to only penetrate pores with a radius greater than 7/~m at atmospheric pressure) [8].

The capillary condensation theory A pore size distribution for the adsorbent can be computed by the application of the Kelvin equation [ 9], which relates the pressure at which condensation or evaporation occurs on a curved surface to the radius of curvature of the surface. Any adsorbate gas may be used, but in practice nitrogen at 77 K is most c o m m o n l y employed (sometimes water at 298.2 K). In the equations which follow, dealing with the use of the capillary condensation theory, any numerical relationships introduced deal with the example of nitrogen adsorption isotherms at 77 K and might need modification when applied to other adsorption isotherms. The Kelvin equation takes the form P -- 27 Vcos 0 tn - - -

Po

rRT

(3)

where 7 is the surface tension of the adsorbate, V the molar volume of the adsorbate (considered as a liquid), 0 the angle of contact between the adsorbate liquid and solid, R the gas constant, T the absolute temperature of adsorption, Po the saturated vapour pressure of the adsorbate at T and P the pressure at which evaporation occurs from a meniscus of radius r. The Kelvin equation was probably first used in this way by Zsigmondy [10], who explained the retention of water by silica gels at reduced water vapour pressures by capillary vapour pressure depression on narrow pores. Huckel [ 11] provides a rigorous derivation of the Kelvin equation which leads to the equation

RTInP°p = 7V(~-ll + -~2) + V(Po--P)

(4a)

234 where R1 and R2 are the principal radii of curvature at right angles. If the second term on the right-hand side of the equation is neglected and the contact angle 0 substituted into the equation it produces the equation RTIn P° -- ~,Vcos 0 P

R-1

+ -R2

(4b)

If R1 and R2 are identical then this is the same equation as that already introduced as the Kelvin equation. Allowance should be made for an adsorbed layer of thickness t, so the Kelvin equation correctly applied is 27 Vcos 0 In Pp o _ R T ( r -- t)

(5)

An alternative equation was proposed by Deryagin [ 12] using the concept of disjoining pressure, which allows for the influence of sorption forces as well as meniscus curvature on capillary evaporation. His equation was

d

_

27V + 2V f PaN s R T l n (Po/Pd) R T l n (Po/Pd) pJ ~- RTdln P

(6)

where d is the pore diameter, Pa the pressure of desorption, Pa the pressure of adsorption and Ns/S denotes the number of moles adsorbed per unit surface area on a completely flat surface o f the same material (evaluated from the t curve), but it has not achieved much prominence. Kiselev [ 13] has proposed that at the pressure that capilla~, evaporation takes place, the change in thermodynamic potential on evaporation of the capillary condensed phase is equal to the change in free energy of the system due to the creation of the liquid vapour interface, i.e. ~dS = A/~da

(7a)

where dS is the magnitude of the created surface, da the number of moles desorbed from the pore and Ap the difference in thermodynamic potential between the capillary condensed phase and the vapour phase. Assuming the potential of the adsorbed phase and bulk liquid phase, and integrating, leads to p I n - da (7b) ? p, P0 In this case as pressure is lowered from P2 to P1 an a m o u n t da desorbs, exposing a surface S. This equation is the basis of the "modelless m e t h o d " of pore analysis due to Brunauer et al. [14]. The Kelvin equation is usually (but not always) applied to those isotherms showing hysteresis (see Fig. 1). A geometrical shape has to be assumed for the pores, and from this the shape of the meniscus at the mouth of the pore is deduced. The Kelvin equation is then applied to the loss of adsorbate from the meniscus by evaporation during desorption. Some workers have preferred to use the adsorption branch of the isotherm [15]. The most S -

--RT

/2

235

Amount Adsorbed

o

P/Po Relative Vapour Pressure

Fig. 1. A typical adsorption isotherm for mesoporous material.

c o m m o n pore shape assumed is cylindrical, which then has a hemispherical meniscus at the pore mouth with R1 = R2. The relationship between pore radius and pressure is then shown schematically in Fig. 2. To enable the calculation of pore size distribution to be made, the normal adsorption isotherm expressed as ml(NTP) g-1 or mg adsorbed g-1 against the equilibrium pressure p is taken through the transformations shown in Fig. 3. This enables a cumulative plot of pore size distribution to be constructed, starting from the highest pressure and working downwards. A typical plot is shown in Fig. 4. A Capillaries (end on view)

v

B O

C

D

~

llllllll

Radius

rI < r2 < r3 < r®

Condensation Pressthre

Pl < P2 < P3 < Po

Fig. 2. Schematic representation of the relationship between capillary radius and equilibrium pressure at which desorption will occur.

236

9

adsorbed per g solid

P v Vol of adsorbate taken up expressed as Vol of liquid per g

k/,../y P

L

,

Po

[2] l

[31

log P

y

%k

[4]

r

Fig.3. Stepsin producinga cumulativeplotofporesizedistribution. Three points remain to be decided in the use of this t y p e o f calculation. (1) Should the adsorption or desorption branch be used? (2) What value should be allocated to 0? (3) Where should the calculation be stopped? The second point was used in some early studies to explain the existence of hysteresis, the desorption and adsorption branches arising out of the existence of different receding and advancing c o n t a c t angles respectively. It is now considered in almost all treatments that the condensation or evaporation p h e n o m e n a occur f r om a surface covered by an adsorbed film, so the contact angle is that for perfect wetting, 0 = 0 o, and cos O = 1. In earlier theories it was assumed t hat this layer of adsorbed material was a monolayer, but it is now usually postulated to be a multilayer whose depth is governed by the BET multilayer t h e o r y of adsorption o n t o planar surfaces. The argument regarding which branch of the hysteresis loop t o use was effectively decided by Cohan [ 16], who was able to show that, on adsorption,

237

800

6O0

/xvp zXr~ 400 m

3

200

I 1.0

2.0

3 .o

4.0

nm

rp

Fig. 4. A typical pore size distribution plot (silica gel [24] ). condensation occurred at a cylindrical meniscus (Fig. 5), and the equat i on then takes the f o r m log

P

Po

-

-- 7 Vcos 0

rRT

(8)

(where R1 = r and the pore radius R2 = oo), whereas desorption takes place from a spherical meniscus (Fig. 5(b)), to which the Kelvin equation applies in an u n m o d i f i e d form. It has already been poi nt ed out, however, t hat this view is n o t universally accepted [ 1 5 ] .

(a)

(b) Fig. 5. The meniscus involved on (a) adsorption (cylindrical meniscus) and (b) desorption (spherical meniscus, cylindrical pore).

Pore size distribution calculation The calculation of pore size distribution is u n f o r t u n a t e l y complicated by the fact t h a t after loss of condensed liquid f r o m pores a multilayer film is left behind. This means t hat a pore is n o t truly emptied in a desorption step and the real pore volume has t o be calculated f r o m the apparent value by allowing for the adsorbed film left. This involves knowing t he thickness of t h e film (which is the standard t curve for the adsorbate, as described in the previous

238 review [1] ) and invoking again the particular pore geometry assumed. The value of r in the Kelvin equation should be regarded as the radius of the " c o r e " condensate inside the multilayer and is given the symbol rk (Fig. 5). After a pore has lost condensed material the multilayers left continue to thin down as the pressure is lowered further. Therefore, in any desorption step, some adsorbate comes from condensate in pores of a particular size and some from multilayer thinning. Due allowance for this effect must also be made. The application of the steps illustrated in Fig. 3 with the corrections described above then results in a pore size distribution. Usually the differential of pore volume with respect to pore radius is plotted against pore radius (see Fig. 4). The fundamental equation to be used [17] is /~ Vp = R n (A y n - - A t n ~ Sp + 2n tnA tnE Lp)

(9)

Before defining these symbols it must be explained that the desorption isotherm is split into a number of steps (about 30 for a " h a n d " calculation). The quantity A Vp is then the volume of pores associated with a step o f desorption P1 to P2. Using the Kelvin equation and the thickness of the remaining multilayers tl and t2, these pressures may be related to the radius of pores rl and r2 in general terms by rk = r -- t -

--

2~V

ln(P/Po)RT

cos 0

(0 = 0, cos 0 = 1)

(10)

where r k is the radius of the core of condensed material inside the multilayers. The average of the radii rl, r2 is ~ and is taken to be the pore size associated with A Vp. Rn is the factor to correct the pore volume for the multilayers left and is given by rp 2

Rn

(11)

-

@-k + ~ tn) 2

where Kk is the average of r~ -- t~ and r 2 -- t 2 and A t n is the decrease in t over the step P1 to P2. A Vn is the desorption given in terms of volume of liquid for a step obtained from the desorption isotherm by multiplying by 0.001555 in the case of nitrogen adsorption if this is in ml (NTP) g-1 or by 0.001244 if it is expressed as mg g-1. The remaining two terms account for the desorption produced by the thinning of multilayers (A tn)- They are determined by summing the surface areas Sp and lengths Lp of all pores involved in previous steps of the calculation. These terms are calculated assuming the geometry of the pores from Sp = 2A Vp/rp

(12)

Lp = Sp/2~%

(13)

and

In each line of calculation the ESp and ELp from the preceding line is used with the terms A t n and t n to calculate the new A Vp. The A Vp is then

239

used to calculate new Sp and Lp values to be added to a running total. The quantity A Vp/Ar ready for plotting is also found. The calculation is thus complex and it is very easy to make a numerical mistake. Therefore, it is usual to use a c o m p u t e r for this purpose. The question of where to stop the calculation remains. It has been suggested t h a t the Kelvin equation cannot apply when the meniscus is only a few molecular diameters wide [18] and a lower limit when rp is about 2 nm has given been. However, if the values of ~ Vp are summed, the value at this point is most often less than the total pore volume given by the total adsorption at or near saturation converted to liquid volume. This must mean that there are more pores to be emptied and the calculation should be continued. The equations described probably do not now apply rigorously; in particular, the z~Vp values obtained will not correspond to the ~ values obtained from the Kelvin equation, and they should be regarded as nominal radii only. When the sum of A Vp equals the total pore volume all the pores should have been emptied and any further desorption should be from multilayer thinning alone. A further step of calculation should result in zero values of A Vp. Unfortunately the calculation is n o t accurate enough for this to happen and finite values of/x Vp still appear and often come out negative. The distribution curve in this region can usually be ignored as inaccurate, b u t the exceeding of total pore volume may be used as the end of calculation of pore area. This is the final ZSp and is called cumulative area or Scum. This should then be the total area of the sample derived geometrically from the pore shape and the Kelvin equation. It is not strictly independent of the BET equation because the calculation depends on values of t and these are in turn derived from Vm and the BET equation for a standard sample. Despite the inaccuracies in calculation and possible faults in the model, much has been made of comparisons of Scum and SB~.T as a check on the pore size calculation method. All this is based on a model of a porous substance that is an artificial unreal model [ 17, 19 - 23], envisaging open-ended cylindrical non-intersecting pores. The use of the BET equation to calculate the thickness of the absorbed layer neglects the point inherent in the proof of the BET equation t h a t the monolayer is n o t complete until P/Po= 1. The ratio V/Vm does not imply the number of layers present; if V~ is the statistical monolayer then V/V~ is the statistical number of layers, n o t the actual number of layers. In spite of this basic objection to the use of the BET equation in such circumstances, m u c h discussion as to the improvement of pore size distribution calculations centre upon the calculation of the thickness of the adsorbed layer by the use of the BET equation, or the use of other measurements of the thickness which ultimately owe their origin to the BET concepts [24]. M e t h o d s o f d e t e r m i n a t i o n o f t values

Early pore size distribution methods used values for the thickness (t) o f the adsorbed film proposed by Shull [ 20] from adsorption on non-porous

240

samples. Other methods for calculating these so-called t values were proposed. It should be noted that according to the original concepts of the BET theory this is to be regarded as a statistical value. Later, Lippens et aL [25] quoted new values from experiments. Theoretical equations such as t h a t due to Halsey (see ref. 17) were used and also in a modified form as suggested by Lippens et al. [25]. Other theoretical equations have been suggested [21, 22, 26], the last mentioned allowing for curvature in the walls of the pore. A comparison of the results [24] shows that some of these methods of t calculation lead to values of A Vp/Ar that oscillate wildly owing to an overestimation of the multflayer correction. Of the other results the best correlation between Sc,m and SBET for a large number of samples was obtained using the Halsey equation with the thickness of a layer of nitrogen equal to 354 pm: t = 354

pm

(14)

Other pore models The cylindrical pore model is not the only one that has been proposed for the analysis of the desorption branch. A slit shape has been implied particularly to alumina for which there is evidence of a stacked plate structure. The Kelvin equation then relates the relative pressure to the width dE of the core of condensate inside the slit and the multilayer-thinning correction is simple because the multilayers lie on a plane surface. A point A on the surface of a meniscus in a slit-shaped pore has two principal radii of curvature R1 and R 2 at right angles. These have the numerical values of re, the radius of the core condensate, and infinity for the direction parallel to the slit (Fig. 6). re,

LL Fig. 6. The meniscus involved on desorption (slit-shaped pore).

Inserting this in eqn. 4(b) leads to In

P

Po

-- 2~V - - cos 0

dKRT

(15)

241

The true slit width d is given by

dK +

2t (16)

Rn = d/dK

Sp-

2AVp d

(17)

Multilayer thinning is t,Y, Sp. Therefore, (18)

h Vp = Rn(h V n - - h t , Z S p )

A more complicated calculation of this type has been given by Broekhoff and de Boer [27]. Other possible shapes for pores have been summarised by Everett [ 28] and the resulting isotherm shapes classified by de Boer et al. [29] (see Fig. 7). From the relative slopes of the adsorption and desorption branches it is possible to deduce information about both the necks of pores and their internal dimensions.

v

Y;

I f

---%0

(a)

PIPo

(c)

¥

¥

I

P/Po

(d)

--'%o

(b)

Y ~

P/Po

(e)

Fig. 7. Adsorption isotherms for various pore shapes: (a) tubular capillaries; (b) slit-shaped capillaries; (c) tapered tubular or wedge-shaped capillaries; (d) wide bodied, narrow necked tubular or non-parallel plates; (e) tubular with short necks, wide sloping bodies or various widths.

Difficulties in deciding between the various models and the unreal nature of the cylindrical approximation most often used has prompted the development of the so-called modelless method [14, 30, 31], in which a hydraulic radius of the condensate core inside the multilayers is related to the core volume. This does not require an assumption of the geometry of the

242 pores or of t values. The Kiselev equation is used to evaluate pore area for a series o f desorption steps and a core volume involved is obtained from the desorption measured. The core volume refers to the condensate lost from the centre o f the pore, ignoring the multilayers left. A hydraulic radius of the core is defined as core volume over area for each group of pores (rh = V / S ) . The core volume is then simply plotted against hydraulic radius. If a t curve is introduced, a " c o r r e c t e d " calculation may be carried out. To do this a pore model has to be assumed (cylindrical or parallel plate). A correction for desorption from the multilayers and for conversion from core volume to pore volume has been applied. The m e t h o d is then no longer modelless and suffers from the same disadvantages as the o t h e r methods. The formulae used for these corrections were, in fact, identical to those for a slit pored model described above. The pore size distribution methods pr oduc e for the majority o f solids a single peak or a binodal distribution, but rarely more peaks than this. In this case to give the information as concisely as possible the data from these pore size distributions may be simply not ed as the value of peak radius, and perhaps also the volume contribution, at these peak values. Such i nform at i on could in fact be obtained directly from the adsorption isotherm in an approximate form w i t h o u t resort to the complicated process of pore size distribution calculation. This is done by taking the steepest part of the desorption curve and relating the pressure to radius by the Kelvin equation.

The packed sphere model An alternative a t t e m p t to postulate a more realistic model for a porous substance has been made by Dollimore and Heal [32, 33]. An adsorbent system was assumed consisting of regularly packed spherical particles (primary particles). This is particularly apt for samples made by gelation of a colloidal suspension such as silica.The simplifying assumption was made t hat there are regions o f cons t a nt sized spheres. This is at least a p p r o x i m a t e d to if the system is mono-disperse with a narrow distribution leading to a single pore size distribution peak. The adsorption on such a surface is considered to be in three forms: multilayers; pendular rings or a toms-shaped condensate held at the points of c o n t act between spheres; and condensate in the cavities between groups of spheres. A further complication is that the spheres may be packed with several possible co o r di na t i on numbers. The simplifying assumption here is that in all regions the spheres pack with the same coordination. This is again reasonable if t h e y are all of the same size and undergo the same mechanism of condensation to gel simultaneously. The pendular ring may be defined as a volume of rot at i on f o r m e d by rotating a curve a b o u t the line of centres of touching spheres. This is illustrated in Fig. 8. The curve is n o t part of a circle, b u t changes its two principal radii o f curvature along its length to keep its harmonic mean radius p constant

243

(a)

(b)

Fig. 8. Adsorption of a pendular ring on two spheres in contact. (b) Elliptical approximation, r represents rt; circular approximation, r represents r v. (see eqn. (21)). T h e value o f p s h o u l d t h e n be related to pressure b y t h e Kelvin e q u a t i o n in t h e f o r m In

P

Po

-

--2~/Vcos0

(19)

pRT

An e x a c t s o l u t i o n o f this p r o b l e m is difficult and a p p r o x i m a t i o n s h a v e b e e n m a d e . A c c o r d i n g to K r u y e r [34] o n e s o l u t i o n is to a s s u m e equal values o f p at t w o p o i n t s only: at t h e m i d - p o i n t o f t h e t o r u s ( a b o v e t h e p o i n t o f c o n t a c t o f t h e spheres) a n d at t h e p o i n t o f c o n t a c t b e t w e e n t h e t o r u s and t h e n o r m a l m u l t f l a y e r s , i.e. t h e edge o f t h e ring. An e q u a t i o n has to be a s s u m e d for t h e curve o f r o t a t i o n a n d an ellipse is a g o o d a p p r o x i m a t i o n . Given t h e s e conditions

b

=

[(l--cos~--(t/R)cos~R rtcos 3 ¢

- - tan 2

~]-~

(20)

where 1

2

1

rt

p

R+t

(21)

and b R ( 1 ' - - cos ~ - -

(t/R)

cos ¢ ~(1 + b 2 t a n 2 ¢)v, 1

(R + t) sin ¢ + ( b 2 R t a n ¢ - 2

bR(1 + b 2 tan 2 ¢)v2

(1 - - cos ¢ - -

(t/R)

cos ~

- 0 (22) P In t h e s e e q u a t i o n s b is t h e s h a p e f a c t o r o f t h e ellipse, R t h e s p h e r e radius a n d r t a n d ¢ are d e f i n e d in Fig. 8(b). T h e s e c o m p l e x e q u a t i o n s can o n l y b e solved b y s t a r t i n g w i t h a l o w value o f ¢ and gradually raising it, until the l e f t - h a n d side o f t h e last e q u a t i o n b e c o m e s zero. T o find t h e v o l u m e o f a p e n d u l a r ring t h e c u r v e c o u l d n o w be a p p r o x i m a t e d as a circle w h i c h w h e n r o t a t e d a r o u n d t h e line o f c e n t r e s gives a v o l u m e o f ( f o r h a l f a ring, i.e. split d o w n t h e c e n t r e )

244 Vt = ~r 2 (R -- h@ )

(23)

where rv- --

R

R -- t

(24)

COS

and 7F

=~- -- ¢

(25)

(rv and $ are defined in Fig. 8(b)). The monolayers cannot be continuous over the whole sphere, and at the points of contact a cap volume Vc is excluded V¢ = ~ ( R t 2 + 2t8/3)

(26)

The multilayer volume, since it is covering a sphere comparable in size with the value of t, is given by (if it covered the whole sphere) Vm = (4/3)Ir{(R + t) 8 - - R 3}

(27)

Combining the adsorption due to multilayers and pendular rings for a coordination number n gives Va = V m - - n V c + n V t

(28)

The cavities left inside the spheres plus the adsorption Va contain condensate which evaporates at various pressures as desorption proceeds. Kiselev [35] has suggested that the cavities e m p t y when the relative pressure is in equilibrium with the meniscus in the m o u t h of the pore. The m o u t h was defined as the narrow portion leading from the cavity to the outside. The radius of curvature of this meniscus is the radius of the inscribed circle (allow. ing for multilayers). The meniscus may be taken to be essentially spherical. This will be true in its centre, but with distortion near the periphery. In this case the normal Kelvin equation applies. The shape and size of the cavity and its m o u t h will vary with coordination number, so each possible value of n has to be considered separately. Likely values are 4, 6, or 8. De Boer e t al. have quoted 6.6 as a practical value [36]. Considering the n value of 6 as an example, geometric considerations give the cavity volume as R3(8 -- 4~/3). A simple way of considering the process at first is to ignore the pendular rings. If the cavity is emptied of condensate the adsorption remaining Vm is given by V~n = 7r(4R2t -- 2 R t 2 - 8t3/3)

(29)

Thus material lost in a desorption step AV~is given by V¢ = volume of cavity -- Vm

(30)

A correction term F to correct ~ V~ to the actual cavity volume A Vp results from this

245

F -

volume of cavity

(31)

volume of cavity -- Vm

or

R3(8 - 4 /3) 4 ~ / 3 ) - - 7r(4R2t - - 2 R t 2 -

F = R3(8 _

8t3/3)

(32)

so that A Vp = FA Vc

(33)

This F term corresponds to Rn for cylindrical pores. The multilayers in previously emptied cavities thin down in each step and must be allowed for in calculating Vc, i.e. (34)

h Ve= A Vi-- A Vm

Where n Vi is the measured desorption and A Vm is multilayer thinning and V m = A t . ~ S p - - 21rtiAti~,L p - - 87rt2AtiY, N p

(35)

ESp is the summation of areas, Lp that of length and Np that of numbers of spheres around emptied cavities. These terms are given by AVp Sp =

AVp -

R(2/~r

--

1/3)

(36) 0.3032R

Sp

Lp -

(37) 2~R

G

N p - 4~rR2

(38)

The m o u t h of the pore has a radius rm related to sphere radius R by rm = 0.414R

(39)

where rm is obtained from the Kelvin radius allowing for a multilayer thickness t. Steps of rm may be conveniently taken to produce a distribution of A V p / A r m (cavity volume) against rm (mouth radius) or against R (sphere radius). The calculation may be made slightly more complicated by allowing for the t o m s in the calculation of F , i.e. F =

volume of cavity volume of cavity -- Va

(40)

where Va replaces V'm. The most complicated type of calculation is where the decrease in volume of the pendular rings left in emptied cavities is added to A Vm. In this case each group of cavity sizes for each desorption step has to be calculated separately.

246

These calculations of cavity volume distribution are complex and tedious in execution. The use of a computer is obviously the only practicable procedure. The results obtained for a narrow pored silica gel and a slightly wider pored one are shown in Figs. 9 and 10 (the differential cavity volume is plotted against m o u t h radius). In the case of Fig. 9 the simplest calculation with no pendular ring or torus allowed for is compared for variation of coordination number. Also the calculation allowing for a torus left in the open cavity is shown again for varying n. The full torus calculation, i.e. shrinking torus value is shown for n = 6 only. Also plotted is the simple cylindrical model result. All the curves are very close to one another and there is no shift in peak radius between the models. There are some vertical shifts but these are only slight. Figure 10 compares only the full torus calculation and cylindrical models but the shifts are again only vertical.

600

Ar

400

m3

m m

×

200

o

I

1.o

~"

I

2~

~

3.0

o

I

~.o

~

I

2.o

.m

I

3.0

4!0

Fig. 9. P o r e a n d c a v i t y size d i s t r i b u t i o n f o r a n a r r o w - p o r e d silica gel ( C r o s f i e l d s A P b r a n d ) : o n o t o r u s , n = 4 ; ~ - t o r u s , n = 4; D n o t o r u s , n = 6 ; - ~ t o r u s , n = 6 ; A n o t o m s , n = 8 , - ~ t o r u s , n = 8; V full t o r u s a l l o w a n c e , n = 6; x c y l i n d r i c a l m o d e l . Fig. 10. P o r e a n d c a v i t y size d i s t r i b u t i o n f o r C r o s f i e l d s A P b r a n d silica gel s o a k e d at p H 11.2: o c y l i n d r i c a l m o d e l ; [] full t o r u s a l l o w a n c e , n = 6.

Obviously if only peak position is required there is no point in using the cavity distribution when the simple cylindrical method is good enough. Pre-

247

sumably the cylindrical m e t h o d applied to cavity type adsorbents is equivalent to finding the distribution of cylinders inscribed through the mouths and into the cavities. The peak positions coincide because the inscribed cylinder and the inscribed circle coincide in the mouth. This result would n o t be true of course if ~ Vp/~R was plotted against R, the sphere radius. The inscribed cylinder does not have the same volume as the cavity, hence the vertical difference in the plots. The full torus calculation would only be justified if very accurate pore volumes are required. Plotting against rm and not R is best because of the analogy to cylinders described, and because the critical dimension is the mouth radius, through which large molecules may have to pass to reach the cavity to adsorb or be catalysed. In the comparison of slit pore and cylindrical models a shift of peak radius as well as large vertical shifts appear, so in this case a definite decision has to be made as to which model to apply. This has to be made on some other evidence, such as electron microscope photographs [29].

The exclusion of adsorbate molecules It has been pointed out that when an adsorbent consists of an assembly of packed spherical particles of size approaching the adsorbate molecule, then the monolayer capacity and hence the BET area will be in error [36]. This is because the adsorbate molecules cannot approach the points of contact. In the symbols used in this paper the area of a sphere radius R touching n others when the adsorbate molecule radius is Ra is given by SOET =

2~[2(R2 +

2RRa)--n{R2+ 2RRa--R(R2+

2RRa)~2 }]

(41)

whereas the area of an isolated sphere (allowing for curvature) would be SBET = 4~(R 2+

2RRa)

(42)

This could be called the apparent area. For example, this means that the measured BET area for a sample with primary particles of 20 nm radius and an adsorbate size of 0.2 nm radius should be multiplied by a factor of 1.05. The correction thus may only be slight and, in any case, the question should be asked: what area is actually required? If it is the area accessible to gas molecules, then the apparent adsorption or reaction area would be the correct one to use. One point that does emerge from a comparison of adsorption isotherms on some porous catalyst support materials (alumina and titania) is t h a t when the adsorption isotherms for nitrogen, oxygen, and carbon monoxide at 77 K are compared on the basis of the volume of liquid adsorbed (cm 3 g-l) against P/Pothen the adsorption isotherms coincide [37]. In the c o n t e x t of the Kelvin equation indications are

N2

CO

248 where 7 is the surface tension, M the molecular weight and p the density of the absorbate in the liquid form. Substitution of values for these absorbates i n t o the above equations indicates that, within the range of experimental accuracy, such a coincidence of data is to be expected. It should be emphasized that this would only be expected for mesopores and that there is great uncertainty in applying bulk liquid properties to adsorbates condensed in micropores or adsorbed in the monolayer [38].

Microporosity This term applies to solid materials with pores of diameter less than 2 nm. In these materials, if the adsorbate can enter the pores or pore entrances then the a m o u n t adsorbed simply rises to a limiting value [39 - 41], usually at a low value of P/Po and then remains almost constant for the remainder of the adsorption isotherm (see Fig. 11). In the limits such an isotherm is associated with an adsorbed layer one molecule thick but, in most cases of physical adsorption which conform to this pattern, it is probably associated with a pore structure where the size of the pores is of the same magnitude as the adsorbate molecule [ 42]. There are two limiting cases. In the first case, the molecule of the adsorbate just fits into the cylindrical pore and we shall demonstrate that the BET transformation of the so-called statistical monolayer into a surface area is too small, provided the system is one of microporous channels. However, in molecular sieves and in many carbons the microporous character arises because of microporous restrictions, on exits, or entrances behind which are cavities of high adsorptive capacity. This is the other limiting case and gives BET surface areas that are too high [43, 44]. The application of the formula Vs/S = F/2

(44)

either to the complete adsorption data as in the above equation (where Vs is the volume uptake near saturation, taken as a liquid, S the surface area and the mean pore radius) or incrementally in a pore size calculation, is suspect if the pores are in the microporous range [45]. On the basis of the conventional model of cylindrical non-intersecting pores we have

(45)

S = hVm

(where h is a constant) This then gives V J h V m = ~-/2

As the knee of the isotherm becomes acute Vm -~ V~ and V,~IV~ ~

1

(46)

249 and T -~ constant minimum value In the case of nitrogen adsorption this is 0.7081 nm. The use of the above formulae therefore fails to give a real value to ~below a certain minimum value. This t y p e of adsorption isotherm is often f o u n d with carbons [ 4 3 ] , and the correct interpretation of the adsorption data can be i m p o r t a n t in following the oxidation of carbon to gaseous products when there is a need to locate the surface where the oxidation process is taking place [ 4 4 ] . An alternative procedure in investigating structures o f this t y p e is to measure the adsorption capacity at a P/Po value near to saturation in terms o f a volume o f condensed liquid (condensed adsorbate) for a series o f different sized adsorbate molecules, and then to interpret this in terms of the pore volume available to differently sized molecules by plotting the available pore volume against the size of the adsorbate molecules. This is shown in Fig. 12 for a microporous silica [42, 46]. The main difficulty is to get a set o f molecular diameters for the adsorbate t hat are consistent. This is n o t always possible but perhaps the best are those calculated from bond angles and atomic radii (see Table 1). The table also lists bond diameters calculated from other data such as liquid densities or van der Waals constant.

300 A ../

O.IC N N2 O O-OS H20

. W; 200 "4~ ..e

v.o.o~

2 °Ar o

O C~H4 OC4%°

ml/g

o.o4

11~

o

o,o~

olz'

~4

o.'6 ' pips

018

1.0

~'o

20

~.o

.! . . . .

diameter of molecules, }~

Fig. 11. Adsorption isotherm for microporous materials. Fig. 12. Adsorption capacity variation with adsorbate molecule size for microporous silica. In a novel m e t h o d e m p l o y e d by Gregg and Langford m i c r o p o r o s i t y is d e t e cted by pre-adsorption of nonane [ 4 7 ] . In this m e t h o d the sample after outgassing was exposed at 77 K to n-nonane vapour, which was t hen outgassed at r o o m tem per a t ur e -- a process which removed the n-nonane from the external surface b u t n o t from the micropores. A nitrogen adsorption isot h e r m was measured before and after the t r e a t m e n t with n-nonane. The difference between the t w o nitrogen adsorption isotherms is then a measure of the microporosity. The m i c r opor ous silicas [46] already m e n t i o n e d are p ro b ab ly examples of a material with microporous channels which can easily be blocked. Such a model would produce an aoparent BET surface area which

250 TABLE 1 Values of adsorbate molecular diameters and liquid densities Adsorbate

Nitrogen Methane Carbon monoxide Ethane Propane Butane Carbon tetrachloride Benzene Cyclopropane Argon Nitrous oxide Carbon dioxide Neopentane

Temperature

Density_

(K)

(g cm -3)

Molecular diameters (nm) a 1

2

3

4

5

0.370 0.414

0.300

0.353 0.430

0.433 0.457

77 90 77

0.808 0.457 0.763

0.315 0.324

195 195 273 298

0.533 0.624 0.601 1.584

0.376 0.406 0.460

0.420 0.489 0.489

0.620

298 195 77 195 195 273

0.874 0.720 1.427 1.201 1.14 0.613

0.451 0.358

0.680 0.475 0.384

0.286

0.592 0.515 0.409

0.454

0.280

0.340

0.295 0.327 0.324

aMolecular diameters: Column 1, Calculated from van der Waals' co-volume; 2, Calculated from viscosity data; 3, Calculated from bond lengths, etc. ; 4, Calculated from heat conductivity; 5, Calculated from liquid density. is l o w e r t h a n t h e r e a l s u r f a c e a r e a . T h i s is s u p p o r t e d b y a d d i t i o n a l o b s e r v a t i o n s o n h y d r o x y l c o v e r a g e f o r t h e s e silicas. T h e B E T s u r f a c e a r e a f o r t h e s e s a m p l e s is c o m p a r a t i v e l y l o w f o r a silica, b e i n g a r o u n d 2 5 0 m u g-1. N o w t h e a c c e p t e d v a l u e f o r t h e n u m b e r o f h y d r o x y l s ( - - O H ) p e r n m 2 is 8 [ 4 2 ] . T h e v a l u e s o b t a i n e d in t h i s w o r k lie b e t w e e n 2 0 t o 3 0 p e r n m 2. T h e m e t h o d o f e s t i m a t i n g t h i s h y d r o x y l c o n t e n t is t h e w e i g h t l o s s b e t w e e n 2 0 0 °C a n d 1 0 0 0 °C a n d w o u l d a p p e a r t o b e c o r r e c t a n d t h e e r r o r lies in t h e c a l c u l a t i o n o f t h e s u r f a c e area. I n t h e e x t r e m e case, t h e m o l e c u l e o f n i t r o g e n j u s t f i t s i n t o t h e p o r e , a n d so o n e m o l e c u l e c o v e r s t h e w a l l s o f a c i r c u m s c r i b e d c y l i n d e r ( s e e F i g . 1 3 ) . T h e a r e a is t h e n g i v e n b y a r e a = 2r2~rr = 47rr 2

(47)

w h e r e r is t h e r a d i u s o f t h e p o r e . T h e B E T v a l u e is t h a t o f a m o l e c u l e c o v e r i n g a p l a n a r s u r f a c e ( s e e F i g . 1 3 ) w h e n t h e a r e a p e r m o l e c u l e is g i v e n b y a r e a = 7rr 2

(48)

We thus have true area

4n r 2 _

apparent area

7Tr2

-

4

(49)

251

Actually the apparent area is a little greater than t h a t shown here, owing to the packing of molecules, and the ratio is really 3.63 [45]. Thus the coverage of h y d r o x y l groups is approximately four times too large because the BET surface area in these cases is probably too small by a factor of between 3 and 4 [45]. If the pore is two diameters wide the factor is again 3.63. For increasing pore size the factor will decrease quickly to unity, but substances with pores approaching molecular size will also show apparent surface areas that are really too low. It can also be shown by argument along these lines that the lowest possible radius is given by ~ 2

(50)

where M is the molecular weight of adroxlate, N the Avogadro number and dL the density of adsorbate in the liquid state. ( M / d ~ ) 1Is can be taken to be approximately equal to the molecular diameter Din. Thus the general pore radius never appears less than twice the adsorbate diameter from use of the calculation using the equation F = 2 Vp/S. Thus the minimum radii for fine pored silicas from these calculations appears between 0.7 and 1,0 nm, whereas molecular sieve effects show the pores to be mainly 0.38 - 0.48 nm in diameter. The lower value to radii calculated from nitrogen adsorption isotherms using this equation has been commented on by others [38]. Such simple geometrical consideration can be extended to materials having pores of more than two or three adsorbate molecular diameters, on the basis that the monolayer capacity calculated by application of the BET equation can be distinguished from the micropore volume estimated by use of the Dubinin-Radushkevich equation or the t-plot m e t h o d [48, 49]. Figure 14 shows a typical pore whose inner surface is covered by a mono layer of m molecules, and nc is the number of molecules that fit into the centre of the pore after its surface has been covered.

Fig. 13. Area o c c u p i e d by adsorbate m o l e c u l e s in small pores: (a) m o l e c u l e just fits into pore (area = 2r × 2?rr = 41rr2); (b) m o l e c u l e o n a plane surface (area = lrr2). Fig. 14. M o n o l a y e r o f adsorbate m o l e c u l e s situated o n the internal surface o f a cylindrical pore. n c is the n u m b e r o f m o l e c u l e s w h i c h can o c c u p y the centre o f the pore after m o n o layer formation.

252 Table 2 shows h o w the ratio o f v o l u m e u p t a k e at s a t u r a t i o n to t h e m o n o l a y e r c a p a c i t y ( V s / V m ) varies with certain values o f nc f r o m 1 to 20 f o r this simple t w o - d i m e n s i o n a l m o d e l , t o g e t h e r with the p o r e radii evaluated w h e n using n i t r o g e n gas as a d s o r b a t e [50, 5 1 ] . TABLE2 Nc

1 3 4 5 6 7 13 15 18 19

20

No. of molecular diameters in pore diameter 1.5 1.5+ 1.5+ 2.5 2.5 2.5 2.5 + 2.5 + 3.5 3.5 3.5

Vs Vm 1.17 1.30 1.36 1.38 1.46 1.54 1.81 1.88 1.95 2.00 2.05

0.5/cos 30 °) 0.5/cos 45 °)

0.5/cos 30 °) 0.5/cos 45 °)

Average pore radius (nm) assuming nitrogen molecular diameter = 0.354 nm 0.531 0.730 0.781 0.885 0.885 0.885 1.085 1.136 1.239 1.239 1.239

The MP m e t h o d The t m e t h o d was used b y de Boer et al. [25] to separate the p o r e systems o f a d s o r b a t e s into n a r r o w and wide. Mikhail et al. [ 5 2 ] have s h o w n t h a t an e x t e n s i o n o f the t m e t h o d is well suited for the v o l u m e analysis o f n a r r o w pores. T h e y n a m e d this t h e " m i c r o p o r e analysis m e t h o d " o r MP m e t h o d , f o r short. It o n l y applies w h e n a V l - t p l o t deviates d o w n w a r d s , as s h o w n in Fig. 15 (instead o f a straight line). Points should be t a k e n f r o m an i s o t h e r m at intervals o f a b o u t P/Po equal to 0.05 and the t values t a k e n f o r a n o n - p o r o u s sample with a b o u t t h e same h e a t o f a d s o r p t i o n as the sample being investigated.

l O.4

o.!

2]45

6 7 8

9

89

} ,

i

i

i

i

,

Fig. 15. A plot of V1 vs. t when the MP method may be applied.

253 Tangents are then drawn to this curve at intervals. Line 1 passes through the origin and its slope gives the total area of the sample St. The n e x t line, 2, deviates downwards as t changes from h to t2, and gives a smaller area $2. A narrow group of pores has become filled with adsorbate and the area of these pores must be St -- $2 = AS1. If the pore walls are visualised as parallel plates, the average statistical thickness of the adsorbed film is (tl + t2)/2 and this is half the distance between the plates. The volume of this group of pores is given by V = (St --$2)(tl + h ) / 2 = ~ S ~ h ~ J 2

(51)

Another tangent can then be taken and this calculation repeated for area Ss and thickness t2 to ts. When there is no more downward deviation the calculation stops. From these results a plot of cumulative pore volume against distance between plates can be made. A good agreement between the total area St and SBET means that all the pores are accounted for in the calculation and that there are no " u l t r a micropores" which are not picked out by this technique. The accuracy of the result depends on the accurate drawing of tangents (not an easy construction) and on having very accurate values of t for the type of solid used.

The Dubinin equation The Polanyi Potential theory of adsorption [53] is mentioned here because Dubinin has applied a modified form of the resultant equation to microporous materials, especially carbons, and an interesting correlation can be made of the constant terms in this equation with the BET constants. The equation is used by Dubinin [3] in the form W = Woe -ke2

(52)

where W is the volume of micropores filled at a particular point on the isotherm, and is obtained by converting the adsorption a to liquid volume using the density of the adsorbate in liquid form. If a is in mol g-1 then W = a V where V is the molar volume in cm 3. W0 is the tbtal volume of all the micropores (not necessarily the adsorption at saturation because there m a y be mesopores as well).: e is the adsorption potential defined as the difference in chemical potential between liquid adsorbate in bulk and in the micropores, b o t h at adsorption temperature. The factor k is a constant for any one solid, but depends on the form of the adsorption space and the distribution of micropore sizes. If W / W o is plotted against e for a series of isotherms then the curves should coincide, and the shape is called the characteristic curve for the system and is a temperature-independent curve. The quantity k varies from one gas to another but if two gases as compared have constants k and kl then the ratio of e values for the same degree

254

of filling are connected by =

=~

(53)

where the constant ~ is called the affinity coefficient. If one gas (giving kl) is taken as a standard then W = W0e-kc~

(54)

For another gas

el = e/~

(55)

Therefore W = Woe-k~/~ (56) Substituting 2.303RTlog Po/P for e and incorporating (2.303R)2k into one constant

l

W = Woexp - - - - ~

log 2

(5"/)

or

BT2

w -- 1OgeWo-- 7

2 P

log Po

(58)

converted to loglo and putting D for BT2/~ 2 log W = log W o - D log u P/Po

(59)

Alternatively, if W is written as a V then log a = log W__o_ Dlog 2 P V Fo

(60)

or

log a = C -- Dlog 2 -P(61) Po In this form a plot of log W against log 2 (Po/P) is linear, the intercept at log 2 (Po/P) = 0 equalling log Wo, the gradient being equal to D [ 3, 48, 54]. Using molecules ranging in size from nitrogen to hexafluoropropylene, Dubinin showed that log Wo was remarkably constant for many activated charcoals. He called these adsorbents of the first type. Other carbons with much wider pores showed a behaviour following an equation W = W~e-m~

(62)

Here W~ is the volume limit o f the adsorption space (not pore volume) and m is characteristic of the non-uniformity of the carbon surface itself. The equation has been used by others [54, 55] for carbons, zeolites, some silicas and other materials with some success, but the plots are sometimes curving or consist of two separate straight lines. If the adsorption is

255 TABLE 3 Values of Vra, V0 and associated data for nitrogen adsorption isotherms on some fibre carbons [ 55 ] Sample Vm number

V0

1a 2a 3b 4c 5b 6c 7b 8c

596 462 149 129 118 97 80 72

388 328 115 100 84 73 59 36

Y°'l

420 380 138 122 84 82 64 32

Y°'9°

640 510 164 146 116 106 96 62

--Vm Vo

V0"I V0.9o

(P~)rn

0.651 0.711 0.771 0.774 0.711 0.759 0.729 0.505

0.654 0.744 0.835 0.836 0.725 0.822 0.666 0.517

0.05 0.07 0.02 0.02 0.05 0.085 0.065 0.105

(P~)

atCvalUevm 0

0.71 0.87 0.48 0.68 0.95 0.49 0.52 0.98

360 177 > 360 > 360 360 116 207 73

aCarbon fibres, commercial preparation. b c a r b o n fibres, activated in CO 2 in crucibles. CCarbon fibres, activated in CO2 in open tubes. Adsorption values are in mI(NTP) g-~.

restricted to a monolayer then the intercept log Wowould be a measure o f micropore area. With carbons in which there is no capillary condensation of nitrogen, the values of surface area from the potential energy equation and the BET equation should agree. Should capillary condensation of nitrogen be occurring then the intercept log Wowould be a measure of micropore volume and would agree with the Vm value calculated using the BET equation, but an area calculation is now unrealistic. The significance of the same data fitting both the BET equation and the Dubinin equation can then be analytically studied. Table 3 indicates BET and Dubinin parameters for a series of fibre carbons [55]. The value of W0 converted back to the adsorption units (to agree with Vm) can be called V0. In all cases Vo > Vm; the ratio Vm/Vo has also been studied by Klemperer for xenon adsorbed at -- 196 °C on metal films [~56]. He found a variation in this ratio but reported that for 25 isotherms the average was 0.975. There was also a tendency for Vra/Voto increase with lower C values. Gottwald [57] also comments on the close correspondence between the values of V0 and Vm. The general interpretation of V0 is that it represents the total volume of all the micropores [7]. Sutherland [58] points out that this is " t h e volume saturation of the carbon which can be determined simply and as accurately from direct inspection of the isotherm". As the BET C values reported in Ainscough's data cause Vm to appear in the P/Poregion 0 . 0 5 - 0.10, it would seem simpler and more direct to take the ratio Vo.1/Vo.sinstead of Vm/Vo. The data in Table 3 suggest this is reasonable. The ratio Vm ]Vo is in fact given by substituting Vm as a special value o f V in the Dubinin equation. This special case gives

256

V m - exp ( _ Dlog~ ~_~o)

(63)

v0 ( C-- 1 ~ =exp --Dloge2 __1+C ~ I

(64)

The inferencefrom Klempererand Gottwald would be that the ratio

Vra/Vo should be unity. A tendencyfor Voto be somewhatlarger than Vo would arise because V0represents an extrapolation to saturatedvapour pressure. Figure 16 represents a plot of D calculatedfrom the Dubinin equation and D calculatedfrom eqn. (64) using BET parameters. The identity of

0.4

0.3

~O.2

E

ooi

O,I D frorrp

o

0.2 Jog 'v'm/V0 tog 2 ( P/Po)rn

0.3

Fig. 16. A plot of D (from Dubinin equation) against D from eqn. (64).

these two calculations is indicated on the graph by the straight line and, bearing in mind the factors involved in the calculations, the results may be thought to be reasonable. The inference however that Vm/Vo should be unity is illusory. It really represents a limiting case, for when Vm = V0 eqn. (64) reduces to 1 = exp {-- Dloge2 (Po/P)m }

(65)

when (P/P0)m = 1 and the BET monolayer Vm occurs at saturation. Kaganer [59] actually t o o k values of Vo and converted them to surface area as if they were Vm values. He then drew comparisons with BET area. His samples were fine pored b u t not exactly microporous. However, this should not be taken as an encouragement to use the method as an alternative to the BET equation for larger pored materials.

257 References 1 D. Dollimore, P. Spooner and A. Turner, Surf. Technol., 4 (1976) 121. 2 I. Langmuir, J. Am. Chem. Soc., 40 (1918) 1361; 38 (1916) 2219. 3 M. M. Dubinin, 2nd Carbon and Graphite Conf., Society of Chemical Industry, London, 1958, p. 219. 4 M. M. Dubinin, J. Colloid Interface Sci., 2 (1967) 487. 5 S. Brunauer, W. E. Deming and E. Teller, J. Am. Chem. Soc., 62 (1940) 1723. 6 G. Pickett, J. Am. Chem. Soc., 67 (1945) 1958. 7 S.J. Gregg and K. S. W. Sing, Adsorption, Surface Area and Porosity, Academic Press, London, 1967, p. 371. 8 S. J. Gregg, The Surface Chemistry of Solids, 2nd edn., Chapman and Hall, London, 1961, p. 268 e t seq. 9 Kelvin (W. Thomson), Philos. Mag., 42 (1871) 448. 10 A. Zsigmondy, Z. Anorg. Chem., 71 (1911) 356. 11 W. Huckel, Adsorption und Kapillar Kondensation, Akademische Verlaggesellschaft, Leipzig, 1928. 12 B. V. Deryagin, J. Colloid Interface Sci., 24 (1967) 357. 13 A. V. Kiselev, Usp. Khim., 14 (1945) 367. 14 S. Brunauer, R. Sh. Mikhail and E. E. Bodor, J. Colloid Interface Sci., 24 (1967) 451. 15 J. C. P. Broekhoff and J. H. de Boer, J. Catal., 9 (1967) 8. 16 L. H. Cohan, J. Am. Chem. Soc., 60 (1938) 433; 66 (1944) 98. 17 D. Dollimore and G. R. Heal, J. Appl. Chem., 14 (1964) 109. 18 S. J. Gregg and K. S. W. Sing, Adsorption Surface Area and Porosity, Academic Press, London and New York, 1967, p. 165. B. P. Bering and V. V. Serpinsky, Russ. J. Phys. Chem., 40 (1966) 644. 19 J. P. Irving and J. B. Butt, J. Appl. Chem., 15 (1965) 139. 20 C. G. ShuN, J. Am. Chem. Soc. 70 (1948) 1405. 21 J. O. Mingle and J. M. Smith, Chem. Eng. Sci., 16 (1960) 31. 22 J. B. Butt, J. Catal., 4 (1965) 685. 23 C. Pierce, J. Phys. Chem., 57 (1953) 64. 24 D. Dollimore and G. R. Heal, J. Colloid Interface Sci., 33 (1970) 508. 25 B. C. Lippens, B. G. Linsen and J. H. de Boer, J. Catal., 3 (1964) 32. 26 A. V. Kiselev and A. P. Karnaukov, Russ. J. Phys. Chem., 34 (1960) 21. 27 J. C. P. Broekhoff and J. H. de Boer, J. Catal., 10 (1968) 391. 28 D. H. Everett, Proc. 10th Symp. Colston Research Soc., Butterworths, London, 1958, p. 95. 29 J. H. de Boer, Proc. 10th Symp. Colston Research Soc., Butterworths, London, 1958, p. 68. 30 S. Brunauer, Chem. Eng. Prog. Symp. Set., 96 (1969) 1. 31 S. Brunauer, in S. Modry (ed.), Proc. Int. Symp. on Pore Structure and Properties of Materials, Prague, Part 1, Academia, Prague, 1973, p. C3. 32 D. Dollimore and G. R. Heal, J. Colloid Interface Sci., 42 (1973) 233. 33 D. Dollimore and G. R. Heal, in S. Modry (ed.), Proc. Int. Symp. on Pore Structure and Properties of Materials, Prague, Part 1, Academia, Prague, 1973, p. A73. 34 S. Kruyer, Trans. Faraday Soc., 54 (1958) 1758. 35 A. V. Kiselev, Russ. J. Phys. Chem., 36 (1962) 1159; Zh. Phys. Khim., 31 (1957) 2635. 36 J. H. de Boer, B. G. Linsen, J. C. P. Broekhoff and Th. J. Osinga, J. Catal., 11 (1968) 46. 37 D. Dollimore and J. Pearee, J. Colloid Interface Sci., 46 (1974) 227. 38 M. R. Harris, Chem. Ind. (1967) 1295. 39 C. Pierce and R. N. Smith, J. Phys. Chem., 57 (1953) 64. 40 M. M. Duhinin, Q. Rev. Chem. Soc., 9 (1955) 101.

258 41 A. V. Kiselev, in D. H. Everett and F. S. Stone (eds.), The Structure and Properties of Porous Materials, Butterworths, London, 1958, p. 195. 42 D. Dollimore and G. R. Heal, Trans. Faraday Soc., 59 (1963) 189. 43 A. N. Ainscough, D. Dollimore and G. R. Heal, Carbon, 11 (1973) 189. 44 D. Dollimore and A. Turner, Trans. Faraday Soc., 66 (1970) 2655. 45 D. Dollimore and G. R. Heal, Nature (London), 208 (1965) 1092. 46 D. Dollimore and T. Shingles, J. Colloid Interface Sci., 29 (1969) 601. 47 S.J. Gregg and J. Langford, Trans. Faraday Soc., 65 (1969) 1394. 48 M. M. Dubinin, Chem. Rev., 60 (1960) 235; J. Colloid Interface Sci., 23 (1967) 487; J. Colloid Interface Sci., 21 (1966) 378; Russ. J. Phys. Chem., 39 (1965) 697. 49 K. S. W. Sing, Chem. Ind. (London), (1967) 829; (1968) 1520. 50 D. DoUimore and A. Turner, Chem. Ind. (London), (1970) 535. 51 D. Dollimore, A. N. Ainscough, B. Holt, W. Kirkham and D. Martin, in J. S. Anderson, M. W. Roberts and F. S. Stone (eds.), Proc. 7th Int. Syrup. on Reactivity of Solids, Chapman and Hall, London, 1972, p. 543. 52 R. Sh. Mikhail, S. Brunauer and E. E. Bodor, J. Colloid Interface Sci., 24 (1967) 451. 53 M. Polanyi, Verh. Dtsch. Phys. Ges., 16 (1914) 1012; Trans Faraday Soc., 28 (1932) 316. 54 T. G. Lamond and H. Marsh, Carbon, I (1964) 281. 55 A. N. Ainscough, Ph.D. Thesis, Salford University, England, 1972. 56 D. F. Klemperer, in D. H. Everett and R. H. Ottewill (eds.), Surface Area Determinations, Butterworths, London, 1970, p. 66. 57 B. A. Gottwald, in D. H. Everett and R. H. Ottewill (eds.), Surface Area Determinations, Butterworths, London, 1970, p. 59. 58 J. W. Sutherland, in R. L. Bond (ed.), Porous Carbons, Academic Press, London, 1967, p. 16. 59 M. G. Kaganer, Russ. J. Phys. Chem., 33 (1959) 352.