Surface Science 52 (1975) 103-119 0 North-Holland Publishing Company

COMPUTED ADSORPTIVE-ENERGY DISTRIBUTION

IN THE MONOLAYER (CAEDMON)

Sydney ROSS and Ian D. MORRISON * Chemistry Department, Rensselaer Polytechnic Institute, Day, New York 12181, U.S.A.

Received 27 March 1975; revised manuscript received 20 May 1975

A computer-based method is developed to perform the numerical analysis of the measured adsorption isotherm in order to determine the quantitative distribution of po- tential energies displayed by a solid substrate for a physically-adsorbed gas. The charac- teristic adsorptive energies of the basal planes of graphite and of boron nitride for argon and nitrogen are determined as examples of the use of the method. As more such values are put on record the method becomes suitable for the qualitative analysis of substrates.

1. Introduction

The heterogeneity of a solid substrate, in terms of a distribution of adsorptive

potentials as encountered by physically-adsorbed molecules, should be ascertainable from equilibrium measurements of monolayer adsorption at known temperatures

and pressures. The treatment of experimental data for the purpose of providing such a description has been approached in different ways. We can at once dispose of those methods that naively suppose that the surface fills up like a tea-cup, i.e., in serial order from the lowest to the highest potential level. The thermal energy of the adsorbate is significant compared to the differences between adsorptive potentials across the surface, and so ensures that adsorbed molecules are distributed through- out all energy sites at any condition of equilibrium. Thus any portion of the adsorp- tion isotherm depends on properties of the entire substrate: if only a small portion of the isotherm is taken it nevertheless reflects the whole range of adsorptive sites, although the more restricted the sampling of data the less precise is the information obtained.

In the present paper we report an attempt to derive the distribution of adsorptive potentials of a substrate from a measured adsorption isotherm. The variation in ad-

* Based on a thesis submitted by Ian D. Morrison in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Colloid and Interfacial Science (School of Science), Rensselaer Polytechnic Institute, June 1975.

104 S. Ross, I.D. Morrison/Adsorptive-energy distribution in monolayer

sorptive potential causes the density of the adsorbed molecules (molecules per unit area) to vary across the surface: hence, the adsorbate cannot be considered as a single surface phase. An analysis can proceed, however, if the adsorbate is treated as a set of independent surface phases, each with its own homogeneous density. This entails that the substrate be regarded as composed of different unisorptic patches.

(A liquid surface is structureless and isoenergetic; a crystalline surface is homotattic; the former of these is unisorptic and so is the latter.)

2. The adsorption isotherm for an unisorptic substrate

The necessary prime requirement before treating an heterogeneous substrate is to establish a valid description of isothermal adsorption on a unisorptic substrate. Experimental observations on such a substrate are rare, as solid surfaces never com- pletely correspond to that condition. The closest to it are the highly graphitized carbon blacks such as P-33 (2700) and Sterling MT (3 100). The careful measure- ments by Sams, Constabaris, and Halsey [ 1 ] of the adsorption of argon on P-33 (2700) in the monolayer region at temperatures of 140-240 K are well described by an adsorption isotherm derived from a virial equation of state for the adsorbate, in which the constants are based on bulk-gas parameters, no “correction” for per- turbation being required [2]. Higher coefficients up to and including the sixth were shown to be necessary. The first virial coefficient, or the Henry’s-law constant as it is sometimes named, depends on the interaction (named “vertical”) between the, gas and the substrate; and that interaction need not, at least as a first approximation, affect any subsequent terms of the power series. The second and third virial coeffi- cients depend on two-body and three-body interactions (named “horizontal”) of the gas molecules, as described by the Lennard-Jones parameters of the bulk (or 3D) gas when re-computed for restriction to two dimensions. At temperatures above the 2D critical temperature, the higher coefficients beyond the third are sufficiently well represented by the hard-disk model of a 2D gas, which has only repulsion inter- actions. Ree and Hoover [3] had calculated the reduced fourth, fifth and sixth viri- al coefficients for a hard-disk 2D gas, and these were the values used.

Morrison and Ross [2] truncated their virialequation description of adsorption at the sixth term because calculated values of still higher coefficients are not avail- able. For data that extend to higher coverages, such as may be anticipated on the higher energy patches of an heterogeneous substrate, it were better to include terms beyond the sixth. We shall therefore approximate the reduced virial coefficients beyond the third by putting them all equal to 2.0, which is an approximation in ac- cord with the findings of Ree and Hoover as far as they go.

The virial equation of state with this approximation is:

7TA e= 1 +B&A +C&A2 t2u6/A3 +2,*/A4 +..., (1)

S. Ross, I.D. MorrisonfAdsorptive-energy distribution in monolayer 105

where u is the Lennard-Jones parameter for the distance of maximum attraction. Because all of the higher virials are equal the infinite series can be summed,

giving:

g = 1 + (&-2$)/A + (C,,-2o4)/A2 + 202’A (l-u2/A)’

The adsorption isotherm is then obtained from the equation of state by means of Gibbs’ adsorption theorem in the form:

jdlnp =g!$jA dn (constant 7).

Solving eq. (2) for dn and substituting in eq. (3) gives:

lnp = ln(l/A) + 2(BzD-2u2)/A + 3(C2D-2u4)/2A2 + -?!? A-u2

- 2ln(l-u2/A)+lnK. (4)

Introducing the measured quantity n = the number of moles adsorbed per gram, gives

l/A = n/z, (5)

where A has the units m2/mole and IZ is the specific area per gram, gives the follow- ing adsorption-isotherm equation based on eq. (4):

In P/K = In [nZ/(Z--u2n)2] + 2(B 2,-2U2)n/~t3(c2D-2d')n2/282t202n z-&l'

(6) To see the effect of including all the higher virials in the adsorption isotherm

equation, we resubmitted the data of Sams, Constabaris and Halsey [l] to a theo-

retical description by means of eq. (6) using the values for BzD, CzD and K/C, the initial isotherm slope, that are reported in table 4 of ref. [2]. The only remaining parameter to be adjusted is Z, the specific surface area of the adsorbent. Fig. 1 shows the close correspondence obtained between calculated and observed isotherms with Z = 14.7 m2/g. The new values of K at the various temperatures lead to the fol-

lowing adsorption parameters:

Vu = 2.02 kcal/mole , v = 1.28 X 1012 set-l .

These latter two values are essentially unchanged from the former analysis. With the higher virials now included, the specific surface area has had to be increased from 14.3 m2/g to 14.7 m2/g, because the adsorbate-adsorbate repulsion that was lacking due to the neglect of the higher virial terms in the former analysis had been compensated by a smaller value for the area.

106 S. Ross, I.D. MorrisonfAdsorptive-energy distribution in monolayer

Argon on P-33(2700)

I40.60?*

/

IO 20 30 40 so 60 70 80

Pressurr in cmr of mrrcury

Fig. 1. The adsorption data [l] of argon on P-33 (2700) at various temperatures. The continu- ous lines are the theoretical descriptions by eq. (6) with the BZD, CUD, K/Z, and o2 as reported in ief. [2] (z: = 14.7 m2/g).

3. Description of adsorption on a polysorptic substrate

Ross and Olivier [4] postulated that a physically realistic model of an hetero-

geneous substrate could be based on the following premises: (1) The whole adsorptive substrate is made up of elemental patches, each with

a different adsorptive energy for the adsorbate, and on each of which monolayer adsorption can be described by an adsorption-isotherm equation, such as eq. (6), which has been established for a unisorptic substrate.

S. Ross, I.D. Mom’sonfAdsorptive-energy dism’bution in monolayer 107

+ 2U2?li/(Zj-U2?Zj) ) (7)

where p is the gas pressure, hi is a characteristic constant, nj is the number of moles adsorbed on the ith patch per gram of adsorbent, Zj is the specific area of the ith patch, and B2,, CzD, . . . etc. are the two-dimensional virial coefficients, the same for each patch, based on bulk-gas parameters. Eq. (7) gives the two-Dimensions density on the ith patch, ni/Zi, implicitly as a function of the ratio p/Ki for the ith patch:

~j~~j=~~/Kj)* 03)

(2) Elemental patches are filled simultaneously, though not to the same density, subject to the condition that the adsorbed phase on each patch has the same chem- ical potential. These elemental patches are assumed to be sufficiently large that boundary effects are unimportant.

(3) The number of moles adsorbed on the whole heterogeneous substrate, rids, is obtained by summing the individual values of the number of moles adsorbed, nil over all the patches:

nads O1) = C nj@) = C ~jf(p/Kj). i i (9)

If the heterogeneity of the substrate can be approx~ated by a continuous func- tion, then:

%&A = q~PQfml% (10)

where P(K) is the probab~i~ of a patch having the Henry’s-law constant K. K is a function of temperature and the adsorptive potential on the ith patch. The form of eq. (10) is a consequence of assuming that the adsorbed molecules form a set of phases in equilibrium with one another. The equation shows that the gas-solid inter- actions for an hetero~neous substrate, which are c~racte~ed by the domain of Kj, do not factor from the adsorption-isotherm equation for the entire collection of patches, as it does for a single patch. Such a separation is not a requirement for our subsequent analysis.

An adsorption isotherm reports me~urements of the number of moles adsorbed per gram of adsorbent at various equilibrium pressures. These equilibria determine a set of equations, one for each of s different pressures:

na&I) = q ~~f~~/K~) >

108 S. Ross, IB. Morrison/Adsorptive-energy distribution in monolayer

(lob)

The problem is to solve the set of eqs. (lOa) for the Ci and the Ki of each patch.

That solution gives the distribution, which is represented by a plot of dZi/RTd In Ki

versus Ki, and also gives the specific surface area, which is the sum of pi. For the con- tinuous distribution, the problem is to solve eqs. (lob) for Z and the parameters of the function P(K).

Two difficulties arise: first, no general closed expression for any other than ex- cessively-simpli~ed equations of state is known for flp/K), and second, even if one were found, eqs. (10) would be simultaneous non-linear equ tions in many variables,

Bt namely, the characteristic constants, Kim The usual simplificat’ons are either to as- sign a range of values of Ki [5], or to assume a particular distribution for them.

Ross and Olivier [4] assumed a continuous Gaussian distribution of values in InKi, and determined the best fit of a calculated isotherm to the data by adjusting the mode and the first moment of the distribution. The strength of this procedure lies in the simplicity of the analysis in that only three variables are adjusted, the two moments of the distribution and the specific surface area. The major weakness is that the real heterogeneous distribution of adsorptive potentials may not be close to Gaussian. Van Dongen [6] extended the assumption of a Gaussian distribution by choosing a distribution function of the form:

P(K)=K-‘exp[CotC1(lnK)t...C,(lnK)~]. (11)

Such a function is theoretically capable of approximating any continuous distribu- tion by means of a sufficient number of adjustable constants and is therefore more flexible. Since ffp/K) has in general no simple analytic form, any continuous distri- bution function has to be integrated numerically. The constants of the function are then determined by the method of least squares by iterating from an initial assumed approximation.

The objection to this method, however, is that the values of the constants,

Co, .--> C, and hence the form of the distribution found by the method of least squares may very well depend on the initial approx~ation, since the deviation of the computed isotherm from the data is a transcendental function of several variables and the minimum obtained is not necessarily unique.

An ideal analysis would determine the distribution of adsorptive potentials direct-

S. Ross, I.D. Morrison/Adsorptive-energy distribution in monolayer 109

ly from the data without any a priori assumption about the shape of the distribu-

tion function, and would include a procedure for finding a unique solution. The method of Adamson and Ling [S] pretends to do just that, but as is well known [6] and as Adamson admits: “Successive approximations do not differ by much and (differ), moreover, nonsystematically so that no continuing trends are apparent” [7]. That particular method is, in fact, analytically unsound as it lacks any provision for leading its approximations on to a unique and optimum solution. The present communication reports how this essential requirement can be met and how to per- form the desiderated analysis.

We start by assigning particular values Of Ki within a range determined by the

data. For temperatures below the 2D Boyle temperature the range of Ki is deter- mined by the lowest and highest data pressures. At any given temperature the sur-

face density of adsorbate borne by any patch is a function only of its p/K ratio. We set the limits on the value Of Ki in terms of the value of p/Ki at the inflection point, i.e.

The lowest value of Ki is chosen to that at the highest data pressure:

Ki(lowest) = p(highest)/(p/K)lnnection .

The highest value of Ki is chosen so that at the lowest data pressure:

Ki(highest) = p (lowest)/(p/K)i,On .

This requirement reads that at the lowest data pressure, the highest energy patch is at or above its inflection point, and the same for the lowest-energy patch at the highest data pressure. The area of a patch can only be determined when the density of adsorbate is manifest, i.e., when the adsorption on that patch is above the Henry’s- law region. The range of values of Ki thus selected should be such that no significant areas lie outside of it. If any do, more experimental measurements at higher and/or lower pressures are required.

The set of eqs. (lOa) now becomes a set of simultaneous linear equations in the only remaining unknowns, Xi* If in this range one merely takes the same number of values of Ki as the number of data points, s, and calculates by eq. (7) all values of f(p/Kj) required for eqs. (1 Oa), one can readily enough solve the s equations for the s unknowns. However, the answers obtained are often physically unreal because some values of Zi come out to be negative. One might suppose that the occurrence of negative roots is due to experimental errors in the data and that if fewer patches than data points were taken only positive roots wouId result. This too was tried by finding the least-squares fit of the calculated isotherm to the data. Each of the equations is formed by putting equal to zero the derivative of the sum of the squares of the deviations with respect to Xi. Some negative values of Zj are still found, how- ever, as this method is no different from solving s equations for s unknowns. We are really as far off as ever from our goal.

110 S. Ross, I.D. Morrison/Adsorptive-energy distribution in monolayer

The source of the problem does not lie in the magnitude of the experimental er- ror but in the fact that the equalities expressed by eqs. (1 Oa) are so interdependent that matrix equations derived from them are ill-conditioned. The successful proce- dure is to assign any initial distribution, say equal areas for all patches in the range of Ki, and to decrease the deviations between calculated and observed data points

by systematic serial adjustment of the independent variables, pi, rather than by a simultaneous adjustment of all of them as dictated by the method of least squares.

For s data points and I patches, the sum of the squared relative deviations is:

(12)

The deviation (12) is a quadratic function of each of the t unknown constants, Xi. The problem is to determine the value of each of the pi that minimizes the deviation (12) of the calculated isotherm from the data. The change in this devia- tion with the change in any one of the pi, say Zm is:

t

ntis@j) - pl xif(piIKi) f@jIKm)

DGZ m =-2,& %ds dtj) I nads@j) ’

(13)

This is a linear equation in each of the pi, including Z, , so that finding the minimum in the deviation (12) with respect to changes in Em, i.e., setting the fust derivative (13) equal to zero, is the same as solving the linear expression (13) for Em. A simple way to find the solution is fust to find the change in the expression (13) with a change in 2, , i.e., the second derivative of the deviation (12) with respect to Em :

(14)

As can be seen from the expression (14), the second derivative of the deviation with respect to any of the Z, is independent of all Zi, so that for computing pur- poses the second derivatives can be calculated once and saved.

For any initial value of Z, , calculating the first derivative (13) at this value of Em, and calculating the second derivative (14) gives EL, the area for minimum deviation, by the substitution:

Z:, = Em - DBi+m lb %#m ’ (15) This I$,, is the value of Z, that minimizes the total deviation with respect to Z, with all the other Xi futed. This adjustment depends on values of all the other Xi, but these in turn are only approximate; therefore, so as not to put the full weight of the correction on one patch when all the others are equally in need of attention,

S. Ross, I.D. MorrisonfAdsorptive-energy distribution in monolayer 111

we adjust Z, only a fraction (one divided by the number of patches) of the way to the minimum. The adjustment for the mth patch is then:

$,, -Xm = -DCif m ltij %#m *

In terms of experimental data, for s data points and t patches, the adjustment for the mth patch is:

1

If(pi/K, )/nd&)] 2 . 1 (17)

The solution sought is the minimum in a quadratic function (12) (the sum of squared deviations or the sum of squared relative deviations) subject to the exclusion of

negative values of Xi. If any adjustment leads to a negative Xm, that Zrn is set equal to zero and iteration continued, so that no unreal solution occurs. Reiteration is continued through all patches until the change in total deviation of the calculated isotherm from the data becomes insignificant compared to experimental error. Since

the total deviation is a quadratic function of all the variables, Zi, it has a unique minimum. The present procedure systematically minimizes deviations so as to ap- proach that minimum.

The answer that gives minimum deviation between the observed and computed data points is the unique solution to the problem, and yields the distribution of ad- sorptive patches that describes the heterogeneity of the substrate, or its polysorp- tivity. The computer program that elicits this answer is designated CAEDMON, the Computed Adsorptive-Energy Distribution in the Monolayer. Such results, where adsorbents rather than adsorbates are analysed, may be regarded as the converse of gas chromatography. The technique is therefore designated Substrate Chromato-

graphy. The CAEDMON program that develops the substrate chromatogram from an

adsorption isotherm analyses each substrate component in terms of its surface area per gram of adsorbent. Substrate chromatography is the only technique yet available that measures this property, which is independent of the concentration of the com- ponent in bulk, e.g., percent by weight, in the sense that one cannot be obtained from the other. Qualitative analysis of substrate components is also an inevitable outcome of this technique, waiting only for standardization by means of the adsorp- tive energies of different crystallographic surfaces of known identity. The basal planes of graphite and of boron nitride are exemplified in the next section of this

paper. For the study of heterogeneous catalysis, a technique that yields qualitative or quantitative changes of surface components cannot but be of interest.

112 S. Ross, I.D. Morrison/Adsorptive-energy distribution in monolayer

4. The analysis of data

The theory developed in the previous section was developed with a view to its ultimate application to experimental observations. Accordingly, we present here the results of this analysis applied to isotherms at 90.1 K of nitrogen and argon on various substrates.

The analysis of each isotherm was made as described above by systematically minimizing the percent deviation of the computed from the observed isotherm. The minimum of the percent deviation is close to that of the standard deviation but they do not always coincide. For small deviations, minimizing the percent deviation is equivalent to minimizing the standard deviation on a log-log plot, which gives

the same emphasis to the low-coverage as to the high-coverage portions of the iso- therm. The standard deviation on the other hand tends to minimize deviations in the high-coverage portion of the isotherm. Our computer program calculates the standard deviation after each iteration through all the values of lui and stops iterating

when the sum of the changes of the standard deviation for 100 successive iterations is less than 10e3 of the value of the lowest standard deviation. These changes are now SO small that they are close to the round-off error of the computer. This crite- rion ensures a good match by both measures of deviation.

Ocassionally the range of the values of Ki:i, determined as described above by the highest and lowest data pressures, is,wider than necessary. This condition is shown when the computed distribution has more than one zero-area root at either one end or the other of the range; it is frequently observed for the more uniform substrates. In such cases the range of values Of Ki is decreased to include only one zero-area root at either extreme of the computed distribution. A new set of values of Ki is taken within the revised limits and the computation continued. For every CAEDMON analysis presented here we have used 20 values of Ki.

Fig. 2 shows the CAEDMON results for nitrogen and argon adsorbed at 90.1 K on the graphitized carbon black, P-33 (2700), based on the data of Ross and Winkler [S]. The points indicated on the distributions (A) represent the values of the fre- quency density of each of the 20 patches within the range of Ki. We have found that the shape of the distribution curve is unchanged if fewer patches are taken, un- less they are too few to determine the distribution. Twenty patches is an arbitrary and satisfactory selection. The solid line through these points is a visual aid. The ad- sorption isotherms (B) show the original data points and also show solid lines that represent the adsorption isotherms computed from the distributions (A) by eqs. (1 Oa). While it is to be expected that the computed isotherms agree with the observed isotherms, the si~i~cant results are the similarity of the two distributions obtained for the two different gases, and the closeness of that description to the known homo- geneity of this substrate. The specific surface areas determined by the analysis are reported in table 1. The slight heterogeneity shown by the width of the peak for argon gives too small a signal to affect the higher-temperature isotherms shown in fig. 1, which were computed for a unisorptic substrate. This statement is based on a

S. Ross/l D. Morrison/Adsorptive-energy distribution in monolayer

P-33 (2700) at 90.1 K

35 I

Nitrogen 0.5

40 -RT In K 1.5 4.0 log P 1.5

125 ‘I

0

Argon

- 2.5 -RT In K o.o-“i

(A)

4 P I.5

(8)

113

Fig. 2. (A) The frequency density, dVads/RTdlnKi, of nitrogen and argon on P-33 (2700) at 90.1 K. The circles indicate pre-assigned values of Ki, uniformly spaced within the range that is itself determined by the measured range of equilibrium pressures of the adsorption isotherm in the monolayer region (Vads in cm3 (STP)/g; p and Ki in torr; RT in kcal/mole). (B) The adsorp- tion data [8] of nitrogen and argon on P-33 (2700) at 90.1 K. The continuous lines are the theoretical descriptions by eqs. (10a) based on the CAEDMON results shown in (A). The iso- therm Parameters are reported in table 1.

recomputation of those higher-temperature isotherms to include the slight hetero- geneity, using the same values for all the former constants, and observing no change.

Fig. 3 shows the CAEDMON results of nitrogen and argon adsorbed at 90.1 K on a sample of boron nitride (Norton Co., Batch 210484) based on the data of Ross and Pultz [9]. The points indicated on the distributions (A) represent the values of the frequency density of each of the 20 values of Ki. The solid line through these points is a visual aid. The adsorption isotherms (B) show the original data points

114 S.Ross, I.D. Mot&on/Adsorptive-energy disrtibution in monokayet

Boron Nitride at SO.\ I(

0 -1.0 -RT In K I-0 -2.5 109 D I*5

0 -I*0 -RT III K I.0 -23 lo9 P I.5

(Al IW

Fig. 3. Same as in fig. 2, mutatis mutandis.

and also show solid lines that represent the adsorption isotherms computed from the distributions (A) by eqs. (1 Qa). The significant results are again the similarity of the two distributions obtained far the two different gases, and the near homo- geneity of the substrate, which corresponds to its known characteristics, The slight contamination of the surface, presumably by boric oxide, which was suspected to exist (ref. 143, p. 203), may show on the argon distribution.

Figs. 4,5,6 and 7 show the C~D~~N distributions, the parent adsorption- isotherm data, and the fit of the computed isotherms drawn for comparison on top of the data points, for nitrogen and argon adsorbed at 90.1 K on a series of thermal- ly-treated carbon blacks, based on the data of Ross and Pultz 19). This series of samples of a carbon black, P-33 (also known as Sterling FT), of surface area of ap

S. Ross, I.D. Morrison/Adsorptive-energy dism’bution in monolayer 115

25.01 Nitrogen on P-33

90.1’ K

untrrotrd

0.0

-1.0 -RT In K I.5

Fig. 4. CAEDMON results for nitrogen adsorbed by carbon blacks that have been thermally con- ditioned (partially graphitized) at different temperatures.

proximately 12 m2/g, had been heat treated for 2 hr periods in an induction furnace at temperatures ranging from 1000-2000 “C. Virtually all the air was driven off in the initial heating stages and excluded for the duration of the heating period. More complete details of the preparation, the gradual perfecting of the graphite lattice as revealed by X-ray diffraction, and other physical properties of the resulting par-

tially-graphitized carbons are reported by Schaeffer, Smith and Polley [lo]. The

present samples are from the same lot for which Polley, Schaeffer and Smith [ 1 l] determined adsorption isotherms. These authors found a gradual development of stepwise isotherms for the progressively more graphitized carbon, which is inter- preted as evidence for a transition from more to less heterogeneous surfaces, as graphitization proceeds. The specific surface areas of the carbon blacks in this series

116 S. Ross, I.D. Morri~on/Adsorptive-energy distribution in monolayer

4&O*

0.0 *-

0.0 --

0.0 --

Argon on P-33

90,l”K

untreated

0.0 I_zzIzk+.-, -2.5 -RT InK 0.0

Fig. 5. Same as fig. 4, mutatis mutandis.

were found to decrease with graphitizing [ 111. The data of Ross and F’ultz shown in figs. 6 and 7 terminate at the upper limit

of the capability of their pressure gauge. The missing hirer-pressure data would have contained the evidence of low energy sites. The CAEDMON distributions show that these low energy sites tend to be progressively lost as graphitization proceeds. The cut-off of the data, therefore, excludes significant low-energy fractions of the total area for the less graphitized samples. The surface areas reported in table 1, which show an apparent increase with ~ap~tization, are properly interpreted as incompletely determined for the samples less perfectly graphitized. The actual change of surface area with the process of graphitizing can only be measured by means of more extensive data, which is currently under investigation in this labora-

tory. In order to make the present record complete, however, the areas as found are reported in table 1.

The significant conclusions to be drawn from figs. 4 and 5 are: fist, the general

S. Ross, I.D. Morrison/Adsorptive-energy distribution in monolayer 117

Fig. 6. The adsorption data [ 91 of nitrogen at 90.1 K on a series of thermally-conditioned carbon blacks. The solid lines are the theoretical descriptions by eqs. (1 Oa) based on the CAEDMON results shown in fig. 4. The isotherm parameters are reported in table 1.

0*5- Nitrogrn on P-33

9 0 . , k K

0.0.

-0.5 -

-3.5 -2.5 -1.5 -0.5

log P

0.5 I.5

agreement of the description given by the two adsorbates, argon and nitrogen; sec- ond, the gradual narrowing of the original distribution toward a limit that is more

and more precisely defined the more pronounced the graphitizing. The limiting val-

ue is characteristic of the basal plane of graphite. This general description given by CAEDMON is completely in accord with all the other physical measurements, which indicate increasing crystallinity of the material.

The value of -RTln K at the position of the maximum in the distribution curve

of P-33 (2700) is a characteristic property of the adsorbate interaction with the basal plane of graphite. The reciprocal of K is the equilibrium constant for the ad- sorption of gas on the substrate:

Ar (gas) at p,(torr) * Ar (ads) at rs(moles/m2).

118 S. Ross, I.D. MorrisonfAdsorptive-energy dism’bution in monolayer

-o-5-

Argon on P-33

SO.1 u 1000° c

reoted

Fig. 7. Same as fig. 6, mutatis mutandis.

The standard free energy (Gibbs) change from the standard state of the gas (pS = 1 torr) to the standard state of the adsorbate (F’s = l/u2 moles/m2 or 8, = 1) is given by:

AGO=RTlnK. (18)

The standard states given above may be replaced by others that are more commonly used [ 121, such asps = 1 atm for the gas and lYS = 1.489 X lo-l1 moles/m2 for the

adsorbate.

AGo = RTln K 2 7600 X1.489X1O-7

AGO = AGO + RTln (1.959 X lo-lo a2). (19)

S. Ross, I.D. MorrisonfAdsorptive-energy distribution in monolayer 119

Table 1 Specific surface areas from CAEDMON (m*/g)

Nitrogen Argon

Untreated P-33 6.16 6.30 P-33 (1000) 10.70 11.04 P-33 (1500) 11.37 12.41 P-33 (2000) 13.35 14.12 Boron nitride 23.6 20.0

Adsorbate parameters

o* (m*/mole) e/K (K) &D (90.1 K) (m*/mole) C2D (90.1 K) (m4/mole2)

8.236 x lo4 6.983 x lo4 95.05 119.8 -1.552 x lo5 -1.624 x lo5

2.293 x 10’0 1.890 x 1O’O

Table 2 Standard free energies of adsorption (kcal/mole) at 90.1 K

Nitrogen Argon

Graphite -1.60 -1.52 Boron nitride -1.27 -1.26

Table 2 reports the values of AGo for the adsorption of nitrogen and argon on the basal plane of graphite and boron nitride as determined from the CAEDMON analysis of the adsorption isotherms.

References

111 [21 I31 141 [51 [61

171

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