on a theory of the van der waals adsroption of gases

7/27/2019 On a Theory of the Van Der Waals Adsroption of Gases

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July, 1940 THEORYF THE VAN DER WAALSADSORPTIONOF GASES 1723with the theoretical. In contrast to tetrabutyl-ammonium picrate, no concentration dependenceof the 60-cycle slope is visible, bu t there is a syste-matic decrease of th e Wien coefficient with in-creasing frequency, which becomes more pro-nounced with decreasing concentration.As stated in the introduction, the Langevintime constant is of the order of milliseconds forthese solutions: at c = 5 X 71. = 0.077 X

sec., and a t c = 1Xsec. With decreasing concentration, TL increases,almost hyperbolically in the region of the mini-mum in equivalent conductance. For the higherconcentrations and low frequency the ionic dis-tribution has time to adjust itself to the field andwe find the normal Wien effect. But when the fre-quency becomes high and the concentration low,the field-sensitive reaction Ion pair+ ree ionsbecomes increasingly less able to follow the field.Consequently the Wien coefficient decreases.

T~ = 0.455 X

The Langevin time constant also depends onfield strength, as Onsager pointed out ; for ourcase, this dependence is a second order effect, be-cause the fields used were far below those corre-sponding to saturation.

Summary1. The conductance of tetrabutylammonium

bromide in diphenyl ether a t 50' has been meas-ured at low voltage over the approximate con-centration range lo+- l o p 3 normal.

Conductances at 60, 600 and 1000 cyclesa t field strengths up to 20 KV/cm. were measuredin the concentration range 0.5-5.0 X3 . For low frequencies, the Wien effect is inagreement with Onsager's theory; when theLangevin time of relaxation becomes comparableto t he period of the impressed field, a dispersion ofthe Wien effect appears.SCHENECTADY,. Y. RECEIVED PRIL20, 1940

2.N .

[CONTRIBUTIONRO M THE BUREAU F AGRICULTURALHEMISTRYND ENGINEERINGND GEORGEWASHINGTONN I-VERSITY ]On a Theory of the van der Waals Adsorption of Gases*

BY STEPHENRUNAUER,OLAS. DEMING,W. EDWARDS EMING ND EDWARDELLERA search of the literature on the van der Waals

adsorption of gases reveals that there exist fivedifferent types of isotherms. To give an ex-ample of each type we may mention the adsorptionof oxygen on charcoal at -183' (Type I),' ni-trogen on iron catalysts a t -195' (Type 11),2bro-mine on silica gel at 79' (Type 111),3 benzeneon ferric oxide gel a t 50' (Type IV),4and watervapor on charcoal at 100' (Type V).5 The fiveisotherms in Fig. 1 llustrate these five types; theyare imaginary, not real isotherms. Type I is thewell-known Langmuir adsorption isotherm, TypeI1 is the S-shaped or sigmoid isotherm, but nonames have been attached so far to the threeother types. Types I1 and I11 are closely re-lated to Types IV and v, only in the former casesthe adsorption increases as the vapor pressure p aof the adsorbed gas is approached, whereas in thelatter cases the maximum adsorption is attained,or almost atta ined , at some pressure lower than thevapor pressure of the gas.

P O P OFig. 1 . -The five types of van der Waals adsorption

An interpretation of the Type I isotherm wasgiven by Langrnuir who in 1915 derived his cele-brated isotherm equation for adsorption in a uni-molecular layer.6 Brunauer, Emmett and Teller'derived an isotherm equation for the multimolec-

isotherms.

(*) Not subject to copyright. ular adsorption of gases th at includes both theLangmuir isotherm and the S-shaped isotherm as1 ) Brunauer and Emmett, Taxs JOURNAL,9, 2682 (1937).(2) Emmett and Brunauer. i b i d . . 59. 1553 (1937).

(3) Reyerson and Cameron, J . P hy s . C he m. , 39, 181 (1936).(4) Lambert and Clark, Proc . Roy. Soc., 8199, 497 (1929).( 5 ) Coolidge, TBISJOURNAL,9, 708 (1927).(6) Langmuir. i b i d . , 37, 1139 (1915).(7) Brunauer, Emmett and Teller, i b i d . , 60, 309 (1938). Thispaper henceforth will be referred o as paper 1.


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1724 STEPHEN RUNAUER,. S. DEMING,W. E. DEMING N D EDWARDELLER Vol. 62special cases, but hitherto no one has derivedequations to cover isotherms of the shape desig-nated as Types 111, IV, and V.It is a fac t tha t the isotherm of paper I includesnot only Types I and 11 for which it was derived,but also Type 111, as will now be shown; it doesnot , however, include Types IV and V. To over-come this deficiency a new equation is now evolvedto cover all five types, and i ts application is shownin the interpretation of some experiments in theliterature. Finally, a comparison is made betweenthe present theory of van der Waals adsorptionand the capillary condensation theory.I. Interpretation of the Type I11 Isotherms

In paper I the following isotherm equation wasderived for multimolecular adsorption on a freesurf ace

where v is the volume adsorbed a t pressure 9 andabsolute temperature T ; Po is the vapor pressureof the gas a t temperature T; v, is the volume ofgas adsorbed when the entire adsorbent surface iscovered with a unimolecular layer; and c is ap-proximately equal to e(E1 EL)'R1', where E1 isthe heat of adsorption of the gas in the firstadsorbed layer, and EL s the heat of liquefactionof the gas. If one plots the function p / v ( p o - )against p / p o one obtains a straight line whose slopeand intercept give the values of o and c.

If the adsorption takes place in a limitedspace, so th at even a t saturation pressure only alimited number of layers can be adsorbed on theadsorbent, one obtains the expression

(B)vncx 1 - (n + l)x* + nxn+l1 - 1 + (c - ) x - C X " + lfJ=-where x = p / p ~ ,, and c have the same meaningas before, and n is the maximum number of layersof adsorbed gas th at can build up on the walls ofthe capillaries, supposedly of plane parallel sides.Equation (B) reduces to (A) when n = 03 ; andin the other extreme case when only one ad-sorbed layer can form on the surface, i. e., whenn = 1 it reduces to the Langmuir type equation

Equation (C) describes a Type I isotherm, andequations (A ) and (B) describe Type I1 isothermswhen G >> 1 or E l > EL,n other words, when theattractive forces between adsorbed gas and ad-sorbent are greater than the attractive forcesbetween the molecules of the gas in the liquefied

P/W = P o / C V m f P / v m (C )

state. All of the cases treated in paper I belongto this class. It should be noted, however, thatwhen the forces between adsorbent and adsorbateare small-4. e., when El < EL-then equations(A ) and (B) describe Type I11 isotherms.

Some of the most interesting Type I11 iso-therms available in chemical literature are thoseof Reyerson and Cameron3 on the adsorption ofbromine and iodine on silica gel. Brunauer andEmmett1r7 nvestigated the adsorption of six differ-en t gases on two samples of silica gel sent to themby Reyerson. They found t ha t t he capillaries insilica gel are relatively large in diameter, n isabout 9, so in fitting th e curves of Reyersonand Cameron we shall assume as an approxima-tion that n = a,whereupon the simpler equa-tion (A ) can be used. Reyerson and Cameroncalculated from the Clapeyron-Clausius equationthat the differential heat of adsorption ofbromine on silica gel is 7700 calories per mole.This value, however, is the heat of adsorption forthe first part of the unimolecular layer covered,so El, the average heat of adsorption for theentire first layer, is most probably smallerthan this. The heat of liquefaction of bromineis 7200 calories per mole, so we assumed as a firstapproximation that El = EL, r c = 1. Underthis condition it follows from (A ) that v = v, whenp = 0.5 PO, nd thus the constant v, is obtainedfrom one of the isotherms. The 79' isothermgives the value 8, = 3 .6 2 millimoles of bromineper gram of silica gel, which corresponds to aspecific surface of 470 square meters per gram,using 21.4 as the area covered by one brominemolecule on the surface. (This area has beencalculated from the density of liquid bromine inthe manner described in reference 2 , page 15%.The density of liquid bromine was taken as 3.06,from the "International Critical Tables," Vol. I,page 102). The average surface area obtainedfrom adsorption isotherms of nitrogen, argon,oxygen, carbon monoxide and carbon dioxide onanother sample of the same silica gel's7 was 507square meters per gram. These five gases yieldType I1 (S-shaped) isotherms on silica gel, andyet the v, value shows a remarkable agreementwith that obtained from the Type I11 bromineisotherm. This we consider to be one of the mostinteresting confirmations of our assumptions.

Having obtained the two constants, c = 1 andv , = 3.62 millimoles of bromine per gram ofsilica gel a t 79O, one can proceed to calculate the


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July, 1940 THEORY OF THE VAN DER WAALS ADSORPTIONF GASES 172.5isotherms by equation (A). The values of c andv, at the three other temperatures were calcu-lated as described in reference 7 , page 316. Figure2 gives the results. The four curves are the cal-culated isotherms ; the circles, squares, and tri-angles are the experimental points of Reyersonand C a m e r ~ n . ~The empty circles, etc., repre-sent adsorption points, the full circles desorptionpoints.) The isotherm equation not only de-scribes the shapes of the curves, but gives also thetemperature dependence, since c and v m were ob-tained from only the one isotherm, namely, thata t 79'.

200 40 0 600 800Pressure, mm .Fig. 2.--Adsorption of bromine b y silica gel (Keyerson

I, = 3.62 i~inioles. er gramrid Cameron: El = EL;at79': n = m .The iodine isotherms on silica gel obtained by

Reyerson and Cameron are striking because ofthe very small adsorption values. Here again weassume as an approximation that n = a, andtherefore that equation (A ) applies. Using thesurface area obtained from the bromine isotherm,and assuming th at t he packing of the iodine mole-cules on the surface is the same as in the liquidstate, one estimates that v,,, = 2.93 millimolesof iodine per gram of silica gel. We have seen tha tfor bromine a t p = 0.5 PO he amount adsorbedwas just sufficient to cover the surface of the en-tire silica gel with a unimolecular layer. Notso with iodine: the 178.4' isotherm shows anadsorption of 0.12 millimole per gram a t p =0.5 PO 330 mm.), which is only one twenty-fourthof the amount necessary to cover the surface withan adsorbed layer of iodine. This indicates thatthe forces of adsorption are considerably weakerthan the forces existing between iodine moleculesin the liquid sta te. Indeed, a value of EL - El =

3500 calories gives a good fit for.the 178.4' iso-therm, as Fig. 3 shows. One can then proceedto calculate the values of vm and c at the othertemperatures (reference 7, page 316) and obtainthe three other isotherms.

100 300 500 700Pressure, mm.Fig. 3.-Adsorption of iodine on silica gel (Reyersonand Cameron): EL- E1 = 3500 cal. ; Vm = 2.93mmoles.per gram at 178.4'; ~t= m .Since we have found tha t El = El. for bromine

on silica gel, and El < EL for iodine, we mightsuspect that El > E l, for chlorine 01 1 silica gel.This means that instead of Type 111 isothermsone would expect the more customary Type 11isotherms. The experiments of Reyerson andWishart8 show that this is actually the case.Since they worked with small p / p o values (theirexperiments extend up to @/ao 0.096), theirisotherms represent the lower region of Type I1curves, the region that is conoave toward thepressure axis. They were not able to fit theircurves with a simple Langmuir equation prob-ably because the heat of adsorption variedstrongly over the surface.

If E1


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1726 STEPHENRUNAUER,. S. DEM ING, . E. DEMINGND EDWARDELLER Vol. 62experimental error of a few molecular layers noappreciable adsorption of toluene took place onvirgin glass. Th e interpretation of their resultsvery close to saturation becomes extremely diffi-cult, since it depends very strongly on the exactvalue of the vapor pressure. In terms of ourtheory these two papers give very interestingexamples of adsorption phenomena where theforces of adhesion (e. g., between glass andtoluene) are very small compared with the forcesof cohesion (between toluene and toluene).

Finally mention should be made of some adsorp-tion measurements of Keesom and Schmidt11v12on helium and neon at temperatures close to theirboiling points. In spite of the fac t th at the ad-sorption took place on free glass surfaces and notin narrow capillaries, a unirnolecular adsorbedlayer was not exceeded even close to saturation.The concave form of the isotherms shows tha t,as should be expected, the forces of a ttract ionbetween the adsorbent and the adsorbed mole-cules are greater than the forces between theadsorbed molecules themselves, so that an ex-planation of t he sort given for the experiments ofMcHaffie and Lenher, and of Frazer, Patrick, andSmith, does not apply here. The experiments,however, might be explained by assuming thatthe heat of adsorption in the second layer issmaller than the heat of liquefaction. This maypossibly be due to a configuration of atoms inthe first layer imposed upon the adsorbed atomsby the surface structure of glass and differing morestrongly than in other cases from the structure of aunimolecular liquid layer.11. Derivation of a More General IsothermEquation that Includ es All of the Five IsothermTypes as Special Cases

As was sta ted earlier, equations (A) and (B) in-clude the first three types of isotherms, but notthe last two. Types IV and V suggest that thecomplete or almost complete filling of the poresand capillaries of t he adsorbent occurs a t a pres-sure lower than the vapor pressure of the gas-indeed, sometimes a t a considerably lower pres-sure. This lowering of the vapor pressure indi-cates tha t as the pressure of the gas is increasedsome additional forces appear that make the hea tof adsorption or energy of binding in some higherlayer to be greater than EL,he heat of liquefac-

(11) Keesom and Schmidt, Proc. A r o d . S c i . Amsterdam, 86 . 82 5(1933).

(12) Keesom an d Sc hmidt . i b i d . , 36, 32 (l!J%3).

tion of the gas. This must necessarily be so,since the last adsorbed layer in a capillary is at -tracted on both sides, and therefore its hea t of eva-poration must be greater than that of any otherlayer (with the possible exception of the firstlayer where the adsorptive forces are very differ-ent) . We shall denote this additional energy ofevaporation by Q, whereupon the rate of eva-poration of the adsorbed gas from the last layerwill not be ke -E L / R T ,but ke - (E L O ) I R T . Whenthe last layer is adsorbed, not only the heat ofliquefaction will be liberated, but in addition alsotwice the surface energy is set free, since twosurfaces disappear. Thus if the last adsorbedlayer fits in exactly and without strain, Q willbe equal to 2aS, where u is the surface tension,and S is the surface area covered by one mole ofthe liquefied gas spread out into a unimolecularlayer. The value of S will naturally depend onthe assumed type of packing. If th e fit of thelast layer is not exact, Q will be smaller.

The derivation of the adsorption isotherm equa-tion is similar to the derivation of equation (B ) inpaper I. Let SO, sl, s2, . . . , si , . . . representthe surface area that is covered by only 0, 1,2, . . .,i, . . ayers of adsorbed molecules. When equilib-rium is established the principle of microscopicreversibility demands that all of the followingevaporation-condensation equilibria be true :

cipq1 = bAle-&I/RTcips, = bs2e-EL/sTaps , = bsae-EL/RT

aps,-2 = bs,-l e - W R Ta p s , - = b s , , d E L + & ) I R T

where p is the pressure, E1 is the heat of adsorp-tion in the first layer, EL is the hea t of liquefac-tion of the gas, E L + Q is the heat of adsorptionin the last layer, and a. and b are constants.The approximations used are the same as inpaper I. It should be noted that EL is not theheat of adsorption of an isolated molecule of thei-th layer condensing on top of the (i - 1)-st ayer,but rather the average energy liberated in thebuilding up of the i- th layer. Thus the averageinteraction of the molecules in the i- th layer isroughly taken into account.

We visualize the adsorption as taking place ina capillary consisting of two plane parallel walls.The maximum number of layers that can fit intothis capillary is assumed to be 2n - 1. We can


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July, 1940 THEORYF THE VAN DER W A A L S ADSORPTIONF GASES 1727obtain the total volume of gas adsorbed by sum-ming up the volume of gas adsorbed on eachsurface element of on e wall of the capillary andmultiplying the result by two. The only excep-tion is the volume adsorbed on the surface ele-ment s,, which appears only once since it is com-mon to both walls of the capillary. The summa-tion, in a manner similar to that given in paper I,gives the equation

1 + ('/zng - n W - 1 - (ng - n + 1)r"+ 1 / 2 n g x n + 11 + (c - ) x + ( l / q cg - c)xr*- ' /acgx"+'

where g = e Q l R T , and all the other terms havethe same meaning as in paper I. The equationdescribes a Type IV or V isotherm, depending onthe value of c; for c >> 1 one obtains a Type IVisotherm; for c < 1,Type V. Equation (D) hasbeen derived for an odd number (2n- ) of layersfitting into the capillary. An analogous equationcan be derived for an even number (2n)of layers,and this lat ter equation reduces to equation (B)if g = 1.

In the derivation of equation (D) it is assumedthat whatever happens on one of the walls of thecapillary the same thing happens on a corre-sponding spot of the opposite wall, so th at the lay-ers building up on the two walls of th e capillaryalways meet in the middle. This may be a fairapproximation, but obviously the actual situationmust be th at when on a surface element of one ofthe walls one layer is adsorbed, there is a certainprobability that on the opposite wall on a corre-sponding surface element there are zero, one, two,three, or more layers adsorbed. Consequentlythe last adsorbed layer may form not only in themiddle of the capillary, bu t in any other part.Th e following derivation takes care of this s itua-tion.

We visualize the capillary again as two planeparallel walls between which the adsorption istaking place. By imaginary partitions we sub-divide the capillary into a large number of narrowprisms, as Fig. 4 shows, and assume tha t t he gasis adsorbed on the base and the top of the prisms.After equilibrium is established there will beprisms whose bases and tops a re both free of ad-sorbed gas, other prisms with one adsorbed layeron the bases and none on the tops, still otherswith one adsorbed layer on the tops and none onthe bases, one adsorbed layer on each, etc. We

0 0 I I 0 3 2 .2 I 0 I 0 0 0 2 I 2 3

Fig. 4.-Multimolecular adsorpt ion in a capillary,shall call the corresponding surface areas soo,sol, s10, ~ 1 1 , oz, . . ., s ; j , . . .,where sij denotes thesum of the areas of the bases and tops of all theelementary prisms that have i adsorbed layers a tone end and j a t the other. The total surface areaof both sides of the capillary is then(1) A = so0 + so1 + SI0 + A-11 + so2 + s2 0 + s 12 + s21 +and the volume adsorbed a t equilibrium is

s 22 + . . ,( 2 ) V = ' / z ~ o [ s O l + Si 0 + 2S11 + S0 2 + S 2 0 + 3312 + S z i +

4szz + . ., + (i + j)sq + . . .where v o is th e volume of gas adsorbed on onesquare centimeter of surface when i t is coveredwith a complete unimolecular layer. It followsthat

-v 2) Z ( i + ) s t j i + = 0, 1, 2, . . .n(3 ) - - = -A V O ~n 22si i, = 0 , 1, 2, . ..n nde-pendentlyThe principle of microscopic reversibility de-mands that when equilibrium is established thefollowing evaporation-condensation equilibriahold true( 4 4 apso0 = b e c E 4 R T s O lWriting

x = % p e E ~ / R Tbandwe obtain(4)Furthermore( 5 )and X S O ~= s02, or(6 )

One can derive by similar argument the equilib-rium adsorption relations between any two sur-face elements. Thus xsll = sal, and, therefore( 7 ) C ~ T ? V , f l = s1 , = S,?

c & E I - E L ) / R T

cxso0 = so1 = s10

SI 1 = cxso1 = c*x2sw

cx*sw = so* = s*o


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172s STEPHENRUNAUER,. S. DEMING, . E. DEMINGN D EDWARDELLER VO l . 62dnd again xs21 = s22, or

(8) c 2 x 4 s ~ SZZ , etc.From the foregoing equations one obtains therule

(12) A = s 1 + 2 6 x 7 + c2x22+ (nca- z + zc)gxiIt has been shown in paper I that(13)whence it follows from (3) that(14)

- dAd x( i + j ) S i j = x-ii = ckx> cz and c >> 1. For the purposeof evaluation of the constants it is useful to putthis equation in the form of a power series

\ n ( Z - )NX - .xz-1+ n x - 2 - 2)jz - 2(1 - x ) 3 x 3 '1%(

' l 2 ( G - 1)(1- ) + 2- n + l ) ( 3 i i - 4 ( 1 - +4 .5(E . + l ) ( Z + 2) (2%- 3)(1 -K 5 x ) S + . .

To illustrate the usefulness of equation (E )we fitted some Type IV isotherms of Lambert andClark4with the results seen in Fig. 5. They repre-sent the adsorption of benzene on ferric oxide gela t 40, 50, and 60'. The experimental isothermsshow very strong hysteresis on desorption ; th epoints given in the figure are all adsorption points.Using the constants v, = 0,OSl g. of benzeneper gram of gel, c = 27 (E1 - EL = 2190 cal.),h = 47,400 (Q = 1690 calories), and f i = 6, weobtained the curve in the middle of the figure(the50' isotherm). Then using the same values ofv , , , n,El - EL, nd Q we calculated the two otherisotherms at 40 and 60'. It will be noted th at thetemperature dependence of the isotherms is


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July, 1940 THEORYF THE VAN DER WAALS DSORPTIONF GASES 1729

Gas A, n = 6 /r1I ; c.20. 9=45 / x

satisfactory. Th at one cannot expect a perfectfit from equation (E) becomes clear if we remem-ber the following facts.

Gas 5. n = 4

U0.2 0.4 0.6 0.8 1.0

Fig. 5.-Adsorption of benzene on ferric oxide gel(Lambert and Clark) : Vm = 0.081g. per g. of gel; ?E = 6 ;E1 - EL = 2190 cal.; Q = 1690 cal.

(1) I n the derivation of the equation we as-sumed th at the capillaries of the adsorbent havea uniform size, whereas actually adsorbentsprobably have capillaries of differing sizes. Thusin our idealized case we derived an expression foran imaginary average sized capillary.(2) We assumed that the walls of the capillaryare parallel, whereas actually the walls may oftenbe convergent (V-shaped pores). To take care ofthis factor one would need a fifth constant, theangle 8, which would make the equation ex-tremely complicated.

(3) We assumed that the adsorption takes placein capillaries open on the sides. If we deal withcompletely enclosed capillaries, then the adsorp-tion in each consecutive layer will be less andless, and the maximum adsorption that can takeplace in each layer will not be a constant v,, butwill be a function of E.

Equations (D) and (E) also can be used toclear up some apparent discrepancies between

I b l I b o ,

0 0.5 1.0 0 0.5 1.0P I P@ PIPO.

Fig. .--Some ext reme cases of Type I V isotherms.(la) Schliiter, Z . p h y s i k . C h e w . , AlS8, 68 (1931).(14) Pidgeon, C a n . J . Research, 12, 41 (1935).


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1730 STEPHENRUNAUER,. S . DEMING,W. E. DEMING ND EDWARDELLER Vol. 62different, i. e., c is about the same for both, butthat the additional adsorption forces appearing inthe last layer are very different, rather small forA, but quite strong for B (g= 5 to 45 for the former,200 to 1000 for the lat ter) . Superficial inspectionof the three isotherms for gas B would suggest th atthey are Type I isotherms, in which a t saturationonly one layer is adsorbed. If one would calculatethe surface of the adsorbent from these pseudo-Langmuir type isotherms and compare it with thesurface of the same adsorbent calculated from theisotherms of gas A, one would find that B givesfour times as large a surface value as A does.Furthermore, the n value for A would come outto be 6, while the apparent n value for B is 1.Naturally the solution of the difficulty is tha t theisotherms of gas B are not Type I, but TypeIV isotherms, which appear to be Langmuir typebecause of the large values of the constant g.The three isotherms of gas A illustrate how aType IV isotherm goes gradually over into a TypeI1 isotherm as the values of g get smaller.

The results of Schliiter and Pidgeon can thus berelatively easily reconciled in terms of the presenttheory. On the other hand, some isothermsof Coolidgels cannot be explained in the samemanner. The two isotherms shown in Fig. 7 weremade on two different samples of identically pre-pared charcoal, therefore the constants obtainedfrom them should be consistent with each other.

A , ;--t--~ Carbon D i so i f i d e of 59 5 C D c o C AI

B

500

300

1 ' 1 - 00I

PIP,.0.2 0.4 0.6 0 8 1.0Fig. 7.-Adsorption of vapors b y charcoal (Coolidge).

The water vapor curve appears to be a Type Visotherm, and indeed one can fit it satisfactorilywith the help of equation (D) using the valuen = 9. On the other hand, the shape of the car-bon disulfide isotherm is of the Langmuir type,indicating unimolecular adsorption. If one would

(15) Coolidge, THIS OURNAL, 46 , 59 6 (1924)

calculate the surface of the charcoal from the twodifferent isotherms, one would find a discrepancytha t is about five-fold. A possible explanation isthat the surface of the charcoal and the size of thecapillaries are both influenced by the adsorbedmolecules. The large surface of charcoal showsthat the material isvery finely divided. It may beth at under the action of water these fine particlescoagulate into coarser ones presenting a smallersurface and a larger pore diameter, whereas underthe influence of carbon disulfide they remain finelydivided with narrow pores between them. Thiswould explain why water gives Type V, while car-bon disulfide gives Type I isotherms on charcoal.

We wish to suggest also an alternative ex-planation. It is possible that both the watervapor and the carbon disulfide isotherms representunimolecular adsorption. Th e carbon disulfidecurve is just a customary Langmuir type iso-therm. The shape of the water vapor isothermshows that at low @/Po values very little ad-sorption takes place owing to the small force ofinteraction between the carbon atoms of the sur-face and the water molecules. However, as soonas some adsorption has taken place, further ad-sorption proceeds more easily because of the largedipole attraction between the water moleculesthemselves. Thus eventually a sharp rise inadsorption takes place, until the capillary isfilled. If this second explanation is the correctone, then the water vapor curve is not a trueType V isotherm and one is not justified in usingequations (D) or (E) to fit it. Apart from theadsorption of water vapor on charcoal we know ofno other adsorption curves that have even thequalitative shape of a Type V isotherm.111. Comparison with the Capillary Condensa-

tion TheoryFor the purpose of comparing the present

theory with the capillary condensation theory,let us imagine th at the adsorbent consists of capil-laries bounded by two plane parallel walls andthat all capillaries have the thickness D. Thencondensation in the capillaries will take placeaccording to the Kelvin equation a t the pressure.where PO, , and V are the vapor pressure, surfacetension and molecular volume of the liquefied gasat temperature T. Below p , no adsorption takesplace, except possibly that of a unimolecular layer,while above 9, the capillaries are filled up com-

pk = p & - - 2 a V / D R T


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July, 1940 THEORYF THE VAN DER WAALSDSORPTION F GASES 17 314

I

I I I I I I 10.2 0.4 0.6 0.8 1 o

P/,/po.Fig. 8.-Comparison between th e present theory an d

the capillary condensation theory: n = 8; c = 150 (El -EL = 757 cal.); g = 17.17 (Q = 440 cal.).pletely. In the present theory multimolecularadsorption sets in below pk, and the capillariesare not filled up completely even above Pk. Thereason for this difference is that in the presenttheory the fluctuations are taken into account byconsidering in detail empty, partly full, and fullcapillaries. In the capillary condensation theory,on the other hand, only the probabilities of emptyand full capillaries are compared, and i t is assumedth at t he state with the greater probability is real-ized even in the pressure ranges where detailed cal-culation shows that the probability of partially fullcapillaries is much greater. Experimentally thesharp rise in adsorption demanded by the capillarycondensation theory has never been found. Thiswas explained by assuming a continuously varyingdistribution of the capillaries. The present theorygives rise to smooth curves, even if the size of thecapillary in the adsorbent is uniform.

In Figs. 8and 9 curves 1and 3 show two adsorp-tion isotherms calculated on the basis of the pres-ent theory, while curves 2 and 4 give the iso-therms according to the capillary condensationtheory. The constants of the four curves arethose of nitrogen at 76'K.; G is taken to be 150,and Z s 8 for Fig. 8 and 15 for Fig. 9. It is as-sumed tha t in the first case 8, and in the secondcase 15 layers of molecules would fill the capillariesexactly (without leaving any gap), consequentlyQ is equal to 2uS, as discussed in the previoussection. On the basis of cubic close packing S =31/'N1/a"a/2"', where N is Avogadro's number.

In the calculation of curves 1an d 3 of Figs. 8and9 we used a modification of equation (E), n the re-

I I I 1 I I I I I 1

0.2 0.4 0.6 0.8 1 oFig. 9.-Comparison between th e present theory an d

the capillary condensation theory: n = 15; c 150(E1 EL = 75 7 cal.); g = 17.17 (Q = 440 cal.).spect tha t hwas set equal to c2 grather than (Zc2-c 2 + 2c)g. This was necessary because in the deri-vation of equation (E) it was assumed tha t the 5ilayers do not fill the capillary exactly, and thereforethere are E + 1 different ways of leaving a gap ifZ layers are adsorbed. If, however, E ayers lill thecapillary exactly, ?idsorbed layers will have thestatistical weight 1. The summation carried outin the same manner as before leads to an equationidentical in form with (E), but with h = c2g.

As the curves show, capillary condensation oc-curs a t pressures a t which the walls of the capil-laries are covered with a unimolecular adsorbedlayer and the rest of the capillary is roughly halffull. Thus for Z = 8, capillary condensationwould occur at a pressure where v/v, is about 2 . 5 ,and for 7i = 15, about 4.25. A further peculiarityof the curves is that the limiting value of v/v,at the saturation pressure is not ' / t7 i , which wouldcorrespond to completely filled capillaries, butless. Actually the value of '/zZis closely ap-proached only if the value of g is high and thatof E s low. For sufficiently high values of Z onlyn/3 layers will be adsorbed near saturation.16

PIP@

-

(16) This can be shown in the following way. Near saturationthe factor xi+i will be independent of i 4- , the number of layersadsorbed, and the weight factor for the adsorptionof a total of Y layerson the two opposite walls ( i 4- - v ) will be Y + 1. For the averagenumber of layers adsorbed near saturation we have therefore.

- G 2%/3S 15 lJ + 1)V S lOn one wall we find therefore an average of z/3or large values of n.

layers.


10/10

1732 STEPHENRUNAUERND P. H. EMMETT Vol. 62In Fig. 9 this limiting value is nearly approachedfor Fi = 15. Here in addition to the unimolecularlayer one-third of the remaining 13 layers is ad-sorbed a t saturation] giving a total of v / v , =5.33. Th at large capillaries according to the pres-ent theory are not completely filled up evennear PO s probably due to the crude manner inwhich the fluctuations were calculated. We ex-pect that one would get completely full capillariesa t po if the interaction of each molecule with itsneighbors were to be taken into account in detail.For smaller capillaries with not too small g valuesthe present theory predicts that the capillariesare practically full a t Po and the above difficultydoes not arise.

We believe that the present theory has twoadvantages over the capillary condensationtheory. In the first place it is a unified theorythat includes unimolecular adsorption, multi-molecular adsorption on free surfaces, and theenhanced adsorption that takes place in capil-laries ; whereas the capillary condensation theoryonly deals with the last of these three phenomena.In the second place the present theory suppliesan isotherm equation that can describe all the

five different types of van der Waals adsorptionisotherms and can be tested by experiment. TheKelvin equation, on the other hand, is not an iso-therm equation; a t best i t allows only the cal-culation of one isotherm from another. A modi-fied form of the capillary condensation theory,proposed by Patrick and McGavack, 7 althoughsuccessful in many cases in fitting experimentaldata, is purely an empirical equation.

SummaryThe isotherm equation derived in a previous

paper for the multimolecular van der Waals ad-sorption of gases has been extended to cover twoadditional cases: (1) when the heat of adsorptionin the first layer is smaller than the heat of lique-faction, ( 2 ) when the capillaries of the adsorbentare completely filled, the heat of adsorption in thelast layer is greater than the heat of liquefaction.The new equation covers all the five types of ad-sorption isotherms found in the literature. Com-parison is made with experimental data and withthe capillary condensation theory.

(17) McGavack and Patrick, THIS OURXAL, 42,946 (1920).RECEIVEDEBRUARY, 1940

[CONTRIBUTIONROM FERTILIZERESEARCHIVISION, BUREAU F AGRICULTURALHEMISTRYND ENGINEERING,u. s. DEPARTMENTF AGRICULTURE]Chemisorptions of Gases on Iron Synthetic Ammonia Catalysts

BYSTEPHENRUNAUERND P. H. EMMEWIntroduction

In several previous publications we reportedthe occurrence of ch emis~rpt ions~f carbon di-oxide, oxygen, carbon m o n o ~ i d e , ~ ~ , ~ydr~gen,~ , and nitrogen8 on various promoted and unpro-moted iron catalysts, and described a meth0d~8~whereby the chemisorptions of carbon monoxideand carbon dioxide in connection with low tem-perature van der Waals adsorption measurements2

(1) Not subject t o copyright.(2 ) Present address. Johns Hopkins University, Baltimore, Md.(3) By chemisorption we refer to adsorptions that involveforces that are greater than those active in physical adsorption or incondensation, and approach in magnitude the forces active in chem-ical reactions (see reference 0 ) . In this sense the term includesactivated adsorption as well as chemical adsorptions that proceedso rapidly that no activation energies can be measured.

(4) Brunauer and Emmett, THIS OURNAL, 67, 754 (1935).(3 ) Emmett and Brunauer, ib id . , 69, 10 (1937).(A ) Emm ett and Brunauer, %bid.,69, 553 (1937).(7 ) Emmett and Harkness, ibid.. 67 , 1631 (1935).( 8 ) Emmett and Brunauer, ibid.,66, 5 (1934).(0 ) Brunauer, Emmett and Teller, ibid.. SO, 300 (1938).

can be utilized to determine the fraction of thesurface covered by the aluminum oxide and thealkali promoter in singly and doubly promotedcatalysts. The present experiments were under-taken to study further the concentration anddistribution of the promoters on the surfaces ofcatalysts, to ascertain the nature of the bindingof the chemisorbed gas to the surface, and in gen-eral to obtain additional information about thenature of the catalyst surface and the mechanismof promoter action.

ExperimentalMost of the chemisorption experiments were perfornird

on three iron synthetic ammonia catalysts: unprornotedcatalyst 973 (containing 0.15% A1208 as impurity), singlypromoted catalyst 954 (containing 10.2% Alaos), antidoubly promoted catalyst 931 (containing 1.59% KrOand 1.3% AlsOa). Whenever in this paper we refer to thethree catalyst types, or t o the pure iron, singly promoted, or

on a theory of the van der waals adsroption of gases

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