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M ark, H . a n d K ra tk y , 0 . 1937 Z . phys. Chem. B, 36, 129.

M ark, H . an d Susich, G. v. 1928 Kolloidzs46, 11.

--- 1929 Z . phys. Chem. 4 , 431.

Meyer, K . H . 1930 Kolloidzschr. 53, 8.

Meyer, K . H . a n d M ark, H . 1928 Ber. dtsch. chem. Ges. 61, 1939.

--- 1930 “ A u fb au der hochpolym eren organischen N atu rsto ffe.” Leipzig.

Nageli, C. 1928 “ Die M icellartheorie.” Leipzig.

S auter, E . 1937 Z . ph ys. Chem. B, 36, 405.

Siefriz, W . 1934 Protoplasma, 21, 129.

Sponsler, O. L. a n d D ore, W . H . 1930 Zellulose Chem. 11, 186.

De s c r ip t io n of pla te 7

Fi g. 5 a a n d 5b. X -ra y p h o to g rap h s of a crepe ru b b e r film ob tain ed from a chloro­

form solution. H igher o rien tatio n is clearly shown.* 700 % stretching. R adiation perpendicular a n d parallel to plane of film respectively.

Fig. 6 a a n d 65. X -ra y interference diagram s for a film of raw ru b b e r (braun-gelb) from a benzene solution. 7 0 0 % stretch in g .

Fi g. 7 a a n d 7 b. P h o to g rap h s show ing higher o rien tatio n of a film of vulcanized ru b b er. 800 % stretch in g .

* Cf. figs, l a a n d 16 o f M ark a n d v. Susich (1928).

Statistical mechanics of the adsorption of gases at solid surfaces

By F. J . Wi l k i n s

Research Department, Imperial Chemical Industries, Billingham (Communicated by R. H. Fowler, F.R.S.— Received 20 September 1937) The object of this paper is to develop a statistical mechanical treatment of adsorption of gases on plane solid surfaces using the method of partition functions. Several authors have approached this problem by way of the Boltzmann equation but without success. In fact K ruyt and Moddermann (1930), who discuss this work in some detail, conclude th a t the equation cannot be satisfactorily applied to the problem of gaseous adsorption at a solid surface and suggest th a t this is because the forces causing adsorption are of a chemical nature.

Recently, Fowler (1936a) has developed the Langmuir isotherm using

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Adsorption of gases at solid surfaces 497

essentially L angm uir’s assum ption of chem isorption on an adsorbing surface similar to a chequer board containing a fixed num ber of points of adsorption, and fu rth er th e assum ption th a t if a molecule is adsorbed on one point it does not influence adsorption on contiguous points. This fu rth er assum ption is relaxed in la te r papers (Fowler 19366; Peierls 1936). The assum ption made by L angm uir th a t adsorption forces are chemical in n atu re is not always true, as it is likely th a t th e forces operative are in m any cases of th e van der W aals’ type. H is other assum ptions are unnecessarily restrictive in view of th e considerable am ount of evidence on lateral m obility in th e adsorbed layer and also of th e certain ty th a t molecules adsorbed con­

tiguously suffer th e norm al m olecular interactions of th e v an der W aals’

type.

A ra th e r more general tre a tm e n t is a tte m p te d therefore in this paper.

The derivation of an adsorption isotherm from th e B oltzm ann equation itself is readily achieved, provided a suitable model of th e adsorbed phase is chosen. I n th e first place it m u st be assum ed th a t th e p o ten tial thro u g h o u t th e adsorption volum e is uniform . I n th e second place, th e adsorbed layer is assum ed to be a fluid phase sim ilar to a compressed perfect gas. The adsorption volum e can be considered therefore to be a p a rt of th e system in which th e gas comes under th e influence of an external field of force.

Fowler (1936a, p. 63) gives th e B oltzm ann equation as axja2 = wx e~eilkT I w2 e~e^JkT.

A pplying this to th e problem under consideration we w rite and as th e average num ber of molecules in equal elem ents of tran slatio n al phase

space which have th e to ta l energies ex and e2: wx and w2 are th e corresponding weights for th e in tern al energies of th e molecules. F o r equal volumes of

physical space accessible to th e system s, in which th eir dynam ical state is th e same, integrating over all possible m om enta, we obtain th e result:

n x/n2 = wx e~w ^kTjw2 e~w */kT, where Wx an d W2 are th e p o ten tial energies of th e molecules in th e two

portions of physical space. This reduces to G JG0 = w2/wxe*IRT,

where Ca and Cgare th e num ber of g. mol. of gas in equal volumes of th e adsorbed and gaseous phases respectively and (J> is th e adsorption potential.

Therefore

x = VR T wx ⁽¹⁾

33-2

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where x is the total number of g. mol. adsorbed per unit area a t pressure

and tem perature T; ocis the fraction of the area available for adsorption and d is the thickness of the adsorbed layer.

I t is im portant to note th a t x is not the number of molecules adsorbed as it is usually determined, since it is the to tal number in the adsorption volume, both bound and free. The am ount determined experimentally is the difference between the total am ount of gas in the adsorbed phase and the am ount in an equal volume of the gas phase. At values of 0 of the order of those considered in this paper and a t small values of x, the difference between the total am ount in the adsorbed phase and the experimental value of x is negligible.

I t is also im portant to note th a t the integration employed to give the adsorption isotherm assumes th a t the adsorbed and gaseous phases have the same dynamical state. The adsorption isotherm will therefore be applic­

able to systems in which the adsorbed phase is similar to the model dis­

cussed above, but it is not applicable in the simple form, as we shall see later, to a system in which the adsorbed gas becomes a p art of the solid j^hase and behaves, for example, as a Planck oscillator.

The simple adsorption isotherm, given by equation (1), gives the amount adsorbed as proportional to the pressure, a result which agrees with experi ­ ment if x is small. The deviations from this equation for larger values of x are quite considerable. The Langmuir adsorption isotherm was one of the first successful attem pts to account for these deviations. I t is important, however, to notice th a t the Langmuir adsorption isotherm can also be obtained if it is assumed th a t the adsorbing surface is uniform, can adsorb only one molecular layer, and th a t the adsorbed molecules are infinitely hard spheres which do not a ttra c t each other. To this extent, therefore, his discussion is equivalent to the assumption th a t the intermolecular energy

of interaction is E(r) = + oo (rccr) and = 0 ( r x r ) , where cr is the molecular diameter and may be considered to take account of the im­

perfection of the adsorbed gas layer. A theoretical derivation of an adsorp­

tion isotherm which takes account of both types of intermolecular forces is given below.

The approach to this problem, using the Boltzmann equation itself, is difficult and cumbersome. Fowler (1936a, p. 253) points out th a t the theoretical treatm ent of imperfect gases in external fields of force is best accomplished by applying the laws of statistical mechanics only to elements of the gas and to supplement these by general thermodynamic theorems.

Following this line of attack we write the free energy for the gaseous and adsorbed phases as Fg and Fa respectively (cf. Fowler 1936a, p. 240).

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A dsorption of gases at solid surfaces

Fa

= ~

b%a(T) + \ogfaiN

and adsorbed phases respectively, and f&,nd f a(T ) are p artitio n functions for m olecular in tern al and tran slatio n al energy; baNg(T ) an d b%a(T ) are th e

corresponding p a rtitio n function per molecule for th e poten tial energy of th e whole gas. The 6’s are related to th e stric t p a rtitio n function for th e potential energy of th e whole gas by {6(T)}iV = B (T ). The condition for equilibrium is th a t

(dFa/dNa)T,v — {dFgldNg)T'V' I f we assum e th e gaseous phase to be perfect, th e n bPNg(T ) = Vg, and th e condition for equilibrium reduces to

, lo g F „ + lo g - ^ = l0%b%J,T) + \o%U p + Nal^-\<>%b%a(T).

The p a rtitio n function b%a(T ) can be evaluated in th e following way.

I f we assume th a t th e adsorption p o ten tial {<f> cal./mol.) is uniform over th e whole of th e adsorbed phase then, neglecting th e im perfection of th e adsorbed layer,

b U T ) = raeV«r (cf. Fow ler 1936a, p. 57).

The im perfection of th e adsorbed layer can be ta k en into account by w riting

b U T ) = F0( l + / ? | !+ y f | + . . . ) e ^ ,

and it will be shown la te r th a t th e coefficients /?,

y,

we have

b%a(T ) = Va B Na (4 )

V M T )

N , VaBf,^ -U T ) e x p [ tf a g L {log B Xa(T)} + 5 4 / i j r ] ,

c a I ( D exp[ </’IR T N“dNa logBN‘{T)~\-

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The general equation assuming imperfection of the gaseous phase is

Equation (1) can be deduced readily from this if we use the same model of the adsorbed layer. Since both gaseous and adsorbed layers are perfect we have

B Ng{T) = B = 1,

(5)

fg{T) = W j fa(T )

and therefore as before v — ^ f/RT P R T w1 '

The process outlined above for evaluating equation (5) is typical of the general method, in which some definite model is chosen for the adsorbed phase and the partition functions corresponding to it evaluated. In the further discussion given in this paper three models of the adsorbed phase will be considered. In the first two the adsorbed gas will be assumed to behave as a highly compressed gas. Proceeding from this, two special cases will be considered. In the first model the adsorbed gas will be assumed to possess three translatory degrees of freedom. This is equivalent to the assumption th a t the amplitude of any vibration of the adsorbed molecules perpendicular to the surface is much larger than the molecular diameter.

In the second model, the adsorbed gas will be assumed to be a two-dimen­

sional gas, in which the molecules moving across the solid surface execute a vibratory motion perpendicular to the surface whose amplitude is of the order of the molecular diameter.

In the third model the adsorbed molecules will be assumed to have no lateral but only vibratory movement. As Lennard-Jones (1932) has shown, the solid surface is a region of varying adsorption potential even on a smooth surface, owing to the arrangem ent of atoms in a lattice structure (cf. Fowler 1932). As the tem perature is raised and the adsorbed molecules obtain sufficient energy to surmount the potential hills in the surface, this model passes over into the second model.

Case I

If the adsorbed phase is a three-dimensional gas, we can write

f g ( T )/f a ( T ) = w l l w 2-

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A dsorption of gases at solid surfaces 501

Assuming th e gaseous phase to be perfect, we have

^ N g i^) — 1.

E quation (5) reduces th e n to

% = e x p [ ~ t l R T - K g ^ l o g i W J ’) ] . S ub stitu tin g B „ .(T ) = ( l + A § + y f | + . A

we get

log ^

j; 217 p > v \

I f x is th e num ber o f g. mol. adsorbed p er sq. cm., d th e thickness of th e

adsorbed layer an d a th e fraction o f th e surface available for adsorption, th e n log- , w<> ad

^ w J i T ^{+ - $ -}R T ad 2 (ad)2 x2

where <fi is th e ad so rp tio n p o ten tial p er g. mol. an d N0 is A vogadro’s num ber.

To p u t th is eq u atio n in to a form suitable for application to experim ental d a ta it is necessary to d eterm ine th e relatio n betw een /? a n d y an d th e virial coefficients for th e gaseous phase. I n order to do th is it will be assum ed th a t th e virial coefficients are th e sam e for th e adsorbed a n d gaseous phases, although la te r th is will be modified to correct for th e tra n sitio n o f molecules from a three-dim ensional to a tw o-dim ensional gas. London (1931) shows th a t dispersion forces betw een molecules are to a first app ro x im atio n ad d i­

tive, an d in those gas-solid interfaces a t which th e v a n der W aals’ forces are predom inantly of th e dispersion ty p e it w ould ap p ear to be reasonable therefore to assum e th a t th e dispersion forces are n o t v ery different in th e adsorbed from those in th e gaseous phase. Such system s are those of argon, nitrogen an d oxygen on p latin u m , which are considered in th e n e x t paper.

Now „ = - ( § ) y We= - ^ 4 [ l o g ^ + lo g /„ (r ) + l ]

k N „ l, fiNa ( Z y - / P ) N l

Va \ Va VI

Ifcr/ /3N0x (2 y —fi2)N

= ~ V a \ K K ~

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B u t th e virial coefficients B C, ... m ay be defined by _ B x 2

1 + x + n + - f i = — B

(2y - f i * ) = - C / N l

1 x ^ w2 ad 6 3 2

l0gP = l° S w 1R T + S f - ^ d - 2- ^ d f - ^{( « )} This equation can be derived equally sim ply b y purely therm odynam ic argum ents, w ith o u t using th e n o ta tio n of p a rtitio n functions. Bradley (1931) has developed an eq u atio n of th e sam e form from such therm o­

dynam ic argum ents. W illiam s (1918) has also deduced an equation of the same ty p e from a m ix tu re of kinetic and therm odynam ic argum ents. He proves th a t

log; A xx,

where A0 an d A x are constants.

I t has been ta citly assum ed th a t th e adsorption p o ten tial <f> is constant th ro u g h o u t th e adsorption volum e. This assum ption will be justifiable if m onom olecular layers exist, w hen ( — 0) will be th e poten tial energy of the adsorbed molecule a t its m ean distance from th e surface. The simplest way o f rem oving this restriction is to sum th e jc’s for each value of <j> which corresponds to each successive adsorption layer. Neglecting th e weight factors, for th e first layer we have

lo g -1 6 V

i a i^ i l o g - ^ +

R T R T a xdx F o r th e second

loga____ . ^22d2

R T ^ a2d2

W 2

2(axdx)*Xl' SC 2 2(a2d2)*X2'

U nfortunately, it is n o t possible to solve these equations in a simple m anner for (xx + x2 + ...) as a function of p , and some device of th e ty p e employed by W hipp (1933) for tw o layer Langm uir adsorption isotherm s will be necessary before th ey can be applied to experim ental results.

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A dsorption of gases at solid surfaces 503

Ca se I I. Ad so r b e d p h a s e. Tw o-d im e n sio n a l la ter a l f r e e motion

TOGETHER WITH VIBRATION PERPENDICULAR TO SOLID SURFACE

*

In this case it is no longer possible to assum e th a t fg(T)/fa(T) = Wf w 2.

We can w rite in stead f a(T) = f'a( T ) f yihXT), where / v m .(T) i s th e p a rtitio n function for th e vibrational energy of th e adsorbed molecule in a direction perpendicular to th e solid surface, and fa (T ) is th e p a rtitio n function for th e tw o-dim ensional tra n slato ry an d ro ta to ry energies. The p a rtitio n function for ro ta tio n has n o t been ab stracted because on th is m odel th e ro ta tio n a l energy in th e gaseous an d adsorbed phase should be equal (cf. L ennard-Jones 1932, p. 340).

F o r a tw o-dim ensional gas

Assuming th e v ibrations are simple harm onic

n - e r ^

Jvib.v-1 ) ~ ^l)

(Fowler 1936a, p. 90). H ere phv is th e m axim um v ib ra to ry energy th e molecule can have w ith o u t leaving th e surface. Since adsorption p otentials

are usually a t least 5000 cal./g. mol. e- can be neglected an d we have

f m . = (1-—r - w w ) - 1*

As before B Ng(T) = 1, b u t B Na(T) is now given by

/ B Na(T) = ( l + ^ + / 3 £ + . . . j ,

where ft' an d y* are th e values of /? and y for th e tw o-dim ensional gas and A is th e area occupied b y one gram molecule of gas. Using th e m ethod of th e Clausius virial th e general equation of sta te of a two-dim ensional im perfect gas can be shown to be (Mitchell 1935)

p A = J ° ° r ( e - W fcT- l ) d r j .

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Assuming th a t E = + oo (r < or)and E = 0 or), where cr is the radius of the molecule, we obtain

1 p A = N k T + -- —

The corresponding equation for th e three-dimensional gas is 2 N 2nkT(r z

p V = N k T + - y — •

From this we see th a t th e equation connecting B and B ', where B ' is the second virial coefficient for a two-dimensional gas, is approxim ately

B = %'.

Similarly we can w rite C = £ ' , where £ is a constant.

Substituting th e above derived values for the various partition functions in equation (5), we have

9 a

c„

2B' 3 C' %

{ Z n m k T f (1 -« -** * * ) e~m T + ^ +2^

and

, x . out w*,

logp = I o g s f ^ - log— i — - \ o g ( l - e - ta,lkT) d> S B

t c

R T 2 our 2(acr)2 (7)

This equation is very similar in type to equation (6) developed from the simpler model.

Case I I I . Ad so r bed gas a Planck oscillator

OE 3 DEGREES OF FREEDOM

Here also it is not possible to write

fg(T)/fa(T ) = •

The adsorbed molecule can be assumed to be a Planck oscillator with three degrees of freedom, and we get

/«(2, )= /vib.(2’) = ( l - e - w *T ) - 3>

f g(T) = (2 rn n kT ^/h 3.

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505 A dsorption of gases at solid surfaces

As before we can w rite

S ubstituting in (5) we obtain

£ s = ^ < 2 - ^ + f •+£-.+•••

Oa w2 h3 K

acr R T wx

, (27m &T )* (1 — e~hv/kT)3

~ l o g ---V--- R T

2B" SC"

A 2 ⁽⁸⁾

where B" an d C" are constants sim ilar in ty p e to virial coefficients.

I t is interesting a t th is stage to derive th e simplified versions of equations (7) and (8) which can be applied if th e adsorbed gas phase is perfect, and compare th e results w ith equation (1). From (7) we get

ad

X = B T w 1

(27rm k T )i (1 — e~hv/kT)

h~e&RT p ,

and from (8) x = — fi^ __e-/u>/fcr^3e<i>iRTp

ill lb

This m akes quite evident th e errors which are obtained if th e B oltzm ann equation is n o t applied w ith sufficient discrim ination.

Sh a pe of th e th eo r etica l a d so r pt io n iso th erm s

The shapes of th e p /x isotherm s given by equations (6), (7) an d (8) are obviously similar. I t is easy to see th a t a t small values of x, x is proportional to p . As x increases th e progress of th e curve depends on th e sign of B , th e

second virial coefficient. I f Bis positive, which corresponds w ith th a t portion of th e p V Ip curve for gases a t which d(pV )/dp is negative, th e n as x

increases th e adsorption isotherm becomes convex to th e pressure axis.

A t this stage th e n e tt interm olecular forces are a ttractiv e. I f B is negative, th en th e isotherm is concave to th e pressure axis and th e n e tt interm olecular force in th e adsorbed layer is repulsive. A t high values of x, th e adsorbed gas will always be a t th a t p a rt of t h e p F jp curve where d(pV )/dp is positive.

I t should, perhaps, be pointed o u t th a t none of th e isotherm s, either (6), (7) or even (8) which approaches m ost nearly to Langm uir’s model, gives a satu ratio n m axim um , because th e adsorbed molecules are no t assumed to be incompressible. F o r all practical purposes, of course, th e isotherm s give a satu ratio n m axim um owing to th e relatively slight com­

pressibility of molecules.

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Th e d if f e r e n t ia l h ea t of a dso rption a n d

THE ADSORPTION POTENTIAL

The m ost commonly determ ined h eat of adsorption is the integral heat of adsorption Q. The differential heat is th en defined as (dQ/dx), which we will write as q. Com paratively few direct m easurem ents of adsorption heats have been m ade and th e norm al m ethod of procedure is to apply the equation

(d lo g p /d T )x = - q J R T2 (9)

to two different <J>X T points on th e same x isostere. The above equation is usually considered to be a corollary to th e Clausius-Clapeyron equation.

Coolidge (1926), after a critical discussion, concludes th a t this equation is satisfactory. McBain (1932), however, draws atten tio n to the difficulty th a t where saturation m axim a v ary w ith tem perature it is impossible to construct the isostere when x approxim ates to xs, the saturation value and th e equation appears to break down. H e suggests th a t this is due to the strains set up in the solid during adsorption and th a t th e extent of this strain varies w ith tem perature. F u rth er difficulties are obtained when differential heats so calculated are com pared w ith directly determined values. Coolidge found th ey agreed only to a first approxim ation, the calculated values being som ewhat less th a n th e experim ental. A similar discrepancy has been reported by Pearce and Taylor (1931).

The above difficulties are entirely rem oved if the equation is applied as

it should be to th e concentration ( x/ad) isosteres instead of x isosteres, for (ad) is not independent of tem perature; in fact, as will be shown in a later

paper,

ad = (ad)0e (10)

I t is not necessary to derive th e exact equation thermodynamically because it can be obtained w ithout trouble from the general adsorption isotherm (equation (5)). In this equation the s determine the potential energy of a molecule due to molecular interaction and we can write

- J f i = log B .v .m + If„ g ^ -lo g

B ^ T ),

- 3 ^ = log B XJ T) + log T),

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507

where Wg and Wa are th e potential energies w ith reference to a perfect gas as the stan d ard state an d are functions of Cg and Ca respectively. Then

, x | x 5 (27rm)5M . (27 logy = l0 g ^ + 2l0 g !r + l0g h* '

+ l o g ( l W ‘ }

where s' and s" are th e appropriate degrees of freedom for translation and vibration in th e adsorbed phase.

In teg ratio n of th e Clausius-Clapeyron equation gives

A dsorption of gases at solid surfaces

log: l n r * . ( Q ) . - ( C ) . s a d

( i i ) Here ( Ct)gand (Ct)a are th e constant p a rt of th e specific h e at for gaseous and adsorbed phases; (7vib is th e specific h e at associated w ith th e vibration

of th e adsorbed molecule an d iis th e chemical constant. The specific heats due to ro ta tio n and in tern al vibrations of th e molecules have been assum ed

to be equal in b o th phases. Comparing (9) and (11)

and

-<f>+Wa+ W a = q0- W g+ W a 3 -s'

i = log^{(27rm) 2 2}

h s~s'

q0 is here th e h e at of adsorption w hen both gaseous and adsorbed phases are perfect and a t absolute zero, and

q -

w„+ W J - T c ^ d T -

T,

where q is th e differential h e at of adsorption an d is given by

( d l o g p / d T \ x/ad) = —q / R T 2.

This gives us th e relation, therefore, betw een 0 and q. We can rew rite (12) as

q = qT ~W g + Wa, (13)

where qTis th e differential h eat of adsorption a t tem perature T under con­

ditions such th a t Wg and Wa are zero, i.e. low concentrations in both adsorbed and gaseous phases.

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The error which arises where xisosteres are used to calculate instead of (x/ad)isosteres can be dem onstrated readily. From (11), (12) and (13)

From (10) we have = (ad)Q e~^T, where j3s a positive quantity, so th a ti

_ 0_

dT +

R T 2’ tf* = £ - / ? •

Values calculated from th e xisosteres are therefore less th a n the true values by the am ount /?. qxis therefore greater th a n q,in agreem ent with the experi­

m ental results of Pearce and Taylor.

Th e d ec r ea se in qw ith in c r ea se of x

As a general rule, th e differential h eat of adsorption decreases with in­

crease in am ount adsorbed. The theoretical relation between x and q can be readily derived. From (9), (11) and (12), we have

.where f \=

log xjp= log ad + T f'( T ) + f"(T ) — q/R T,

(5 s e^- hv^/k T^-

\ 2 7 + (e- W ’_ l j W * ) ’

f ( T ) = [ | l o g r + log _ lo g (2y - + log(l - g ^ w y r ] . There are no satisfactory d ata existing which enable a test to be made of this equation, for no one has measured heats of adsorption on plane surfaces.

The calculated values reported in th e literature are erroneous for the reasons mentioned above. The direct relation between qand x is given by (13). I f as usual Wg = 0,

4 = gT - j j y ( ^ + 2 r ^ - a+ . . . ) .

I f this decrease is due to molecular interaction and not to heterogeneity of the surface field /? m ust be positive. Therefore B, the second virial coefficient, m ust be negative, indicating a n e tt repulsive force between molecules in the adsorbed layer.

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Adsorption of gases at solid surfaces 509

Su m m a r y

The generalized adsorption isotherm is shown to be

where Ca and Cg are th e molecular concentrations in the adsorbed and gaseous phases respectively; <f> is the adsorption potential and f g{T),

f a{T), BNg{T) and B Na(T) are p artitio n functions. These functions have been evaluated for three special cases in which the adsorbed phase is (a) a

three-dimensional gas, (6) a gas of two-dim ensional lateral m obility whose molecules v ib rate in a plane perpendicular to th e plane of lateral m obility, and (c) a group of Planck oscillators w ith three degrees of freedom.

The relation between <j> and q, th e differential h eat of adsorption, is deduced and it is shown th a t th e common practice of calculating from th e equation

(0 log pjdT)x = - 2

is erroneous. The correct equation is

(0 log p /d T \xlad) = - q /R T 2, where (x/ocd) is th e concentration in th e adsorbed layer.

My th an k s are due to Professor R. H. Fowler for m uch kind advice and assistance during th e preparation of this paper.

Re f e r e n c e s

B radley, R . S. 1931 P h il. M a g . Ser. 7, 11, 690—6.

Coolidge, A. S. 1926 J . A m er. Chem. Soc. 48, 1795—814.

Fow ler, R . H . 1932 T ra n s. F araday Soc. 28, 419—20.

— 1936a ‘‘Statistical M echanics.” Cambridge, 2nd ed.

— 19366 P roc. Camb. P h il. Soc. 32, 144—51.

K ruyt and Moddermann 1930 Chem. Rev. 7, 329.

London, F . 1931 Z . ph ys. Chem. B , 11, 222—51.

Lennard-Jones, J . E . 1932 Trans. F araday Soc. 28, 333—59.

M cBain, J . W . 1932 “ Sorption o f G ases” , p. 141.

M itchell, J . S. 1935 Trans. F araday Soc. 31, 980—6.

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