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Adsorption and surface heterogeneity
SURFACE SCIENCE 24 (1971) 391-403 8 North-Holland Publishing Co.
ADSORPTION
AND SURFACE
HETEROGENEITY
*
G. F. CEROFOLINI Uficio Consulenze Scientifiche, via Pier della Francesca 74,201W Milano, Italy
Received 5 June 1970 The problem, of finding the energy distribution function of a heterogeneous surface, is considered here. In order to solve the problem, we approximate the effective local isotherm (supposed to be Langmuir) with the nearest one, in the Lagrangian sense, in a given functional class. With this local isotherm, and Dubinin-Radushkevich (DR) as the overall one, we compute the energy distribution function. With this result, we are able to compare the developed theory with the Polanyi one. 1. Introduction A surface, heterogeneous with respect to physical adsorption of a given gas, can be considered as consisting of a family of non-interacting, energetically homogeneous, zones. If B(p, q) is the local isotherm, i.e. the law that relates the coverage 8 to pressure p in the zone with binding energy q, then the isotherm for the whole surface is given by +a, WP) =
s
ebdcP(ddq,
0
where cp(q)dq is the fraction
of surface with binding
energy between
q and
q+dq. The most important problem arising from eq. (1) is the following: “given the local isotherm 6(p, q) and the overall isotherm 9(p), determine the distribution function that satisfies eq. (1)“. In the following, we shall consider only coverages less than one monolayer, for which lim
O(p,q)=l,
p++C0
lim
9(p)
= 1.
p++a,
* Part of this work has been supported by a SAES Getters SpA grant. 391
(2)
(3)
392
G. F.
Substituting relationships condition for q(q),
CEROFOLINI
(2) and (3) in eq. (1) we get the normalization +m
s
cP(ddq= 1,
0
which suggests that eventual solutions of eq. (1) belong to the functional space L’ (0, + 00)~ consisting of the summable functions on the positive semiaxes (0, + a). If no solutions exist in this space, they must be looked for in the larger distribution space, as shown by a simple example: if 9 (p) = 0 (p, 4), then the solution of eq. (1) is cp(q) = 6 (q-q) which corresponds a homogeneous surface with binding energy q. In this paper we shall consider as local isotherm the Langmuir one
O(P,4) = ~~
P
p+PLev(-q/W’
and as overall isotherm a(P)=
the DR onei-3)
expC- B(RT In(P~/P>>“I =P i
]
G p. y
-=P’PO?
where the symbols have the usual meaning. Both eqs. (4) and (5) satisfy relationships (2) and (3); furthermore Langmuir local isotherm satisfies the following asymptotic condition
(5)
the
i.e., Langmuir’s isotherm behaves at low pressures as Henry’s isotherm. In contrast with this, the DR isotherm does not have this property; the meaning of this is discussed elsewhere4). The problem of finding the distribution function q(q) satisfying eq. (1) when the local isotherm is the Langmuir isotherm, and the overall isotherm is DR isotherm, has not yet been solved and in order to find approximated solutions, two methods have been developed. In the first method, the overall isotherm is approximated by anothers), obeying analytical requirements such that the computation technique described by Sipsc) can be used. In the second method, the local isotherm is replaced by a kernel which, considered as a function of p, contains a discontinuity or an angular point 4,‘~ 8) (i.e., a point where the first derivative is discontinuous). So doing, the first kind Fredholm integral equation (1) is easily reduced to a Volterra integral equation. whose solution method is well knowng).
393
ADSORPTION AND SURFACEHETEROGENEITv
In this paper
we prefer the second method
since the overall
isotherm
is
the only sure datum* of our problem. As a consequence of our choice a preliminary problem raises up, i.e., to choose - in a given functional class - the kernel which is the best approximation
to the Langmuir
local isotherm. 2. The variational problem
Let us consider ditions :
a family
(9
of local isotherms
H'(p,q)=
satisfying
the following
con-
H(P, 4)-=P G P'(4),
I 1
GP>
p'(4);
p’(q) is a “condensation pressure”, and is - for physical reasons - a monotonically decreasing function of q. The variational problem we consider is the following: “find the function p=p’ (q) such that the Lagrangian distance between H’(p, q) and 0 (p, q):
(ii)
d [H’, O] =
sup
HO. +=Jj
W’(PT 4) - B(P, 41
for all q”.
is minimum,
2.1. CONDENSATION LOCAL ISOTHERM The condensation
local
isotherm
K(PY 4) = 40
F
Oc=P G Pc(d,
0
%(p,q)
r____________________________-___
-P
(7)
1 l~P>Pc(q).
:
0
Fig. 1.
is the following*)
-
(-$)
P/P,
sy-&)
The Langmuir isotherm compared with the nearest condensation
local isotherm.
* The DR isotherm has been tested for various gas-surface systems and for coverages 9 between 10e6 and 10-r; for lower values of 9 the isotherm should behave as the Henry isotherm*); for greater values of 9 the isotherm falls in the BET range, i.e., on the surface there is a multi-layer adsorption. As a general rule, the DR isotherm holds true for heterogeneous surfacesla~4.11-1*) and this is the experimental background of our considerations.
394
G. F. CEROFOLINI
From fig. 1 it is easily seen that the distance
between
8 (p, q) which are continuous
(for all local isotherms
I-I, and 0 is minimum
* with respect top) when
e(Pc(4)t4) = 4. Replacing
in the implicit
equation
PC(q)= so that the best approximating K(P,
pL
(8) the expression
O-=P
(4) we obtain
exp(-- q/W, is ** :
local isotherm
4) =
(8)
< pLexp(- q/RT),
(7’)
i l-=p>PLexp(-q/W.
2.2. ASYMPTOTICALLYCORRECTLOCALISOTHERM Now we consider
the following
H,(PT4) =
adsorption
isotherm
PIPLexp(-q/RT)~pdP,(4),
1 l
which is called “asymptotically
-=P ’
correct”
P,(4),
since it satisfies conditions
(2) and
(6).
Fig.2. The Langmuir isotherm compared with the nearest asymptotically correct local isotherm. * If the local isotherm is the Hill-De Boer isotherm, then the value of the condensation pressure is suggested by different considerations*). ** Hobson and Armstrong4) considered the following condensation pressure P (4) =
PO em
(-
q/RT).
(9’)
With this choice (which shows the correct functional dependence on q) the condensation on the zone with zero binding energy occurs at the vapour pressure of the adsorbate (it is to be remembered that the DR isotherm holds true at temperatures lower than that of vapour condensation). In such a manner, with the choice (9’), Hobson and Armstrong have also taken into account the imperfect nature of the gas.
ALISORPTTON
AND SURFACE
395
HETEROGENEITY
From fig. 2 it is easily seen that the function which solves the variational problem is implicitly given by Pa(q) ~ exp(q/RT) PL
=
1,
from which we obtain pa(q) = pL exp (-
q/RT).
(11)
Under condition (1 l), eq. (10) becomes (P/P,) exp(qlRT)
* P 6 pL exp (-
-=P>PLexp(-q/RT).
q/RT),
(10’)
An isotherm of type (10) has already been considered by Hobson’), who also imposed a condition of type (11) for condensation pressure. However, the choice (11) in the work of Hobson is an “ad hoc” hypothesis and is not upheld by considerations of variational nature. The condensation pressure pa(q) [=pc(q)] satisfies the requirement (ii), so that no other conditions must be imposed. 3. Distribution function In this section we shall compute the energy distribution function corresponding to local isotherms RC(p, q) and ii,(p, q) 3.1. CONDENSATIONLOCALISOTHERM
Let us consider the inverse function of pc (q) Q(P) = RT In(p,/p),
and express the overall isotherm as a function of Q. From eq. (5) we obtain
exp[I-
B(Q
-
QJ'l-=Q
2 Qo,
*Q
(12)
where
Q. =RTln(pL/po).
(13)
Replacing in eq. (1) the best condensation local isotherm, eq. (7’), and the DR isotherm, eq. (12), we obtain +CO
s
4oc(ddq = g(Q),
Q
396
0.F.
which can be solved by partial
the distribution
.~. 0
function
CEROFOLINI
differentiation
is reported
with respect
to Q:
in fig. 3.
._I_._
2
I
3
4
5
Fig. 3. The distribution function according to the condensation approximation: the function is compared with a Gaussian one (dashed line). This is constructed according to the following rules: the maximum of the Gaussian function is the same of the Maxwellian one, and the width of the curve computed at a half of the maximum height is the same of the Maxwellian function.
3.2. ASYMPTOTICALLYCORRECTLOCALISOTHERM From eqs.(l)and(1O’)we have
where Q(p) is the inverse function of p, (q). Expressing, by means of eq. (1 l), p as a function
of Q we obtain
exp(-~~)jexp(~)U.~q)dq + ? rp,(cddq=g(Q), (14) 0
Q where ,i?(Q) is given by eq. (12). By partial differentiation of eq. (14) with respect following first kind Volterra integral equation exp(-,e,)jexp(~T),.(q)dq=-RT~, 0
to Q we obtain
the
ADSORPTION
AND SURFACE
which reduces to a second kind entiation
Volterra
391
HETEROGENEITY
integral
equation
by partial
differ-
with respect to Q:
This equation
is solved in Appendix (Q)
cp
=
_
RT
B
A and its solution
is
a’So_ aS(Q)
(16)
x’
aQ2
i.e., for eq. (12)* - 2B(Q- Q~)"lewCB(Q- Qo>'l +2~(Q-Qo>ex~[-~(Q-Qo)21-=Q~QO~ 0 *Q
4. Average binding energy In this section, we shall limit our attention to the condensation local isotherm, since results achieved with this method are much simpler than those of the asymptotically correct local isotherm method. In particular we will focus our attention on the average binding energy that in previous works495) has been found to be Cj=(*n)+B-% For the first model considered
(17)
here the average
& =
Q,, it is
qC =
(Qo+u)2Buexp(-Bu2)du=Qo
+m
q4oo(q) dq = j-
0
energy is
Qo+(h)*@,
in fact, if we put p=q+CC s
binding
2Bu s
0
0 -l-co
x exp(-
Bu’) du +
2~2~’ exp(-
Bu’) du = Q. + (@)‘B-*
.
s
0
We can evaluate
Q, as follows.
The expression
p,_ = n,(2rcmk,T)+
of pL is
v. ,
where its is the number of sites per true unit area, m is the ad-atom mass, constant, and v. is the vibration frequency of the ad-
k, is the Boltzmann
* It is to be noted that the problem of finding the distribution function is simply solved in both cases, and the solution methods can be applied to overall isotherms different from the DR. An example of such application will be considered in Appendix B.
398
G. F. CEROFOLINI
atom perpendicular of p. as follows:
to the surface. We can give an approximated
= pL !L’ 5’ n,
expression
exp
~0
where vb, ni and qs represent respectively the vibration frequency perpendicular to the surface, the surface density and the heat of sublimation, all referred to the adsorbate in the bulk (=liquid) phase. Since it is ni/n,- 1 we obtain , which, compared
with eq. (13), gives Q. = qs + RT ln(v,/vb).
If we suppose that the adsorbate is in the liquid phase we obtain Then the expression of the average binding energy becomes
Q, =qs.
& = qs + ($T)+B-f.
(18)
But, if the adsorbate can be considered as a liquid, we can compare our result with those of the Polanyi potential theoryls). According to this theory, by using the Clausius-Clapeyron equation, we obtain the isosteric heat of adsorptionl6) : 4: = qs + [ln(1/9)]2 from which, by repeated of adsorption :
*, we obtain
integrations qt=
the average isosteric
qs + 2B-f.
From fig. 4 it is seen that the following qst > & )
Bet,
qst N qc )
(19)
relationships (because
$’ 9 $RT),
must hold: both are satisfied, as it is seen by comparing Let us compare now in greater detail the advantages respect to the Hobson
and Aimstrong
eqs. (18) and (19). of our model with
one, and to the Polanyi
one.
* In fact, if we put y = l/9, we obtain 1 [In (l/@12 d9 = / - i2 (111y)~dy = [b (lny)2 1 + 2 / _ $ lny dy = +m +CO 0 +CC
=2)[+yj +m
+ j-i2dd=2. +m
heat
ADSORPTION
AND SURFACE
399
HETEROGENEITY
The latter theories both predict an adsorption binding energy which depends on the nature of the adsorbent surface only through B-*, a parameter which - according to our considerations (see comments to fig. 3) gives the energetic heterogeneity of the surface. In particular, if we consider
Fig. 4.
Comparison
between isoteric heat of adsorption
and binding energy.
the adsorption of argon, we have B -*N 1 kcal/mole for practically all adsorbent solids: this implies (in those theories) binding energies practically constant for all systems; this fact is hard to believe. Furthermore two simple experimental evidences regret the quoted theories. The first evidence is the following: the Hobson and Armstrong choice of the condensation pressure (9’) implies the expression (17) for the average binding energy; the experimental value of Bw3 is, however, too low to give a reasonable accord with computer determinationsr7), at least in one case: i.e.14) argon adsorption on nickel. The second evidence is the following: the Polanyi theory gives an average adsorption binding energy always greater than q,, while BET measurements report that the adsorption binding energy can be lower or greater than qs. In our model, for a suitable choice of v,, we are able to remove all these difficulties. Appendix A
A second kind Volterra integral equationg) Q
s(Q)-~SF(Q,q)e(q)dq=g(q) 0
400
G. F. CEROFOLINI
has the solution
0
where
G(Q,~.)= +f ~"-1F,(Q,~), ll=l
(A21
F,(Q,q)=F(Q,q),
I
Q
F,(Q, 4)=
s 4
F,-,(Q, t)F(t,q)G
n> 1
From eq. (15) it is seen that if
(A3)
g(q)~-~Td2gce, aQ2
’
A = l/(RT)) we have the equation which interests us. For eq. (15) evaluating the sum (A2) with the conditions
(A3) we obtain
+g[fT+&(g)2+...}=
G(Q,q;rl)=exp(-CR?){1
1,
for which, from eq. (Al), we have eq. (16):
q(Q)=-RT-+--
’ _RTa2~(Q')dQ, =RTaZg(Q)
a's(Q) 1 aQ RT s
ag(Q>
aQ' -aQ*
,Q'z
0
OBSERVATION
If we consider
the functions O(P) = 1 - W),
K(PT 4) = 1 - B(PT 419 from eq. (I), under
normalization +m
condition
for q(q), we have
s
K(PY4)cP(d& = O(P).
(A4)
0
Now we approximate
K (p, q) with the kernel K(P,
K'(p,q)= () i
4) -z=P G JYq) v -=p>P(q),
(A9
ADSORPTION
AND SURFACE
401
HETEROGENEITY
where P (q) is a monotonically decreasing function of q, and then invertable. If Q=Q(p) is th e inverse function of P=P(q), eq. (A4) with kernel (A5) becomes Q(P) K(P,
4) v(q)
dq
ew
= O(P).
s
0
Considering Q as independent eq. (A6) becomes
variable
and expressing
p as a function
of Q,
Q
s
&Q, 4)cp(4)dq = e(Q)1
(A71
0
where
e(Q)=+(Q)).
R(Q,4)= K@'(Q),41,
By partial differentiation with respect to Q eq. (A7) becomes second kind Volterra integral equation Q
&Q,
Q)
cp(Q)+
s
R,(Q,
4) cp(4) dq =
the following
36(Q) mae-
>
0
whose solution method has been described before. It is to be noted that the two models considered in this work derive from particular applications of the general method described here. This method seems to us the most general one able to give approximated solutions of eq. (1). Appendix B It is claimedi*~4~is~s0) that in the low pressure limit the DR isotherm also reduces to the Henry isotherm. In this Appendix we suppose that effectively the DR isotherm reduces to the Henry one, and considering the asymptotically correct local isotherm, we shall determine self-consistently the pressure at which there is the transition between the DR and Henry isotherms. The overall isotherm we consider is the following &.cxp(Q/W c=Q>QH9 g(Q)= exp[-B(Q-Qo)Z1~QH>Q~Qo, -+Qo>Q, i 1 where k and QH are the parameters which we want to determine. Applying eq. (16) (which has been deduced independently by the overall
402
isotherm
G. F. CEROFOLINI
expression)
in the Henry range, we obtain q,a (Q)
So the distribution
= _
function
a’s(Q) _aS(Q! =
RT
aQ'
aQ
0.
becomes
0
( cP’(Q)=
-+Q~QH, 2B(Q - Qo)‘l+ 2B(Q xexp[-B(Q-QQ,)‘]+Q,>Q>Q,, 0 -=Qo'Q-
@BRTEl-
But (Pi must be normalized near p=O:
independently
Qo)}
from the behaviour
of 9 (p)
QH
l=
s
~~ce,dQ=[-RT~~~]Q=Q~-ia(Q,)-I)
0
=
2BRT(Q,
_
and this is just an equation 2BRT(Q,
,-B(QH-Q@
Q,)
{,-'J(QH-Qo)*
_
I},
for Qu _
Qo)
e-B(Q~-Qo)2 = ,-WQH-QO)~,
which is easily solved, and whose solutions
QH=m, The last result
_
is of present
are
QH - Q. = 1/2BRT. interest
and can be written
as a function
of
&~=Q~-Qo=RTln(p,/po): &H= 1/2BRT. The result (Bl) has been proposed by Hobson and Armstrong4) and was used by Schramls) in thermodynamical computations: strated that eq. (Bl) strictly follows from our formalism.
(Bl) as an ansatz, we demon-
References 1) 2) 3) 4) 5) 6)
M. M. Dubinin and L. V. Radushkevich, Dokl. Akad. Nauk SSSR 55 (1947) 331. M. G. Kaganer, Dokl. Akad. Nauk SSSR (1957) 251. M. M. Dubinin, J. Am. Chem. Sot. 81 (1959) 235. J. P. Hobson and R. A. Armstrong, J. Phys. Chem. 67 (1963) 2000. D. N. Misra, Surface Sci. 18 (1969) 367. R. Sips, J. Chem. Phys. 16 (1948) 490; R. Sips, J. Chem. Phys. 18 (1950) 1024. 7) J. P. Hobson, Can. J. Phys. 43 (1965) 1934; J. P. Hobson, Can. J. Phys. 43 (1965) 1941. 8) L. B. Harris, Surface Sci. 10 (1968) 129;
ADSORPTION
9) 10) 11) 12) 13) 14) 15) 16) 17)
18) 19) 20)
AND
SURFACE
HETEROGENEITY
L. B. Harris, Surface Sci. 13 (1969) 377; L. B. Harris, Surface Sci. 15 (1969) 182. see e.g.: F. G. Tricomi, Znregrul Equations (New York, 1957); A. E. Taylor, Introduction to Functional Analysis (New York, 1958). N. Hansen, Vakuumtechnik 11 (1962) 70. N. Endow and R. A. Pastemak, J. Vacuum Sci. Technol. 3 (1966) 196. R. Haul and B. A. Gottwald, Surface Sci. 4 (1966) 334. F. Ricca, R. Medana and A. Bellardo, Z. Physik. Chemie NF, 52 (1967) 276. A. Schram, Suppl. Nuovo Cimento 5 (1967) 276. M. Polanyi, Verh. Deut. Physik. Ges. 10 (1916) 55; M. Polanyi, Trans. Faraday Sot. 28 (1932) 316. A. Schram, Suppl. Nuovo Cimento 5 (1967) 309. The adsorption binding energy can be evaluated with the aid of potential reported in: F. Ricca, Suppl. Nuovo Cimento 5 (1967) 339; using the value of binding energy of the pair Ni-Ar reported in: J. Lorenzen and L. M. Raff, J. Chem. Phys. 52 (1967) 339. T. L. Hill, Advan. Catalysis 14 (1952) 211. J. P. Hobson, J. Vacuum Sci. Technol. 3 (1966) 281. S. E. Hoory and J. M. Prausnitz, Surface Sci. 6 (1967) 377.
403
maps
ADSORPTION
AND SURFACE
HETEROGENEITY
*
G. F. CEROFOLINI Uficio Consulenze Scientifiche, via Pier della Francesca 74,201W Milano, Italy
Received 5 June 1970 The problem, of finding the energy distribution function of a heterogeneous surface, is considered here. In order to solve the problem, we approximate the effective local isotherm (supposed to be Langmuir) with the nearest one, in the Lagrangian sense, in a given functional class. With this local isotherm, and Dubinin-Radushkevich (DR) as the overall one, we compute the energy distribution function. With this result, we are able to compare the developed theory with the Polanyi one. 1. Introduction A surface, heterogeneous with respect to physical adsorption of a given gas, can be considered as consisting of a family of non-interacting, energetically homogeneous, zones. If B(p, q) is the local isotherm, i.e. the law that relates the coverage 8 to pressure p in the zone with binding energy q, then the isotherm for the whole surface is given by +a, WP) =
s
ebdcP(ddq,
0
where cp(q)dq is the fraction
of surface with binding
energy between
q and
q+dq. The most important problem arising from eq. (1) is the following: “given the local isotherm 6(p, q) and the overall isotherm 9(p), determine the distribution function that satisfies eq. (1)“. In the following, we shall consider only coverages less than one monolayer, for which lim
O(p,q)=l,
p++C0
lim
9(p)
= 1.
p++a,
* Part of this work has been supported by a SAES Getters SpA grant. 391
(2)
(3)
392
G. F.
Substituting relationships condition for q(q),
CEROFOLINI
(2) and (3) in eq. (1) we get the normalization +m
s
cP(ddq= 1,
0
which suggests that eventual solutions of eq. (1) belong to the functional space L’ (0, + 00)~ consisting of the summable functions on the positive semiaxes (0, + a). If no solutions exist in this space, they must be looked for in the larger distribution space, as shown by a simple example: if 9 (p) = 0 (p, 4), then the solution of eq. (1) is cp(q) = 6 (q-q) which corresponds a homogeneous surface with binding energy q. In this paper we shall consider as local isotherm the Langmuir one
O(P,4) = ~~
P
p+PLev(-q/W’
and as overall isotherm a(P)=
the DR onei-3)
expC- B(RT In(P~/P>>“I =P i
]
G p. y
-=P’PO?
where the symbols have the usual meaning. Both eqs. (4) and (5) satisfy relationships (2) and (3); furthermore Langmuir local isotherm satisfies the following asymptotic condition
(5)
the
i.e., Langmuir’s isotherm behaves at low pressures as Henry’s isotherm. In contrast with this, the DR isotherm does not have this property; the meaning of this is discussed elsewhere4). The problem of finding the distribution function q(q) satisfying eq. (1) when the local isotherm is the Langmuir isotherm, and the overall isotherm is DR isotherm, has not yet been solved and in order to find approximated solutions, two methods have been developed. In the first method, the overall isotherm is approximated by anothers), obeying analytical requirements such that the computation technique described by Sipsc) can be used. In the second method, the local isotherm is replaced by a kernel which, considered as a function of p, contains a discontinuity or an angular point 4,‘~ 8) (i.e., a point where the first derivative is discontinuous). So doing, the first kind Fredholm integral equation (1) is easily reduced to a Volterra integral equation. whose solution method is well knowng).
393
ADSORPTION AND SURFACEHETEROGENEITv
In this paper
we prefer the second method
since the overall
isotherm
is
the only sure datum* of our problem. As a consequence of our choice a preliminary problem raises up, i.e., to choose - in a given functional class - the kernel which is the best approximation
to the Langmuir
local isotherm. 2. The variational problem
Let us consider ditions :
a family
(9
of local isotherms
H'(p,q)=
satisfying
the following
con-
H(P, 4)-=P G P'(4),
I 1
GP>
p'(4);
p’(q) is a “condensation pressure”, and is - for physical reasons - a monotonically decreasing function of q. The variational problem we consider is the following: “find the function p=p’ (q) such that the Lagrangian distance between H’(p, q) and 0 (p, q):
(ii)
d [H’, O] =
sup
HO. +=Jj
W’(PT 4) - B(P, 41
for all q”.
is minimum,
2.1. CONDENSATION LOCAL ISOTHERM The condensation
local
isotherm
K(PY 4) = 40
F
Oc=P G Pc(d,
0
%(p,q)
r____________________________-___
-P
(7)
1 l~P>Pc(q).
:
0
Fig. 1.
is the following*)
-
(-$)
P/P,
sy-&)
The Langmuir isotherm compared with the nearest condensation
local isotherm.
* The DR isotherm has been tested for various gas-surface systems and for coverages 9 between 10e6 and 10-r; for lower values of 9 the isotherm should behave as the Henry isotherm*); for greater values of 9 the isotherm falls in the BET range, i.e., on the surface there is a multi-layer adsorption. As a general rule, the DR isotherm holds true for heterogeneous surfacesla~4.11-1*) and this is the experimental background of our considerations.
394
G. F. CEROFOLINI
From fig. 1 it is easily seen that the distance
between
8 (p, q) which are continuous
(for all local isotherms
I-I, and 0 is minimum
* with respect top) when
e(Pc(4)t4) = 4. Replacing
in the implicit
equation
PC(q)= so that the best approximating K(P,
pL
(8) the expression
O-=P
(4) we obtain
exp(-- q/W, is ** :
local isotherm
4) =
(8)
< pLexp(- q/RT),
(7’)
i l-=p>PLexp(-q/W.
2.2. ASYMPTOTICALLYCORRECTLOCALISOTHERM Now we consider
the following
H,(PT4) =
adsorption
isotherm
PIPLexp(-q/RT)~pdP,(4),
1 l
which is called “asymptotically
-=P ’
correct”
P,(4),
since it satisfies conditions
(2) and
(6).
Fig.2. The Langmuir isotherm compared with the nearest asymptotically correct local isotherm. * If the local isotherm is the Hill-De Boer isotherm, then the value of the condensation pressure is suggested by different considerations*). ** Hobson and Armstrong4) considered the following condensation pressure P (4) =
PO em
(-
q/RT).
(9’)
With this choice (which shows the correct functional dependence on q) the condensation on the zone with zero binding energy occurs at the vapour pressure of the adsorbate (it is to be remembered that the DR isotherm holds true at temperatures lower than that of vapour condensation). In such a manner, with the choice (9’), Hobson and Armstrong have also taken into account the imperfect nature of the gas.
ALISORPTTON
AND SURFACE
395
HETEROGENEITY
From fig. 2 it is easily seen that the function which solves the variational problem is implicitly given by Pa(q) ~ exp(q/RT) PL
=
1,
from which we obtain pa(q) = pL exp (-
q/RT).
(11)
Under condition (1 l), eq. (10) becomes (P/P,) exp(qlRT)
* P 6 pL exp (-
-=P>PLexp(-q/RT).
q/RT),
(10’)
An isotherm of type (10) has already been considered by Hobson’), who also imposed a condition of type (11) for condensation pressure. However, the choice (11) in the work of Hobson is an “ad hoc” hypothesis and is not upheld by considerations of variational nature. The condensation pressure pa(q) [=pc(q)] satisfies the requirement (ii), so that no other conditions must be imposed. 3. Distribution function In this section we shall compute the energy distribution function corresponding to local isotherms RC(p, q) and ii,(p, q) 3.1. CONDENSATIONLOCALISOTHERM
Let us consider the inverse function of pc (q) Q(P) = RT In(p,/p),
and express the overall isotherm as a function of Q. From eq. (5) we obtain
exp[I-
B(Q
-
QJ'l-=Q
2 Qo,
*Q
(12)
where
Q. =RTln(pL/po).
(13)
Replacing in eq. (1) the best condensation local isotherm, eq. (7’), and the DR isotherm, eq. (12), we obtain +CO
s
4oc(ddq = g(Q),
Q
396
0.F.
which can be solved by partial
the distribution
.~. 0
function
CEROFOLINI
differentiation
is reported
with respect
to Q:
in fig. 3.
._I_._
2
I
3
4
5
Fig. 3. The distribution function according to the condensation approximation: the function is compared with a Gaussian one (dashed line). This is constructed according to the following rules: the maximum of the Gaussian function is the same of the Maxwellian one, and the width of the curve computed at a half of the maximum height is the same of the Maxwellian function.
3.2. ASYMPTOTICALLYCORRECTLOCALISOTHERM From eqs.(l)and(1O’)we have
where Q(p) is the inverse function of p, (q). Expressing, by means of eq. (1 l), p as a function
of Q we obtain
exp(-~~)jexp(~)U.~q)dq + ? rp,(cddq=g(Q), (14) 0
Q where ,i?(Q) is given by eq. (12). By partial differentiation of eq. (14) with respect following first kind Volterra integral equation exp(-,e,)jexp(~T),.(q)dq=-RT~, 0
to Q we obtain
the
ADSORPTION
AND SURFACE
which reduces to a second kind entiation
Volterra
391
HETEROGENEITY
integral
equation
by partial
differ-
with respect to Q:
This equation
is solved in Appendix (Q)
cp
=
_
RT
B
A and its solution
is
a’So_ aS(Q)
(16)
x’
aQ2
i.e., for eq. (12)* - 2B(Q- Q~)"lewCB(Q- Qo>'l +2~(Q-Qo>ex~[-~(Q-Qo)21-=Q~QO~ 0 *Q
4. Average binding energy In this section, we shall limit our attention to the condensation local isotherm, since results achieved with this method are much simpler than those of the asymptotically correct local isotherm method. In particular we will focus our attention on the average binding energy that in previous works495) has been found to be Cj=(*n)+B-% For the first model considered
(17)
here the average
& =
Q,, it is
qC =
(Qo+u)2Buexp(-Bu2)du=Qo
+m
q4oo(q) dq = j-
0
energy is
Qo+(h)*@,
in fact, if we put p=q+CC s
binding
2Bu s
0
0 -l-co
x exp(-
Bu’) du +
2~2~’ exp(-
Bu’) du = Q. + (@)‘B-*
.
s
0
We can evaluate
Q, as follows.
The expression
p,_ = n,(2rcmk,T)+
of pL is
v. ,
where its is the number of sites per true unit area, m is the ad-atom mass, constant, and v. is the vibration frequency of the ad-
k, is the Boltzmann
* It is to be noted that the problem of finding the distribution function is simply solved in both cases, and the solution methods can be applied to overall isotherms different from the DR. An example of such application will be considered in Appendix B.
398
G. F. CEROFOLINI
atom perpendicular of p. as follows:
to the surface. We can give an approximated
= pL !L’ 5’ n,
expression
exp
~0
where vb, ni and qs represent respectively the vibration frequency perpendicular to the surface, the surface density and the heat of sublimation, all referred to the adsorbate in the bulk (=liquid) phase. Since it is ni/n,- 1 we obtain , which, compared
with eq. (13), gives Q. = qs + RT ln(v,/vb).
If we suppose that the adsorbate is in the liquid phase we obtain Then the expression of the average binding energy becomes
Q, =qs.
& = qs + ($T)+B-f.
(18)
But, if the adsorbate can be considered as a liquid, we can compare our result with those of the Polanyi potential theoryls). According to this theory, by using the Clausius-Clapeyron equation, we obtain the isosteric heat of adsorptionl6) : 4: = qs + [ln(1/9)]2 from which, by repeated of adsorption :
*, we obtain
integrations qt=
the average isosteric
qs + 2B-f.
From fig. 4 it is seen that the following qst > & )
Bet,
qst N qc )
(19)
relationships (because
$’ 9 $RT),
must hold: both are satisfied, as it is seen by comparing Let us compare now in greater detail the advantages respect to the Hobson
and Aimstrong
eqs. (18) and (19). of our model with
one, and to the Polanyi
one.
* In fact, if we put y = l/9, we obtain 1 [In (l/@12 d9 = / - i2 (111y)~dy = [b (lny)2 1 + 2 / _ $ lny dy = +m +CO 0 +CC
=2)[+yj +m
+ j-i2dd=2. +m
heat
ADSORPTION
AND SURFACE
399
HETEROGENEITY
The latter theories both predict an adsorption binding energy which depends on the nature of the adsorbent surface only through B-*, a parameter which - according to our considerations (see comments to fig. 3) gives the energetic heterogeneity of the surface. In particular, if we consider
Fig. 4.
Comparison
between isoteric heat of adsorption
and binding energy.
the adsorption of argon, we have B -*N 1 kcal/mole for practically all adsorbent solids: this implies (in those theories) binding energies practically constant for all systems; this fact is hard to believe. Furthermore two simple experimental evidences regret the quoted theories. The first evidence is the following: the Hobson and Armstrong choice of the condensation pressure (9’) implies the expression (17) for the average binding energy; the experimental value of Bw3 is, however, too low to give a reasonable accord with computer determinationsr7), at least in one case: i.e.14) argon adsorption on nickel. The second evidence is the following: the Polanyi theory gives an average adsorption binding energy always greater than q,, while BET measurements report that the adsorption binding energy can be lower or greater than qs. In our model, for a suitable choice of v,, we are able to remove all these difficulties. Appendix A
A second kind Volterra integral equationg) Q
s(Q)-~SF(Q,q)e(q)dq=g(q) 0
400
G. F. CEROFOLINI
has the solution
0
where
G(Q,~.)= +f ~"-1F,(Q,~), ll=l
(A21
F,(Q,q)=F(Q,q),
I
Q
F,(Q, 4)=
s 4
F,-,(Q, t)F(t,q)G
n> 1
From eq. (15) it is seen that if
(A3)
g(q)~-~Td2gce, aQ2
’
A = l/(RT)) we have the equation which interests us. For eq. (15) evaluating the sum (A2) with the conditions
(A3) we obtain
+g[fT+&(g)2+...}=
G(Q,q;rl)=exp(-CR?){1
1,
for which, from eq. (Al), we have eq. (16):
q(Q)=-RT-+--
’ _RTa2~(Q')dQ, =RTaZg(Q)
a's(Q) 1 aQ RT s
ag(Q>
aQ' -aQ*
,Q'z
0
OBSERVATION
If we consider
the functions O(P) = 1 - W),
K(PT 4) = 1 - B(PT 419 from eq. (I), under
normalization +m
condition
for q(q), we have
s
K(PY4)cP(d& = O(P).
(A4)
0
Now we approximate
K (p, q) with the kernel K(P,
K'(p,q)= () i
4) -z=P G JYq) v -=p>P(q),
(A9
ADSORPTION
AND SURFACE
401
HETEROGENEITY
where P (q) is a monotonically decreasing function of q, and then invertable. If Q=Q(p) is th e inverse function of P=P(q), eq. (A4) with kernel (A5) becomes Q(P) K(P,
4) v(q)
dq
ew
= O(P).
s
0
Considering Q as independent eq. (A6) becomes
variable
and expressing
p as a function
of Q,
Q
s
&Q, 4)cp(4)dq = e(Q)1
(A71
0
where
e(Q)=+(Q)).
R(Q,4)= K@'(Q),41,
By partial differentiation with respect to Q eq. (A7) becomes second kind Volterra integral equation Q
&Q,
Q)
cp(Q)+
s
R,(Q,
4) cp(4) dq =
the following
36(Q) mae-
>
0
whose solution method has been described before. It is to be noted that the two models considered in this work derive from particular applications of the general method described here. This method seems to us the most general one able to give approximated solutions of eq. (1). Appendix B It is claimedi*~4~is~s0) that in the low pressure limit the DR isotherm also reduces to the Henry isotherm. In this Appendix we suppose that effectively the DR isotherm reduces to the Henry one, and considering the asymptotically correct local isotherm, we shall determine self-consistently the pressure at which there is the transition between the DR and Henry isotherms. The overall isotherm we consider is the following &.cxp(Q/W c=Q>QH9 g(Q)= exp[-B(Q-Qo)Z1~QH>Q~Qo, -+Qo>Q, i 1 where k and QH are the parameters which we want to determine. Applying eq. (16) (which has been deduced independently by the overall
402
isotherm
G. F. CEROFOLINI
expression)
in the Henry range, we obtain q,a (Q)
So the distribution
= _
function
a’s(Q) _aS(Q! =
RT
aQ'
aQ
0.
becomes
0
( cP’(Q)=
-+Q~QH, 2B(Q - Qo)‘l+ 2B(Q xexp[-B(Q-QQ,)‘]+Q,>Q>Q,, 0 -=Qo'Q-
@BRTEl-
But (Pi must be normalized near p=O:
independently
Qo)}
from the behaviour
of 9 (p)
QH
l=
s
~~ce,dQ=[-RT~~~]Q=Q~-ia(Q,)-I)
0
=
2BRT(Q,
_
and this is just an equation 2BRT(Q,
,-B(QH-Q@
Q,)
{,-'J(QH-Qo)*
_
I},
for Qu _
Qo)
e-B(Q~-Qo)2 = ,-WQH-QO)~,
which is easily solved, and whose solutions
QH=m, The last result
_
is of present
are
QH - Q. = 1/2BRT. interest
and can be written
as a function
of
&~=Q~-Qo=RTln(p,/po): &H= 1/2BRT. The result (Bl) has been proposed by Hobson and Armstrong4) and was used by Schramls) in thermodynamical computations: strated that eq. (Bl) strictly follows from our formalism.
(Bl) as an ansatz, we demon-
References 1) 2) 3) 4) 5) 6)
M. M. Dubinin and L. V. Radushkevich, Dokl. Akad. Nauk SSSR 55 (1947) 331. M. G. Kaganer, Dokl. Akad. Nauk SSSR (1957) 251. M. M. Dubinin, J. Am. Chem. Sot. 81 (1959) 235. J. P. Hobson and R. A. Armstrong, J. Phys. Chem. 67 (1963) 2000. D. N. Misra, Surface Sci. 18 (1969) 367. R. Sips, J. Chem. Phys. 16 (1948) 490; R. Sips, J. Chem. Phys. 18 (1950) 1024. 7) J. P. Hobson, Can. J. Phys. 43 (1965) 1934; J. P. Hobson, Can. J. Phys. 43 (1965) 1941. 8) L. B. Harris, Surface Sci. 10 (1968) 129;
ADSORPTION
9) 10) 11) 12) 13) 14) 15) 16) 17)
18) 19) 20)
AND
SURFACE
HETEROGENEITY
L. B. Harris, Surface Sci. 13 (1969) 377; L. B. Harris, Surface Sci. 15 (1969) 182. see e.g.: F. G. Tricomi, Znregrul Equations (New York, 1957); A. E. Taylor, Introduction to Functional Analysis (New York, 1958). N. Hansen, Vakuumtechnik 11 (1962) 70. N. Endow and R. A. Pastemak, J. Vacuum Sci. Technol. 3 (1966) 196. R. Haul and B. A. Gottwald, Surface Sci. 4 (1966) 334. F. Ricca, R. Medana and A. Bellardo, Z. Physik. Chemie NF, 52 (1967) 276. A. Schram, Suppl. Nuovo Cimento 5 (1967) 276. M. Polanyi, Verh. Deut. Physik. Ges. 10 (1916) 55; M. Polanyi, Trans. Faraday Sot. 28 (1932) 316. A. Schram, Suppl. Nuovo Cimento 5 (1967) 309. The adsorption binding energy can be evaluated with the aid of potential reported in: F. Ricca, Suppl. Nuovo Cimento 5 (1967) 339; using the value of binding energy of the pair Ni-Ar reported in: J. Lorenzen and L. M. Raff, J. Chem. Phys. 52 (1967) 339. T. L. Hill, Advan. Catalysis 14 (1952) 211. J. P. Hobson, J. Vacuum Sci. Technol. 3 (1966) 281. S. E. Hoory and J. M. Prausnitz, Surface Sci. 6 (1967) 377.
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