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Plane monolayer adsorption with area changes
JOURNAL OF COLLOID AND INTERFACE SCIENC~ 21, 358-366 (1966)
PLANE MONOLAYER ADSORPTION WITH AREA CHANGES G. D. Halsey, Jr. Department of Chemistry, University of Washington, Seattle, Washington 98105 Received April 2, 1965; revised January 3, 1966 ABSTRACT A specific thermodynamic treatment of adsorption at a plane surface, with and without surface lattice structure, is presented. Equations based on the Flood-Farhan treatment are given, and the Langmuir equation is used as an example although the treatment is applicable to any monolayer equation that can be expressed as a virialcoefficient expansion. The question of the natural variation of surface area with temperature and spreading pressure is emphasized. No general thermodynamic results are obtained, but lattice-spaced and self-spaced adsorbates are contrasted.
In the application of the Clausius-Clapeyron equation to adsorption equilibrium, as typified by the Langmuir equation, McBain noted a difficulty if the apparent adsorption maxima varied with temperature (1). For adsorption on a heterogeneous surface, this objection can be met (2) with the observation that the maxima are only apparent; nevertheless, it is incorrect to dismiss McBain's objection in every case. The current treatments of adsorption thermodynamics (3, 4), certainly with some justification, observe that for actual cases the true variation of monolayer capacity with temperature is not large enough to be important. In what follows an attempt is made to treat this problem and associated ones from a more formal point of view. EQUATIONS FOR A LIQUID OR STRUCTURELESS SOLID SURFACE
The thermodynamics of adsorption on a plane surface is usually developed in terms of variables primarily suited to a free liquid surface characterized by a surface tension or associated spreading pressure and the area of the interface. The spreading pressure and area are, at least in principle, viewable as external variables. Current treatments (5) adopt this formulation for treatment of adsorption on a solid surface. For a simple isotropic surface the area of which is independent of temperature and amount adsorbed, the treatment is straightforward. Here we wish to consider cases in which the area is not constant. If we follow the development of surface thermodynamics given by Guggenheim (6), the usual Gibbs dividing surface is introduced, and the surface thermodynamic quantities may be defined. For a two-component 358
PLANE MONOLAYER ADSORPTION WITH
AREA CHANGES
359
system, made up of two bulk phases with a single dividing surface, we may start with his equation 751: [1]
d E s = T d S ~ -~- 7 d A + ~2 d'n2.
Guggenheim then defines, in addition to the four usual thermodynamic functions E, H, F, and G, four other functions, indicated in his book by a bar, but here by a tilde, of the type = G-
~A
[2]
with three similar equations that define E , /t, and F. Hill (5), using different notation, has shown the utility of G a n d / t in developing expressions for heats and entropies calculated at constant spreading pressure. For these and many other calculations, it is convenient to arbitrarily divide E ~ and the other quantities into two terms, one assigned to the bare surface with the other accounting for the adsorbed material. For the bare surface, we have [3]
dEo ~ = T dSo ~ -[- "Yo d A .
If this quantity is subtracted from d E s we obtain the energy equation for the two-dimensional surface film dE ~=
dE ~-
dEo ~ = T ( d S ~ -
dSo ~) 4-
('7 - -
70) d A
+
#2dn2;
[4
= T d S ] - - ¢ p d A -{- g2dn2 ;
where S I = S ~ - So~, the entropy of the adsorbed film, and ¢ = "y0 - 7, the spreading pressure. This arbitrary separation of the surface contribution to the total value of d E into two parts is made in order to introduce the surface spreading pressure, which has the advantage of going to zero as n~ goes to zero. Otherwise a completely equivalent treatment could be made by retaining - ~ , for ¢. Similarly the introduction of S f for S" avoids the problem of the residual entropy of the bare surface, which for a solid adsorbent is not readily accessible to measurement. Note that G~ = Gt -]- cA, etc. With Eq. [4] as a start, the usual thermodynamic treatment of adsorption on a liquid surface can be made. For example, dF I = dG s = --S ] dT-
CdA
[5]
+ ~2dn~
and dlF] = dG ~" = - - S ~ d T
-[- A d4, + u2 dn2
;
[6]
from which, by integration at constant intensive variables, we have F s = GI = - C A + u2n~
[7]
360
HALSEY, fiR.
and
: '•
P
[S}
= O: = . : ~ .
(These equations are a reason for calling F:-the Helmholtz free energy of the surface, and/W-the Gibbs free energy of the surface; this leads to an unfortunate conflict with the equation F + P V = G, and so we avoid this simplification of notation.) We also find, by the equivalent of the Gibbs-Duhem equation 0 =
--S:dT
+ A d¢ -- n~dt~2,
[9]
which leads directly to the Gibbs adsorption isotherm. At constant temperature d~ = r dt~2,
[10]
where F = n 2 / A . THE
F L O O D - F A R H A N T R E A T M E N T OF A S O L I D - S u P P O R T E D S U R F A C E F I L ~ [
The only work term in the surface free energy (Eq. [5]) is --4~ dA. However, a solid can have its surface properties changed by strain, even at constant area. Thus an added work term is necessary. This work has been formally symbolized by Flood and Farhan (7) as - ~ dq. It is important to note that the displacement q is an intensive quantity, such as a lattice parameter, and thus the generalized potential ~ is an extensive variable. It is quite possible to have a number of terms of this type, but one will be sufficient for our purpose. With the inclusion of t h i s work term, Eq. [4] becomes dE: = T dS: -- e: d A -
• dq + ~2 dn~
[11]
with similar inclusion of the extra work term in Eqs. [5] and [6]. Since q is intensive, Eqs. [7] and [8] remain the same. Equation [9] becomes 0 = --S:dT
+ A de -- ,~dq -- n~dtt2.
[12]
This equation leads to Flood's variation of the Gibbs isotherm for solids dep = ~8 dq + r d ~ ,
[13]
where ~ is the intensive quantity ~ / A . It also provides the specification for the potential eb~ : At constant temperature and spreading pressure, ¢~ = - F(O#2/Oq),,~ .
[14]
Not only can the area be changed at constant lattice parameter, by manipulation of the usual barrier, but the chemical potential can be changed by stretching or compressing the lattice underneath the film, even at constant spreading pressure.
PLANE MONOLAYER ADSORPTION WITH AREA CHANGES
36i
THE GAS PHASE In the gas phase, which contains only component 2, we have a number of standard expressions among which we will need E ~ = /~ g and H g = Hg = E +
[15]
PV;
[16]
d~2 = --Sg d T + fz~ dP,
Where S~ and ~o are the usual molar quantities. THE
VARIATION OF CHEMICAL POTENTIAL WITH TEMPERATURE
The chemical potential of component 2 in the surface can be developed in terms of G or 0. Thus at constant n2, from Eq. [5], with an added term in dq, as in Eq. [11], we have dG s = - - S / d T -
[17]
~ d A -- ~ dq.
Thus, as a function of T, A, and q, at constant n~ d ~ ( T , A , q) = d(OGS/On2).4,r,q ; =
(OSf/On2).~,r,q d T -- (O,~/On2).~,r,q gA
-
- -
[18]
(O~/On2)~,r,q dq.
Although the first coefficient is sometimes called the partial molar entropy of the film, these coefficients are not true partial molar quantities. However, if we work with G, and start with Eq. [6], with the added term in dq, we find d i n ( T , ¢, q) = -- (OS/On~.)T,¢,q d T
[191 + (OA/On2)r,¢,q d4~
- -
(O~/On2)r.~,,~ dq.
Comparison of this equation with Eq. [12] allows the coefficients to be evaluated d ~ ( T , ~, q) = --(S~/qn2) d T + ( l / F ) d~ -
( ~ / F ) dq.
[20]
We see that in this system of variables the coefficients have the partial molar quality; not only does 0 = n2~2, but (recall~ = (~/A and F~ = n 2 / A ) 5 s = n2(OSI/On~)r,~,q
[21]
A = n~(OA/an:)r,~,q
[21]
= n~(O~/On2)r,~,,q.
[23]
and and
There is no reference to component 1, for by the definition of the Gibbs dividing surface it is excluded from the film.
362
HALSEY, JR. THE LANGMUIR EQUATION AS EXAMPLE
Various properties of the Langmuir equation can be used to illustrate the working use of the thermodynamic relationships. I t can be written P / P o = 0/(1 - 0) = 0 ~- 02 ~- 03, etc. = Y,a~Oi.
[24]
A more general equation can be written where the coefficients depend on temperature, or at least differ from unity. But in general the expansion is in terms of the variable which we will write explicitly [25]
0 = n~4_q/A.
In these terms the Langmuir equation has the form P / P o ( q , T ) = 1/[(A/Aqn~) -- 1],
[26]
which takes specific account of the three possible variables: the total area A, the minimum area per mole A q , which as indicated m a y depend on q, and n2, the number of moles adsorbed. We have also indicated that the reference pressure P0 may depend on q as well as T, although we shall not consider such cases here. If the gas phase is perfect, the chemical potential of component 2 is given by tL2 - ~2° = R T In Po - R T In [(A/A~n2) -- 1].
[27]
The spreading pressure, measured at constant q, from Eq. [10], is dp/RT = - ( 1 / A q )
In [1 -
(Aqn2/A)],
[28]
which reduces to the perfect gas law in two dimensions at small n2 or large A ; ¢p/RT = n2/A.
[29]
In general, the spreading pressure can be written as ch/RT = ( 1 / A q ) Y,c~O~.
[30]
From this result, it is apparent that the q~A product can be expressed ¢ A / n 2 R T = f(O)
[31]
so that this product remains constant at constant n2, T, and 0, even though and A may change. We have concentrated attention on the Langmuir equation because it is simple to handle; in particular, Eq. [28] can be solved for the 0 variable in terms of spreading pressure 0 = Aqn2/A = 1 -
e-~A~/Rr.
[32]
This result can conveniently be substituted into the Langmuir equation,
PLANE MONOLAYER ADSORPTION WITH AREA CHANGES
363
and its associated chemical potential, to give tL~ -
~2° = R T in Po -t- R T In [exp ( r h A q / R T )
-- 1],
[33]
which is suitable for differentiation at constant 4i. TYPES OF EQUATIONS
The Langmuir equation is used to stand formally for any isotherm equation that can be expanded in a dimensionless 0 variable, such as the Vohner equation, or any two-dimensional virial-eoefficient equation. Ordinarily any area change actually taking place in a reversible manner will, in the case of a solid, be accomplished by a change in a lattice dimension rather than an increase in lattice sites. However, there are two distinct types of isotherm models. In the first the parameter A~ is lattice or adsorbent determined, as in the localized Langmuir model. Secondly, Aq may be adsorbate determined as in nonideal mobile adsorption. In the second case, q may be treated as a constant in that it does not affect A q . USES WHERE q IS CONSTANT
The various analogous forms to the Clapeyron equation are all obtained by equating d~2 in the gas to d ~ on the surface. In particular, we observe the simplest results by equating Eq. [16] and Eq. [18]: (OP/OT)n,A,q = (1/Vg)[S " -
(OSf/On~)T,~];
[34]
and similarly, from equating Eq. [16] and Eq. [20] (OP/OT)~,¢,q = (1/P")[S" -
(S//n2)]
[35]
The particular utility of Eq. [34] comes from the association of it with heats of adsorption. If, in Eq. [34], we use the perfect gas law for ~'g and observe that since the chemical potentials for component 2 in the gas and on the surface are the same tt~ = I7tg -
TS g = (OHJ/On2)r,~,q-
T(OSJ/On2)r,~,q,
[36]
we find [0 i n P / O ( 1 / R T ) ] n , A , q -= (OHZ/On2)r,x,q -- Iri g.
[37]
The right-hand side of this expression is often called the differential enthalpy of adsorption, and can be identified with isothermal calorimetric measurements. Similar results are obtained with Eq. [35], except that the differential coefficient in Eq. [37] is replaced by (HI~n2), which is not an enthalpy. The two types of differentiation are discussed at length in references 3, 4, and 5. Equation [37] is the natural differentiation to make, if the area of the crystal is constant, since n2, one of the primary measured quantities, is
364
HALSEY, JR.
easily held constant also. Even if the area is constant, the application of Eq. [35] requires a computation of spreading pressure in order that the differentiation can be carried out. Similarly, if the area is not constant, can be carried out. Similarly, if the area is not constant, computations are required to apply either type of differentiation. If the area is determined by thermal expansion of the underlying crystal, we may assume a relationship of the form dA = coast. × dT.
[38]
I t would seem that we could no longer apply Eq. [36] to evaluate the enthalpy of adsorption without including a correction term. However, if we made a change of variable from A to F, the surface concentration, and observe that at constant n and T dA = --
(m/r ~) dr,
[39]
we can write
(n/r~)(O~/dn~)~,.~ dr.
d#2 = - (osf/On~)r,A d T +
[40]
From this expression at constant F and from Eq. [16] we obtain (OP/OT)~,r,q = (OP/OT)~,A,q = ( O P / O T ) r . q .
[41]
Thus, one way of taking the correct derivative at constant A is to take a derivative at constant r . This procedure is essentially that suggested b y McBain. Now Eq. [35] for S f / n : formally applies at constant ~ whether or not area is a function of temperature. But since the Gibbs adsorption isotherm, which must be integrated to find ~, is expressed entirely in intensive variables, including F, a knowledge of the changing area is necessary in order to carry out an actual differentiation at constant ~. .
°
CASES WHERE q IS NOT CONSTANT The simplest case where q must be considered as a variable arises in the Langmuir equation when the variation of area with temperature is due solely to the change in the lattice constant or unit area Aq in the plane o~ the surface. I n such a case dA/A
= dAq/A~.
[42]
If we write Eq. [18] in terms of the variables T, A, and Aq, at constant n2 d ~ ( T , A , Aq) = -- (OSf/Or~)r,~,A, d T
[43] However, if we note that the Langmuir equation has a chemical potential
PLANE
MONOLAYER
ADSORPTION
WITH
AREA
CHANGES
365
that is a function of the ratio n2Aq/A = 0 and not of these quantities independently, we find (cf. Eq. [27]) (O¢~/On2)r,~,~ = (Og2/OA)r,~ = -- (0t~2/00)(n2Aq/A ~)
[44]
( OO/On2)r,~,A~ = ( &~/OAq) ~,~ = ( 0~2/00) (n2/A ).
[45]
and
When Eq. [42] holds, these results ensure that the last two terms in Eq. [43] cancel, so that with the use of Eq. [16] we find the simple result d P / d T = (1/Vg)[S g -
(OSS/On~)r,~].
[46]
No such simple result is obtained with the derivative at constant spreading pressure. However, if the expansion of the lattice with temperature is ignored, and a false spreading pressure is calculated, the equivalent of Eq. [35] will result. Isothermal Dimensional Changes. Flood and Farhan (7) have pointed out that an equation of the type of Eq. [12] is necessary to treat the general case of adsorption accompanied by dimensional changes. In the absence of a term in dq, Eq. [13] can be integrated to give the spreading pressure =
r d~.
[47]
Now, if as is usually the case, the distortion is small, it will be linear with spreading pressure and proportional to this integral. However, such a treatment is predicated on the absence of a change in a parameter q; specifically, if ~ is a function of 0 = nAq/A, then Aq must be a constant independent of A. This is a characteristic of the mobile-gas type of equation of state on the surface. On the other hand, if we must include a variation in the parameter Aq, for example, of the type given by Eq. [42], we find that the spreading pressure is still given by Eq. [47] if a barrier is moved across the solid at constant Aq. However, the work done by the film is done on the underlying solid by a lattice expansion rather than a barrier motion and is the sum at constant T, dA ~ = dW = - - ¢ d A - - ~ d A q = --d(C~A) + A d C - - O d A q .
[48]
If we apply the complete Gibbs isotherm, Eq. [13], to this result, we find dW = - d ( ~ A )
+ n2 d#2.
[49]
Now for an expansion at constant n2 and constant ratio Aq/A, 0 is constant and, if we recall Eq. [31] and Eqs. [42] through [45], so are g~ and 6 A P V . Therefore, no work is done to stretch the lattice. If we rearrange Eq. [48]
366
HALSEY, JR.
using the condition Eq. [42], we find d W = -[,~ + (Aq/A)~] dA = O.
[50]
The two terms in the bracket cancel one another, and thus there is no force available to distort the crystallite. Because of the fact that the coefficients c~ in Eq. [30] have been assumed independent of q, this result (Eq. [50]) and the conclusion drawn from it, apply only in the absence of lateral forces between the immobile adatoms. CONCLUSION
It appears that in order to calculate isothermal data (heats of adsorption) with the temperature coefficient of adsorption isotherms (dP/dT), specific information about mechanicM changes in the adsorbate is necessary. Guggenheim (8) has objected to requirements for information of this type, and proposed a "black box" approach that avoids mention of area and spreading pressure. However, we have not found specific examples of such relationships and further consideration of more general results will not be found here. It is interesting that the results given here may serve to distinguish in yet another way between mobile, or imperfect gas-like adsorption, on the one hand, and immobile, or site-determined adsorption, on the other. This would be based on whether Aq, the steric parameter in the isotherm equation, was independent (mobile) or strictly associated with (immobile) the lattice parameter, q. Explicit intermediate isotherms, perhaps containing two steric factors, one dependent and the other independent, are as yet an unrealized possibility but have some interest. A specific purpose of this paper was to correct the impression (2) that l~/IcBain's suggestion about Langmuir maxima was always wrong. It clearly is not. REFERENCES 1. McBAIN, J. W., "Sorption of Gases and Vapours " Routledge, London, 1932. 2. HALSEY, G., J. Chem. Phys. 16, 931 (1948). 3. YounG, D. M., AND CnOW~.LL, A. D., "Physical Adsorption of Gases." Butterworths, Washington, 1962. 4. Ross, S. AND OLIVER, J. P., "On Physical Adsorption." Interscience Publishers, New York, 1964. 5. HILL, T. L., Advan. Catalysis 4, 211 (1952). 6. GCGGENHEIM, E. A., "Modern Thermodynamics by the Methods of Willard Gibbs." Methuen, London, 1933. 7. FLOOD, E. A., AND FA~HAN, F. M., Can. J. Chem. 41, 1703 (1963). 8. GUGGENHEIM,E. A., Proc. Conf. Interfaclal Phenomena Nucleation Boston 1951, 3, 61 (1955). AFCRC-TR-55-211B.
PLANE MONOLAYER ADSORPTION WITH AREA CHANGES G. D. Halsey, Jr. Department of Chemistry, University of Washington, Seattle, Washington 98105 Received April 2, 1965; revised January 3, 1966 ABSTRACT A specific thermodynamic treatment of adsorption at a plane surface, with and without surface lattice structure, is presented. Equations based on the Flood-Farhan treatment are given, and the Langmuir equation is used as an example although the treatment is applicable to any monolayer equation that can be expressed as a virialcoefficient expansion. The question of the natural variation of surface area with temperature and spreading pressure is emphasized. No general thermodynamic results are obtained, but lattice-spaced and self-spaced adsorbates are contrasted.
In the application of the Clausius-Clapeyron equation to adsorption equilibrium, as typified by the Langmuir equation, McBain noted a difficulty if the apparent adsorption maxima varied with temperature (1). For adsorption on a heterogeneous surface, this objection can be met (2) with the observation that the maxima are only apparent; nevertheless, it is incorrect to dismiss McBain's objection in every case. The current treatments of adsorption thermodynamics (3, 4), certainly with some justification, observe that for actual cases the true variation of monolayer capacity with temperature is not large enough to be important. In what follows an attempt is made to treat this problem and associated ones from a more formal point of view. EQUATIONS FOR A LIQUID OR STRUCTURELESS SOLID SURFACE
The thermodynamics of adsorption on a plane surface is usually developed in terms of variables primarily suited to a free liquid surface characterized by a surface tension or associated spreading pressure and the area of the interface. The spreading pressure and area are, at least in principle, viewable as external variables. Current treatments (5) adopt this formulation for treatment of adsorption on a solid surface. For a simple isotropic surface the area of which is independent of temperature and amount adsorbed, the treatment is straightforward. Here we wish to consider cases in which the area is not constant. If we follow the development of surface thermodynamics given by Guggenheim (6), the usual Gibbs dividing surface is introduced, and the surface thermodynamic quantities may be defined. For a two-component 358
PLANE MONOLAYER ADSORPTION WITH
AREA CHANGES
359
system, made up of two bulk phases with a single dividing surface, we may start with his equation 751: [1]
d E s = T d S ~ -~- 7 d A + ~2 d'n2.
Guggenheim then defines, in addition to the four usual thermodynamic functions E, H, F, and G, four other functions, indicated in his book by a bar, but here by a tilde, of the type = G-
~A
[2]
with three similar equations that define E , /t, and F. Hill (5), using different notation, has shown the utility of G a n d / t in developing expressions for heats and entropies calculated at constant spreading pressure. For these and many other calculations, it is convenient to arbitrarily divide E ~ and the other quantities into two terms, one assigned to the bare surface with the other accounting for the adsorbed material. For the bare surface, we have [3]
dEo ~ = T dSo ~ -[- "Yo d A .
If this quantity is subtracted from d E s we obtain the energy equation for the two-dimensional surface film dE ~=
dE ~-
dEo ~ = T ( d S ~ -
dSo ~) 4-
('7 - -
70) d A
+
#2dn2;
[4
= T d S ] - - ¢ p d A -{- g2dn2 ;
where S I = S ~ - So~, the entropy of the adsorbed film, and ¢ = "y0 - 7, the spreading pressure. This arbitrary separation of the surface contribution to the total value of d E into two parts is made in order to introduce the surface spreading pressure, which has the advantage of going to zero as n~ goes to zero. Otherwise a completely equivalent treatment could be made by retaining - ~ , for ¢. Similarly the introduction of S f for S" avoids the problem of the residual entropy of the bare surface, which for a solid adsorbent is not readily accessible to measurement. Note that G~ = Gt -]- cA, etc. With Eq. [4] as a start, the usual thermodynamic treatment of adsorption on a liquid surface can be made. For example, dF I = dG s = --S ] dT-
CdA
[5]
+ ~2dn~
and dlF] = dG ~" = - - S ~ d T
-[- A d4, + u2 dn2
;
[6]
from which, by integration at constant intensive variables, we have F s = GI = - C A + u2n~
[7]
360
HALSEY, fiR.
and
: '•
P
[S}
= O: = . : ~ .
(These equations are a reason for calling F:-the Helmholtz free energy of the surface, and/W-the Gibbs free energy of the surface; this leads to an unfortunate conflict with the equation F + P V = G, and so we avoid this simplification of notation.) We also find, by the equivalent of the Gibbs-Duhem equation 0 =
--S:dT
+ A d¢ -- n~dt~2,
[9]
which leads directly to the Gibbs adsorption isotherm. At constant temperature d~ = r dt~2,
[10]
where F = n 2 / A . THE
F L O O D - F A R H A N T R E A T M E N T OF A S O L I D - S u P P O R T E D S U R F A C E F I L ~ [
The only work term in the surface free energy (Eq. [5]) is --4~ dA. However, a solid can have its surface properties changed by strain, even at constant area. Thus an added work term is necessary. This work has been formally symbolized by Flood and Farhan (7) as - ~ dq. It is important to note that the displacement q is an intensive quantity, such as a lattice parameter, and thus the generalized potential ~ is an extensive variable. It is quite possible to have a number of terms of this type, but one will be sufficient for our purpose. With the inclusion of t h i s work term, Eq. [4] becomes dE: = T dS: -- e: d A -
• dq + ~2 dn~
[11]
with similar inclusion of the extra work term in Eqs. [5] and [6]. Since q is intensive, Eqs. [7] and [8] remain the same. Equation [9] becomes 0 = --S:dT
+ A de -- ,~dq -- n~dtt2.
[12]
This equation leads to Flood's variation of the Gibbs isotherm for solids dep = ~8 dq + r d ~ ,
[13]
where ~ is the intensive quantity ~ / A . It also provides the specification for the potential eb~ : At constant temperature and spreading pressure, ¢~ = - F(O#2/Oq),,~ .
[14]
Not only can the area be changed at constant lattice parameter, by manipulation of the usual barrier, but the chemical potential can be changed by stretching or compressing the lattice underneath the film, even at constant spreading pressure.
PLANE MONOLAYER ADSORPTION WITH AREA CHANGES
36i
THE GAS PHASE In the gas phase, which contains only component 2, we have a number of standard expressions among which we will need E ~ = /~ g and H g = Hg = E +
[15]
PV;
[16]
d~2 = --Sg d T + fz~ dP,
Where S~ and ~o are the usual molar quantities. THE
VARIATION OF CHEMICAL POTENTIAL WITH TEMPERATURE
The chemical potential of component 2 in the surface can be developed in terms of G or 0. Thus at constant n2, from Eq. [5], with an added term in dq, as in Eq. [11], we have dG s = - - S / d T -
[17]
~ d A -- ~ dq.
Thus, as a function of T, A, and q, at constant n~ d ~ ( T , A , q) = d(OGS/On2).4,r,q ; =
(OSf/On2).~,r,q d T -- (O,~/On2).~,r,q gA
-
- -
[18]
(O~/On2)~,r,q dq.
Although the first coefficient is sometimes called the partial molar entropy of the film, these coefficients are not true partial molar quantities. However, if we work with G, and start with Eq. [6], with the added term in dq, we find d i n ( T , ¢, q) = -- (OS/On~.)T,¢,q d T
[191 + (OA/On2)r,¢,q d4~
- -
(O~/On2)r.~,,~ dq.
Comparison of this equation with Eq. [12] allows the coefficients to be evaluated d ~ ( T , ~, q) = --(S~/qn2) d T + ( l / F ) d~ -
( ~ / F ) dq.
[20]
We see that in this system of variables the coefficients have the partial molar quality; not only does 0 = n2~2, but (recall~ = (~/A and F~ = n 2 / A ) 5 s = n2(OSI/On~)r,~,q
[21]
A = n~(OA/an:)r,~,q
[21]
= n~(O~/On2)r,~,,q.
[23]
and and
There is no reference to component 1, for by the definition of the Gibbs dividing surface it is excluded from the film.
362
HALSEY, JR. THE LANGMUIR EQUATION AS EXAMPLE
Various properties of the Langmuir equation can be used to illustrate the working use of the thermodynamic relationships. I t can be written P / P o = 0/(1 - 0) = 0 ~- 02 ~- 03, etc. = Y,a~Oi.
[24]
A more general equation can be written where the coefficients depend on temperature, or at least differ from unity. But in general the expansion is in terms of the variable which we will write explicitly [25]
0 = n~4_q/A.
In these terms the Langmuir equation has the form P / P o ( q , T ) = 1/[(A/Aqn~) -- 1],
[26]
which takes specific account of the three possible variables: the total area A, the minimum area per mole A q , which as indicated m a y depend on q, and n2, the number of moles adsorbed. We have also indicated that the reference pressure P0 may depend on q as well as T, although we shall not consider such cases here. If the gas phase is perfect, the chemical potential of component 2 is given by tL2 - ~2° = R T In Po - R T In [(A/A~n2) -- 1].
[27]
The spreading pressure, measured at constant q, from Eq. [10], is dp/RT = - ( 1 / A q )
In [1 -
(Aqn2/A)],
[28]
which reduces to the perfect gas law in two dimensions at small n2 or large A ; ¢p/RT = n2/A.
[29]
In general, the spreading pressure can be written as ch/RT = ( 1 / A q ) Y,c~O~.
[30]
From this result, it is apparent that the q~A product can be expressed ¢ A / n 2 R T = f(O)
[31]
so that this product remains constant at constant n2, T, and 0, even though and A may change. We have concentrated attention on the Langmuir equation because it is simple to handle; in particular, Eq. [28] can be solved for the 0 variable in terms of spreading pressure 0 = Aqn2/A = 1 -
e-~A~/Rr.
[32]
This result can conveniently be substituted into the Langmuir equation,
PLANE MONOLAYER ADSORPTION WITH AREA CHANGES
363
and its associated chemical potential, to give tL~ -
~2° = R T in Po -t- R T In [exp ( r h A q / R T )
-- 1],
[33]
which is suitable for differentiation at constant 4i. TYPES OF EQUATIONS
The Langmuir equation is used to stand formally for any isotherm equation that can be expanded in a dimensionless 0 variable, such as the Vohner equation, or any two-dimensional virial-eoefficient equation. Ordinarily any area change actually taking place in a reversible manner will, in the case of a solid, be accomplished by a change in a lattice dimension rather than an increase in lattice sites. However, there are two distinct types of isotherm models. In the first the parameter A~ is lattice or adsorbent determined, as in the localized Langmuir model. Secondly, Aq may be adsorbate determined as in nonideal mobile adsorption. In the second case, q may be treated as a constant in that it does not affect A q . USES WHERE q IS CONSTANT
The various analogous forms to the Clapeyron equation are all obtained by equating d~2 in the gas to d ~ on the surface. In particular, we observe the simplest results by equating Eq. [16] and Eq. [18]: (OP/OT)n,A,q = (1/Vg)[S " -
(OSf/On~)T,~];
[34]
and similarly, from equating Eq. [16] and Eq. [20] (OP/OT)~,¢,q = (1/P")[S" -
(S//n2)]
[35]
The particular utility of Eq. [34] comes from the association of it with heats of adsorption. If, in Eq. [34], we use the perfect gas law for ~'g and observe that since the chemical potentials for component 2 in the gas and on the surface are the same tt~ = I7tg -
TS g = (OHJ/On2)r,~,q-
T(OSJ/On2)r,~,q,
[36]
we find [0 i n P / O ( 1 / R T ) ] n , A , q -= (OHZ/On2)r,x,q -- Iri g.
[37]
The right-hand side of this expression is often called the differential enthalpy of adsorption, and can be identified with isothermal calorimetric measurements. Similar results are obtained with Eq. [35], except that the differential coefficient in Eq. [37] is replaced by (HI~n2), which is not an enthalpy. The two types of differentiation are discussed at length in references 3, 4, and 5. Equation [37] is the natural differentiation to make, if the area of the crystal is constant, since n2, one of the primary measured quantities, is
364
HALSEY, JR.
easily held constant also. Even if the area is constant, the application of Eq. [35] requires a computation of spreading pressure in order that the differentiation can be carried out. Similarly, if the area is not constant, can be carried out. Similarly, if the area is not constant, computations are required to apply either type of differentiation. If the area is determined by thermal expansion of the underlying crystal, we may assume a relationship of the form dA = coast. × dT.
[38]
I t would seem that we could no longer apply Eq. [36] to evaluate the enthalpy of adsorption without including a correction term. However, if we made a change of variable from A to F, the surface concentration, and observe that at constant n and T dA = --
(m/r ~) dr,
[39]
we can write
(n/r~)(O~/dn~)~,.~ dr.
d#2 = - (osf/On~)r,A d T +
[40]
From this expression at constant F and from Eq. [16] we obtain (OP/OT)~,r,q = (OP/OT)~,A,q = ( O P / O T ) r . q .
[41]
Thus, one way of taking the correct derivative at constant A is to take a derivative at constant r . This procedure is essentially that suggested b y McBain. Now Eq. [35] for S f / n : formally applies at constant ~ whether or not area is a function of temperature. But since the Gibbs adsorption isotherm, which must be integrated to find ~, is expressed entirely in intensive variables, including F, a knowledge of the changing area is necessary in order to carry out an actual differentiation at constant ~. .
°
CASES WHERE q IS NOT CONSTANT The simplest case where q must be considered as a variable arises in the Langmuir equation when the variation of area with temperature is due solely to the change in the lattice constant or unit area Aq in the plane o~ the surface. I n such a case dA/A
= dAq/A~.
[42]
If we write Eq. [18] in terms of the variables T, A, and Aq, at constant n2 d ~ ( T , A , Aq) = -- (OSf/Or~)r,~,A, d T
[43] However, if we note that the Langmuir equation has a chemical potential
PLANE
MONOLAYER
ADSORPTION
WITH
AREA
CHANGES
365
that is a function of the ratio n2Aq/A = 0 and not of these quantities independently, we find (cf. Eq. [27]) (O¢~/On2)r,~,~ = (Og2/OA)r,~ = -- (0t~2/00)(n2Aq/A ~)
[44]
( OO/On2)r,~,A~ = ( &~/OAq) ~,~ = ( 0~2/00) (n2/A ).
[45]
and
When Eq. [42] holds, these results ensure that the last two terms in Eq. [43] cancel, so that with the use of Eq. [16] we find the simple result d P / d T = (1/Vg)[S g -
(OSS/On~)r,~].
[46]
No such simple result is obtained with the derivative at constant spreading pressure. However, if the expansion of the lattice with temperature is ignored, and a false spreading pressure is calculated, the equivalent of Eq. [35] will result. Isothermal Dimensional Changes. Flood and Farhan (7) have pointed out that an equation of the type of Eq. [12] is necessary to treat the general case of adsorption accompanied by dimensional changes. In the absence of a term in dq, Eq. [13] can be integrated to give the spreading pressure =
r d~.
[47]
Now, if as is usually the case, the distortion is small, it will be linear with spreading pressure and proportional to this integral. However, such a treatment is predicated on the absence of a change in a parameter q; specifically, if ~ is a function of 0 = nAq/A, then Aq must be a constant independent of A. This is a characteristic of the mobile-gas type of equation of state on the surface. On the other hand, if we must include a variation in the parameter Aq, for example, of the type given by Eq. [42], we find that the spreading pressure is still given by Eq. [47] if a barrier is moved across the solid at constant Aq. However, the work done by the film is done on the underlying solid by a lattice expansion rather than a barrier motion and is the sum at constant T, dA ~ = dW = - - ¢ d A - - ~ d A q = --d(C~A) + A d C - - O d A q .
[48]
If we apply the complete Gibbs isotherm, Eq. [13], to this result, we find dW = - d ( ~ A )
+ n2 d#2.
[49]
Now for an expansion at constant n2 and constant ratio Aq/A, 0 is constant and, if we recall Eq. [31] and Eqs. [42] through [45], so are g~ and 6 A P V . Therefore, no work is done to stretch the lattice. If we rearrange Eq. [48]
366
HALSEY, JR.
using the condition Eq. [42], we find d W = -[,~ + (Aq/A)~] dA = O.
[50]
The two terms in the bracket cancel one another, and thus there is no force available to distort the crystallite. Because of the fact that the coefficients c~ in Eq. [30] have been assumed independent of q, this result (Eq. [50]) and the conclusion drawn from it, apply only in the absence of lateral forces between the immobile adatoms. CONCLUSION
It appears that in order to calculate isothermal data (heats of adsorption) with the temperature coefficient of adsorption isotherms (dP/dT), specific information about mechanicM changes in the adsorbate is necessary. Guggenheim (8) has objected to requirements for information of this type, and proposed a "black box" approach that avoids mention of area and spreading pressure. However, we have not found specific examples of such relationships and further consideration of more general results will not be found here. It is interesting that the results given here may serve to distinguish in yet another way between mobile, or imperfect gas-like adsorption, on the one hand, and immobile, or site-determined adsorption, on the other. This would be based on whether Aq, the steric parameter in the isotherm equation, was independent (mobile) or strictly associated with (immobile) the lattice parameter, q. Explicit intermediate isotherms, perhaps containing two steric factors, one dependent and the other independent, are as yet an unrealized possibility but have some interest. A specific purpose of this paper was to correct the impression (2) that l~/IcBain's suggestion about Langmuir maxima was always wrong. It clearly is not. REFERENCES 1. McBAIN, J. W., "Sorption of Gases and Vapours " Routledge, London, 1932. 2. HALSEY, G., J. Chem. Phys. 16, 931 (1948). 3. YounG, D. M., AND CnOW~.LL, A. D., "Physical Adsorption of Gases." Butterworths, Washington, 1962. 4. Ross, S. AND OLIVER, J. P., "On Physical Adsorption." Interscience Publishers, New York, 1964. 5. HILL, T. L., Advan. Catalysis 4, 211 (1952). 6. GCGGENHEIM, E. A., "Modern Thermodynamics by the Methods of Willard Gibbs." Methuen, London, 1933. 7. FLOOD, E. A., AND FA~HAN, F. M., Can. J. Chem. 41, 1703 (1963). 8. GUGGENHEIM,E. A., Proc. Conf. Interfaclal Phenomena Nucleation Boston 1951, 3, 61 (1955). AFCRC-TR-55-211B.