Crider, W. L., J . Am. Chem. SOC.78, 925 (1956). Gilles ie, M. D., thesis, University of California, Berkeley, 1967. Hirscifelder, J. O., Curtiss, C. F., Bird, R. B., “Molecular Theor of Gases and Liquids,” Wiley, New York, 1954a. Hirschferder, J. O., Curtiss, C. F., Bird, R. B., “Molecular Theory of Gases and Liquids,” p. 987, Wiley, New York, 1954b.
Hudson, G. H., McCoubrey, J. C., Trans. Faraday SOC.66, 761 (1960).
Kihara, T., Rev. Mod. Phys. 26, 831 (1953). Monchick, L., Mason, E. A., J . Chem. Phys. 36, 2746 (1962). Monchick, L., Yun, K. S., Mason, E. A., J . Chem. Phys. 89, 854 (1982). \ - - - - I -
Nelson, E. T., J . A p p l . Chem. 6 , 286 (1956). O’Connell, J. P., dissertation, University of California, Berkeley,
O’Connell, J. P., Prausnitz, J. M., IND.ENQ.CHEM.FUNDAMENTALS
8, 453 (1969).
O’Connell, J. P., Prausnitz, J. M., “Thermodynamic and Transgort Properties at Extreme Temperatures and Pressures,’’ . Gratch, ed., p. 19, ASME, ,New York, 1965. Olds, R. H., Sage, B. H., Lacey, W. N., Znd. Eng. Chem. 34, 1223 (1942).
Prausnitz, J. M., Benson, P. R., -4.Z.Ch.E. J. 6, 161 (1959). Reamer, H. H., Olds, R. H., Sage, B. H., Lacey, W. N., Znd. Eng. Chem. 36, 790 (1943). Rigby, M., Prausnitz, J. M., J . Phys. Chem. 72, 330 (1968). Schwerz, F. A,, Brow, J. E., J . Chem. Phys. 19, 640 (1951). Walker, R. E., Westenberg, A. A., J . Chem. Phys. 32, 436 (1960). RECEIVED for review January 29, 1968 ACCEPTEDDecember 5, 1968
1967.
ADSORPTION OF GASES ON SOLIDS Review of the Role of Thermodynamics H.
C. V A N N E S S
Rensselaer Polytechnic Institute, Troy, N . Y . 12181 It is the function of thermodynamics to relate those properties of a system required for practical or theoretical purposes to the parameters that are most readily measured, and thus to provide the maximum return of information for any investment in experiment. This paper explores how thermodynamics may most effectively serve this end in mixed-gas adsorption.
THEpurpose of this paper is to exploit the analogy between
the thermodynamics of solutions and the thermodynamics of mixed adsorbates. The study of vapor-liquid equilibrium plays an important part in solution thermodynamics, and this has a strong analogy with adsorbate-mixed gas equilibrium. There is an important difference, but this does not destroy the essential similarity. The framework of thermodynamics is a set of differential equations which interrelate the properties of a carefully defined system. Most of the properties are thoroughly abstract, but the equations provide a few connections with quantities regarded as having physical reality and which can be measured. The function of thermodynamics is to maximize the return in useful information for any investment in experiments which provide entry into the thermodynamic network of equations. It is necessary a t the outset to have a specific system to which the equations apply. This poses an immediate problem for the thermodynamicist, because the interfacial region is ill defined. This difficulty is circumvented by a trick devised by J. Willard Gibbs. The gas phase does not extend unchanged all the way to the solid surface. In the neighborhood of the solid, the gas-phase properties change, but they do not change abruptly. There is a region of change, and although the gradients in the properties with distance from the surface may be large, they are not infinite. Thus we cannot know precisely the extent of the interfacial region nor the exact distance into the gas phase that the solid makes its influence felt. Were we to compare actuality with a simple-minded picture of a gas phase that persisted unchanged all the way to the solid surface, we could say that the solid induces abnormalities in the gas phase close to the interface. The 464
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FUNDAMENTALS
trick therefore is to replace the real situation with a hypothetical one, which for purposes of thermodynamic analysis is intended to be equivalent. Thus in our minds we transform the actual interfacial region into one where we imagine that the gas phase persists unchanged up to the solid surface, and then we attribute the abnormalities in the properties of the real interfacial region to an imaginary mathematical surface, which we treat as a two-dimensional phase with its own thermodynamic properties. At the very least, this procedure provides us with precisely defined systems to which we may apply the equations of thermodynamics. Not only have we defined a two-dimensional phase to account for the abnormalities of the interfacial region, but we have also extracted these abnormalities from the three-dimensional gas phase so that this phase too may be treated precisely. Of course, abnormalities also exist within the solid, both a t and close to its surface. These are usually presumed not to change during an adsorption process, and on this basis are omitted from consideration. This presumption would seem unnecessary for most thermodynamic applications, for such applications depend on experimental measurements which cannot separate the observed results into parts attributed to changes on the gas side and on the solid side of the interface. Any property changes attributed to a twodimensional phase as a result of experiment reflect changes in the abnormalities of the entire surface region. Since the results of any thermodynamic analysis apply to exactly the same region, the cause of the property changes should be immaterial. The really important assumption is that although the twodimensional phase is presumed to be in thermal and mechani-
cal equilibrium with both the solid and the gas phases, it is in phase equilibrium only with the gas phase. We assume that phase equilibrium is always established between the adsorbate, considered as a two-dimensional phase, and the bulk gas phase. Clearly, our use of the term “two-dimensional phase” in no way restricts treatment to what is termed monolayer adsorption. We make the usual assumption that the interfacial area is an independent variable, uninfluenced by temperature, pressure, composition, or the amount of material adsorbed. It is clear that the two-dimensional phase must be treated as an open system, because of our assumption that it is always in equilibrium with the gas phase. For a homogeneous three-dimensional fluid phase treated as an open system, the fundamental therniodynainic property relation is:
d ( n C ) = T d ( n S ) - Pd(nV)
+C
(/A
dn,)
where U , S, and V are the molar internal energy, entropy, and volume, and the summation is over all chemical species. The chemical potential of species i is denoted by pa, and n, is the number of moles of species i. The total number of moles, n, is given by n = C nl. For a two-dimensional phase we have an entirely analogous expression of the fundamental property relation. The only difference is that pressure and volume are not appropriate coordinates for a two-dimensional system. Thus pressure is replaced by the spreading pressure, a,and the molar volume by the molar area, e:
+C
d ( n U ) = T d ( n S )- ad(%&)
(/L$ dna)
CI-JL,d
( ~ i )=] 0
If we expand the differentials and collect like terms, we get: n(dU - TdS
+ adQ. -
p,
dx,)4-
dn(U - T S
+ aa -
pa%,)=
0
Since n and dn are independent and arbitrary, the terms in parentheses must separately be zero. Thus dU = TdS - ad@
+ C (p, dx,)
U = T S - dt+
C(/.L,X,)
U
u + *a U
(2 )
(3 )
Equations 1 and 2 are similar, but there is an important difference. Equation 1 applies to a system of n moles, where n may vary; Equation 2 applies to a system where n is unity and is invariant. Thus Equation 2 is subject to the x, = 1 or that dxi = 0. restraint that Equation 3 dictates the possible combinations of terms which may be defined as additional primary thermodynamic
- TS
U $-
- TS
U
Internal energy
H F G
Enthalpy Helmholtz function Gibbs function
+ +- r a
Z(rizi) TS - u a Z(rtzi) TS Z(/.tiZi)
Z(/.tizi)
functions. There are just eight possible distinct combinations, only four of which have been given names and symbols and are in common use. These four are displayed in Table I. Unfortunately, both the Helmholtz function, F , and the Gibbs function, G, have been called “free energy” in the literature, and both have been given the symbol F. This has led to much confusion. I n some papers no explicit definition has been given for the particular “free energy” used, and the reader has been left to guess which function the author had in mind, if indeed the author knew. I make here a strong plea for the use of an explicit terminology, and recommend a t the very least the use of the modifiers Helmholtz and Gibbs to distinguish these functions. From Table I we may write general expressions for H , P , and G in accord with their definitions. For example, H= U+dt
or n H = n U + s ( n @ )
Thus
(1)
The product ( n a ) is, of course, the total area A . The fipreading pressure is not subject to direct experimental measurement, but must be calculated, as shown later. This significantly complicates the treatment of a two-dimensional phase in comparison with a three-dimensional phase. However, any proper and effective thermodynamic treatment of a two-dimensional phase requires that the spreading pressure be accorded its full significance as a thermodynamic coordinate. Equation 1 is called the fundamental property relation because all other equations interrelating thermodynamic properties of the two-dimensional phase are derived from it. The most direct procedure is as follows. Since n, = ma, where x, is the mole fraction of i in the two-dimensional phase, we may rewrite Equation 1: d ( n U ) - Td(nS) f a d ( n a ) -
Table I. Primary Thermodynamic Functions Primary Alternative Grouping Symbol Name Grouping
d(nH) = d ( n U )
+ ad(na) + (na)da
Substitution for d ( n U ) by Equation 1 then gives a general expression for the total differential, d (nH). Equations for d (nF) and d (nG) are derived in the same way. These three differential equations together with Equation 1 represent a set of basic relations interconnecting the primary thermodynamic variables: d ( n u ) = Td (nS) - r d ( n a ) d (nH) = Td (nS)
+
+ (na)da4-
d ( n F ) = - (nS)dT - ad ( n a ) 4d (nG) =
- (nS)dT +
(pidni)
(1)
(pidni)
(4)
(pidni)
(5)
(na)da-I- C (pidnc)
(6)
An analogous set of equations could be developed from Equation 2. They would be valid for a system where n is always unity. The dni’s would become dxi’s and would be subject to the restraint that C dxi = 0. In such equations the xi’s must never be treated as though they were all independent variables. Many mathematical operations may be carried out with these equations. For example, it is clear from Equation 6 that
where Gi is the partial molar Gibbs function and is equal to the partial derivative by definition. Subscript i may refer to any component, and subscript n, denotes all mole numbers except the ith. From Table I it is seen that the Gibbs function, G, i s also related to the chemical potentials by or
VOL.
8
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3 AUGUST
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465
Equations 7 and 8 express the complete formal relationship of the chemical potentials to the Gibbs function. The Gibbs function can be nondimensionalized by dividing it by R T -i.e., GIRT is dimensionless, and is a useful function. The following is a mathematical identity:
By definition, nG = nH - T ( n S ) . Substitution for nG by this expression and for d (nG) by Equation 6 in the preceding equation gives:
nH
X(2dni)
(9)
(-&dxi)
(10)
For a system of one mole, this becomes:
H -dT+ R T2
U
d - = -dr(:T) RT
If the gas phase is ideal-Le., that (Van Ness, 1964) :
an ideal gas-it
can be shown
dpp = RT d In y i p
where P is the pressure and yi is the mole fraction of i in the gas phase. For this case the Gibbs adsorption isotherm becomes : -Udn
+ RT
(const. T )
(xid In yip) = 0
It is not essential to assume ideality for the gas phase; corrections for nonideality can readily be made. However, it effects a considerable simplification, and because it represents a close approximation for many adsorption processes of practical interest, we assume throughout this paper that the gas phase is ideal. Thus we take an alternative form of the above equation as our general expression for the Gibbs adsorption isotherm:
The Gibbs-Helmholtz equation results immediately from Equation 10 when it is restricted to constant spreading pressure and composition:
If the gas phase is held at constant composition in an experiment where the gas pressure is increased from zero to P and the spreading pressure increases from zero to 7r, Equation 14 may be integrated to give: Gibbs Adsorption Isotherm
7r
Another very useful equation is obtained when two different general expressions for the total differential of any of the primary thermodynamic functions are compared. For example, Equation 6 gives the total differential, d(nG). If we differentiate Equation 8, we get another equation for d (nG):
d(nG) =
C (Cc,dn,)
+ C (n,dCc,)
Comparison with Equation 6 shows that
+ SdT - U d r +
(nS)dT - (na)d7r or
(n,dp,) = 0 (z&)
=0
1
(const. T )
(13)
Prg
dPa= dPP
and we can write the Gibbs adsorption isotherm in the form:
466
I&EC
+ C (z,dp,g) = 0
FUNDAMENTALS
(const. T )
=
d In P
(const. T and y . . . )
A / n and d In P = dP/P, we have:
?r
=
[F
dP
- (%)dr+
where superscript g denotes a gas-phase property. For any change in the equilibrium conditions, we must have:
-Udr
Since
[d
(12)
This famous equation has a valuable use that has never been exploited. It is explained in what follows, and provides a perfect example of the role of thermodynamics. Our assumption is that the two-dimensional phase representing an adsorbate is always in equilibrium with a gas phase. It is this equilibrium which allows us to determine the properties of the adsorbate, and which we exploit in order to employ adsorption processes to effect chemical separations. The condition of equilibrium is that the chemical potential of each species present be the same in both the adsorbate and the gas phase-that is, Pl =
RT
(const. T and y.. . )
(15)
Equation 15 provides the means for the evaluation of 7r from experimental data. Consider now a binary system of components 1 and 2, for which Equation 14 becomes:
In either form Equation 12 is the Gibbs-Duhem equation for the two-dimensional phase. If the second form is restricted to constant temperature, we have the equation for the Gibbs adsorption isotherm:
+ C (z,dp,) = 0
-=
Since 21
+
=
--(--&)dir+dlnP+
dln P +
21
Y1 dyl+
$2 Yz
dyz = 0
1 and 1 ~ 1 + y2 = 1, rearrangement gives:
"- y1 dyl = Yl(1 - Y1)
o
(const. T ) (16)
This equation is the Gibbs adsorption isotherm for a binary adsorbate in equilibrium with an ideal gas. If T / R T is evaluated from Equation 15 for various values of y1, but all for the same T and P, we may set d In P = 0 in Equation 16 and solve for 21:
The sort of data needed for use of Equations 15 and 17 is shown schematically in Figure l(left), which is a graph of n / A us. P, showing isotherms a t constant gas composition. For the evaluation of the spreading pressure, or of ir/RT, by Equation 15, it is more direct to plot the data as shown in Figure l(right). The intercepts on this graph, marked K , are the terminal slopes a t P = 0 on the graphs of Figure 1 (left). Evaluation of n / R T by Equation 15 up to a specific pressure P, as illustrated by the dashed line in Figure 1 (right), allows one to plot T / R T us. y1 a t constant T and P, and so to determine the partial derivative in Equation 17.
CONST.
T
,y . = I
SLOPE = K , n A
P
Figure 1.
Adsorption isotherms at constant gas composition
analog of the ideal-gas equation, sa = RT. This equation is the simplest possible equation of state connecting T , a, and T for the two-dimensional phase, and its use provides convenient “base” values for the various thermodynamic properties, such as a’. Similarly, H’, S’, and G’ are the molar enthalpy, entropy, and Gibbs function that the twodimensional phase would have were the two-dimensional analog of the ideal-gas equation the correct equation of state. Residual Functions. I t is useful to define several quantities which represent the differences between a property that would be obtained if the ideal-gas analog equation were valid and the actual property. These are called residual functions or configurational properties. Thus we have:
left. Adsorption isotherms for constant gas composition Right. Determination of K from adsorption data In some cases curves become very steep as P+O and this makes determination of ihe intercept most difficult.
Thus the calculation of the adsorbate composition is direct, and the amount of experimental information required is minimized. In mixed-gas adsorption it is a tremendous saving of experimental effort to avoid the measurement of adsorbate composition. This is exactly the purpose of the procedure proposed here. Of course, one must design his experiments so as to allow data of the required form to be taken, and this may well require the exercise of experimental ingenuity. The method proposed here requires adsorption isotherms measured at constant gas composition, and one must find a way to accomplish this. n / A may be measured by volumetric methods, but it may prove difficult to hold the gas composition constant. A gravimetric method might be more promising. However, a gravimetric method introduces the complication that n, the number of moles adsorbed, must be determined from the mass adsorbed by dividing by the molecular weight, M. But M is the average molecular weight of the adsorbate, calculated from
Ji = ZlMl+ (1 - X 1 ) M * This clearly requires knowledge of 21, the quantity to be determined. This does not invalidate the method, but it does mean that an iterative scheme carried out by computer must be devised. Fortunately, 21 is usually a weak function of P. Moreover, a “starting value” of 21 a t P - 0 can be determined from the pure-component isotherms, as shown below. I have been unable to find a set of binary adsorption data adequate for testing the procedure just described.
Fugacity. The fugacity is defined in direct relation to the Gibbs function, and there are in fact three fugacities-the fugacity of a pure adsorbate, f,; the fugacity of a mixed adsorbate, f ; and the fugacity of a component in a mixed adsorbate, fi. The fugacity of a pure adsorbate is defined so as to satisfy the following equations: dGi
which may also be written nQ w=-=--
RT
a _ -a R T / n a’
a’ is the molar area as given by the two-dimensional
(const. T )
(18)
The definition of the fugacity of a mixed adsorbate is analogous : dG = RTd In f (const. T ) (20 1
f
lim-=
I
u-4
For a component in a mixed adsorbate, we have: dpi =
dGi
=
RTd lnfi ji
7 T 4
(const. T )
(22)
1
lim-=
XiT
I would make several important observations about these fugacities and their definitions. First, one must never overlook the restriction to constant temperature imposed on Equations 18, 20, and 22. Within this restriction, these equations merely provide a change of variable. Thus general integration at constant T always gives:
AS =
TO. = wRT
where
RTd In fi
=
Auxiliary Functions
There is a considerable advantage to be gained by the introduction of certain auxiliary thermodynamic functions. The definitions are arbitrary and are conditioned solely by their subsequent utility. They parallel the auxiliary functions used for three-dimensional P-V-T systems (Van Ness, 1964). Compressibility Factor. The compressibility factor, w , is defined by the equation
A@,’ = a’- a AH’= H I - H A S ’ = 8’- S AG‘ = G’ - G
Residual area Residual enthalpy Residual entropy Residual Gibbs function
RTAln$
or Sfinal
- $initial
=
$final
RT In $initial
where the script letters represent any of the three cases considered above. Clearly, integration a t constant T permits changes in both spreading pressure and composition_ between the initial and finalstates. As xi- 1, both G and Gi become Gi and both f andfi become fi. When the analog of the ideal-gas law applies, the fugacities become equal to appropriate spreading pressures:
f becomes equal to T , the spreading pressure of the adsorbed mixture. fi becomes equal to ~ i the , spreading pressure of pure i. f i becomes equal to Z ~ T ,the partial spreading pressure of i in the adsorbed mixture. VOL.
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NO.
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A U G U S T
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467
The fugacity fi of a component in an adsorbed mixture is not a partial molar propert? with respect to f. However, there is a relation between fi and f, shown as follows. By Equation 20 for a mixed adsorbate: dG = RTd In j Integration a t constant temperature and adsorbate composition from a* to a gives: G-G*=
RTlnf-
RTlnf*
Two General Equations
By Equation 20 for a mixed adsorbate: dG = RTd In f
(const. T )
Restricting Equation 6 as written for one mole to constant temperature and composition, we have: dG = a d a Combination of these two equations gives:
If a*-+ 0, then by Equation 21, f * = a*. Therefore,
G - G*= R T l n f - R T l n a * For n moles we have: nG - nG* = n R T In f
- n R T In a*
This is a general equation, which may be differentiated a t constant T and a with respect to ni:
If we imagine a mixed adsorbate to change a t constant T, a,and composition into a state such that it obeys the ideal-gas analog equation, we may integrate Equation 20 for this change to give: G-Gf=
a (nG*)
a (nG) -[
[ x ] T , r , n j
F ] T , r , n j
RTlnf-
RTlnf'= RTlnf- RTlna
or
=
G G' In f = - - RT RT or
-
[-I
-
+lna
Differentiating,
Gi - Gi* = R T a ( n 1 n f ) ani
- RTlna*
alnf
(24)
T.a,nj
For component i in the mixed adsorbate, Equation 22 gives:
By Equation 11 this reduces to:
d a i = RTd lnfi Integration a t constant temperature and adsorbate composition from a* to ?r gives:
ai- a,* = R T 1nfi - R T 1nfi* If a*-+ 0, we have from Equation 23 thatfi*
-
-
=
xis*.
fi
Gi - Gi* = R T In - - R T In a* Xi
where AH' is the residual enthalpy. The quantity n In j is an extensive thermodynamic p r o p erty, and as such is a function of T, a, nl, m,123. . Thus,
..
Thus,
(25)
Comparison of Equations 24 and 25 shows that:
or Since this is exactly the equation wbich defines a partial molar property, we conclude that In (fi/zi) is a partial molar property with respect to In f. In view of the general relation between partial properties and mixture properties (Van Ness, 1964), we may write immediately: fi
Inj= CxiIn-
alnf d(n1nf) = n(=) *,Z...
d T + n(-)
aa
T+
da
+
The three partial differential coefficients are given by Equations 26, 29, and 30. Thus we have the general equation:
Xi
Activity Coefficient. The activity coefficient, yi, is defined only for a component of a mixed adsorbate:
alnf
d(n1n j )
=
?lAH' RT2
ne RT
-d T + - d a +
(In-dni )
(31)
It is often advantageous to deal with the ratio f/a rather than with j itself. Equation 31 can be transformed through use of the following mathematical identity: where j i is the fugacity of pure adsorbed i a t the same temperature and spreading pressure as the mixed adsorbate. The limiting value of the activity coefficient as a - + O is unity:
d(n1na) = n d l n a + lnadn Since n = Thus,
nil dn =
dni, and In adn =
da d(nlna)= n-+ ?r
468
I&EC
FUNDAMENTALS
Clnrdni
In adni.
Subtraction of this equation from Equation 31 gives:
(
f)
d nln-
nAH' RT2
=-
The quantity RT/T is the molar area, ideal-gas analog equation. Thus
(a -
y)
= (@,-
a', as given by
the
a') = -Aha'
where A@' is the residual area. Thus we have for a second general equation:
(
i)
d nln-
=
very low pressures for pure i. Fortunately, such data are required only for the pure components. This is one of the contributions of thermodynamics that aid experimentalists in the field of adsorption. An adsorption isotherm for pure i is illustrated in Figure 1 (left) (curve denoted by y1 = 1 ) . The isotherm necessarily starts a t the origin, and a tangent drawn to the curve a t the origin clearly approximates the isotherm for a t least a short distance. The slope of the tangent is designated Ki (K1in Figure 1, left), and is given by:
At very low pressures the ideal-gas analog equation becomes valid. Thus, a;*@,* = R T , or alternatively:
ai*A = ni*RT
nAH' nA@' R T 2 " - -R T
or
Equations 31 and 32 have a large number of potential applications, and a few examples of their use are given below.
ni* ai* _ -A
RT
Thus,
Required Data for Very l o w Spreading Pressures
Equilibrium between an adsorbate and a gas phase requires: pi = paQ and
Equations 34 and 35 now become:
dpa = dplQ
f i (a)=
For each phase we have the expressions: dp, = RTd lnfa
KiRTyiP (a)
(36 )
f; (a)= K;RTPi (a)
(const. T )
(37 )
K ; is a function of temperature only for a given adsorbent and substrate. It is a quantity which characterizes the specific interaction between a particular adsorbed species and a particular substrate. A different, but related, K is often found in treatments of monolayer adsorption, defined as
drag = RTd lnfaQ (const. T )
Thus we may write: d lnfl = d hfaQ(const. T ) Integration a t constant temperature from an equilibrium state of pure i a t very low spreading pressure at* in the adsorbate and very low pressure Pa* in the gas to the equilibrium state of interest where i has mole fraction za at spreading pressure a in the adsorbate and mole fraction yl at pressure P in the gas gives:
K = lim (P/8) 0-4
where 8 is the fractional surface coverage. I have avoided the use of 8 so as not to restrict this treatment to monolayers. Substitution of Equations 36 and 37 into Equation 28, the defining equation for the activity coefficient, gives:
or
If we let ai* and Pi* approach zero, then f i (ai*)* ai* and (Pi*)-+ Pi*, and the relation between fugacities at equilibrium becomes :
fig
(33 1
If the gas is ideal, thenf;Q(P) = yip, and (34)
P ( a ) is the mixed-gas pressure and P i ( a ) is the pure-gas pressure that produce the same spreading pressure (a) in the respective adsorbates. Equation 38 provides the means for calculating activity coefficients from adsorption isotherms. Alternatively, if y; values can be predicted or determined in some other way, Equation 38 allows the calculation of 2;. For example, yi is always unity for an ideal solution. Thus the assumption of ideal-solution adsorbates allows the prediction of mixed gas adsorption equilibria from pure-component isotherms. In this case Equation 38 with yi = 1 becomes the adsorption analog of Raoult's law. Myers and Prausnitz (1965) have given a thorough treatment of this topic. In the very-low-pressure region (see Figure 1, right) where
For the special case of pure i a t the same temperature and spreading pressure, this becomes:
(35)
n*/-A -K
P*
we may substitute
/A K for nP
The ratio (ni*/Pi*) which appears in Equations 33, 34, and 35 must be evaluated from experimental isotherms a t VOL.
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NO.
3
AUGUST
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469
in Equation 15: (const. T and y . . . )
KdP = KP*
RT
(39)
For different compositions at the same spreading pressure K* and temperature T , .rr*/RT is constant. Thus
K,P,*(n*)
=
KP*(a*)
dG,
=
dG,Q =
+ V,QdP
- S,QdT
+ RT(dP/P)
= - S,QdT
For a differential change in the equilibrium conditions, dG,Q = dG,, and therefore
Xi p * (I E *.
(S,Q- S,)dT
=
R T ( d P / P ) - @,dK
Since G,B = H,Q - TS,Qand G,
Substituting in Equation 40, we get:
(411
+ @,da
- S,dT
For one mole of the pure ideal gas:
(40 1
The activity coefficient becomes unity a t very low pressures, and for this case Equation 38 becomes:
p i * ( K * )=
We now relate the enthalpy of a n adsorbate to the enthalpies of the pure gases from which it is formed. Consider the equilibrium between a pure adsorbate i and pure i as a n ideal gas. By Equation 6 for one mole of pure adsorbate:
=
H , - TS,,
GIB - G, = HtQ- H , - T ( S , Q- S,) But for equilibrium G,Q = G,.
Furthermore,
S'Q-
Therefore
Hi s , - HiQT 1 -
and Hi@- Hi
T This equation shows that the terminal slope of a n adsorption isotherm for constant mixed-gas composition may be calculated from the terminal slopes of the isotherms for the pure components a t the same temperature. Thus it is not necessary to take data for mixed-gas isotherms in the very-lowpressure region. For a binary system Equation 42 is:
K = YiKi
+ YZKZ
dT= RTdlnP-
If we consider changes at constant K , we may write:
/a
H? - Hi
In P\
From Equation 37, f, = K , R T P , and therefore In P = lnf,
- In K , - In R - In T
Thus, d In P
Substitution into Equation 39 gives:
-* Equation 31 for one mole of pure i becomes: AH,'
Differentiating,
RT2 As a result of these equations, we have: Substitution of this result into Equation 17 gives: 21 = y1+
Yl(1 - Y 1 ) (Ki - Kz)P* n*/ A
(43)
HSQ- H,
AH2'
RT2
RT2
By definition, AHI' = H,'
- HI. Therefore
H,Q - H,' = - - -d-l n K , R T2 dT
Since
1 T
dlnK, dT
1
T
or Hi' = Hi0
Equat'ion 43 reduces to:
This result could also have been obtained from Equation 41. The asterisk has been added to x1 to emphasize that it represents the adsorbate composition at very low pressures, strictly speaking, at a pressure approaching zero. Thus we have a very simple equation to give the limiting adsorbate composition from pure-component data. Such values were referred to in connection with Equation 17 as "starting values" for a possible iteration scheme. 470
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FUNDAMENTALS
+ RT (T d;TKi + 1)
(45)
For a mixed adsorbate that obeys the analog of the ideal-gas equation the mixed gas enthalpy is simply the molar average of the pure-component enthalpies-that is,
For the mixed-gas adsorbat,e H = H'
H
=
+
x~H~Q RT2
- AH'.
Therefore
d In Ki xi dT R T - AH'
restricted to one mole of a constant-composition adsorbate:
Finally,
A H’ a d l n j = -dT+-da R T2 RT This equation relates the enthalpy of a mixed adsorbate to the enthalpies of the pure ideal gases a t the same temperature and tJo the effects of temperature on the Ki’s for the pure components. The value of AH’ can be determined by use of an equation of state, as shown below.
An equation of state describing the a - a - T - x behavior of an adsorbate can often be employed very effectively. Consider the calculation of the fugacity of a mixed adsorbate. For one mole of adsorbate at fixed temperature and composition, Equation 32 becomes:
f
(50) The first term on the right is determined by differentiation of Equation 49 :
Use of an Equation of State
dln-= a
If we divide this equation through by dT and restrict the changes to constant a, we obtain:
-A@’ RT
-da
1
alnf Since a a = wRT,
Substitution of the last two equations into Equation 50 gives:
Integration from n*-+ O to a gives:
f
l n - - I n - =f * n a*
(const. T a n d x ...)
- [ s d a
or
By Equation 21, f*/n* = 1, and A@’ can be related to w , the compressibility factor:
a’
A@,’ RT
RT/a RT
_---=
RT
RT
w a
(1-
W )
a
Thus
This equation is convenient when the equation of state expresses w as an explicit function of a. For the more common case of an equation which gives w as an explicit function of a, we transform Equation 47 as follows. Since a& = wRT, then a t constant T and x.. . :
.
where the integral is evaluated a t constant T and x . . . We now illustrate the use of Equations 49 and 51 by a consideration of specific examples. First, if w = l-i.e., if the adsorbate obeys the ideal-gas analog equation-Equation 49 reduces to f = a and Equation 51 gives AH’ = 0, as one would expect. The two-dimensional analog of van der Waals’ equation has found wide application in adsorption studies (Ross and Olivier, 1964) : RT a a=---
a-p
where a and /3 are constants. From this we get:
*a
ad@,+ ada = RTdw
or
a a-P
cd=-=---
RT
da T _ -- -R& a aa
dB dQ. - = dw ---
a
w
a2
a @RT
and
a
Thus, (w-
I ) -da = n
( w - 1 ) dw --
( w - 1 ) da -
a
W
Also, a
and
=
aRT2
Substitution into Equations 49 and 51 leads directly to:
RT lnj=ln---+-
or
CL-P
da (w-1)-+w-1-lnw
AH’ - - 2a
Alternatively,
1
m
a
da (w- ~ ) - + w -
a
1+1n-
P a-/3
(52)
and
a
1nf=
2a BRT
RT
a
The integral in Equations 48 and 49 is evaluated at constant T and 2.. .. To obtain an expression for AH’ we make use of Equation 31
P a-p
(53) RT CtRT The equilibrium pressure, P , of the gas phase is related to the adsorbate fugacity by Equation 37, f = K R T P , and this may be combined with Equation 52 to give: 2a l n P = -1nK-ln (@-P)--+/3 (54) CiRT a - 8 VOL.
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471
The use of this equation for pure adsorbates has been illustrated by Hoory and Prausnitz (1967b). For mixtures, the value of K is determined from data for the pure components by Equation 42, and constants a and /3 for the mixed adsorbate may be calculated from the constants for the pure components by appropriate “mixing rules,” as discussed later. Equation 53 may be combined with Equation 46 to allow determination of the adsorbate enthalpy: We consider now the evaluation of fi, the fugacity of a component in the adsorbate. The starting point is Equation 31 restricted to constant temperature:
ji
A d(nlnj)=-da+ RT
multiply by n : nu-
n=
n2,8 na-np
n2a naRT
- ~n2P _ n2a _ _ A-nP
ART
(const. T )
xln-dn; Xi
where we have set n a = A , the surface area. Division by dni and restriction to changes a t constant T , A , and n,, where j # i, leads to: r e ] T , A , n j
-
=
a
P -+--a-p
Since SS = wnRT, a=
a(np) (a-p)2 ani
a(n2a)/dni naRT
To evaluate the derivatives on the right we need a pair of “mixing rules”-i.e., equations expressing a and 0 as functions of composition. One pair of rules which has proved useful for a binary system is:
RT A
- (m)
and
a = x?a1+
221x20112
+ x22a2
where 011, a2, &, and /32 are the constants for the pure materials and a12 is an “interaction” constant. All are true constants. The above rules may also be written:
Thus,
nP =
or
@I+
n2B2
+
n2a = n&1+ 2n17~12
+ In xi
n?a2
The required differentiations give:
T,A,ni
The partial derivative on the right can be determined from Equation 49, which for n moles may be written:
dQ.
RT
a
a
(nu - n ) - - n + n l n Q
Thus,
Therefore
I =
r$]T,A,nj
T,A,ni
a (m)
-1
[{[=]T,A,nj
dQ. -+ln-
}a
RT
-
=
+--( @ap,- P I 2
P a-P
2a1x1+ 2al252 &RT
and
Q.
where use has been made of the fact that R T / a = n R T / A . Finally, or
.
where the integral is evaluated a t constant T and x. . . The two-dimensional analog of van der Waals’ equation again provides a suitable example. As before, we have:
P a-p
u-l=---
a @RT
where a and ,8 are functions of composition only. We may 472
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FUNDAMENTALS
Similarly,
If we eliminate fl and f2 by Equation 36, fi
=
KiRTyiP, we
F = molar Helmholtz function
get for equilibrium:
8 = a fugacity, in general f =
fugacity of a mixed adsorbate i as a pure adsorbate a component in a mixed adsorbate molar Gibbs function G . = partial molar Gibbs function of component i = pi molar Gibbs function, in general H = molar enthalpy K = slope of adsorption isotherm a t P = 0 (Figure 1 ) Ji = molecular weight n = number of moles P = pressure R = universal gas constant S = molar entropy T = absolute temperature U = molar internal energy V = molar volume xi = mole fraction of component i in adsorbate yi = mole fraction of component i in gas phase
= fugacity of f i = fugacity of fi
e=
21
+
s = 52
= 1,
y l + yz = 1
If one can assign values to the constants LYI, a ~CYU, , PI,Pz, K1,and K2, these equations can be solved for XI, x ~ and , B for any values of y1, y2, and P . The use of these equations has been illustrated by Hoory and Prausnitz (1967a). They used constants for the pure components as provided by Ross and Olivier (1964), and took = dG. The two-dimensional analog of van der Waals’ equation is probably best suited to monolayer adsorption. It is limited to this case when the values of pi are determined from molecular dimensions and the assumption that /3 is equal to the molar area a t 100% surface coverage. An alternative procedure has been illustrated by Trotta and Myers (1968). They employ the excess Gibbs function of mixing and its relation to the activity coefficients. The procedure is entirely analogous to the usual treatment of vapor-liquid equilibria. Conclusions
I have tried to illustrate the uses of classical thermodynamics in the adsorption of gases on solids. I have concentrated in particular on equations applicable to mixed-gas adsorption. This is a very difficult area from the experimental point of view, and this accounts for the paucity of mixed-gas adsorption data. The methods of thermodynamics, properly used, can greatly lighten the burden on the experimentalist and open the way to far more productive experimental study of mixed-gas systems. Acknowledgment
Acknowledgment is made to the donors of The Petroleum Research Fund, administered by the American Chemical Society, for partial support of this research. Nomenclature
A = total surface area B = molar surface area
GREEKLETTERS a = a constant = a constant yi = activity coefficient of component i in a mixed adsorbate 8 = fractional surface coverage ~ . l i = chemical potential of component i = di T = spreading pressure (J = compressibility factor for an adsorbate SUPEHSCRIPTS = gas phase * = value a t a very low pressure ’ = value for the ideal-gas state a t T and T
Q
SUBSCRIPT i = identifies quantity as applying to species i literature Cited
Hoory, S. E., Prausnita, J. M., Chem. Eng. Sci. 22, 1025 (1967s). Hoory, S. E., Prausnitz, J. M., Trans. Faraday SOC.65, 455 (196713). Myers, A. L., Prausnita, J. M., A,I.Ch.E.J. 11, 121 (1965): Ross, S., Olivier, J. P., “On Physical Adsorption,” Interscience, New York, 1964. Trotta, F. J., Myers, A. L., personal communication, 1968. Van Ness, H. C., “Classical Thermodynamics of Non-Electrolyte Solutions,” Pergamon Press, London, 1964. RECEIVED for review June 17, 1968 ACCEPTEDDecember 26, 1968 I&EC Summer Symposium on Interfaces, Washington, D. C., June 1968.
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