3986
J. Phys. Chem. 1988, 92, 3986-3988
Enthalpy of Immersion of a Microporous Solid M. Jaroniect and R. Madey* Department of Physics, Kent State University, Kent, Ohio 44242 (Received: August 25, 1987; In Final Form: December 18, 1987)
Presented here is a general relationship between the enthalpy of immersion of a microporous solid in a liquid and the average adsorption potential. Special cases of this relationship are discussed for uniform as well as nonuniform microporous solids.
Introduction Experimental studies of gas adsorption on microporous solidsl.2 showed that the characteristic adsorption curve 6 versus A is frequently independent of temperature, i.e. (dA/dT), = 0 Here 6 denotes the volume fraction of the micropores filled by an adsorbate, T i s the absolute temperature, and-A denotes the change in the Gibbs' free energy AG taken with a minus sign, Le. A = -AG = R T In ( p o / p ) (2) Here p is the equilibrium pressure, p o is the saturation vapor pressure, and R is the universal gas constant. For adsorption systems satisfying the condition given by eq 1, Bering et aL2s3 showed that the Of adsorption AH is expressed by AH = -A + a T ( a A / a In (3) where
6 = W/Wo
(4)
a = -(d In Wo/dT)
(5)
Here W denotes the volume of the adsorbate condensed in the micropores at temperature T and relative pressure p / p o , Wo is the maximum volume of the adsorbate in the micropores at temperature T , and a as defined by eq 5 is the negative thermal coefficient of the logarithm of the maximum adsorption. The differential molar entropy of adsorption A S is equal to2s3 AS = a ( a A / a In W), (6) Previously, Jaroniec4 showed that eq 3 and 6 for A H and A S may be expressed in terms of the characteristic adsorption curve 6 = 4 ( A ) and the adsorption potential distribution X ( A ) : AH = -A - a T $ ( A ) / X ( A ) (7) AS = CY^ (A) /X(A )
(8) The adsorption potential distribution X ( A ) that appears in eq 7 and 8 is defined in terms of the condensation approximation as follows:5 X ( A ) = -d$(A)/dA (9) It was shown4 that eq 7 and 8 are useful for deriving expressions that represent A H and A S for energetically heterogeneous microporous solids characterized by various distributions of the adsorption potential A. It is noteworthy that eq 7 and 8 are valid for uniform as well as nonuniform (Le., structurally heterogeneous) microporous s01ids.~ In this article a general expression for the enthalpy of immersion AHl, for a microporous solid in a liquid is derived and discussed in relation to the parameters obtained from the characteristic curve 0 = 4 ( A ) for the adsorption of a vapor. For the characteristic adsorption curve 6 = $(A) given by the Dubinin-Radushkevich (DR) equation,'-3 we show that this expression leads to another Permanent address: Institute of Chemistry, M. Curie-Sklodowska University, 2003 1 Lublin, Poland.
0022-3654/88/2092-3986$01.50/0
equation for AHi,, which was studied by Stoeckli and Kraehenbueh16s7and also by Radeke.8 A special case of the above general expression for Mi, is discussed for a y-type distribution, which represents the structural heterogeneity of a microporous solid.4 General Expression for Immersion Enthalpy of a Microporous Solid According to earlier s t u d i e ~ > ~the * ' enthalpy ~ of immersion AHh for completely microporous solids is related to the differential molar enthalpy of adsorption A f t AH,,(T) = l l A H ( T , O )d6 0
(10)
It is noteworthy that the differential molar enthalpy AH is equal t ~ ~ . ~
Here qne'is the differential net heat of adsorption, which is equal to the differential molar heat of adsorption (Le., the isosteric heat of adsorption), @', minus the molar heat of condensation, @. For a derivation of an equation for AHi,( r ) by means of the integral eq 10, it is necessary to present the differential enthalpy of adsorption AH (eq 3) as a function of 0. Let $*(e) denote the function inverse to the characteristic adsorption curve $ ( A ) , i.e. A = $*(6)
for 0 = $(A)
(12)
Then eq 3 may be rewritten as follows: A H = -$*(e)
+ aT6[&$*(0)/aO],
Substitution of eq 13 into the integral eq 10 gives AHi,(T) = -J1$*(6)
d0
+ CYTJ'O[a$*(O)/a6],d6
(14)
Taking into account the limiting values of the functions $ ( A ) and $*(e) and performing a partial integration of the second integral in eq 14, we obtain AHi,(T) = -(1
+ a r ) J 1 4 * ( 0 ) d6
(15)
(1) Dubinin, M. M. Prog. Surf. Membrane Sci. 1975, 9, 1. (2) Bering, B. P.; Dubinin, M. M.; Serpinsky, V. V. J . Colloid Interface Sci. 1972, 38, 185. (3) Bering, B. P.; Dubinin, M. M.; Serpinsky, V. V. J . Colloid Inferface Sci. 1966, 21, 378. (4) Jaroniec, M. Langmuir 1987, 3, 673. (5) Jaroniec, M.; Piotrowska, J. Monatsh. Chem. 1986, 117, 7. (6) Stoeckli, H. F.; Kraehenbuehl, F. Carbon 1981, 19, 353. (7) Stoeckli, H.F.; Kraehenbuehl, F. Carbon 1984, 22, 297. (8) Radeke, K. H. Carbon 1984, 22, 473. (9) Barton, S. S.; Boulton, G. L.; Dacey, J. R.; Evans, M . J. B.; Harrison, B. H . J . Colloid Interface Sci. 1973, 44, 50. (10) Clint, J . H. J . Chem. SOC.,Faraday Trans. 1 1973, 73, 1320.
0 1988 American Chemical Society
The Journal of Physical Chemistry, Vol. 92, No. 13, 1988 3987
Enthalpy of Immersion of a Microporous Solid Now, we will show that the integral in eq 15 is the average adsorption potential A of the distribution function X ( A ) (eq 9 ) , which is associated with the characteristic adsorption curve. The average adsorption potential A is defined as follows:
A = J m A X ( A ) dA
(16)
A!A = PEo(l
#(a)
By letting r = 2 in eq 22-25, we obtain the quantities p R ( A ) , To express the immersion enthalpy AHi,,, in terms of the DA adsorption parameters, we should replace the average adsorption potential in eq 19 by eq 23: AfeA(T) = - ( W o / r ) F ( l / r ) ( l
W$(T) = -(pEo/2)*'/2(1
A = L'4*(0)d0
(18)
Equations 15 and 18 yield the following simple relation between the enthalpy of immersion AHi, and the average adsorption potential A:
+ CYT)
(19)
Equation 19 is valid for energetically heterogeneous microporous solids, which may possess uniform as well as nonuniform micropores. Enthalpy of Immersion for a Uniform Microporous Solid Experimental showed that adsorption on uniform microporous solids is represented by the Dubinin-Radushkevich (DR) equation:
ODR = +DR(A) = e ~ p [ - B ( A / p ) ~ = ] exp
+ UT)
[-(&)I
+ aT)
Equation 27 was derived by Stoeckli and Kraehenbueh16 and verified experimentally by using adsorption and calorimetric measurements for microporous activated carbons.6-8 Enthalpy of Immersion for a Nonuniform Microporous Solid Previously, Jaroniec4 presented equations to describe adsorption on a nonuniform microporous solid. For such solids, the characteristic adsorption curve is given by5,",12
0" = 1mexp[-B(A/p)2] F(B) dB
where r is a parameter with a value usually greater than unity.' For r = 2, the DA eq 21 reduces to the DR eq 20. The adsorption potential distribution X ( A ) corresponding to the DA eq 21 may be calculated according to eq 9 : P A ( A ) = r(@E0)' A"
e~p[-(A(/3E~))~] ( 2 2 ) The average value ADAfor the P A ( A )distribution (eq 22) may be calculated according to eq 16 or 18: ADA = (PEo/r) r(1/ r ) (23) To characterize completely the PA@) distribution (eq 22), we calculated the dispersion a:" and the point AEA where the P A ( A ) function has a maximum:
[
=
ADA[ 2 r r ( 2/ r ) -1
(28)
where the superscript n refers to a nonuniform microporous solid, and F ( B ) is the distribution of the structural parameter B normalized to unity. For slitlike micropores, the half-width x of the micropores is associated with the structural parameter B as fol12,1 IOWS:
B = cx2
(29)
where c is a constant for microporous solids with relatively large micro pore^.'^^'^*'* Equation 29 permits transforming the F(B) distribution to the micropore-size distribution J(x):I2 J(x) = 2cx F(Bx)
with B = cxz
(30)
The adsorption potential distribution associated with eq 28 is4,5 P ( A ) = J m 2 ( B / p 2 ) A e ~ p [ - B ( A / p ) ~F] ( B ) dB
(31)
This distribution can be expressed in terms of the micropore-size distribution J(x):
P ( A ) = S m 2 m x 2 Aexp(-mx2A2) J ( x ) dx
(32)
m = c/p2
(33)
0
= J m ( A - ADA)2PA@) dA]'"
(27)
(20)
where Eo is the characteristic adsorption energy, B is the structural parameter that characterizes the uniform microporous structure of a solid, and p is the similarity coefficient that characterizes the adsorbate.' Some author^'^-'^ used also the Dubinin-Astakhov (DA)' equation for representing adsorption in uniform micropores. For the majority of the activated carbons, it is worth noting that the DR eq 1 gives a better description of the volume filling of uniform micropores than the DA e q ~ a t i o n . ~ ~ "Because ~ ~ ~ - ' the ~ DA equation is more general than the DR eq 1 , we will discuss here an expression for the immersion enthalpy that is associated with the DA equation:'
,2A
(26)
For r = 2, eq 26 reduces to the expression that defines AHi,,, in terms of the DR parameters:
= 0, we have
AHim(T) = -A(1
(25)
ADR, a;R, and AER associated with the DR eq 20.
Taking into account in the integral eq 16 that
and that 4(0) = 1 and
- l/r)'/r
where
The average adsorption potential 2" associated with the adsorption potential distribution P ( A ) (eq 31 or eq 32) is given by
Combination of eq 19 and 34 gives the following general equation for the immersion enthalpy AHi,:
]'Iz (24)
/r) (1 1) Stoeckli, H.F.J. Colloid Interface Sci. 1977, 59, 184. (12) Dubinin, M.M.;Stoeckli, H. F. J. Colloid Interface Sci. 1980, 75, 34. (13) Dubinin, M.M.Carbon 1985, 23, 373. (14) Suzuki, M.;Sakoda, A. J. Chem. Eng. Jpn. 1982, 25, 279. (15) Rozwadowski, M.;Wojsz, R. Carbon 1984, 22, 363. (16) Wojsz, R.; Rozwadowski, M. Carbon 1984, 22, 431.
Equation 35 defines AHi, for an arbitrary distribution F ( B ) or (17) Dubinin, M.M.Carbon 1981, 19, 321. (18) Dubinin, M.M.;Kadlec, 0. Carbon 1987, 23, 321.
3988 The Journal of Physical Chemistry, Vol. 92, No. 13, 1988 J ( x ) , which characterizes the structural heterogeneity of a microporous solid. In an earlier article: the following y-type distribution was used to represent the structural heterogeneity of a microporous solid: 4"+*
P ( B ) = -Bn
r(n + 1)
exp(-qB)
(36)
where q and n are parameters greater than zero, and the superscript y refers to the y-type distribution. Substitution of eq 36 into eq 28 gives
e'=[
]
4
Jaroniec and Madey offers the possibility of correlating the R"' value with independent calorimetric measurements of the immersion enthalpy Wh.This correlation was studied for the DR equation only;* it would be interesting to verify eq 45, which is valid for nonuniform microporous solids. On the basis of the studies of Dubinin,I7 the geometric surface area Smiof the nonuniform micropores characterized by the micropore-size distribution J ( x ) may be expressed as follows:
n+ 1
(37)
4+
Equation 37 was studied by Jaroniec and Choma.I9 The average adsorption potential 1 7 for nonuniform microporous solids characterized by the P ( B ) distribution (eq 36) may be calculated according to eq 34:
where V, denotes the total volume of the micropores. Substituting into eq 46 the micropore-size distribution J ( x ) obtained from eq 30 and 36, we have (47) Dividing
where (39) Equation 38 combined with eq 19 gives the following expression for AHyi,(T):
Equation 40 defines the immersion enthalpy in terms of the parameters q and n, which characterize the structural heterogeneity of a microporous solid. The immersion enthalpy M y i , ( T)may be correlated with the average micropore dimension x and the geometric surface area Smiof the micropores. The average value f is defined R =
J ( x ) dx
(41)
For the micropore-size distribution J ( x ) obtained according to eq 30 and 36, the average value 3 is given by
by
Symi,
we obtain
This ratio does not depend on the type of the micropore-size distribution J ( x ) . Comparison of eq 46 with eq 34, which defines the average adsorption potential in terms of J ( x ) ,shows that the ratio A"/S, = A y / F , for an arbitrary micropore-size distribution J ( x ) . The combination of eq 49 and eq 19 gives the following relationship between AHYimand S y m i :
Equations 45 and 49 can be used to evaluate .fy and S Y m i , respectively, from calorimetric measurements of the immersion enthalpy. For solids that possess both micropores and mesopores, the total enthalpy of immersion contains a contribution for the immersion of the surface area of the mesopores.' Let AEi(g)im,tdenote the total enthalpy of immersion of a microporous solid expressed in J/g. This total enthalpy may be ~ r i t t e n : ~ All(g)jm,t= AH(g)j,
+ Ah*S,,
where is the enthalpy of immersion for the micropores in J/g, Ah is the enthalpy of immersion of the mesopores expressed in J/m2, and S,, is the surface area of these mesopores. The relationship between AE-l(glim and AHimis7 AHig)im= (V,~/U,)AH,,
where (43) It is easy to show that
=
2Y.fY
E(2
T)1'2y2(
c
1
+
&)
(44)
The average value 747 may be calculated from eq 44 and substituted into eq 19; the result of this operation yields the following relationship between AWi, and 3': AHyi,(r) =
"( 2
T)"2yA2( 1
c
+ &)(1 + an--1
(45)
RY
It is noteworthy that the average micropore dimension f Y (eq 42) may be obtained from the adsorption isotherm 0 versus p / p o , which yields values for the parameters n and q (eq 37). Equation 45 (19) Jaroniec, M.; Choma, J. Muter. Chem. Phys. 1986, 15, 521
(50)
(51)
where u, is the molar volume of the liquid filling the micropores, and Vmiis the total volume of the micropores, which is usually identified with the parameter Wo (eq 4).
Conclusions A general equation (viz., eq 19) is derived for the enthalpy of immersion of a microporous solid in a liquid; this equation shows that the immersion enthalpy is proportional to the average adsorption potential. The average adsorption potential may be calculated from the parameters that characterize the energetic heterogeneity of microporous solids (e.g., eq 23 for an uniform microporous solid and eq 34 for a nonuniform microporous solid). Special equations that correlate the immersion enthalpy with the average micropore dimension and the geometric surface area of the micropores were obtained for the y-type micropore distribution. These equations allow correlations between quantities obtained in adsorption and calorimetric measurements. Acknowledgment. This work was supported in part by the Division of Chemical Sciences, Office of Basic Energy Sciences, Department of Energy.
