Langmuir 1990, 6, 1734-1738
1734
Adsorption of C02 and Xenon by Porous Glass over a Wide Range of Temperature and Pressure: Applicability of the Langmuir Case VI Equation Christopher G. V. Burgess,? Douglas H. Everett,* and Stuart Nuttallt Department of Physical Chemistry, University of Bristol, Bristol BS8 1 TS, U.K. Received February 5, 1990. I n Final Form: May 15, 1990 Data on the adsorption of COZ and Xe by Vycor porous glass have been analyzed in terms of Langmuir's case VI equation. Measurements over a wide range of temperature and pressure enable a thermodynamic analysis to be made which identifies the parameters of the equation with those postulated in its theoretical derivation. The monolayer capacities in both instances are independent of temperature over a range of 100 K. The curves of isosteric enthalpy against coverage can be accounted for quantitatively in terms of the shift from monolayer to multilayer adsorption as the pressure is increased, and for the two adsorptives the curves can be correlated by a corresponding states procedure. The enthalpy and entropy curves show that the grounds on which case VI has been rejected in the past are false. A striking empirical relation is found between one of the parameters of the equation and the molar volume of the bulk liquid.
Introduction
choice of Va is largely arbitrary, but for porous solids a reasonable, though still approximate, choice is the pore volume of the solid. In the present instance, taking the pore volume of Vycor porous glass as 0.200 cm3g-l, the correction term varies from zero a t low pressures to between 10% and 15% a t pressures approaching the bulk critical pressure. It is not necessary to include the third virial coefficient in making this correction.
The preceding paper in this issue' examined the applicability of Langmuir's case VI equation2 t o the adsorption of Nz, Ar, and neopentane by nonporous solids. Here the analysis is extended to the adsorption of COz and Xe by Vycor porous glass over a wide range of pressures and temperatures approaching the critical points of the bulk adsorptives. Attention is directed to the lower portions of the isotherms before capillary condensation and hysteresis ensue, corresponding to adsorptions up to about twice the nominal monolayer coverage. We examine the applicability of the equation and in particular test the physical significance of the parameters by means of a thermodynamic analysis. This shows that previous criticisms of the model leading to the case VI equation are invalid. The two systems are found to conform to a "corresponding states" correlation.
Experimental Section The experimental technique3-'used will be described in detail elsewhere but followed the principle of the volumetric method described by Dubinin et al.6 Two different samples of Vycor 7930 were used in this work. That used for the COz experiments had a conventionalsurface area (BET, Nz)of 177 m2 g-l, while the Xe work employed a sample of surface area 166 m2 g-l. The experimental results are shown in Figures 1 and 2.
Application of Langmuir Case VI to Adsorption at High Pressures by Porous Solids
Isotherm Analysis Tables I and I1 summarize the results of analysis in terms of the equation
Two modifications have to be made to the analysis used in the preceding paper. First, the pressure, p , has to be replaced by the fugacity, f , of the gas phase:
where nIt is the total amount adsorbed in the first layer, nmtthe total monolayer capacity, and f* a parameter chosen both to give the best fit to eq 4 and to lead to values of nlt from eq 3 which tend toward nmtas f f*. C is the analogue of the C parameter of the BET equation. Figures 3 and 4 illustrate the application of these equations. In all cases, good linear graphs (correlation coefficients greater than 0.9999) are obtained up to the fugacity corresponding roughly to the lower limit of the hyst,eresis loop or to a relative fugacity, flfo,of 0.750.80, whichever is the lower. The deviations from eq 4 then indicate a larger adsorption than that predicted by the equation, which we attribute to the onset of capillary condensation, leading to hysteresis. As found previously,'
where B and C are respectively the second and third virial coefficients, R the gas constant, and T the temperature. Second, since the experimental isotherms yield only the Gibbs relative adsorption na,while the case VI equation relates to the total adsorption nt, the following correction is necessary: n t = n a + p e V B = n a +[ p / ( R T + B p ) ] V
-
(2)
where pg is the density of the equilibrium gas phase, expressed in mol volume-', and Va is the "volume of the adsorbed phase". In the case of nonporous solids, the + Present address: BP International, Sunbury Research Centre, Sunbury-on Thames, TW16 7LN U.K. Present address: The Royal Grammer School, Guildford, Surrey, GU1 3BB, U.K. (1)Everett, D. H. Langmuir,preceding paper in this issue. (2) Langmuir, I. J.Am.Chem. SOC.1918,40,1361 and references given in ref 1.
0743-7463f 90/2406-1734$02.50f 0
(3) Burgess, C. G.V. PbD. Thesis, Bristol, 1971.
(4) Nuttall, S. Ph.D. Thesis, Bristol, 1974.
(5) Dubinin, M.M.;Bering, B. P.; Serpinekii, V. V.; Vasil'ev, B. N. In Surface Phenomena in Chemistry and Biology;Danielii, J. F.,Pankhurst, R. G. A., Riddiford, A. C., Eds.; Pergamon Press: London, 1958; p 172. (Q
1990 American Chemical Society
Langmuir, Vol. 6, No. 12, 1990 1735
Applicability of Langmuir's Case VI d
I
t"
InJ/atm
.
Figure 1. Adsorption of COZ by Vycor porous glass from 173 to 273 K plotted as na against In f/atm at the following temperatures (K):(a) 184.7, (b) 195.0, (c) 201.5, (d) 207.7, (e) 213.9, (f) 217.5, (g) 224.5, (h) 233.2, (i) 241.8, 6)253.0, (k)258.9, (1) 273.1. For clarity, the desorption curves are omitted, but the points of onset and closure of the loops are indicated by arrows.
that adsorption occurs on sites whose number is independent of temperature. If this were so, however, one would expect the monolayer capacities for COZand Xe (expressed as mol m-2) to be the same. Although the conventional surface areas of the two Vycor samples may not have absolute significance, it is probably justifiable to use them for comparative purposes, leading to monolayer capacities per unit area, n,, of 8.53 f 0.20 pmol m-2 for COz and 5.66 f 0.05 pmol m-2 for Xe, the ratio between them being 1.50 f 0.15. Thus, if sitewise adsorption occurs, the sites active for the adsorption of C02 must differ from those adsorbing Xe. Assuming the N2-BET surface areas to be correct, the areas of these sites would be 0.195 nm2 for C02 and 0.294 nm2 for Xe (ratio 1.50), whereas if one applies the conventional formula6 to the liquid densities a t a reduced temperature ( T / T , ) of 0.7, values of the molecular areas of 0.169 and 0.205 nm2, respectively, are obtained (ratio 1.21). A more likely interpretation is that the adsorbate molecules in a complete monolayer pack in essentially the same way as in the solid, since the ratio of the molar volume of solid Xe to that of COz, raised to the 2/3 power, is 1.44, close to the ratio of molar areas. Secondly, we draw attention to the remarkable empirical observation that for both systems vl/(fr/p)is independent of temperature in the temperature range in which hysteresis is observed: at higher temperatures there seems to be a slight increase. The mean values are 30.13 f 0.17 cm3mol-' for C02 and 44.6 f 0.40 cm3 mol-' for Xe. The ratio of these figures is 1.48 f 0.04, close to the ratio of the reciprocals of the monolayer capacities. Thermodynamic Analysis The isosteric enthalpies of adsorption, calculated from the equation
a,were
a,
SinceBp is typically about 1 5% of this correction term was ignored. The results derived from the adsorption curves are illustrated in Figure 6. The behavior of the desorption curves in the hysteresis region will be discussed elsewhere. There was some evidence of slight curvature of the isosteres, but the data are not precise enough to The figdetermine the temperature dependence of ures given, therefore, correspond to the slopes in the middle of the temperature range. The enthalpies of condensation of the pure liquids, Go, were calculated from the observed saturation fugacities f' and are shown in the figures. The form of the isosteric enthalpy curves can be interpreted in terms of the case VI equations. First, the isosteric enthalpy tends, as the limit of the applicability of the equation is approached, to a value greater (Le., more negative) than the enthalpy of condensation. Since this limit correspondsto adsorption into multilayers, this result invalidates the criticism of the case VI model; namely, it implies that the enthalpy of adsorption into multilayers is numerically less than the enthalpy of condensation: the opposite is the case. The difference between the limiting isosteric enthalpy and Acho can be calculated from the temperature dependence off". Thus, according to the statistical mechanical interpretation of case VI, f* is the fugacity of adsorbed multilayers. Consequently, when f* is inserted intoeq 5, one obtainsh,h*. Alternatively, from graphs of In f*/P against 1/T, one obtains a,h* - Ach"
a.
I
I
I
I
I
I
111/'/ritm
Figure 2. Adsorption of Xe by Vycor porous glass from 183 to 273 K plotted as na against In f/atm at the following temperatures (K):(a) 183.5, (b) 193.2, (c) 201.9, (d) 210.4, (e) 221.1, (f) 232.8, (g) 238.4, (h) 241.9, (i) 252.6, (i) 273.1. For clarity, the desorption curves are omitted, but the points of onset and closure are indicated by arrows. these deviations occur when the total adsorption reaches about twice the monolayer capacity. The first important result of this analysis is that for both systems the total monolayer capacities (but not the Gibbs monolayer surface excesses) remain constant to better than f 1 % over the whole liquid range (and in the case of COZ for a short distance below the triple point temperature) (Figure 5 ) . At lower temperatures, for COz/Vycor, there is evidence for a small increase of about 1% in nmt. This observation is clear proof that the area occupied by a molecule in the first layer is independent of temperature. Consequently, the widely used estimates of molecular area based on the molar volumes of the liquid are inapplicable, at least in the present systems. On the contrary, it suggests
a*
(6) Brunauer, S. The Adsorption of Gases and Vapours; Clarendon Presn: Oxford, 1945.
1736 Langmuir, Vol. 6, No. 12, 1990
Burgess et al.
Table I. Parameters of Ea 4 and Characteristics of the Adsorption Isotherms for COdVycor.
T,K
f*, atm
173.17
0.435
state 1 s
184.70
1
1.06
s
194.96
1
2.13
s
201.45
1
2.90
s
207.70
1
3.70
s
f*/p
C
0.874 3.20 1.024 2.50 1.187 2.13 1.185 1.73 1.144 1.40
42.06
1.537
2.1
3..4
1.538
2.4
35.8
1.517
2.7
21.5
1.539
(35.85)
(30.25)
3.3
22.3
1.526
(36.27)
(31.70)
2.9
0.53
nmt,mmol g-l
mean: 213.90
d/(f*/p),mol-'
n$i,, mmol gl
(f/fo)h
(f/p)lim 0.27 0.98 0.39 0.97 0.48 0.93 0.64 0.93 0.57 0.70
1.531 f 0.010
1
5.17
s
217.45 224.45 233.26 241.78 252.98 258.88 273.15
vi, cm3 mol-'
6.02 8.10 11.60 15.70 22.7 26.70 41.0
1.218 1.29 1.225 1.28 1.31 1.348 1.412 1.414 1.494
20.62
1.519
(36.7)
30.13
3.00
0.59
0.60
19.77 18.30 17.35 15.60 14.98 13.90 12.97
1.518 1.505 1.520 1.521 1.511 1.517 1.519
37.2 38.2 39.5 40.7 42.5 43.7 47.4
30.37 29.84 30.15 30.19 30.10 (30.90) (31.73)
3.00 2.90 2.97 3.00 3.10 2.95 2.75
0.63 0.66 0.72 0.76 0.84
0.65 0.65 0.69 0.72 0.75 0.75
30.13 f 0.17
2.97 f 0.06
1.516 f 0.005
means:
0.77
0 cf/fo)h is the relative fugacity a t the lower limit of the hysteresis loop. (f/p)lim and n$imare the relative fugacity and total adsorption where the deviations from eq 4 appear. Below the triple point (216.5 K), two values of f*/pand (f/P)limare given corresponding to the fugacity of the solid and liquid (extrapolated). Values in parentheses have been omitted from the means.
Table 11. Parameters of Eq 4 and Characteristics of the Adsorption Isotherms for Xe/VycoP
T,K
f*, atm
f*/p
C
183.45 193.18 201.87 210.36 221.14 232.84 238.35 241.89 252.64 273.15
2.47 3.80 5.50 7.50 11.00 15.30 18.00 20.00 25.50 39.70
1.029 1.058 1.106 1.130 1.191 1.211 1.237 1.264 1.275 1.328
12.05 10.48 10.36 8.92 8.70 7.83 7.49 7.34 6.87 5.91
ul, cm3 mol-'
46.3 47.8 49.2 50.5 52.2 54.1 54.9 56.3 59.7 66.5
0.934 f 0.008
means: 0
nmt,mmol g-' 0.926 0.927 0.933 0.934 0.945 0.928 0.940 0.949 0.927 0.933
uI/(f*/p),cm3 mol-'
n$im,mmol g-1
45.0 45.2 44.5 44.7 43.8 44.7 44.4 44.5 (46.8) (50.1)
1.90 2.06 1.93 2.03 2.08 2.10 2.08 1.92 1.84 1.98
44.6 f 0.4
1.99 f 0.09
(f/P)h
(f/P)lh
0.55 0.59 0.63 0.69 0.76 0.83 0.85 0.84
0.56 0.61 0.61 0.66 0.70 0.74 0.75 0.73 0.70 0.74
See notes for Table 1.
directly. We find a,h* = -17.13 f 0.50 k J mol-' for COz and -13.7 f 0.50 k J mol-' for Xe, which agree within the experimental uncertainty with the limiting enthalpies of -17.4 and -13.4 k J mol-' shown in Figure 6. For the differences - Ache, we find 1.65 f 0.02 kJ mol-' for COz and 1.27 f 0.02 kJ mol-' for Xe. That the fugacity f* of the multilayers is higher than that of the bulk liquid, while energetically the molecules are more tightly bound, is a consequence of the entropy term, as implied by A n d e r ~ o n .Figure ~ 7 shows the differential entropies of adsorption as a function of amount adsorbed. The limiting values at high adsorptions are close t o those calculated directly from the temperature coefficients off*. For both systems, is more negative than bo, the entropy of condensation to bulk liquid. Thus, as anticipated,' the entropy of molecules in the second and higher layers is less than that in the liquid, and this entropy effect outweighs the effect of the higher enthalpy of adsorption. The initial decrease in can also be given a quantitative interpretation in terms of the case VI equations. Thus, the parameter C is defined in the statistical mechanical theory by
a*
G*
(7) Anderson, R. B.J. Am. Chem. SOC.1946, 68,686.
c = Ql/Q* where q1 is the molecular partition function of molecules adsorbed in the first layer and q* that for molecules in higher layers. Consequently - -
(7)
-
G*
where Aahl and are the isosteric enthalpies of adsorption in the first and subsequent layers. Figure 8 shows the graphs for the two systems from which we derive
- -
Aahl - Aah* = -4.24 k J mol-' for CO,
- Aahl - Aah* = -3.16
(8a)
k J mol-' for Xe (8b) In each case, these values are found to be equal to the between the extrapolated value a t zero decrease in coverage, where adsorption occurs into the monolayer, and the limiting value L\clh*,where the monolayer is virtually full and adsorption occurs mainly into higher layers. Furthermore, a quantitative check on the shape of the curve of against nt is possible. Thus
a
where nIt and nzt are the amounts adsorbed in the first
Langmuir, Vol. 6, No. 12, 1990 1737
Applicability of Langmuir's Case VI
I:
T/K 1go
I
2?0,
, , 290,
I
f/atm
Figure 3. Application of eq 4 to the adsorption of COz by Vycor: plotted as f/nlt against f. Different scales are used on the abscissa as indicated by the vertical lines on the curves. For clarity, results at only a selection of temperatures are shown.
-A,h/k
J mol -I
-
y ;.< -
(a) 0,
'. - -
-
-A,hX
!L -AT!
bo
-@(C 0,)
(b) Xe
-
-A;,h*
1
2
I
7 A&'(Xe) 1 ,
f/a t m
Figure 4. Application of eq 4 to the adsorption of Xe by Vycor: see caption of Figure 3.
and higher layers respectively. It then follows that
- -
- -
S,h - A,h* = (A,h, - A a h * ) ( ~ n ~ t / ~ ~ (10) /(~nt/~~ By use of eqs 3 and 4, it is readily shown that
- -
Ash - A,h*
- -
+
= (dahl- A,h*)[(l - f / f * ) 2 / ( 1 (C- 1 ) X
(f/f*)2)1
(11) Taking values of C near the middle of the experimental calculated from eq 1 1 temperature range, values of for the two systems are shown as dashed curves in Figure 6. They follow the pbserved isosteric enthalpy curves to within the experimental uncertainty. The isosteric enthalpy curves for COz and Xe can be correlated closely by dividing the isosteric enthalpy of adsorption by the enthalpy of condensation and expressing this ratio as a function of the fractional coverage, 8, by using the monolayer capacities derived from the case VI analysis.
0
I h
-A$'
n ' h m o l g-1
Figure 7. Differential entropies of adsor&on as a function of nt for (a) CO*/Vycorand (b) Xe/Vycor. Aes* is the entropy of adsorption at the second and higher layers and Q0the entropy of condensation from the vapor at unit fugacity.
The points for both adsorptives then fall, within experimental uncertainty, on the same curve (Figure 9). We thus arrive a t a "corresponding states" type correlation of the adsorption data. A similar, though somewhat less precise, correlation is found for the entropies. We conclude that the parameters n,, f*,and C have the
1738 Langmuir, Vol. 6, No. 12, 1990
Burgess et al. Table 111. Values of u1 at
T,
un at T, u* independent of T
Figure 8. In C as a function of 1/T for (a) COz/Vycor and (b) Xe/Vycor.
I
1
2
(I
9
Figure 9. Reduced enthalpies of adsorption A.h/A,ho as a function of fractional coverage, 8: ( 0 )COs/Vycor; (0) Xe/ Vycor.
thermodynamic significance ascribed to them by the statistical mechanical theory and are not just empirical factors. Comparison may also be made with the Frenkel-HalseyHill equation? The present experimental data for Xe, for example, lead to a variation of the parameter B from 1.19 at 183 K to 2.20 at 273 K. Thus, the simple model in which B is related to the exponent of the distance in the attractive term in the adsorption potential cannot be realistic, since there is no theoretical basis for a variation of this exponent with temperature. I t would indeed be surprising if the "slab model" of the adsorbed phase on which the FHH equation is based applied to the region of the isotherm where most of the adsorption occurs in the first and second layers. Whether or not the FHH equation becomes valid at higher relative pressures where the case VI equation breaks down cannot be tested with the present data, since in porous media deviations from this equation are masked by capillary condensation. We note that the variation of B with f* indicated in Figure 3 of ref 1 is confirmed by the present calculations. Finally, we speculate on the meaning of the empirical observation that, for each system, u'/(f*/P)is independent of temperature, at least in the range in which hysteresis occurs. (8) For example: Hill, T. L. Adu. Catal. 1952,4, 236 and references given in ref 1. (9) International Thermodynamic Tables of the Fluid State 3, Carbon Dioxide; Pergamon IUPAC: London, 1976. (10)Street, W. Sagan, L. S.;Staveley, L. A. K. J. Chem. Thermodyn. 1973,5,633. (11)Ruhemann, B.; Simon, F. 2.Phys. Chem. 1931,815,369.
d.;
v (cm3 mol-')
coo
XP
37.49 28.B9 30.13 f 0.17
44.21° 36.9 f 0.4" 44.6 f 0.4
Since fD and f* refer to the bulk liquid and multilayer states, respectively, it seems reasonable to write u* = ul/ ( f * / f o ) , where u* is a molar volume related to (or perhaps equal to) that of the multilayer state. The constancy of u* would then suggest that the structure of the multilayer is temperature invariant. Taken with the temperature independence of the monolayer capacity, this would then support the view that the structure of the whole adsorbed phase remains constant. Table I11 compares the values of the molar volumes of liquid and solid COZand with the values of Xe at the triple point temperature (Tt) u*. If u* may be taken as the molar volume of adsorbate in the multilayer, then it would appear that in the case of COZthe multilayers resemble the bulk solid, while for Xe they have a structure more closely similar to that of the liquid at the triple point. It is more difficult to find rational explanation of the temperature independence of u*. In the crudest of cell theories of the liquid state,lZ the molecular partition function is of the form q1 = u: exp(-$'/kT),
(12) where ud is the "free volume" of a molecule in its cell and $I the energy change per molecule when N molecules are brought together from infinite separation to the liquid state with each molecule in the center of its cell. If a similar equation can be used for the multilayer, and since according to the statistical mechanical derivation of the case VI equation
f*lP= Q1/Q*
(13)
we have
f*lP = (u:/uf*) em[($* - $')/kT] (14) Now the free volume will be some fraction of the molar volume u: = klu'; uf* = k*u*
(15)
so that
u ' / ( P / P ) = u*[(k*/k')exPC-($* - $')/kT)I (16) There is no theoretical reason to suppose that k' and k* are independent of temperature nor that the temperature dependence of their ratio cancels that of the exponential term. If, however, the term in square brackets were independent of temperature and approximatelyunity, then the identification of u1/V*/fD) with the molar volume u* of the multilayer state would justified, and its constancy would imply the constancy of the structure of the multilayer. A more sophisticated theoretical approach is clearly needed to explain the observed behavior more convincingly. (12) For example: Pryde, J. A. The Liquid State; Hutchineon: London, 1966.
