these qualitative predictions are verified should be neutral with respect to adsorption from liquid shown in Figure 6. res of l ~ ~ ~ and~ heptane ~ ~ because o ~ the ~ isoe ~ mental ~ ~ data e therms cross o w another; preferential adsorption of Acknowledgment. Financial support by t,he Kational beptme a t 1 ow p wssure i s cancelled by preferential Science Foundation is gratefully acknowledged. a ~ ~ s o r ~crf ~ ~~~~~~~~~~~~n~ ~on at high pressure. All of
Analogy between Adsorption from Liquids and Adsorption from Vapors y A, ‘L.Myers* and S. Sircar School of Chemical Engineering, University of Pennsylvania, Philadelphia, Pennsylvania (Received M a y 16, 19%)
19104
Publication costs assisted by the National Science Foundation
The relation between adsorption from liquids and adsorption from unsaturated vapors is derived. The surface excess for adsorption from liquids can be calculated from equilibrium data on adsorption of the unsaturated gases. Equations for the surface excess are derived from simple type I and type I1 models of gas adsorption.
Theories of adsoqAion from liquids are made difficult by the chemical heterogeneity and structural irregularity of the solid surface as well as the fact that a t least two cornponcnts are involved in competitive adsorption a t the solid 9urface.l The simplest, case is the interaction of a dense fluid mixture with a solid surface. Lack of specific information about the composition and structure of the surface also complicates theories of adsorption from tingle gases, Several theories that nore the detailed structure of the surface have been proposed, tiit: Lsngmuir,2 Brunauer, Emmett, and Teller (BET) , 8 6teele and E I a l s e ~ ,Frenkel-Halsey~ ill slab theory,&eta:. Although these and other theories are usefgl to a degree for fitting observed data, they contain constants that cannot be independently measured. Thus there is no Fdly adequate theory of adsorption from gases, nor ia there a satisfactory theory of adsorption from IiyJids. Howevcr, the two phenomena are related : adeorptiom from liquids can be fully explained in terms ol adrjorption from their unsaturated vapors. The object of this work is to study the analogy between adsorption from liquids and adsorption from unsaturated vapors.
Equilibria for adsorption from a binary mixture mast obey the foilowing thermodynamic consistency test6
JPP P=O
g d p
p
$‘ nl’ -dP P
I
P=O
+
The first two integrals refer to the pure, unsaturated vapor adsorption isotherms and the last integral refers to the isotherm for adsorption from the liquid mixture. nleis the surface excess of component no. 1; the activity coefficients (y1,y2) and mole fractions (zi,xZ)are those in the equilibrium bulk liquid. Define the integral for adsorption of a pure vapor by
where rpis is called the free energy of immersion of the ith adsorbate. This terminology is appropriate because &* is related to the heat of immersion of the adsorbent in the liquid by a Gibbs-Helmholtz type of equation? With this definition eq b becomes (1) A. C. Zettlemoyer and F. J. Micale, Croat. Chem. Acta, 42, 247 (1970). (2) I. Langmuir, J. Amer. Chem. Soc., 40, 1361 (1918). (3) S. Brunauer, P. H. Emmett, and E. Teller, ibid., 60,309 (1938). (4) W. A. Steele and G . D. Halsey, Jr., J . Chem. Phys., 22,979 (1954). (5) D. M. Young and A. D. Crowell, “Physical Adsorption of Gases,” Butterworths, London, 1962, p 167. (6) A. L. Myers and S. Sircar, J.Phys. Chem., 76,341 2 (1972). (7) 8 . Sircar, J. Novosad, and A. L. Myers, f n d . EWJ.C h m . , Fundam., 11,249 (1972).
The Journal of Physical Chemistry, Vol. 78, N o . $3, 1972
3416
0.7
Aceording tcl. eq 3 there is no difference between adsorption irom saturated vapors and the corresponding liquids; the thermodynamic properties (in this case, free energy) of the pure adsorbates are identical. The surface exceqs for adsorption from a binary liquid mixture C.nle>ir, given by8
nl?=
n’(2I1 - 2 1 ) =
nyxp - Zl)
(4)
0
a
Experimental doia of Arnold
0.6 E
e -$ 0.5
T
u
1--
e
78.2*K
P’* 296
mm of
HQ
‘E 0.4
iR
0.3
0
n’ is the total number of moles of adsorbate in the adsorbed phase and 2 1 ’ is the mole fraction of component E in the adsorbed! phase. x? and x1 are the mole fraction of component J in the bulk liquid before and after contact with the adsorbent, respectively. The second expression, .n3(zr0-- xl)? is used to make experimental measurements of adsorption from liquids.8 rile may also be obtained from experimental data on the unsaturated vapor mixture by determining the limiting value of the quantity n‘(xl’ - xl) a t saturation pressure
c
1 0.2
a
0.1
01
‘0
0.2
03
0.4
0.5 P/
06
0.7
OB
0.3
IO
p’
Figure 1. Adsorption from gas mixture of nitrogen and oxygen on Ti02 at 78.2”K.
80,
1
P -
0th n’ and xI’are a function of pressure. At any equilibrium point, let the mole fraction of component 1 in the unsaturated vapor be y1; then XI is the mole fraction in the liquid that would be in equilibrium with vapor of composition y1 a t saturation. Thus x1 = xl(ylj is the liquid--vapor equilibrium which is measured independently of the adsorption experiment. Both n’ znd (Q’ .- xl)have finite values a t saturation for adsorption on a microporous adsorbent. However, n‘ terds to infinity and xl’approaches x1 as P approaches Ps if the adsorption takes place on a nonmicroporous adsorbent (see, for example, Figure 1); the product n’(5,’ -- xl)has the form m X 0 but the limit is finite. Thus the limit in eq 5 is finite whether the adsorbent i s nleroporous or nonmicroporous. Ei
QQ ~
x p e ~ ~ r n Data ~nta~
Equation 5 applies to adsorption on any highly dispersed (or microporous) adsorbent with a large surface area (e.g., I m2/g or more), Consider, for example, the experimend data of Arnoldg for the adsorption of mixtures of nitrogen and oxygen vapor on titanium dioxide (surface area 13.8 m2/g) a t 78.2”K. Figure 1 shows Arnold’.. data Sor a vapor mixture containing 49.8 mol yo nitrogen, The surface excess of nitrogen for adsorptilon from the saturated vapor, n’(x1’- xl) was calculated from these data and is plotted in Figure 8. For this v:tpor composition the equilibrium liquid contains 15.1 rnol t% nitrogen a t saturation ( P = 296 mm). The limit in e q 5 is nle = 70 ,umol/g and is found most convenieritlp at (I/%’) = 0 on a plot of n’(xl’ x,) V P . (l/n,’)~ The complete isotherm for the liquid mixture, show. in Figure 3 , was obtained by repetition The JozirizaE of Ph&ai
C.hemistry, VoL. 76, No. 29, 1972
E“
w z 0
t U v)
w x w 3
TOTAL AMOUNT ADSORBED,
pmcla/g
Figure 2. Surface excess of nitrogen adsorbed on TiQzfrom vapor mixture of 50.2 yooxygen and 49.8 u/c nitrogen at 78.2”K.
of the preceding calculation a t the other vapor compositions studied by Arn01d.~ There are two interesting features of Figure 2. First, the surface excess adsorbed from the vapor approaches the surface excess for adsorption from the liquid smoothly and asymptotically, in spite of the fact that the individual isotherms (Figure 1) intersect : nitrogen is preferentially adsorbed a t low coverage but oxygen is preferentially adsorbed at, high coverage. Second, the surface excess at BET monolayer coverage is 85% of the value for adsorption from the liquid solution. This suggests that the monolayer theory of adsorption from liquids,8for which the entire contribution to the surface excess comes in the first molecular layer, is reasonable as a first approximation. (8) J. J. Kipling, “Adsorption from Solutions of Non-Electrolytes,” Academic Press, New York, N. Y., 1985,gp 28,40. (9) J. R. Arnold, J.Arne?. Chem. Soc., 71, 104 (1949).
ble P: Surface Excess Quantities and Their Limits at Saturation Quantity
RET model
Langmuir model
mi In (I 4-ciP)
XI'
- XI
0
(1
+ (C, - Il")(1
Yl -P -- Pyz P,g
P l S
Pp5
m
The Langmuir and BET equations for adsorption of pure gases are given in Table I. The mixed-gas isotherms are based upon the assumption that the adsorbed phase and the bulk liquid phase obey Raoult's law; for the bulk liquidlo
and for the adsorbed phase1'
MOLE FRACTION NITROGEN IN LIQUID
Figure 3. Surfsee excest, of nitrogen adsorbed on TiOz from liquid solution of oxygen and nitrogen a t 78.2'K.
Py1 =
PlQXlI
(8)
Py, =
PzoX2f
(9)
Pi0 is the sorbate vapor pressure of the pure ith component. The values of PIQand Pzo are measured at) equal values of spreading pressure'l so that
erivaliom of Equations for Adsorption from Liquids Analytic expressions for adsorption from liquids can be derived from equations for adsorption from pure gases using eq 5, Here we shall derive equations for rile based upon the Eangmuir2 and BET3 equations 01 gas adsorption. These gas isotherms are chosen because they are the simplest examples of the two extremes merttioried previously : the amount adsorbed at saturation presawe (P) is finite for the Langmuir model but tends to infinity for the BET model. (%le)
The total amount adsorbed (n') from an ideal adsorbed solution isll 1
XI'
-
_
n'
I
n,?
322'
i_
n20
(10) K. Denbigh, "The Principles of Chemical Equilibrium," Cambridge University Press, London, 1966, p 250. (1 1) A. L. Augers and J. M. Prausnita, AIChEJ., 11, 121 (1965).
The Journal of Phyeical Chemistrp, Val. 76,N o .
1979
3418
A. E. MYERSAND S.SIRCAR
It is assumed that the amount adsorbed a t monolayer coverage 11s the s:me for both adsorbates ml = m2 Since ( X I ’
4-XZ‘)
=
(12)
1, we obtain from eq 8 and 9
Elimination of PZo between eq 10 and 13 gives an equation for P? as a function of pressure ( P ) and vapor composition (yl> the result is shown in Table I. The composition of the adsorbed phase (x~’),the number of moles In the hdsorbed phase (n’),and the surface excess are calculated from eq 6-13* The equations for i hwe qmntities and their limiting values at saturation are given in TabIe I. The variables must be expres,ced in terms of P and y1 before taking the limit P+P :it constant yl. Both n’ arid (xl‘ - XI) are finite a t saturation according to the Lsngmuir model and therefore there is XKI difficulty in calculating the surface excess using eq ET model, on the other hand, n’ approaches infinity and (zl’- XI) approaches zero a t saturation This 1s the expected behavior for a type $1. isotherm. The surface excess at saturation for the BET model 1s derived by means of L’HBpital’s rule
of immersion is finite so that the BET’ model does satisfy eq 3. Equation 15 is identical with the result obtained from the monolayer cell model of adsorption from ideal, binary liquid solutions’3 and is considered the basic equation describing adsorption from 1 i q ~ i d s . l ~However, it has never been derived before from models for adsorption of gases; eq 16 and 17 are new. The agreement of the derivation with accepted theories of adsorption from liquids confirms the validity of our basic equations (3 and 5). At first sight it is unclear why the B E T model (a type I1 isotherm) and the Langmuir model (a type I isotherm) should give the same equation (15) for adsorption from liquids. The BET model does allov for multilayer formation but the properties of the second and higher layers are assumed to be the same as that of the bulk liquid. Therefore the surface excess of any component must be concentrated in the first layer and, a t least with respect to the surface excess, the BET model is a disguised monolayer model. It would be interesting to derive expressions for the surface excess using more realistic models of multilayer adsorption. Equations 15 and 16 may be tested using the B E T constants (Ct,m,) found by Arnold;O these are given in Table 11. Equation 1.5 used with the average value ~
Jim P-Pq
The result is given in Table I . A comparison of the equations for surface excess in Table I s h a m that both are of the form
where
K
==
C2/G (BET model)
(16)
and
k‘
-+
TI=
~
~~
Table 11: Constants of BET Equation for Adsorption of Nitrogen and Oxygen on Anatase a t 78.2”Kg
1 czP2fl (Langmuir model) I c c*Pp
_____
(17)
For the Langmuir model the free energy of immersion (see eq 2 and Table I) is =
-rn,ln
(1
+ czP;)
(1s)
Equations 15, 17, and 18 for the Langmuir model satisfy the thermodynamic consistency test, eq 3 . For the BET model, the right-hand side of eq 3 is of the form ( m -- cc) because the free energy of immersion approaches infinity at saturation. This incorrect limiting behziior for the free energy of immersion is one of the flaws of the B E T theory.12 However it can be shown thal the limit of the diflereizce in free energies The Journal o,f Plysical Chernistrv, Vol. 76, No. 23, 197.2
Gas
C
w, mmol/g
Nz
143 72
0.141 0.131
0 2
of m (0.136 mmol/g) underestimates the experimental values shown in Figure 3 by a factor of about 2 . The reason for this poor agreement is that the B E T equation, which fits Arnold’s data reasonably well in the region of high surface coverage, does not fit the lowcoverage data where the adsorbent has a strong preference for nitrogen.
Summary This work shows how adsorption from liquids is related to adsorption isotherms for the unsaturated vapors. Here \%-ehave restricted ourselves to simple but representative models of gas adsorption in order t o illustrate the method but the theory can be applied to more realistic models of adsorption. Another interest,ing possibility is to predict adsorption from liquids in terms of the experimental isotherms for adsorption (12) T.L. Hill, “An Introduction to Statistical Mechanics,” AddisonWesley, Reading, Mass., 1960, p 135. (13) S. Sircar and A. L.Myers, J.Phgs. Chem., 74,2828 (1970). (14) D. H. Everett, Trans. Faraday Sac., 60,1803 (1964).
CALCULATIQN or’ DQFFUSION
3449
COEFFICIENTS FOR KBr-NBr-M@
from pure vapors, without recourse to any particular model of adsorption,
n‘ = total number of moles adsorbed from mixture per gram of adsorbent np surlace excess of ith component ndwoTbed from liquid pep gram of adsorbent no = total number of moles in bulk liquid per gram of adsorbent (beforecontact;with adsorbent) P = pressure Pi0 = vapor pressure of sorbate of ith ~ O I X I ~ Q I X Y ~ ~ Po = vapor pressure of bulk liquid R = gasconstauit T = absolute temperature z i = mole fraction of ,ith component in bulk liquid phase xi’ = mole fraction of ith component in adsorbed phase yi = mole Iract,iori oi itb componentin vapor phase ziO = mole fraction of itb component’in bulk liquid phase (before contact with adsorbent) y ; = activity coefficient of ith component in bulk liquid phase cpz8 = free energy of immersion of adsorbent in ith liquid7eq 2 =i
~ c ~ ~ , ~ o wfiiriancial l ~ ~ support ~ ? ~ ~ by?the ~ National ~ . Soirnca Foundarior; is ~ r a t e f u ~acknowledged. ~y
=- conJtant ~nLangmmx equation (see Table I ) I: = constant in BET equation (see Table I) K = conatant, eq 16 and 17 m -= number of rioFe6 adsorbed a t monolayer surface coverage G
per gram of otkorhent of ntoles 31 piire Jthcomponent adsorbed per gram of adsorbtnt nz’ = number of 11101e3 of zth component adsorbed from mixture per gram of arlec1rhent npO --
riiindm
ure for Calculating the Four Diffusion Coefficients
Ternary Systems from Gouy Optical Data. Application e System KBr-HBr-H,O at 25”1)2
Deportm,eizt of Chemistry and The Institute for Enzyme Research, University of Wisconsin, Madison, Wisconsin 68706 (Receiaed April 17, 1972)
k new method is presented for calculating the four diflusion coefficients, Dij, for ternary systems from Gouy optical data. I n this approach, the very accurately measurable reduced height-area ratios, DA,are first fit directly to the precisely known values of the solute refractive index fractions, a l , using the method of least squares. Then the experimentally measured reduced fringe deviations, 9, are fit to the corresponding reduced fringe numbers, f ({), by a least-squares procedure using the theoretical expression for i2 us. f ({). This calculation procedure, which is a theoretical improvement over previously used methods, was applied to new data for the system KBr-HBr-H20 a t 25’. The new method gives values for Dij for this part’icular system which are slightly better than those computed using the older “area” method. Gosting’s approximate theory for predicting L ) i j in electrolyte systems gives values for the diffusion coefficients which agree semiquantitatively with the measured Dij, indicating that the interaction between solutes in this system is due mainly Lo the eiectric field created by the rapid diffusion of H+ion relative to Br-- ion. The Onsager reciprocal relation was t’estedand found to hold within experimental error.
~~~~~~~~~i~~ 4
is then possible to derive expressions for each of the Calculation of tbe four diffusion coefficient^,^ D,,, four Dtl in terms of a n y four such “convenient conibinafor ternary liquid systems from Gouy optical data pre(1) This investigation was supported, in parr,, by research Grant sents a very int eresting problem in the treatment of exNo. AM-05177-02 from NIAMD, National Institutes of Health. The author was the recipient of predoctorai fellowships from the perimental results A difficulty arises because it is not Wisconsin Alumni Research Foundation a.nd from the NationaI Infeasible to derive closed-form analytical expressions for stitutes of Health. the D,,in terms of measured quantities. However, (2) *tiom of this work were submitted to the Graduate School of t,he bniversity of Wisconsin in partial fulfillment of the requirements the theoretical equations relating the D,, to experimenfor the Ph.D. degree. tal parameters do contain certain recurring combinat mas (3) Present address: Institute of Molecular Biology, University of of t h e D2,, which can be determined from the data. It Oregon, Eugene, Oregon 97403. The Journal of Physical Chemistru, Vol. 7 8 , N o . 23, 1972
