May, 1960
‘YHEORY OF DIFFUSION CONTROLLEU hBSORl’T1ON
As the transference numbers of E(+ and C1- vary leks than 1% over the entire range of H20-Dz0 mixtures a t 2 5 O Z 3 the two types of diffusion potentials encountered in Table I are almost completely determined by the isotopic composition and the KC1 concentration of the two solutions forming the diffusion boundary. Application of this principle to the chain consisting of 4 cells of type I, encountered in Table I and having a total e.mf. = AE, shows that 4E should in fact be very nearly zero as found experimentally. We shall finally examine the 0.44 correction in equation XI1 from the point of view of Haugaard’s theory of the glass electrodez4in moderately acid €Is20.According to this theory the proton has the same electrochenaicnl potential in the surface layer of the glass electrode as in the solution X bathing the glass electrode, the chemical potential or the actiTity of the proton in the glass electrode surface layer being held constant by a large number of acid ai,d basic groups located there. The 0.44 correction in equation XI1 is defined, e . g . , by the cell gli-lsl 10 01 df HCI in I120 IIiCl(batd. in HA)) I 0 01 d l (HC1 DC1) in 98yGD20 IIglaas 2
+
(XV)
( 2 3 ) L. G.Longsworth a n d D. A. iilaelnnes, J. A m . Chena. Sac., 69, llj(i6 (1837). ( 2 4 ) C;. TIitii!!mrcI, CunLr,l. rcnd. ( m i , . l a b . Cndsbi,rg. 2 a , I90 (1038).
KINETICS
637
the e.m.f. r2-r1, of which (right side positive) is 0.44 X 0.0586 = 0.025 v. I n order to apply Haugaard’s theory to our data we shall make the reasonable assumption that the two diffusion potentials in cell XV cancel approximately. It then follows from Haugaard’s theory applied to cell XV that 0.44 = T ! ! = log(aa)l 0.05860
-
where ( u H ) ~is the constant hydrogen ion actil-ity i n the outer surface layer of glass electrode 1 and (CLH ~ D ) is Z the constant hydrogen ion activity in the outer surface layer of glass electrode 2. The right side of equation XTTI corresponds to the change in p(DH) which the buffer mixture in the glass electrode surface layer suffers when taken from H20to 98% DzO. I n view of the fact that the pK values of a number of acids dissolyed in HzO are increased 0.40-0.52 unit when dissolved in D2OZ5 it would appear that the 0.44 correction in equation XI1 tentatively may be accounted for in terms of Haugaard’s theory of the glass electrode. Acknowledgment.--We are grateful to Profeijsor I?. Lumry for a review of this manuscript and to Professor 11. Ottesen for generous help in various
+
TVilJ%. ( 2 5 ) C . Iheoretical interpretation of Hommelen’s ‘1) Drpartment of Chemistry. Iowa State University. Ames, I o n a . P;s.tional Scienrr Foundation Senior Postdoctoral Fpllolr- at TT. S C . ‘2) A . 1:. II. Ward and 1., Tordai, J . Cheni. I ’ h u s . . 63, 485 (19.H;). 8:i) Ii. Siitlieilanil, X u , ! . .J. S c i . R e s . , AS, 683 (1952). 1 4 ) P.Di.1dliny a ~ i i lC . T . Fikr, J . A m . C h r m . S O C . , 8 0 , 2ti28 (1lGS). ( 5 ) J. R . Iluiuuiclcn. J . Culliizd Y c i . , 14, 385 (1059).
result. Incident to this it mill be shown that the theory of diff uaion controlled adsorption kinetics is not made appreciably more complex if evaporation of solute from the interface is also considered. New general results, applicable whethpr or not evaporation occurs and for general form of adsorption iqotherm will be developed. The ,qxcial case of adsorbates satisfyirig the Langmuir adoorption isotherm nil1 also be treated. Theoretical Let r ( c ) be the isotherm for adsorption of solute a t the liquid-vapor interface, C,(z,t) and Cz(z,t) the concentration of solute in mole~,/cm.~ in liquid and gas phaqe, respectively, a t position z and time t , D, and DZ the diffusion roefficients of wliite in liquid and gaq phase, reqpectirely, and K thr equililrinni roii~taiitfor diqtribution of solute lietn ecn liquid aiid gas phases. The following 1 )oiiidary value l)rol~lcniis gciirrated
ROBERTS. HANSEN
638 Illclxx(x,t)= C1t(z,t), x > 0, t CI(5,O) = co, z > 0 =O,z=O Lim Cl(x,t) = C,, t > 0 '
Y
>0
m
D:C%,(s,t) = C:t(s,fj, 5 < 0 , t > 0 CL(5,Oj = 0, z Q 0 Lim C,(z,t) = 0, t > 0 ' 2 -m Cl(0,t) = liCl(O,t), t 2 0 = rt ~ ~ c ~ , ( ot ,>t io,
wIx(o,c
+
(1) (21
(3)
I't
) + I-2 [KC,(O,ti - P O ' J b
(9)
which implies that C,(O,t) must satisfy the integral equation
(5) (5a)
(6)
(7)
(8d)
Since I'(t)=T' [C1(0,7)1, i.e., the surface excess a t any instant j s in equilibrium with the subsurface concentration, the integral equations 8 suffice in principle to determine Cl(O,t) if r ( c ) is given. If Dz or K are zero (Le., loss of solute neglected) eq. 8b reduces to a result given by Ward and Tordai.2 The integral equations cannot be solved to give an arialytic expression for C,(O,t) for arbitrary choice of I'(c), but they can nevertheless be compared with experiment numerically; thus r(c), the equilibrium adsorption isotherm, and r(t) can be measured, Cl(O,t) inferred from r(c) and r(t),and the integrals evaluated numerically. Thus eq. 8b, for example, asserts that the ratio ~ ( t ) / [ C d ' / 2 (1 K d D T G C i ( O , t ) d ( t - 7)'"l is coustsiit arid equal to 2 2 / D l / n and this is subject to experimental verification. The boundary value problem set forth in eq. 1-7 would not represent measurements of adsorption kinetics obtained by the vibrating jet method, for diffiision in the gas phase would then occur both in the direction of the jet axis and perpendicular to it. Possibly such a physical situation may be reprmented by boundary layer diffusion in the gas phase. Let h be the thickness of the boundary layer, atid Co' the concentration of solute 011 the gas phaqe side of the boundary layer. Eq. 4 aiid 5 of the origiiial hoimdary value problcni )%illnot apply, aud initcad oi cq. 7 n c nil1 h a w
+ +
D,c;,(0,t) =
i.1)
(subscripts 3: and t denote partial differentiation after the indicated variables, e.g., Cixx = b2C1/ ax2). It may be shown (very readily by Laplace transformation techniques) that the concentration of solute a t the interface must satisfy an integral equation which may be expressed in any of the equivalent forms
Cl(O,t sin2 0) sin 0 do]
Vol. 64
Other equivalent forms similar to those ill eq. 8 can he derived readily. This result could also be compared with experiment numerically; the comparison could be carried out using b and Co' a$ adjustable parameters or, more satisfactorily, by devising an independent estimation of them. Since such estimates do not appear possible for data presently available we shall not pursue this aspect of the problem further a t this time. Strictly, it must be noted that loss of solute (and, in general, solvent) to the gas phase must result in displacement of the liquid-gas interface, so that one must consider a problem of diffusion with moving boundary of the type treated generally by Danckwerts.6 The error resulting from neglect of this boundary motion can he bounded as follon 5: the amount of material transferred to the interface as a result of boundary motion a t time tis not greater greater than XCo,n-here X is the distance the boundary moTres in time t. X is not grcater thaii the distance the boundary would move if the diffusing components maintained their saturated vapor pressures a t the boundary ( L c . , no depletion a t liquid side of boundary due to loss of material to the gas phase). But for a single component in this case X t
=
-z'D2C, (0,t)
=
(p,v/RT)
e)'''
in which I,' is the component molar volume in the liquid state and po is its saturated vapor pressure. Hence X = 2 ( p o u / R T ) m . Suppose we consider loss of water to be the principal cause of boundary displacement. At 20" pa = 17.5 mni., u = 18 cc., Dz = 0.2 cm.2/sec., and so XCO= 9 X 10-6Cot'/z. On the other hand, for small t the leading term in r(t) is 2 2 / D T C 0 t ' / z , so for Dl = 8 x cm.2/sec., this term is about 3 X Cot'/*,or about 300 times as great as XCyo. Hence neglect of boundary displacement appears xell justified a t least for surface excesses nppreciahly less than their equilibrium values. Certain consequences of eq. 8 c a n be deduced for arbitrary form of the adsorption iwtherm ; additional consequences can be deduced if the ihotherm is specified. We therefore discuss thew consequences in two xctions, using the Langniuir isotherm to illustrate the second group. A. General Remarks on Diffusion-controlled Adsorption Kinetics.-On consideration of eq. 8d it is apparent that the limit a t large t of Cl(O.t) is not c0,imt c,)'(1 K ~ D 'D~). ? It iq therefore of iiiterest to coilcider the magnitude of the factor 1+KdD,!D1) for materials of the type frequently sith,jec*ted to kinetic study. Organic acids and alcohols of intermediate chain lmgth, slightly soluble in water, in aqueous solutioii are ;.tich nin-
+
( 0 ) 1'
\'.
~ ~ d r l L ~ ~ l r L YI ltUbT .l s . F U l U d l L U
, 46, 701 l l n j O j
THEORY OF DIFFUSION COSTROLLKD ABSORPTION KIXETICS
May, 1960
639
terials. Let p o be the saturated vapor pressure of the organic compound in mm., moits saturation concentration in water (molesjliter). Then k' = (po,/+i60),/(moR2'). Choosing as representative values of the diffusion coefficients D, = 0.1 cm.*/sec. sec. (ethanol in air a t 9.5"), D, = (phenol in waterat 18"'onehasat2O0 K d m = 5.1. X 10-3 (po,'.mo). For pentanol-1 at 20°, p o = 2.8 inin., = 0.22 mole/liter, and therefore li. t/D2!D1 = 0.07, i.e., in this system Cl(O,t+m) will be less than Co by 7%. At half coverage, r = 4.0 X mole/em.2, approximately, and tE,e predictrd shift in surface tension is 0.66 dyne/ ein. This is to be compared with the shift of 1.0 dyne/cm. observed by Hommelen in the aqueous hc~sanol-Isystem. Let be the limiting surface excess, corresponding to Cl(o,t) = C,[l K dm'1-I. Then the aiyniptotic behavior of Cl(O,t) can also be deduced by consideration of eel. 8d; it is
r
(',(O.t) =
-_-
+
1
+
Po f i \ m
-
r
dxt
(1 $. K d m ) ( t large)
(11)
For small l , and therefore for small values of C1(O,t), we may develop Cl(O,t) in poFver series in substitute this series in the Alaclauriii exp:msion of r(c)and establish the coefficients in the power series by means of eq. 8d. In this way we ohtain ?'/2,
(LT, -
B. Diff usion-controlled Adsorption Kinetics in Systems Obeying Langmuir's Adsorption Isotherm. -- Let r ( c ) = I'n,c c / ( l + (10 (13) 11c tlleii iiit rotiwc riev varia1)lcq ___ = ( / ( ' l ( O , t i , ii = lmi c = ~ c , / ( + I fidf12/Dl)
z-m-ll
(1'))
which ha;, the form of a recurrence wlatioii pvrmitting evaluation of Ir, in terms of L-n-l, I n--l . . . ., C0; since 7 - 0 = 0, VI, 7,-2, . . . ., eaii be evaluated sequentially by means of eq. 19. Thi> procedure is exact in the limit as az + 0, aiicl mi be tested for accuracy of approximation by Fpeatiiig the calciilatioii with smaller iiiercnieiiti t+ Az uiitil variation is satisfactorily miall. Equaticii: (ecc cq. 11) 19 m s programmed for cloinputer s(:luti~iiawi \olved in this manlier for U = 16, S, C, 2 . 1, G .i. z = ( 2 I ',~ d&t/7-: ~ qj = i x L , r m ) d D 1 7 , Tlicii cq. 8a-d can he reduced to integral equatioc< 0 25 and 0.125 t o a precision of 15, or hettc~: e~tablishing T7(:) Tvith a single parameter I=; Recults are presented graphically in Fig. I . From a coniparison of results obtninctl t).v nwnci*the follon iiig are reprcwntatix e ical integration with those obtained hy t1.e :i\yn:ptotic formula eq. 16. the latter formnla 11 :i\ ludjir.tl to yicld a value accurate to nithin I C , if 7,'f 1. greater than 0 7. Similarly the w i i ( i f the f1r.t three terms in the power series iii eq. 17 was foiuicl to differ from 7 2 by not more than thc third terni. The functioii [ ' ( r ) having 1)ecii oiitaiiicd, t h c ~ I *
m
Discussion
1.o
0.5
2.
Fig. 2.-Dcpcndence of reduced surface cxcess r/r, 011 reduced square root of time for various values of limiting reduced hubsurface concentration.
1.5
2'.
T1.1 I 0
c
v
0. 5
0.5 1.0 22. Fig. 3 -1kpendencc of reduced spreading prtssurc. (70,- I/) P,ET on reduced time 2 2 = (2Co/rm)2D,t/?r for various limiting reduced subsurface concentrations.
dependence of surface excess and boundary tensioii on z can be obtained from eq. 20 and 21 (YO
r/rm= u/(i + u) - y)/rmRT = In (1 + U )
(20) (21)
Dependcme of reduced surface excess r/r, on t/?) is presented in Fig. 2; dependence of reduced qpreading pressure (yo - y) r,RT (11 z 2 (proportional to t ) is presented 111 Fig. 3. x (proportional to
'
The asymptotic expression eq. 11 provides a simple estimate of the time required for approximate equilibration of a new surface. Thus if Dt = cm.2/sec., f = 4 x 1 0 - 1 0 mole/cm.? (about half coverage in the case of straightchain organic molecules) C,(O,t) will differ from its limiting value by less than 10% if t > 5.1 X lO-'/CoZ sec., €0 being the bulk concentration in nioles/liter. Further, this equation can be used with the Gibbs adsorption theorem to derive the following limiting laws for dependence of boundary tension on t'ime y - ym = - f'R7.111(1 - F/Cod/;;7,Tt) (22) = Fz ~~,~d\/?rL>it ( t large! ("3) For the same-choices of and DI, we find y - y m = '7.0 x 10-4/'COt1/2. This simple formula represents u-ithin a factor two the difference hetn-een boundary tension and its equilibrium value near equilibrium (within about 3 dynes/cm.) as observed for aqueous solutions of aliphatic alcohols and acids of from 5 to 11 carbons by a variety of techniquess-I1; the time scales over these systems vary by lo5and this agreement is probably as cloEe as can be expected for an arbitrarily selected I?. At sufficiently low surface ages (ages siifiiciently low that back diffusion is negligible) the amount of material transported to the interface is the same as that which would have been transported into an infinite medium , initially empty; the leading term in eq. 12 corresponds to this amount, and this Equation 12 permit's esresult is well tension of the range in which surface acciimulat~ion a t early ages can he calculated for arbitrary adsorp t,ion isotherms to t'hird-order ternis. A consequence of the t'/? dependence of thc leading terni is that the initial rate of adsorptioii is iiifiiiite ill the diffusion-controlled model, and coiiscqueiitly the initial rate of surface tension depresPion is also infinite. Figure 3 clearly s h o w t,his hchnvior for a solute ~atisfyingthe Langmuir isothcirni! aiid it, should be emphasized that this re+:ilt is gcneral. The comment by Defay and t i o n i n i ~ ~ l ethat ii "the surface tension does liot decrea,
