J. Php. Chem. 1983,87,3988-3990
3988
2TTF+----* TTF + TTF2+ TTF2+f 20HTTF-(OH)z -+
(20) (21)
Conclusions Dimer cations of large aromatic compounds such as naphthalene and its derivatives, anthracene, and pyrene have been demonstrated to exist. The majority of these studies were pursued in glassy matrices a t low temperatures. Subsequent warming of the matrices led to detection of dimeric and even tetrameric species. Often this warming of the matrix caused cracking of the glass and was, thus, undesirable. Some of the dimers so formed were the product of hole capture of preexisting neutral dimer molecules and, consequently, in an excited-state configuration, i.e., “unrelaxed” dimer cations. The studies presented herein give the possibility of easily studying dimer cation formation resulting from the reaction of a monomer cation and a neutral molecule within the lipoidic core of
the microemulsion droplet. The dimer cation, thus formed, is “relaxed” and can maintain its “normal” configuration (probably sandwichlike). Extremely interesting is the finding of a tetrathiafulvalene dimer cation. Although its formation should not be surprising due to the facility with which it stacks in organic conducting salts,28its spectral characteristics have not previously been reported. Microemulsions appear to be excellent media for examining such reactions as they give a relatively viscous environment which enhances agglomeration to form multimers without restricting the core solubilized molecules to a too great extent. The importance of the microemulsion droplets’ ability to induce charge separation and to enhance substantial product ion formation should not be left unnoticed. Registry No. 2$-Dimethylnaphthalene, 581-42-0; tetrathiafulvalene, 31366-25-3; duroquinone, 527-17-3;sodium hexadecyl sulfate, 1120-01-0;2,6-dimethylnaphthalenedimer cation, 38071-82-8;tetrathiafulvalene dimer cation, 70257-88-4.
Diffusion-Controlled Adsorption. Concentration Kinetics, Ideal Isotherms, and Some Applications Harry L. Frlsch Department of Chemistry, State University of New York at Albany, Albany, New York 72222
and Karol J. Mysels” Research Consulting, La Jolla, California 92037 (Received: December 3, 1982)
Changes of concentration c(n,t) occurring in a quiescent solution, initially having a uniform concentration C, during adsorption on a surface of arbitrary shape can be analyzed by using the method of reflections and superpositions. Diffusion to the surface, considered as a sink, yields one component of the total expression and diffusion from the surface, considered as a source, yields the other one. If the former is given by C,$(n,t), the latter becomes .f:(l- $ ( X , t - T ) ) dc(0,T)where J. is often well-known and c(O,T) is the subsurface concentration at time T. Each increment dc(0,T)is considered as a source acting for its lifetime t - T . The same kind of argument has previously yielded an expression relating c(0,t) to the amount adsorbed and the adsorption isotherm. All these results involve integral equations, but, when the adsorption isotherm is ideal, Laplace transform methods can be used to obtain equations which can be solved explicitly for specific geometries. Application to the plane and the sphere gives such explicit solutions. One. of these agrees with one pertinent specific solution found in the literature.
The kinetics of adsorption can be of intrinsic interest as providing clues to the molecular mechanism of this process or to the nature of the participating species; their understanding can help guide equilibrium adsorption studies; and they can be important in practical applications such as catalysis, flotation, detergency, or chromatography. Yet few theoretical models exist to help in designing or interpreting experiments. Even in one of the simplest cases, diffusion-controlled adsorption, the few equations available are presented as purely mathematical consequences of appropriate differential equations without providing insight into their physical basis. We shall return to one of these prior results, dealing with the rate of adsorption on a plane,’ in connection with eq 16. In a recent paper,2 one of us has shown that by using the simple method of reflections and superpositions a physically meaningful general expression for the kinetics of diffusion-controlled adsorption could be obtained for *Address correspondence to the author at Research Consulting, 8327 La Jolla Senic Dr., La Jolla, CA 92037. 0022-365418312087-3988$0 1.50/0
any shape for which the law of diffusion to a sink was known. We now present an analogous expression for the kinetics of concentration changes in the solution as adsorption proceeds. For the special case of an ideal adsorption isotherm, explicit solutions are also presented for both adsorption and concentration kinetics for the plane and for the sphere in a quiescent liquid. Future papers will be concerned with ideal adsorption when a stagnant layer is effective, which involves more complicated calculations, with Langmuirian adsorption, which requires numerical computations, and with asymptotic behavior, the analysis of which involves both of the above.
Concentration Changes A solution of concentration C,, in equilibrium and bounded by a surface, may be considered as being the site of two opposite fluxes. One is transport of solute toward (1) Bakker, C. A. P.; Van Buytenen, P. M.; Beek, W. J. Chem. Eng. Sci. 1960, 22, 1039. (2)Mysels, K. J. J . Phys. Chem. 1982, 86, 4648.
0 1983 American Chemical Society
The Journal of Physical Chemistry, Vol. 87, No. 20, 1983 3989
Diffusion-Controlled Adsorption
this surface considered as an ideal sink from a solution initially at uniform concentration C,. The other is diffusion from the solution in immediate contact with that surface considered as a source of constant concentration C, into a solution of initially zero concentration. The second flux may be considered as the reflection of the first one at the impervious surface. The two fluxes exactly cancel each other and the concentration remains unchanged. This is a simple illustration of the method of reflections and superpositions for treating diffusion problems which is valid as long as the diffusion coefficient is c o n ~ t a n t . ~Physically ,~ it corresponds to the fact that, when the diffusion coefficient is independent of concentration, each molecule, and therefore also each group of molecules, diffuses independently of othersa2 More specifically, if the first flux results in a concentration c ( x , t ) a t a point distant by x from the surface at time t which is given by C,$(x,t), then the effect of the second flux must be given by C , ( l - $ ( x , t ) ) so that their sum gives the undisturbed original concentration C,. The exact form of $ depends on the shape of the surface5 and may be found for simple cases in books on diffusion3p4 or heat conduction.6 Obtaining it does not involve time-dependent boundary conditions. As in the previous paper2 we apply this method to the case when the surface adsorbs the solute. Now the second flux depends on the momentary subsurface concentration c(0,t). It can be considered as the sum of fluxes due to each increment d c ( 0 , ~of ) this concentration, each beginning to contribute a t a time 7 when c(0,7) is reached. A t time t each of these increments has a lifetime t - 7 , generating a contribution to the concentration equal to 1- $(x,t-7) d c ( 0 , ~ ) .Hence, the concentration is given by c ( x , t ) = c,$(x,t)
+
1
t
0
(1- $ ( X , t - 7 ) )
dC(0,7)
(1)
Transposing the variables gives the simpler form c(x,t) = C,$(x,t) -
1 t
0
C(O,7)
d$(x,t-7)
(2)
The same result could be obtained more directly by the application of Duhamel’s theorem7 but would provide less physical insight. The value of c(O,T),which appears in the above equations, is given as shown in the previous paper,2 by a similar reasoning, by
where rt is the amount adsorbed (surface concentration) in time t and C,f(t) is the amount transported in time t from a solution of initially uniform concentration C, to a sink. The relation between f ( t )and $ ( x , t ) is t
f ( t )= D S0 ( W ( x , t ) / a x ) , = ~dt
(4)
General Solutions Equation 3 involves two unknowns: c(0,t) and rt. When equilibrium between surface and subsurface is sufficiently rapid so that the adsorption rate is only diffusion con(3) Crank, J. ‘The Mathematics of Diffusion”; Oxford University Press: London, 1957. (4) Joost, W. “Diffusion in Solids, Liquids, and Gases”; Academic Press: New York, 1960. (5) Specifically J. is the solution of the following diffusion problem: a$/at = DV2;initial condition, J.(x,O) = C, ( x b 0); boundary condition, $(O,t) = 0 ( t > 0). (6) Carslaw, H. S.; Jaeger, J. C. ‘Conduction of Heat in Solids”, 2nd ed.; Oxford University Press: London, 1947. (7) Duhamel, J. Ec. Polytech. Paris 1833, 14, Cah. 22, 20; cf. ref 6, p 30-1.
trolled, these two are related by the equilibrium adsorption isotherm applicable to the system and presumed to be known. With this information, eq 2 and 3 can be solved numerically by iterative procedures. In some situations, however, the method of Laplace transforms can lead to more general results. Equation 3 can then be applied and solved independently of eq 2 or it can be combined with eq 2 in in the transform stage and the same solution obtained as the limit for x = 0 of the more general solution based on eq 2. The former procedure is much simpler, but we will use the latter alone for the sake of brevity as we wish to obtain the more general results anyhow. Laplace transforms are particularly applicable to this problem when the adsorption isotherm is ideal, i.e., when r = ac (5) The proportionality constant a has the dimensions of length and equals the depth of solution containing, a t equilibrium, the same amount of solute as is adsorbed on the underlying surface. Thus, a is a natural unit of length in these systems and will be used to scale other distances. The Laplace transform of eq 2 is E(x,s)/C, = $b,s) - (E(o,s)/C,)(x$(x,s) - 1) (6) and, in view of eq 5, that of eq 3 is E(O,s)/C, = f ( x ) / ( a + s f ( s ) ) (7) where s is the Laplace variable and the superscript bar indicates the tranformed equation. Combining eq 6 and 7 and rearranging gives for ideal adsorption the subsidiary equation for the concentration ~ ( x , s ) / c ,= $ ( x , s ) - f(s)(s$(x,s)- l ) / ( a + 8 s ) ) (8) Use of this equation reduces the problem of defining the concentration kinetics to finding $ ( x , t ) , f ( t ) , and their transforms, and, after substitution into eq 8, finding the inverse transform of the resulting expression. We shall illustrate this below for the plane and the sphere in a quiescent liquid. Simple Geometries Plane in a Quiescent Liquid. For a plane in a quiescent liquid, the basic diffusion equations are given by eq 3.15 and 3.13 of ref 3 as f(t) = 2(Dt/x)’l2
$ ( x , t ) = erf [ ~ / ( 2 D ’ / * t ’ / ~ ) ] Their Laplace transforms are f(s) = l / s q
(9)
(10)
$ ( x , s ) = (1 - e - * , ) / s
(12) where q ( S / D ) ’ / ~ . For ideal isotherms, we obtain the complete rate expression from eq 8 which now gives c(x,s)/C, = l / s - e - Q X / s e-qx/(as(q- l / a ) ) (13) Upon inversion and introduction of the reduced time T D t / a 2 and the reduced distance from the surface X E x l a , we obtain c(x,t)/C, = 1 - e - X 2 / 4 T E ( X / 2 T 1+/ 2‘ P I 2 ) (14)
+
For brevity we defined
E(m)= em’ erfc m
(15)
which is a tabulated8 function ubiquitous in these calculations. (8).References 3 and 6; more extended values are in tables of the error function of the imaginary argument in ref 6 and 9 (for x = 0 and y = m).
3990
The Journal of Physical Chemistty, Vol. 87, No. 20, 1983
The limit as x as
-
0 gives the subsurface concentration
c(O,t)/C, = 1 - E(T'/')
(16)
This expression has been previously obtained by Bakker et al.' starting with the basic differential equations and boundary conditions. Sphere in a Quiescent Liquid. We consider a sphere of radius p . The distance, r, from the center of this sphere is frequently used in lieu of x which would be the distance from its surface. We shall use the reduced radius R = p / a and the reduced distance from the surface X = ( r - p ) / a and note that r / p = (R + X)/R. For our case eq 6.60 of ref 3 is applicable and can be rewritten as
Frisch and Mysels
Here G (1- 4/R)'I2. This quantity becomes zero when the radius of the sphere becomes 4 times the characteristic depth a, i.e., is relatively large. For smaller spheres, G becomes imaginary and an alternative expression is needed to utilize tabulated functions. Tables in ref 9 and 6 give values of the real part 3 and the imaginary part 3 of w ( z ) - e-L2 erfc (-iz). By taking =
+ i(X/2T1I2+ T1I2/2)
z = GT1I2/2
eq 22 reduces to c(r,t)/C, =
The limits of eq 22 and 23 as x concentration as Using relation 4 above, this gives f ( t ) = Dt/p + 2(Dt/7r)lI2
(18)
The Laplace transforms of these two functions are (19) f ( s ) = ( l / s q ) ( l / p q + 1) $(r,s) = (l/s) - (p/rs)e-q(('-p)
c ( O , t ) / C , = 1 - (1/2G)
(1 - G)E(
[
T'i2(l - G)
c(O,t)/C, = 1 - 3
0 give the subsurface
( )]
(1 + G)E T'/'(;+
(20)
For ideal adsorption isotherms, eq 8 now gives
-
(G imaginary) (23)
+ 3/G
G))
-
(G real) (24)
(G imaginary) (25)
with 3 and 3 defined as above but for z = (GT'J2 + iT'9/2. After inversion and conversion to reduced quantities, this gives c ( r , t ) / C , = 1-
GI") 2
-
Re-X2/4T
2(R
+ X)G
(1 - ,E(
[
(1
+ G)E( 2T1i2+ (1 t
x 2T'I2 +
(1 - G)")] 2
(22)
Acknowledgment. H.L.F. acknowledges the donors of the Petroleum Research Fund, administered by the American Chemical Society, and the US. Army Research Office (Grants DAAG2981K0040 and DAAG2981D0093 for partial support of this work. (9) Abramowitz, M.; Stegun, I. 'Handbook of Mathematical Functions"; Dover Publications: New York, 1965.
