3297
TRANSFER DIFFUSION
Transfer Diffusion. I.
Theoretical
by I. Ruff* and V. J. Friedrich Institute of Inorganic and Analutical Chemistry, L. Eotvds University, Budapest, Hungary
(Received January 18, 1971)
Publication costs borne completely by The Journal of Physical Chemiatry
In a system in which AX diffuses in the presence of A, transfer diffusion has been defined as the apparent diffusion due to the jump of X from an A it was bound to, to another, free A. This process results in the displacement of AX without translating A’s. Theoretical equations have been given for such types of transport phenomena, particularly for isothermal diffusion, electric conduction, and thermal diffusion. The correlations obtained are suitable to calculate the second-order rate constant of the exchange reaction AX A* A A*X. The conditions of the application of this effect for kinetic measurements of very fast processes are discussed in various cases. It is shown that the diffusion limit is also affected by transfer diffusion and results in a concentration dependence of the diffusion limit.
+
+ *
the experimental parts; nevertheless, in many points of view, it gives the first detailed description of the I n previous communications‘ a new method has been phenomenon itself. suggested for the investigation of very fast exchange reactions which is based on the measurement of the 2. General Theory diffusion coefficient apparently increased by the exLet us consider an exchange reaction in which A and change reaction in a system where both reactants are A* are chemically identical particles and X is an atom, present and-at least-one of those diffuses according radical, ion, electron, or an energy quantum that can to a concentration gradient. Some aspects about this transfer between them phenomenon in electron or proton transfer reactions have been discussed theoretically by Levich,2 D a h m ~ , ~ AX 4-A* A -k A*X (1) and Wyatt,4 but they considered only special cases with where the asterisks serve only for distinguishing between respect to either the type of transport or the type of rethe two sides of eq 1; they do not mean, however, any action. physical or chemical differences except that A is The methods, applied so far for the determination another particle in the space than A*. of the rate of exchange reactions, were isotopic exchange Let a system consist of several AX’S and A’s which and nmr line broadening techniques. The upper limit can equally be a mixture of the gases or their solution of the first one is related to the time consumed for in either liquid or solid state. If the gradient in their mixing the reactants and for separating them, while chemical potential is not zero all over the space conthat of the latter one corresponds to the ratio of the sidered, they would move according to this gradient. time of interaction of the electromagnetic quantum with In course of their translation two processes are in operathe species and the lifetime of that species in the initial tion: (i) the conventional migration of the species, or product state. These technical and principal restricand (ii) the exchange reaction which causes an apparent tions caused the upper limit to be about lo7 and-in displacement of, say, A by every effective collision with successful cases-lo9 ll.l-l sec-l for the second-order AX that results in the transfer of X, since, after such a rate constant, respectively. collision, A can continue its movement from the point It will be shown below that the transfer diffusion a t which the species AX has situated. method, as we intend to call it further, gives the possiIt is again due to the gradient in the chemical pobility to determine the rate of the exchange reactions tential that the jump of X is favored in the direction of within about one order of magnitude below the diffusion the gradient; thus, in addition to the conventional limit. Thus, it has a significant importance in intransport, the exchange reaction increases the flux of vestigating the processes which proved to be too fast AX. for the methods known so far. As some simple examples, the effect of the exchange The purpose of the studies to be published in the series, started with the present paper, is to show the (1) (a) I. Ruff, Electrochim. Acta, 15, 1059 (1970); (b) I. Ruff and applicability of the effect of transfer diffusion for I. Korosi-bdor, Inorg. Chem., 9, 186 (1970). kinetic investigations of exchange reactions near the (2) V. G. Levich, Advan. Electrochem. Electrochem. Eng., 4, 314 (1966). diffusion limit and to gain some information about the ( 3 ) H. Dahms, J . Phys. Chem., 7 2 , 362 (1968). mechanism of these very fast processes. The theoret(4) P. A. H. Wyatt, 1st Interam. Conference on Radiochemistry, ical discussion given here serves partly as a basis for Montevido, 1963.
1. Introduction
The Journal of Physical Chemistry, Vol. 76,N o . 91,,1971
I. RUFFAND V. J. FRIEDRICH
3298
dbt
5tep
2nd
step
3rd
step
m . 0
4th step
stands for “more terms” which influence the chemical potential in particular cases not considered here, e.g., pressure, magnetic field, etc. Some of the quantities listed above as T,c, E, can depend on the position coordinates x, y, and x, but this will not be marked separately. It seems to be useful for the sake of simplicity to exclude the conditions, when the directions of the two gradients are not parallel a t any point of the system. This restriction involves that along any surface where PA is constant, PAX is also constant. The parallelism of the gradients does not mean any additional restriction except with respect to the concentration gradients, since any gradient in the intensive parameters ( T , E, etc.) gives evidently parallel gradients in the chemical potential of both reactants. Let us define the flux of X , disregarding whether it is carried along by the migration of an AX or it jumps from an AX to an A, over a surface where the chemical potentials are constant. Thus
Figure 1.
reactions of iron(I1) with iron(II1) and iodide with triiodide is shown schematically in Figure 1. In the iron( 11)-iron(II1) exchange A corresponds to iron( 111) and X stands for the electron, while in the other reaction A is the iodide ion and X is 12. In Figure 1, four steps of the collision process are shown: (1) the approach of the reactants, (2) the moment of collision before X is transferred, (3) that after the transfer, and (4) the separating products. It can be seen that the distances 8.4 and AX which should not be moved along by A and AX, respectively, are always the distance between the mass centers of the initial and final A or AX. They can be incidentally equal t o each other, if X is a pointlike particle as it is in electron exchange reactions; however, in general, their length is different. I n general, the directions of the jumps do not lie in the same line, but they are always parallel, since the mass center of the activated complex, as a whole, does not change. To formulate the effect of the exchange reaction increasing the transport, the quantities involved should be defined. For the chemical potentials we have PA =
H A - TSA
+ RT In + ZAFE+ m.t. CA
(2)
and PAX
=
+
HAX- TXAX RT In CAX
+ XAXFE+ mat.
and
where PA,^ and AX,^ are the chemical potentials a t the surface (constant all over it) and for the meaning of the distances 8A’ and AX' we refer to Figure 2. On the other hand, the chemical potentials a t the points of the mass centers of the final states, marked by parentheses in Figure 2, are
(3)
where H = enthalpy of formation, S = entropy of formation, T = temperature in OK, R = universal gas constant, c = concentration (supposing ideal behavior), z = electric charge in atomic units, F = Faraday constant, E = electric potential, and the abbreviation m.t. The Journal. of Physical Chemistry, Vol. 76,N o . 81, 1971
where nx is the moles crossing a unit area of this surface, pax is the general transport coefficient, and the v’s are the rates of the exchange reaction between the layers of the two sides of the surface in forward (positive flux) and backward (negative flux) directions. Consider an elementary area of the surface mentioned above with an AXA activated complex whose axis AX crosses the surface with an angle cp. Introducing the local coordinate X which is perpendicular to the plane of the elementary surface area and its direction points to that of the positive flux of AX, for the chemical potentials at the points of the mass centers of the initial AX and A one has
and
The rate of a chemical reaction, in general, can be
3299
TRANSFER DIFFUSION
activated complex is asymmetric. The less the gradients, the less this asymmetry. Thus, if the gradients are small enough in comparison to the interaction listed under ii, the change in the chemical potential of the activated complex due to the gradients can be approximated by the arithmetic mean of those of the initial and final states PAXA
:[so
PA,s
SA’ sin
+
PAX,s
+
+
APs*
SAX’ sin
‘p
grad PAdh
t
‘p
grad PAxdh
-
According to eq 10, we have
c
-A
=
+A
GAXA- P A RT
Figure 2.
- PAX
and given as the product of the concentration of the activated complex and the frequency factor 2 of its decomposition to the products. Using the well-known correlation of the thermodynamics
G*-Gl-G2=
-RTIn-
C ’ CiCa
(9)
Substituting eq 5-8 and 11 and 12 into eq 13 and 14, one obtains from eq 4 dnx -= dt
-PAX
grad
PAX
-
one obtains
v
=
~ c = * zexp(-
G*
- PI - p2 RT
where G* is the free energy of the activated complex and G1 and G2 are those of the reactants. I n this way, the chemical potential of the activated complex for the case under discussion should consist of three parts: (i) the sum of the chemical potential of the reactants in a system in which there is no gradient and their value is equal t o that of PA,^ and pax,s; (ii) the increase in the chemical potential ApS* due to the interactions between the reactants that is characteristic t o the particular reaction, but independent of the gradients; and (iii) the effect of the gradients on the activated complex. The determination of this latter quantity needs some approximation. If the gradients were zero, the structure of the activated complex would corresponld to t’hat X would be at the half of its pathway between the initial and final states; i.e., one-half of the part,icles AX which gained the necessary free energy of activation are still in their initial position and one-half of them already transferred. The same is valid for A. If, however, the gradients have finite values, the potential barrier to be overcome by X is more or less distorted, its maximum is not in the middle of the pathway of X, and so the
( 6 ~ - 6 ~ ’ sin )
(o
grad p~dX)]dhdp
(14)
where v is the same as in eq 10, but with PI and p2 equal to PA,^ and PAX,*, respectively. Integration over cp takes into account that any case from BAX parallel to the surface to EjAX perpendicular to that can equally influence the flux, while the second integration over X from zero to 8 A X sin cp accounts for the fact that all positions of the activated complex affect the flux, if BAX crosses the surface or-at least-reaches it by one of the ends. (It is mentioned, however, that integration over X-between 0 and B A sin cp-must not be done, since displacement of the mass center of A along BA does not directly influence the flux of X.) The integrations of eq 14 can be performed, if the function of the gradients with respect to the position coordinates-and so to X-is known. For the most of the practical cases the approximation grad PA = constant and grad PAX = constant, within the ranges the integration is done, seems to be satisfactory. Hence The Journal of Physical Chemistry, Vol. 76, No. 81, 1071
I. RUFFAND V. J. FRIEDRICH
3300
[l
--
grad ]}grad C AX grad PAX
PAX
(15)
It is seen that the term involving v in this equation depends on the particular structure of the activated complex and on the gradients in the chemical potential of the reactants, i e . , on the experimental conditions. The latter can certainly be varied so as to get effect in addition to PAX on the basic transport process, and in this way v =
(16)
~CA.~CAX,~
can be determined, if the 6’s can be calculated from other sources. The main restriction that still remains in the experimental application is that the term including v (i.e.l “transfer diffusion term”) should exceed the error in PAX. These errors result in a lower limit in k. Nevertheless, an experimental method can be suggested on the basis of the correlation in eq 15 by measuring the increase in the transport due to exchange reactions fast enough. Equation 15 becomes less complicated in particular cases, when the concrete form of the gradients in the chemical potentials are substituted and, e.g., the 6’s are equal. These examples will be discussed below.
3. Special Cases A . Isothermal Diflusion. one has
For isothermal diffusion,
In this case the transfer diffusion term becomes independent of x, and so a normal diffusion is observed in the presence of A too. Thus DAXshould be determined in the absence of A, while in its presence the enhanced diffusion constant DAX’ can be obtained, since
The difference DAX’- DAXserves for the calculation of k , if AX is known, e.g., from crystallographic data on A and AX, or-reversed--bAx can be calculated, if k is known from an independent source. An experimentally determined value of AX is of importance to the mechanism of the exchange reaction. To observe any difference between DAXand DAX‘, the second term in eq 22 must exceed twice the errors in those, ADAX,hence
Because the order of magnitude of ADAXis about lo-* em2sec-l and that of 6 ~ isx10-16 ~ cm2,k must be about 5 X ~O’/CA J1-l sec-l or larger. The lower limit of the observation of the effect of transfer diffusion is quite high, but often just this region of fast exchange reactions cannot be studied by other methods. Case 2 . Consider again a linear diffusion, but let the reaction be electron exchange ( 6 ~ x= BA) with the initial conditions
RT
0 CAX
grad p = - grad c C
and hence the flux in eq 15 can be rewritten as
grad
CA
grad
CAX
= c0 and
CAX
(18)
= CA =
CA
=
0, if x
(24)
c0/2, if x = 0
This is a case of “counterdiffusion,” when AX diffuses into a space there was only A at the beginning and vice versa. Thus, for the flux across a plane at a certain x one has r
if the diffusion coefficient
RT
D=-p
C
is introduced. Let us simplify eq 18 according to the following conditions. Case 1. If the diffusion is linear, say, in the direction x grad c Let the concentration Thus
CA
dc dx
= -
be constant all over the space.
The Journal of Physical Chemistry, Vol. ‘76,N o . 2 l 9 1971
L
This is the particular equation Dahms obtained3 for “electronic conduction in solution” due to concentration gradients. Because the term in the brackets of eq 25 depends on x, the diffusion experiment performed in the presence of A should be evaluated from point t o point along x
TRANSFER DIFFUSION
3301
by a suitable iteration procedure. In this case, both DAX and DA must be determined in the absence of A and AX, respectively. If, however, D A is incidentally equal to D A Xor sufficiently near to it, a relation similar to that in eq 22 can be obtained except that the sum CA C A X appears instead of CA. (It is to be noticed that here the gradients are of opposite directions.) If the latter simplification is valid, an indirect method seems t o be applicable also: any parameter that depends on the diffusion coefficient can be investigated as a function of the sum of the concentrations, and there should be a deviation from the usual dependence of that parameter on concentration due t o the transfer diffusion term, if it is high enough. Yamely, voltametric limiting or peak currents vary usually as linear functions of the concentration of the electroactive species; transfer diffusion, however, would result in a deviation upwards from this linearity according to the diffusion coefficient which depends itself on the concentration. The "counterdiff usion" would be developed in the vicinity of the electrode by the electrochemical reaction itself. B. Electric Conductivity. Assuming that no concentration gradient is caused by an alternating current of sufficient frequency and the system is otherwise homogeneous, the gradient in the chemical potential becomes grad 1.1 = xF grad E (26)
+
If not, the weight average of the conductivity of the cations and anions with respect t o the transference numbers should be introduced in the well-known way as it has to be done for electrolytes. An unsuccessful choice of the counterions (when they diffuse much faster than those responsible for the transfer diffusion) can depress the whole effect expected. To avoid this complication, counterions of small mobility should be chosen, if possible. C . Thermal Diffusion. The result of the temperature gradient is, in general, a gradient in concentration; thus grad p should be transformed with respect to these two parameters grad
p =
( R In c - 8 ) grad T
+ (RT/c) grad c
(28)
By introducing eq 28 into eq 16, both the flux of AX and A can be obtained. Combining these two equations in order to eliminate the concentration gradient of A, one gets a relation between concentration gradient and temperature gradient for the stationary case, when the fluxes of both species are zero d- In DA'(RIn C A X - S A X ) - CAX d In T RDA'DAx' - K'CACAXQAQAX
(Dax"
DAX" - "AX') DA'
+ DA"CA
(29)
where
Since the flux times z gives the current density, and the total current density is the sum of that for A and AX, we have
&A
=
8A
8AX
Y.4
=
R In C A - SA R In C A X - SAX
(33)
and Q A X , D A ~ ' 'and , Yax are similar quantities with the labels exchanged. The right-hand side of eq 29 is the Soret coefficient in its extended form due to the transfer diffusion. If no transfer diffusion exists, it is
It is seen that the increase of the conductivity, when both reactants are present, in comparison to the sum of their own conductivity measured in the absence of the other reactant, is caused by transfer diffusion. Taking d In_ CAX AX _ _--In C A X - S__ (34) into account that the terms in the brackets are in the dln T R order of magnitude of one, the reaction must be again as Thermodiffusion also seems suitable to determine the fast as given in eq 23 to exhibit any unusual behavior. effect under discussion, although the complicated This is why no effect could be observed by Wyatt4in the feature of eq 29 with respect t o the parameters dependconductivity of the systems Fez+ Fe3+,hexacyanoing on the individual behavior of the system does not ferrate(I1) and -(III), and RIn042- Rho4-,since the allow any general conclusion in addition to that the second-order rate constants for these exchanges are effect appears in this case too. about 1, lo3, and lo2M-' sec-I, respectively. Equation 27 does not include the conductivity due to 4. Effect on the Diffusion Limit the other ions certainly present as counterions B""- of Smoluchowski6 treated the problem of the diffusion the electrolytes AzBzA+BzAzBand AXZBZAX+ZBAyZB-. limit of chemical reactions by the "sink model" calIf these counterions diffuse much slower than those giving the transfer diffusion, eq 27 still remains valid. ( 5 ) M.V . Smoluchowski, 2.Phys. Chem. ( L e i p z i g ) , 92, 129 (1917).
+
+
The Journal of Physical Chemistry, Vol. 76, N o . $1, 1971
3302
I. RUFFAND V. J. FRJEDRICH
culating the flux of the reaction partners B over the surface of the sphere of radius 6 around one of the other reactants A. 6 is the sum of the radii of A and B. Summation over all A gives the number of collisions in unit time. If every collision is effective in resulting in the products, i.e., the free energy of activation is very small in relation to RT, the rate constant of the reaction proceeding as fast as possible is
kl = 4?fNA6(DA
+ DB) x lo-’
(35)
in M-’sec-l units. Here N A is the Avogadro number. Though several modifications in the treatment have been made (see ref 6 and the references therein), Smoluchowski’s result proved to be correct. Transforming eq 35 for the reaction in eq 1, if no transfer diffusion operates, one gets a similar relationship to that in eq 35 with B replaced by AX. This equation holds only when the effectivity of the collision is equal at every point of the spherical A and AX. If not, the sensitivity of the collision to its direction would appear as a configurational entropy of activation and so the free energy of activation would not be negligibly small. Including this latter case, Noyes7deduced the relation
k = kl/(l
+
kl/k2)
(36)
where IC is the rate constant that can be observed as a resultant one, if both diffusion and activation control the reaction, and k2 is the rate constant for the purely activation controlled reaction. When transfer diffusion iduences the diffusion, it should enhance the value of the diffusion limit as well. To obtain the correlation for such a case, let us follow Smoluchowski’s deduction. Consider a spherical surface with a radius 6 around the center of an A and define the flux in eq 18 for that surface. Every AX crossing this surface “sinks” into the sphere, Le., CAX.. = 0. On the other hand, nothing happens to an A, if it comes close to the one in the center of the sphere; thus grad C A = 0. Hence the flux is given by eq 21, while for the reversed case-the flux of A across a similar sphere around an AX-& described by an equation identical with eq 21, but with the subscripts exchanged. For the diffusion limit one obtains in this way kl’
=
kl
+
Cb(CA8AX2
+ cAX6A2)k
where
The Jou~nalof Physieal Chemistry, Vol. 76,No. $1, 1971
(37)
a Substituting
kl‘
=
+ N A ~x 10-3
instead of
kl
(38)
into eq 36 it is altered to
+ CAX6A2)ak2+ [kl + - a(CA6AX2 + CAX6A2)k2]k-
(cA6AX2
k2
klk2
=
0 (39)
which can be easily solved with respect to IC. The important feature of k is that it slightly depends on the concentrations, if it i s close to the difusion limit and this is just the case, when transfer diffusion can be observed at all. This subsidiary dependence of the transfer diffusion terms written in the previous sections should be always accounted for. It leads, however, to a more general consequence in reaction kinetics, too. Let us formulate a chemical reaction of finite net free energy change as
A X + B - -kAXB +BX+A
+
(40)
+
If at least one of the exchanges AX A or BX B is fast enough to approach the diffusion limit of the exchange, the rate of the reaction in eq 40 would be increased also due to the transfer diffusion term appearing in it like in eq 37. This would affect ~ A X Beven if the cross reaction is slow, since the resultant rate constant is the product of the diffusion limit and exp( - (AF*/RT)) (see eq 36, if kz
