J. DEMAN AND W. RIGOLE
1122
Chemical Reaction during Electromigration of Ions by J. Deman and W. Rigole Laboratorium voor Fysische Scheikunde, University of Ghent, Belgium,
(Received June 23,1969)
It is shown what happens when reactive ions of opposite charge, present in different regions of a conducting medium, meet each other in the course of electromigration. If the computations are confined to precipitation reactions, theory predicts that a reaction region will narrow till a sharp reaction front is formed. The direction of movement of this front and its velocity are calculated, and the correctness of the formulas is briefly checked by some experiments.
It is possible to prepare an electric conductor which consists of a sequence of electrolytic solutions of different composition. Electrophoresis describes the change in concentration distributions accompanying application of direct current. However, so far no attempt has been made to investigate what happens when ions which react with each other, and which initially were present in separate regiong, are brought together by electromigration. In our treatment, we have chosen conditions as simple as possible. First it is supposed that the reactive ions are of opposite sign, and that the regions containing them are arranged so that mere application of current will bring them together. Further, the calculations are carried out on the assumption that a variation of the concentration of the reaction products does not influence the concentration of the reactants to a noticeable degree. Consequently, all kinds of reactions are included of which the equilibrium constants are small enough. The reaction products may be either charged or not. However, for the sake of brevity, we confine our present computations to the uncharged case. I n the theory, therefore, only precipitation reactions are explicitly treated. This reaction type fullfills the requirements mentioned above. Moreover, experience has shown that with precipitates the effects that are to be described are easily observed visually. Diffusion is left out of consideration together with the influence of products originating in electrode reactions. The field strength is supposed to be constant with time and to be uniform over the whole conductor. As will be pointed out further on, this condition is fulfilled when the conducting medium contains a nonreactive electrolyte in a uniform concentration largely exceeding the concentrations of the reactive ions. It should be realized that owing to this assumption the connection between the present theory and general electrophoresis theory is seriously loosened. It is clear that in experimental work, a means has to be devised to impede relative displacements which are not caused by the applied direct current. Agar gel proved to be most suitable for this purpose. The Journal of Physical Chemistry
In this medium, convection may be neglected and the precipitate remains a t the place where it was actually formed. Hermans4 has taken diffusion as the driving force for the ionic displacement and arrived a t conclusions which are in good analogy to ours. While moving through a region in which reaction products are present, ions in solution will exchange with those bound in the products. In another paper it will be shown that owing to these exchange reactions a separation between different ionic species can be obtained.
Preliminary Remarks Suppose the conducting medium is an aqueous gel column. At the anodic side, a part of the column is made to contain the positive ionic species &+. In another region nearer to the cathodic compartment the anionic species L1- is placed. When direct current is sent through, these ionic regions will move closer to each other and will produce the precipitate AIL, at the section of meeting. We introduce the notation mA, mL: effective electrophoretic mobilities of A”+ respective to L1- (the term “effective” is used to emphasize that sorption of the ions to the gel molecules results in a decrease of apparent mobility) ; CA, CL: concentrations expressed in g equiv/ cm3; E: the electric field strength supposed to be uniform over the whole gel column. (Not only the column has a constant cross-sectional area but there is also a large excess of nonreactive electrolyte. As a consequence, it can be assumed that E is not affected by precipitation and also that the constraint that otherwise would be imposed on the concentrations and mobilities of the reactive ions in their respective regions, falls away) ; v = mE: the ionic velocity; P: in the gel medium the precipitate is very finely dispersed, therefore the term “concen(1) E’. Kohlrausch, Ann. Phys. Chem., 6 2 , 209 (1897). ( 2 ) D. C. Henry and J. Brittain, Trans. Faraday Soc., 29, 798 (1933). (3) V. P.Dole, J . Amer. Chem. Soc., 67, 1119 (1945). (4)J. J. Hermans, J . Colloid Sci., 2,387 (1947).
CHEMICAL REACTION DURING ELECTROMIGRATION OF IONS tration of precipitate" appears allowable. This concentration is denoted by the symbol P and is expressed in g equiv of either the cationic or anionic species present in the precipitate per em3. Ideal behavior with respect to constant activity coefficients and constant mobilities is assumed. Variation of effective mobility with sorption is neglected. The positive 5 axis points in the direction in which the positive ions move. That part of the gel column in which precipitation is actually taking place is called the "reaction." A reaction region narrows itself to a "reaction front." That part of the column in which precipitate is present is called the "product region." Of the general precipitation reaction ]A'+
+ aL1- = AIL,
1123
Let us take an infinitesimal seotion dydx of the conductor. Ions As+ originally present a t this section will have covered a distance mAE dt after a time dt. I n this way a volume mAEdy dzdt is filled by A'+ which has passed the section. Hence the amount of As+ which passed the section is mAcAEdydzdt
(4
The amount of A which leaves the volume a t the opposite section is
The increase in A is (a)
- (b) ,or
the solubility product is
K,
=
[Aa+]'[L1-Ia
With square brackets, the concentrations are expressed in mol/l. When expressed in g equiv/cm* (symbols CA and CL), the solubility product becomes KAL
=
c A ~ c L= ~
alP(lO-a)l+"K.
When we divide by the volume dxdydx and time dt, we obtain the increase of total amount per unit of volume and of time
(1)
KALis, as is K , , constant a t a given temperature. Formation of a Reaction Region Suppose an experiment in which the regions A"+ and L1- are initially separated. Because of diffusion, concentration gradients arise a t the edges of these regions (Figure 1). Because of their greater concentration, &+ions which are present a t point p are able to precipitate the L1- ions which a time before were in equilibrium with the As+ ions of point q. At the same moment, the A"+ ions of q react with the L1ions which a time before were in equilibrium with the A"+ ions of point r . Consequently, A"+ ions are precipitated a t points p and q simultaneously. It follows that a reaction region is established.
This increase is also equal to the sum of the changes in concentrations CA and P per unit time
For L we obtain aCL - + bP - = mdat at ax acL
P is eliminated by subtraction
can be expressed in function of KALand CA by means of (1). CL
Figure 1. Ionic regions with their initial concentration distributions.
After integration we arrive a t
The Differential Equation In the reaction region, ions are disappearing during their movement, by precipitation. Exact mathematical description requires the formulation of the material balance.
1
E(amAcALiA
- 1mLKALi)t
= (p(cA) (3)
~ ( c A is ) a function which can be determined with the aid of the concentration distribution a t t = 0. Volume 74,Number 6 March 6,10'70
J. DEMAN AND W. RIGOLE
1124
Evolution of the Concentration Gradients in the Reaction Region From eq 2 and 1 we can derive voA i.e., the velocity with which a given CA is displaced in the reaction region
OCA
=
(%)
- -- amyIcA
(3 -
acA
- ImLCL
+ 1CL
E
(4)
Since the denominator is positive, a given CA will move in the direction of the positive x axis if amilcg > ImLcL. When amAcA < ImLcL, the direction of movement is reversed. Because of the CA gradient in the reaction region, a CA exists which does not move. The condition is amAcA = lmLcL or
It can be demonstrated that the reaction region will narrow itself around the fixed section zet until the reaction front is formed. Indeed, when eq 4 is differentiated with respect to CA, (with CL treated as a function of CA according to the constraint of eq l), then the result is
-=
-n
dc A
(ac,
+ 1CLY
Both numerator and denominator are positive. It follows that with greater concentration CA, the velocity v, increases. When vCA is positive (movement to the cathode), then greater CA values will move faster than smaller ones. If v, is negative, the velocity in the
c4rCL
I
direction of the anode decreases with greater CA value (Figure 2a). The cL-values follow the cA-values t o which they are linked by eq 1. The argument makes itself clear, when we substitute CA in eq 2 by CL by means of eq 1. We obtain a?)
(aKnLt+ 1cL
bCL
at = -
Since
It follows
VCL
=
amAcA M A
- ImLcL E
+
1CL
the same equation as eq 4. Summarizing, we state that the reaction region is narrowing and the concentration gradients are growing more steep a t the fixed section Xst. We have seen earlier that when the ionic regions first make contact with each other, a reaction region will develop because concentration gradients exist a t the edges of the regions (Figure 1). However, the bulk of the ions AB.+and L1- is present in uniform concentration CA,O and CL,O in their respective regions. Since uniform concentrations give no rise to a reaction region (this will become evident in the following paragraph) , it is obvious that the narrowing tendency of the reaction region will result in a discontinuity in the concentrations (reaction front), when contact is made between CA,O and CL,O (Figure 2b). At the section of discontinuity 1 c A , ~falls
KAL~
to concentration aCL,Ol‘ 1 KAL~ CL,~ falls to concentration 7 CA .Oa
Another Approach
b
i I ......
......
Equation 3 can be written as f X
Figure 2. a, Formation of a reaction front; directions and lengths of arrows indicate directions and magnitudes of velocities (schematical). b, Concentration distribution after formation of the front. The Journal of Phosical Chemistry
1125
CHEMICAL REACTION DURING ELECTROYIGRATION OF IONS
L
t
Figure 3. Concentration distributions at t
X
= 0.
att = 0
The inverse function of S~(CA) is denoted by C A = S ~ ( X ) . It describes the concentration CA in function of x at t = O . There are two possibilities since CA may either increase or decrease with increasing abscissa (Figure 3) * Equation 3’ is composed by the initial function SO(CA) to which is added a term which describes the change with time. Partial differentiation with respect to CA leads to
in Figure 1. It is evident that the uniform concentration CA,O does not exert any influence on the discontinuity after its formation. Because the front moves and regions A”+ and L1will be gnawed off (see following paragraph), we can discern four stages: 1, formation of a reaction region; 2, formation of a discontinuity in the ionic concentrations (reaction front) ; 3, movement of the front till the concentration gradient a t the rear end of the ion region is reached; and 4, breakdown of the front and reinstallation of a reaction region.
Displacement of the Reaction Front It is assumed that at both sides of the front a uniform concentration exists (Figure 2b). The velocity of the front is denoted as vf. The amount of an ionic species precipitated a t the front per unit time is equal to the difference between the velocity of the ion and that of the front, multiplied by the fall in ionic concentration. Thus
whence
A discontinuity in CA wiil develop if b c ~ / d x+ a V f = ___ L e . , ~ Z / ~ C+ A 0. The second term at the right-hand CA.0 side is always positive. As a consequence, ( ~ z / ~ c A ) , can never be zero when S0’(c~)is positive (curve 11). A curve of type I has a negative SO’(CA)value and ( ~ z / ~ c Aapproaches ) zero with time. The concentration of the precipitate is Figure 4 shows what happens. We have taken separately the concentration gradients of region A”+
[
cA
1
(5)
t.0
cA’oJ -7
(6) When ~ A , Ois the initial length of the A”+ region and d, the length of the product region after all the A&+ions have made contact with L1-, then
-dP- -d A ,o
cA,O
-
($)’ P
Substitution of P yields
Figure 4. Evolution of the concentration gradients in a reaction region.
(7 ) Volume 7.4, Number 6 March 6,1970
J. DEMAN AND W. RIGOLE
1126 I n most cases ( K A L / c L , ~ and & ) ~ ( K A L / C L , O I ) ‘ are negligible with respect to C A , ~and CL,O, respectively. With good approximation, we may then write
Some cases of practical importance can be discerned, We confine ourselves to the simple formulas 5‘ and 6‘. 1’. When ~ A C A , =~ m L c L , o , the front does not move (vf = 0). All the precipitate accumulates a t one definite section of the gel column (P --t a). In practice the width of the product region is not infinitesimally small because diffusion of the reactant ions has been neglected. 2.‘ When m A c A , O m L c L , O , the part of the column containing precipitate increases steadily. The product region is bordered sharply at both sides. One border remains stationary; this is the section where the reactant ions met each other initially. The other border is the reaction front. When m A C A , Q > m L c L , O , then the front moves in the direction of the cathode. The velocity incremes when C A , O / C L , ~is taken greater. When mACA,O < m L c L , O 1 then vf is negative and the front moves in the opposite direction. 3’. When simultaneously m A >> m L and CA,O >> CL,O then vf and P are approaching to their respective ,limiting values vcA and CL,O. On the other hand, vf approaches to vcL and P to CA,O when at the same time mL>> m A and CL,O >> CA,O. The theory of diffusion of ions into gels containing precipitating ions has been given by Hermans. He comes to the conclusion that also in that case a reaction front will be formed. This front will be displaced due to a difference between d 2 m A / d x 2 and b2mL/dx2. It is obvious that displacement of the front by diffusion can be taken into account only if we refrain from our assumption of constant mobilities.
+
Discussion We have verified the conclusions of the theory by experiments. A column of agar gel a t 1% (wt/vol) containing the indifferent electrolyte KC1 (0.2 N ) was used. The precipitation reaction studied was Co2+ 20H- -+ Co(OH)%,pK, = 15.6 at 25’. At first
+
The Journal of Phusical Chemistrg
the concentrations were chosen so that the bluegreen precipitation band extended in the direction of the cathode. We found that the moving side of the band was very sharply bordered (reaction front). When the hydroxyl concentration is enhanced, keeping the cobalt concentration constant, the precipitate becomes more dense and the velocity of the front decreases. Finally, with hydroxyl concentrations which are high enough, the direction of movement of the front is reversed. The ratio [Co2+l/[OH-] at which the front does not move is found to be 2.11 (Co2+ = 10-2 N , OH- = 4.73 X 10+ N). When computed theoretically with the aid of equation 5’ using the effective mobilities mco2+= 45 and m o H - = 125.9 determined in separate experiments, we found a ratio of 2.79. The agreement is to be called satisfactory bearing in mind that in the computation, no account has been taken of the nonuniformity of the electric field strength. Moreover, the exact temperature at the reaction front was not known. As already pointed out, the theory is applicable as such to other ionic reactions with uncharged reaction products provided the equilibrium constant is small enough. If the latter is not the case, then the concentrations of the reactants are to be regarded as dependent on the concentration of the product. Consequently, an attempt to an exact treatment will have to surmount some serious mathematical difficulties. It is our opinion that also in this case formation of an abrupt reaction front may be expected. Indeed a reaction region will continue to narrow as long as concentration gradients exist, the greater concentrations moving faster than the smaller ones. When the reaction products are charged as, e.g., in a complex-forming reaction + 4L1- = A L ( P a - d ) * P
P
or in a redox reaction
AB.++ Ll-
=
A(a-r)* + L(I--r)T
then many possibilities are open depending on the direction of movement and the velocity which the products assume. Due to this movement, the length of the product region can be either shortened or extended. It is conceivable that a product, say Fe(CNS)63-, moves more quickly in the direction of the anode than the reaction front between CNSand Fe3+. The front, as we have defined it, will then be situated behind the place where the victims of the battle are found.
