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liquid interfaces
J. Electroanal. Chem., 143 (1983) 23-36
23
Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands
ELECTROCHEMICAL IRREVERSIBILITY OF ION TRANSFERS AT LIQUID/LIQUID INTERFACES P A R T III. EQUILIBRIUM CONDITIONS AND TIME DEPENDENCES FOR TWO, OPPOSITE-CHARGE (OR SAME-CHARGE) ION TRANSFERS
OWEN R. MELROY and R I C H A R D P. BUCK *
William R. Kenan Laboratories of Chemistry, The University of North Carolina, Chapel Hill, NC 27514 (U.S.A.) (Received 23rd March 1982; in revised form 28th May 1982)
ABSTRACT Constant-current transport of oppositely-charged ions across liquid/liquid interfaces is a common occurrence because of the narrow potential 'window' and the limited range of single-ion partition coefficient values. Transition times and potential-time curve shapes are derived for fixed ratios of the two ions' forward rate constants. Effects of two-ion transport on determination of kinetic parameters for the first ion, using the log analysis plot method, are illustrated. Experimental examples cover the range of behavior from two isolated, irreversible waves to merged, single waves as forward rates approach a common value.
INTRODUCTION
In Part II of this series, single-ion, irreversible transport at liquid/liquid interfaces was analyzed by the quasi-thermodynamic method [1,2]. In response to the recent view of Koryta [3], effects of distributed potential through diffuse and compact regions at interfaces were considered. Three cases of potential-dependent interracial ion transport kinetics were devised: all interfacial pds occur through doubly-diffuse space charge regions; pds are found in both doubly-diffuse and doubly-compact layers; and the final case is the analog of conventional metal/electrolyte electrochemical kinetics for which the overpotential occurs mostly in compact layers. This paper will deal with the transport of two, oppositely-charged ions (moving in opposite directions), under the assumption that a significant fraction of the overvoltage falls within the two compact layers, and that the potential drop within the diffuse layers remains constant for low current densities and high mass transport rates. Evidence to support this assumption has previously been presented
* To whom correspondence should be addressed. 0022-0728/83/0000-0000/$03.00
© 1983 Elsevier Sequoia S.A.
24 [1]. In addition, the present paper compares oppositely-charged ion transport with transport of two ions of the same sign in the same direction. We will show that these cases are virtually equivalent. In the previous paper [1], rate constants of ion transfers were expressed in several ways depending on whether rates were computed in the space charge region or in the bulk, and for unit activity or for equilibrium activities, as they may exist. Expressions were given for rate constants at the interface (in the space charge region) as well as expressions for rate constants applying to bulk activities, and formal rate constants applying to b u l k concentrations. In terms of the potential-independent "chemical" rate constants k c and k c, individual ionic fluxes can be written: J_+ =/~c_+ (FC)a(bulk) exp[-T-azfAq,]-/¢~ + (FC)~(bulk) exp[ _+ (1 - a)zfA~]
(1)
with =
q,
FC = the Frumkin correction
(2)
(3)
derived in Part II, for ion transfers at doubly-diffuse interfaces. The simplification in terms of bulk concentrations gave approximate formal expression for fluxes: J+ = k , + c exp[ gazfAq,] - kb+_? exp[ _+ (1-a)zfAdp]
(4)
Signs apply to mobile positive and negative ions. A positive current is a positive flux (plus charge moving in the + x direction, which is from aqueous to organic phases). When two ions of opposite charge are free to partition and to transport under applied potential, the chemical system is not "simply adjustable" in the sense that individual ion activities on each side of the interfaces cannot be arbitrarily selected from stoichiometric addition of salts. The initial condition is determined by the concentration of added salts after consideration of electroneutrality and the partitioning equilibria, according to eqn. (6) of Part II. In practical systems, there are both transferring and non-transferring (or blocked) species initially present. Blocked species have zero rate constants for interfacial transport and they do not partition or carry current across the interface. Typically, aqueous supporting electrolytes are comprised of blocked ions, e.g., LiF, while the substance under study, MX, can contain reversible, partially blocked, or kinetically-limited ions, e.g., a tetraalkylammonium bromide. In the organic phase, a very hydrophobic salt NY or MY is usually present to increase conductivity and to provide a common ion when required. A typical example is a tetraalkylammonium tetraphenylborate. The presence of one salt NY or MY that is very much favored in the organic phase and a second salt MX which is favored in the aqueous phase is necessary to develop a range of interfacial pds. The activity of MX is the variable in the aqueous phase. In the next section it is shown that the interfacial pd between water and an immiscible phase containing a single equilibrated simple salt, MX, is nearly constant and independent of salt activity. When the two salts are present, a potential "window" is created that can be spanned by variation of aqueous salt activities.
25 EQUILIBRIUM POTENTIALDIFFERENCES [4-7] To avoid extremely complicated sets of simultaneous equilibria at interfaces, experiments can be designed to contain a maximum of four exchangeable species. There are then only four single-ion partition coefficients, defined in eqn. (2) of Part II, and two salt partition equilibrium constants, expressible in terms of products of single-ion coefficients. Even three ions M + , X - , Y , are sufficient to duplicate the transition from two salt MX, MY equilibrium extraction to a liquid ion exchanger for M ÷ . Only one ion, Y - , must change from reversible to blocked behavior to effect this response change. In this paper, because of the need to calculate time responses, as well as steady-state responses, we have simplified the system still further in the kinetics section, so that only one salt, MX, can partition. A more complete discussion of Galvani potential differences has been presented by Hung [7]. The activities and concentrations for aqueous and organic phases are designated a +_, a _+, c _+, ~ _+ for general univalent ions. Bars denote the organic phase. In the reversible limit for two partitioning salts, the interfacial pd is generally written ~-q~ = (I/2f)In[
KMaM/?tM+---KNaN/?t--'--~N] Kxax/?tx + Kvav/a v ]
(5a)
with
f= F/RT
(5b)
The values of activities to be used are those calculated from equilibrium partitioning. But, when single salts MX and NY are used, one has: ~-q~ = (1/2f)In[
Kr~ar~/~M+KNaN/ + Kvav/~'v ~'N]Kxax/~/x
(6)
For one salt (say MX) with NY blocked,
~-q,= (1/2f ) ln[ K+ K_ a+/yt+ a_/~_ ]
(7)
This equation is suitable even in the case of additional NX or MY. However, if only MX is present, a simplification is possible
- q~= ( 1 / 2 f ) In
K_a_/~,_
= (I/2f)
In
K_y_y+
(8)
These values are, in some sense derived from corrosion theory, " m e a n " or "mixed" potentials because they are determined by exchange of two charged species. It should be noted that the values computed from eqns. (5)-(8) must also be given by the individual equations for each partitioning and potential-determining ion: - ep = ( l / f )
ln[K+a+/a+ ]
(9)
26
and ln[fi _ / K _ a _
- ~b = ( l / f )
]
(10)
ANTICIPATED EQUILIBRIUM POTENTIAL " W I N D O W " FOR TWO SALT SYSTEMS: MX, MY
For a typical two phase, w a t e r / n i t r o b e n z e n e system containing equilibrated salts M X and MY, some M X and M Y will be found in each phase. The concentrations in each depend on the salt partition coefficients K M K x and K M K v. W h e n M + is a short chain t e t r a a l k y l a m m o n i u m ion T A A + a n d X = C 1 - , w h i l e Y = tetraphenylborate ( T P B - ) , M X is predominately in water while T A A T P B is predominately in nitrobenzene. This result, determined experimentally, can n o w be predicted from tabulated values [8] or K i or AG (trans. w ---, o). For this example we have chosen K v >> K x and from the relation A G = - R T l n Ki, the transfer free energy for X is more positive than the value for Y. The interfacial pd for an entirely equilibrated system containing constant [MY] = 10 -2 M in nitrobenzene with variable added concentration of MX, can be calculated. For illustration, consider a system with K v = 10 6, K M = 10 2 and K x = 10 -2. The interfacial pd Aq~ = f f - ~ is shown in Fig. 1. The potential " w i n d o w " is determined by the pds for excess M X (point A) and pure M Y (point E). The limiting pd at point A is given by: In[ K ] KMx' TYM MY ~ xX ] = 118 m V a t 2 5 ° C
Adp = - 2 f
120
I
]
t
80-
I ~
(11)
.( / / " f ' 'Donnon "-F/iluree A CM=Cx; ~M~X > ~Y-
/ 40-
/2" -40 -
D/
-80 Exlr~OcCti°n c of M y ~ Nernstion Response M ~Y~X~
-Izo
E,L-""f
iO-'t
i
10-6
to OM+ t
10-5
i
t
10-4 rio--3 Added CM= cx/M
t
10--2
t
I0--1
Fig. l. An example of the potential "window". Computations for two salts: MX, mainly water-soluble and MY, mainly oil-soluble. Concentration of MY = 10 -2 M; single-ion partition coefficients K v = 106, K M = 102 and K x = 10 -2. The computed potential window is 336 mV using eqns. (l l) and (12). Overall curve obeys eqns. (11-14). c M and c x are added concentrations.
27 while at point E A4~=~fln[g~a3'M]=-l18mVat25°CKv,TM3,v]
(12)
(activity coefficient values have been assumed to be unity). At point A, Donnan exclusion by Y is violated, while at infinite dilution of MX the interfacial pd is determined by extraction equilibrium of MY at point E. In the midrange of the figure, point C, the response approaches Nernstian behavior that is most easily expressed by eqns. (9) or (10). In the transition regions, points B and D, the pd equations are quadratic:
1
7 In
[--Y+(I'2+4KMKxaMax/'YMYX)
]Yx
(13)
and for point D, A , = -~ In
2y.yM
where a N is the added activity of MX. The potential "window" depends upon the ratios K M / K x and K ~ / K v. The response along the activity axis depends on the extraction coefficient products KMK x and KMK v. STEADY-STATEAND TRANSIENT IRREVERSIBLEMIXED POTENTIALDIFFERENCES For two univalent, irreversibly exchanging species M + and X - , the steady state interfacial pd depends on the applied current. The form of the interfacial pd expression, excluding IR drop, is determined by the "regime": whether surface rate (for i < io+ and io) or mass transport (for i > io+ and i o_ ) is controlling. Well-stirred, imperfectly-stirred or static solution conditions figure in deriving specialized equations for the cases. In the following section only approximations can be derived in the usual electrochemical manner [2], because of the exponential terms in the equations. Writing the separate ion fluxes as: j + = kr+,-.kb+c + ~ l-~c+ (exp[ - a f q ] - exp[ (1 - a)fB] )
(15)
J - = ,.t-~z.a c,_-~cP_( e x p [ f l / . ]
(16)
- exp[ (1 -- fl)f ] )
with ~/= A4~ - ACq
(17)
and A~eq given by eqn. (8), one can express formally
I = r ( J + - J_ ) = f(r/)
(18)
by subtracting eqn. (16) from eqn. (15). The exchange fluxes in eqns. (15) and (16) are related to the exchange current densities io+ and i o_ by the Faraday constant, F. At small overpotentials, with stirring or rapid mass transport, expansion of the
28 exponentials gives: I=
(19)
- (io+ + i o _ ) f i l
and at higher overpotentials, but still with fluxes below mass transport control, ~ is c o m p u t e d from eqn. (18) with eqns. (15) and (16). W h e n the system is not stirred, the bulk concentrations and activities cannot be maintained at the interfaces. This is the usual situation. Then, I = io+
( c + ( x =c+O , t )
+ i o_
exp[-c~fij]
~+(x=0, ?+
{ c_(x=O,t) exp[ fifo ] + c_
t)
exp[ (1 -
a)fil]
)
t) exp[ - (1 - fl)fil] }
c_
(20)
and the time-dependences enter through the dependences of c+(x = O, t) and c (x = 0, t) on time. The c o m m o n irreversible case, e.g., io+ and i o_ < 10 2 A / c m 2, but i o_ < 1 0 - 2 i o + , provides two chronopotentiometric waves. H o w e v e r as i o_ approaches equality with io+, only one wave appears. The total transition time is the same in b o t h cases. In the first instance, a positive current carries only cations from aqueous to organic phase until t = ~1,
,t/2 = F( D +~r)1/2c+/2I
(21)
T h e second transition time (,~ + "2) obeys the equation
('1 +*2)l/2=(r(Tr)l/Z/2I)(Dl+/2c+
+ ~1/2~
)
(22)
In the limit of equal exchange current densities, there is only one transition time = "1 + "2. Eqns. (21) and (22) are of the same form as the description of two different cations moving from water to organic phase [9]. The wave shapes follow from conventional analysis for these two limiting cases. F o r the two step behavior, c+ (0, t)/c+ is replaced by 1 - (t/~'l)l/z,t < *l and is zero thereafter. ~_(0, t ) / ~ (or the ratio for second aqueous cation) is replaced by 1 - [(t - * 1 ) / ( * 1 -~ ']'2)] 1 / 2 for t > 'r 1 and is unity prior *l- This substitution is only valid for total irreversibility with aq. cation transporting more readily than oil-soluble anion (or second aq. cation). Corrections for the back reactions can be m a d e by using ~+(0, t)/~+ equal to 1 + (t/'r;) 1/2 for t < zl, and c (0, t)/c_ replaced by 1 + [(t - ~-;)/(*( + ~.~)]1/2 for t > ~'1. The transition times are approximately
, ,tl/2
= r ( D -+- ~ r )
1/2
~+/2I
(23a)
,~/2 = F( D_ ~r)1/2c_/21
(238)
In the single wave limit, io+ = i0_, c+ (0, t)/c+ and ~ (0, t ) / { are replaced by 1 - [t/('q + ~-2)]1/2. Another practical case involves one reversible and one irreversible ion from the same salt. For a typical reversible cation and irreversible anion, using high loading
29 of cation in the organic phase, I = ( 1 / 2 ) F ( D +~r)~/2c +[ t '/2 + ~-'I/2 ( D + / D + )1/2 exp( fAq~ -- fAqT')] -1 + i o_
{
exp[flfq] -} -
c
exp[ - (1 - fl)f~]
c
}
(24)
The first term determines the chronopotentiogram shape for t < ~'~ because the anion carries no current. Eventually anion depletion occurs and a second wave appears. The substitutions for ~ (0, t)/~_ and c (0, t ) / c , given above, are appropriate in this case, also. When mass transfer intervenes but does not completely control fluxes, as in imperfectly stirred solutions, surface concentrations may differ from bulk values according to the Nernst diffusion layer thickness. Then c+ (0, t)/c+ is replaced by 1--_i+/id+ ; c+(O, I)/C+ replaced by 1 + i+/id+ ; c_(O, t ) / c _ replaced by 1 + i / i d and ~ (0, t)/~_ replaced by 1 - i _ / i d _ . The large limiting currents are given approximately by the expression id,j = DjFc3/3
(25)
for each ion in each phase. 3 is the steady state Nernst layer thickness. In the low overpotential region, one can account for partial mass transport control in the current-voltage expression eqn. (19) by:
(1 1)
I+io+i + . +ld+ /d+
+io_i_
1)
ia - + id-
~ - ( i o + +io_)fil
(26)
The added terms cause ~ to move more negatively for positive current flow in comparison with the prediction of eqn. (19). Equations (19) and (26) are significant for liquid/liquid ion transfer kinetics because of the narrow range of transport free energies. The experimental consequence is that most practical systems for investigation show a narrow potential range for single-ion transport [10]. Allowance must be made, generally, for the possible reverse transfer of an oppositely-charged ion, or simultaneous transport of an ion of same charge, over some part of the accessible potential window. CURRENT PARTITIONING
The contributions of ionic currents i+ and i_ to the total constant current I, do not generally remain constant over time. For irreversible two-ion transport in stirred solutions, instantaneous and steady-state partitioning can remain fixed for a few instances. At small overvoltages,
i + / I = i o + / ( i o + +io_);
i / I = i o /(io+ +io_ )
(27)
At higher overpotentials, but for fluxes well below mass transport involvement,
i + / I = i o + e x p [ - a f q ] / ( i o e x p [ - a f i / ] +io+ exp[ - (1 - f l ) f l / ] }
(28)
30
/
a
~3
/
U_
CO
° 0
b
/
LI_
12.
2q-.
12.
TIME/S
2~.
TIME/S
C
/ /
I
0
12.
o
2q.
/
j / P
12.
0
TIHE/S
2~.
TIHE/S
Fig. 2. D i g i t a l l y - s i m u l a t e d c h r o n o p o t e n t i o g r a m s for irreversible two-ion t r a n s p o r t in a q u e o u s s u p p o r t i n g electrolyte. T r a n s p o r t e d ions are u n i v a l e n t with opposite charge. Positive current carries c a t i o n s from aqueous to organic phase, a n d anions from organic to aqueous phase. R a t e c o n s t a n t s are selected to allow t h e cation to m o v e first in Figs. 2 a - 2 c . D = D = 3 × 10 - 5 c m 2 / s ; a = fl = 0.5; i = 1 0 0 / ~ A / c m 2. k b _ / c m s - l : 10 - 7 (a), 10 - 6 (b), 10 -S (c), 10 - 4 (d).
Ion
Charge
c/M
~/M
k//cm s -
1
kb/Cm s- 1
1
1
5 × 10 - 4
1 X 10 - 2
1 X 10 - 4
1 X 10 - 4
2
- 1
1 X 10 - 2
5 X 10 - 4
1 × 10 - 4
variable
3 4
1 - 1
1 0.99
0.99 1
0 0
0 0
and there is an equivalent expression for i_/I. Only in the event that a = 1 - fl will the current remain partitioned at a constant value throughout a wide potential range. For unstirred solutions, because of different ionic concentrations and diffusion coefficients, current partitioning ratios will generally vary during the chronopotentiogram.
31 Aqueous Cation Profile ~LS
Organic Anion Profile
~"~..
,,,"
~ .~.. " "
~'.
f.
~-, "S
4--
\\\\\\. ~
3
,,..\ \
°
8
/
/
//
/
\1/ "\1
I0
I 9
I 8
I 7
I 6
I 5
I 4
I 3
I 2
I I
0
..I I
I 2
I 3
I 4
I 5
I 6
I 7
I 8
I
9
I0
Distance from Interface x IOZ/cm
Fig. 3. Concentration-distance profiles for positive current carrying cations (aqueous, ion 1, left) and anions (organic, ion 2, r i g h t ) tabulated properties of ions 1-4 in Fig. 2 legend apply here. k f _ / c m s - i: (------)
10-8; (
) 10-7; ( . . . . . .
DIGITAL
SIMULATION
) 10-6; ( . . . . .
) 10 - 5 .
RESULTS
The principal results of the theory using eqns. (15)-(19), with the continuity equation in the digital simulation mode [1] are hypothetical chronopotentiograms and log analytical plots. A selection of results for two-ion transport are given in Figs. 2a-d. The characteristic behavior from distinguishable two-wave, kinetically-limited cases to merged, undistinguishable, single-wave behavior is illustrated. When both ions are slow (10 -l] < k f < 10 -2 cm//s), but one of the ions has a forward rate constant four orders of magnitude or more larger than the other, two distinct waves are predicted and shown in Figs. 2a and b. As ke for each ion approach a common value, 10 -4, a single, totally irreversible wave is found in Fig. 2d. An indistinguisha-
TABLE
1
1st Transition times for two-ion processes, k I = k I = k z = 10 - 4 c m 2 / s . D] = D 2 = D ] = D 2 = 3 × 10 - 5 c m 2 / s , c = surface concentration of ion 1 ; c o = bulk concentration of ion I k ~/k 2 104 103 102 10 1 By
"rI ( o b s e r v e d ) / s '~ 6.30 6.35 6.55 24.0 24.0
(one wave only) (one wave only)
rj/s b
~'j/S ~
6.15 6.25 9.65
6.35 8.00 17.8
20.2 23.5
23.5 23.9
inflection point method, b When c°/cl = 100, c When c°/cl = 1000.
32
(a)
(b)
oh.
C~
G:). L
tu
co
°0
6.0
3.0 LN [1-(t/l") 1/2]
o 0
3.0
6.0
-LN [ 1 - ( t / ' r ) 1/2]
Fig. 4. Log analytical plots for Fig. 2. (a) Analysis of the first wave in Fig. 2b; (b) analysis of the only wave in Fig. 2c.
ble merging of two waves into an unusual single-transition time chronopotentiogram is the intermediate result in Fig. 2c. Merging of two irreversible waves into a single, reversible wave can also occur when the rate constants are allowed to increase. Two, oppositely-charged ions, moving at constant current to yield a single transition time were illustrated in a previous paper [11]. Appearance of a separate or merged wave in two-ion chronopotentiometry depends on both concentrations and forward rate constants. For ions with nearly equal a's, the two oppositely charged ions will carry nearly equal amounts of current when ( k f C ) l = ( k b C ) 2 ; C1 and C2 are surface concentrations. A normal first transition
500
t
i
i
I I0
I 20
I 30
i
t
[
[
I 60
I 70
4OO
3OO "~/mV 2OO
I00
O~
I I 40 50 Time / s
I 80
90
Fig. 5. Two-step, two opposite-charge ion transport. First process: I - ( n i t r o b e n z e n e ) ~ I (aq.). Second process TEA + (aq.) ~ T E A + (nitrobenzene). i = 11.0/xA/cm2; [ I - ] org. = 6.6 × l0 - 4 M as tetraheptyla m m o n i u m iodide; [TEA + ] aq. = 2.8 × 10 - 4 M as tetraethylammonium chloride; 0.1 M LiCI (aq.) and 0.0074 M tetraheptylammonium tetraphenylborate (org.).
33 180
1
160
//
//
//
140 120 F
,ool
y
///
-
~/mV 8O 60
40 20 0V 0
I I
I 2
-hn [I-(t/r) U2]
I 3
Fig. 6. Log analytical plots for tetrabutylammonium transport from water to nitrobenzene, i = 19 /~A/cm2; 8 x 10-4 M TBAC1(aq.); 0.02 M TBATPB (nitrobenzene). Curve a: 0.1 M LiC1; Curve b: 0.1 M CsC1.
time will occur for ion 1 if its forward rate constant is large enough to permit c~ to decrease to about l0 -3 c o before significant current partitioning b y the second ion occurs. The quantity c o is the initial bulk concentration of ion 1. Figure 3 shows the concentration profiles of ion 2 at the first transition time (for ion 1), as a function of kf2 relative to k n for solutions in which c o = ~2°. The first transition time can be erroneously long, without appearance of a developed second wave for the interfering species, ion 2. Table 1 shows h o w the first transition time can be lengthened by the presence of the interference at equal concentrations (c o = c °) and the time at which the surface concentration has decreased to selected fractions of its bulk concentration for the rate constant ratio kfl/kf2. The effect of back rates of ion transfer of two ions, is not significant in terms of the observed transition times. Log analytical plots of separated waves in Fig. 2a give expected irreversible slopes, RT/aF. However, analysis of the first wave from Fig. 2b shows severe deviation from linearity during the latter half of the wave. The sub-Nernstian curvature at times near the transition time is characteristic of interference as shown in Fig. 4a. A second ion is already carrying a significant portion of applied current
34 500
I
I
I
I
I
[
I a b
c
400
3O0 "r//mY 200
I00 0 0
I 5
I I0
I 15
I 20 Time / s
I 25
I 30
I 35
40
Fig. 7. Chronopotentiograms for 9× 10-4 M tetrabutylammonium chloride (aq.). i = 19 /LA/crn2; 0.2 M TBATPB (nitrobenzene). Curve a: 0.1 M NaC1; Curve b: 0.1 M NaCI + 1.4 X 10-2M NaI; Curve c: 0.1 M NaCI+0.1 M NaI.
in the vicinity of the apparent first transition time. Analysis of well-merged waves from Fig. 2c show even greater non-linearity in the log plot, Fig. 4b. If the initial slope of Fig. 4b is taken to be RT/aF, observed et's will be less than the true value for either ion, because of early co-transport. We have briefly investigated two other two-ion cases that give apparently one transition time: (1) equal forward rate constants but unequal backward rate constants, and (2) equal rate constants but differing a-values. Both cases give non-linear log analytical plots. The latter can show two regions of linearity with slopes R T / % F and 2RT/(% + ct2)F for cation and 2 R T / ( a + fl)F for cation and anion transport. EFFECTS OF LIMITED POTENTIAL "WINDOW" ON CHRONOPOTENTIOGRAMS Use of tetraphenylborate salts in the organic phase, and supporting aqueous electrolytes containing alkali metal ions, produces a limited potential window, i.e., there is a negative-going potential limit (~org.- ~aq..... ) imposed by transport of T P B - or M ÷ , whichever happens more easily, e.g., at less negative potentials. By selecting LiC1 as supporting electrolyte, two-wave chronopotentiograms can be observed. An example of this situation is provided by transport of tetraethylammonium (TEA +) and I - at H 2 0 / n i t r o b e n z e n e . Figure 5 shows the results of transporting similar concentrations of TEA ÷ and I - in aqueous LiCI and TEATPB as the organic supporting electrolyte. The largest TAA + ion which can be transported from water to nitrobenzene without complications from adsorption is tetrabutylammonium (TBA+). The relatively large difference in free energies of transport from water to nitrobenzene
35 between TBA + and T P B - gives a large potential window for the previously discussed phenomenon to be observed. When transport of TBA ÷ alone is desired, LiC1 provides the optimum electrolyte. As shown in Fig. 6, curve a, the log analytical plot for TBA ÷ is linear over two decades. On the other hand, an example of massive interference by a second ion is found in TBA ÷ transport at water/nitrobenzene when CsC1 is the supporting electrolyte. The potential window is diminished and the log analytical plot, curve b, shows the apparent interference as a sub-Nernstian deviation. Interference by supporting electrolyte components is common. When two waves are expected to occur (from estimated relative rates or free energies of transfer), only one wave is sometimes seen. The second process appears only as a potential plateau reached at the end of the first transition. The principal cause is the high concentration of the second transporting ion, frequently a component Of the aqueous supporting electrolyte. When the forward rate constants of two ions differ by one order of magnitude or less, only one wave is predicted. This behavior is seen in the co-transport of TBA ÷ and I - . Figure 7 shows the chronopotentiograms for a 9 x 1 0 - a M solution of TBAC1 with combinations of NaC1 and NaI used as the aqueous supporting electrolyte. When no I - is present, curve a, all current is carried by TBA ÷ . For relatively small amounts of NaI in the supporting electrolyte (1.4 x 10 - 2 M), curve b, the transport involves both TBA ÷ from water to nitrobenzene and I - from nitrobenzene to water. At higher concentrations of NaI, curve c, all current is carried by I - . A much more extensive series of cation responses (both tetraalkyl cations and a selection of supporting electrolytes) will be included in our subsequent, "experimental" paper. CONCLUSIONS Two-ion responses in liquid/liquid ion transfer studies are expected to be common. Because of the narrow potential window, determined by the extraction coefficients of two-salt systems, good separation of chronopotentiometric waves is not expected except for the few salts that allow forward rate constants ratios of 103 to 104. The use of very hydrophobic ions, such as tetraphenylarsonium or crystal violet for cations and dicarbollyl cobaltate, as the anion of the organic supporting electrolyte can enlarge the potential window. The fundamental problem however, still remains. Gavach's published chronopotentiograms and log analytical plots show sub-Nernstian slopes at times approaching the transition time expected for the two-ion transport situation. Other cases show super-Nernstian slopes characteristic of quasi-reversibility at short times for systems that become increasingly irreversible during the course of the experiment (because of concentration polarization). Other researchers have not analyzed their data over a sufficiently wide range of t/~ to show these effects which occur principally near to the transition time. Our simulated and experimental data have shown linearity (by log plot analysis) over only 60% of ~.
36
The region between 60% and 95% is most discriminating with respect to occurrence of two-ion transport, as deviations become readily apparent. For two-ion transport in supporting electrolyte, there is little practical difference between the transport of ions of opposite or the same charge. There is, however, a theoretical difference by virtue of the potential barrier symmetry. When transport occurs in little or no supporting electrolyte, the two cases can be dramatically different. These changes will be the topic of the final communication in this series. ACKNOWLEDGEMENT
This work was supported by National Science Foundation grant CHE81-03334. REFERENCES 1 0 . R . Melroy and R.P. Buck, J. Electroanal. Chem., 136 (1982) 19. 2 P. Delahay, Double Layer and Electrode Kinetics, Wiley, New York, 1965, pp. 154-158. 3 J. Koryta in E. Pungor (Ed.), Ion-Selective Electrodes, Vol. 3, Akademiai Kiado, Budapest, 1981, pp. 53-72. 4 R.P. Buck, Crit. Revs. Anal. Chem., 5 (1975) 323. 5 R.P. Buck in H. Freiser (Ed.), Ion-Selective Electrodes in Analytical Chemistry, Vol. 1, Plenum Press, New York, 1978, pp. 1-141. 6 J. Koryta and P. Vanysek in H. Gerischer and C. Tobias (Eds.), Advances in Electrochemistry and Electrochemical Engineering, Vol. 12, Wiley, New York, 1981, p. 113. 7 L.Q: Hung, J. Electroanal. Chem., 115 (1980) 159. 8 J. Koryta, P. Vanysek and M. Brezina, J. Electroanal. Chem., 75 (1977) 211. 9 T. Berzins and P. DelahaY, J. Am. Chem. Soc., 75 (1953) 4205. 10 D. Homolka, L.Q. Hung, A. Hofmanova, M.W. Khalil, J. Koryta, V. Maracek, Z. Samec, S.K. Sen, P. Vanysek, J. Weber, M. Brezina, M. Janda and I. Stibor, Anal. Chem., 52 (1980) 1606. 11 O.R. Melroy, R.P. Buck, F.S. Stover and H.C. Hughes, J. Electroanal. Chem., 121 (1981) 93.
23
Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands
ELECTROCHEMICAL IRREVERSIBILITY OF ION TRANSFERS AT LIQUID/LIQUID INTERFACES P A R T III. EQUILIBRIUM CONDITIONS AND TIME DEPENDENCES FOR TWO, OPPOSITE-CHARGE (OR SAME-CHARGE) ION TRANSFERS
OWEN R. MELROY and R I C H A R D P. BUCK *
William R. Kenan Laboratories of Chemistry, The University of North Carolina, Chapel Hill, NC 27514 (U.S.A.) (Received 23rd March 1982; in revised form 28th May 1982)
ABSTRACT Constant-current transport of oppositely-charged ions across liquid/liquid interfaces is a common occurrence because of the narrow potential 'window' and the limited range of single-ion partition coefficient values. Transition times and potential-time curve shapes are derived for fixed ratios of the two ions' forward rate constants. Effects of two-ion transport on determination of kinetic parameters for the first ion, using the log analysis plot method, are illustrated. Experimental examples cover the range of behavior from two isolated, irreversible waves to merged, single waves as forward rates approach a common value.
INTRODUCTION
In Part II of this series, single-ion, irreversible transport at liquid/liquid interfaces was analyzed by the quasi-thermodynamic method [1,2]. In response to the recent view of Koryta [3], effects of distributed potential through diffuse and compact regions at interfaces were considered. Three cases of potential-dependent interracial ion transport kinetics were devised: all interfacial pds occur through doubly-diffuse space charge regions; pds are found in both doubly-diffuse and doubly-compact layers; and the final case is the analog of conventional metal/electrolyte electrochemical kinetics for which the overpotential occurs mostly in compact layers. This paper will deal with the transport of two, oppositely-charged ions (moving in opposite directions), under the assumption that a significant fraction of the overvoltage falls within the two compact layers, and that the potential drop within the diffuse layers remains constant for low current densities and high mass transport rates. Evidence to support this assumption has previously been presented
* To whom correspondence should be addressed. 0022-0728/83/0000-0000/$03.00
© 1983 Elsevier Sequoia S.A.
24 [1]. In addition, the present paper compares oppositely-charged ion transport with transport of two ions of the same sign in the same direction. We will show that these cases are virtually equivalent. In the previous paper [1], rate constants of ion transfers were expressed in several ways depending on whether rates were computed in the space charge region or in the bulk, and for unit activity or for equilibrium activities, as they may exist. Expressions were given for rate constants at the interface (in the space charge region) as well as expressions for rate constants applying to bulk activities, and formal rate constants applying to b u l k concentrations. In terms of the potential-independent "chemical" rate constants k c and k c, individual ionic fluxes can be written: J_+ =/~c_+ (FC)a(bulk) exp[-T-azfAq,]-/¢~ + (FC)~(bulk) exp[ _+ (1 - a)zfA~]
(1)
with =
q,
FC = the Frumkin correction
(2)
(3)
derived in Part II, for ion transfers at doubly-diffuse interfaces. The simplification in terms of bulk concentrations gave approximate formal expression for fluxes: J+ = k , + c exp[ gazfAq,] - kb+_? exp[ _+ (1-a)zfAdp]
(4)
Signs apply to mobile positive and negative ions. A positive current is a positive flux (plus charge moving in the + x direction, which is from aqueous to organic phases). When two ions of opposite charge are free to partition and to transport under applied potential, the chemical system is not "simply adjustable" in the sense that individual ion activities on each side of the interfaces cannot be arbitrarily selected from stoichiometric addition of salts. The initial condition is determined by the concentration of added salts after consideration of electroneutrality and the partitioning equilibria, according to eqn. (6) of Part II. In practical systems, there are both transferring and non-transferring (or blocked) species initially present. Blocked species have zero rate constants for interfacial transport and they do not partition or carry current across the interface. Typically, aqueous supporting electrolytes are comprised of blocked ions, e.g., LiF, while the substance under study, MX, can contain reversible, partially blocked, or kinetically-limited ions, e.g., a tetraalkylammonium bromide. In the organic phase, a very hydrophobic salt NY or MY is usually present to increase conductivity and to provide a common ion when required. A typical example is a tetraalkylammonium tetraphenylborate. The presence of one salt NY or MY that is very much favored in the organic phase and a second salt MX which is favored in the aqueous phase is necessary to develop a range of interfacial pds. The activity of MX is the variable in the aqueous phase. In the next section it is shown that the interfacial pd between water and an immiscible phase containing a single equilibrated simple salt, MX, is nearly constant and independent of salt activity. When the two salts are present, a potential "window" is created that can be spanned by variation of aqueous salt activities.
25 EQUILIBRIUM POTENTIALDIFFERENCES [4-7] To avoid extremely complicated sets of simultaneous equilibria at interfaces, experiments can be designed to contain a maximum of four exchangeable species. There are then only four single-ion partition coefficients, defined in eqn. (2) of Part II, and two salt partition equilibrium constants, expressible in terms of products of single-ion coefficients. Even three ions M + , X - , Y , are sufficient to duplicate the transition from two salt MX, MY equilibrium extraction to a liquid ion exchanger for M ÷ . Only one ion, Y - , must change from reversible to blocked behavior to effect this response change. In this paper, because of the need to calculate time responses, as well as steady-state responses, we have simplified the system still further in the kinetics section, so that only one salt, MX, can partition. A more complete discussion of Galvani potential differences has been presented by Hung [7]. The activities and concentrations for aqueous and organic phases are designated a +_, a _+, c _+, ~ _+ for general univalent ions. Bars denote the organic phase. In the reversible limit for two partitioning salts, the interfacial pd is generally written ~-q~ = (I/2f)In[
KMaM/?tM+---KNaN/?t--'--~N] Kxax/?tx + Kvav/a v ]
(5a)
with
f= F/RT
(5b)
The values of activities to be used are those calculated from equilibrium partitioning. But, when single salts MX and NY are used, one has: ~-q~ = (1/2f)In[
Kr~ar~/~M+KNaN/ + Kvav/~'v ~'N]Kxax/~/x
(6)
For one salt (say MX) with NY blocked,
~-q,= (1/2f ) ln[ K+ K_ a+/yt+ a_/~_ ]
(7)
This equation is suitable even in the case of additional NX or MY. However, if only MX is present, a simplification is possible
- q~= ( 1 / 2 f ) In
K_a_/~,_
= (I/2f)
In
K_y_y+
(8)
These values are, in some sense derived from corrosion theory, " m e a n " or "mixed" potentials because they are determined by exchange of two charged species. It should be noted that the values computed from eqns. (5)-(8) must also be given by the individual equations for each partitioning and potential-determining ion: - ep = ( l / f )
ln[K+a+/a+ ]
(9)
26
and ln[fi _ / K _ a _
- ~b = ( l / f )
]
(10)
ANTICIPATED EQUILIBRIUM POTENTIAL " W I N D O W " FOR TWO SALT SYSTEMS: MX, MY
For a typical two phase, w a t e r / n i t r o b e n z e n e system containing equilibrated salts M X and MY, some M X and M Y will be found in each phase. The concentrations in each depend on the salt partition coefficients K M K x and K M K v. W h e n M + is a short chain t e t r a a l k y l a m m o n i u m ion T A A + a n d X = C 1 - , w h i l e Y = tetraphenylborate ( T P B - ) , M X is predominately in water while T A A T P B is predominately in nitrobenzene. This result, determined experimentally, can n o w be predicted from tabulated values [8] or K i or AG (trans. w ---, o). For this example we have chosen K v >> K x and from the relation A G = - R T l n Ki, the transfer free energy for X is more positive than the value for Y. The interfacial pd for an entirely equilibrated system containing constant [MY] = 10 -2 M in nitrobenzene with variable added concentration of MX, can be calculated. For illustration, consider a system with K v = 10 6, K M = 10 2 and K x = 10 -2. The interfacial pd Aq~ = f f - ~ is shown in Fig. 1. The potential " w i n d o w " is determined by the pds for excess M X (point A) and pure M Y (point E). The limiting pd at point A is given by: In[ K ] KMx' TYM MY ~ xX ] = 118 m V a t 2 5 ° C
Adp = - 2 f
120
I
]
t
80-
I ~
(11)
.( / / " f ' 'Donnon "-F/iluree A CM=Cx; ~M~X > ~Y-
/ 40-
/2" -40 -
D/
-80 Exlr~OcCti°n c of M y ~ Nernstion Response M ~Y~X~
-Izo
E,L-""f
iO-'t
i
10-6
to OM+ t
10-5
i
t
10-4 rio--3 Added CM= cx/M
t
10--2
t
I0--1
Fig. l. An example of the potential "window". Computations for two salts: MX, mainly water-soluble and MY, mainly oil-soluble. Concentration of MY = 10 -2 M; single-ion partition coefficients K v = 106, K M = 102 and K x = 10 -2. The computed potential window is 336 mV using eqns. (l l) and (12). Overall curve obeys eqns. (11-14). c M and c x are added concentrations.
27 while at point E A4~=~fln[g~a3'M]=-l18mVat25°CKv,TM3,v]
(12)
(activity coefficient values have been assumed to be unity). At point A, Donnan exclusion by Y is violated, while at infinite dilution of MX the interfacial pd is determined by extraction equilibrium of MY at point E. In the midrange of the figure, point C, the response approaches Nernstian behavior that is most easily expressed by eqns. (9) or (10). In the transition regions, points B and D, the pd equations are quadratic:
1
7 In
[--Y+(I'2+4KMKxaMax/'YMYX)
]Yx
(13)
and for point D, A , = -~ In
2y.yM
where a N is the added activity of MX. The potential "window" depends upon the ratios K M / K x and K ~ / K v. The response along the activity axis depends on the extraction coefficient products KMK x and KMK v. STEADY-STATEAND TRANSIENT IRREVERSIBLEMIXED POTENTIALDIFFERENCES For two univalent, irreversibly exchanging species M + and X - , the steady state interfacial pd depends on the applied current. The form of the interfacial pd expression, excluding IR drop, is determined by the "regime": whether surface rate (for i < io+ and io) or mass transport (for i > io+ and i o_ ) is controlling. Well-stirred, imperfectly-stirred or static solution conditions figure in deriving specialized equations for the cases. In the following section only approximations can be derived in the usual electrochemical manner [2], because of the exponential terms in the equations. Writing the separate ion fluxes as: j + = kr+,-.kb+c + ~ l-~c+ (exp[ - a f q ] - exp[ (1 - a)fB] )
(15)
J - = ,.t-~z.a c,_-~cP_( e x p [ f l / . ]
(16)
- exp[ (1 -- fl)f ] )
with ~/= A4~ - ACq
(17)
and A~eq given by eqn. (8), one can express formally
I = r ( J + - J_ ) = f(r/)
(18)
by subtracting eqn. (16) from eqn. (15). The exchange fluxes in eqns. (15) and (16) are related to the exchange current densities io+ and i o_ by the Faraday constant, F. At small overpotentials, with stirring or rapid mass transport, expansion of the
28 exponentials gives: I=
(19)
- (io+ + i o _ ) f i l
and at higher overpotentials, but still with fluxes below mass transport control, ~ is c o m p u t e d from eqn. (18) with eqns. (15) and (16). W h e n the system is not stirred, the bulk concentrations and activities cannot be maintained at the interfaces. This is the usual situation. Then, I = io+
( c + ( x =c+O , t )
+ i o_
exp[-c~fij]
~+(x=0, ?+
{ c_(x=O,t) exp[ fifo ] + c_
t)
exp[ (1 -
a)fil]
)
t) exp[ - (1 - fl)fil] }
c_
(20)
and the time-dependences enter through the dependences of c+(x = O, t) and c (x = 0, t) on time. The c o m m o n irreversible case, e.g., io+ and i o_ < 10 2 A / c m 2, but i o_ < 1 0 - 2 i o + , provides two chronopotentiometric waves. H o w e v e r as i o_ approaches equality with io+, only one wave appears. The total transition time is the same in b o t h cases. In the first instance, a positive current carries only cations from aqueous to organic phase until t = ~1,
,t/2 = F( D +~r)1/2c+/2I
(21)
T h e second transition time (,~ + "2) obeys the equation
('1 +*2)l/2=(r(Tr)l/Z/2I)(Dl+/2c+
+ ~1/2~
)
(22)
In the limit of equal exchange current densities, there is only one transition time = "1 + "2. Eqns. (21) and (22) are of the same form as the description of two different cations moving from water to organic phase [9]. The wave shapes follow from conventional analysis for these two limiting cases. F o r the two step behavior, c+ (0, t)/c+ is replaced by 1 - (t/~'l)l/z,t < *l and is zero thereafter. ~_(0, t ) / ~ (or the ratio for second aqueous cation) is replaced by 1 - [(t - * 1 ) / ( * 1 -~ ']'2)] 1 / 2 for t > 'r 1 and is unity prior *l- This substitution is only valid for total irreversibility with aq. cation transporting more readily than oil-soluble anion (or second aq. cation). Corrections for the back reactions can be m a d e by using ~+(0, t)/~+ equal to 1 + (t/'r;) 1/2 for t < zl, and c (0, t)/c_ replaced by 1 + [(t - ~-;)/(*( + ~.~)]1/2 for t > ~'1. The transition times are approximately
, ,tl/2
= r ( D -+- ~ r )
1/2
~+/2I
(23a)
,~/2 = F( D_ ~r)1/2c_/21
(238)
In the single wave limit, io+ = i0_, c+ (0, t)/c+ and ~ (0, t ) / { are replaced by 1 - [t/('q + ~-2)]1/2. Another practical case involves one reversible and one irreversible ion from the same salt. For a typical reversible cation and irreversible anion, using high loading
29 of cation in the organic phase, I = ( 1 / 2 ) F ( D +~r)~/2c +[ t '/2 + ~-'I/2 ( D + / D + )1/2 exp( fAq~ -- fAqT')] -1 + i o_
{
exp[flfq] -} -
c
exp[ - (1 - fl)f~]
c
}
(24)
The first term determines the chronopotentiogram shape for t < ~'~ because the anion carries no current. Eventually anion depletion occurs and a second wave appears. The substitutions for ~ (0, t)/~_ and c (0, t ) / c , given above, are appropriate in this case, also. When mass transfer intervenes but does not completely control fluxes, as in imperfectly stirred solutions, surface concentrations may differ from bulk values according to the Nernst diffusion layer thickness. Then c+ (0, t)/c+ is replaced by 1--_i+/id+ ; c+(O, I)/C+ replaced by 1 + i+/id+ ; c_(O, t ) / c _ replaced by 1 + i / i d and ~ (0, t)/~_ replaced by 1 - i _ / i d _ . The large limiting currents are given approximately by the expression id,j = DjFc3/3
(25)
for each ion in each phase. 3 is the steady state Nernst layer thickness. In the low overpotential region, one can account for partial mass transport control in the current-voltage expression eqn. (19) by:
(1 1)
I+io+i + . +ld+ /d+
+io_i_
1)
ia - + id-
~ - ( i o + +io_)fil
(26)
The added terms cause ~ to move more negatively for positive current flow in comparison with the prediction of eqn. (19). Equations (19) and (26) are significant for liquid/liquid ion transfer kinetics because of the narrow range of transport free energies. The experimental consequence is that most practical systems for investigation show a narrow potential range for single-ion transport [10]. Allowance must be made, generally, for the possible reverse transfer of an oppositely-charged ion, or simultaneous transport of an ion of same charge, over some part of the accessible potential window. CURRENT PARTITIONING
The contributions of ionic currents i+ and i_ to the total constant current I, do not generally remain constant over time. For irreversible two-ion transport in stirred solutions, instantaneous and steady-state partitioning can remain fixed for a few instances. At small overvoltages,
i + / I = i o + / ( i o + +io_);
i / I = i o /(io+ +io_ )
(27)
At higher overpotentials, but for fluxes well below mass transport involvement,
i + / I = i o + e x p [ - a f q ] / ( i o e x p [ - a f i / ] +io+ exp[ - (1 - f l ) f l / ] }
(28)
30
/
a
~3
/
U_
CO
° 0
b
/
LI_
12.
2q-.
12.
TIME/S
2~.
TIME/S
C
/ /
I
0
12.
o
2q.
/
j / P
12.
0
TIHE/S
2~.
TIHE/S
Fig. 2. D i g i t a l l y - s i m u l a t e d c h r o n o p o t e n t i o g r a m s for irreversible two-ion t r a n s p o r t in a q u e o u s s u p p o r t i n g electrolyte. T r a n s p o r t e d ions are u n i v a l e n t with opposite charge. Positive current carries c a t i o n s from aqueous to organic phase, a n d anions from organic to aqueous phase. R a t e c o n s t a n t s are selected to allow t h e cation to m o v e first in Figs. 2 a - 2 c . D = D = 3 × 10 - 5 c m 2 / s ; a = fl = 0.5; i = 1 0 0 / ~ A / c m 2. k b _ / c m s - l : 10 - 7 (a), 10 - 6 (b), 10 -S (c), 10 - 4 (d).
Ion
Charge
c/M
~/M
k//cm s -
1
kb/Cm s- 1
1
1
5 × 10 - 4
1 X 10 - 2
1 X 10 - 4
1 X 10 - 4
2
- 1
1 X 10 - 2
5 X 10 - 4
1 × 10 - 4
variable
3 4
1 - 1
1 0.99
0.99 1
0 0
0 0
and there is an equivalent expression for i_/I. Only in the event that a = 1 - fl will the current remain partitioned at a constant value throughout a wide potential range. For unstirred solutions, because of different ionic concentrations and diffusion coefficients, current partitioning ratios will generally vary during the chronopotentiogram.
31 Aqueous Cation Profile ~LS
Organic Anion Profile
~"~..
,,,"
~ .~.. " "
~'.
f.
~-, "S
4--
\\\\\\. ~
3
,,..\ \
°
8
/
/
//
/
\1/ "\1
I0
I 9
I 8
I 7
I 6
I 5
I 4
I 3
I 2
I I
0
..I I
I 2
I 3
I 4
I 5
I 6
I 7
I 8
I
9
I0
Distance from Interface x IOZ/cm
Fig. 3. Concentration-distance profiles for positive current carrying cations (aqueous, ion 1, left) and anions (organic, ion 2, r i g h t ) tabulated properties of ions 1-4 in Fig. 2 legend apply here. k f _ / c m s - i: (------)
10-8; (
) 10-7; ( . . . . . .
DIGITAL
SIMULATION
) 10-6; ( . . . . .
) 10 - 5 .
RESULTS
The principal results of the theory using eqns. (15)-(19), with the continuity equation in the digital simulation mode [1] are hypothetical chronopotentiograms and log analytical plots. A selection of results for two-ion transport are given in Figs. 2a-d. The characteristic behavior from distinguishable two-wave, kinetically-limited cases to merged, undistinguishable, single-wave behavior is illustrated. When both ions are slow (10 -l] < k f < 10 -2 cm//s), but one of the ions has a forward rate constant four orders of magnitude or more larger than the other, two distinct waves are predicted and shown in Figs. 2a and b. As ke for each ion approach a common value, 10 -4, a single, totally irreversible wave is found in Fig. 2d. An indistinguisha-
TABLE
1
1st Transition times for two-ion processes, k I = k I = k z = 10 - 4 c m 2 / s . D] = D 2 = D ] = D 2 = 3 × 10 - 5 c m 2 / s , c = surface concentration of ion 1 ; c o = bulk concentration of ion I k ~/k 2 104 103 102 10 1 By
"rI ( o b s e r v e d ) / s '~ 6.30 6.35 6.55 24.0 24.0
(one wave only) (one wave only)
rj/s b
~'j/S ~
6.15 6.25 9.65
6.35 8.00 17.8
20.2 23.5
23.5 23.9
inflection point method, b When c°/cl = 100, c When c°/cl = 1000.
32
(a)
(b)
oh.
C~
G:). L
tu
co
°0
6.0
3.0 LN [1-(t/l") 1/2]
o 0
3.0
6.0
-LN [ 1 - ( t / ' r ) 1/2]
Fig. 4. Log analytical plots for Fig. 2. (a) Analysis of the first wave in Fig. 2b; (b) analysis of the only wave in Fig. 2c.
ble merging of two waves into an unusual single-transition time chronopotentiogram is the intermediate result in Fig. 2c. Merging of two irreversible waves into a single, reversible wave can also occur when the rate constants are allowed to increase. Two, oppositely-charged ions, moving at constant current to yield a single transition time were illustrated in a previous paper [11]. Appearance of a separate or merged wave in two-ion chronopotentiometry depends on both concentrations and forward rate constants. For ions with nearly equal a's, the two oppositely charged ions will carry nearly equal amounts of current when ( k f C ) l = ( k b C ) 2 ; C1 and C2 are surface concentrations. A normal first transition
500
t
i
i
I I0
I 20
I 30
i
t
[
[
I 60
I 70
4OO
3OO "~/mV 2OO
I00
O~
I I 40 50 Time / s
I 80
90
Fig. 5. Two-step, two opposite-charge ion transport. First process: I - ( n i t r o b e n z e n e ) ~ I (aq.). Second process TEA + (aq.) ~ T E A + (nitrobenzene). i = 11.0/xA/cm2; [ I - ] org. = 6.6 × l0 - 4 M as tetraheptyla m m o n i u m iodide; [TEA + ] aq. = 2.8 × 10 - 4 M as tetraethylammonium chloride; 0.1 M LiCI (aq.) and 0.0074 M tetraheptylammonium tetraphenylborate (org.).
33 180
1
160
//
//
//
140 120 F
,ool
y
///
-
~/mV 8O 60
40 20 0V 0
I I
I 2
-hn [I-(t/r) U2]
I 3
Fig. 6. Log analytical plots for tetrabutylammonium transport from water to nitrobenzene, i = 19 /~A/cm2; 8 x 10-4 M TBAC1(aq.); 0.02 M TBATPB (nitrobenzene). Curve a: 0.1 M LiC1; Curve b: 0.1 M CsC1.
time will occur for ion 1 if its forward rate constant is large enough to permit c~ to decrease to about l0 -3 c o before significant current partitioning b y the second ion occurs. The quantity c o is the initial bulk concentration of ion 1. Figure 3 shows the concentration profiles of ion 2 at the first transition time (for ion 1), as a function of kf2 relative to k n for solutions in which c o = ~2°. The first transition time can be erroneously long, without appearance of a developed second wave for the interfering species, ion 2. Table 1 shows h o w the first transition time can be lengthened by the presence of the interference at equal concentrations (c o = c °) and the time at which the surface concentration has decreased to selected fractions of its bulk concentration for the rate constant ratio kfl/kf2. The effect of back rates of ion transfer of two ions, is not significant in terms of the observed transition times. Log analytical plots of separated waves in Fig. 2a give expected irreversible slopes, RT/aF. However, analysis of the first wave from Fig. 2b shows severe deviation from linearity during the latter half of the wave. The sub-Nernstian curvature at times near the transition time is characteristic of interference as shown in Fig. 4a. A second ion is already carrying a significant portion of applied current
34 500
I
I
I
I
I
[
I a b
c
400
3O0 "r//mY 200
I00 0 0
I 5
I I0
I 15
I 20 Time / s
I 25
I 30
I 35
40
Fig. 7. Chronopotentiograms for 9× 10-4 M tetrabutylammonium chloride (aq.). i = 19 /LA/crn2; 0.2 M TBATPB (nitrobenzene). Curve a: 0.1 M NaC1; Curve b: 0.1 M NaCI + 1.4 X 10-2M NaI; Curve c: 0.1 M NaCI+0.1 M NaI.
in the vicinity of the apparent first transition time. Analysis of well-merged waves from Fig. 2c show even greater non-linearity in the log plot, Fig. 4b. If the initial slope of Fig. 4b is taken to be RT/aF, observed et's will be less than the true value for either ion, because of early co-transport. We have briefly investigated two other two-ion cases that give apparently one transition time: (1) equal forward rate constants but unequal backward rate constants, and (2) equal rate constants but differing a-values. Both cases give non-linear log analytical plots. The latter can show two regions of linearity with slopes R T / % F and 2RT/(% + ct2)F for cation and 2 R T / ( a + fl)F for cation and anion transport. EFFECTS OF LIMITED POTENTIAL "WINDOW" ON CHRONOPOTENTIOGRAMS Use of tetraphenylborate salts in the organic phase, and supporting aqueous electrolytes containing alkali metal ions, produces a limited potential window, i.e., there is a negative-going potential limit (~org.- ~aq..... ) imposed by transport of T P B - or M ÷ , whichever happens more easily, e.g., at less negative potentials. By selecting LiC1 as supporting electrolyte, two-wave chronopotentiograms can be observed. An example of this situation is provided by transport of tetraethylammonium (TEA +) and I - at H 2 0 / n i t r o b e n z e n e . Figure 5 shows the results of transporting similar concentrations of TEA ÷ and I - in aqueous LiCI and TEATPB as the organic supporting electrolyte. The largest TAA + ion which can be transported from water to nitrobenzene without complications from adsorption is tetrabutylammonium (TBA+). The relatively large difference in free energies of transport from water to nitrobenzene
35 between TBA + and T P B - gives a large potential window for the previously discussed phenomenon to be observed. When transport of TBA ÷ alone is desired, LiC1 provides the optimum electrolyte. As shown in Fig. 6, curve a, the log analytical plot for TBA ÷ is linear over two decades. On the other hand, an example of massive interference by a second ion is found in TBA ÷ transport at water/nitrobenzene when CsC1 is the supporting electrolyte. The potential window is diminished and the log analytical plot, curve b, shows the apparent interference as a sub-Nernstian deviation. Interference by supporting electrolyte components is common. When two waves are expected to occur (from estimated relative rates or free energies of transfer), only one wave is sometimes seen. The second process appears only as a potential plateau reached at the end of the first transition. The principal cause is the high concentration of the second transporting ion, frequently a component Of the aqueous supporting electrolyte. When the forward rate constants of two ions differ by one order of magnitude or less, only one wave is predicted. This behavior is seen in the co-transport of TBA ÷ and I - . Figure 7 shows the chronopotentiograms for a 9 x 1 0 - a M solution of TBAC1 with combinations of NaC1 and NaI used as the aqueous supporting electrolyte. When no I - is present, curve a, all current is carried by TBA ÷ . For relatively small amounts of NaI in the supporting electrolyte (1.4 x 10 - 2 M), curve b, the transport involves both TBA ÷ from water to nitrobenzene and I - from nitrobenzene to water. At higher concentrations of NaI, curve c, all current is carried by I - . A much more extensive series of cation responses (both tetraalkyl cations and a selection of supporting electrolytes) will be included in our subsequent, "experimental" paper. CONCLUSIONS Two-ion responses in liquid/liquid ion transfer studies are expected to be common. Because of the narrow potential window, determined by the extraction coefficients of two-salt systems, good separation of chronopotentiometric waves is not expected except for the few salts that allow forward rate constants ratios of 103 to 104. The use of very hydrophobic ions, such as tetraphenylarsonium or crystal violet for cations and dicarbollyl cobaltate, as the anion of the organic supporting electrolyte can enlarge the potential window. The fundamental problem however, still remains. Gavach's published chronopotentiograms and log analytical plots show sub-Nernstian slopes at times approaching the transition time expected for the two-ion transport situation. Other cases show super-Nernstian slopes characteristic of quasi-reversibility at short times for systems that become increasingly irreversible during the course of the experiment (because of concentration polarization). Other researchers have not analyzed their data over a sufficiently wide range of t/~ to show these effects which occur principally near to the transition time. Our simulated and experimental data have shown linearity (by log plot analysis) over only 60% of ~.
36
The region between 60% and 95% is most discriminating with respect to occurrence of two-ion transport, as deviations become readily apparent. For two-ion transport in supporting electrolyte, there is little practical difference between the transport of ions of opposite or the same charge. There is, however, a theoretical difference by virtue of the potential barrier symmetry. When transport occurs in little or no supporting electrolyte, the two cases can be dramatically different. These changes will be the topic of the final communication in this series. ACKNOWLEDGEMENT
This work was supported by National Science Foundation grant CHE81-03334. REFERENCES 1 0 . R . Melroy and R.P. Buck, J. Electroanal. Chem., 136 (1982) 19. 2 P. Delahay, Double Layer and Electrode Kinetics, Wiley, New York, 1965, pp. 154-158. 3 J. Koryta in E. Pungor (Ed.), Ion-Selective Electrodes, Vol. 3, Akademiai Kiado, Budapest, 1981, pp. 53-72. 4 R.P. Buck, Crit. Revs. Anal. Chem., 5 (1975) 323. 5 R.P. Buck in H. Freiser (Ed.), Ion-Selective Electrodes in Analytical Chemistry, Vol. 1, Plenum Press, New York, 1978, pp. 1-141. 6 J. Koryta and P. Vanysek in H. Gerischer and C. Tobias (Eds.), Advances in Electrochemistry and Electrochemical Engineering, Vol. 12, Wiley, New York, 1981, p. 113. 7 L.Q: Hung, J. Electroanal. Chem., 115 (1980) 159. 8 J. Koryta, P. Vanysek and M. Brezina, J. Electroanal. Chem., 75 (1977) 211. 9 T. Berzins and P. DelahaY, J. Am. Chem. Soc., 75 (1953) 4205. 10 D. Homolka, L.Q. Hung, A. Hofmanova, M.W. Khalil, J. Koryta, V. Maracek, Z. Samec, S.K. Sen, P. Vanysek, J. Weber, M. Brezina, M. Janda and I. Stibor, Anal. Chem., 52 (1980) 1606. 11 O.R. Melroy, R.P. Buck, F.S. Stover and H.C. Hughes, J. Electroanal. Chem., 121 (1981) 93.