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Electron and ion transport in membranes
Journal of Membrane Science, 2(1977)23-38 o Elsevier Scientific Publishing Company, Amsterdam -Printed
in The Netherlands
ELECTRON AND ION TRANSPORT IN MEMBRANES
D. WALZ and 0. KEDEM* Department of Pharmacology,
Biorentrum,
University
of Busel, Base/ {Switzerland}
(Received June 7,1976)
Summary A formal description of electron flow throughmembranes is given,including the winition of the electron transport number, re. The effect of electron transfer on the membrane potential is worked out and the relevance of the membrane potential to the redox potentials in the aqueous phase is discussed. Conditions for the maintenance of a stationary state as well as for conversion of free energy between electron and ion transport are analysed. The relevant points are exemplified with a simple, but fully accessible, experimental system. Introduction Some synthetic membranes are able to transfer electrons from one aqueous phase into another thus exhibiting a phenomenon which is well known for many biological membranes. This electron transport involves redox reactions between membrane components and solutes in the aqueous solution and, in the stationary state, the number of electrons entering the membrane is equal to that given off. The resulting “flow of electrons” in the membrane can interact with ion flows, either by a direct coupling of the two processes or due to the electric potential difference established across the membrane. Electron flow has theretire to be included in the quantitative description of flows in the system. In the following, a discussion of redox processes across membranes and experimental results on an illustrative system are given. Energy dissipation in the presence of electron transfer For the solution components not participating in the redox reactions, the conservation of matter can be written as
where n: and nf’ are the number of mols of component i in solutions I and II, respectively, Ji is the flow of component i per unit area into solution I, A the * Permanentaddress:Laboratoryof Membranesand Bioregulation, Weizmann Institute of
Science, Rehovot (Israel).
membrane area, and the dot indicates the time derivative. The material balance for the redox components must allow for electron transfer: the electron does not exist as a species in any of the phases, but at the interfaces it is a separately transferred entity. Within the membrane the mechanism of conductance may be either electron transfer between fixed components, or counter-diffusion of reduced and oxidized species, or a combination of both. Whatever the mechanism, we assume that all processes in the membrane are fast compared to the changes in the aqueous solutions, and a stationary state of the membrane is maintained. The process may then be characterized by an electron flow, J, , per unit membrane area. We assume that the redox components in the solutions cannot pass the membrane and consider only one redox-pair in each solution, not necessarily identical in both. For simplicity, a single-electron transfer is chosen. Thus oxl+e+redl ox2+~=+red2
in solution I in solution II
(2)
and the material balance equation becomes: -1 =%dl
*II -%d2
’ II =nox2=-noxl
-1
=
AJ,
(3)
The dissipation of free energy per unit time and area is given by [ 1,2]
(4) where fis I1 denotes the electrochemical potential of component i in solutions I Ad II, respectively. Introducing the flows according to eq. 1 and 3, @ may be rewritten as (5)
1
II
where Api = /-Li- pf. The driving force for J, in this equation may be recognized as the difference in electrochemical potential of the electron across the membrane. Assuming equilibrium at the membrane surface, the electrochemical potential of the electron in solution I, $ and in the membrane near the su:pe contiguous to this solution are equal and a similar condition applies for EC, in solution II. Since (cf. eq. 2)
the dissipation function becomes @ = CJi
A;;i’
+
J,AEc
(7)
25
Flow equations All flows are in general functions of all forces, and in the near-equilibrium range linear flow equations may be expected to hold: Lik Ack + LiEATe
Ji =C
(8)
k
J, =CL,zi
A’& + L, AT;
(9)
with chX%Iger’S relation Lik = Lki, L, i = Lie. The straight coefficient, L,, reflects the electron conductance and the Li, ‘s indicate a possible coupling between ion flow and electron transfer. Difference in redox potential across the membrane The electrochemical potential of an ion, rl, may be regarded as the sum of two components, viz. the composition-dependent part and the term containing the electrostatic potential, \k. Hence A& = Afii +z,FA\k; A\k = ‘I?ll-@l
(10)
where Zi is the valency of component i. Strictly speaking, the electrostatic potential difference between two phases is not accessible to measurement and, by the same token, single ion activities cannot be determined. However, the separation of Ap, into its components is essential for many measurements. The standard calomel electrode (with salt bridge) gives a sufficiently good and much-tested approximation to electrostatic potential differences and thereby provides the basis for the potentiometric determination of parameters such as pH or redox potential. In analogy to eq. 10 and from eq. 6, with a single-electron transfer for the redox pairs AFC = ApE-FA\Ir
(11)
where A& =P~~-P::~-(P~I-P:~~)
(12)
The difference in chemical potential of the redox components can be assessed by the redox potential, & , which, for experimental convenience, may be written as c & = Em +Flogz (13) %ed where c xScred denote the concentrations of the redox species. & is the potential determined by the standard chemic~p&,enti& ad the activity coefficients of the redox components. The difference in
so-CakfmlciPoint
26
redox potential between the two compartments, A&, then becomes At = C:II--t‘l = - Aiu /F r
(14)
Membrane potential If an external electric potential is applied across the membrane, electron transfer as well as ion flows will contribute to the electric current through the membrane and the total current density is I=F(CZiJi-J,)
(15)
i
The contribution of each component, when all AMi and ApE are zero, is defined as the corresponding transport number 71. With the flow eq. 8 and 9
(16)
where L=
C i,k
ziZkLik_2
CziLi,
+L,
i
In the absence of an electric current, different solutions on both sides of the membrane give rise to a membrane potential. Introducing eq. 8 to 11 and 16 into 15 yields, for the condition of vanishing current (171 In this expression for the membrane potential, ~~ appears in addition to the standard expression for A@m [ 31. The meaning of eq. 17 may be illustrated by introducing a platinum sheet as a “membrane”. Only electrons can then pass, i.e. T, =l,ri=O,andA+m= A&F. As expected, -A9,is equal to the difference in redox potential, AK (cf. eq. 14), and the system may be regarded as two redox measurements in series: (calomel I/Pt) + (Pt/ca.lomelll) giving -L1 + Erl = -A*,. Conversion between redox and osmotic free energy To show the interdependence between ion and electron flows, let us con-
27
sider a system which contains only one uni-univalent salt and one redox system, at a much lower concentration than the salt. The concentration of permeable anions and cations in the solutions cannot remain exactly equal when a redox reaction takes place across the membrane. But since we assumed that the salt is in large excess compared to the redox components these differences in ion concentration remain relatively small, and to a good approximation (18)
AY, = AIL = A~,12 Introducing this and eq. 15 into eq. 7 yields for the dissipation function
(19) In these terms we consider an electron transfer across the membrane driven by a difference in redox potentials, an average ion or salt flow driven by the chemical potential difference of the salt, and an electric current driven by an electric potential difference. Under this condition and in the absence of an electric current, salt diffusion and electron transfer are coupled even if all LIE are zero. From the flow eq. 8 and 9 with Lie = Lik = 0 together with eq. 10, 11 and 18, the electron flow and salt flow as a function of A& and ALL,become J, = LT,(~FT,)
J+ +J_ Js =2
r+---7_
A/_L~ + LrE ~
AP,
2
Ape
+T+
+T_-(T+-T-)~]
Aus
With a permselective membrane 7 + # 7__,and a salt concentration can build up a redox gradient and vice versa.
gradient
A simple but illustrative system Two solutions containing a uni-univalent electrolyte in different concentrations and a redox pair at different ratios of oxidized to reduced species are separated by an anion-exchange membrane. The membrane prevents the transfer of the di- and trivalent, positively charged redox species between the solutions. An “electron-transfer mechanism” is created by two large platinum electrodes, one in each solution, which upon short-circuiting allow the electrons to pass from one solution to the other independent of the ion flows through the membrane. Hence Li, = 0. The current through the short-circuit, I,, monitors the electron flow. For consistency, J, is defined per unit plembrane area as are the ion flows and I, = -AF
J,
(21)
The electron transfer process can also be followed by the change in redox composition in both solutions or by the concomitant change in redox poten-
28
tials. A calomel electrode and a (measuring) platinum electrode in both solutions enable us to determine all relevant parameters. The potential difference between the two calomel electrodes is the membrane potential, A 9,, while the redox potential, L, in each solution is measured between the platinum and calomel electrodes, Finally, by virtue of eq. 11 and 14 A c‘ + A\k, = -APE/F, which means that the potential difference between the two platinum electrodes indicates -Aj&/F. Characterizing parameters pertinent to the state before the short-circuit of the large platinum electrodes by the superscript” and remembering Li, = 0, it is easily verified by means of eq. 16 and 17 that ?“i/riO= l-7,
(22)
and hence (cf. eq. 14) (23) In this equation A@& and A*, denote the membrane potential just before and after short-circuiting, respectively. Experimental The experimental set-up consisted of two cylindrical cells (volume 50 ml) with the anion exchange membrane (F. de KBriisy, US. Pat. 3,388,080,1968) sandwiched in between. A platinum electrode (type P 1312) and a calomel electrode (type K 4112, both from Radiometer) as well as a large platinum electrode (surface 6.5 cm* ) were fitted through holes into each cell. The solutions in both compartments were continuously flushed with water-saturated N2 gas. The cells were completely closed except for a small hole which served as the gas outlet and was also used for taking small samples from the solutions. Stirrers were fixed in each compartment and were driven by a magnetic stirring motor below the cells. The temperature was kept at 21 * 1°C. Potentials were measured with a Radiometer pH meter, type PHM 26c (accuracy < + 0.5 mV), and the current running through the short-circuited platinum electrodes was monitored with a recording microammeter (Elaviscript 3). The readings of this instrument were calibrated by measuring the potential drop over a 1 kR resistor (1% precision). Before use, calomel electrodes were placed overnight in a 0.1 M KC1 solution and short-circuited. Small asymmetry potentials which occasionally occurred were corrected in the potential readings. Platinum electrodes were electrolytically cleaned in concentrated HN03. The redox pair used, i.e. Fe3’/Fe2’ in acidic solution, equilibrates with the platinum electrode within a minute and gives stable and reliable potential readings. Low pH prevents hydrolysis and slows down autoxidation. A plot of redox potential versus redox composition according to eq. 13 is shown in Fig. 1 for the data obtained from both compartments during the experiments.
29
FmV
720
Fig.1. Plot of redox potential versus redox composition compartment I ( l ) and compartment II (A ).
according
to eq.13.
Data from
The slope of both straight lines is 58.3 mV and equal to the ideal value, incheating no significant change in activity coefficients over the whole concentration range. The difference in intercepts, however, is due to different activity coefficients in the two compartments. Thus Fe3*/Fe2+ is a fully reversible redox pair with midpoint potentials of 742.4 mV and 731.9 mV in solutions I and II, respectively. 45 ml of a 0.11 M KNOB solution and an equal volume of Hz0 were pipetted into compartment II and compartment I, respectively, and 0.5 ml of a 1.03 M HN03 solution were added to each compartment. NO; was chosen as the anion since Cl- in excess of the Fe3+ concentration yields negatively charged complexes, FeC!ln3-n with IZ= 4 to 6, which would permeate through the membrane. N2 gas was bubbled through both solutions for about an hour. Then the gas inlet tubes were fixed just above the surface of the solutions and the stirrers switched on. (A stream of gas bubbles severely interfered with the potential readings, but uniform stirring was of little influence). When the potential difference between the calomel electrodes was stable, 1 to 3.5 ml of both a concentrated FeClz and FeCl, solution were added to each compartment. These solutions were freshly prepared under N2 and contained about 50 m&f of the iron salt and 10.3 mM HN03. The ratios Fe3’/Fe2’ varied, but the total volume added was always 4.5 ml, corresponding to a final concentration of total iron of about 5 mM in each solution. Potential differences were periodically measured between the calomel electrodes, the platinum electrodes and the platinum vs. calomel electrodes in each compartment. From time to time samples were taken from both compartments and analysed for Fe” and Fe3+ as follows. The samples (25 or 50 ~1) were added to 2.95 ml of a 25 mM dipyridyl solution and the optical density was read at 523 nm (the peak in the spectrum of the red Fe2’-dipyridyl complex where the brown Fe3+ complex has essentially no absorption; the extinction
30
caefficient is 8.78 r&f-’ cm-’ as determined with standards of analytical grade ferrous ammonium sulfate). Then 10 ~1 of a 1 M sodium ascorbate solution were added and the optical density was read again thus yielding the total iron concentration. Fe3+ was calculated as the difference between total iron and Fe’+. Results and discussion The time course of the four parameters: membrane potential, electrochemical potential difference of the electron across the membrane, and redox potential in both solutions are shown in Fig.2 for experiment 1. A membrane potential arises when two solutions of a salt of different concentrations are separated by an anion selective membrane. It decreased upon addition of iron salts due to the changes in anion concentration. Short-circuiting the large platinum electrodes caused an instantaneous drop in membrane potential which indicates that electron transfer contributes significantly to A*, (cf, eq. 17). The electron transfer as monitored by the short-circuit current 1s is shown in Fig. 3. The rising curve in the same figure gives the change in redox com-
80
Fig.2 Time course of the four measured parameters for experiment 1. Potential difference between calomel electrodes, A Cal (o), and platinum electrodes, Apt (A) in compartment II and I (upper part); redox potential, E1 (a) and &*I (A), in solutions I and II, respectively (lower part). The curves were calculated as described in the Appendix. The first abrupt change in A Cal was due to the addition of iron salts, the second one (with a corresponding drop in A Pt) occurred upon short-circuiting of the large platinum electrodes.
31
2
1
6
a
10
days
Fig. 3. Short-circuit current, Is, and concomitant change in redox composition for experiment 1. Is measured (heavy line) and calculated from JE (see Appendix) and eq. 21 (broken line). The rising curve indicates the number of “mols of electrons”, n,(t), transferred by Je from solution I into II during the period from short-circuiting (t = 0) up to
t
time t (=
s 0
Is dt’/F).
The changes
of Fe3+ concentration
in solution
I, &$,M
(e), and of
Fe*+ concentration in solution II, Acgez+ (A), during the same time period were calculated with eq.13, the measured redox potentials, and with the total iron concentrations for both solutions which were found to be constant within experimental error. V denotes the equal and constant volume of both solutions (see Appendix).
position expected from the time integral of the current, while the points were calculated from the redox potential measurements. The consistency of these measurements indicates that neither Fe” nor Fe3’ was permeable. By virtue of eq. 9 with Li, = 0 and 21 L, = --I,/(A&AF)
(24)
and L, thus calculated was found to be independent of ion composition in the solutions. T, was determined according to eq. 23 which together with L, then yields L (cf. eq. 16 with Lie = 0).A simplifying assumption is necessary in order to estimate the ion transport numbers. H’ and K’ as well as Nq and Cl- were treated together as one fictitious permeable cation (subscript +) and anion (subscript -), respectively. Hence from eq. 17 o _ Ap_-FA
7+---
&-G
r” =
\k” m
+ Act_
A/L++ FA’P;
(25)
AP+ + ALL
and the two transport numbers could be determined from the initial composition of the solutions (cf. eq. A2)and the membrane potential at the beginning of the experiment. Finally T, and T_ were calculated according to eq. 22.
32
With the data for ri and L (see Table 1) and from the initial composition of both solutions, the time course of all measured parameters can be calculated (cf. Appendix) and the curves shown in Fig. 2 were thus obtained. The fit between experimental data and calculated curves is satisfactory; the deviations observed with increasing time were to be expected in view of the simplifying assumptions made, and as a whole we can conclude that the system behaved as expected. TABLE 1 Transport numbers and sum of permeability coefficients
1 273
0.32 16.8
9.97 8.45
0.155 0.0026
0.015 0.0184
0.830 0.979
‘A denotes the membrane area. ‘L = T Li (i = +,-,E), the permeabilities Li can be calcu-
lated from Ti and eq. 16 with Lik = Lk = 0. For Ti see eq. 22. Two main points are emphasized by the experiment recorded in Fig. 2: (1) Equilibration for the electron (Ap, = 0) does not lead to equal redox potentials in both solutions, but A& = -A*, (cf. eq. 11 and 14); (2) The electron approaches equilibrium long before the system as a whole is equilibrated. Relaxation times are determined by kinetic parameters (in our system, the permeabilities) and capacities (which are related to the concentrations). Although the permeability to the anion was more than 5 times larger than that to the electron (cf. Table l), the much lower concentrations of Fe” and Fe3’ compared to the salt concentration in one of the solutions lead to a decay of A;;- faster than the equilibration of the whole system. The approach to and maintenance of a stationary non-equilibrium state is seen in the dissipation function. Fig. 4 gives the time course of @ determined according to eq. 7 and with the flows and forces calculated as described in the Appendix. The sharp rise of Q, upon short-circuiting is followed by a decrease to a level which then remains constant for a long time: a constant non-vanishing rate of energy dissipation is the clearest characterization of a stationary state. Strictly speaking, this flat portion of @ represents a quasi-stationary state in which the salt concentrations change very slowly (see Fig. 7A) and the redox compositions continuously adjust themselves so that the deviation from equilibrium state for the electron remains very small. Of course, after a much longer period of time the concentrations in both solutions become equal for all components and Cpvanishes, i.e. the whole system reaches equilibrium. A 60-fold smaller 7, and somewhat larger T+ and 7_ result when the same
33
days
0
I
8
12
t
Fig.4. Time course of the dissipation function, @. Solid line: experiment 1 (lower and left hand scale), broken line: experiment 2 (upper and right hand scale), A denotes the membrane area.
large platinum electrodes are used but the membrane area is increased about 50 times (experiments 2 and 3). In this case, the contribution of electron transfer to the membrane potential is too small and no drop in A\kmwas observed upon short-circuiting of the platinum electrodes (Fig. 5 and 6). HOWever, L, and TO+, r” couid still be determined according to eq. 24 and 25, respectively, while 7;f, T+ and T_(see Table 1) were inferred by means of the L, and L_ from experiment 1 which apply because the same membrane material was used. The relaxation times for electron transfer and salt diffusion now become comparable. With an initial redox composition closer to the equilibrium state for the electron than in experiment 1, the system proceeded towards this state and eventually reached it but only transiently (experiment 2, Fig. 5). No quasi-stationary state occurred as indicated by the continuously decreasing dissipation function (Fig. 4). In experiment 3 the initial A& was chosen to be of approximately the same magnitude as in experiment 2 but with opposite sign and equilibration of the electron was only achieved when the whole system reached equilibrium (Fig. 6). The last point which can be demonstrated is the conversion of osmotic into redox free energy. Though the iron concentrations in solution I were comparable to the salt concentration, eq. 18 is still a reasonable approximation and Fig, 7 then presents the dependence of electron flow, J,, and salt flow, J, = (J.+ + L)/2, as a function of the conjugated forces, i.e. the differences in chemical potential for the electron, Ape, and for the salt, AH, = A/J+ + AM(cf. eq. 20). For convenience, the straight coefficients are abbreviated by Lr E L 7, (l-7,) and L, z L [T+ + T- - (T+ - T-)‘I/4 while the coupling coefficient is L, G LT-~(T+ -G)/2 (see eq. 20 and Table 2).
34
mV
ii
4
15 UA
b
a
, 10
0 Q
0
_1
0
1
2
3 days
Fig.5. Time course of A Cal, & Pt,L dition, the short-circuit current, I,, and the broken line (calculated). A ever, A Cal and A Pt did not change
1
5
6
t
I and Err for experiment 2, plotted as in Fig.2. In adis shown in the upper part as the heavy line (measured) drop in A Cal occurred on addition of iron salts; howupon short-circuiting (t = 0) since TV<< 7_.
760 =iA
710
-2
720 7cn
Fig.6. Time course of measured parameters for experiment 3, plotted as in Fig.5. The curves were calculated for a longer time period than that of the actual experiment to show that the whole system approaches equilibrium. Note the break in the time scale.
35
A
AJJ~
-2
-L
> *w 7
0
2
B
IJhmolei 1
1.5
2
2.5
5 f)______
___
9.0
9.4
9.8 ~~~
5
7
9
(J/m mole)
Fig.7. Electron flow, JE, and salt flow, J,, as a function of the conjugated forces, A& and Aps (difference in chemical potential between solutions II and I), according to eq. 20. A: experiment 1, B: experiment 2. Heavy lines represent data calculated as described in the Appendix, with arrows indicating the chronology. Thin lines are Je= Lr~pe and J, = LsAps, respectively, for the values of L, and L, given in Table 2. Points mark experimental data derived from redox potentials (eq. 14) and from the short-circuit current (eq.21). For L, S and S’ see Fig.8.
TABLE 2 Parameters for coupling of electron flow and salt flow (eq. 20) ___ ~_ ___L,x 10TC qd L,, x 10’ b L,X 10’a Exp.
e z
1 2,3
1.31 0.022
aL, = L7e (1~~ ‘According
4.63 -0.010
0.45 0.158
), bLrs = L7e(7+-~_)/2.
-0.82 -0.17
1.70 0.37
‘L, = L[T+ + T- - (T+ -T_)’ ] /4 (see eq. 20).
to Kedem and Caplan [4] the degree of coupling, 4, is equal to LrsdL,L,.
“The “mechanistic”
stoichiometry,
.a, is equal to JL,/L,.
At the beginning of experiment 1, both flows had the same sign as the conjugated forces but were larger than expected from these forces due to coupling (Fig. 7A). The system only dissipated energy as can be seen from the plot of flow ratios versus force ratios (Fig. 8) according to Kedem and Caplan [4]. Note that the flows were negatively coupled (I&., < 0, cf. Table 2) because the electron and the more permeable ion of the salt have the same sign. At the point marked by L in Fig. 7A and 8, which corresponds to the cross-over point of redox potentials in Fig. 2, the system reached the state of
36
J, ‘J,
Fig.8. Plots of flow ratio, (JdJ& z, and efficiency of energy conversion, n = J,ApE/J&ks versus force ratio z(A~.+/ALI~),according to Kedem and Caplan [4]. Heavy lines represent data calculated as described in the Appendix and with the z values given in Table 2. Thin lines are theoretical curves [4] for 9 = -0.16, -0.69 and -0.82 (top to bottom), respectively. The apparent variation of the degree of coupling, 4, for experiment 1 and the lower value of q than that in Table 2 for experiment 2 are caused by deviations from the condition Ap+ = AL (eq. 18) which become larger the more the redox compositions in the two solutions are changed. The value and the location of the maximum of the efficiency [4] yield q = -0.75 and -0.16 for experiments 1 and 2, respectively (note that 20~ is plotted for experiment 2). L denotes the state of level flow, while S and S’ are static head conditions. Energy is converted when the flow ratio versus force ratio curve goes through the lower right quadrant but energy is only dissipated when this curve is in the lower left and upper right (not shown) quadrants [4].
level flow (AP, = 0). This state, however, cannot be maintained in a closed system, and the system proceeded into the phase where J, was driven by ApcLs against Ape, with J, being therefore smaller than L,Ap,. Besides dissipation of energy, osmotic free energy was converted into redox free energy during this phase with an efficiency 77(see Fig.g).until the static head condition J, = 0 was attained (S in Figs. 7A and 8). The system then remained for about 8 days in this strictly speaking “close to static head” state (J, Y 0, a quasistationary state) with little change of All, (see range marked in Fig.7A) before it entered the phase where both flows again had the same sign as the conjugated forces and which eventually led, solely with dissipation of energy, to the equilibration of the whole system. The initial redox composition of experiment 2 let the system start from an en$rgy converting state (Fig. 8). However, due to the small value of TV,the degree of coupling 4 (see Table 2) and hence the efficiency t) were low, while J, accordingly differed little from LsAps (Fig.7B). The static head condition (J, = 0, S’ in Figs. 7B and 8) could not be maintained for the reasons given above and the system immediately entered the phase where energy is only dissipated on the way to overall equilibration. Finally, the initial redox composition of experiment 3 caused the system to start in the latter phase and no energy converting state could thus occur.
37
Acknowledgements This investigation was begun when one of us (O.K.) was on a sabbatical year at the Biozentrum. It is a pleasure to thank Prof. F. Gri.in for his efforts which made this stay possible, as well as for his interest and for helpful discussions. We are indepted to Prof. F. de K&-&y for kindly supplying the anion exchange membrane. The technical assistance of Miss A. Kr%uchi is gratefully acknowledged. References 1 I. Prigogine, Introduction to Thermodynamics New York, London, Sydney, 1967. 2 S.R. de Groot and P. Mazur, Non-Equilibrium
of Irreversible
Processes,
Thermodynamics,
Amsterdam, 1962. 3 A.J. Stavermann, Trans. Faraday Sot. 48 (1952) 4 0. Kedem and S.R. Caplan, Trans. Faraday Sot.
176. 61 (1965)
Interscience
Publ.
North-Holland,
1897.
Appendix The following assumptions were made in addition to the postulate of only one permeable cation and anion: The flows of ions are not coupled to each other, i.e. Lik = 0 while Lie = 0 due to the experimental set up. Hence from eq, 8 and 9 Ji = LiA~i
(i = +,-,E)
(Al)
with Li being independent of concentration (as verified for L,). The activity coefficients of the permeable ions were taken to be constantnt in each solution and the differences in these coefficients between the two solutions were neglected. The difference in chemical potential of the two ions at time t then becomes A/J-i(t)= RT In ~fI(t)/~f(t)
(i = +,-)
(A21
where cl;I1 (t) d enotes the concentration of these species in solutions I and II, respectively. The changes in volume due to the ion flows, the concomitant solvent flow, and the removal of small samples for assay of iron composition were neglected, i-e. the volume, V, is at any time the same for both solutions. If the concentrations of ions and redox species are known at time t, all relevant parameters can be calculated: The redox potentials, C1pl’, are given by eq. 13 and they determine the difference in chemical potential of the electron (cf. eq. 14). Apu, and A,u+, ACL according to eq. A2 yield the membrane potential A*, (eq. 17) and by this in turn the corresponding electrochemical potential differences (eq. 10 and 11). The forces Ace, Ai;, and A?_ then determine the conjugated flows (eq. Al).
38
The flows J, , J+ and J change the concentrations of ions and redox species in both solutions. Since c = n/V for all species, we obtain, when using a finite difference approximation for derivatives, from eq.1 +j $
= 6ct=AJi
st/V
t-43)
and from eq. 3 (-44) for the changes in concentration after a small time increment 6 t. Thus, the concentrations at time t + 6 t can be determined and the calculations of relevant parameters repeated which then gives the concentrations at time t + 26 t and so on. Starting from the initial concentrations of ions in both solutions and taking into account the addition of iron salts at a later time, the time course of all measured parameters was calculated with a computer as outlined above and the results are plotted in the pertinent figures. This computer simulation also provided the data for @ (Fig. 4) and the flow-force relations shown in Figs. 7 and 8. The values used for the coefficients Li and ri are given in Table 1; for CL” see Experimental.
in The Netherlands
ELECTRON AND ION TRANSPORT IN MEMBRANES
D. WALZ and 0. KEDEM* Department of Pharmacology,
Biorentrum,
University
of Busel, Base/ {Switzerland}
(Received June 7,1976)
Summary A formal description of electron flow throughmembranes is given,including the winition of the electron transport number, re. The effect of electron transfer on the membrane potential is worked out and the relevance of the membrane potential to the redox potentials in the aqueous phase is discussed. Conditions for the maintenance of a stationary state as well as for conversion of free energy between electron and ion transport are analysed. The relevant points are exemplified with a simple, but fully accessible, experimental system. Introduction Some synthetic membranes are able to transfer electrons from one aqueous phase into another thus exhibiting a phenomenon which is well known for many biological membranes. This electron transport involves redox reactions between membrane components and solutes in the aqueous solution and, in the stationary state, the number of electrons entering the membrane is equal to that given off. The resulting “flow of electrons” in the membrane can interact with ion flows, either by a direct coupling of the two processes or due to the electric potential difference established across the membrane. Electron flow has theretire to be included in the quantitative description of flows in the system. In the following, a discussion of redox processes across membranes and experimental results on an illustrative system are given. Energy dissipation in the presence of electron transfer For the solution components not participating in the redox reactions, the conservation of matter can be written as
where n: and nf’ are the number of mols of component i in solutions I and II, respectively, Ji is the flow of component i per unit area into solution I, A the * Permanentaddress:Laboratoryof Membranesand Bioregulation, Weizmann Institute of
Science, Rehovot (Israel).
membrane area, and the dot indicates the time derivative. The material balance for the redox components must allow for electron transfer: the electron does not exist as a species in any of the phases, but at the interfaces it is a separately transferred entity. Within the membrane the mechanism of conductance may be either electron transfer between fixed components, or counter-diffusion of reduced and oxidized species, or a combination of both. Whatever the mechanism, we assume that all processes in the membrane are fast compared to the changes in the aqueous solutions, and a stationary state of the membrane is maintained. The process may then be characterized by an electron flow, J, , per unit membrane area. We assume that the redox components in the solutions cannot pass the membrane and consider only one redox-pair in each solution, not necessarily identical in both. For simplicity, a single-electron transfer is chosen. Thus oxl+e+redl ox2+~=+red2
in solution I in solution II
(2)
and the material balance equation becomes: -1 =%dl
*II -%d2
’ II =nox2=-noxl
-1
=
AJ,
(3)
The dissipation of free energy per unit time and area is given by [ 1,2]
(4) where fis I1 denotes the electrochemical potential of component i in solutions I Ad II, respectively. Introducing the flows according to eq. 1 and 3, @ may be rewritten as (5)
1
II
where Api = /-Li- pf. The driving force for J, in this equation may be recognized as the difference in electrochemical potential of the electron across the membrane. Assuming equilibrium at the membrane surface, the electrochemical potential of the electron in solution I, $ and in the membrane near the su:pe contiguous to this solution are equal and a similar condition applies for EC, in solution II. Since (cf. eq. 2)
the dissipation function becomes @ = CJi
A;;i’
+
J,AEc
(7)
25
Flow equations All flows are in general functions of all forces, and in the near-equilibrium range linear flow equations may be expected to hold: Lik Ack + LiEATe
Ji =C
(8)
k
J, =CL,zi
A’& + L, AT;
(9)
with chX%Iger’S relation Lik = Lki, L, i = Lie. The straight coefficient, L,, reflects the electron conductance and the Li, ‘s indicate a possible coupling between ion flow and electron transfer. Difference in redox potential across the membrane The electrochemical potential of an ion, rl, may be regarded as the sum of two components, viz. the composition-dependent part and the term containing the electrostatic potential, \k. Hence A& = Afii +z,FA\k; A\k = ‘I?ll-@l
(10)
where Zi is the valency of component i. Strictly speaking, the electrostatic potential difference between two phases is not accessible to measurement and, by the same token, single ion activities cannot be determined. However, the separation of Ap, into its components is essential for many measurements. The standard calomel electrode (with salt bridge) gives a sufficiently good and much-tested approximation to electrostatic potential differences and thereby provides the basis for the potentiometric determination of parameters such as pH or redox potential. In analogy to eq. 10 and from eq. 6, with a single-electron transfer for the redox pairs AFC = ApE-FA\Ir
(11)
where A& =P~~-P::~-(P~I-P:~~)
(12)
The difference in chemical potential of the redox components can be assessed by the redox potential, & , which, for experimental convenience, may be written as c & = Em +Flogz (13) %ed where c xScred denote the concentrations of the redox species. & is the potential determined by the standard chemic~p&,enti& ad the activity coefficients of the redox components. The difference in
so-CakfmlciPoint
26
redox potential between the two compartments, A&, then becomes At = C:II--t‘l = - Aiu /F r
(14)
Membrane potential If an external electric potential is applied across the membrane, electron transfer as well as ion flows will contribute to the electric current through the membrane and the total current density is I=F(CZiJi-J,)
(15)
i
The contribution of each component, when all AMi and ApE are zero, is defined as the corresponding transport number 71. With the flow eq. 8 and 9
(16)
where L=
C i,k
ziZkLik_2
CziLi,
+L,
i
In the absence of an electric current, different solutions on both sides of the membrane give rise to a membrane potential. Introducing eq. 8 to 11 and 16 into 15 yields, for the condition of vanishing current (171 In this expression for the membrane potential, ~~ appears in addition to the standard expression for A@m [ 31. The meaning of eq. 17 may be illustrated by introducing a platinum sheet as a “membrane”. Only electrons can then pass, i.e. T, =l,ri=O,andA+m= A&F. As expected, -A9,is equal to the difference in redox potential, AK (cf. eq. 14), and the system may be regarded as two redox measurements in series: (calomel I/Pt) + (Pt/ca.lomelll) giving -L1 + Erl = -A*,. Conversion between redox and osmotic free energy To show the interdependence between ion and electron flows, let us con-
27
sider a system which contains only one uni-univalent salt and one redox system, at a much lower concentration than the salt. The concentration of permeable anions and cations in the solutions cannot remain exactly equal when a redox reaction takes place across the membrane. But since we assumed that the salt is in large excess compared to the redox components these differences in ion concentration remain relatively small, and to a good approximation (18)
AY, = AIL = A~,12 Introducing this and eq. 15 into eq. 7 yields for the dissipation function
(19) In these terms we consider an electron transfer across the membrane driven by a difference in redox potentials, an average ion or salt flow driven by the chemical potential difference of the salt, and an electric current driven by an electric potential difference. Under this condition and in the absence of an electric current, salt diffusion and electron transfer are coupled even if all LIE are zero. From the flow eq. 8 and 9 with Lie = Lik = 0 together with eq. 10, 11 and 18, the electron flow and salt flow as a function of A& and ALL,become J, = LT,(~FT,)
J+ +J_ Js =2
r+---7_
A/_L~ + LrE ~
AP,
2
Ape
+T+
+T_-(T+-T-)~]
Aus
With a permselective membrane 7 + # 7__,and a salt concentration can build up a redox gradient and vice versa.
gradient
A simple but illustrative system Two solutions containing a uni-univalent electrolyte in different concentrations and a redox pair at different ratios of oxidized to reduced species are separated by an anion-exchange membrane. The membrane prevents the transfer of the di- and trivalent, positively charged redox species between the solutions. An “electron-transfer mechanism” is created by two large platinum electrodes, one in each solution, which upon short-circuiting allow the electrons to pass from one solution to the other independent of the ion flows through the membrane. Hence Li, = 0. The current through the short-circuit, I,, monitors the electron flow. For consistency, J, is defined per unit plembrane area as are the ion flows and I, = -AF
J,
(21)
The electron transfer process can also be followed by the change in redox composition in both solutions or by the concomitant change in redox poten-
28
tials. A calomel electrode and a (measuring) platinum electrode in both solutions enable us to determine all relevant parameters. The potential difference between the two calomel electrodes is the membrane potential, A 9,, while the redox potential, L, in each solution is measured between the platinum and calomel electrodes, Finally, by virtue of eq. 11 and 14 A c‘ + A\k, = -APE/F, which means that the potential difference between the two platinum electrodes indicates -Aj&/F. Characterizing parameters pertinent to the state before the short-circuit of the large platinum electrodes by the superscript” and remembering Li, = 0, it is easily verified by means of eq. 16 and 17 that ?“i/riO= l-7,
(22)
and hence (cf. eq. 14) (23) In this equation A@& and A*, denote the membrane potential just before and after short-circuiting, respectively. Experimental The experimental set-up consisted of two cylindrical cells (volume 50 ml) with the anion exchange membrane (F. de KBriisy, US. Pat. 3,388,080,1968) sandwiched in between. A platinum electrode (type P 1312) and a calomel electrode (type K 4112, both from Radiometer) as well as a large platinum electrode (surface 6.5 cm* ) were fitted through holes into each cell. The solutions in both compartments were continuously flushed with water-saturated N2 gas. The cells were completely closed except for a small hole which served as the gas outlet and was also used for taking small samples from the solutions. Stirrers were fixed in each compartment and were driven by a magnetic stirring motor below the cells. The temperature was kept at 21 * 1°C. Potentials were measured with a Radiometer pH meter, type PHM 26c (accuracy < + 0.5 mV), and the current running through the short-circuited platinum electrodes was monitored with a recording microammeter (Elaviscript 3). The readings of this instrument were calibrated by measuring the potential drop over a 1 kR resistor (1% precision). Before use, calomel electrodes were placed overnight in a 0.1 M KC1 solution and short-circuited. Small asymmetry potentials which occasionally occurred were corrected in the potential readings. Platinum electrodes were electrolytically cleaned in concentrated HN03. The redox pair used, i.e. Fe3’/Fe2’ in acidic solution, equilibrates with the platinum electrode within a minute and gives stable and reliable potential readings. Low pH prevents hydrolysis and slows down autoxidation. A plot of redox potential versus redox composition according to eq. 13 is shown in Fig. 1 for the data obtained from both compartments during the experiments.
29
FmV
720
Fig.1. Plot of redox potential versus redox composition compartment I ( l ) and compartment II (A ).
according
to eq.13.
Data from
The slope of both straight lines is 58.3 mV and equal to the ideal value, incheating no significant change in activity coefficients over the whole concentration range. The difference in intercepts, however, is due to different activity coefficients in the two compartments. Thus Fe3*/Fe2+ is a fully reversible redox pair with midpoint potentials of 742.4 mV and 731.9 mV in solutions I and II, respectively. 45 ml of a 0.11 M KNOB solution and an equal volume of Hz0 were pipetted into compartment II and compartment I, respectively, and 0.5 ml of a 1.03 M HN03 solution were added to each compartment. NO; was chosen as the anion since Cl- in excess of the Fe3+ concentration yields negatively charged complexes, FeC!ln3-n with IZ= 4 to 6, which would permeate through the membrane. N2 gas was bubbled through both solutions for about an hour. Then the gas inlet tubes were fixed just above the surface of the solutions and the stirrers switched on. (A stream of gas bubbles severely interfered with the potential readings, but uniform stirring was of little influence). When the potential difference between the calomel electrodes was stable, 1 to 3.5 ml of both a concentrated FeClz and FeCl, solution were added to each compartment. These solutions were freshly prepared under N2 and contained about 50 m&f of the iron salt and 10.3 mM HN03. The ratios Fe3’/Fe2’ varied, but the total volume added was always 4.5 ml, corresponding to a final concentration of total iron of about 5 mM in each solution. Potential differences were periodically measured between the calomel electrodes, the platinum electrodes and the platinum vs. calomel electrodes in each compartment. From time to time samples were taken from both compartments and analysed for Fe” and Fe3+ as follows. The samples (25 or 50 ~1) were added to 2.95 ml of a 25 mM dipyridyl solution and the optical density was read at 523 nm (the peak in the spectrum of the red Fe2’-dipyridyl complex where the brown Fe3+ complex has essentially no absorption; the extinction
30
caefficient is 8.78 r&f-’ cm-’ as determined with standards of analytical grade ferrous ammonium sulfate). Then 10 ~1 of a 1 M sodium ascorbate solution were added and the optical density was read again thus yielding the total iron concentration. Fe3+ was calculated as the difference between total iron and Fe’+. Results and discussion The time course of the four parameters: membrane potential, electrochemical potential difference of the electron across the membrane, and redox potential in both solutions are shown in Fig.2 for experiment 1. A membrane potential arises when two solutions of a salt of different concentrations are separated by an anion selective membrane. It decreased upon addition of iron salts due to the changes in anion concentration. Short-circuiting the large platinum electrodes caused an instantaneous drop in membrane potential which indicates that electron transfer contributes significantly to A*, (cf, eq. 17). The electron transfer as monitored by the short-circuit current 1s is shown in Fig. 3. The rising curve in the same figure gives the change in redox com-
80
Fig.2 Time course of the four measured parameters for experiment 1. Potential difference between calomel electrodes, A Cal (o), and platinum electrodes, Apt (A) in compartment II and I (upper part); redox potential, E1 (a) and &*I (A), in solutions I and II, respectively (lower part). The curves were calculated as described in the Appendix. The first abrupt change in A Cal was due to the addition of iron salts, the second one (with a corresponding drop in A Pt) occurred upon short-circuiting of the large platinum electrodes.
31
2
1
6
a
10
days
Fig. 3. Short-circuit current, Is, and concomitant change in redox composition for experiment 1. Is measured (heavy line) and calculated from JE (see Appendix) and eq. 21 (broken line). The rising curve indicates the number of “mols of electrons”, n,(t), transferred by Je from solution I into II during the period from short-circuiting (t = 0) up to
t
time t (=
s 0
Is dt’/F).
The changes
of Fe3+ concentration
in solution
I, &$,M
(e), and of
Fe*+ concentration in solution II, Acgez+ (A), during the same time period were calculated with eq.13, the measured redox potentials, and with the total iron concentrations for both solutions which were found to be constant within experimental error. V denotes the equal and constant volume of both solutions (see Appendix).
position expected from the time integral of the current, while the points were calculated from the redox potential measurements. The consistency of these measurements indicates that neither Fe” nor Fe3’ was permeable. By virtue of eq. 9 with Li, = 0 and 21 L, = --I,/(A&AF)
(24)
and L, thus calculated was found to be independent of ion composition in the solutions. T, was determined according to eq. 23 which together with L, then yields L (cf. eq. 16 with Lie = 0).A simplifying assumption is necessary in order to estimate the ion transport numbers. H’ and K’ as well as Nq and Cl- were treated together as one fictitious permeable cation (subscript +) and anion (subscript -), respectively. Hence from eq. 17 o _ Ap_-FA
7+---
&-G
r” =
\k” m
+ Act_
A/L++ FA’P;
(25)
AP+ + ALL
and the two transport numbers could be determined from the initial composition of the solutions (cf. eq. A2)and the membrane potential at the beginning of the experiment. Finally T, and T_ were calculated according to eq. 22.
32
With the data for ri and L (see Table 1) and from the initial composition of both solutions, the time course of all measured parameters can be calculated (cf. Appendix) and the curves shown in Fig. 2 were thus obtained. The fit between experimental data and calculated curves is satisfactory; the deviations observed with increasing time were to be expected in view of the simplifying assumptions made, and as a whole we can conclude that the system behaved as expected. TABLE 1 Transport numbers and sum of permeability coefficients
1 273
0.32 16.8
9.97 8.45
0.155 0.0026
0.015 0.0184
0.830 0.979
‘A denotes the membrane area. ‘L = T Li (i = +,-,E), the permeabilities Li can be calcu-
lated from Ti and eq. 16 with Lik = Lk = 0. For Ti see eq. 22. Two main points are emphasized by the experiment recorded in Fig. 2: (1) Equilibration for the electron (Ap, = 0) does not lead to equal redox potentials in both solutions, but A& = -A*, (cf. eq. 11 and 14); (2) The electron approaches equilibrium long before the system as a whole is equilibrated. Relaxation times are determined by kinetic parameters (in our system, the permeabilities) and capacities (which are related to the concentrations). Although the permeability to the anion was more than 5 times larger than that to the electron (cf. Table l), the much lower concentrations of Fe” and Fe3’ compared to the salt concentration in one of the solutions lead to a decay of A;;- faster than the equilibration of the whole system. The approach to and maintenance of a stationary non-equilibrium state is seen in the dissipation function. Fig. 4 gives the time course of @ determined according to eq. 7 and with the flows and forces calculated as described in the Appendix. The sharp rise of Q, upon short-circuiting is followed by a decrease to a level which then remains constant for a long time: a constant non-vanishing rate of energy dissipation is the clearest characterization of a stationary state. Strictly speaking, this flat portion of @ represents a quasi-stationary state in which the salt concentrations change very slowly (see Fig. 7A) and the redox compositions continuously adjust themselves so that the deviation from equilibrium state for the electron remains very small. Of course, after a much longer period of time the concentrations in both solutions become equal for all components and Cpvanishes, i.e. the whole system reaches equilibrium. A 60-fold smaller 7, and somewhat larger T+ and 7_ result when the same
33
days
0
I
8
12
t
Fig.4. Time course of the dissipation function, @. Solid line: experiment 1 (lower and left hand scale), broken line: experiment 2 (upper and right hand scale), A denotes the membrane area.
large platinum electrodes are used but the membrane area is increased about 50 times (experiments 2 and 3). In this case, the contribution of electron transfer to the membrane potential is too small and no drop in A\kmwas observed upon short-circuiting of the platinum electrodes (Fig. 5 and 6). HOWever, L, and TO+, r” couid still be determined according to eq. 24 and 25, respectively, while 7;f, T+ and T_(see Table 1) were inferred by means of the L, and L_ from experiment 1 which apply because the same membrane material was used. The relaxation times for electron transfer and salt diffusion now become comparable. With an initial redox composition closer to the equilibrium state for the electron than in experiment 1, the system proceeded towards this state and eventually reached it but only transiently (experiment 2, Fig. 5). No quasi-stationary state occurred as indicated by the continuously decreasing dissipation function (Fig. 4). In experiment 3 the initial A& was chosen to be of approximately the same magnitude as in experiment 2 but with opposite sign and equilibration of the electron was only achieved when the whole system reached equilibrium (Fig. 6). The last point which can be demonstrated is the conversion of osmotic into redox free energy. Though the iron concentrations in solution I were comparable to the salt concentration, eq. 18 is still a reasonable approximation and Fig, 7 then presents the dependence of electron flow, J,, and salt flow, J, = (J.+ + L)/2, as a function of the conjugated forces, i.e. the differences in chemical potential for the electron, Ape, and for the salt, AH, = A/J+ + AM(cf. eq. 20). For convenience, the straight coefficients are abbreviated by Lr E L 7, (l-7,) and L, z L [T+ + T- - (T+ - T-)‘I/4 while the coupling coefficient is L, G LT-~(T+ -G)/2 (see eq. 20 and Table 2).
34
mV
ii
4
15 UA
b
a
, 10
0 Q
0
_1
0
1
2
3 days
Fig.5. Time course of A Cal, & Pt,L dition, the short-circuit current, I,, and the broken line (calculated). A ever, A Cal and A Pt did not change
1
5
6
t
I and Err for experiment 2, plotted as in Fig.2. In adis shown in the upper part as the heavy line (measured) drop in A Cal occurred on addition of iron salts; howupon short-circuiting (t = 0) since TV<< 7_.
760 =iA
710
-2
720 7cn
Fig.6. Time course of measured parameters for experiment 3, plotted as in Fig.5. The curves were calculated for a longer time period than that of the actual experiment to show that the whole system approaches equilibrium. Note the break in the time scale.
35
A
AJJ~
-2
-L
> *w 7
0
2
B
IJhmolei 1
1.5
2
2.5
5 f)______
___
9.0
9.4
9.8 ~~~
5
7
9
(J/m mole)
Fig.7. Electron flow, JE, and salt flow, J,, as a function of the conjugated forces, A& and Aps (difference in chemical potential between solutions II and I), according to eq. 20. A: experiment 1, B: experiment 2. Heavy lines represent data calculated as described in the Appendix, with arrows indicating the chronology. Thin lines are Je= Lr~pe and J, = LsAps, respectively, for the values of L, and L, given in Table 2. Points mark experimental data derived from redox potentials (eq. 14) and from the short-circuit current (eq.21). For L, S and S’ see Fig.8.
TABLE 2 Parameters for coupling of electron flow and salt flow (eq. 20) ___ ~_ ___L,x 10TC qd L,, x 10’ b L,X 10’a Exp.
e z
1 2,3
1.31 0.022
aL, = L7e (1~~ ‘According
4.63 -0.010
0.45 0.158
), bLrs = L7e(7+-~_)/2.
-0.82 -0.17
1.70 0.37
‘L, = L[T+ + T- - (T+ -T_)’ ] /4 (see eq. 20).
to Kedem and Caplan [4] the degree of coupling, 4, is equal to LrsdL,L,.
“The “mechanistic”
stoichiometry,
.a, is equal to JL,/L,.
At the beginning of experiment 1, both flows had the same sign as the conjugated forces but were larger than expected from these forces due to coupling (Fig. 7A). The system only dissipated energy as can be seen from the plot of flow ratios versus force ratios (Fig. 8) according to Kedem and Caplan [4]. Note that the flows were negatively coupled (I&., < 0, cf. Table 2) because the electron and the more permeable ion of the salt have the same sign. At the point marked by L in Fig. 7A and 8, which corresponds to the cross-over point of redox potentials in Fig. 2, the system reached the state of
36
J, ‘J,
Fig.8. Plots of flow ratio, (JdJ& z, and efficiency of energy conversion, n = J,ApE/J&ks versus force ratio z(A~.+/ALI~),according to Kedem and Caplan [4]. Heavy lines represent data calculated as described in the Appendix and with the z values given in Table 2. Thin lines are theoretical curves [4] for 9 = -0.16, -0.69 and -0.82 (top to bottom), respectively. The apparent variation of the degree of coupling, 4, for experiment 1 and the lower value of q than that in Table 2 for experiment 2 are caused by deviations from the condition Ap+ = AL (eq. 18) which become larger the more the redox compositions in the two solutions are changed. The value and the location of the maximum of the efficiency [4] yield q = -0.75 and -0.16 for experiments 1 and 2, respectively (note that 20~ is plotted for experiment 2). L denotes the state of level flow, while S and S’ are static head conditions. Energy is converted when the flow ratio versus force ratio curve goes through the lower right quadrant but energy is only dissipated when this curve is in the lower left and upper right (not shown) quadrants [4].
level flow (AP, = 0). This state, however, cannot be maintained in a closed system, and the system proceeded into the phase where J, was driven by ApcLs against Ape, with J, being therefore smaller than L,Ap,. Besides dissipation of energy, osmotic free energy was converted into redox free energy during this phase with an efficiency 77(see Fig.g).until the static head condition J, = 0 was attained (S in Figs. 7A and 8). The system then remained for about 8 days in this strictly speaking “close to static head” state (J, Y 0, a quasistationary state) with little change of All, (see range marked in Fig.7A) before it entered the phase where both flows again had the same sign as the conjugated forces and which eventually led, solely with dissipation of energy, to the equilibration of the whole system. The initial redox composition of experiment 2 let the system start from an en$rgy converting state (Fig. 8). However, due to the small value of TV,the degree of coupling 4 (see Table 2) and hence the efficiency t) were low, while J, accordingly differed little from LsAps (Fig.7B). The static head condition (J, = 0, S’ in Figs. 7B and 8) could not be maintained for the reasons given above and the system immediately entered the phase where energy is only dissipated on the way to overall equilibration. Finally, the initial redox composition of experiment 3 caused the system to start in the latter phase and no energy converting state could thus occur.
37
Acknowledgements This investigation was begun when one of us (O.K.) was on a sabbatical year at the Biozentrum. It is a pleasure to thank Prof. F. Gri.in for his efforts which made this stay possible, as well as for his interest and for helpful discussions. We are indepted to Prof. F. de K&-&y for kindly supplying the anion exchange membrane. The technical assistance of Miss A. Kr%uchi is gratefully acknowledged. References 1 I. Prigogine, Introduction to Thermodynamics New York, London, Sydney, 1967. 2 S.R. de Groot and P. Mazur, Non-Equilibrium
of Irreversible
Processes,
Thermodynamics,
Amsterdam, 1962. 3 A.J. Stavermann, Trans. Faraday Sot. 48 (1952) 4 0. Kedem and S.R. Caplan, Trans. Faraday Sot.
176. 61 (1965)
Interscience
Publ.
North-Holland,
1897.
Appendix The following assumptions were made in addition to the postulate of only one permeable cation and anion: The flows of ions are not coupled to each other, i.e. Lik = 0 while Lie = 0 due to the experimental set up. Hence from eq, 8 and 9 Ji = LiA~i
(i = +,-,E)
(Al)
with Li being independent of concentration (as verified for L,). The activity coefficients of the permeable ions were taken to be constantnt in each solution and the differences in these coefficients between the two solutions were neglected. The difference in chemical potential of the two ions at time t then becomes A/J-i(t)= RT In ~fI(t)/~f(t)
(i = +,-)
(A21
where cl;I1 (t) d enotes the concentration of these species in solutions I and II, respectively. The changes in volume due to the ion flows, the concomitant solvent flow, and the removal of small samples for assay of iron composition were neglected, i-e. the volume, V, is at any time the same for both solutions. If the concentrations of ions and redox species are known at time t, all relevant parameters can be calculated: The redox potentials, C1pl’, are given by eq. 13 and they determine the difference in chemical potential of the electron (cf. eq. 14). Apu, and A,u+, ACL according to eq. A2 yield the membrane potential A*, (eq. 17) and by this in turn the corresponding electrochemical potential differences (eq. 10 and 11). The forces Ace, Ai;, and A?_ then determine the conjugated flows (eq. Al).
38
The flows J, , J+ and J change the concentrations of ions and redox species in both solutions. Since c = n/V for all species, we obtain, when using a finite difference approximation for derivatives, from eq.1 +j $
= 6ct=AJi
st/V
t-43)
and from eq. 3 (-44) for the changes in concentration after a small time increment 6 t. Thus, the concentrations at time t + 6 t can be determined and the calculations of relevant parameters repeated which then gives the concentrations at time t + 26 t and so on. Starting from the initial concentrations of ions in both solutions and taking into account the addition of iron salts at a later time, the time course of all measured parameters was calculated with a computer as outlined above and the results are plotted in the pertinent figures. This computer simulation also provided the data for @ (Fig. 4) and the flow-force relations shown in Figs. 7 and 8. The values used for the coefficients Li and ri are given in Table 1; for CL” see Experimental.