J. Phys. Chem. 1981, 85,3461-3468
t r o n i and, ~ ~ ~in suitable cases, vibrational excitation43are important additional applications. The 1-ms time scale associated with sampling radicals in the present experiments is not a serious limitation. By generating reactive (42) Jonathan, N.; Morris, A.; ROSS,K. J.; Smith, D. J. J. Chem. P h p . 1971,54,4954. Jonathan, N.; Smith, D. J.; Ross, K. J. Ibid. 1970,53,3758. (43) Dyke, J.; Jonathan, N.; Morris, A,; Sears, T. J. Chem. SOC.,Faraday Trans. 2 1976, 72, 597.
3481
intermediates directly in the 21-eV beam, it should be possible to examine species with lifetimes in the range of 10-9-104 s.
Acknowledgment. This research has been supported in by a grant from the Department of Energy, Grant No. EX-76-G-03-1305- F.A*H*thanks IBM for a (1977-1978).
Mixing and Relative Stabilities of Pumped Stationary States K. Bar-Eli* Chemistry Department, Tei-Aviv University, Tel-Aviv, Israel
and W. Geiseler Institute of Technical Chemistry, Technical Unlversity of Berlin, West Berlin, West Germany (Received: April 7, 198 1; In Final Form: June 22, 1981)
Two coupled continuously stirred tank reactors (CSTR) containing cerium bromate and bromide ions in sulfuric acid medium were examined for oscillations and transitions between one steady state and another. The CSTRs were coupled in two different configurations,namely, in parallel and in series. No oscillations were found under all tested contraints and configurations. The transitions between one steady state and the other follow quite accurately the used Noyes-Field-Thompson (NFT) mechanism. The particular steady state to which the system will transfer depends on the particular set of constraints. A dividing line following a “cusp” catastrophe is observed between these two possible transitions. Examining the free-energy difference between the two states shows that, because one state is always at a higher free energy than the other, the free-energy difference cannot serve as a criterion for stability.
Introduction An open chemical system in a continuously stirred tank reactor (CSTR) can exist in more than one steady state. A system that shows such behavior is the cerous oxidation by bromate in a sulfuric acid solution. Such a system was shown by Geiseler and Follnerl to exist under certain experimental conditions in two stable steady states, while under other conditions only one steady state is possible. Bar-Eli and no ye^,^,^ using a detailed chemical mechanism4p5coupled to the input and output flow, modeled these results numerically. Recently Geiseler and Bar-Eli6 calculated and measured the hysteresis limits of this system and obtained accuracy limits for some of the rate constants. When two cells, each containing a different stable steady state, are mixed, by pumping material from one cell to the other and vice versa, new situations are in principle possible. Smale,7following an idea of Turing,s claimed that (1) W. Geiseler and H. Follner, Biophys. Chem., 6, 107 (1977). (2) K. Bar-Eli and R. M. Noyes, J. Phys. Chem., 81, 1988 (1977). (3) K. Bar-Eli and R. M. Noyes, J. Phys. Chem., 82, 1352 (1978). (4) R. M. Noyes, R. J. Field, and R. C. Thompson, J. Am. Chem. SOC., 93, 7315 (1971). (5) S. Barkin, M. Bixon, R. M. Noyes, and K. Bar-Eli, Int. J. Chem. Kinet., 11, 841 (1977). (6) W. Gieseler and K. Bar-Eli, J. Phys. Chen., 85, 908 (1981). (7) S. Smale in “Lectures in Mathematics in Life Sciences”, Vol. 6, J. D. Cowan, Ed., American Mathematics Society, Providence, RI, 1974, p 17. (8) A. Turing, Philos. Trans. R. SOC.London, Ser. B , 237, 37 (1952). (9) R. M. Noyes, J. Chem. Phys., 71, 5144 (1979). (10)R. M. Noyes, J. Chem. Phys., 72, 3454 (1980). 0022-3654/81/2085-3461$01.25/0
such a coupled system of two cells may be driven into oscillations. The possibility of oscillations produced by coupling of two cells, neither of which is oscillatory by itself, was discussed also by other authors.12J3 Tpon13examines two cells, each containing an “~regonator”’~ in a nonoscillatory steady-state situation. Coupling the cells through a membrane permeable to bromous acid only can produce oscillations. Such a selective membrane is rather difficult to realize experimentally, whereas the mixing of material from one cell to the other is a relatively easy procedure. Another possibility is that the two cells will, a t a certain critical coupling, go over to the same steady state. If this possibility prevails, an idea as to the relative stability of the two steady states can be obtained by their measurement~.~JO Experimental Section The system contains two cells each of which is a continuously stirred tank reactor (CSTR) into which bromate, bromide, and cerous ions in sulfuric acid solution are pumped, while the reaction products are pumped out. (11) (a) R. J. Field, Koros, and R. M. Noyes, J. Am. Chem. SOC., 94, 8649 (1972); (b) W. M. Latimer, “Oxidation Potentials”, 2nd ed., Prentice Hall, New York, 1952. (12) V. I. Krinskii, A. M. Pertsov, and A. N. Reshetilov, Biophysics . . (Engl. Trans!.), 17, 282 (1972). (13) J. J. Tyson, Ann. N.Y. Acad. Sci., 316, 279 (1979). (14) (a) R. J. Field and R. M. Noyes, J. Chem. Ph-ys.,60, 1877 (1974): (b) R. M. Noyes, Symp. Faraday soc., 9, 21 (1974); (15) T. Poston and I. N. Stewart, “Taylor Expansions and Catastrophes”, Pitman Publishing, London, 1976.
0 1981 American Chemical Society
3462
,
The Journal of Physical Chemistry, Vol. 85, No. 23, 1981
Bar-Eli and Geiseler
Br03- + Br-
j“p
+ 2H+ F! HBrOz + HOBr
kl = 2.1 M-3 s-l
kl= 1 X lo4 M-I s-l HBrOz + Br- + H+ e 2HOBr
kz = 2 X lo9 MT2 HOBr
lko
iko SERIES
(1)
k-z = 5
Br03- + HBrOz
Calculations and Results The mechanism of cerous oxidation by bromate ions in sulfuric acid is made up of the following seven reversible reactions: 4-5
k-3 = 110 s-l (4)
lo7 M-l 8-l
k4= 2 X
+
Ce3++ BrOz. + H+ e Ce4+ HBrOz
PARALLEL
Each cell can be operated in such a way as to arrive at and remain in a particular steady state, depending on the initial conditions and the external constraints, as described in detail earlierS6 The external constraints are the input concentrations, namely, [Br0;l0, [Br-Io, [Ce3+lo,[H+lo,and the flow rate (reciprocal of residence time), i.e., ko. The cells are now coupled by transferring material from one cell to the other at a desired rate, with the aid of a peristaltic pump. At each coupling rate, a typical concentration was measured in both cells. In most cases it was bromide ion, with the aid of a bromide-selective electrode by Orion, Inc. In some cases a Pt electrode was used (giving a rough measure of [Ce4+]/[Ce3+]), while in others [Ce4+]was measured directly by the optical absorbance at 317.5 nm. The molar extinction coefficient of ceric ions was measured to be 5.8 X lo3 M-l cm-’. After a few residence times, the system is established at a new state. These new steady states differ slightly from the previous ones. The coupling rate was slightly increased and the experiment repeated. At a certain coupling rate, depending on the various constraints, the two cells go over abruptly to one of the two original (i-e.,without coupling) steady states (Appendix A). During the time that the materials spend in the connecting tubes, their concentrations change slightly. Therefore, the concentrations that enter one cell are not exactly equal to those that leave the other cell. In order to measure this effect, we measured the critical coupling rate with various tube lengths. The same results were obtained regardless of the tube lengths. This conclusion was also verified by calculations which will be described in Appendix B. Two coupling configurations were tested: (a) Cells in parallel; i.e., each cell has an independent inflow and outflow stream, and in addition, there is a flow of material from one to another. This configuration was tested with varibus constraints and with ceric ions flowing into one cell while cerous ions were flowing into the other. (b) Cells in series; i.e., the outflow of the first (denoted “upper”) cell serves as an inflow to the second cell (denoted “lower”) cell. In addition a second flow of material from the upper to the lower cell and vice versa serves as the coupling. Figure 1shows schematicallythese two configurations. In all experiments and calculations a search for oscillations and examinations of the transitions between the steady states were conducted.
(3)
+ H+ F! 2Br02-+ HzO
k4 = 1 X lo4 M-2 s-l
Flgure 1. Schematic drawing of the two coupling configurations: (left) series configuration; (right) parallel configuration.
X
+ Br- + H+ e Br2 + HzO
k3 = 8 X lo9 M-2 s-l
Po
(2)
M-’ s-l
k5 = 6.5 X lo6 M-’
8-l
k-6
= 2.4 X lo7 M-l
Ce4++ BrOz. + HzO e Ce3+ + Br03+ k6
= 9.6 M-l
8-l
k+ = 1.3 X k7.= 2.1
+ 2H+
(6)
M-3 S-’
2HBrOZF! BrOC + HOBr
k7 = 4 X lo7 M-’ s-l
(5) 8-l
+ H+
X
(7)
10-lo M-2 s-l
The various reaction rate constants were deduced by Noyes et al.4and confirmed and slightly revised by Barkin et ala5and recently by Geiseler and Bar-EliS6 To these chemical reactions, terms describing the flow rates and coupling rates were added. Turning our attention first to the parallel configuration, we obtain the following rate equations: dCi,/dt R(Ci,) + ko(Ci0 - Cia) + k,(Cjg - Cia) (8)
dCig/dt = R(CiB) + ko(Ci0 - Cia) + k,(Ci,
- Cia)
(9)
where Cia and C, are the concentrations of the various species i in cells (Y and 0,respectively, Cio is the input concentration of species i, KO is the reciprocal of residence time ko = V / VR where V R is the reaction cell volume and V is the flow rate, and R(Ci,) and R(Cig) describe the chemical mass action reaction rates derived from the above mechanism. k, is the rate of coupling between the cells. When k, equals zero, both eq 8 and 9 become identical and are the same as the ones for a single CSTR cell. Three steady-state (SS) solutions, ci = 0 with k, = 0 for the differential equation 8 or 9, are possible: SSI, SSII, and SSIII. The first two are stable while the third is unstable. As t OD, the system will go into either SSI or SSII, while SSIII, being unstable, cannot, of course, be attained experimentally. The d e s i g n a t i ~ n ~of*the ~ l ~steady states is as follows. SSI is the steady state in which relatively high bromide and low ceric ion concentrations prevail. SSII is the steady state in which relatively low bromide and high ceric concentrations prevail. The concentrations of the various species in the unstable SSIII are roughly the geometric average of those of the two stable states. A typical plot of the experimental and calculated steady states is shown in Figure 2.6 The system, being at a certain steady state, can go over to the other one by changing slowly [Br-lo and traversing a hysteresis loop. The approach to equilibrium, and the details of the hysteresis limits as a function of the various constraints, are described in detail e l s e ~ h e r e . l - ~Introducing ?~ the coupling, i.e., putting k, # 0 in the equations, will cause variations in the steady-state concentrations. When k, is small, Ciawill be near CjI and Cigwill be near CiII,the deviations from the unconnected steady states will increase, and then both cells will jump to either SSI or SSII, Le., Cia = C, = Cqcn, (see Appendix A). The particular steady state to which the transfer will occur depends on the set of constraints
-
376
Stabilities of Pumped Stationary States
The Journal of Physical Chemlstty, Vol. 85, No. 23, 1981 3463
r n -
-5 -6
-
-7
-
z-8
-
-
transition to
m
Y
-
/
-9-
-I
-10 -
-1'
' ' a
-8
-7
-6
-5
-4
lg b - 1 , Figure 2. Plot of log [Br-]= vs. log [Br-1, in a single cell without coupling, kx = 0 (solid Ilne) calculated stable steady states SSI and SSII; (dashed line) calculated unstable steady state SSIII; (circles) experimentally measured steady states. Other constraints: k o = 4 X s-', [H'], = 1.5 M, [Ce3+l0= 1.5 X lo4 M, [BrO3-Io = 2 x 10-3 M.
used. Typical plots of steady-state concentrations are shown in Figure 3. Experimental points are also shown. Both experiments and calculations were done by slowly increasing the coupling rate k, from zero to the transition point and beyond. At each rate the computed or measured bromide concentration is plotted (see also Appendix A). With increasing k, the concentration of bromide ion at SSa,[Br-I,, decreases, while a very small increase is obtained for [Br-],. For the critical k,, both cells (in case of plots B and C) go over to SSII, with cell CY having a large fall, and cell P a small one, exactly as predicted by the calculation. In both Figures 2 and 3, the concentrations of bromide ion at the steady states differ quite appreciably from the calculated ones, although the agreement with the hysteresis limit is quite satisfactory. The experimental concentrations of bromide ion at the SS, [Br-lss, were calculated from the measured voltage with the use of the Nernst equation by an appropriate calibration curve. This procedure has a few drawbacks: (a) The calibration is done without the presence of bromate and cerium ions. (b) Concentrations below the square root of the solubility product of silver bromide (7.7 X M2)are unreliable. The values given are those obtained from the Nernst equation16without regard to the above-mentioned facts. However, the exact values of the bromide ion concentration at the SS, [Br-Iss, are unimportant in obtaining the hysteresis limits or the critical k, of transformation to the uncoupled steady states, since in these cases only the "jump" from one state to the other is relevant. Our results, therefore, depend only on the knowledge of the flow rate and inflow concentrations and not on the actual steadystate concentrations of Br- ion in the solution, which are difficult to measure. Indeed, the agreement between the measured and calculated limits is remarkably good. Moreover, the same limits, within the experimental error, (16)J. H.Woodson and H.A. Liebhafsky, Anal. Chern., 41, 1894 (1969).
[Br-lo k x , calcd 1 x 10-6 2.8 x 10-4 2 x 10-6 4 x 10'6 3.03 x 10-3 1 x 10-5 2 x 10-5 3.16 x 10-5 1.45 X SSI 3.3 x 10-5 1.08 x 10-2 3.5 x 10-5 5.5 x 10-3 3.6 x 10-5 3.8 x 10-5 3.2 x 10-3 " Other constraints: [ B r 0 3 - ] , = 2 x 1.5 x M, ["I, = 1.5 M, k , = 4 X
k x , exptl 1.8x 10-4 3.6 x 8.8 x 10-4 2.1 x 10-3 4.0 x 10-3
SSII
'L
CII
TABLE I: Change in k, vs. [Br-1, for Parallel Configuration"
9.2 x 10-3
M, [Ce3+],= T = 25 "C.
S-',
TABLE 11: Change in k, vs. [ BrO,-], for Parallel Configuration" transition to [Br03-], k x , calcd SSI 5 x 10-4 1.15 x 10-3 9.4 x 10-3 6X 6.5 X lo-* 1.15 X 10'' SSII 7 x 10-4 9.9 x 10-3 1 x 10-3 5.9 x 10-3 1.4 x 10-3 2 x 10-3 3.03 x 10-3 3 x 10-3 2.25 x 10-3 4 x 10-3 5 x 10-3 8 x 10-3 1x
k x . exDtl
3.6 x 3.1 x 2.7 x 2.1 x 1.7
1.47 x 10-3 6.4 x 10-4 7.2 x 10-5
10-3 10-3 10-3 10-3
x 10-3
8.0 x 10-4
" Other constraints:
1.5 x
[Br-1, = 1 X lO-'M, [Ce3+],= M, [H'], = 1.5 M, k , = 4 X s-', T = 25 "C.
TABLE 111: Change ink, vs. [Ce3+],for Parallel Configuration" transition to ice3+], k x , calcd SSI 3.0 x 10-5 9.5 x 10-3 SSII 3.5 x 10-5 8.5 x 10-3 4 x 10-5 6.5 x 10-3 5 x 10-5 5.5 x 10-3 1.5 x IO-^ 3.03 x 10-3 3.75 x 1.5 x 10-3 5 x 10-3 1.5 x 1 0 - 2
" Other constraints:
1X
lo-'
8.45 x 10-4 2.7 x 3.95 x 10-5
[ B r 0 3 - ] ,= 2 X M, [H+], = 1.5 M, k , = 4 X
k x , exptl
3.8 x 2.1 x 1.2 x 5.5 x 1.9 x
10-3
10-3 10-3 10-4
M, [Br-1, = s - ' , T = 25 "C.
are obtained with the use of a Pt electrode (the potential of which is roughly proportional to log [Ce4+]/[Ce3+]), giving further evidence that both the experimental procedure and the mechanism are essentially correct. When ceric ion concentration is measured, very good agreement is obtained between the calculated and measured concentrations (see below). The general behavior of the experimental and computed curves agrees very well: SSa goes down while SSP goes slowly up, i.e., ICi, - C,,l < lCor - CiIIl. At the critical coupling rate both experimental and computed curves of both SSa and SSP show the drop to SSII. This agreement up to small details lends further support to our model. Under no circumstances, neither experimental nor computational, did the system in either coupling configuration show oscillations. Our system does not, therefore, follow Smale's' predictions. As the constraints of the system are changed, similar behavior is seen, but the critical coupling rate will, however,
3464
The Journal of Physical Chernistty, Vol. 85, No. 23, 1981
Bar-Eli and Gelseler I
IO-&
I
I
I
1
i IO- 3
10-4
10-2
kX
Flgure 5. [BrO,-] vs. k,: (line) calculated; (circles) experimental. Constant constraints: [Br-1, = 1 X M, [Ce3+], = 1.5 X lo4 M, [H'], = 1.5 M, ko = 4 X lov3S-'.
I0-
5
15
IO kX(1O3)
Flgure 3. Plots of [Br],, in cells CY and 0as a function of the coupling rate, k,. The computed and measured values of [Br-1, were obtained by slowly increasing the coupllng rate from zero to the marked value. At k, = 0, SSCY= SSI and SSP = SSII. Parallel configuration: (A, solld lines) [BrO3-Io = 6 X lo-' M, transfer to SSI (lower abscissa); (B, dashed lines) [Br03-], = 2 X M, transfer to SSII (upper abscissa); (C, dash-dot lines) [Br03-] = 1.4 X M, experlmental points, transfer to SSII (upper abscissa). Constant constraints In all plots, [Br-1, = 1 X lod M, [Ce3+], = 1.5 X 10" M, [H'], = 1.5 M, and k , = 4 X s-'. I
I
I ss Io-
I
I
10-4
10-3
10-2
kX
I
I
Flgure 6. [Ce3+], vs. k,: (line) calculated; (circles) experimental. Constant constraints: [BrO,-] = 2 X M, [Br-1, = 1 X M, [H'], = 1.5 M, ko = 4 X lo4 s-'. The right-hand edge in each figure marks the transition point between the system jumping to SSI or to SSII. See text for more details. Figures 4-6 are for a system in a parallel conflguration.
fluctuations are, however, absent in the completely deterministic equations of the model. The transition to SSI is more difficult to achieve experimentally than the one to SSII since the graph is flatter in the region of transition to this state. Thus, slight experimental fluctuations can bring the system to the point where transfer to SSII instead of SSI will occur. Even the one case in which it was attained (see the uppermost point of Figure 4) is doubtful, since the two steady states are too near each other, so that it is difficult to tell experimentally to which steady state the transition takes place. Each of the Figures 4-6 is clearly divided into two regions: the one in which the transition at the critical coupling is to SSI and the other where the transition is the SSII. The behavior of the system very near this dividing line is shown in Figure 7, which is in fact an enlargment of Figure 4 near the largest coupling point. It is clearly seen that a "cusp" catastrophe15 is obtained. In other words the [Br-1, vs. k, plot is made up of two separate portions in each of which d[Br-],/dk, = 0 near the dividing point, which happens to be [Br-1, = 3.263 X 10" M under the particular constant constraints used. At bromide concentrations larger than 3.263 X 10" M, the system will, at high enough coupling rates, end at SSI. At
Stabilities of Pumped Stationary States
I
I
The Journal of Physical Chemistry, Vol. 85,No. 23, 1981 3465
I
I
I
I
I
I
I
I
I 14
I
1 1.6
I
I
I
I
I
I
I
1
1
’1 3.1
3.01
IO
I
12
1.8
2 0
k,(10‘1
Flgure 7. [Br-1, vs. k, in parallel configuration near the critical point, Le., the translion between SSI to SSII. Above [Br], = 3.263 X M the system will go over to SSI, while below this value it will go to SSII. Constant constraints: [BrO3-Io = 2 X M, [Ce3+Io= 1.5 X s-’. M, [H’], = 1.5 M, ko = 4 X
lower concentrations, increasing 12, will bring both cells to SSII. Unfortunately, this behavior cannot be tested experimentally, because of the rather small region of parameters that it covers. The “cusp” behavior is, however, clearly seen in the other figures, too. It is important to realize that Figures 4-7 were calculated and measured at constant [Br-1, and slowly varying k, (see also Appendix A). Completely different behavior is expected if k, is kept constant and [Br-], is varied. If both cells are at the same SS, e.g., SSII, i.e., at low values of [Br-I,, then increasing [Br-1, will cause both of them to M, regardless of k,. transfer to SSI at [Br-1, = 3.84 X This transfer will occur at the upper hysteresis limit of the single-cell cases6 A similar situation will occur when both cells are at SSI and [Br-1, is lowered. We have neither calculated nor measured the case where the cells are at different SS at k, # 0 and [BT], is changed until both cells go over to the same SS. This transfer will not necessarily occur at the lines shown in Figures 4-7. After one such transition, both cells will be at the same SS and further hysteresis cycles will be the same as for the single CSTR. We can look at this from a slightly different point of view. Let us define the system by two indexes giving the steady states of each cell. Thus at k, = 0 the system can be in one of the three states: (I,I), (II,II), and (1,II). At small values of k,, the system can be at (I,I), (II,II), and (a,@).The system being at one of the two former states will remain so at any value of k,. The system at the state (a,@) will go to either (1,I) or (I1,II) at the appropriate k,, depending on whether [Br-1, is above or below 3.263 X
M. When the system is at (1,I) or (I1,II) and [Br-1, is changed, a transfer to the other state will occur at the appropriate hysteresis limit. This hysteresis limit will be the same regardless of the value of k,, since both cells are at the same concentrations. When the system is at (a,@)at some k, value and [Br-1, is changed, the system will end at either (1,I) or (11,II);the transition points are, however, not necessarily those given in Figures 4 and 7. We have not measured or calculated this situation. Further cycles will, however, be the same as for the single cell. Parallel Configurationwith One Cell Having Ceric Instead of Cerous Ions. In the previous system cerous ions at concentration [Ce3+Jowere introduced into the two cells. We have also looked into the situation where ceric ions at concentration [Ce4+Iowere introduced into one (left-hand) cell, while cerous ions were introduced into the other (right-hand) cell. In these experiments two situations may arise; namely, the right-hand cell can be in either SSI or SSII when the coupling rate, i.e., k,, is 0. The left-hand cell will always be at a state near SSII since ceric ions are
I ”
-4
-3
- 2 ” -5
19 k x
I
I
-3
-4
I
-2
19 k x
Flgure 8. Change of ceric ion concentration vs. coupling rate, [Ce4+] vs. k,, in parallel configuration. Constant constraints: [Br03-] = 2 X lov3M, [Br-1, = 1 X M, [H’], = 1.51 M, k, = 4 X los s-l. Right-hand cell: [Ce3+], = 1.5 X lo-‘ M. Left-hand cell: [Ce4+l0 = 1.5 X lo4 M. Right plots: At k, = 0 right-hand cell is at SSII. Left plots: At k, = 0 right-hand cell is at SSI. a denotes left cell; p denotes right cell. Line: calculated. Points: experimental.
introduced from the start and bromate ions are used mainly to oxidize bromide ions, the concentration of which at the steady state will be 10% of the [Br-1, in the input. Both possibilities were investigated. Again, no oscillations were obtained under all tested constraints. These experiments were done by measuring ceric ion concentration directly and [Br-] by a bromide-sensitive electrode. In Figure 8 typical calculated and measured [Ce4+]values are shown for various coupling rates. The measured ceric ion concentrations match perfectly the calculated data. Since the method of measurement of the ceric ions is reliable (optical density), the agreement with the computed values gives further assurance that both the model and the particular rate constants used are correct. On the right-hand side of the figure, the right-hand cell is initially (i.e., at k, = 0) at SSII. The cerium in the left-hand cell will be nearly totally oxidized, its concentration being very near [Ce4+Io,i.e., 1.5 X M. As k, increases, the [Ce4+Issvalues of both cells approach each other. The [Ce4+]ssof the left-hand cell decreases (upper curve), while that of the right-hand cell increases (lower curve). A sharp transition to lower values of [Ce4+Issis obtained at k, = 4.3 X s-l while the measured value s-l. On further is slightly higher, namely, k, = 5.75 X increase of k,, the concentrations of both cells will continue to approach each other. On the left-hand side of the figure, the right-hand cell is initially at SSI, while the left-hand cell has its cerium nearly totally oxidized as before. Again, as k, is increased, the [Ce4+]ssvalues of both cells approach each other; that of the right-hand cell increases while that of the left-hand cell decreases. There is no sharp transition as observed on the right-hand side of the figure. Similar behavior is observed when bromide ion concentration is measured. When the right-hand cell is initially at SSII, a sharp transition occurs at the abovementioned, k,, while a smooth approach of the concentrations of both cells occurs when the right-hand cell is at SSI. At coupling rates above the transition, the concentrations of the various species at the right-hand cells as well as the left-hand cells in the two cases are equal regardless of whether the cell was originally at SSI or at SSII. The left-hand cells and the right-hand cells will, of course, differ since they differ in the initial inflow concentration. As k, values continue to increase, the concentrations of both cells will continue to approach each other. Free Energy. In closed systems the free energy is the criterion for the stability of the system. A system will tend
-
3466
The Journal of Physical Chemistry, Vol. 85, No. 23, 7981
-2
Bar-Eli and Geiseier
Y
IO-
23.0
r
5-
0-
I
IO-^ Flgure 9, The difference in free energy between two uncoupled cells, each at a different steady state vs. [Br],. Insert: detail of the re ion of maximum AG. Constant constraints: [BrO,-], = 2 X I O - M, [Ces+]o= 1.5 X I O 4 M, [H'], = 1.5 M, ko = 4 X I O 3 s-', k, = 0. Arrow shows the transition point between SSII and SSI (compare Figure 7).
!
to minimize its free energy. It is interesting to find out whether the free energy can serve as a stability indicator in open systems, too. To this end we have calculated the difference in free energy between the two parallel cells: AG = G(SSa)- G(SSO),where G = C#PCi RTCi In Ci is the free energy of formation of 1 L of solution. Standard free energies of the various species were obtained from literature measurements and estimates.ll The following conclusions are obtained: (a) AG is always positive (or zero in the range where only one SS exists); in other words the free energy of SSa (derived from SSI) is always greater than that of SSP (derived from SSII). This is clearly seen in Figure 9, in which AG is plotted vs. [Br-lo for k, = 0, Le., for the uncoupled case. Similar figures are obtained for k, different from zero. In all cases the free energy of SSa will be greater than that of SSP. Obviously, below [Br-Io = 4.25 X lo-' M or above [Br-Io = 3.84 X 10" M, AG = 0 since only one steady state exists in these regions. Between these limits AG is positive, increasing and passing through a maximum at [Br-1, = 3.49 X 10" M. The point [Br-lo= 3.263 X lo" M (marked with an arrow in the insert) is the transition point. Below this concentration the system will eventually end at SSI at high coupling rates. We note that the maximum of AG and the cusp point (arrow in Figure 9) do not coincide. (b) As the coupling rate k, increases, the concentrations of the various species approach each other, and this results in a smooth decrease of AG. The latter will, of course, drop abruptly to zero at the critical coupling rate, since the two cells become essentially one. I t is interesting to note that SSII is the one that evolves into the equilibrium state as the flow rate is reduced to zero, i.e., KO = 0, in which case a closed system is obtained. Although AG is always positive, we cannot use it as a criterion for stability for a particular steady state. The steady state toward which the coupled system will evolve depends, as we see from the figures, on the particular constraints used. At certain values of constraints, the system will transfer to SSI, while at others it will transfer to SSII. The arrow in Figure 9 and the cusps in Figures 4-7 clearly show this
+
IO-^
IO-^
IO-^
IO-^
1Br-I,
Figure I O . log [&-Iss vs. [Br-lo in a series configuration. Constant constraints: [BO,-1, = 2 X IO-, M, [Ce3+Io= 1.5 X I O 4 M, [H+Io = 1.5 M, ko = 4 X lo-, s-', k, = 0, Le., cascade of two CSTRs. Hysteresis limits for different k, are shown in Table IV.
dividing point. In other words, depending on the constraints, one or the other steady states will become dominant at a sufficiently high coupling rate. The fact that the arrow and the maximum of the graph of Figure 9 do not coincide tells us that even the derivative of AG cannot be used as a stability indicator. If the maximum of AG did coincide with the arrow in Figure 9, then positive values of dAG/d[Br-l0 would mean that the state (1,111 must end at (1,I)at high enough k,. A negative value for dAG/d[Br-], would mean that (1,II) will end at (11,II). The fact that the maximum of AG and the arrow do not coincide proves that neither AG nor its derivative can be used as a stability criterion. Series Configuration. This configuration is equivalent to a cascade of two reactors when the coupling, Le., k,, is 0. In other words the lower (1) cell (Figure 1) will receive material not at concentration Cio,as the upper (u) cell does, but at concentration Ciu,i.e., the outflow of the upper cell. Introducing the coupling k, has the effect of introducing a certain feedback, and thus we thought that this configuration might produce oscillations where the other-the parallel one-did not. As it turned out, no oscillations were obtained in this configuration, either. The relevant equations will therefore be dCiu/dt = R(Ciu) + ko(Ci0 - Ciu) + k,(Cil- Ciu)
(10)
dCil/dt = R(Ci1) + ko(Ci, - Cil) + k,(Ci, - Cil) (11) The main difference between the parallel configuration and the present one is the term koCiu in eq 11. Since the equations are not symmetrical, it is obvious that the cells can never attain the same concentration in the steady state; however, as k, increases, the concentrations of both cells will approach each other. When k, 0, the upper cell can be in either SSI or SSII and go from one to the other via a hysteresis cycle exactly as if it were alone (compare the upper portion of Figure 10 and Figure 2). The lower portion of Figure 10 shows that the lower cell follows a similar hysteresis cycle shifted to the right, i.e., to a higher [Br-1, concentration. The shift toward higher [Br-lo is explained by the fact that the input to the lower cell contains bromide ion a t
The Journal of Physical Chemistry, Vol. 85, No. 23, 198 1 3467
Stabilities of Pumped Stationary States
T A B L E IV: Hysterisis L i m i t s of Two Cells in a Series Configuration As a Function of kx and [Br-1,
transition to SSII
transition to SSI calcd 3.84 x 10-5 9.75 x 1 0 - ~ 3.84 x 10-5 9.25 x 1 0 - ~
umer cell G&er cell upper cell lower cell upper cell lower cell upper cell lower cell upper cell lower cell upper cell lower cell upper cell lower cell upper cell lower cell upper cell lower cell upper cell lower cell 22
exptl 3.5 x 10-5 4.25 x 10-4
calcd 4.25 x 10-7 1.73 X 6.15 x 10-7 1.75 X
exp tl 2.75 X l o - ’ 2.5 X 1.63 X
3.94 x 7.72 x 4.15 x 5.58 x 4.15 x 4.15 x
4 x 10-5 4.2 X 4.25 x 10-5 2.75 x 10-4
10-5 10-5 10-5 10-5 10-5 10-5
1.47 X 1.47 X 1.25 X 1.25 X 1 0 - 6 1.2 x 10-6 1.2 x
1.90 x 10-4 4 x 10-5 9.3 x 10-7 9.3 x 10-7
4.23 x 10-5 4.23 x 10-5
2.25 X 2.25 X 2.15 X 2.15 X
2.5 X 2.5 X
lo-“
kX
0 0
3.6 X 3.6 x 10-4 10-3 10-3 3 x 10-3 3 x 10-3 5 x 10-3 5 x 10-3 5.2 x 10-3 5.2 x 10-3 6.7 x 10-3 6.7 x 10-3 8X
lo-’
8 x 10-3 8.8 X
1.3 x 10-4 I
I
I
I
I
I
20 -
-
18-
SSI
16 -
f(ciil
\
-
e
-
-
I4 e
E
-
; ; 122
-
10-
i!
6
X
0
-
7SSll 0
2
4
6
Figure 12. Schematic plot of the reaction rate (chemical and flow) vs. concentration (Appendix A).
I 8
1 0 1 2
14
k, (sec-l)(103)
Flgure 11. [Br-Io vs. kxin a series configuration. Constant constraints: [Br03-] = 2 X M, [Ce3+], = 1.5 X lo-‘ M, [H+l0 = 1.5 M, ko =4X s-l. Upper cell starts at SSI, and lower cell at SSII. k, changes slowly until the transition. Line: calculated. Points: experimental.
lower concentration than [Br-1,; in order to achieve the necessary transition, a still higher [Br-1, has to be supplied to the upper cell. As k, is introduced, one obtains similar hysteresis cycles for each coupling rate, the limits of which are shown in Table IV. It is seen that the hysteresis limits of the two cells, although different a t zero and low coupling rates, approach each other and finally coincide at high coupling rates. A very good agreement between calculated and experimental results is obtained for the transition to SSII in both the upper and lower cells, and for the transition to SSI in the upper cell, for all values of 112,. The transition to SSI in the lower cell occurs, however, a t values of [Br-1, about 4 times those calculated. We note that, even at very high coupling rates, two steady states will exist between certain values of [Br-1,. In fact, we expect that at infinitely
t (sec)
Figure 13. Ceric ion concentration vs. the “time in the tube”. The initial concentrations entering the tube are those of SSI correspondin to the following constraints: [BrOC], = 2 X M, [Br-1, = 10M, [Ce3+], = 1.5 X lo4 M, [H’], = 1.5 M, k, = 4 X s-’. Note the sharp transition to near equilibrium and the slow approach to it. Compare with the results of Barkin et aL5 (Appendix B).
Q
high coupling rates the two cells will behave as if they were one, and thus can show bistability and hysteresis limits as for a single cell. Transitions were also measured and calculated for constant [Br-],, when the upper cell is at SSI and the lower one at SSII, while k, increases slowly. Figure 11, which corresponds to Figures 4 and 7 of the parallel configuration, shows the results of such measurements and calculations. Again, a very good agreement is obtained for the transition to SSII, while the transitions to SSI are off by a factor of 4. As in the case of the parallel configuration, the stability of the system depends on the constraints: a t some con-
3468
The Journal of Physical Chemistty, Vol. 85, No. 23, 1981
straints SSI will eventually dominate, while at others SSII will be finally approached. The transitions shown in this figure and the transitions given in Table IV are not the same. The latter are the hysteresis limits obtained at constant k, and varying [Br-],, while the former are obtained by slowly varying k,, keeping [Br-lo constant. Acknowledgment. The fruitful discussions with Professor M. Bixon are kindly appreciated. The authors are indebted to Mrs. H. Schenke for her technical assistance in conducting part of the experiments.
Appendix A It is desirable to show that, as k, increases, one arrives at a critical 12, in which both cells jump to one and the same steady state, either SSI or SSII, of the uncoupled system. Let us write eq 8 and 9 for parallel configuration in the following form: Cia
= f ( c i a ) + kx(Ci, - Cia)
(AI)
where f contains the chemical reaction term R and the inflow and outflow term ko. When k, = 0, the only possible solutions to the equations Cia = Ci = f(Ci) = 0 are Ci = Cg or Ci = CiIP A third Ci = CiIIIsofution is also possible, but it is not stable and therefore not physically attainable. A schematic plot of f(Ci) vs. Ci is shown in Figure 12. Note that Ci is, of course, multidimensional, and therefore the plotted line is in fact a surface. The function f crosses zero at three points marked I, 11, and 111, correspondingto SSI, SSII and SSIII, and, being a continuous function of Ci, must have the described shape. The two lines equidistant from the f = 0 line at a distance of k,(C, = c,) cross the f line at the marked points, which are possible solutions to eq A1 and A2, corresponding to a particular coupling rate k,. Points C and C’, being near 111,have also at least one positive eigenvalue and are therefore unstable. Points A’ and B correspond to the physically meaningless negative k,. The only two physically possible solutions are, therefore, points A and B’ at the concentrations Cia and Cia,respectively. As the coupling k, increases, the lines will eventually reach either point D or D’ beyond which no two different Cia or Cia values are possible as solutions to the equations. The only solutions will then be when both cells are at CiI or at CiII to which the system will suddenly transfer. This conclusion is seen to be verified by both computation and experiment, as seen in Figure 3. Note that this jump can occur before point D (or D’), at, e.g., point E (or E’), if an eigenvalue becomes positive at this point. The particular steady state (SSI or SSII) to which the system evolves after passing the critical coupling depends, of course, on the rate equations, i.e., on the differential equations and on the initial conditions. The final state may be different depending on whether k, is increased slowly or abruptly. This point was discussed recently by N o y e ~ . ~For J ~ example, if we start with k, = 0 and initial concentrations CiI and CiII and increase k, slowly, the system will move to the nearby concentration Cia and Ci, until it reaches the critical k, at point E or (E’), from which both cells will transfer to either SSI or SSII. However, one can start from CiI and CiIIas initial concentrations and increase k, suddenly to a large value corresponding, e.g., to point E (or E’). Since the initial concentrations will now be different from those of the
Bar-Eli and Geiseler
previous method, the final SS need not be the same, as indeed happens. In order to avoid confusion, we always conducted both experiments and calculations in the previous “slow” method. Similar arguments hold for a system in a series configuration.
Appendix B In the problem of the two coupled cells, the term k,(Ci, - Cia) was introduced into the rate equations. I t means that cell /3 obtains material from cell a at the rate k,Cia and that materials from this cell leave it at the rate k,CW Similar terms exist for the other cells a. However, an implicit assumption is that this transfer occurs instantaneously. Since the connecting tube is not infinitesimally short, this transfer takes some time, during which materials of concentrations Cia (and Cia) obey the equation Cia = R(Cia)without the terms of KO and k,, with similar equations for C,. Under these conditions the reactants in both tubes will approach equilibrium, which is fairly near SSII. The transition to the state near equilibrium will be abrupt (Figure 13) although the final approach is very slow, as is known from the work of Barkin et al.5 and is seen in the figure. This sharp transition occurs at t = 1428 s for the case [BrO
