J. Phys. Chem. lB83, 87,1352-1357
1352
sorption cross section has been published. We used the following formula derived by F O r ~ t e rto~ calculate ~ a radiative lifetime:
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1 / =~ 2.88
X
1 0 ~ 9 n 2 ( g l / g , , ) ~ ~ -( 2v) -~ oc (-v ) / D )dD
(8)
where iio is the wavenumber of the 0-0 transition and n is the refractive index of the surrounding media. go and g, represent the degeneracy of lower and upper states, respectively. There appeared to be some disagreement in four quantitative s t ~ d i e s ~ Jof~ the J ' ~absolute ~~ absorption cross section of NO, regarding the magnitude and shape. Marinelli, Swanson, and Johnston" showed that the integrated absorption cross section of the four studies are coincident with each other within 20% and recommended that ref 5 and 10 give the best estimate of the integrated absorption. If we assume that the excited state is degenerate (2E'),the radiative lifetime derived from the cross sections of Graham and Johnston (400-704 nm)5 is about 1.4 ps, which is shorter than our extrapolated fluorescence lifetime (2.8 ps) by a factor of 2. This fact clearly indicates that the nonradiative process including the predissociation to NO + O2 does not dominate the fate of NO3 excited at 662 nm. Magnotta and Johnston' directly measured the wavelength dependence of photoproducts (NO and 0) by resonance fluorescence coupled with the pulsed photolysis of NO3. They reported that the quantum yield for NO2 + 0 was ca. 0.6 and that for NO + O2 was ca. 0.3 at 600 nm. These yields decreased with increasing wavelength and (23) Th. Forster, "Fluoreszenz Organischer Verbindunger", Vanderhoeck and Ruprecht, GBttingen, 1951. (24) H. S. Johnston and R. A. Graham, Can. J. Chem., 52,1415 (1974).
only slight photolysis was observed at 623 nm. Further, the strong 0-0 band at 662 nm was photochemically inactive with respect to dissociation. Our result that the nonradiative process does not proceed in NO, excited at 662 nm is consistent with their observation. The contribution of dissociation processes becomes dominant at wavelengths shorter than 662 nm. However, it is difficult to measure quantitatively this process since the apparent lifetimes are affected by the appearance of NO + 0, chemiluminescence as well as the occurrence of a vibrational relaxation. We plan further investigations to obtain spectroscopic and photochemical information about NOg. For example, a high-resolution spectroscopic study of the infrared absorption at 1480 cm-' is in progress from which detailed information about the structure of the ground state NO3 will be obtained.25 Also a molecular beam experiment coupled with laser-induced fluorescence is attempting to examine the radiationless processes in the excited state under the collision-free condition. Moreover, a simplified spectrum of NO, in a cold jet would be extremely useful for the analysis of the excited state. Acknowledgment. This research was performed under a joint program with the Institute for Molecular Science. We thank Dr. J. R. McDonald for a helpful discussion and showing us his results prior to publication. We are also grateful to Drs. H. Akimoto and N. Washida for the loan of an ozonizer. Registry No. NO3, 12033-49-7. (25) T. Ishiwata, K. Kawaguchi, E. Hirota, and I. Tanaka. D, symmetry of NO3 in the ground state was indicated by infrared diode laser spectroscopy of the NO3 v3 band.
Perturbations around Steady States in a Continuous Stirred Tank Reactor K. Bar-Ell" Department of Chemistry, Tel-Aviv University, Tel A viv 69978, Israel
and W. Geiseler Institute of Technical Chemistry, Technical University of Berlin, Berlin, West Germany (Received: June 6, 1982)
Perturbations around steady states of a system of cerium or manganese bromide and bromate ions in a sulfuric acid solution, in a continuous stirred tank reactor (CSTR), are investigated. Two kinds of perturbations are discussed, namely, small-speciesand large-constraint ones. The feasibility of measuring a particular rate constant by following the system behavior after a small perturbation is analyzed. Some general theorems regarding this behavior are obtained. Large perturbations, namely, those that cause a transition of the system from one steady state to another, are analyzed and an approximate relationship between the perturbations' intensity and length is developed. The results are in good agreement with the experimental data.
Introduction open systemin a continuous~ystirred tank reactor (CSTR) can exist in more than one steady state, depending on the conditions in which it is operated. A typical such systemwhich shows this steady-state mulor manganous ion oxidation by tiplicity is the bromate ions in sulfuric acid solution. Such a system was shown by Geiseler and Follner' to exist under certain ex-
perimental conditions in two stable steady states, while under other conditions only one steady state is possible. In this paper we want to examine the dynamical behavior of a system initially in a steady state under the influence of perturbations, and to study the feasibility of such experiments in order to get a better understanding (1) W. Geiseler and H. Follner, Biophys. Chem., 6, 107 (1977).
0022-3654/83/2087-1352$01.50/00 1983 American Chemical Society
The Journal of Physical Chemistty, Vol. 87,No. 8, 1983
Perturbations around Steady State in a CSTR
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of the chemistry of the present and similar systems. The studied perturbations are of two kinds: (a) The steady-state concentrations of the species was perturbed by injection of a certain species directly into the CSTR. The concentration of the injected species will increase momentarily. These perturbations are usually small and will be denoted small-species perturbations. (b) The external constraints were perturbed by changing a concentration or the flow of the feed for a certain period of time. The perturbation may be either positive or negative. These perturbations are usually large and will be denoted large-constraints perturbations.
Experimental Section The reaction system under investigation is the stirred flow oxidation of cerous or manganeous ions by bromate in sulfuric acid solution in the presence of bromide. Bromate, bromide, and cerous or manganeous ions dissolved in sulfuric acid are fed continuously from outside containers into the reactor, the temperature of which is kept constant, and the concentration of bromide and ceric ions is monitored. The details of the procedure and measurements are given elsewhere.2 Large-constraint perturbations are performed by changing one of the containers for another one with a different concentration for a certain period of time, and thus a square pulse of different concentration is delivered to the reactor. Small-species perturbations can be delivered by injection of a small amount of a certain species, directly into the reaction vessel. The concentration of this species will momentarily increase. Kinetic Model The Noyes, Field, and Thompson3 (NFT) mechanism is made up of seven chemical reactions (reactions 1-7), Br03-
+ Br- + 2H+ e HBr02 + HOBr k1= 1 X lo4 M-’
kl = 2.1 M-3 s-l HBrOz
k2 = 2 x 109 M-2 HOBr
(1)
s-]
+ Br- + H+ e 2HOBr s-1
(2)
tz-z = 5 x 10-5 M-1
9-1
+ Br- + H+ e Brz + H 2 0
k3 = 8 X lo9 M-2 s-l
k-3
(3)
= 110 s-1
Br03- + HBrOZ + H+ e 2Br02. + HzO
k4 = 1 x lo4 M-2 s-l Ce3+ + BrOz.
~ z= - ~2 x 1 0 7 ~ s-1 4
+ H+ F! Ce4++ HBr02
k5 = 6.5 x 105 M-2 s-1
(5)
lt-5 = 2.4 x 107 M-’ s-1
+
+
Ce4+ + BrOz- + HzO F! Ce3+ Br03- 2H+ M-3 s-] k, = 9.6 M-’ s-l k4 = 1.3 X 2HBrOZ e Br03- + HOBr
k7 = 4 x IO7 M-’ s-l
(4)
k-, = 2.1
+ H+
X
(6)
(7) M-2 s-l
the rate constants of which are shown, assuming the concentration of water is constant. When the manganese system was used, Ce3+and Ce4+must be changed to Mn2+ and Mn3+,respectively. The necessary changes in the rate constants, reactions 5 and 6, are discussed below. (2) W.Geiseler and K. Bar-Eli, J. Phys. Chem., 85, 908 (1981).
(3) R.M.Noyes, R. J. Field, and R. C. Thompson, J.Am. Chem. SOC., 93,7315 (1971).
1353
Equation 8 describes the dynamics of the system. Fi(C) dCi/dt = Fi(C) + ko(C,i - Ci)
(8)
describes the mass action chemistry (eq 1-7) and kO(Coi - Ci) describes the inflow and outflow of concentrations
COiand Ci, respectively. Equation 8 was solved by Newton’s method for a steady state, i.e., for C = 0. At a certain region of constraints only one steady state exists, while at other regions three steady states exist. Two of them, SSI and SSII, will be stable and the third, SSIII, will be unstable and will not be found experimentally. The steady-state concentrations of the various species at typical constraints are given in the supplementary material (see paragraph at end of text regarding supplementary material). Geiseler and Bar-Eli2 have measured and calculated the bistability domains in this system for many constraints. Geiseler4has also measured the bistability domains for a similar system in which the cerium ions are replaced by manganese ions. Small-Species Perturbations To a system which is present a t a steady state a small amount of a certain species is injected very rapdily, thus increasing its concentration abruptly. The dynamics of the system immediately after the injection involves only the reactions in which the perturbed species participates. It is possible, in principle, that a simple kinetics will evolve, which will depend on a few (preferably one) rate constants only. The feasibility of such experiments will be investigated in this section. Using the equations in Appendix A (supplementary material) and noting that only one species was injected, Le., all AC’s are zero, except one, ACj, we obtain ACi=($-
ko6ij
)
ACj
(9)
Note that no summation is done, since only one AC is different from zero. By examining the chemical equations, we notice that aFi/aCj is negative for i = j and positive, negative, or zero, for i # j. This means that, immediately after the perturbation, the concentration of the injected species will start falling, while the concentrations of all the other species will either rise, fall, or remain unchanged. The first-order rate constants calculated from the approximate eq 9, namely, the term in parentheses, is compared with the pseudo-first-order rate computed from the exact solution of eq 8 by the Gear’s a l g ~ r i t h m . ~ For this equation to be useful, four conditions must be fulfilled: (1)The flow term in the Jacobian matrix -ko must be small compared to the chemical term aFi/aCj. (2) AC should be small enough so that the simplified linear eq 9 will be a good approximation to the full eq 8. (3) The measurement time must be short enough so that no other species except the injected one will develop, and the simple eq 9 will transform to the more complicated equation given in the Appendix. (4) An accurate measurement of the species concentration should be available. By examining a few examples, it will be realized that no useful chemical information can be retrieved for this (4) (a) W.Geiseler, J. Phys. Chem., 86, 4394 (1982); (b) W.Geiseler in “Non Linear Phenomena in Chemical Dynamics”, C. Vidal and A. Pacault, Ed., Springer-Verlag, Berlin, 1981, p 261. (5) (a) C. W. Gear, “Numerical Initial Value Problems in Ordinary Differential Equations”, Prentice Hall, Engelwood Cliffs, NJ, 1971, pp 209-29. (b) A. C. Hindmarsh, Gear: ‘Ordinary Differential Equations System Solver”, VCID 2001, Rev. 3, Dec 1974.
1354
The Journal of Physical Chemistty, Vol. 87, No. 8, 1983
particular system unless fast mixing and measurement techniques are used together with the small perturbation. The examples are calculated with ceric ion being the measured species. The latter is most easily and accurately measured optically due to its high ( t = 5.44 X lo3 M-' cm-l at 317 nm) extinction coefficient. Bromide ions6and BrO,. radicals7 can also be measured giving, however, the same results obtained with ceric ions. Ce4+Perturbations. The appropriate term of the Jacobian matrix is r3[Ce4+]/d[Ce4+] = -k-,[HBr02] - k6[Br02-]- ko Using values of HBrO, and Br02 at SSI (supplementary material) one obtains -((2.4 X 107)(3.226X 10-l2) + 9.6(1.971 X lo-',) + 4 X 10-3) = -4.08 x 10-3 s-l and for SSII -((2.4 X 107)(1.975X 9.6(1.604 X 4X 10-3) = -4.78 x 10-1s-1 The main contribution to the return to SSI will be, therefore, the flow rate with very little chemistry involved. M will make any The small value of [Ce4+IssI= 9.2 X measurement of this species in this region an impossibility even if we could have learned some chemistry from it. The contribution to the initial return to SSII comes mainly from reaction -5 while reaction 6 and the flow contribute very little. In order to examine the validity of the approximation and its time length, we carried out a complete solution of the rate equations and compared the calculated first-order rates, Le., A[Ce4+]/A[Ce4+],to the approximated one. The calculation was done for a 10% increase of [Ce4+]sSII,i.e., from 3.695 X to 4.067 X M, while all other constraints were as in Table A1 (supplementary material). The results reveal that the approximated equation is good for a period of about 1.5 X s. The rate falls by a factor of 20 in a few milliseconds and in a few minutes, the main factor contributing to the return to the steady state will be the flow rate and not the relevant chemistry. A smaller change, e.g., by 1%, gives roughly the same results. A twofold perturbation is already too large and brings the system from SSII to SSI. The reason for this discrepancy is fairly clear. Immediately after the injection of ceric ions, their concentration starts to fall a t the expected rate via reaction -5; this pushes reaction 5 toward the left, decreasing the bromous acid concentration and causes further changes in other species concentrations. The simple eq 9, which depends on one concentration change only, does not hold any more, although the linear approximation, eq A12, still does. However, solving (A12) is not easier than solving the complete set, and no simple chemical information can thus be retrieved. Ce3+Perturbation. The relevant terms of the Jacobian matrix are r3[Ce4+]/d[Ce3+] = k5[Br02.][H+] + k-,[Br03-] [H+] which for SSI are equal to (6.5 X 10b)(1.971X 10-12)1.5+ (1.3 X 10-4)(1.998 X 10-3)1.5 = 2.31 X s-' and for SSII (6.5 x 105)(1.604x 10-7)1.5 + (1.3 X 10-4)(1.986 X 10-3)1.5 = 0.156 s-l
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+
+
-
(6) N. Ganapathisubramanian and R. M. Noyes, J. Phys. Chem., 86, 3217 (1982). (7)H.D.Forsterling, H. Labbey, and H. Schreiber, 2. Nuturforsch. A, 35, 329 (1980).
Bar-Eli and Geiseler
Again only perturbations at SSII might give some relevant information. Immediately after injection of cerous ions, the concentration of ceric ions start to rise. The rate of ceric ion increase deviates from the expected value by a factor of 10 even after 10 ms. The rate of increase will decrease after some time, since the ceric ions concentration must eventually return to its original value. The maximum concentration will occur after about 100 s. Bromous Acid Perturbation. Within 10 ms after the injection, about 90% of the extra bromous acid disappears. The effect on the concentration of ceric ions will therefore be minimal. Indeed bromous acid participates in many reactions, thus its effect on a particular rate of interest is therefore insignificant. The results of the above computations and of many others of similar nature point to the same conclusion. Unfortunately in the system under consideration, the initial return to the steady state after a small perturbation is too fast to be measured by usual experimental procedures. A t times in which measurements can be done, the flow rate will be, in general, the main contributor toward the return to the steady state. Thus no information regarding a particular rate constant can be deduced from these measurements. In Appendix A of the supplementary material, the problem is analyzed again from a more general point of view. The main results obtained in the Appendix will be summarized here. (1) There are only N = 9 species in the CSTR, since water is taken as R' = 7 reactions but only R = 5 independent ones, i.e., two reactions depend on the other five. (2) There are N - R = 9 - 5 = 4 relationships among the species, thus the number of independent species is only 9 - 4 = 5. Three of these relationships can be taken as conservation of bromine atoms, cerium ions, and charge. The choice of the four relationships is somewhat arbitrary, provided that they are independent of each other and orthogonal to the columns of the stoichiometric matrix. (3) There are N = 9 negative eigenvalues for SSI and SSII, while SSIII has one positive eigenvalue. Out of the N = 9 eigenvalues, N - R = 9 - 5 = 4 will be equal to -Ao. Accidental degeneracy is possible. These eigenvalues will, in general, be the smallest ones and thus the final approach to the steady state will be a t this rate, i.e., the flow rate. This agrees with the particular calculations in the above examples. Very near the bistability limits, an eigenvalue approaches zero and will be smaller than the flow. In these cases the final approach to the steady state will be slowed. This was found experimentally by Heinrichs and Schneider8 in studies of RNA replication in a CSTR. (4) The eigenvectors associated with the degenerate eigenvalues X = -k, are true eigenvectors and not pseudoeigenvector~.~ (5) The initial behavior of the system after the perturbation will have contributions from all eigenvectors in time scales which depend on the eigenvalues. Due t o the large, absolute magnitude of some of the eigenvalues, the contribution of the associated eigenvectors will die away very fast, in about 1 s. It is thus obvious why all the above calculations deviated very quickly from the simple eq 9. (6) The eigenvectors associated with the eigenvalues A # -ko will be a linear combination of the columns of the stoichiometric matrix. As it turns out the vector associated (8) (a) M. J. Heinrichs and F. W. Schneider, Ber. Bunsenges. Phys. Chem., 84,417 (1980);(b) ibid.,84,857 (1980);(c) J . Phys. Chem., 85,
2112 (1981). (9)M. C.Pease, "Methods of Matrix Algebra", Academic Press, New York, 1965,p 77 ff.
Perturbations around Steady State in a CSTR
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,
o
-
I
0
200 400 600 000
The Journal of Physical Chemistry, Vol. 87, No. 8, 1983 1355
1
tis)
I000
t (sec)
Figure 1. Bromide concentration vs. time at pulses of 20-150 (the system reverts to SSII)and 20-160 s. (The system goes over to SSI.) Constant constraints: [Br03-], = 2 X lo3 M, [Ce3+], = 1.5 x M, [H+], = 1.5 M, k , = 4 X s-’, [Br-1, = 1 X M, [Br-Ip =6X M.
with the largest eigenvalue, i.e., the fastest process, is parallel to reaction 3, Le., bromine hydrolysis, and its reverse will be the first to die out. The second largest eigenvalue, of SSII, is associated with an eigenvector which is nearly parallel to reaction 5, i.e., the oxidation of cerous ions. These reactions, when perturbed, return to their original state which is very near equilibrium in times too short to be measured by the present techniques of CSTR. In general, however, the method is not devoid of merit. A system with slower reactions, but considerably faster than the flow rate, may be amenable to measurement by the present methods. So that the present system can be studied, fast techniques should be developed. These fast techniques call, however, for injection, mixing, and measuring the components in time scales of less than 10 ps which is not feasible a t the present.
Large-Constraint Perturbations These are affected by letting the concentration of a certain species in the flow deviate, either upward or downward, from the usual flow for a certain period of time. A square pulse of concentration C , for a period of time will occur. If the pulse is small, it is reasonable that the point in the phase space will remain near the original one and, as the pulse ends, the system will return to the original steady state. If the pulse is large, however, the movement of the point in the multidimensional phase space, as dictated by eq 8, can be such as to leave the attraction basin of the original steady state and go over to the attraction basin of another steady state, where the system will be finally located. There is no mathematical proof that such may not be the case. However, in all the examined cases the system returns to the original steady state if the pulse is smaller than the bistability limit. In other words, the same limits are obtained whether one moves slowly along the steady-state line, or whether one suddenly changes the constraints of the system. If the pulse is greater than the hysteresis limit, the system will go over to another steady state only if the pulse
Figure 2. Measured potential of a bromide-sensitive electrode vs. time for various pulse lengths: (1) 35 s, (2)45 s, (3)55 s,(4)58 s, (5)200 s. Constant constraints as in Figure 1. [Br-1, = 10 X M. The system is originally at SSII.
.
16
I
I
I
I
I
I
4
1
I
I I
0
I 400
1
I 800
1
I
1200
t p(sec)
Flgure 3. Concentration of bromide ion at the pulse ([Br-I,] vs. the pulse duration (t,) that causes transitions from SSII to SSI. Constant contstraints as in Figure 2: line, calculated; circles, measured.
lasts long enough. If the pulse is rather short, the system will go back to its original steady state. In Figure 1 a typical calculation is shown. If a pulse of bromide ion 130 s long is given, the bromide ion concentration rises and then falls back to its original SSII value. If the pulse is longer, e.g., 140 s, the bromide ion concentration continues to rise, even after the termination of the pulse and finally stabilizes a t much higher value corresponding to SSI. A very similar behavior is seen experimentally in Figure 2, in which a series of pulses of the same concentration but with different durations are shown. For the short pulses the system ends a t the same steady state as it starts, i.e., SSII, while for the longer ones it ends a t SSI. It should be noted that the concentration of Br- beyond which the system goes over to SSI is =1 X lo-@M, which is very near the bromide ion concentration of SSIII. The points near SSIII thus serve as a “watershed” between the attraction basins of SSI and SSII. In figure 3 a plot of bromide ion concentration in the pulse -[Br-1, vs. t, (i.e., the duration of the shortest pulse that causes transition from SSII to SSI) is shown for the
The Journal of Physical Chemistry, Vol. 87,No. 8, 1983
1356
r
TABLE I : Pulse Duration ( f p , s ) for Transition from SSII to SSI 10s [Br-].,M 15.0 12.0
10.0 8.0 7.5 6.0 5.0 4.5 3.25 4.0 3.95 3.9 3.85 3.0 2.6 2.25
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1.95
Ce system
5 01
exptl
calcdC
exptl I
34.5 37.5 47.5 72.5
136.5 225 325 435 675
56 78 143.5 222 33 2.5
-625
,’/
I 20
Mn system
...____I___.
calcdb
Bar-Eli and Geiseler
19.5
17.5
28
24
51
43
74
52
0~
95-
/
,
c,?
975 2275
’ 135.5 221.5
318 593 74 2
107.5 172.5 222 -480 c,?
Constant constraints as in Figure 1. For the manganese system [ M n 2 + I o= 1 . 5 x M. Calculated hysteresis limit for cerium system [Br-]R = 3 . 8 4 8 5 5 x l o T 5M. k , = 1.8 x l o 5M-Zs-’ instead of 6 . 5 x l o 5 M-2 s-’. Hysteresis limit [ B ~ - ] R= 1 . 8 7 2 3 x lo-’ M. a
cerium system. Table I shows calculated and experimental results for both cerium and manganese systems. In both systems, t, increases very sharply as [Br-1, decreases. There is a fairly good agreement between the experimental and calculated data. It is seen that the lower asymptote is the same as the hysteresis limit -[B~-]Robtained by Geiseler and Bar-Elia2The equality between the hysteresis limit, obtained by moving slowly along the steady-state line, and the minimal pulse needed to achieve transition is by no means trivial. The pulses needed to invoke transitions are much shorter in the manganese system. The only difference between the two systems can be in either reaction 5 or 6 . Reaction 6 was shown earlier2J0 to be of rather limited importance. We are left, then, with either k6 or k-, or both to be changed. Indeed, decreasing k , to 1.8 X lo5 M-2 s-l gives a fairly good agreement with the experimental pulses, as can be seen in Table I. A similar agreement could be achieved by increasing k-5 by roughly the same proportion. We can therefore deduce that the ratio k5/k-, is approximately 3.6 times smaller in the manganese system than in the cerium system. In other words, the free energy of oxidation of manganeous to manganic ions is about 0.77 kcal/mole higher than that of the oxidation of cerous to ceric ions. This is lower than the value of 1.6 kcal/mol obtained from the difference of the oxidation potentials.” The measured upper hysteresis limits (Le., the disappearance of SSII) obtained with the manganese system were shown by Geiseler4 to be at lower [Br-lo values, in complete agreement with the results obtained here, and can likewise be explained by either lowering k5 or increasing k-5 as described above. The dynamics of approach to the new steady state, dictated by eq 8, is complicated. The point in the phase space moves until it reaches the “watershed” between the attraction basins of two stable steady states. The pulse length is, therefore, the time needed to reach the (10)S. Barkin, M. Bixon, R. M. Noyes, and K. Bar-Eli, Int. J.Chem. Kinet., 11, 841 (1977). (11) W . A. Latimer, ‘Oxidation Potentials”, 2nd ed, Prentice Hall, New York, 1952.
i
i log (Cp-Co)/(cp-
CR)
Flgure 4. k,t, vs. log [(C,- Co)/(C, - CR)] for the cerium system. Transitions from SsII to SSI: k, = 1 x 1 0 - ~ s-l ([Br-IR = 5.464 x M), k, = 4 X s-‘ ([Br-IR = 3.847 X M), k o = 10 X S-‘ ( [ k - ] ~= 2.944 X lo-‘ M), [BrO3-I0 = 10 X M ([Br-1, = 11.3515 X M); solid line, calculated bromide pulse; 0 , experimental points. Transitions from SSI to SSII: pointed line, negative bromic% ion pulses (supplementary material data);dashed line, positive bromate ion pulses (supplementary material data). When not indicated, constant constraints are as in Figure 1.
“watershed”, which is located near SSIII, i.e., cross the separating surface between the basins of attractions of the two steady states. In spite of this complexity it can be shown (Appendix B of supplementary material) that there is a rather simple approximate relationship between t, and C,, i.e., between the pulse intensity. This relationship is given by
where ACR = CR- Co is the minimum change in concentration needed to bring about the transition. CR is found, in all tested cases, to be the concentration at the hysteresis limit. A similar equation was derived by Blair12to describe the relationship between the intensity and time length of an electric pulse stimulating a living tissue. This simplified equation cannot be completely accurate since it does not include the detailed kinetics of approaching the vicinity of SSIII as described above. Figure 4 shows the calculated and experimental results of various pulses under a variety of constraints as given in the legend. The data are plotted on a log-log scale in order to provide for a span of three orders of magnitude in the coordinates. All the plots are fairly near each other and the slope is not too far from unity indicating, at a first glance, that eq 10 is approximately correct. The bromate pulses which cause a transition from SSI to SSII obey eq 10 exactly, with 0.8ko instead of ko on the left-hand side. The bromide pulses which cause transitions from SSII to SSI are approximated by a similar equation in which the left-hand side is 1.55k0 and the right-hand side is raised to the power of 1.2. The data fall on the same line regardless of the value of other constraints such as [BrO,-], or k,,. The experimental points fall also on the same plot. This is not exactly the Blair equation but is not too far from one. Similar results are also obtained for the data of the manganese system given in Table I. The Blair equation can thus be used to obtain a rough estimate of the pulse length-pulse intensity relationship.
J. Phys. Chem. 1983, 87,1357-1361
1357
The biological systems discussed by Blair12" and by RashevskylZbare a far cry from the chemical system discussed here. However, both can exist in two possible states and can go from one state to the other by a strong enough perturbation. The fact that the perturbation time-in-
tensity relationship in such different systems is so similar may indicate a deeper connection between seemingly unrelated phenomena. Registry No. Ce, 7440-45-1; Mn, 7439-96-5; bromate, 1554145-4.
(12)(a) H.A. Blair, J.Gen. Physiol., 15,709(1932);(b) N. Rashevsky, "Mathematical Biophysics", The University of Chicago Press, Chicago, 1948,p 279. (13)P. H.Richter, I. Procaccia, and J. Ross in "Advances in Chemical Physics", Vol. XLIII, I. Prigogine and S. A. Rice, Ed., Wiley, New York, 1980,p 217 ff.
Supplementary Material Available: Appendix A concerns the problem of small perturbations to a steady state in an open system and Appendix B shows why the Blair equation gives a rough approximation to the pulse intensity-pulse time relationship (15 pages). Ordering information is available on any current masthead page.
Kinetics of Formation and Dissociation of the Cryptates Ag(2,2,2)+ and K(2,2,2)+ in Water Mixtures at 25 OC Acetonitrile Downloaded by UNIV OF GLASGOW on September 12, 2015 | http://pubs.acs.org Publication Date: April 1, 1983 | doi: 10.1021/j100231a017
+
B. G. Cox, Department of Chemistry, Universtty of Stirling, Stir/ing, FK9 4LA Scotland
C. Gumlnski, University of Warsaw, Institute of Fundamental Problems of Chemistry, Warsaw, Poland
P. Firman, and H. Schnelder' Max-Planck-Institut fur biophysikaiische Chemie, 0-3400 GGttingen, West Germany (Received: September I , 1982)
The dissociation rates of Ag(2,2,2)+and K(2,2,2)+in the acetonitrile (AN) + water system show a quite different dependence upon solvent mole fraction. The dissociation rate constant, kd, of Ag(2,2,2)+is almost independent of solvent composition and the rapid decrease of the stability constant, K,, near 3cm = 0 is determined completely by the variation in the formation rate constant, k p The constant value for k d for Ag(2,2,2)+in the mixtures indicates that in the transition state the silver ion is strongly bonded to the (2,2,2) nitrogen atoms in a manner typical of the partially covalent interaction of monovalent dO ' ions with nitrogen donors (e.g., in nitrilic solvents). This result, and the very similar variation of the Gibbs free energies of the transition state and stable cryptate complex with solvent composition, suggests that the transition state is very close to that of the products. The result is striking because for alkali-metal cryptates, particularly in nonaqueous solvents, a very similar solvent dependence is found for the reactants and transition state. The interaction between K+ and (2,2,2) is also found to be very different in this binary solvent system from that of Ag(2,2,2)+. Both k d and kf contribute similarly to the increase of the stability constant of K(2,2,2)+with increasing mole fraction of acetonitrile, and comparisons of the Gibbs free energies of reactants, transition state, and product do not indicate any simple correlations between the solvation behavior of the three over the whole range of solvent composition.
Introduction Most macrocyclic ligands are characterized by their ability to distinguish between metal ions and to form the most stable inclusion complex within a series of related cations, such as the alkali or alkaline-earth metal ions, with the cation that fits optimally into the cavity of the ligand.'-' Rate measurements have shown that the variation (1)J. J. Christensen, D. J. Eatough, and R. M. Izatt, Chem. Rev., 74, 351 (1974). (2)J. M.Lehn, Stmct. Bonding (Berlin), 16,l(1973);Acc. Chem. Res., 11, 49 (1978);Pure Appl. Chem., 50, 871 (1978). (3)B. G. Cox, J. Garcia-Rosas,and H. Schneider, J.Am. Chem. SOC., 103,1384 (1981). (4)W. Burgermeister and R. Winkler-Oswaitsch, Top. Curr. Chem., 69,91 (1977). (5) B. G. Cox, J. Garcia-Rosas, and H. Schneider, J. Am. Chem. Soc., 103,1054 (1981).
in stability of the complexes is almost entirely reflected in the dissociation rates, with the most stable complex having the lower dissociation rate. The change in formation rate with ionic radius is generally small.4 This behavior has been demonstrated most clearly for cryptands? i.e., macrobicyclic diazapolyethers, first synthesized by Lehn et a1.: which form exceptionally stable complexes with metal iom2 For these ligands it has been shown that changes in solvent also normally influence the dissociation rates much more strongly than the formation rates.5 Recently complex formation between the cryptand (2,2,2)-N( (CH,CH,0)2CH,CH2J3N-and K+ and Ag+ has (6)J. M.Lehn and J. P. Sauvage, Tetrahedron Lett., 2885, 2889 (1969). (7) B. G.Cox, C. Guminski, and H. Schneider,J.Am. Chem. Soc., 104, 3789 (1982).
0022-3654/83/2087-1357$01.50/00 1983 American Chemical Society
