Traveling Waves in the Arsenite-Iodate System Irving R. Epstein Brandeis University, Waltham, MA 02254
The investigation of a variety of nonlinear dynamic phenomena such as chemical oscillations (1,2), bistability ( 3 ) , chaos ( 4 ) ,and traveling waves (5)has become one of the most I~scinntin,:;md r.tpidly grwing .lrr.ls i l l ph\si;nl d~emiatry 161. Study or the tempmal hehav~wniiord~a deeper u~idcriti~lldillpi ~ i t h ekhetics 0 1 wmplrx reactions than can I)? CAItained &om the more traditional examples usually found in textbooks. Spatial waves provide both an opportunity to observe the detailed interaction between reaction and diffusion and an introduction to the important notion of symmetry breaking (71, in which structure may appear spontaneously in an initially homogeneous medium. The reaction between arsenite and iodate in acidic solution offers an excellent pedagogic introduction to a number of exciting dynamic phenomena, particularly traveling waves. The net reaction involves two component processes which are themselves of historic and pedagogic interest, and a simple model can he solved numerically to give quantitative insight into the origin of the waves. The Component Reactions The reaction between arsenite and iodate is one of the family of Landolt reactions (8) in which a reducing agent reacts with iodate in an autocatalytic process to yield iodide or iodine. The autocatalysis was demonstrated some sixty who suggested that the years ago by Eggert and Scharnow (9), reaction involves two component processes1
Time
Time Figure 1. Schematic traces of lodine concentration as a function of time in the arsenitpiodate reaction with excess lodate (top)or excess arsenite (bottom) is significantly faster than reaction (1). Under these circumstances, the net reaction is (1) 3(2) or
+
51-
+ IOF + 3H3As03+61- + 3H3AsOd
(3)
There is net production of iodide, but the rate-determining process (1) accelerates with [I-], so the reaction is autocatalytic. With excess arsenite in a stirred solution, the brown color appears only for a brief moment before the iodine is consumed, Reaction (1) is the well known Dushman reaction (101, and we have a rather dramatic clock reaction (see Fig. 1). The which forms the basis of at least one standard laboratory exRoebuck reaction is of critical importance in the history of periment in physical chemistry (11). It proceeds rapidly once chemistry, because it provided the first experimental proof significant amounts of iodide have been produced. In a stirred solution, if iodate is in excess over arsenite ([IO;]O/[H~ASO~]~ (12) that the ratio of the forward and reverse rates in a com> 1:2.5), the reaction ends, as shown in Figure 1,with the sudden appearance of the dark brown color typical of 1 2 or The DK values for the first dissociation of arsenious and arsenic . ...-.--,13. ac~dare 9 2 and 3 6, respect.vely Slnce our so 4t ons are bufferedat If arsenite is in excess, then the iodine produced in reaction PH's s~gnlfcanlly oelow lhese ralues. me wr te ooln arsenlle and arsenate in the fully protonated forms. (1) is consumed by process (2), the Roebuck reaction, which
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Journal of Chemical Education
plex reaction is equal to the equilibrium constant given by the Law of Mass Action. Traveling Waves I t was reported some 25 years ago (13)that if the arseniteiodate reaction is carried out in a thin tuhe with appropriate concentrations and without stirring. a single wave of hrown cdoratiun may propngate rhrough thr in~ti:illylionio~enei~us sulution. .A recent studs I 141in which thr..;~waves were looked a t in greater detail suggests conditions appropriate for demonstration of the phenomenon. The reaction is carried out in test tubes about 10 cm in length and 1cm in diameter. Sulfate-bisulfate buffers of pH 1.80 (A) and 1.50 (B) are prepared, and the arsenite and iodate solutions are made up in these.2 We require four solutions in all: 0.18 M solutions of NaIOs in each buffer, a 0.05 M solution of NaAsOa in A and a 0.115 M solution of Na4sOz in B. To this last solution we also add enough soluble starch to give about a 0.5% solution. The two solutions in buffer A are mixed in a beaker in the ratio of one volume of iodate to five volumes of arsenite, stirred thoroughly, and then poured into atest tube. After a somewhat variable period of about a few minutes, a thin hrown ring is observed at the top of the solution. The ring gradually grows, and the wave proceeds down the tuhe until the entire solution is brown. If the buffer B solutions are combined in the same ratios, the ring that forms is fainter (the starch is added to make it more visible) and instead of growing, it moves down the tube maintaining its width and leaving a colorless area behind it. The two types of wave are shown in Figure 2. The different behavior of the rings in the two cases follows directly from the reaction of the stirred solutions. With excess iodate (A), the final product is 12, so as a region of the tuhe undergoes reaction, i t turns and stays brown. In the excess arsenite case (B),the I2 is converted to colorless I almost as soon as i t is formed, so only a t the edge of the wave can the color be seen. ~
~
Mathematical ( 14) While no complete explanation of spatial wave propagation has yet been developed, it appears that a crucial element is the The time required to initiate the waves is quite sensitive to pH.
couulina . . between an autocatalytic reaction and the diffusion or the autwatalytic apcuies S: I f thr re;irliun and difiuiitm rates arr in t h wprupri,i~e ~.. . reliitionship, then when S exceeds a threshold value somewhere in the solution as the result of a fluctuation, its concentration builds up there at an increasing rate, X diffuses into neighboring regions where its concentration again grows rapidly, and the wave propagates. A simple model for such a system is provided by the set of equations
where A may be identified with 103, X with I-, Y with I2 and Z with arsenite (see eqns. (1) and (2)).If reaction (5) is faster than reaction (4), the overall (4) (5), A Z X, is autocatalytic in X. Diffusion may he introduced into the ahove scheme by dividing the system into N compartments as in Figure 3. The effects of diffusion are taken into account by adding to the rate of reaction for species U(U = A, X, Y or Z) in compartment
+
+
-
i, dud'" L- Hu[(U,+l- U,) + (U,-I - U,)] dt
(6)
a term where Hu is proportional to the diffusion coefficient of species U and the fust set of terms in parentheses is omitted if i = N, while the second set is omitted for i = 1. Thus, for example the differential equation for species X in compartment 2 ( N > 3) is given by
Equation (6)represents a crude numerical approximation to the one-dimensional diffusion equation given by Fick's law. Qualitatively informative results may he obtained with as few as 4 compartments in the model. Further simplification can be gotten by fixing the values of either A or Z in the excess iodate or excess arsenite cases, respectively. In the tahle we give the results of numerically integrating the ahove model
Figure 2. Chemical waves in a testtube with excess iodate (a,b)or excess arsenite(c,d). Dark areas are brown or, in the presenceof starch indicator,blue. Pictures are taken roughly (a)9 min, (b) 12 min. (c)4 min, (d) 7 min aner mixing the solutions as described in the text. Volume 60
Number 6
June 1983
495
Times of I-(X) Peaka and M Y ) Color Appearance in a SixCornDartrnent Model of a Tube Reaction = Compartment
4-($1
thl
t,,(sl
1 2 3
221 223 226
208 232 232
225 227
229
212 235 235
T i m e at which [XI reaches a maximum. Time at which IY] reaches lo-' M, the concenvatiotion at which l2 is assumed to become visible. Assumed rate parameters in eqns. 14-61, k A = 3 X i0-2 s-', k' = 3 X 10' M' s-'. H* * M. = 12 X i0-3 .,-', t6 = 3 X 10W SF'. Times with zero ruperscrlpis are for the same calculation in the absence of dinusion. H* = M. = & = 0. initial concentraiions. [XI = 10-' M, [Z] = to-' Min all compartments. [Y] = 5 X 10.' Min compartment 1 and [Y]= 0 in all other compartments.
F g.re 3 Scnema1.c a aSlbm GI:ne h - c o m ~ a n m e n m l o d e % , s t e m A r r o r s ina!cale 0 I f s o n 1 on xtreen c o n l p a n m e n l s a n 0 :ermr t o t n r r gnt of c o m panments are Vle conv o ,t uns c i u f f ~ on s to ire r a e of c n r n g e of conccnva4on of s p e c i e s U in that compartment
with six compartments (N = 6) and an initial Y (iodine) M for the excess iodate case. Note how the fluctuation of introduction of diffusion delays the appearance of color in the first compartment and accelerates it in the others, producing a wave which travels with nearly constant velocity. The progress of the wave can also be followed by monitoring when [XI reaches a maximum in each compartment. This can he accomplished experimentally, at least in principle, by situating several iodide sensitive electrodes along the tube. Related Phenomena
The demonstrations described ahove can easily he elahorated into a full laboratory experiment. Students may wish to study the source of the wave by introducing various species-acids, iodine, iodide, arsenite-at different points in the solution. Does the wave always start at the top of the tube? If so, why? Does excluding oxygen have any effect on the waves? What is the concentration denendence of the wave velocity and the time required for initiatibn of the wave on pH, temoerature. reactant concentrations. etc.? The mathematical model also lends itself well to variation and elaboration,
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Journal of Chemical Education
Finally, it may he noted that in addition to exhibiting spatial waves, the arsenite-iodate reaction shows histahility (15, 16)and plays a major role in a recently discovered oscillating chemical reaction (17) involving chlorite, arsenite, and iodate. Acknowledgment
This work was funded by a grant (CHE-7905911) from the National Science Foundation. The demonstrations described ahove were developed hy Debra Banville, and discussions with Patrick De Kepper, Kenneth Kustin, and Kenneth Showalter provided many insights into the system. Literature Cited (1) Fie1d.R. L a n d Noyes,R. M.. Accountso/Chsm.Res.. 10.214 (1977). (2) Noyes. R. M.,andFidd, R. J.,Accounir o/Chem. R r s , LO, 213 11977). (3) Nirran. A,, Ortoleva. P.,Deutch, J.,andRoaa,J.. J. Chem. Phys., 61,1056 119741. (4) Rdssler, O.E.,and Wegmann,K.,Naiuis.271,89(1978). (6) Fie1d.R. J.. and Winfree,A. T.,J. CHEM EDUC., 56,764 (1979). (6) Pacault, A , and Vidd, C., (Editors)."SynergeLies: Far Crom Equilibrium? springer^ Verlsg, Berlin. 1979. (7) Nicolis, G. and Prigogine. I.. "SeKOrgsnizstion in Non~EquilibriumSystems? Wiiw New York, 1977. (8) Landolt, E.. Bwichtr Deutsch. Cham. Gas., 19,1317 (18%). (9) Eggert, J., and Scharnaw. B., .7 Eldtiochem., 27,465 (1921). (10) Dushman, S. J.. J. Phys. Chem.,8,4M (1904). (11) Shoemaker,D. P., Garland, C. W.,and Steinfeld. J.,"Erperiments inPhysical Chem~ istry: 3rd Ed., MeGraw~Hili;New York, 1974, pp. 273-283. (12) Roebuck, J. R., J.Phyr. Chrm., 6,365 (1902). (13) Epik,P.A.. and Shub, N. S.,Dakl Akod. NoukSSSR, 100,503 (1855). (14) G~ibschaw,T. A , Sehowslter, K., Bsnville, D. L., and Epstein, 1. R.. J . Phys. Chem., 85,2162 l19811. (15) De Kepper. P., Epswin, I. R., and Kustin. K., J. A m w Chem. Soc., 101, 6121 ,,O",,
(16) Papsin. G., Hsnns. A. and Showalter, K., J . P h y s . Chem., 85,2576. (17) De Kepper, P.. Epstein.1.R. andKurtin,K., J Amri. C h r m Sac.. 103.2133 (1981).
