A Simple Demonstration of Convective Effects on Reaction-Diffusion Systems: A Burning Cigarette John A. Pojman Brandeis University. Waltham, MA 02254 There is ereat interest in noneauilihrium chemical svstems (1.4)cecause of the wide range of spatial and temporal behavior they exhibit. A characteristic of many of these systems is that they contain autocatalytic reactions in which a reactant is produced that speeds up the rate of its own production. Exothermic reactions can he autocatalytic if sufficient heat is produced to accelerate the rate of reaction. In well-stirred reactors such systems exhibit histability and sometimes oscillatory behavior. When allowed to proceed in unstirred reactors, then chemical reaction fronts may develop. The simplest example to consider is that of burning paper: The system consists of a high-free-energy material (paper) and oxygen. The system is not a t equilihrium, which would instead be carbon dioxide, water, and heat. However, the confieuration is metastable in that a "push" is necessary to get tKe reaction started, after which i t will proceed toward eouilihrium. This can be accomplished with a match, which p;o\.i(lessuffirient energy to heyin thereaction.The hurning is itseli exothermic so that it produces heat, u,hich diffuses by heat conduction to adjacent and unreacted paper, stimulating that material toreact. This process proceeds in achain reaction, and a reaction front is observed moving along the material. The rate at which this front propagates is a function of the rate of chemical reaction (burnine) and the ease by which the heat is conducted through the &per. Directlv analoeous hehavior can be observed in autocatalytic solution reactions. If a reaction mixture is in a metastable state and can react toward equilihrium by producing products that catalyze the reaction (autocatalysis), then chemical wa\.efront may occur. Simple systems have been found in which a rearriun produces hydrogen ions IH'I and the reaction is acid catalyzed. Here, the H- plays the same role as the heat in our hurning paper example. The reaction spreads l~ccausethe H7 diffuses, stimulating neighboring regions to react. Diffusion reolaces heat conduction. Like the paper-and-oxygen system, the solution requires initiation with the autocatalytic species. Such reactions are described in one dimension by the so-called reaction diffusion equation, which applies to any unstirred reaction (5).
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The equation says that the rate of change of the concentration of the ith chemical species a t position x is affected by two factors. The first term describes the change from diffusion in which D;(cm2/s) is the diffusion coefficient, which is multiplied timds'the rate of change of the gradient of the concentration at the ~osition.(This second derivative with respect to position iscalled the one-dimensional Laplacian and is a measure of how sharply the concentration changes with position.) The second term indicates that there are chemical reactions that also are occurring and affect the concentration of each species. These reactions may be very complex. The same mathematical formalism is used to describe the 792
Journal of Chemical Education
tem~eraturein burnine DaDer. . Comdex reactions produce heai, raising the temperature, which-affects the ratk of further heat production. The transport term is similar except the diffusion coefficient is replaced with a,the thermal diffusivity (cm2/s). Reaction terms are multiplied by the heat capacity of the material to relate the amount of heat produced to the corresponding temperature. This equation is a simplification because the heat loss through radiation is
Systems with acidic frontsarereadily observed u,hcn a pH indicator isadded. Numerous svsrems uf this kind have been found (5). Other systems with-autocatalytic reactions have also been studied, including the Beleusov-Zbabotinskii reaction, which is an oscillating reaction that also exhibits wave formation in unstirred reactors ( I ) . I t differs from the other wave-suooortine reactions in that it does not make a simple transit& f r o g its initial state to equilibrium as the front passes. Instead, the system is called excitable because the wavefront consists of a region in which the state is not at equilihrium hut is still very different from its initial state. After a period of time, the system returns to its initial condition (actually, to a state slightly closer to equilihrium) and can support another wavefront after a refractory period of time. This fascinating system can be prepared from a recipe . developed by winfree (6). As if this self-organizing behavior were not remarkable enough, these systems can exhibit tremendous sensitivity to external fields, including gravity. Systems a t equilibrium are normallv immune to the effect of eravitv. Given the extremely small masses of molecules, we would not expect that gravity would play a role in the rate at which the chemical wavefronts propagate. Yet, this is not the case. One of the first systems that demonstrated this sensitivity is the autocatalytic oxidation of iron(I1) to iron(II1) by HNOa (7). The reaction consists of iron(I1) sulfate in nitric M hydrazine (which removes trace amounts of acid with HN02).A drop of nitrite added to the solution in aglass tube will initiate a wavefront consisting of a ring of FeN02+complex. This reaction is also used for the brown ring test of qualitative analysis (8). The reaction-diffusion wavefront has a velocity of 0.7 mmlmin. However, if the tube is held vertical and the wave allowed to propagate downward, the velocity increases to as much as 60 mmlmin! Moreover, the effect of eravitv is not the same on an ascending wavefront whose vefocity is between 6 and 10 times less than the downward wavefront but still greater than that from reaction and diffusion alone. What causes this dramatic increase of the wavefront velocity? During the course of the reaction at a wavefront, the density can change because of an increase in temperature as well as because of the change in chemical composition. This change in density can initiate the macroscopic (large scale) transport of fluid. This macroscopic movement of a fluid due ~
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to density differences is called "free or natural convection" (9).Bowers and Soltzherg have described a system for ohserving convection in a chemical system (10). Convection is a much more efficient process for transoorting heat and matter and is largely thidriving force hehind weather. That convection is a more effective mode of material transport one only needs to consider what would happen if smoke in a fireplace were removed solely by diffusion. Quickly the room would fill with smoke. Instead, the smoke goes up the chimney because the exothermic combustion reactions in the fire produce heat, whichdecreases the density of the gases, allowing them to "float" up the flue through the force of huoyancy. We understand this idea of huoyancy intuitively when we say that "heat rises". Other reactions have been studied in which the oooosite .. behavior is observed; namely, the ascending wavefronta proceed more rapidlv than the descendine ones. Such is the case in the reaction of chlorite and thiosuifate in basic solution, which produces an acidic wavefront when initiated with a drop of acid (11). This reaction is extremely exothermic (AH0 = -1,077 KJfrnol), and this large amount of heat decreases the density of the solution causing upward convective flow (12). But is the ironfnitric acid reaction endothermic? No, in fact i t is also exothermic hut only to a lesser degree (AH0 = -36 KJImol) ~. (11). . . There is a comnetine . " effect uwon the density from an isothermal contraction of the reaction mixture caused by the change in chemical composition. Alternately, the sum of the partial molal volumes of the products is greater than the wartial molal volumes of the reactants (1:). Simply, a solution of the products is less dense than a solution of the reactants a t the same temperature. To explain quantitatively the fast descending wavefront of the ironlnitric acid system requires an understanding of a complicated form of convection "donhle-diffusive" or multicomponent convection (14). This process is important in ocean layers that differ in temperature and concentration of salt. Nonetheless, the forces a t work are the same ones that cause smoke to rise. The chlorite-thiosulfate reaction exhihits a similar isothermal contraction. If the rate at which heat is dissipated from the system (e.g., by surrounding the tuhe in which the reaction is oerformed with water instead of air), the descending waves proceed more rapidly than the ascending ones. This graphicallv demonstrates the extreme sensitivity of reactions pkrformkd in open systems far from equilibrium. A third oossihilitv occurs in the iodate-arsenous acid svstern, whose wavefionts have been elegantly studied -by Showalter and co-workers (15). In this system, the density decreases both from the exothermicity of the reaction and from the change in composition (16). Both the ascending and horizontally propagating waves always have greater velocities than the descending ones. Yet, the waves propagating in a horizontal tube may proceed as rapidly as the ascending ones. Convection is possible in a horizontal configuration as we11 (17). The role of convection is further demonstrated by considering the effect of the tube diameter. The more narrow a tuhe, the greater the frictional force upon a flow, and the greater the density difference required to initiate convection. In fact, a threshold is present for tube diameter, below which no convection can ensue for a given densitv eradient (18). For sufficiently narrow tubes no difference hitween the ascendine and descendine wavefront velocities is observed. ~oreove;, the anisotrop; in velocities decreases with decreasing tuhe diameter. Convection can also he eliminated by performing the reaction in a tube filled with silicagel (11). The gel prevents any macroscopic movement of fluid such that mass transport can only occur via diffusion as described in eq 1. ~
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While these are exciting phenomena to study, two complications limit their use in an undergraduate laboratory: The ironfnitric acid reaction is difficult to run, requiring extremely pure nitric acid. The chloritelthiosulfate reaction is potentially explosive due to its extreme exothermicity, and the iodatefarsenous acid system involves toxic compounds. Nonetheless, a simple reaction-diffusion system exists that exhibits sensitivitv to orientation with resaect to the force of gravity-a burning cigarette. A cigarette is a system away from equilibrium that can support a traveling chemical reaction front if properly stimulated, in other words, if it is lit. The heat of the match orovides the energy to push the high-free-energy tobacco o;er its activation Lrrie; toward the equilibrium configuration of Con and HzO, liberating additional heat. This heat diffuses to unreacted neighboring sections, causing them to react, and a wavefront propagates along the cigarette. The natural convection arising from the exothermic nature of the chemical wavefront in a ciearette has a laree effect on the rate of propagation. This can he readily demonstrated in a laboratorv and orovides students an oooortunitv to directly observing the coupling of hydrody&ics and chemistrv. As in the liquid chemical reactions, the chemical reaction causes a chanee in density. The heat produced exuands the gas mixture, Jecreasing &s density, thus subjecting it to a buoyant force in the upward direction. This is how the smoke rises from a hurning cigarette-the combustion products are lighter than the surrounding air and rise like a hot air balloon.
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Three filtered eiearettes (the filter is helnful in orovidine an insulated section to mount the ciearette hut is not essentiall were
filter,another wassuspended inaglass jar withaclothespin,and the last mounted horizontally. The experiments were conducted away from large air currents, which could affect the natural convection. The cigarettes were lit, and the time at which the burning zone reached each measured mark was noted. The inhomogeneous packine of the tobacco resulted in some uneven burnine. which sometmes mnde it diiiirult 10 drlinr the positmn uf the iruni. .Iplot c,f d~rlnnrea l m g the rlgarrtlr WTSIIS time pro\ idcd a line whose h p r was the xwrfnmt prbpagntion wlorit).. Thi5 u a i repeated t w o times and the average of the slopes determined.
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Dlscusslon The figure shows the least-square fit lines for the upward and downward cases. The wavefronts propagate with nearly
0
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time (minutes) The position of the burning front as a function of time is ploned fortwo different orientatia of e cigarette.The slopes of the lines provide the propagation velocities: 0.49 0.02 cmlmin (downward). 0.74 i 0.04 cmlmin (upward).
Volume 67
Number 9
September 1990
793
constant velocities with the slowest being the downward moving front with a velocity of 0.49 f 0.02 cmlmin. The front
