Gas-phase reactions .sx The existence of oscillations and the potential for chaotic behaviour in gas-phase reactions have received a much less widespread coverage than their solution-phase counterparts .sx An introduction to this topic can be found in Gray and Scott ( 1985 ) , Griffiths ( 1985 a ) , and in the companion to this book ( Gray and Scott 1990) .sx Recent reviews have also been given by Griffiths ( 1985 b , 1986 ) and Griffiths and Scott ( 1987) .sx Perhaps the simplest non-linear behaviour of interest here shown by gas-phase systems is that of thermal explosion .sx In many ways this is the equivalent of the solution-phase clock reaction .sx The reactants are mixed and a relatively slow evolution begins , but after some ( usually short ) time there is a rapid acceleration in rate leading to virtually complete conversion .sx This process relies on the exothermic nature of many chemical processes and their sensitivity to temperature through the reaction rate constant .sx Gas-phase systems are much more prone to significant departures from isothermal operation , and temperature plays the role of 'autocatalyst' .sx There can also be genuine chemical autocatalysis in gas-phase reactions , particularly oxidation processes involving oxygen .sx Such branded-chain reactions can also provide spectacularly non-linear responses .sx If self-heating and chemistry conspire together , almost anything seems possible .sx 9.1. Transient oscillations from self-heating :sx an experimental realization of the Salnikov model .sx Di-tertiary butyl peroxide ( DTBP ) is an organic species that undergoes a simple first-order decomposition reaction .sx formula .sx with a significant rate at temperatures above about 400 K. The change in number of moles gives an increasing pressure under constant volume conditions and this allows the extent of reaction to be monitored relatively easily as a function of time .sx At low temperatures within this range and especially if the reactant is heavily diluted with an inert gas such as N 2 , the reaction is very well behaved and is widely used to demonstrate first-order kinetics in undergraduate courses .sx graph&caption .sx The detailed kinetic mechanism is not of great importance here , but is believed to involve the unimolecular breaking of the O-O bond , followed by elimination of methyl radicals which may then combine :sx formulae .sx leading to the stoichiometry ( 1) .sx Step ( 2 ) is usually rate determining , so the overall kinetics are insensitive to any changes brought about in steps ( 3 ) and ( 4) .sx The overall process ( 1 ) is exothermic :sx formula and the overall reaction rate constant k has been found to have an activation energy of 152 kJ mol -1 .sx If O 2 is added to the reacting vapour , the effect is to increase the exothermicity ( by changing the final products through steps that compete with ( 4 ) above ) but this does not affect the kinetics which remain first order with the same Arrhenius parameters .sx At sufficiently high reaction temperatures , the exothermic heat release accompanying reaction can lead to self-heating .sx The primary requirement for self-heating is that the reaction rate and , hence , the heat release rate should at some stage become sufficiently high so as to exceed the natural heat transfer rates ( conduction , Newtonian cooling , ) .sx The evolution of self-heating in an experiment can be followed using a coated fine-wire thermocouple positioned at some point within the reacting gas ( usually close to the centre) .sx With a reference junction on the outside wall of the vessel , the thermocouple records directly the temperature difference or extent of self-heating DELTA T. Figure 9.1(a ) shows a typical trace appropriate to a relatively low partial pressure of the reactant or a low ambient temperature .sx The reacting gas temperature ( i.e. that inside the vessel ) here increases above that of the surroundings , but the temperature excess remains small - the maximum DELTA T in this particular example is about 8 K. The corresponding pressure and mass spectrometer records show a steady consumption of the reactant .sx Because of the self-heating , the reaction rate initially increases with increasing extent of reaction - the indication of an acceleratory process , here through thermal feedback .sx The maximum temperature excess observed increases if either the ambient temperature of the initial partial pressure of the DTBP are increased ( both of these increase the rate of this first-order process with a positive activation energy) .sx For small changes , the response changes smoothly , but if the increase in T a or p DTBP is to sic !sx large , there is a qualitative change in the reaction evolution .sx Figure 9.1(b ) shows a typical response for a 'thermal explosion' .sx There is an induction period in which the temperature excess increases to a value not much different from that observed for the previous subcritical behaviour , but this is now followed by a rapid acceleration during which the gas temperature increases swiftly to a high transient excess .sx There is virtually complete removal of DTBP during this ignition pulse and the temperature then falls back to ambient .sx The conditions separating these two forms of response ( subcritical and supercritical ) give rise to a sharp p-T a ignition boundary as shown in Fig. 9.2. The exact location of this limit depends on many factors , such as the vessel size and geometry and the presence or absence of O 2 and inert diluents ( Egieban et al. 1982 ; Griffiths and Singh 1982 , Griffiths and Mullins 1984 ) , but can be relatively successfully predicted from classical thermal explosion theory ( Boddington et al. 1977 , 1982 a , b ) .sx The length of the induction period can also be estimated , and expressed in terms of the 'degree of supercriticality' .sx This latter can be represented as the difference between the actual operating pressure and the corresponding 'critical' pressure ( i.e. the pressure on the limit ) for the operating ambient temperature .sx Then .sx graphs&captions .sx formula .sx i.e. the period gets shorter as the system moves further into the ignition region above the limit , but can become ( arbitrarily ) long close to the limit .sx The application of such theories and interpretations implicitly requires that the reactants should be admitted to the vessel in as short a time as possible ( preferably instantaneously ) and the above experiments come close to that .sx Of more interest in the present context , however , is an experimental situation in which the admission process is not fast , but becomes comparable or even slower than the reaction timescale .sx Griffiths et al. ( 1988 ; Gray and Griffiths 1989 ) employed a reactor into which the reactants were admitted via a capillary tube .sx In this way it was possible to obtain a series of " repetitive thermal explosions " .sx Example time series are shown in Fig. 9.3. A relatively simple explanation for the oscillatory waveforms observed under these conditions of gradual admission can be given .sx As the gas enters the vessel at some fixed ambient temperature , the partial pressure of the reactant increases and so we move up a vertical line in the p DTBP -T a parameter plane ( the ignition diagram , Fig. ) .sx Once the pressure increases beyond the critical pressure for that ambient temperature there will be a thermal ignition ( with perhaps some induction period) .sx This ignition decreases the partial pressure of the reactant below the critical pressure ( and probably close to zero ) , although the total pressure will actually increase .sx Continued inflow of fresh reactant on the slower timescale through the capillary increases p DTBP again , and a second ignition can occur if the critical pressure is reached again .sx This process may repeat a number of times until the pressures inside the vessel and in the reservoir ( i.e. at each end of the capillary ) become equal .sx This simple interpretation is of some value , but does not deal too easily with all the experimental observations .sx For instance there are both upper and lower limits to the temperature for which multiple ignitions ( oscillations ) are found for any given final pressure and inflow characteristics .sx Equivalently , there are upper and lower final pressure limits for oscillations at any fixed ambient temperature .sx The lower limits correspond simply to conditions for which the system does not reach the ignition limit .sx The upper limits are less intuitive , but relate to the matching of inflow and the supercritical reaction rate , so the ignition becomes sustained and steady rather than quenching due to reaction consumption after each excursion .sx Fortunately there is a simple quantitative model for this system - one we have already seen .sx The experiments described constitute a realization of the Salnikov scheme of Section 3.5 , with some minor modification .sx In particular , the role of the first step formula , which was taken to be a chemical reaction with zero exothermicity and no activation energy , is now played by the continuous slow inflow .sx The DTBP decomposition reaction then provides the exothermic first-order step formula .sx The train of oscillations must now be finite , as the inflow only continues for a finite time and is decreasing in magnitude as the pressure difference decreases .sx Numerically computed traces using the known thermokinetic parameters are shown in Fig. 9.4 and successfully reproduce the experiments .sx This design of transient oscillations can clearly be extended to many other exothermic processes .sx Coppersthwaite and Griffiths ( Coppersthwaite 1990 ) have produced similar behaviour in the H 2 + Cl 2 system .sx As in the above case , the oscillatory excursions can lead to the attainment of dramatically higher temperatures than those that occur in neighbouring pseudo-steady-state responses and so there may be considerable hazards associated with the onset of this mode of reaction .sx 9.2. Clock reactions and the pic d'arr e-circ t .sx The idea of a thermal explosion as a fast-timescale clock reaction may seem to push the unified view of solution-phase and gas-phase non-linear behaviour to an extreme .sx There is , however , a second situation in which this analogy is clearly justified to the full .sx The oxidation of hydrocarbons will feature strongly in later sections of this chapter , but here we can consider one aspect - the evolution of reaction for relatively fuel-rich mixtures .sx Snee and Griffiths ( 1989 ) studied the oxidation of cyclohexane under such conditions , at atmospheric pressure in a wide range of vessel sizes .sx Their investigation was primarily concerned with investigating 'minimum ignition temperatures' - the lowest temperature of a preheated flask for which ignition of a 1 cm 3 volume of cyclohexane would occur after injection of the fuel .sx More standard kinetic studies showed that the reaction could exhibit 'quadratic auto - catalysis' in the absence of any self-heating , i.e. the dependence of the reaction rate on the extent of reaction ( measured in terms of the limited oxygen concentration ) has a simple parabolic form as shown in Fig. 9.5. In the presence of self-heating which inevitably accompanied reaction at higher temperatures , this parabolic dependence became skewed , as is also shown in the figure .sx graph&caption .sx Griffiths and Phillips ( 1989 ) found similar autocatalysis in the oxidation of n-butane , again under fuel-rich conditions ( formula ) at reduced pressure ( 500 Torr ) over the range 589-600 K. The fuel-air mixture is admitted to a heated vessel and the reaction can be monitored by a mass spectrometer ( m/e = 32 for O 2 ) , photomultiplier ( chemiluminescence from electronically excited formaldehyde CH 2 O* ) and a fine-wire thermocouple as described above .sx Figure 9.6 shows four different O 2 concentration time series for temperatures within this range .sx With T a = 589 K , no self-heating is observed at any stage during the reaction .sx The time trace shows a classic autocatalytic or clock reaction form , with a long induction period followed by an accelerating rate and then slowing due to reactant consumption .sx A plot of rate ( i.e. slope of the time trace ) against the extent of conversion is again of simple parabolic form appropriate to quadratic autocatalysis .sx At higher temperatures , the induction periods shorten , and the increase in magnitude of the rate at the end of this period becomes more marked .sx Significant transient temperature rises ( self-heating ) accompany the maximum reaction rate , up to 15 K for T a = 600 K. An additional feature here is that the reaction produces a chemiluminescent intermediate CH 2 O* at appreciable concentrations during the period of high rate and this emission can be observed just before the reaction stops due to ( virtually ) complete O 2 consumption .sx This phenomenon has been termed the pic d'arr e-circ t ( D e chaux and Lucquin 1968 , 1971 ; D e chaux et al. 1968) .sx graph&caption .sx Griffiths and Phillips were able to match the form of their experimental traces using a simple model based on quadratic autocatalysis and self-heating ( Kay et al. 1989) .sx They included a modification that allowed a surface - controlled termination step , reflecting the sensitivity of their results to pretreatment and ageing of their vessel .sx The model , then , has the form .sx formulae .sx along with the appropriate heat-balance equation .sx