Full Arrhenius rate law

The multiplicity analysis described above can be extended to cope with the exact form of the Arrhenius temperature dependence (non-zero y). The stationary-state condition has a slightly more cumbersome form  

The corresponding residence times can be obtained by substituting this result into eqn (7.31). More importantly, the requirements on the values of the two parameters y and 0ad can be obtained by the condition that the term under the square root sign must be positive for real solutions. This gives 

This requires at least that 0ad must be larger than 4 and that the group y, which is inversely proportional to the activation energy, must be less than i- Multistability is thus favoured by high exothermicity and by a high temperature coefficient for the rate constant. 

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FONI model (a) unique (b) single hysteresis loop or breaking wave (c) isola (d) isola + hysteresis loop (e) mushroom. With the full Arrhenius rate law and the provision pre-heating or cooling, two additional patterns are found (f) reversed hysteresis loop and (g) reversed hysteresis loop + isola. Also shown are various degenerate loci corresponding to parameter values on the boundaries or special points in the parameter plane (see Fig. 7.5). 

With the full Arrhenius rate law, an extra unfolding parameter y is introduced. Even then, however, the appropriate stationary-state condition and its derivatives for the winged cusp cannot be satisfied simultaneously (at least not for positive values of the various parameters). Thus we do not expect to find all seven patterns. 

We may also note, for the special case / = 1, that the locus described by eqns (10.58) and (10.59) is exactly that corresponding to the boundary between unstable focus and unstable node for the well-stirred system. This seems to be a general equivalence between the existence of unstable nodal solutions in the well-stirred system and the possibility of diffusion-driven pattern formation in the absence of stirring. We have seen in chapter 5 that unstable nodes are not found in the present model if the full Arrhenius rate law is used and the activation energy is low, i.e. iff <4 RTa. In that case we would also not expect spatial instability. 

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