Instabilities in Coupled Chemical Reactions with Pressure-Dependent Rate Coefficients

Abstract

Instabilities are investigated in a model system consisting of several coupled chemical reactions; one reaction step is autocatalytic, and in another partial molar volume changes and pressure dependent rate coefficients are considered. The hydrodynamic equations of motion (for density, mass velocity, and concentration of chemical species) for this system are linearized in small deviations from steady state. The behavior of the system is analyzed by means of a set of hydrodynamic mode coordinates and their corresponding eigenvalues; the system is shown to be unstable under certain conditions not only to perturbations in concentrations but also to other hydrodynamic variables. For spatially homogeneous modes (k=0) the stable, marginally stable, unstable, oscillatory, and nonoscillatory regions of the eigenvalue spectrum are calculated for several limiting cases of the various experimental parameters. The normal modes are also calculated in the same limiting cases. A phase diagram is constructed for the stability of the system to homogeneous perturbations in terms of reduced parameters. For k≠0 there appear the expected acoustic modes but their form is altered by the presence of the reactions. Some of the reaction modes at k≠0 become unstable closer to chemical equilibrium than the corresponding modes in the homogeneous case (k=0). This result holds even in the limit of no diffusion. In that limit there exists a new mechanism of symmetry breaking which establishes inhomogeneous reaction modes due to the dynamical coupling of scalar equations of reaction (with pressure-dependent rate coefficients) and the vector equation of conservation of momentum. Possible biological applications of the theory are mentioned.