Tham khảo tài liệu 'understanding non-equilibrium thermodynamics - springer 2008 episode 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 108 4 Chemical Reactions and Molecular Machines Autocatalytic Reactions and Diffusion Morphogenesis Another consequence of the coupling between chemical reactions and diffusion is the occurrence of patterns . spatial inhomogeneities in systems far from equilibrium. Although this topic is studied in detail in Chap. 6 we find it interesting to present here an introductory version based on a model proposed by Turing 1952 who settled the foundations of morphogenesis . the transition of an originally homogeneous system into an inhomogeneous one. This is a characteristic feature of non-equilibrium thermodynamics because near equilibrium transport phenomena matter diffusion thermal conduction tend to make the system spatially homogeneous in the long run. Consider two fluid subsystems at the same pressure and temperature which are separated by a permeable membrane. In each of them the following autocatalytic reaction will take place between substances A and X A 2X 3X. In this reaction a substance A is transformed into the substance X at a rate which depends on the concentration of X. The molar concentrations of X in both subsystems cX1 and cX2 respectively tend to become equal due to diffusion of X across the membrane. The diffusion flux of X from subsystem 1 to subsystem 2 is given by J 2 a cX1 - ỖX2 where a is a constant proportional to the permeability and the area of the membrane. We will show that under some conditions the difference of concentrations cx between both subsystems will increase because of the autocatalytic reaction . First we note that the rate of production of substance X is higher where the molar concentration cx is higher as it follows from the mass action law jX kcA cx 2 where k is a positive kinetic constant. Taking diffusion into account the evolution of cX1 as a function of time in subsystem 1 is given by dX1 -a cXi - CX2 kcA cXi 2 whereas the evolution of cX2 in subsystem 2 is dt2 a cX1 cX2 I kcA cX2 2. By
