Chemical reaction network theory

Chemical reaction network theory is an area of applied mathematics that attempts to model the behaviour of real-world chemical systems. Since its foundation in the 1960s, it has attracted a growing research community, mainly due to its applications in biochemistry and theoretical chemistry. It has also attracted interest from pure mathematicians due to the interesting problems that arise from the mathematical structures involved. 2 H 2 + O 2 ⟶ 2 H 2 O C + O 2 ⟶ CO 2 {displaystyle {egin{aligned}{ce {{2H2}+ O2}}&{ce {-> 2H2O}}\{ce {{C}+ O2}}&{ce {-> CO2}}end{aligned}}}     (reaction 1) A 2 + 2 Z ↽ − − ⇀ 2 AZ {displaystyle {ce {{A2}+2Z <=> 2AZ}}}     (reaction 2) B + Z ↽ − − ⇀ BZ {displaystyle {ce {{B}+Z <=> BZ}}}     (reaction 3) AZ + BZ ⟶ AB + 2 Z {displaystyle {ce {{AZ}+BZ -> {AB}+2Z}}}     (reaction 4) B + Z ↽ − − ⇀ ( BZ ) {displaystyle {ce {{B}+ Z <=> (BZ)}}}     (reaction 5) Chemical reaction network theory is an area of applied mathematics that attempts to model the behaviour of real-world chemical systems. Since its foundation in the 1960s, it has attracted a growing research community, mainly due to its applications in biochemistry and theoretical chemistry. It has also attracted interest from pure mathematicians due to the interesting problems that arise from the mathematical structures involved. Dynamical properties of reaction networks were studied in chemistry and physics after the invention of the law of mass action. The essential steps in this study were introduction of detailed balance for the complex chemical reactions by Rudolf Wegscheider (1901), development of the quantitative theory of chemical chain reactions by Nikolay Semyonov (1934), development of kinetics of catalytic reactions by Cyril Norman Hinshelwood, and many other results. Three eras of chemical dynamics can be revealed in the flux of research and publications. These eras may be associated with leaders: the first is the van 't Hoff era, the second may be called the Semenov–Hinshelwood era and the third is definitely the Aris era. The 'eras' may be distinguished based on the main focuses of the scientific leaders: The mathematical discipline 'chemical reaction network theory' was originated by Rutherford Aris, a famous expert in chemical engineering, with the support of Clifford Truesdell, the founder and editor-in-chief of the journal Archive for Rational Mechanics and Analysis. The paper of R. Aris in this journal was communicated to the journal by C. Truesdell. It opened the series of papers of other authors (which were communicated already by R. Aris). The well known papers of this series are the works of Frederick J. Krambeck, Roy Jackson, Friedrich Josef Maria Horn, Martin Feinberg and others, published in the 1970s. In his second 'prolegomena' paper, R. Aris mentioned the work of N.Z. Shapiro, L.S. Shapley (1965), where an important part of his scientific program was realized. Since then, the chemical reaction network theory has been further developed by a large number of researchers internationally. A chemical reaction network (often abbreviated to CRN) comprises a set of reactants, a set of products (often intersecting the set of reactants), and a set of reactions. For example, the pair of combustion reactions form a reaction network. The reactions are represented by the arrows. The reactants appear to the left of the arrows, in this example they are H 2 {displaystyle {ce {H2}}} (hydrogen), O 2 {displaystyle {ce {O2}}} (oxygen) and C (carbon). The products appear to the right of the arrows, here they are H 2 O {displaystyle {ce {H2O}}} (water) and CO 2 {displaystyle {ce {CO2}}} (carbon dioxide). In this example, since the reactions are irreversible and neither of the products are used up in the reactions, the set of reactants and the set of products are disjoint. Mathematical modelling of chemical reaction networks usually focuses on what happens to the concentrations of the various chemicals involved as time passes. Following the example above, let a represent the concentration of H 2 {displaystyle {ce {H2}}} in the surrounding air, b represent the concentration of O 2 {displaystyle {ce {O2}}} , c represent the concentration of H 2 O {displaystyle {ce {H2O}}} , and so on. Since all of these concentrations will not in general remain constant, they can be written as a function of time e.g. a ( t ) , b ( t ) {displaystyle a(t),b(t)} , etc.