## Chemical Field

### Chemical Kinetics and Its Mathematical Description

Chemical kinetics or reaction kinetics investigates the rates of chemical processes and the influence of experimental conditions on the rates. For the chemical field in numerical simulation, the information needed from chemical kinetics is the mathematical models for describing the characteristics of a chemical reaction. The determination of reaction rates including both rate laws and rate constants can be fulfilled by means of experiments, theories, or ab initio simulations.

To understand chemical kinetics, let us consider a typical chemical reaction: \[a{\rm A}+b{\rm B}\to c{\rm C}+d{\rm D},\] where the lowercase letters ($a$,$b$,$c$, and $d$) represent the stoichiometric coefficients and the capital letters represent the reactants ($A$ and $B$) and the products($C$ and $D$). The reaction rate r for a chemical reaction occurring in a closed isochoric system such as the following equation (no reaction intermediates) is defined as [IUPAC's Gold Book], \[r=-\frac{1}{a} \frac{d\left[{\rm A}\right]}{dt} =-\frac{1}{b} \frac{d\left[{\rm B}\right]}{dt} =\frac{1}{c} \frac{d\left[{\rm C}\right]}{dt} =\frac{1}{d} \frac{d\left[{\rm D}\right]}{dt} ,\] where $\left[{\rm X}\right]$ denotes theconcentrationof the substance X. For the chemical reaction $a{\rm A}+b{\rm B}\to c{\rm C}+d{\rm D}$, the rate equation is of the following form: \[r=k\left[{\rm A}\right]^{a} \left[{\rm B}\right]^{b} , \] where $k$ is the reaction rate coefficient or rate constant. However, this parameter is not necessarily a constant. The reaction rate coefficient allows for the influences of all the parameters except concentration. We then can obtain the following first-order ordinary differential equation: \[\frac{d\left[{\rm A}\right]}{dt} =-ak\left(T\right)\left[{\rm A}\right]^{a} \left[{\rm B}\right]^{b} . \] This can be viewed as the governing equation for the chemical field, though a comprehensive description of the field requires the mass transport as well. A theoretical solution is possible for simple cases such as simple reactions. Taking the reaction aA$\mathrm{\to}$cC+dD for example, the reaction rate of this first-order reaction can be obtained as \[r=-\frac{d\left[{\rm A}\right]}{dt} =k\left[{\rm A}\right]. \] The integration of the above equation yields \[\left[{\rm A}\right]=\left[{\rm A}\right]_{0} \cdot e^{-kt} \] where $\left[{\rm A}\right]_{0} $ is the concentration of A at the beginning of the chemical reaction.