Ivan A Gargurevich^{*}  
Chemical Engineering Consultant, Combustion and Process Technologies, 32593 Cedar Spring Court, Wildomar, CA 92595, USA  
Corresponding Author :  Ivan A Gargurevich Chemical Engineering Consultant, Combustion and Process Technologies 32593 Cedar Spring Court, Wildomar, CA 92595, USA Tel: 9516759455 Email: [email protected] 
Received: February 10, 2016; Accepted: February 27, 2016; Published: February 29, 2016  
Citation: Gargurevich IA (2016) Chemical Kinetic Modeling and its Principles. J Chem Eng Process Technol 7:281. doi:10.4172/21577048.1000281  
Copyright: © 2016 Gargurevich IA. This is an openaccess article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. 
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The emphasis will be to demonstrate the fundamental mathematical concepts in calculating rates of reactions and the meaning of chemical reaction mechanisms. Reactions as mentioned in a previous chapter can be frequently classified as homogeneous or heterogeneous. Homogeneous reactions occur in a single phase for example, the gas phase reaction involving NO and Oxygen
Keywords 
Chemical kinetics; Polymerization 
Introduction 
Chemical kinetics is the study of the rates of chemical reactions, the means by which reaction rates may be controlled, and the ways in which reactions proceed on the atomicmolecular level [1]. 
2NO (g) +O_{2} (g) <==> 2NO_{2} (g) 
Heterogeneous reactions take place on a phase boundary, e.g., the formation of solid Magnesium oxide, 
2Mg (s) +O_{2} (g) <= => 2M_{g}O(_{s}) 
Reactions are also classified as reversible or irreversible. The reaction to produce ammonia below is an example of reversible reaction or, 
N_{2} (g) + 3H_{2} (g) < = => 2NH_{3} (g) 
Depending on conditions of temperature and pressure the reaction leads to the formation of Ammonia, if conditions are suddenly changed such as lowering the reaction pressure substantially, ammonia dissociates back into Nitrogen and Hydrogen. 
Reaction mechanisms and elementary reactions 
Other important concepts to understand are reaction mechanisms and elementary reactions. This can be demonstrated by considering the gasphase decomposition of N_{2}O_{5} (g) [2], The overall reaction is 
N_{2}O_{5} (g) <= = > 2NO_{2} (g) + 1/2O_{2} (g) 
The actual decomposition of N_{2}O_{5} occurs via three “elementary” steps or “molecular events”, 
N_{2}O_{5} <= => NO_{2} + NO_{3} 
NO_{2} + NO_{3} < = = > NO + O_{2} + NO_{2} 
NO + NO_{3} < = => 2NO_{2} 
Another important example are “chain reaction mechanisms” since they provide an explanation for important process such as photochemical, combustion, and polymerization. 
Chain reactions consist of three kinds of elementary steps initiation (or activation) step, propagation, and termination. As an example consider the gasphase chlorination of propane (PrH) [2]. 
Initiation 
Cl_{2} (g) <==> 2Cl 
II. Propagation 
Cl + PrH <= => Pr + HCl 
Pr + Cl_{2} < = => PrCl + Cl 
III Termination 
Cl + Cl < = => Cl_{2} HOMOGENEOUS 
Cl + Pr < = => PrCl 
Cl + W ==> End Products HETEROGENOUS 
Pr + W ==> End Products 
In the initiation step Cl atoms can be generated thermally at high temperatures. The propagation step produces unstable radical intermediates such as Pr. Both Pr and Cl in the Propagation step are considered chain carriers. The Propagation step is responsible for the high rate of reaction since the chain carriers are produced. The Termination step reduces the overall rate of reaction and results in the formation of the end products homogeneously or heterogeneously (where W, reactor wall surface). 
Concentration and reaction rates 
The chemical law of mass action states that, in general, for any single step or elementary reaction (or for one step of a multistep mechanism) such as the rate equation [3] is 
wW + xX = => Products 
Reaction rate = k[W]^{w}[X]^{x}.= d[Products]/dt 
where k is the rate constant. The order of the reaction is given by the sum of the exponents of the concentrations appearing in the rate equation or 
Reaction order = w + x +… 
For example, The experimentally determined rate law for the overall reaction (Gardiner), rate, 
H_{2} + D_{2} <= => 2 HD 
R= ½ d[HD]/dt = k(T) [H2]^{0.38}[D2]^{0.66}[Ar]^{0.98} 
The order or sum of exponents is 2.0 and the reaction is 0.38 and 0.66 order with respect to Hydrogen and deuterium respectively. 
If the reaction is an elementary reaction, the sum of the exponents is then the molecularity of the elementary reaction: one for a unimolecular reaction, two for a bimolecular reaction, `and three for a termolecular reaction. 
Temperature and reaction rates 
Theoretically (Kinetic Theory of Reactions), the temperature dependence of the reaction rate is due to two important factors: the rate of molecular collisions (proportional to T^{1/2}) and the increasing probability of high energy encounters (varies as exp (const/T). Based on experimental evidence Savante Arrhenius arrived at the following expression for the temperature dependence of the rate constant, k (T) 
K (T) = A exp (E_{a}/RT) 
The exponential factor is the most important factor, although the constant A is also known to have temperature dependence. The A factor is related to the molecular encounter frequency as well as encounter geometry. The activation energy factor E_{a} is related to the strength of the chemical bonds that are broken and formed, and is the most important factor in the Arrhenius equation. 
As noted above, the activation energy factor can be derived from the MaxwellBoltzmann Distribution Law of molecular energy in the Kinetic Theory of Gases. 
Mathematical description of chemical kinetics in solution (Batch) 
The mathematical treatment of rate equations lead to integral forms as shown in Table 1 [2]. For example, for a zero order elementary reaction in a batch and isothermal system, the integration of the rate equation leads to: 
D [A]/dt = k_{o} 
[A] = [A]_{0}kt 
Where [A]_{0} is the initial concentration of species [A]and t is time. 
Other types of reactions can be similarly analyzed mathematically and analytical solutions found for composition of the solution in a batch reactor as shown in Table 1. 
A more complete treatment of closed isothermal systems can be found in “Mathematical description of Chemical Kinetics in Solutions” by Capellos and Bielski [3]. 
In Table 1 for the 2^{nd} order reversible reaction the constant q, α, β, γ are as follows: 
And K is the reaction equilibrium constant. 
References 

Table 1 