Starting from the general idea of reaction kinetics, their classification, concentrations, and chemical equilibrium, we will focus on their activation energy and complexity arising during the chemical reaction. As in complex and higher-dimensional chemical problems, we need special arrangements, specifically, in the case when a system attains different completion paths or several routes. The stiffness of the system can be removed if we distinctly measure their available reaction routes and get a comparison between them and overall reactions. Secondly, the construction and comparison of the invariant region of the manifold based on the modern decomposition techniques in different available reaction routes allow us to discuss the dynamical properties of the system.
- chemical equilibrium
- detailed mechanism
- model reduction
- reaction routes
- invariant manifold
The chemical kinetics or reaction kinetics is the branch of physical chemistry that deals with the study of chemical processes, their rates, rearrangement of atoms, the effect of various variables, the formation of intermediates, etc. In fact, the chemical kinetics is the study of different factors affecting the speed of a chemical process and gives information about the mechanism of reaction and transition states. At the macroscopic level, the chemical kinetics deals with the study of amount reacted, formed, and the rates of their formation. While at the microscopic or molecular level, we study the mechanism of a chemical reaction, i.e., atomic collision, activation energy at different stages during the reaction.
The chemical kinetics is classified into three types, mathematical, detailed, and applied kinetics, while their elementary reactions are described as unimolecular, bimolecular, and termolecular reactions.
The construction of the mathematical model is the key part of chemical kinetics, which gives a complete description of reaction mechanism and its rates. It provides a working tool to better understand and design chemical processes, i.e., food decomposition and the complex chemistry of biological systems, etc. These models are also used in designing the fast and slow trajectories of complex chemical reactions and modification of chemical reactors to optimize product yield, more efficiently separate products, and eliminate environmentally harmful by-products.
The goal of
The reactions in which a single molecule rearranges itself to make one or more products are called unimolecular reactions or a first-order reaction (
The chemical reactions in which the reactants are in the same phases are called homogeneous reactions, i.e., the reaction between two gases, two solids, or two liquids. Let us consider a reversible chemical reaction represented as
The reactions in which the reactants are in different phases and their rates are affected by surface areas are called heterogeneous reactions, i.e., the reaction between gases and liquids, solids and liquids, etc. As in the case of gas solid catalytic reactions, reactants at elementary steps will be gas phase component or surface intermediate. Thus, Eq. (1) can now be written as
With an assumption that
The concentration of the involved species can be measured as (single step reaction);
|At initial space:||0|
|time t > 0:||(|
The reaction rates measured on either side are (forward rate of reaction) and (backward rate of reaction) and the product formation is the difference between the rate of forward and backward reactions, i.e.,
The equilibrium is a dynamical process, and when a system goes to an equilibrium, the left-hand side will become equal to the right-hand side, i.e.,
while the conversions of products to reactants and reactants to products are still going on, although there is no net change in the number of reactant and product molecules.
The system becomes complex when a reaction undergoes more than one pathways when more than one products are formed from the same reactants or different reactants produce the same products. Such types of reactions are called parallel reactions or side reactions, i.e.,
or a system is reversible at different stages. In all these cases, we need to follow all their paths to get the detailed mechanisms,
The rate of reaction (
There may be a few reasons behind its ineffectiveness , i.e.:
The reactant molecules may attain insufficient energy (<
The molecules may not get aligned properly or orientate during the collision (depending upon the geometry of the particles and kind of reaction that is taking place), etc.
If Z is the effective collision in which molecules have energy ≥
In a complex chemical reaction, the reactant molecules (intermediates, complex Θ
The energy required to pass the reactant
The activation energy during the forward and backward reactions must be the same or different depending on the type of reactions. In thermos, the neutral reaction and
The activated complex is a separate entity and there exists an equilibrium between reactants (products, under reversible reactions) and activated complex (Figure 1). Thus, a reaction mechanism can be defined as
But still, there is some activation going on between the activated complexes , i.e.,
Therefore, a complete complex chemical reaction mechanism can be defined as .
In case of reversible complex chemical reactions,
The concentration of activated complex can thus be obtained by applying the equilibrium conditions, i.e.,
where ∓ refers to the activated complex.
2. Reaction rate
A stoichiometric vector
The chemical composition of the substances is given by the molecular matrix
The total number of any moles of
The dynamics of the involved concentration species can be measured when we measure the rate of formation of the products or deformation and disappearance of the reactants.
Finally, the rate of reaction will take a form
3. Linear algebra and graph theory
In chemical engineering, the mathematical methods of graph theory have found wide applications in complex chemical reactions and in a sequence of uni (or multi) or parallel reacting events. A graph is a combination of nodes (points) and edges (lines) , while a cyclic graph involves finite sequences of edges with the single node (from where it begins and ends).
Similarly, related to any combination of reaction, a tree can be defined as a sequence of noncyclic graph edges. In a spanning tree, certain intermediate may form from other intermediates after a sequence of transformations but does not agree to counter any two reactions with the same step (e.g., +1 and −1) nor two reactions started with the same intermediates (e.g., −1 and +2, or +1 and −3),
Spanning trees can be described in terms of “forward” (generated by a sequence of forwarding reactions), “backward” (generated by a sequence of reverse reactions), and “combined” spanning trees (generated by a sequence of both forward and backward reactions). A single-route, n-steps
In a chemical reaction, the overall reaction can be found by multiplying the reactions with certain coefficients, the so-called Horiuti numbers
Horiuti number allows us to distinguish the short-lived intermediate and long-lived components, i.e., to eliminate the intermediates using an RREF of the stoichiometric matrix
The numbers of key components
and the number of key components equals the number of key reactions. Also, the number of key components + number of nonkey reactions = number of reactions
In Figure 2, their curves represent two different solution curves of their respective reaction routes lying at different phase space, i.e., one lies in 2D while the second lies within 3D.
Now the question arises, if a complex reaction adopts different completion routes before giving the product, then how can one relate (or distinguish) such available routes and why they are important to be measured?
For this, the reaction route
To answer the second part of the above question, we need to consider all its available routes to get the detailed reaction mechanism. Then a comparison of these route solutions with the whole reaction mechanism allows us to give any concluding remarks, but we believe that the result obtained through different routes may be similar or vary depending on the type of reactions.
4. Multiroute reactions mechanism
To understand this idea, let us discern the
Atomic balance constraints are given by Eq. (11)
This means we can reduce this system into three components
Hence, this reaction mechanism has two independent routes
First-route: two-step mechanisms (Figure 4).
Second-route: three-step mechanisms (Figure 5).
5. The measuring methods
The kinetic equations of the above reaction mechanism (
Thus, a single reaction route is available. The kinetic equations for the involved species are given by Eq. (13)
Figures 6 and 7 clear the idea of the slow invariant manifold (SIM), i.e., decomposing the system into their fast and slow motion. Their solution trajectories (during their relaxation time) quickly move toward the low-dimensional manifold and after that start moving along it [4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. That is the easy way of getting an idea of the SIM. Otherwise, by using the different available methods of SIM, i.e., [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27] we will get their initial approximations lying on it or near to it. For comparison, we refer the readers to .
6. The routes comparison
In Figure 9, the curves lie in the plane c1 and c4 are not the projected image of the above curves. Instead, it is the behavior of the species measured near the equilibrium point in the first reaction route mechanism, whose
Now, the overall reaction mechanism involves no intermediate, and the variations of the concentration of involved chemical species are given in Figure 10.
In this chapter, both the physicochemical conceptual assumptions (used for species behavior and activated complex) and a set of mathematical tools (for their dynamical behavior and simplification) are presented. Mathematically, simplification can be done by “model reduction,” that is, the rigorous way of approximating and representing a complex model in simplified form.
Here, we have considered a complex problem having a common step: conferred their available routes then allied graphically. Although we have not applied any numerical or analytical technique to measure the SIM but one can easily examine (by applying such techniques) that their solution trajectories will also lie in the same invariant regions that can also be correlated with each other and even with the whole reaction mechanism.
Thus, the idea initiated here can easily be correlated with the method used for the construction of slow manifold in a complex chemical reaction based on the decomposition techniques of entropy maximum along with certain constraints (lies on the manifold or given by slowest eigenvectors) at the equilibrium point. This will allow us to bring together different available mathematical ideas and methods, commonly used to transform the complex chemical problems from one way to the other, to enhance progress in understanding.