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Whence Dynamical Systems
Published in LM Pismen, Working with Dynamical Systems, 2020
Chemical kinetics. The mass action law assumes that the reaction rate is proportional to the product of reactant concentrations. This, indeed, should be true if reactant molecules, present in macroscopic amounts, are uniformly distributed in space, and the reaction might occur when they collide at random. The probability of the reaction upon collision is measured by the reaction rate constant k. Thus, the change of the concentration ci of a chemical species due to a reversible reaction defined by (1.31) is c˙i=vi(k+∏i∈I−ci|vi|−k−∏i∈I+civi), where k± are rate constants of the direct and reverse reactions.
Oragnic Chemicals in the Environment
Published in Richard A. Larson, Eric J. Weber, Reaction Mechanisms in Environmental Organic Chemistry, 2018
Richard A. Larson, Eric J. Weber
Once a reaction mechanism consisting of a sequence of individual elementary reactions has been proposed it is possible to develop rate equations, which predict the dependence of the observed reaction rate on concentration. The principle of mass action, which states the rate at which an elementary reaction takes place is proportional to the concentration of each chemical species participating in the molecular event, is used to write differential rate equations for each elementary reaction in the proposed reaction mechanism. The goal is then to obtain explicit functions of time, which are referred to as integrated rate laws, from these differential rate equations. For simple cases, analytical solutions are readily obtained. Complex sets of elementary reactions may require numerical solutions.
Principles of Chemistry
Published in Arthur T. Johnson, Biology for Engineers, 2019
Before two compounds can react chemically, they must contact each other. Collision theory has been formulated to explain various factors on chemical reaction rates. Part of this theory is the Law of Mass Action, which states that a chemical reaction will occur at a rate proportional to the concentration of its reactants. Thus, anything that can be done to concentrate reaction precursors in the presence of each other will favor the desired chemical combination to occur.
Michaelis–Menten kinetics as a model of doctoral supervisor–supervisee relationship
Published in International Journal of Mathematical Education in Science and Technology, 2023
First, consider that case K = 1; that is: which corresponds to the original scheme proposed by Michaelis and Menten (Johnson & Goody, 2011; Michaelis & Menten, 1913; Suzuki, 2019). The law of mass action states that the rate of a chemical reaction is proportional to the product of the concentrations of the reacting substances, with each concentration raised to a power equal to the coefficient that is used to balance such a chemical reaction (Voit et al., 2015). By applying this law, the rates of the reactions shown in Equation (3) are written as: Since , then increases as time progresses. Notice that the variable does not affect the time evolutions of and ; hence, Equation (6) can be neglected in the analysis of Equations (4)–(5). Notice also that . Consequently, ; that is, the total number of professors with or without students remains constant and equal to N. By taking , Equation (4) can be rewritten as: The case K = 2 corresponds to: This case is described by the differential equations: Since , then and Equations (10)–(11) can be rewritten as: In the next section, the long-term behaviours of the cases with K = 1 and K = 2 are analysed. Their dynamics are governed by Equation (7) and by Equations (14)–(15), respectively. The results can give hints for the cases with .
A novel kinetic model for a cocoa waste fermentation to ethanol reaction and its experimental validation
Published in Preparative Biochemistry & Biotechnology, 2023
Eduardo Alvarado-Santos, Ricardo Aguilar-López, M. Isabel Neria-González, Teresa Romero-Cortés, Víctor José Robles-Olvera, Pablo A. López-Pérez
Generally, many fermentation studies have confirmed that the unstructured models poorly describe experimental data in which initial conditions, operating conditions, time range, and microorganism’s type change.[41] In contrast, the use of a more detailed approach to cell metabolism, aimed at better describing the dynamic behavior that includes intra and extra-cellular concentrations, can be applied to the so-called structured models.[42–44] However, parameter estimation can become a difficult task due to the ample experimental data generated and the need to apply complex numerical methods, leading to parameter values without physical meaning. The latter frequently occurs in the parameters of fits to unstructured models that do not represent a biological interpretation.[45] A class of structured models that are potentially useful is formed by simply applying the structured formulation, which allows describing the quantity of biomass and its properties by using two or three variables, resulting in the so-called compartment models or phenomenological model.[42–46] For example, a lin-log model[47] was derived based on an already established mechanistic model of the central carbon metabolism in Escherichia coli,[30] and was found to give similar simulation results despite its simpler structure and fewer parameters. In three parallel models of sphingolipid metabolism in yeast,[48] the power-law formats, GMA and S-systems, were compared to Michaelis–Menten kinetics. These models combine a better description of the system's behavior with reasonable mathematical complexity and a smaller number of parameters.[45] In biochemical reaction networks, for example, ethanol production is multi-step; these processes may follow simple laws. Rate law could directly result from the sequence of elementary steps that constitute the global reaction mechanism. As such, it could determine an unknown mechanism within the case of “n-elementary reactions.”[47] The reaction mechanism is based on the so-called law of mass action. Kinetic rate expressions which are the symbolic expressions describing the reactions and interactions between elements of the general reaction mechanism were considered. The law of mass action states that the reaction rate is proportional to the concentrations of the reactants (substrate, biomass) or of the products (metabolites) for a unimolecular reaction, and is used as a description for elementary reactions (reactions with one step). Kinetics of multi-step reactions can be derived by combining the mass action kinetics of their elementary reactions.[30,33,48,49]