Chemical kinetics rate law

Complex chemical mechanisms are written as sequences of elementary steps satisfying detailed balance where tire forward and reverse reaction rates are equal at equilibrium. The laws of mass action kinetics are applied to each reaction step to write tire overall rate law for tire reaction. The fonn of chemical kinetic rate laws constmcted in tliis manner ensures tliat tire system will relax to a unique equilibrium state which can be characterized using tire laws of tliennodynamics. 

Solution of the coupled mass-transport and reaction problem for arbitrary chemical kinetic rate laws is possible only by numerical methods. The problem is greatly simplified by decoupling the time dependence of mass-transport from that of chemical kinetics the mass-transport solutions rapidly relax to a pseudo steady state in view of the small dimensions of the system (19). The gas-phase diffusion problem may be solved parametrically in terms of the net flux into the drop. In the case of first-order or pseudo-first-order chemical kinetics an analytical solution to the problem of coupled aqueous-phase diffusion and reaction is available (19). These solutions, together with the interfacial boundary condition, specify the concentration profile of the reagent gas. In turn the extent of departure of the reaction rate from that corresponding to saturation may be determined. Finally criteria have been developed (17,19) by which it may be ascertained whether or not there is appreciable (e.g., 10%) limitation to the rate of reaction as a consequence of the finite rate of mass transport. These criteria are listed in Table 1. 

Interestingly, in the experiments devoted solely to computational chemistry, molecular dynamics calculations had the highest representation (96-98). The method was used in simulations of simple liquids, (96), in simulations of chemical reactions (97), and in studies of molecular clusters (98). One experiment was devoted to the use of Monte Carlo methods to distinguish between first and second-order kinetic rate laws (99). One experiment used DFT theory to study two isomerization reactions (100). 

Maja-Perez, F. and Perez-Benito, J.F. (1987) The kinetic rate law for autocatalytic reactions. Journal of Chemical Education, 64 (11), 925-7. 

Another consideration in choosing a kinetic method is the objective of one s experiments. For example, if chemical kinetics rate constants are to be measured, most batch and flow techniques would be unsatisfactory since they primarily measure transport- and diffusion-controlled processes, and apparent rate laws and rate coefficients are determined. Instead, one should employ a fast kinetic method such as pressure-jump relaxation, electric field pulse, or stopped flow (Chapter 4). 

The transformation from reactants to products can be described at either a phenomenological level, as in classical chemical kinetics, or at a detailed molecular level, as in molecular reaction dynamics.1 The former description is based on experimental observation and, combined with chemical intuition, rate laws are proposed to enable a calculation of the rate of the reaction. It does not provide direct insight into the process at a microscopic molecular level. The aim of molecular reaction dynamics is to provide such insight as well as to deduce rate laws and calculate rate constants from basic molecular properties and dynamics. Dynamics is in this context the description of atomic motion under the influence of a force or, equivalently, a potential. 

Any surface reaction that involves chemical species in aqueous solution must also involve a precursory step in which these species move toward a reactive site in the interfacial region. For example, the aqueous metal, ligand, proton, or hydroxide species that appear in the overall adsorption-desorption reaction in Eq. 4.3 cannot react with the surface moiety, SR, until they leave the bulk aqueous solution phase to come into contact with SR. The same can be said for the aqueous selenite and proton species in the surface redox reaction in Eq. 4.50, as another example. The kinetics of surface reactions such as these cannot be described wholly in terms of chemically based rate laws, like those in Eq. 4.17 or 4.52, unless the transport steps that precede them are innocuous by virtue of their rapidity. If, on the contrary, the time scale for the transport step is either comparable to or much longer than that for chemical reaction, the kinetics of adsorption will reflect transport control, not reaction control (cf. Section 3.1). Rate laws must then be formulated whose parameters represent physical, not chemical, processes. 

So far we have shown how multivariate absorbance data can be fitted to Beer-Lamberf s law on the basis of an underlying kinetic model. The process of nonlinear parameter fitting is essentially the same for any kinetic model. The crucial step of the analysis is the translation of the chemical model into the kinetic rate law, i.e., the set of ODEs, and their subsequent integration to derive the corresponding concentration profiles. 

The power-law formalism was used by Savageau [27] to examine the implications of fractal kinetics in a simple pathway of reversible reactions. Starting with elementary chemical kinetics, that author proceeded to characterize the equilibrium behavior of a simple bimolecular reaction, then derived a generalized set of conditions for microscopic reversibility, and finally developed the fractal kinetic rate law for a reversible Michaelis-Menten mechanism. By means of this fractal kinetic framework, the results showed that the equilibrium ratio is a function of the amount of material in a closed system, and that the principle of microscopic reversibility has a more general manifestation that imposes new constraints on the set of fractal kinetic orders. So, Savageau concluded that fractal kinetics provide a novel means to achieve important features of pathway design. 

Lichtner (2001) developed the computer code FLOTRAN, with coupled thermal-hydrologic-chemical (THC) processes in variably saturated, nonisothermal, porous media in 1, 2, or 3 spatial dimensions. Chemical reactions included in FLOTRAN consist of homogeneous gaseous reactions, mineral precipitation/dissolution, ion exchange, and adsorption. Kinetic rate laws and redox... 

Arrhenius law (1889) describing the dependence of a chemical reaction rate constant on temperature T is one of the most fundamental laws of chemical kinetics. The law is based on the notion that reacting particles overcome a certain potential barrier with height E , called the activation energy, under the condition that the energy distribution of the particles remains in Boltzmann equilibrium relative to the environment temperature T. When these conditions are satisfied, the Arrhenius law states that the rate constant K is proportional to exp[ —E /Kgr], where Kg is the Boltzmann constant. It follows that, for E > 0, K tends to zero as T 0. 

Then Eq. (151) simply relates the overall time dependence of the concentration to the algebraically additive mass transfer and chemical components. In this respect it is important to point out that the chemical term is identical to that obtained under homogeneous conditions for the same chemical sequence. Thus Eq. (151) constitutes a generalization of the usual kinetic rate laws to conditions in which concentrations are not homogeneous. Although developed here in the context of electrochemical techniques, it is valid in any kind of chemical situation in which concentrations are not uniform, as nearly any heterogeneous reactant or phase-transfer boundaries. 

The kinetic rate law for each elementary irreversible chemical reaction is written in terms of gas-phase molar densities (A, B, C, D, where A = Ca, etc.) as follows ... 

Sequential application of the steady-state design equations is required when multiple chemical reactions occur in a series configuration of well-mixed tanks. If temperature, residence time, kinetic rate laws, and the characteristics of the feed to the first reactor are known, then it is possible to predict molar densities in the exit stream of the first reactor, which represent the feed to the second reactor, and so on. Subscripts are required to monitor ... 

For a particular liquid-phase chemical reaction, the kinetic rate law is zeroth order ... 

Now the total differential of specific enthalpy contains a chemical reaction contribution via the kinetic rate law 3R ... 

The reversible kinetic rate law for nth-order chemical reaction is... 

If the reactor is well stirred, then the molar densities of reactant A and product B in the kinetic rate law are expressed in terms of conversion / via stoichiometry and the steady-state mass balance with convection and chemical reaction ... 

Illustrative Problem. Consider a spherical solid pellet of pure A, with mass density pa, which dissolves into stagnant liquid B exclusively by concentration diffusion in the radial direction and reacts with B. Since liquid B is present in excess, the homogeneous kinetic rate law which describes the chemical reaction is pseudo-first-order with respect to the molar density of species A in the liquid phase. Use some of the results described in this chapter to predict the time dependence of the radius of this spherical solid pellet, R(t), (a) in the presence of rapid first-order irreversible liquid-phase chemical reaction in the diffusion-limited regime, and (b) when no reaction occurs between species A and B. The molecular weight of species A is MWa. 

If diatomic A2 participates in a catalytic surface reaction by dissociating and adsorbing atomically prior to reacting with B, then the modified Langmnir isotherm that describes snrface coverage by atomic A is apparent in the denominator of the kinetic rate law. For example, the overall chemical reaction is... 

Total pressure analysis of the initial reactant product conversion rate can distinguish between these two mechanisms, provided that rates of conversion can be measured at sufficiently high pressure. The rate expressions given by equations (14-188) and (14-191) have units of mol/area-time for surface-catalyzed chemical reactions. However, rate data obtained from heterogeneous catalytic reactors are typically reported in units of mol/time per mass of catalyst. One obtains these units simply by multiplying the kinetic rate law (i.e., mol/area-time) by the internal surface area per mass of catalyst (i.e., S ), which is usually on the order of 100 m /g. If the feed stream to a packed catalytic reactor contains pure ethanol, then the initial reactant product conversion rate for the four-step mechanism is... 

Derive the Hougen-Watson kinetic rate law for the generic chemical reaction... 

This is a mathematical expression for the steady-state mass balance of component i at the boundary of the control volume (i.e., the catalytic surface) which states that the net rate of mass transfer away from the catalytic surface via diffusion (i.e., in the direction of n) is balanced by the net rate of production of component i due to multiple heterogeneous surface-catalyzed chemical reactions. The kinetic rate laws are typically written in terms of Hougen-Watson models based on Langmuir-Hinshelwood mechanisms. Hence, iR ,Hw is the Hougen-Watson rate law for the jth chemical reaction on the catalytic surface. Examples of Hougen-Watson models are discussed in Chapter 14. Both rate processes in the boundary conditions represent surface-related phenomena with units of moles per area per time. The dimensional scaling factor for diffusion in the boundary conditions is... 

Most important, heterogeneous surface-catalyzed chemical reaction rates are written in pseudo-homogeneous (i.e., volumetric) form and they are included in the mass transfer equation instead of the boundary conditions. Details of the porosity and tortuosity of a catalytic pellet are included in the effective diffusion coefficient used to calculate the intrapellet Damkohler number. The parameters (i.e., internal surface area per unit mass of catalyst) and Papp (i.e., apparent pellet density, which includes the internal void volume), whose product has units of inverse length, allow one to express the kinetic rate laws in pseudo-volumetric form, as required by the mass transfer equation. Hence, the mass balance for homogeneous diffusion and multiple pseudo-volumetric chemical reactions in one catalytic pellet is... 

The latter two conditions indicate that reactant concentration within the catalyst vanishes at the critical spatial coordinate when 0 < criticai < H and it does so with a zero slope. Conditions 2a and 3 are reasonable because reactant A will not diffuse further into the catalyst, to smaller values of r), if it exhibits zero flux at ]criticai. When / critical < 0, couditiou 2b must be employed, which is consistent with the well-known symmetry condition at the center of the catalyst for kinetic rate laws where lEl constant. Zeroth-order reactions are unique because they require one to implement a method of turning ofF the rate of reaction when no reactants are present. Obviously, a zeroth-order rate law always produces the same rate of reaction because reactant molar densities do not appear explicitly in the chemical reaction term. Hence, the mass balance for homogeneous onedimensional diffusion and zeroth-order chemical reaction is solved only over the following range of the independent variable criticai < < 1. when Jiciiacai is... 

Hence, it is not possible to redefine the characteristic length such that the critical value of the intrapellet Damkohler number is the same for all catalyst geometries when the kinetics can be described by a zeroth-order rate law. However, if the characteristic length scale is chosen to be V cataiyst/ extemai, then the effectiveness factor is approximately A for any catalyst shape and rate law in the diffusion-limited regime (A oo). This claim is based on the fact that reactants don t penetrate very deeply into the catalytic pores at large intrapellet Damkohler numbers and the mass transfer/chemical reaction problem is well described by a boundary layer solution in a very thin region near the external surface. Curvature is not important when reactants exist only in a thin shell near T] = I, and consequently, a locally flat description of the problem is appropriate for any geometry. These comments apply equally well to other types of kinetic rate laws. 

Consider the following nine examples of diffusion and chemical reaction in porous catalysts where the irreversible kinetic rate law is only a function of the molar density of reactant A. Identify the problems tabulated below that yield analytical solutions for (a) the molar density of reactant A, and (b) the dimensionless correlation between the effectiveness factor and the intrapellet Damkohler number. 

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