Rate law, defined

Haim11 has said, Although the form of the rate law defines the composition of the activated complexes, the rate law does not specify (a) the order of formation of the activated complexes, (b) the species (reactants or intermediates) which generate the activated complexes, or (c) the decomposition products (intermediates or products) of the activated complexes. ... 

At equilibrium, the left side of Eq. 4.26 vanishes and the rate law defines an adsorption isotherm equation ... 

The reaction rate was found to obey the rate law defined by kobs = kx + k2/[H+] [54]. The acetamidate-bridged [Pt225]8 octamer [Pt8(NH3)16(acet-amidato)8]10+ was also reported to exhibit similar behavior after dissolution in aqueous media (Eqn. 2) [55]. 

Aii radioactive decay processes foiiow first-order kinetics. What does this mean What happens to the rate of radioactive decay as the number of nuciides is haived Write the first-order rate law and the integrated first-order rate law. Define the terms in each equation. What is the half-life equation for radioactive decay processes How does the half-life depend on how many nuclides are present Are the half-life and rate constant k directly related or inversely related ... 

So far, we have considered a simple reaction with only one reactant. How is the rate law defined for reactions with more than one reactant Consider the generic reaction ... 

Unlike the case we have just discussed, in most cases the integrals catmot be calculated algebraically and we must be content with rate laws defined by the integrals that are calculated numerically. 

Neither (A3.4.15) nor (A3.4.17) is of the fonn (A3,4,10) and thus neither reaction order nor a unique rate codficient can be defined. Indeed, the number of possible rate laws that are not of the fonu of (A3.4.10) greatly exceeds those cases following (A3.4.10). However, certain particularly simple reactions necessarily follow a law of type of (A3.4.10). They are particularly important from a mechanistic point of view and are discussed in the next section. 

It is convenient to analyse tliese rate equations from a dynamical systems point of view similar to tliat used in classical mechanics where one follows tire trajectories of particles in phase space. For tire chemical rate law (C3.6.2) tire phase space , conventionally denoted by F, is -dimensional and tire chemical concentrations, CpC2,- are taken as ortliogonal coordinates of F, ratlier tlian tire particle positions and velocities used as tire coordinates in mechanics. In analogy to classical mechanical systems, as tire concentrations evolve in time tliey will trace out a trajectory in F. Since tire velocity functions in tire system of ODEs (C3.6.2) do not depend explicitly on time, a given initial condition in F will always produce tire same trajectory. The vector R of velocity functions in (C3.6.2) defines a phase-space (or trajectory) flow and in it is often convenient to tliink of tliese ODEs as describing tire motion of a fluid in F with velocity field/ (c p). 

The relationship between the bore fluid pressure drop, AP and its flow rate is defined by Poiseuike s law ... 

The IUPAC has provided a convention used by many authors, but not all. It specifies that the elementary reaction written in Eq. (1 -9) has associated with it the defining rate law in Eq. (1-10). This removes any ambiguity. 

Several additional forms will be considered. Each of these represents a case in which the rate law does not show a defined order in [A]. We shall consider three situations. In addition to this treatment one should consider the numerical solutions in Chapter 5. 

Making the steady-state approximation for [PFe], derive the rate law. Next, repeat the derivation including the reverse step with k-2. If [CO] and [02] are s> [PFe(O2)]0, what is the expression for ke, as defined in Chapter 3 ... 

The route from kinetic data to reaction mechanism entails several steps. The first step is to convert the concentration-time measurements to a differential rate equation that gives the rate as a function of one or more concentrations. Chapters 2 through 4 have dealt with this aspect of the problem. Once the concentration dependences are defined, one interprets the rate law to reveal the family of reactions that constitute the reaction scheme. This is the subject of this chapter. Finally, one seeks a chemical interpretation of the steps in the scheme, to understand each contributing step in as much detail as possible. The effects of the solvent and other constituents (Chapter 9) the effects of substituents, isotopic substitution, and others (Chapter 10) and the effects of pressure and temperature (Chapter 7) all aid in the resolution. 

We have noted previously that the forward and reverse rates are equal at equilibrium. It seems, then, that one could use this equality to deduce the form of the rate law for the reverse reactions (by which is meant the concentration dependences), seeing that the form of the equilibrium constant is defined by the condition for thermodynamic equilibrium. By and large, this method works, but it is not rigorously correct, since the coefficients in the equilibrium condition are only relative, whereas those in the rate law are absolute.19 Thus, if we have this net reaction and rate law for the forward direction,... 

The rate law of a reaction is an experimentally determined fact. From this fact we attempt to learn the molecularity, which may be defined as the number of molecules that come together to form the activated complex. It is obvious that if we know how many (and which) molecules take part in the activated complex, we know a good deal about the mechanism. The experimentally determined rate order is not necessarily the same as the molecularity. Any reaction, no matter how many steps are involved, has only one rate law, but each step of the mechanism has its own molecularity. For reactions that take place in one step (reactions without an intermediate) the order is the same as the molecularity. A first-order, one-step reaction is always unimolecular a one-step reaction that is second order in A always involves two molecules of A if it is first order in A and in B, then a molecule of A reacts with one of B, and so on. For reactions that take place in more than one step, the order/or each step is the same as the molecularity for that step. This fact enables us to predict the rate law for any proposed mechanism, though the calculations may get lengthy at times." If any one step of a mechanism is considerably slower than all the others (this is usually the case), the rate of the overall reaction is essentially the same as that of the slow step, which is consequently called the rate-determining step. ... 

The simplest form of a physicochemical reaction takes place when one species simply changes to another. This can be written in a general way as A B. The rate of such a reaction is defined as the amount of reactant (the reacting species, A, in this case) or equivalently the product (B) that changes per unit time. The key feature here is the form of the rate law, i.e., the expression for the dependence of the reaction rate on the concentrations of the reactants. For a first-order reaction... 

This simple example of a non-catalytic reaction demonstrates how a reaction rate law may be comprehensively defined in two substrates by just two reaction progress experiments employing two different values of excess [e]. A classical kinetics approach using initial rate measurements would require perhaps a dozen separate initial rate or pseudo-zero-order experiments to obtain the same information. 

For part (a), assume that the system is at equilibrium and that the law of mass action holds. Use the procedures described in Chapter 1 to derive an expression forpAB, at equilibrium. At equilibrium, the forward and backward rates for each reaction in the mechanism must be equal. The forward and backward rates are defined using the law of mass action ... 

Now that we have a definition of rate, we need to develop a basic model to describe the rates of chemical reactions. The basic model takes the form of a rate law. A general rate law allows us to relate the rate of a reaction to the concentration of the reactants present. It makes sense, intuitively, that at low reactant concentrations the rate of the reaction will be quite slow. Also, at high reactant concentrations in the same reaction we would expect a faster rate. The general rate law reflects this intuition by stating that, in simple reactions, such as the one in Eq 4.1, we define the rate according to Eq. 4.4. 

Once we have a defined rate law, we can predict the concentration of any species in the reaction at any time in the reaction by integrating the rate law equation with respect to time. By doing this, the concentration of a species at some time, f, relative to its initial concentration at time zero can be determined. 

In this work, a detailed kinetic model for the Fischer-Tropsch synthesis (FTS) has been developed. Based on the analysis of the literature data concerning the FT reaction mechanism and on the results we obtained from chemical enrichment experiments, we have first defined a detailed FT mechanism for a cobalt-based catalyst, explaining the synthesis of each product through the evolution of adsorbed reaction intermediates. Moreover, appropriate rate laws have been attributed to each reaction step and the resulting kinetic scheme fitted to a comprehensive set of FT data describing the effect of process conditions on catalyst activity and selectivity in the range of process conditions typical of industrial operations. 

This equation is known as a rate law. It tells you how the rate of the reaction depends on the concentration(s) of the substrate. The order of the reaction is defined as the power to which the substrate concentration is raised when it appears in the rate law. In the preceding case, [A] is raised to the first power ([A]1), so the reaction is said to be first-order with respect to the A concentration, or simply first-order in A. The rate constant k is a proportionality constant thrown in so that the equation works and so that the units work out. Since v must have units of molar per second Mls) and [A] has molar units (M), then k must have units of reciprocal seconds (1/s or s ). 

In this chapter we consider how to construct reaction models that are somewhat more sophisticated than those discussed in the previous chapter, including reaction paths over which temperature varies and those in which species activities and gas fugacities are buffered. The latter cases involve the transfer of mass between the equilibrium system and an external buffer. Mass transfer in these cases occurs at rates implicit in solving the governing equations, rather than at rates set explicitly by the modeler. In Chapter 16 we consider the use of kinetic rate laws, a final method for defining mass transfer in reaction models. 

To run the simulation, we decouple acetate from carbonate, and sulfide from sulfate, and suppress the iron sulfide minerals pyrite and troilite (FeS), which are more stable than mackinawite, but unlikely to form. We set the fluid composition, including an amount of HS small enough to avoid significantly supersaturating mackinawite, and define the rate law for the sulfate reducers. The procedure in REACT is... 

A parabolic rate law will also be obtained if part or even all, of the diffusion through the product layer is by grain boundary diffusion rather than diffusion through the volume of each grain. The volume diffusion coefficient is quite simply defined as the phenomenological coefficient in Fick s laws. The grain boundary diffusion must be described by a product, DbS, where S is the grain... 

Outer-sphere (OS) reaction rates and rate laws can be defined for solvolysis of a given complex. Complex formation is defined as the reverse reaction—that is, replacement of solvent (S) by another ligand (L )- Following the arguments of... 

Note that the rate of formation of A is rA, as defined in section 1.4 for a reactant, this is a negative quantity. The rate of disappearance of A is (-rA), a positive quantity. It is this quantity that is used subsequently in balance equations and rate laws for a reactant. For a product, the rate of formation, a positive quantity, is used. The symbol rA may be used generically in the text to stand for rate of reaction of A where the sign is irrelevant and correspondingly for any other substance, whether reactant or product. 

The concentration c, in equation 4.1-3, the rate law, is usually expressed as a molar volumetric concentration, equation 2.2-7, for any fluid, gas or liquid. For a substance in a gas phase, however, concentration may be expressed alternatively as partial pressure, defined by... 

Central to catalysis is the notion of the catalytic site. It is defined as the catalytic center involved in the reaction steps, and, in Figure 8.1, is the molybdenum atom where the reactions take place. Since all catalytic centers are the same for molecular catalysts, the elementary steps are bimolecular or unimolecular steps with the same rate laws which characterize the homogeneous reactions in Chapter 7. However, if the reaction takes place in solution, the individual rate constants may depend on the nonreactive ligands and the solution composition in addition to temperature. 

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