Mass action rate expressions

These simplifying assumptions allow elimination of some reaction steps, and representation of free radical and short-lived intermediates concentrations in terms of the concentration of the stable measurable components, resulting in complex non-mass action rate expressions. 

Assuming the first two reactions are in equilibrium, expressions are found for the concentrations of the free radicals Cl and COCl in terms of the species CO, Cl2, and COCL2, and when these are substituted into the mass action rate expression of the third reaction, the rate becomes... 

A macroscopic, deterministic chemical reacting system consists of a number of different species, each with a given concentration (molecules or moles per unit volume). The word macroscopic implies that the concentrations are of the order of Avogadro s number (about 6.02 x 10 ) per liter. The concentrations are constant at a given instant, that is, thermal fluctuations away from the average concentration are negligibly small (more in section 2.3). The kinetics in many cases, but far from all, obeys mass action rate expressions of the type... 

Fig. 4.1 Schematic of reaction mechanism with elementary steps given in eq. (4.1) and mass action rate expressions in eq. (4.2).

Most analyses of kinetic data have the object of identifying the constants of a rate equation based on the law of mass action and possibly some mass transfer relation.. The law of mass action Is expressed In terms of concentrations of the participants, so ultimately the chemical composition must be known as a function of time. In the laboratory the chemical composition Is determined by some instrument that is suitably calibrated to provide the needed information. Titration, refractive index, density, chromatography, spectrometry, polarimetry, conductimetry, absorbance, magnetic resonance — all of these are used at one time or another to measure chemical composition. In some cases, the calibration to chemical composition is linear with the reading. 

Usually kc is the specific rate that is sought because the law of mass action is expressed in concentrations and is the basis of rate equations... 

Despite its limitations, the reversible Random Bi-Bi Mechanism Eq. (46) will serve as a proxy for more complex rate equations in the following. In particular, we assume that most rate functions of complex enzyme-kinetic mechanisms can be expressed by a generalized mass-action rate law of the form... 

Lattice Points positions in a unit cell occupied by atom, molecules, or ions Law of Definite Proportion law that states that different samples of the same compound always contain elemental mass percentages that are constant Law of Mass Action mathematical expression based on the ratio between products and reactants at equilibrium, an equation to determine the equilibrium rate constant Law of Multiple Proportions law that states when two elements combine to form more than one compound that the mass of one element compared to the fixed mass of... 

We are now able to obtain the collision theory approximation to the bimolecular rate constant k (T). Recall that the mass-action kinetics expression for the reaction rate q is... 

Numerous investigators have examined the kinetics of metal removal from petroleum fractions. The majority of studies have used conventional hydrotreating catalysts, Mo03 supported on y-Al203 and promoted with CoO or NiO. Metal removal rates have generally been interpreted using simple mass action kinetic expressions of the form... 

Analyses of kinetic data are based on identifying the constants of a rate equation involving the law of mass action and some transfer phenomena. The law of mass action is expressed in terms of concentrations of the species. Therefore, the chemical composition is required as a function of time. Laboratory techniques are used to determine the chemical composition using an instrument that is suitably calibrated to give the required data. The techniques used are classified into two categories, namely chemical and physical methods. 

Often the key entity one is interested in obtaining in modeling enzyme kinetics is the analytical expression for the turnover flux in quasi-steady state. Equations (4.12) and (4.38) are examples. These expressions are sometimes called Michaelis-Menten rate laws. Such expressions can be used in simulation of cellular biochemical systems, as is the subject of Chapters 5, 6, and 7 of this book. However, one must keep in mind that, as we have seen, these rates represent approximations that result from simplifications of the kinetic mechanisms. We typically use the approximate Michaelis-Menten-type flux expressions rather than the full system of equations in simulations for several reasons. First, often the quasi-steady rate constants (such as Ks and K in Equation (4.38)) are available from experimental data while the mass-action rate constants (k+i, k-i, etc.) are not. In fact, it is possible for different enzymes with different detailed mechanisms to yield the same Michaelis-Menten rate expression, as we shall see below. Second, in metabolic reaction networks (for example), reactions operate near steady state in vivo. Kinetic transitions from one in vivo steady state to another may not involve the sort of extreme shifts in enzyme binding that have been illustrated in Figure 4.7. Therefore the quasi-steady approximation (or equivalently the approximation of rapid enzyme turnover) tends to be reasonable for the simulation of in vivo systems. 

Composition The law of mass action is expressed as a rate in terms of chemical compositions of the participants, so nltimately the variation of composition with time mnst be fonnd. The composition is determined in terms of a property that is measnred by some instm-ment and calibrated in terms of composition. Among the measnres that have been used are titration, pressure, refractive index, density, chromatography, spectrometry, polarimetry conductimetry absorbance, and magnetic resonance. In some cases the composition may vary linearly with the observed property, but in every case a cahbration is needed. Before kinetic analysis is undertaken, the data are converted to composition as a function of time (C, t), or to composition and temperature as functions of time (C, T, t). In a steady CSTR the rate is observed as a function of residence time. 

Species rate expressions are assumed to obey mass action kinetics, expressed in terms of concentration as follows ... 

It is obvious to the user at this juncture that the subject of environmental chemical fate models enjoys many individual mass transfer processes. Besides this, the flux equations used for the various individual processes are often based on different concentrations such as Ca, Cw, Cs, and so on. Since concentration is a state variable in all EC models, the transport coefficients and concentrations must be compatible. Several concentrations are used because the easily measured ones are the logical mass-action rate drivers for these first-order kinetic mechanisms. Unfortunately, the result is a diverse set of flux equations containing various mechanism-oriented rate parameters and three or more media concentrations. Complications arise because the individual process parameters are based on a specific concentration or concentration difference. As argued in Chapter 3, the fiigacity approach is much simpler. Conversions to an alternative but equivalent media chemical concentration are performed using the appropriate thermodynamic equilibrium statement or equivalent phase partition coefficients. The process was demonstrated above in obtaining the overall deposition velocity Equation 4.9. In this regard, the key purpose of Table 4.2 is to provide the user with the appropriate transport rate constant compatible with the concentration chosen to express the flux. Eor each interface, there are two choices of concentration... 

The constants k1 and k 1 are, respectively, the forward and backward rate constants and their ratio can be expressed by the law of mass action as... 

In these equations the independent variable x is the distance normal to the disk surface. The dependent variables are the velocities, the temperature T, and the species mass fractions Tit. The axial velocity is u, and the radial and circumferential velocities are scaled by the radius as F = vjr and W = wjr. The viscosity and thermal conductivity are given by /x and A. The chemical production rate cOjt is presumed to result from a system of elementary chemical reactions that proceed according to the law of mass action, and Kg is the number of gas-phase species. Equation (10) is not solved for the carrier gas mass fraction, which is determined by ensuring that the mass fractions sum to one. An Arrhenius rate expression is presumed for each of the elementary reaction steps. 

In the following discussion, we continue with the simple model for the combination of a ligand with its binding sites that was introduced in Section 1.2.1 (Eq. (1.1)). Assuming as before that the law of mass action applies, the rate at which receptor occupancy (pAR) changes with time should be given by the expression ... 

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 ... 

For reversible reactions one normally assumes that the observed rate can be expressed as a difference of two terms, one pertaining to the forward reaction and the other to the reverse reaction. Thermodynamics does not require that the rate expression be restricted to two terms or that one associate individual terms with intrinsic rates for forward and reverse reactions. This section is devoted to a discussion of the limitations that thermodynamics places on reaction rate expressions. The analysis is based on the idea that at equilibrium the net rate of reaction becomes zero, a concept that dates back to the historic studies of Guldberg and Waage (2) on the law of mass action. We will consider only cases where the net rate expression consists of two terms, one for the forward direction and one for the reverse direction. Cases where the net rate expression consists of a summation of several terms are usually viewed as corresponding to reactions with two or more parallel paths linking reactants and products. One may associate a pair of terms with each parallel path and use the technique outlined below to determine the thermodynamic restrictions on the form of the concentration dependence within each pair. This type of analysis is based on the principle of detailed balancing discussed in Section 4.1.5.4. 

The rate of gradual destruction of active sites and pore structure can be expressed as a mass action relation, for instance a second order reaction,... 

It should be emphasized that Eq. (22) is already based on a number of preconditions. In particular, the intracellular medium may significantly deviate from a well-stirred ideal solution [141 143], While the use of Eq. (22) is often justified, several authors have suggested to allow noninteger exponents in the expression of elementary rate equations [96,142,144] corresponding to a more general form of mass-action kinetics. A related concept, the power-law formalism, developed by M. Savageau and others [145 147], is addressed in Section VII.C. 

In the case of classic chemical kinetics equations, one can get in a few cases analytical solution for the set of differential equations in the form of explicit expressions for the number or weight fractions of i-mcrs (cf. also treatment of distribution of an ideal hyperbranched polymer). Alternatively, the distribution is stored in the form of generating functions from which the moments of the distribution can be extracted. In the latter case, when the rate constant is not directly proportional to number of unreacted functional groups, or the mass action law are not obeyed, Monte-Carlo simulation techniques can be used (cf. e.g. [2,3,47-52]). This technique was also used for simulation of distribution of hyperbranched polymers [21, 51, 52],... 

The kinetic behavior of the reductive dissolution mechanisms given in Figure 2 can be found by applying the Principle of Mass Action to the elementary reaction steps. The rate expression for precursor complex formation via an inner-sphere mechanism is given by ... 

In the absence of an enzyme, the reaction rate v is proportional to the concentration of substance A (top). The constant k is the rate constant of the uncatalyzed reaction. Like all catalysts, the enzyme E (total concentration [E]t) creates a new reaction pathway, initially, A is bound to E (partial reaction 1, left), if this reaction is in chemical equilibrium, then with the help of the law of mass action—and taking into account the fact that [E]t = [E] + [EA]—one can express the concentration [EA] of the enzyme-substrate complex as a function of [A] (left). The Michaelis constant lknow that kcat > k—in other words, enzyme-bound substrate reacts to B much faster than A alone (partial reaction 2, right), kcat. the enzyme s turnover number, corresponds to the number of substrate molecules converted by one enzyme molecule per second. Like the conversion A B, the formation of B from EA is a first-order reaction—i. e., V = k [EA] applies. When this equation is combined with the expression already derived for EA, the result is the Michaelis-Menten equation. 

What factors afiect the rate of homogeneous and heterogeneous chemical reactions What is called the order of a reaction Give-examples of first- and second-order reactions. Write a mathematical expression of the law of mass action for a first- and second-order irreversible reactions. 

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