Stoichiometric coefficient, various

The effectiveness of inhibitors is measured in terms of the rate constant ratio kz/kp and the stoichiometric coefficient. The stoichiometric coefficient is the moles of radicals consumed per mole of inhibitor. These parameters may be determined by various methods. A brief description of the classical kinetic treatment for evaluating k7/kp follows. Consider the reaction scheme shown which describes ideal inhibition and retardation (Scheme 5.11). 

The choice is a matter of personal convenience. The essential point is that the ratios of the stoichiometric coefficients are unique for a given reaction i.e., vC0/v02 = ( —2/—1) = [ —1/( —1/2)] = 2. Since the reaction stoichiometry can be expressed in various ways, one must always write down a stoichiometric equation for the reaction under study during the initial stages of the analysis and base subsequent calculations on this reference equation. If a consistent set of stoichiometric coefficients is used throughout the calculations, the results can be readily understood and utilized by other workers in the field. 

As equation 2.4.8 indicates, the equilibrium constant for a reaction is determined by the temperature and the standard Gibbs free energy change (AG°) for the process. The latter quantity in turn depends on temperature, the definitions of the standard states of the various components, and the stoichiometric coefficients of these species. Consequently, in assigning a numerical value to an equilibrium constant, one must be careful to specify the three parameters mentioned above in order to give meaning to this value. Once one has thus specified the point of reference, this value may be used to calculate the equilibrium composition of the mixture in the manner described in Sections 2.6 to 2.9. 

The reaction rate is properly defined in terms of the time derivative of the extent of reaction. It is necessary to define k in a similar fashion in order to ensure uniqueness. Definitions in terms of the various rt would lead to rate constants that would differ by ratios of their stoichiometric coefficients. 

The rate expressions have been written in generalized fashion with the terms fp, fb, fst, and fgt containing the reaction rate constants, stoichiometric coefficients, and concentrations of the various stable species present in the reaction mixture. If one also wished to consider bi-molecular radical processes, these could also be lumped into the / parameters. 

Dialkyl thiophosphates can also terminate chains by reacting with peroxyl radicals, as follows from their ability to inhibit the initiated oxidation of hydrocarbons and to extinguish CL in oxidizing hydrocarbon [66,68], The stoichiometric coefficient of inhibition/for various dialkyl thiophosphates is close to 2, as follows from the data for diisopropyl dithiopho-sphates ... 

Numerous investigations have shown the existence of the heptamolybdate, [Mo7024]6 , and octamolybdate, [Mo8026]4, ions in aqueous solution. Potentiometric measurements with computer treatment of the data proved to be one of the best methods to obtain information about these equilibria. Stability constants are calculated for all species in a particular reaction model, which is supposed to give the best fit between calculated and experimental points. In the calculations the species are identified in terms of their stoichiometric coefficients as described by the following general equation for the various equilibria... 

The reactions that are possible in a closed vessel are restricted by conservation laws for the atoms involved. Let a label the various kinds of atoms and suppose Xj contains m atoms of kind a, where m = 0,1,2,. Then the stoichiometric coefficients of (1.1) obey for each a... 

The first term on the right is the total standard molar entropy of the products and the second term is that of the reactants n denotes the various stoichiometric coefficients in the chemical equation. Example 7.7 illustrates how this expression is used. 

Thus the presence of steps for the interaction between various intermediates in the detailed mechanisms is only a necessary condition for the multiplicity of steady states in catalytic reactions. A qualitative analysis of the dynamic system (5) for mechanism (4) showed that the existence of several stable steady states with a non-zero reaction rate needs the following additional conditions (a) the stoichiometric coefficients of intermediates must fit definite relationships ensuring the kinetic competition of these substances [violation of conditions (6)] (b) the system parameters must satisfy definite inequalities. 

In the usual case, t and ain will be known. Equation (1.49) is an algebraic equation that can be solved for aout. If the reaction rate depends on the concentration of more than one component, versions of Equation (1.49) are written for each component and the resulting set of equations is solved simultaneously for the various outlet concentrations. Concentrations of components that do not alfect the reaction rate can be found by writing versions of Equation (1.49) for them. As for batch and piston flow reactors, stoichiometry is used to relate the rate of formation of a component, say Sl-c, to the rate of the reaction SI, using the stoichiometric coefficient vc, and Equation (1.13). After doing this, the stoichiometry takes care of itself. 

Whenever the kinetics of a chemical transformation can be represented by a single reaction, it is sufficient to consider the conversion of just a single reactant. The concentration change of the remaining reactants and products is then related to the conversion of the selected key species by stoichiometry, and the rates of production or consumption of the various species differ only by their stoichiometric coefficients. In this special case, the combined influence of heat and mass transfer on the effective reaction rate can be reduced to a single number, termed the catalyst efficiency or effectiveness factor rj. From the pioneering work of Thiele [98] on this subject, the expressions pore-efficiency concept and Thiele concept have been coined. 

Relative Rates of Reaction. The relative rates of reaction of the various species involved in a reaction can be obtained from the ratio of stoichiometric coefficients. For Reaction (2-2),... 

The form of equation (5.2) is useful in that it relates the rates of removal or production of the species in the stoichiometric equation. To specify how this rate depends on the concentrations of the various species involved in the reaction, we must return to the individual elementary steps that comprise the mechanism for this reaction. When dealing with a set of elementary steps, we can use stoichiometric coefficients of the form i>,j that specify the coefficient for species j in step i. The overall rate equation will then involve the rates of the individual elementary steps summed, as they involve the stoichiometric coefficients of the species in terms of which the overall rate has been expressed. 

Equations of this type are the simplest to deal with. The quantities a and b are termed the orders of the reaction with respect to A and B as mentioned previously while their sum, (a+b), is termed the total order of the reaction. The equation can be generalized without difficulty to cover the case in which the concentrations of more than two reactants enter the rate equation. It must be emphasized that the various orders cannot be equated to or deduced from the stoichiometric coefficients but must be obtained from the kinetic data by the methods to be explained. 

The form of the equilibrium condition is distinctly nonlinear because it asserts that the product of the concentrations raised to the power of the stoichiometric coefficients equals some constant depending on temperature and pressure. In the last chapter we saw that the various measures of concentration varied linearly with the extent or as the quotient of two linear expressions, and we might expect that the interaction of linear and nonlinear conditions would tend to make the calculations difficult. Let us look first at the simple reaction Ai — Aq — As = 0 whose stoichiometry we studied in Sec. 2,4, We shall use the molar concentrations, though any other measure of concentration could be used. 

It is made clear in Equations (25-57) and (25-58) by the stoichiometric coefficients x and y, that FeO in equilibrium with iron has a stoichiometry different from FeO in equilibrium with Fe304. In the region of stability of the FeO phase the stoichiometry varies from jr to with change in Pq. Thus, conditions can be adjusted by oxygen sensors to produce various defect concentrations. 

Thus the vector of the differences of concentrations is a product of a matrix for the general stoichiometric coefficients and the vector of the linear independent degrees of advancements. This relationship is valid for each time t,. For various times given as f (/ = 0,1,2,...,/n) these measured differences in concentration can be arranged in a matrix... 

Mass action forms of rate correlation, often referred to as power law correlations, are widely applied, particularly for reactions in homogeneous phases. Table 1.1 gives a representative selection of correlations for various reactions. It is seen in several of the examples that the reaction orders are not those to be expected on the basis of the stoichiometric coefficients. Since these orders are normally established on the basis of experimental observation, we may consider them correct as far as the outside world is concerned, and the fact that they do not correspond to stoichiometry is a sure indication that the reaction is not proceeding the way we have written it on paper. Thus we will maintain a further distinction between the elementary steps of a reaction and the overall reaction under consideration. The direct application of the law of mass action where the orders and the stoichiometric coefficients correspond will normally pertain only to the elementary steps of a reaction, as will the dependence of rate on temperature to be discussed in the next section. Also,... 

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