Stoichiometric reactions generalized equation

SIT ion interaction coefficient between substance B and substance 62 stoichiometric coefficient of substance B (negative for reactants, positive for products) stoichiometric coefficient general equation for a chemical reaction equilibrium constant... 

For more complicated redox reactions, a general fonn of the Nemst equation may be derived by analogy with A2.4.113. If we consider a stoichiometric reaction of the following type ... 

Equation (2.39) is a generalization to M reactions of the stoichiometric constraints of Equation (2.35). If the vector e is known, the amounts of all N components that are consumed or formed by the reaction can be calculated. 

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 general equation, above, is no longer accurate above an x of 1. Beyond x = [n - (p/2)] = 1 (when p = 0), where water is a product, the heat of reaction is determined by the phase of the product water. At still higher values, the excess oxygen oxidizes the hydrogen to produce water. Finally, at stoichiometric combustion, all carbon and hydrogen are converted to carbon dioxide and water. 

The electrochemical formation of a radical ion from an aromatic compound or other highly conjugated species is, generally, fast and, therefore, the kinetics of the heterogeneous electron transfer process usually do not interfere with the kinetics of the follow-up reactions to be studied. For species with only one or two double bonds, the initial electron transfer process is often slow and may even be rate determining. In such cases, the kinetics of the follow-up reactions may be studied only with some difficulty. One method is to use a so-called mediator (Med) which serves to shuttle electrons between the substrate and the electrode. Thus, the slow electron transfer between the substrate and the electrode is replaced by two fast electron transfer processes, between the mediator and the electrode, and between the oxidised or reduced mediator and the substrate. In this event, the single reaction of Equation 6.4 is replaced by the two reactions in Scheme 6.8 [32 ]. It is seen that the mediator is recycled and consequently needs be present in only small, non-stoichiometric amounts. 

Equation 5-60 is the generalized equation, which can be used to determine the reaction time of an nth order reaction with b, the stoichiometric coefficient of the product component B. 

The general equations for chemical reaction in a turbulent medium are easy to write if not to solve (2). In addition to those for velocities (U = U + uJ and concentrations (Cj = Cj + Cj), balance equations for q = A u, the segregation ( , and the dissipations e and eg can be written (3). Whatever the shape of the reactor under consideration (usually a tube or a stirred tank), the solution of these equations poses difficult problems of closure, as u S, 5 cj, cj, and also c cj, c Cj in the reaction terms have to be evaluated. The situation is even more complicated when the temperature and the density of the reacting mixture are also fluctuating. Partial solutions to this problem have been proposed. In the case of instantaneous reactions (t << Tg) the "e-quilibrium assumption" applies the mixed reactants are immediately converted and the apparent rate of reaction is simply that of the decrease of segregation, with Corrsin s time constant xs. For instance, with a stoichiometric proportion of reactants, the extent of reaction X is given by 1 - /T ( 2), a simple result which can also be found by application of the IEM model (see (33)). 

We distinguish integral and differential characteristics of catalytic properties. One of the integral characteristics is the extent of reaction Generally, any chemical reaction, whether overall or an elementary step, can be represented by a stoichiometric equation ... 

We see from Eq. (15.3) that either the v, s or e must be expressed in molest and that the other quantity must be a pure number. As a matter of convenience we choose to express the reaction coordinate e in moles. This allows one to speak of a mole of reaction, meaning that e has changed by a unit amount, i.e., by one, mole. When Ae = 1 mol, the reaction proceeds to such an extent that the chaise in mole number of each reactant artd product is equal to its stoichiometric number. When two or more independent reactions proceed simultaneously, we let subscript j be the reaction index, and associate a separate reaction coordinate with each reaction. The stoichiometric numbers are doubly subscripted to identify , their association with both a species and a reaction. Thus designates th stoichiometric number of species i in reaction j. Since the number of moles of i species n,- may change because of several reactions, the general equation analogous to Eq. (15.3) includes a sum ... 

If liquid ethyl bromide is shaken with water at 25 C, no appreciable reaction takes place even after several days. The aciiieous phase will not show a Br test with Ag+, and the original reactants may be recovered unchanged. With -butyl bromide, on the other hand, one finds a fairly vigorous reaction with water at 25°C, accompanied by the liberation of heat, to produce -butyl alcohol and HBr. With bcnzhydryl bromide, (CeH6)2CHBr, the hydrolysis reaction appears to be almost immeasurably fast. Although all of these reactions can be represented stoichiometrically by the same general equation,... 

Wlien two or more independent reactions proceed simultaneously, subscript j serves as the reaction index. A separate reaction coordinate j then applies to each reaction. The stoichiometric numbers are doubly subscripted to identify their association with botli a species and a reaction. Thus vij designates the stoichiometric number of species / in reaction j. Since the number of moles of a species n, may change because of several reactions, tlie general equation analogous to Eq. (13.3) includes a sum ... 

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 1,3-disubstituted phosphinite (171) (AP) 149 ppm) can be prepared by reaction of the precursor diol with two equivalents of Pr 2PCl in THF with DMAP (Equation (42)).398 The unsymmetrical tertiary phosphite (172) was synthesized by reaction of two equivalents of phenol with the dichloroprecursor (173) (Equation (43)).399 Similar to (173) is (174) (Scheme 12), prepared by stoichiometric reaction of the trisubstituted phenol and PC13. 0 This general approach was also employed in the preparation of the Q-symmetric phosphite (175). 1 The cyclic phosphite (175) is remarkably stable with respect to oxidation. Even after reflux in toluene or acetone/water in the presence of air for 24 h, no decomposition was observed, strongly contrasting with the behavior of P(OPh)3. Phosphite (175) displays very good stability to hydrolysis. 

Remember that in the laws of physical chemistry, stoichiometric numbers are algebraic coefficients. The general equation that is obtained as a result can equally be applied to a half-reaction occurring either in the direction of oxidation (Ve = -i-n) or reduction (Vg = - n) R provided that the reaction rate is also taken in algebraic terms following the direction of the redox reaction. 

Another class of hydrates is gas hydrates. Unlike the stoichiometric inorganic salt hydrates, gas hydrates are nonstoichiometric clathrate soUd crystalline substances consisting of a lattice formed by water molecules (host) and entrapped gas molecules (guest). They are stable under high pressure and low temperature. The hydrate formation reaction can be described by the following general equation ... 

If the process of chemical transformation goes in more than one stage, than such reaction is complex. Generally for complex reaction there is inconsistency between stoichiometric and kinetic equations. In the equation for the law of mass action for the complex reaction the exponents of the concentrations are some numbers, defined experimentally, and in most cases are not equal to the stoichiometric coefficients. 

An analytically tractable approximation of the general equation is used to exemplify this method for a network of chains with trifunctional branch points. In addition, it is shown how actually obtained conversions of the reactive groups in non-stoichiometric compositions may be estimated, and that T reaches a maximum at the stoichiometric composition provided n side reactions occur. 

The general rate functional in the reaction velocity equation is a power function of the concentrations of reactants Cj whose powers oj correspond to the absolute values of the integer coefficients v of the relevant stoichiometric equation... 

In the liquid hydrocarbon mixture, a great variety of sulfur compounds is present. Due to this and because only bulk concentrations are measured for a single lump, for example, total sulfur content, it is not possible to know a priori the value of stoichiometric coefficients for the chanical reactions that are taking place. To overcome this disadvantage, the following generalized equation has been proposed in the literature (Korsten and Hoffmann, 1996) ... 

Consider, for example, the general equilibrium reaction shown in equation 6.1, involving the solutes A, B, C, and D, with stoichiometric coefficients a, b, c, and d. 

Rates determined by monitoring different species in a chemical reaction need not have the same value. The rate R in equation A5.2 and the rate R in equation A5.3 will have the same value only if the stoichiometric coefficients of A and C in reaction A5.1 are the same. In general, the relationship between the rates R and R is... 

One molecule (or mole) of propane reacts with five molecules (or moles) of oxygen to produce three molecules (or moles) or carbon dioxide and four molecules (or moles) of water. These numbers are called stoichiometric coefficients (v.) of the reaction and are shown below each reactant and product in the equation. In a stoichiometrically balanced equation, the total number of atoms of each constituent element in the reactants must be the same as that in the products. Thus, there are three atoms of C, eight atoms of H, and ten atoms of O on either side of the equation. This indicates that the compositions expressed in gram-atoms of elements remain unaltered during a chemical reaction. This is a consequence of the principle of conservation of mass applied to an isolated reactive system. It is also true that the combined mass of reactants is always equal to the combined mass of products in a chemical reaction, but the same is not generally valid for the total number of moles. To achieve equality on a molar basis, the sum of the stoichiometric coefficients for the reactants must equal the sum of v. for the products. Definitions of certain terms bearing relevance to reactive systems will follow next. 

The kinetics of a coupled reacting system consisting of n stoichiometrically simple reactions is described generally by a set of n differential equations... 

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