Rewriting reactions

Each time we change the basis, we must rewrite each chemical reaction in the system in terms of the new basis. This task, which might seem daunting, is quickly accomplished on a computer, using the transformation matrix. Consider the reaction to form an aqueous species, 

The new reaction coefficients for the species Aj, then, are simply the matrix products of the old coefficients and the transformation matrix  

Following our example from above, we write the reaction to form, as an example, carbonate ion, 

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To account for the decomposition of TiCuH, we proceed as follows although both x and y in Reaction 7 vary with temperature, AH remains reasonably constant with temperature and composition, as seen by the straight line in Figure 2. Such behavior is, in fact, observed experimentally in most metal hydrides (14). This being the case, we approximate the enthalpy of formation of TiCuH by rewriting Reaction 7 (Reaction 8) and assume AH for this reaction to be —75... 

The Lindemann treatment for association reactions is analogous to the theory just given for unimolecular reactions. For convenience, rewrite reactions 9.100 and 9.101 in the reverse directions... 

This result is the major clue to understanding the process the order shift reveals the specific N-reactant, and the reaction rate demonstrates that the arene radical cation is a key intermediate. The rationale goes as follows First, the addition of NO drives the N(IV) virtually fully to N203 and the N204-N02 balance shifts to favor the dioxide. N203 is not expected to be a reactant. As discussed earlier, the shift from first order to second order dictates that the tetroxide or 2N02 accompanies pyrene in the transition state. Because the N02 case would be the result of a trimolecular reaction, we conclude that the tetroxide is the specific species reacting with pyrene. Thus, we rewrite reactions 5 and 6 as follows ... 

Once again, these fluxes are not all independent and some care must be taken to rewrite everything so that syimnetry is preserved [12]. Wlien this is done, the Curie principle decouples the vectorial forces from the scalar fluxes and vice versa [9]. Nevertheless, the reaction temis lead to additional reciprocal relations because... 

In order to segregate the theoretical issues of condensed phase effects in chemical reaction dynamics, it is usefiil to rewrite the exact classical rate constant in (A3.8.2) as [5, 6, 7, 8, 9,10 and U]... 

But the key to success Is getting the right form of HBr for the reaction. A lot of people start off with the CA abstract that uses aqueous 70% HBr as the reagent (this abstract is essentially a rewrite of Merck s original patent) [59], The following is the write up ... 

Determining the Standard-State Potential To extract the standard-state potential, or formal potential, for reaction 11.34 from a voltammogram, it is necessary to rewrite the Nernst equation... 

The reverse reaction can be excluded if the value of K53 is very large. This characteristic is easily seen by rewriting the rate law as... 

How can one formulate a mechanism for a chain reaction when the rate law does not provide the composition of the transition state The process is an inexact one, but this guidance may be helpful. Factor the rate law mentally into components that suggest the multiplication of one reagent by one intermediate. Thus, for Eqs. (8-20)-(8-23), we might rewrite Eq. (8-24) to read... 

Rewrite the weathering reactions shown in Section 8.3.2.2 using HNO3 in place of H2CO3. 

Since enthalpy is a state variable, the integral on the right side of equation 10.2.6 is independent of the path of integration, and it is possible to rewrite this equation in a variety of forms that are more convenient for use in reactor design analyses. One may evaluate this integral by allowing the reaction to proceed isothermally at the initial temperature from extent 0 to extent and then heating the final product mixture at constant pressure and composition from the initial temperature to the final temperature. 

In the case of an irreversible reaction, therefore, two parameters are now defined with respect to the number of electrons transferred in the reaction n now refers to the electrons transferred overall, while na indicates the number of electrons participating in the rds. Thus, for example, we can rewrite equations (2,144) and (2.146) as ... 

Taking into consideration the acid-base dissociation equilibrium reaction of the ligand, one can obtain an expression for [L" ] and rewrite equation (61) as follows ... 

The example reactions considered in this section all have the property that the number of reactions is less than or equal to the number of chemical species. Thus, they are examples of so-called simple chemistry (Fox, 2003) for which it is always possible to rewrite the transport equations in terms of the mixture fraction and a set of reaction-progress variables where each reaction-progress variablereaction-progress variable —> depends on only one reaction. For chemical mechanisms where the number of reactions is larger than the number of species, it is still possible to decompose the concentration vector into three subspaces (i) conserved-constant scalars (whose values are null everywhere), (ii) a mixture-fraction vector, and (iii) a reaction-progress vector. Nevertheless, most commercial CFD codes do not use such decompositions and, instead, solve directly for the mass fractions of the chemical species. We will thus look next at methods for treating detailed chemistry expressed in terms of a set of elementary reaction steps, a thermodynamic database for the species, and chemical rate expressions for each reaction step (Fox, 2003). 

In Chapter 16, we wrote rate laws for simple dissolution and precipitation reactions, such as those for the silica minerals forming from SiC>2(aq). Rewriting Equation 16.22 in terms of volumetric concentration C , assuming the activity coefficient Yi does not vary over the reaction, gives the rate law,... 

A quantitative description must account for the band structure of the electrode, and can be formulated within the theory of electron-transfer reactions presented in Chapter 6. We start from Eq. (6.12) for the rate of electron transfer from a reduced state in the solution to a state of energy e on the electrode, and rewrite it in the form ... 

In contrast to the rotating disc electrode, mass transport to the ring is nonuniform. Nevertheless, the thickness of the diffusion layer Spj, which depends on the coordinate x in the direction of flow, and the rate of mass transport can be calculated. We consider a simple redox reaction, and rewrite Eq. (14.5) in the form ... 

But we need to be careful when talking about the magnitudes of Consider the case of sodium ethanoate dissolved in dilute mineral acid the reaction occurring is, in fact, the reverse of that in Equation (4.45), with a proton and carboxylate anion associating to form undissociated acid. In this case, = 1 mol before the reaction occurs, and its value decreases as the reaction proceeds. In other words, we need to define our reaction before we can speak knowledgeably about it. We can now rewrite our question, asking Why is < 1 for a weak acid ... 

The bracketed term is the reaction quotient, expressed in terms of pressures, allowing us to rewrite the equation in a less intimidating form of Equation (4.49) ... 

We conventionally cite the oxidized form first within each symbol, which is why the general form is o,r> so pb4+ Pb + is correct, but 2+ 4+ is not. Some people experience difficulty in deciding which redox state is oxidized and which is the reduced. A simple way to differentiate between them is to write the balanced redox reaction as a reduction. For example, consider the oxidation reaction in Equation (7.1). On rewriting this as a reduction, i.e. Al3+(aq) + 3e = A Em, the oxidized redox form will automatically precede the reduced form as we read the equation from left to right, i.e. are written in the correct order. For example, o,r for the couple in Equation (7.1) is Ai3+,ai-... 

Step 2 Rewrite the balanced equation at equilibrium for the reaction. 

Because the numerator contains the products of the reaction and the denominator contains the reactants, it is clear that the reaction needs to proceed more towards the reactants. We shall equilibrate this equation, by introducing a variable x, to represent the extent of the reaction to proceed to the right and rewriting the equilibrium equation as ... 

It has been shown in the previous chapters that the product of the electrochemical reactions in the sulphide flotation system is determined by the mixed potential of the flotation pulp. The value of the potential is dependent on the equilibrium of anodic and cathodic process existing in the pulp. In general, the most important cathodic reaction existing in the pulp is the oxygen reduction. To rewrite Eq. (1-1) as the following ... 

Here, r is the vector of all relevant particle coordinates and x is one coordinate chosen to define the reaction coordinate. The trick to rewriting this rate in a usable way is to treat both of the integrals that appear in it with the harmonic approximation we used in Section 6.1. For the denominator, we expand the energy surface around the energy minimum at r r ... 

In this section we discuss the thermochemistry of some species with the general formulas, RC(=0)00H, RC(=0)00R and RC(=0)00C(=0)R. These species have the generic names peracids (peroxycarboxyUc acids), peresters (percarboxylate esters) and acyl peroxides. The enthalpy of formation values are in Table 3. Three formal reactions that are discussed here are conceptually the same as in the earlier sections. Because now there is a carbonyl group present, we rewrite equations 5, 6 and 9 as equations 14, 15 and 16. 

Mathematical Analysis. Reactions 3 to 5 are first-order or pseudo-first-order reactions. Thus, the pseudo-first-order constant for Reaction 3 is ksE. For brevity we rewrite the two intermediates (E2+ 02H A )2 and (+E—E+ 02H A )g as Ci and C2, respectively. We assume that the light intensity is proportional to C2 during any one run. Thus, dl/dt = dC2/dt and the calculated decay of C2 can be related to k0. 

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