Reaction vector

For simplicity we speak of a mechanism or a reaction, rather than a mechanism vector or reaction vector. The distinction lies in the fact that a reaction r (or mechanism) is essentially the same whether its rate of advancement is p or a, whereas pr and or are different vectors (for p a). Therefore, a reaction could properly be defined as a one-dimensional vector space which contains all the scalar multiples of a single reaction vector, but the mathematical development is simpler if a reaction is defined as a vector. This leaves open the question of when two reactions, or two mechanisms, are essentially different from a chemical viewpoint, which will be taken up... 

In treating systems of biochemical reactions it is convenient to use the fundamental equation for G in matrix form (Alberty, 2000b). The extent of reaction t for a chemical reaction was discussed earlier in Section 2.1. For a system of chemical reactions, the extent of reaction vector , is defined by... 

Fig. 11.6. Reaction vectors in exergy-enthalpy diagrams (a) a non-spontaneous process giving a single reaction vector, (b) coupling and coupled processes leading to a composite reaction vector.

The right-hand side of Eqs. (3) and (4) consist each of two terms. The first term represents the effect of distillation (separation vector), and the second the effect of the chemical reaction (reaction vector). For nonreactive systems only the separation vector plays a role. For kinetically controlled systems, both vectors can dominate the system behavior, depending on the value of Da. For Da —> °°, the liquid mixture approaches chemical equilibrium. 

Equation (20) must be satisfied for all possible reaction vectors hence, = 0. Now, given any C-dimensional vector B, the distribution a(x) B is orthogonal to a(y,u) = B = 0. Since the set a(jc) is complete, only distributions expressible as p(x) B are orthogonal to a(y,u). Now suppose that, from some independent argument, one has ascertained that some distribution j(jc), otherwise unknown, is orthogonal to a(y,u) we will see that, under appropriate qualifying conditions, the equilibrium distribution of chemical potentials, //.(x), falls into this category. Then there must exist a C-dimensional vector B such that (x) = a(x) B. Thus the search for an unknown function j(x) (an infinite-dimensional vector) is reduced to the search for a C-dimensional vector. 

As the chemical reactions proceed, q changes in time its rate of change is called the rate of reaction vector r ... 

Again, I is a vector to which all reaction vectors mj — m2 are orthogonal. However, in this case the total mass of odd-valued x-olefins is also constant, and hence one has also a second basis vector in the orthogonal complement ... 

No reaction vector in the AR boundary can point out of the AR. If this were the case, the AR could be extended further by PFR reactors, which have trajectories that are always tangent to the rate vectors. 

No reversed reaction vector in the complement of the AR can point back into the AR. Note that a CSTR can be represented in the AR by a line with ends at the feed and outlet concentrations, with the rate vector at the CSTR outlet collinear with this line. Thus, this condition ensures that the AR cannot be extended further by a CSTR. 

For clarity, the index i will hereafter be used for steps, the index j for species, and the index k for mechanisms. A reaction vector r, associated with each mechanism, describes the net transformation that is accomplished by the mechanism ... 

A O and AD values of vent fluids (Table 1) can best be understood in terms of water-rock interaction within the ocean crust. Field studies, experimental studies, and isotopic exchange computations (Muehlenbachs 1972 Stakes and O Neil 1982 Bowers and Taylor 1985 Cole et al. 1987 Bowers 1989 Bohlke and Shanks 1994 Shanks et al. 1995) have clearly shown that both A 0 and AD increase due to water-rock interaction with igneous crust. Oxygen and hydrogen isotope values of end-member vent fluids (Fig. 6) follow a calculated seawater-basalt reaction vector (within AD error of 1.5%o ), due to fluid evolution to decreasing water/rock mass ratios (Shanks et al. 1995). 

CSTR. The mixing vector v is collinear with the scaled reaction vector because c-c = rr(c). 

This gives the chemical length of the reaction vector R, so it represents the number of "electron pushing arrows in the synthesis L,R) of Z thus, it represents the "complexity of chemical operations."... 

By increasing the Damkohler number, the influence of the chemical reaction is fortified. For Da = 10 (Fig. 5.16b) the shape of trajectories starting with a relatively high mole fraction of MeOH is still similar to those of distillation without reaction, so that pure MeOH remains a stable node in the system. Near the MTBE vertex, the conditions change remarkably. The reaction vector is pointed towards the chemical equilibrium line (dashed curve). As a consequence, the stable node moves from... 

Further enhancement of Damkohler number to Da = 2 X 10 (Fig. 5.16c) leads to the situation in which the lower stable node coincides with the saddle point so that they extinguish each other. Hence the trajectories run into pure MeOH. At Da = 1 (Fig. 5.16d) the reaction vector is dominant in relation to the separation vector. Thus every residue curve is dominated by the stiochiometric restriction when moving towards the curve of chemical equilibrium. Because of the dominance of chemical reaction, the trajectories do not pass this line, but remain on it until they reach the pure MeOH vertex. 

The difference between the liquid composition vectors x and x is given in terms of a separation vector x— y) and a reaction vector Ya = i Vk )- Given a value for F, a choice of (p sets all of the flows and Da determines the reactor volume. The yield of the desired product B and the conversion of A is found from the combined streams L and Y, and the overall outlet composition vector is given by ((1 - [Pg.149]

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