Independent reactions, determination

Figure 1.8. Schematic frequency distributions for some independent (reaction input or control) resp. dependent (reaction output) variables to show how non-Gaussian distributions can obtain for a large population of reactions (i.e., all batches of one product in 5 years), while approximate normal distributions are found for repeat measurements on one single batch. For example, the gray areas correspond to the process parameters for a given run, while the histograms give the distribution of repeat determinations on one (several) sample(s) from this run. Because of the huge costs associated with individual production batches, the number of data points measured under closely controlled conditions, i.e., validation runs, is miniscule. Distributions must be estimated from historical data, which typically suffers from ever-changing parameter combinations, such as reagent batches, operators, impurity profiles, etc.

Once the number of independent reactions has been determined, an independent subset can be chosen for subsequent calculations. 

When no observable changes of pressure were observed (equilibrium), the catalyst was added by rotating the glass jar. The catalyst dissolved in 10-30 sec, depending on olefin and catalyst concentrations. The pressure drop was recorded as a function of time by the pressure transducer and a recorder. The reaction rate was determined by measuring the slope of the tangent to the curve of the pressure drop at the point corresponding to the desired H2-partial pressure and the vapor pressure of the reaction mixture. The variation between two independent rate determinations at the same conditions was always less than 10% of the absolute value. 

Remark 7. The number of independent reactions in a given set lr is the rank of the stoicheiometric matrix and may be determined by elementary row operations. 

In the s dimensional space of concentrations the state of the system can be represented by the point c = (c1,. .., cs). Evidently this point can only move in an r dimensional subspace if the composition changes are due to only r independent reactions. The orientation of this subspace is given by the matrix of stoicheiometric coefficients, but its position is fixed by the initial composition. Indeed the stoicheiometry of the possible reactions can be determined from the atomic matrix by the following theorem this does not of course imply that all, or even any, of the possible reactions do take place. 

Such a definition can, evidently, be extended to any number of routes. It is clear that if A(1), A(2), A<3) are routes of a given reaction, then any linear combination of these routes will also be a route of the reaction (i.e., will produce the cancellation of intermediates). Obviously, any number of such combinations can be formed. Speaking in terms of linear algebra, the reaction routes form a vector space. If, in a set of reaction routes, none can be represented as a linear combination of others, then the routes of this set are linearly independent. A set of linearly independent reaction routes such that any route of the reaction is a linear combination of these routes of the set will be called the basis of routes. It follows from the theorems of linear algebra that although the basis of routes can be chosen in different ways, the number of basis routes for a given reaction mechanism is determined uniquely, being the dimension of the space of the routes. Any set of routes is a basis if the routes of the set are linearly independent and if their number is equal to the dimension of the space of routes. 

The mechanism of these reactions has been the subject of considerable speculation. In the first place, it is necessary to determine if 2 and 3 are independent reactions or if 3 gives rise to a hot cyclobutane molecule (denoted by an asterisk in 3a) which can decompose as in 3b unless the excess energy is removed by collisions. 

When the equilibrium constants for the reactions (A) and (B) are expressed in terms of the partial pressure of the various species (in atm), the equilibrium constants for these reactions have the values KpA = 0.046 and KpB = 0.034. Determine the number of independent reactions, and then determine the equilibrium composition of the mixture, making use of a simple MATLAB program that you develop for this purpose. 

A graph circle is a final sequence of the edges in which no node except the starting point occurs twice. A graph for the isomerization reaction has one circle, whereas that for vinyl chloride synthesis contains two circles. Every route of a chemical reaction corresponds to a graph s circle and vice versa. The number of independent reaction routes is equal to the number of elements in the basis of circles. It permits us to determine independent reaction routes from the graph type. 

Not all strained compounds are necessarily more reactive than less strained analogs. Reactivity will always depend on the type of reaction under scrutiny, and if the rate determining step of a given reaction is not accelerated by strain, the rate of reaction of strained and unstrained compounds will be similar. One example of such strain-independent reaction rates is the hydrolysis of lactams under basic reaction conditions (Scheme 3.8). Although /3-lactams are more strained than six-membered lactams, both are hydrolyzed at approximately the same rate, presumably because the rate determining step is the addition of hydroxide to the amide bond, and not... 

The kinetics of the inverse reaction (67) is determined by the concentration of polymers Pol(7V + 1) and the number of units which have one single bond with the basis part of the polymer, i.e., extreme units. Every extreme unit has m-vacancies, where m is the maximum number of possible branches on every unit. The number of extreme units on the kth level is denoted as Uk(N). Then, the independent reactions are written as ... 

If chemical reactions occur, then we must introduce a new variable, the i coordinate e for each independent reaction, in order to formulate the mate balance equations. Furthermore, we are able to write a new equilibrium rela [Eq. (15.8)] for each independent reaction. Therefore, when chemical-rea equilibrium is superimposed on phase equilibrium, r new variables appear r new equations can be written. The difference between the number of va and number of equations therefore is unchanged, and Duhem s theorem originally stated holds for reacting systems as well as for nonreacting syste Most chemical-reaction equilibrium problems are so posed that it is 1 theorem that makes them determinate. The usual problem is to find the corn-tion of a system that reaches equilibrium from an initial state of fixed an of reacting species when the two variables T and P are specified. 

A chemically reactive system contains the following spedes in the gas phase NH3, NO, N02, 02, and H20. Determine a complete set of independent reactions for this system. How many degrees of freedom does the system have ... 

Calculate AG° and Kfor each independent reaction. This may be done as in the relevant examples earlier in this section, with determination of AG° as a function of temperature. An easier route, however, is to use the standard Gibbs free-energy change of formation A Gy for each compound at the temperature of interest in the relationship... 

When the reactivity of the centre is determined not only by the last added unit but also by the last but one unit, we speak of the penultimate effect. Merz et al. treated this problem using eight independent reactions [200, 201 ]. 

The reactions in a set are independent if none of the reactions can be obtained by adding and subtracting other reactions in the set. (In Chapter 7 we will see that there is a more operational method for determining the number of independent reactions by use of linear algebra.) Experimental data at a specified T and P can be interpreted by use of... 

Once a generation or consumption term has been calculated for a species in a given reaction, the generation and consumption terms for all other species in that reaction may be determined directly from the stoichiometric equation. (We will shortly illustrate this determination.) One generation or consumption term must therefore either be specified or calculated for each independent reaction, which is why each reaction adds a degree of freedom to the system. 

If r independent reactions occur among the system components and the reactions proceed to equilibrium, then the right-hand side of this equation must be reduced by r. [Note Perry s Chemical Engineers Handbook (see footnote 1), p. 4-24, presents a proof of the phase rule and outlines a method for determining how many independent reactions may occur among the components of a system.]... 

However, there is one fly in the ointment here It may not be possible to determine the rate laws for each of the reactions. In this case it may be necessary to work with the minimum number of reactions and hope that a rate law can be found for each reaction. That is, you need to find the number of linearly independent reactions m your reaction set. In Example 6-8 just discussed, there are four reactions given [(E6-8.5) through (E6-8.8)]. However, only three of these reactions are independent, as the fourth can be formed from a linear combinafion of the other three, Tcchttiques for determining the number of independent reactions are given by Aris. ... 

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