Algebraic stoichiometric coefficients

This example shows that the usuai form of writing the balanced reaction uses absolute values for the stoichiometric numbers. In some books, one finds a more mathematical form of these equations, in which algebraic stoichiometric coefficients are used directly ... 

In order to answer this question, let us construct the matrix N x R) of the algebraic stoichiometric coefficients each component k in each... 

We verify the independence of these three equilibrium states. Indeed, on the basis of the matrix of the algebraic stoichiometric coefficients, which is written as ... 

Reactant A is likely to intervene in various elementary steps with the algebraic stoichiometric coefficient, therefore we will have ... 

If we indicate the concentration of the main component j of the total reaction by Cj, affected by its algebraic stoichiometric coefficient vj, and reveahng supersaturation that was previously definite (section 3.10.1), we can write ... 

The new coefficients 2, 1, -2 and -2 are ealled algebraic stoichiometric coefficients. It should be noted that they are positive for a produet of the reaction and negative for a starting compound. 

Generalization.- A reaction between tlie eompounds A, 2, [ ]> producing species A, A 2 [...], A j will be generally written as [1.R3], with the arithmetic stoichiometric coefficients P/t or as [1.R4] with the algebraic stoichiometric coefficients v ... 

If several independent reactions are now carried out in our reactor that involve component as the reactant or the product, with the algebraic stoichiometric coefficient v p in the p-th reaction, the balance of this component - considering only the reactions - will be ... 

The source terms that are provided during the time At by the speeds of chemical reactions that involve X in the slice. In each reaction p in which X is involved, its algebraic stoichiometric coefficient is Vp (which might be null if X is not involved in the corresponding reaction). 

In fact there is an iniinite series of multiplying coefficients, all proportional to each other. For the main component A whose algebraic stoichiometric coefficient in the overall reaction is v, the following relation should be respected ... 

In this case, a linear combination of the elementary steps must ehminate the intermediary species. In other words, for a main reactant of reaction k (reactant or product) involved in step p with the algebraic stoichiometric coefficient v p, the following relationship can be written ... 

With perfect gases, if j is the sum of algebraic stoichiometric coefficients of the activated complex reaction formation, we will have ... 

The coefficients v then make sense of algebraic stoichiometric coefficients. This reaction may take place in one direction or another. 

Xj mean number of active centers formed from one active center vj algebraic stoichiometric coefficient of A V, algebraic stoichiometric coefficient of A,... 

Note that the matrix of stoichiometric coefficients devotes a row to each of the N components and a column to each of the M reactions. We require the reactions to be independent. A set of reactions is independent if no member of the set can be obtained by adding or subtracting multiples of the other members. A set will be independent if every reaction contains one species not present in the other reactions. The student of linear algebra will understand that the rank of v must equal M. 

We believe that it is not necessary to consider the overall kinetic order of steps above three in mechanism (4). We have analyzed comprehensively [97, 102, 103] all the possible versions for mechanism (4) assuming that the stoichiometric coefficients n, m, p, and q can amount to 1 or 2, p + q < 3, and k 3 = 0. The principal results of this analysis are listed in Table 2. By using the method of general analysis and the Sturm and Descartes theorem concerning the number of positive roots in the algebraic polynomial (ref. 219, pp. 248 and 255), we could show that there exist four detailed mechanisms providing the possibility of obtaining three steady states with a non-zero catalytic reaction... 

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. 

Calculate the algebraic sum of the stoichiometric numbers for the reaction Asdiscussed in Example 4.4, stoichiometric numbers for the components in a given reaction are numerically equal to the stoichiometric coefficients, but with the convention that the reactants get minus signs. In the present case, the stoichiometric numbers for H20, H2, and 02 are, respectively, —1, +1, and +V2-Their sum is + V2 ... 

The kinetics of a multistep reaction is described by a set of simultaneous rate equations, one for each participant. The equations are independent of one another in their algebraic forms and values of their coefficients. Each equation is the summation of the contributions from all steps in which the respective species participates. Each such contribution is the product of the stoichiometric coefficient of the species, the rate coefficient of the step, and the concentration (or concentrations) of the reactant (or reactants) of the step, raised to the power corresponding to the molecularity. 

The algebraic equation that relates - to the species concentrations is called the kinetic expression or rate law. The specific rate of reaction, > ike the reaction rate - / a, is always refeped to a particular species in the reactions and normally should be subscripted with respect to that species. However, for reactions in which the stoichiometric coefficient is 1 for all species involved in the reaction, for example. 

The species stoichiometric coefficients are Sa= —a, Sg= —b, Sc = c, s = d, and, for any inert species I, sj = 0. By defining the stoichiometric coefficients in this manner, chemical reactions are expressed as homogeneous algebraic equations. For example. Reaction 2.2.4 is expressed as... 

Let s first try balancing this equation using the method in Skill 2.1b. There is one Cu on both sides, but none of the other atoms are balanced. If we assume that Cu(N03h has a stoichiometric coefficient of one, then Cu(s) will also have a coefficient of one, but the other coefficients are difficult to balance right away. We could use algebra, and assign variable names to the other coefficients ... 

The term stoichiometric coefficients for reaction j, and since the stoichiometric coefficients for products are positive while those of the reactants are negative, there is a fair degree of cancelling inherent in f7 i= ii However, AU j = AH j only when the number of kilogram-moles of products formed by the reaction equals the number of kilogram-moles of reactants used up in doing so, i.e. when the reaction exhibits neither a... 

The burning will take place at a high temperature, and so that the physical state of the water produced will be steam, so that both reagents and products will be gaseous. Summing the stoichiometric coefficients algebraically gives ... 

The algebraic sum of the stoichiometric coefficients used in chemical equation (13.56) is —1 (see equation... 

Here are involved four chemical compounds with totally four atoms. In Eq. (13.7), we had three compounds with totally two atoms. As we can see, for the solution of the stoichiometric coefficients, a system of linear equations arises. Thus, certain rules from algebra are applicable. 

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