Matrix formula

The entries in A are the subscripts to the elements in the molecular formulas of the substances (in an arbitrary order). Each column is a vector of the subscripts for a substance, and A is called a formula matrix. 

A, molecular formula of species f pre-exponential factor with respect to species i A system formula matrix with entries aki... 

Let us consider a mixture Ay of S chemical species involving N elementary components . By setting the atoms in rows and the species in columns, a formula matrix B can be built ... 

Smith and Missen [20] exemplified the solution to obtain the stoichiometric coefficients of this equation by the matrix method [21], A reacting system consisting of a set of chemical reaction equations is represented by a formula matrix A = (Oki), where the element Uki of this matrix is the subscript of the chemical element k in the compound i occurring in the reaction equation. Consider a matrix N = (v,j) of stoichiometric coefficients, where Vij is the coefficient of the species i in the chemical equation j. The matrix N is obtained by solving the matrix equation... 

One important operation allows us to identify the number of independent reactions. In the traditional approach, we must reduce the proposed reactions to an independent set, but when many reactions occur, finding an independent set can be tedious and prone to error. However, in (7.4.4) the number of independent reactions (R is simply related to the rank of the formula matrix A specifically. 

Consider the formation of synthesis gas (CO and H2) by incomplete combustion of methane in oxygen. Let be the initial number of moles of methane and let N°2 be the initial number of moles of oxygen. Assume the feed ratio is N /Nj =3/2. We expect the products will be CO2, H2O, CO, and H2. So we have C = 6 species formed from nig = 3 elements (C, H, and O), and the formula matrix looks like this ... 

Here A is an (m x C) formula matrix, N is a (C x 1) vector of mole numbers, and b is an (mg x 1) vector of constant elemental abundances. (Basics of linear algebra are reviewed in Appendix B.) The matrix A is known from the chemical formulae of the species present, and the abundances b are known from the amounts initially loaded into the reactor. But the mole numbers N are unknown. Moreover, the sets N that satisfy the balances (7.4.2) are not unique many different combinations of amounts of the given species (N) can produce the same elemental balances (b). This means that the formula matrix A is singular. 

Therefore, to find all possible sets of mole numbers that satisfy the elemental balances (7.4.2) for a reacting system, we need the basis vectors for both the range and the nullspace of the formula matrix A. This part of the problem is solved in the remainder of this section. With all possible N known, we would then search among... 

To summarize, we perform a singular value decomposition of the augmented formula matrix to obtain the matrices U, W, and V. With these, we use (11.2.10) to obtain a particular basis vector N for the range. From V, we form P and then use (11.2.7) to obtain all sets of stoichiometric coefficients Vy. Then we combine N and Vy into (11.2.5) to determine all sets of mole numbers that satisfy the elemental balances. Therefore, a singular value decomposition provides the number of independent reactions 91, all sets of 91 independent stoichiometric coefficients Vy, and all possible combinations of mole numbers N that satisfy the elemental balances. A computer program for performing the decomposition is contained in the book by Press et al. [9] routines for performing the decomposition are also available in MATLAB and in Mathematica . 

How can stoichiometric coefficients be obtained from a singular value decomposition of the formula matrix ... 

Here the % are elements in the formula matrix A each % represents the number of atoms of element A on a molecule of species i. The Fj. are elemental abundances each bj is the total number of atoms of element fc in the system. From the known initial amounts of species loaded into the reactor, we can determine each then the guesses for the final mole numbers must satisfy (11.2.25) for each element fc. This completes the initialization stage of the algorithm. 

We have C = 6 spedes (CH4, O2, CO2, H2O, CO, H2) and nig = 3 elements (C, H, O). In 11.2.2 the formula matrix A for this situation was constructed and the singular value decomposition performed. That decomposition gave H = 3 independent reactions, with implicit stoichiometric coeffidents contained in matrix P of (11.2.14). Choosing a basis of 1 mole of O2 fed, the elemental abundances are... 

We have C = 5 species composed of JWg = 3 elements, and the formula matrix is Eton HAc EtAc HOH(HAc)2... 

Starting by determining the number of independent constituents, the formula matrix can be written as follows ... 

In these equations G(n) is the Gibbs energy function for the mixture and n e TZ is a column vector (sometimes called the species-abundance vector) containing the unknown variables representing the number of moles of each of the q species (molecules) in the mixture. A e is the predefined formula matrix. This coefficient matrix defines how the m elements in the mixture are distributed within the q species (molecules) that can exist in the system, m is the total number of elements in the mixture, b e 7 is a column vector (sometimes called the element-abundance vector) containing the known values for the total amount of the different elements in the mixture. 

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