Vectors column

Below we use explicit form of K left and right eigenvectors. Let vector-column r and vector-row f be right and left eigenvectors of K for eigenvalue —fc,. For coordinates of these eigenvectors we use notation r and /. Let us choose a normalization condition r — — It is straightforward to check that r = 0... 

For right eigenvectors, columns r we have the following asymptotics (we write vector-columns in rows) ... 

If, inverse, k steady state is accumulated inside A. After restoring the cycle Ai A2 A3 A we find that in the steady state almost all concentration is accumulated in A2 (the component at the beginning of the limiting step of this cycle, A2 A3). Finally, the eigenvector for zero eigenvalue is estimated as the vector column with coordinates (0,1,0,0,0,0). 

The right eigenvectors r are (we represent vector columns as rows) ... 

The ith column of this matrix coincides with the vector column jy Other columns are equal to zero. For each r we define a set of matrices Jir — wJn li e /, and Ji — The reaction network is solvable if and only if the finite set of... 

Here M is the vector column of the molecular masses of the substances, A is the molecular matrix and MA is the vector column of the atomic masses. The size of matrix A is N-by-m, where N is the number of reactants in the system and m the number of chemical elements entering into the composition of reactants. 

Here a is the vector column of the reactants. Thus, the stoichiometric equations (22) can be obtained by multiplication... 

The size of the matrix TA obtained by multiplying matrices T (s-by-lV) and A (N-by-m) is (s-by-m). If this matrix is multiplied by the vector column of atomic masses MA then, taking into account eqn. (21), we obtain... 

Here c is the vector column of substance concentrations, T is the transposed stoichiometric matrix, and w(c) is the vector column of reaction rates determined from eqns. (6) (8). It is an unsteady-state kinetic model. 

After multiplying matrix T by the vector column of the reaction rates, w, we obtain an unsteady-state kinetic model... 

The vector column for the matrix of stoichiometric numbers v (s-by-P) is called the route of a complex reaction. The rank of matrix Tint cannot be... 

Here v(p) is the vector column of the rates over the reaction routes and w the vector column for the rates of the steps. So the rate of every step is represented as a linear combination of the rates over the reaction routes. 

Let Nt be the content (mole) of substance A in the system, 2 the vector column with components N,. Similarly, let bt be the content (mole) of By in the system and b the vector column with components bj. They are related by matrix A (A transposed)... 

Matrix A will be used more often than A. Therefore it would be more correct to introduce this matrix immediately and to designate it as "atomic rather than "molecular , but we will adopt the conventional approach. Historically, the introduction of the designations and terminology used is substantiated by the relationship between vector columns of molecular M and atomic Ma weights... 

Kinetic equations can be reduced to a more compact form using a stoichiometric matrix and writing the rates for the various steps as a vector column. Then... 

This can also be said about the Horiuti number (or, as Horiuti called it himself, the stoichiometric number) discussed previously. The Horiuti numbers are the numbers chosen such that, after multiplying the chemical equation for every step by the appropriate Horiuti number and subsequent adding, all intermediates are cancelled. The equation thus obtained is the stoichiometric (brutto) equation. Each set of stoichiometric numbers leading to the elimination of intermediates is called a reaction route. In the general case, the Horiuti numbers form a matrix and its vector columns are the routes. 

The multiplication of the matrices vT(P x S) and fint(S x 7tot) gives the matrix vTrint whose size is (P x /tot). The vector column of the matrix for the... 

Here Z is the active site of (HgCl2-HCl). Vector-columns of the stoichiometric numbers are given to the right of the equations of the steps. This mechanism corresponds to the graph formed by two cycles having one common node, i.e. the intermediate Z [Fig. 3(d)]. 

Here v is the matrix of the Horiuti (stoichiometric) numbers and v and w the vector-columns of the rates along basic routes and of the step rates, respectively. Thus the rate of every step is represented as a linear combination of the rates along the basic routes. Here it is recommended that a simple hydrodynamic analogy be used. The total liquid flow along the tube (step) is the reaction rate. This flow consists of individual streams which are the rates along the routes. 

A set of quasi-steady-state equations for a linear mechanism is of the form 6(e)tf = 0, where x and c are the vector-columns of the concentrations for the intermediates and observed substances (those participating in the brutto-reaction, i.e. initial substances and products) and b(c) is the matrix of the reaction weights... 

Here A and X are the vector-columns of the observed and intermediate substances, respectively, and rA and T the matrices of their stoichiometric coefficients. 

It should be mentioned that the diagonal components of this matrix contain the theoretical variances of coefficients Pj,j = 1,..N. Moreover, these variances are necessary to test the significance of the coefficients of the model. Indeed, when matching a model with an experimental study, matrix (5.90) is fundamental for testing the significance of the coefficients. Now, we have to consider the differences between the measured y ,i = 1,. .N and the expected mean values of the measurements introduced through the new vector column (Yo, ) ... 

As I and U play a particular role in an electrochemical system, and both cem be considered to be either input or output quantities, according to the type of regulation of the polarization (potentiostat or galvanostat), I and U appear in either the input or the output vector columns. According to equation (14.16)... 

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