Brinkleys criterion

Let us assume that the c constituents of a system are formed from m elements E(1), E(2),. . . , E(m) of the periodic table. The chemical formula of constituent C may be written 

The law of conservation of elements requires that the following relationships hold. 

This is a homogeneous system of m linear equations in the c unknowns vn vn vic Let us denote by c the rank of the matrix of element indices akJ in the constituent formulae. The number, m, of elements is less than or equal to the number, c, of the constituents i.e. m c. Furthermore, since all constituents may be formed from the m elements, the number, c, of independent constituents is, at the most, equal to m i.e. c m. c may be less than m if some constituents may be obtained by chemical reaction between other constituents. Thus, c m c. 

The statement of Brinkley s criterion is, therefore the number, c, of independent constituents is equal to the rank of the matrix of element indices in the constituent formulae. The number, s, of independent stoichiometries to be written is s = c — c.  

This is a homogeneous system of m linear equations in the c unknowns ii , Vjc. Let us denote by c the rank of the matrix of element 

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Very often, it happens that some , are nearly zero (or even weakly negative) and, therefore, the corresponding stoichiometric equations can be considered as of negligible importance. Thus, the reacting system is described by a number, s, of stoichiometric equations less than the number, s = c — c, given by the Brinkley criterion. 

Using the Brinkley criterion, there are three equations for the reaction ... 

Moreover, by combining the Brinkley s [19] criterion, m rt-if/, one can calculate the number of independent stoichiometric equations. In this criterion the number y/ of independent constituents of a chemical system is equal to the rank of the matrix of the indexes of the elements in the formula of the constituents, hence cd = n-y/ = 9-2 = 7. Kinetic considerations have led to write 11 stoichiometric equations. It must be checked by Jouguet s criterion [18] that these 11 stoichiometric equations are independent. In the case of cycohe.xane the stoichiometric equations are 2, 3, 4, 8, 9, 11, and 12. Thus these seven stoichiometric equations are independent and describe the decomposition of cyclohexane. 

In a recent paper, S. R. Brinkley, J. Chem. Phys., 14, 563, 686 (1946) has proposed another criterion for determining the number of independent components c —c-r starting from the stoichiometric coefficients of the elements in each compound. For a criticism of this method c/. I. Prigogine and R. Defay, J. Chem. Phys., 15, 614 (1947). For a discussion of the relation between Brinkley s criterion and that of Jouguet cf. A. Penecloux, C. R. 228, 1729 (1949). 

First system of a linear homogeneous equations in an expression (1.22) has m-rk(0 ) fundamental solutions to which m-rk(0 ) of linear independent vectors V/ correspond. Each of these vectors determines a stoichiometric note of the / of the final reaction. That is why, in accordance with the criterion of Brinkley, we have... 

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