Jouguets criterion

We recall first the definition of independent reactions introduced in chapter I, 6. If we have a system containing c constituents, which can undergo r reactions other than transfers of molecules from one phase to another, then we have (c/. 1.62) for each reaction a stoichiometric equation  

The reactions 1. .. / are said to be independent if none of the stoichiometric equations can be derived by linear combination of any of the others. 

This definition can be expressed in an alternative form by noting that the equations (29.1) can be considered as r homogeneous equations in the variables. .. M. The condition that these equations are linearly independent is that, from the table of coefficients (called a matrix) 

An alternative form of this criterion may be obtained by multiplying the first column of (29.2) by the second by ilfg, etc  

All the determinants formed from this new matrix will be zero or not in the same way as the corresponding determinant derived from (29.2). We can say therefore that the r reactions are distinct if one can form from (29.3) at least one determinant of order r which is not zero. 

Two processes may be used to write s stoichiometric equations between c constituents. The first process consists of writing the equations in their usual form 

After s stoichiometric equations have been written, Jouguet s criterion allows us to ensure that the equations are actually independent. Let us assign by S the ith stoichiometry, viz. 

If the stoichiometries are not independent, numbers X, exist, which are not all equal to zero, and such that 

This is a homogeneous system of c linear equations in the s unknowns Xj. Let s be the rank of the matrix of stoichiometric coefficients. If the stoichiometric equations are independent, the X, s are all zero, and thus s = s. 

The statement of Jouguet s criterion is, therefore s stoichiometric equations are independent if the rank of the matrix of the stoichiometric coefficients is equal to s. 

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In order for all the balance equations to be independent, no rows of the above matrix should be obtainable by a linear combination of other rows. This means that this matrix must contain at least one determinant of a nonnull rank R. This condition is called the Jouguet criterion or criterion of independence of the reactions. 

E. Jouguet also proved in a general form that this thermodynamic criterion of the possibility of compressive shock waves... 

Cyclohexane is rly easily dehydrogenated into benzene, and even at very low extents of reaction, stoichiometry reaction (6) can be replaced by the secondary stoichiometry reaction (12). For cyclohexane, the constituents are (apart from cyclohexane) hydrogen, methane, ethane, ethylene, acetylene, propene, 1-butene, 1,3-butadiene, cydohexene, and benzene [3]. However, one can check that the equations written are independent, using the Jouguet [IS] criterion, (t.e., if/ = n-o)). In this criterion, the number of the independent constituents, 1//, for a chemical system is equal to the required constituents. n, (i e., H3, Cl, C3, CsHe, Q, C4, Cj. c-Q, CaJ, subtracting the number of independent stoichiometric equations. q>. 

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 each of these equations r is determined by Jouguet s criterion. 

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). 

Each linear combination of stoichiometric equations is itself a stoichiometric equation. This is the reason why the number of stoichiometric equations is fixed, but not their nature. The writing of a complex stoichiometric system is therefore likely to lead to equations which are not independent. One must make certain that this is not the case. JOUGUET s criterion indicates that the equations are independent if the rank of the matrix of the stoichiometric coefficients is equal to I. 

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