Independent elementary reactions

Note that this constraint implies that the (/) columns of Y are orthogonal to the (E) rows of A. Thus, since each column of Y represents one elementary reaction, the maximum number of linearly independent elementary reactions is equal to K — E, i.e., N-f = rank(Y) < K — E. Formost chemical kinetic schemes, Ny = K — E however, this need not be the case. 

In a chemical system with S steps and a maximum of Q linearly independent elementary reactions every set of Q steps whose reactions are linearly independent constitutes a cycle-free subsystem. It is apparent that, if R(Sj),..., R(sq) are linearly independent, then no cycle can be formed with the steps... 

Let the g + 1 linearly independent elementary reactions that are involved in a RR be where the subscripts >L>Vi r pr sent an ordered set... 

Matrix of stoichiometric coefficients. Five components participate in three independent elementary reactions. Hence, three extents of reaction are required. The kinetic rate law for each elementary step is included in the following table. 

Kinetic rate laws for three independent elementary reactions in the first CSTR ... 

Analyze the transient startup behavior of a train of two liquid-phase CSTRs that operate isothermally at the same temperature. Four components participate in two independent chemical reactions. In the first independent elementary reaction, 1 mol of reactant A and 2 mol of reactant B reversibly produce 1 mol of intermediate product D ... 

Solution. Four unsteady-state mass balances are written and solved numerically to characterize the composition of the exit stream for each reactor. Initially, all of the parameters are declared. The average residence times are ti = 15 min and T2 = 10 min. The third-order kinetic rate constant for the forward step in the first independent elementary reaction is ki = 0.5 L / mol min. The equilibrium constant, based on molar densities, for the first independent elementary reaction is eq.c = 10 L /mol. The second-order kinetic rate constant for the second independent elementary reaction is kj, = 0.2 L/ mol min. The molar densities of all four components in the inlet stream to the first CSTR, for a stoichiometric feed of reactants A and B, are... 

The kinetic rate laws for both independent elementary reactions in the second CSTR that operates at temperature T2 = Ti are. represented by equation (2-11). Hence, the kinetic rate constants and the eqniUbrium constant are the same in both reactors ... 

Let us consider the second method of assigning in advance a dynamics of composition, namely via the vector of elementary reaction rates. Since every element v of the vector v represents by itself one of the possible elementary reactions of a system, the number of independent elements of vector v and a number s of independent elementary reactions are the same. 

Note that not all vectors in the quasistationary system are lineally independent. Just as the number of independent equations of connection in equation (1.83) is equal to rk( the number of independent elements of the vector v = (v ) is equal to s-rk(d ). It exactly determines the number of independent elementary reactions in the quasistationary system 5 ... 

What is the number of independent elementary reactions in this complex chemical reaction ... 

A complex is called short, if it is not longer than two. A mechanism is a second order mechanism, if all the reactant complexes are short and if there exists at least one of length two. A set of elementary reactions is said to be independent if there is no way of expressing any of the elementary reaction vectors as a linear combination of the others. In the opposite case the elementary reactions are said to be dependent. From this definition it is clear that the number of independent elementary reactions is the number of independent columns of y. But this number is called in linear algebra the rank of y rank(y). This number is usually denoted by S and is considered as the dimension of the stoichiometric space, i.e. the dimension of the linear... 

It may be worth emphasising that if one only considers different independent elementary reactions then an elementary reaction and its multiples are considered to be the same. 

The following question arises. Given a set of atomic components how many independent elementary reactions are possible Using the notations introduced above the question becomes how many linearly independent solutions g are there to the equation... 

The first important contribution to atomic stoichiometry in this century seems to be provided by Brinkley (1946). He has shown the importance of the rank of the atomic matrix and presented a proof of the phase rule of Gibbs (1876). A systematic outline of stoichiometry was presented by Petho (sometimes Petheo) and Schay (Petheo Schay, 1954 Schay, Petho, 1962). They gave a necessary and sufficient condition for the possibility of calculating an unknown reaction heat from known ones based upon the rank of the stoichiometric matrix. They introduced the notion of independence of components and of elementary reactions, the completeness of a complex chemical reaction (see the Exercises and Problems) and gave a method to generate a complete set of independent elementary reactions with as many zeros in the stoichiometric matrix as possible (see Petho, 1964). 

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